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+{"text":"\\section{Introduction}\n\nAfter great success of the three-flavor mixing scheme of neutrinos describing almost all data available to date, the neutrino experiments entered into the era of precision measurement and paradigm test. Here, it may be interesting to pay attention to the mutually different roles played by the appearance and the disappearance channels. The appearance channel $\\nu _\\mu \\rightarrow \\nu _e$ (or its T-conjugate) can play an important role to signal new effects, such as giving the first indication of nonzero $\\theta_{13}$ \\cite{Abe:2011sj}, which would also offer the best chance for discovering lepton CP violation in the future \\cite{Abe:2015zbg,Acciarri:2015uup}. \nOn the other hand, precision measurements of the mixing parameters to date are carried out mostly by using the disappearance channels $\\nu _\\alpha \\rightarrow \\nu _\\alpha$ ($\\alpha = e, \\mu$ including antineutrino channels). It includes tens of experiments using the atmospheric, solar, reactor and the accelerator neutrinos as in e.g., \\cite{SK-neutrino-2016,Aharmim:2011vm,Abe:2016nxk,Gando:2013nba,An:2016ses,RENO:2015ksa,Abe:2014bwa,Adamson:2014vgd,Abe:2015awa,NOvA-neutrino-2016}, which are on one hand complementary to each other, but on the other hand are competing toward the best accuracy. \nTo quote another example of their complementary roles, precision measurement of the survival probabilities in the channels $\\nu _{e} \\rightarrow \\nu_{e}$ and $\\nu _\\mu \\rightarrow \\nu _\\mu$ is essential for accurate determination of $\\theta_{13}$ and $\\theta_{23}$, respectively, while the appearance channel helps as a degeneracy solver \\cite{Minakata:2002jv,Hiraide:2006vh}. Therefore, precise knowledge of the disappearance channel oscillation probability in matter could be of some help for a better understanding of the data with diverse experimental settings. \nHereafter, we refer $\\nu _\\alpha$ disappearance channel oscillation probability $P(\\nu _\\alpha \\rightarrow \\nu _\\alpha)$ as the $\\nu _\\alpha$ survival probability. \n\nIn this context, it is noteworthy that the authors of ref.~\\cite{Nunokawa:2005nx} presented effective two-flavor description of the three-flavor neutrino survival probability in vacuum. They introduced an effective $\\Delta m^2$ to describe superposition of the atmospheric-scale oscillations with the two different frequencies associated with $\\Delta m^2_{32}$ and $\\Delta m^2_{31}$. Interestingly, the effective $\\Delta m^2$ is channel dependent: $\\Delta m^2_{ee} = c^2_{12} \\Delta m^2_{31} + s^2_{12} \\Delta m^2_{32}$ and $\\Delta m^2_{\\mu \\mu} = s^2_{12} \\Delta m^2_{31} + c^2_{12} \\Delta m^2_{32}$, respectively, to zeroth order in $\\sin \\theta_{13}$. It suggests a possibility that the effective $\\Delta m^2$ measured in the reactor $\\bar{\\nu}_{e}$ \\cite{An:2016ses,RENO:2015ksa} (see \\cite{An:2013zwz} for the first measurement) and the accelerator $\\nu_{\\mu}$ \\cite{Adamson:2014vgd,Abe:2015awa,NOvA-neutrino-2016} disappearance experiments can have a tiny difference of the order of $\\Delta m^2_{21}$. If observed, the difference between $\\Delta m^2_{ee}$ and $\\Delta m^2_{\\mu \\mu}$ could have an important implication because the sign of $\\Delta m^2_{ee} - \\Delta m^2_{\\mu \\mu}$ will tell us about which neutrino mass ordering is chosen by nature \\cite{Nunokawa:2005nx,Minakata:2006gq}. \n\nIn this paper, we investigate the question of whether the similar effective two-flavor description of the three-flavor neutrino survival probability is viable for neutrinos propagating in matter. We emphasize that it is a highly nontrivial question because the structure of neutrino oscillations is drastically altered in the presence of Wolfenstein's matter potential $a$ in the Hamiltonian \\cite{Wolfenstein:1977ue}. It also brings a different (not in the form of $1\/E$) energy dependence into the Hamiltonian. Using perturbative expression of the survival probability $P(\\nu_\\alpha \\rightarrow \\nu_\\alpha)$ in matter, and by introducing the similar ansatz for the effective two-flavor form of the probability as in vacuum, we will give an affirmative answer to the question to first order in the small expansion parameter $\\epsilon \\equiv \\Delta m^2_{21} \/ \\Delta m^2_{31}$. The ansatz includes the effective two-flavor $\\Delta m^2_{\\alpha \\alpha} (a)$ ($\\alpha=e, \\mu, \\tau$) in matter as a natural generalization of $\\Delta m^2_{\\alpha \\alpha}$ in vacuum. \n\nBut, then, it turned out that $\\Delta m^2_{\\alpha \\alpha} (a)$ becomes a dynamical quantity, which depends on neutrino energy $E$. It may be inevitable and legitimate because the energy dependence comes in through the matter potential $a \\propto E$. However, a contrived feature appears in $\\Delta m^2_{\\mu \\mu} (a)$ that it depends on $L$, the baseline distance. This feature does not show up in $\\Delta m^2_{ee} (a)$. Thus, while the effective two-flavor description of the three-flavor neutrino survival probability in matter seems to be possible, the resultant effective $\\Delta m^2_{\\alpha \\alpha} (a)$ does not appear to possess any fundamental significance as a physical parameter. We will argue that this feature is not due to the artifact of the perturbative treatment. \n\nLet us start by refreshing our understanding of the effective two-flavor description of the three-flavor neutrino survival probability in vacuum. \n\n\\section{Validity of the effective two-flavor approximation in vacuum}\n\\label{sec:validity}\n\nSuppose that one can measure neutrino energy with an extreme precision, $\\frac{\\Delta E}{E} \\ll \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}}$.\\footnote{\nThis condition is derived by requiring uncertainty of the kinematical factor $\\frac{\\Delta m^2_{31}L}{4E}$ of $\\Delta m^2_{31}$ wave due to energy resolution $\\Delta E$ is much smaller than the difference between the $\\Delta m^2_{31}$ and $\\Delta m^2_{32}$ waves, $\\frac{\\Delta m^2_{21}L}{4E}$.\n}\nLet us then ask a question: \nCan one observe two dips in the energy spectrum of $\\nu_{\\mu}$ in muon neutrino disappearance measurement due to two waves modulated with two different frequencies associated with $\\Delta m^2_{32}$ and $\\Delta m^2_{31}$? In vacuum and at around the first oscillation maximum (i.e., highest-energy maximum) of the atmospheric scale oscillation, $\\frac{ \\Delta m^2_{31} L }{ 2E} \\simeq \\pi$, we can give a definitive answer to the question; one never. It will be demonstrated below. If the same feature holds in matter, it provides us the raison d'~\\^etre for the approximate effective two-flavor form for the survival probability in matter in the three-flavor mixing scheme. \n\nIn the rest of this section, we start from the ``proof'' showing that in vacuum the $\\Delta m^2_{32}$ and $\\Delta m^2_{31}$ waves always form a single collective wave and has no chance to develop two minima in the energy spectrum of survival probability $P(\\nu_\\alpha \\rightarrow \\nu_\\alpha)$, where $\\alpha$ is one of $e$, $\\mu$, or $\\tau$. \nThen, we formulate an ansatz for the effective two-flavor approximation of the three-flavor probabilities in vacuum, which in fact gives a premise for the similar treatment in matter. \n\n\\subsection{Two waves form a single collective wave in vacuum}\n\\label{sec:two-wave}\n\nWe discuss the $\\nu_{\\alpha}$ survival probability $P(\\nu_\\alpha \\rightarrow \\nu_\\alpha)$ ($\\alpha=e, \\mu, \\tau$) in vacuum to understand the reasons why we expect that the effective two-flavor approximation is valid. Using unitarity, it can be written without any approximation as \\cite{Minakata:2007tn}\n\\begin{eqnarray}\nP(\\nu_\\alpha \\rightarrow \\nu_\\alpha) &=& 1- \n4\\vert U_{\\alpha 3}\\vert^2 \\vert U_{\\alpha 1}\\vert^2 \\sin^2 \\Delta_{31} -\n4 \\vert U_{\\alpha 3}\\vert^2 \\vert U_{\\alpha 2}\\vert^2 \\sin^2 \\Delta_{32}  -\n4\\vert U_{\\alpha 2}\\vert^2 \\vert U_{\\alpha e1}\\vert^2 \\sin^2 \\Delta_{21}, \n\\nonumber \\\\\n&=&\n1 - 4\\vert U_{\\alpha 2}\\vert^2 \\vert U_{\\alpha e1}\\vert^2 \\sin^2 \\Delta_{21} \n\\nonumber \\\\\n&-&\n2 \\vert U_{\\alpha 3}\\vert^2 \n\\left( \\vert U_{\\alpha 1}\\vert^2 + \\vert U_{\\alpha 2}\\vert^2 \\right) \n\\left[\n1 - \\sqrt{1-\\sin^2 2 \\chi \\sin^2 \\Delta_{21}} \n~\\cos (2 \\Delta_{\\alpha \\alpha} \\pm \\phi) \n\\right]\n\\label{P-alpha-alpha-vac}\n\\end{eqnarray}\nwhere the sign $\\pm$ in the cosine function at the end correspond to the mass ordering, $+$ for the normal and $-$ for inverted orderings. $U_{\\alpha j}$ $(j=1,2,3)$ denotes the MNS matrix elements \\cite{Maki:1962mu}. The kinematical factor $\\Delta_{j i}$ used in eq.~(\\ref{P-alpha-alpha-vac}) is defined as \n\\begin{eqnarray}\n\\Delta_{j i} \\equiv \\frac{\\Delta m^2_{j i} L }{4E}, \n\\hspace{10mm}\n(i, j = 1, 2, 3), \n\\label{Delta-ji-def}\n\\end{eqnarray}\nwhere $E$ is neutrino energy and $L$ the baseline distance. $\\Delta m^2_{ji}$ denote neutrino mass squared differences, $\\Delta m^2_{ji} \\equiv m^2_{j} - m^2_{i}$ $(i, j = 1,2,3)$.\n\n\nThe angle $\\chi$ in the square root in (\\ref{P-alpha-alpha-vac}) are defined  as\n\\begin{eqnarray}\n\\cos \\chi = \\frac{ \\vert U_{\\alpha 1} \\vert }{ \\sqrt{ \\vert U_{\\alpha 1}\\vert^2 + \\vert U_{\\alpha 2}\\vert^2  } },\n\\hspace{10mm}\n\\sin \\chi = \\frac{ \\vert U_{\\alpha 2} \\vert }{ \\sqrt{ \\vert U_{\\alpha 1}\\vert^2 + \\vert U_{\\alpha 2}\\vert^2  } }.\n\\label{chi-def}\n\\end{eqnarray}\nNow, $\\Delta_{\\alpha \\alpha}$ in the argument of the cosine function in (\\ref{P-alpha-alpha-vac}) is defined as follows: \n\\begin{eqnarray}\n\\Delta_{\\alpha \\alpha} \\equiv \\frac{\\Delta m^2_{\\alpha \\alpha} L }{4E}, \n\\hspace{10mm}\n\\Delta m^2_{\\alpha \\alpha} \\equiv \\cos^2 \\chi \\vert \\Delta m^2_{31} \\vert + \\sin^2 \\chi \\vert \\Delta m^2_{32} \\vert.\n\\label{dm2-mumu}\n\\end{eqnarray}\nFinally, the phase $\\phi$ is defined as\n\\begin{eqnarray}\n\\cos \\phi & = & \n\\frac{\\cos^2 \\chi \\cos \\left( 2 \\sin^2 \\chi \\Delta_{21} \\right) + \\sin^2 \\chi \\cos \\left( 2 \\cos^2 \\chi \\Delta_{21} \\right) }\n{ \\sqrt{1-\\sin^2 2 \\chi \\sin^2 \\Delta_{21}}},\n\\nonumber \\\\\n\\sin \\phi  & = & \n\\frac{ \\cos^2 \\chi \\sin \\left( 2 \\sin^2 \\chi \\Delta_{21} \\right)\n- \\sin^2 \\chi \\sin \\left( 2 \\cos^2 \\chi \\Delta_{21} \\right) }\n{\\sqrt{1-\\sin^2 2 \\chi \\sin^2 \\Delta_{21}}}. \n\\label{phi-def}\n\\end{eqnarray}\nNotice that $\\phi$ depends only on the 1-2 sector variables, or the ones relevant for the solar-scale oscillations.\n\nThanks to the hierarchy of the two $\\Delta m^2$, \n\\begin{eqnarray}\n\\epsilon \\equiv \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}} \\approx 0.03 \\ll 1, \n\\label{epsilon-def}\n\\end{eqnarray}\none can obtain a perturbative expression of $\\sin \\phi$, \n\\begin{eqnarray}\n\\sin \\phi = \\frac{ \\epsilon^3 }{3} \\sin^2 2\\chi \\cos 2\\chi (\\Delta_{31})^3 + \\mathcal{O} (\\epsilon^5), \n\\label{phi-approx}\n\\end{eqnarray}\nwhich shows that $\\sin \\phi$ is extremely small, $\\sin \\phi \\lsim 10^{-5}$, at around the first oscillation maximum of atmospheric scale oscillations, $\\Delta_{31} \\sim 1$. (The similar argument applies also to the second oscillation maximum.) Notice that at $\\Delta_{21} = \\epsilon \\Delta_{31} \\sim 1$, the perturbative expansion breaks down. \n\nThus, the superposed wave in the last line in (\\ref{P-alpha-alpha-vac}) can be well approximated by a single harmonic and there is no way that $\\Delta_{31}$ and $\\Delta_{32}$ waves develop two minima inside the region of interest, $0 < \\Delta_{31} \\sim \\Delta_{32} < \\pi$. Notice that the modulation due to the solar $\\Delta m^2_{21}$ term in (\\ref{P-alpha-alpha-vac}) does not alter this conclusion because of its much longer wavelength by a factor of $\\sim 30$.\n\nOne may argue that were the baseline $\\Delta_{21} \\sim 1$ is used instead, then one can distinguish between oscillations due to $\\Delta m^2_{31}$ and $\\Delta m^2_{32}$ waves, thereby could see the double dips. Despite that the former statement is in a sense true, the latter is not. In other word, what happens is different in nature. The feature that the superposed two waves behave as a single harmonics prevails. The difference between $\\Delta m^2_{31}$ and $\\Delta m^2_{32}$, $\\vert \\Delta m^2_{31} \\vert > \\vert \\Delta m^2_{32} \\vert$ (normal mass ordering) or $\\vert \\Delta m^2_{31} \\vert < \\vert \\Delta m^2_{32} \\vert$ (inverted mass ordering), entails advancement or retardation of the phase of the single wave formed by superposition \\cite{Minakata:2007tn}. Therefore, it appears that the property of no double dip generically applies even in the case $\\Delta_{21} \\sim \\Delta_{31}$. However, we did not try to make the statement of no double dip in $P(\\nu_\\alpha \\rightarrow \\nu_\\alpha)$ in vacuum at all energies and the whole parameter regions a rigorous theorem.\n\n\\subsection{Effective two-flavor approximation in vacuum}\n\\label{sec:2flavor-vac}\n\nIn this section, we try to provide the readers a simpler way of understanding the results obtained in ref.~\\cite{Nunokawa:2005nx}. We postulate the following ansatz for an effective two-flavor form of the three-flavor $\\nu _\\alpha$ survival probability $P(\\nu _\\alpha \\rightarrow \\nu _\\alpha)$ ($\\alpha= e, \\mu, \\tau$) in vacuum which is valid up to order $\\epsilon$, \n\\begin{eqnarray} \nP(\\nu _\\alpha \\rightarrow \\nu _\\alpha) &=&\nC_{\\alpha \\alpha} - A_{\\alpha \\alpha} \\sin ^2 \\left(\\frac{\\Delta m^2_{\\alpha \\alpha} L}{4E} \\right). \n\\label{P-alpha-alpha-vacuum-ansatz}\n\\end{eqnarray}\nIn principle, it is also possible to seek the effective two-flavor form which is valid to higher order in $\\epsilon$ by adopting more complicated ansatz. But, we do not try to pursue this line in this paper to keep the simplicity of the resultant expressions. We remark that, throughout this paper, we limit ourselves into the region $\\Delta_{31} \\sim \\Delta_{32} \\sim \\Delta_{\\alpha \\alpha} \\sim 1$ for the effective two-flavor formulas to work both in vacuum and in matter. Therefore, $\\Delta_{21}$ is of the order of $\\epsilon$. \n\nFor clarity we discuss here a concrete example, $P(\\nu _\\mu \\rightarrow \\nu _\\mu)$ in vacuum. In this paper we use the PDG parametrization of the MNS matrix. We keep the terms of order $\\Delta_{21}^2 \\sim \\epsilon^2$, the only exercise we engage in this paper to examine the order $\\epsilon^2$ terms. It is to give a feeling to the readers on how the two-flavor ansatz could (or could not) be extended to order $\\epsilon^2$. \nThe $\\nu_{\\mu}$ survival probability $P(\\nu _\\mu \\rightarrow \\nu _\\mu)$ in vacuum can be written to second order in $\\epsilon$ as \n\\begin{eqnarray}\n&& P(\\nu _\\mu \\rightarrow \\nu _\\mu) \n\\nonumber \\\\\n&=& \n1 - 4 \\epsilon^2 \\left(s^2_{12} c^2_{23} + c^2_{12} s^2_{23} s^2_{13} + 2 J_r \\cos \\delta \\right) \\left(c^2_{12} c^2_{23} + s^2_{12} s^2_{23} s^2_{13} - 2 J_r \\cos \\delta \\right) \\Delta_{31}^2\n\\nonumber \\\\\n&-& 4 s^2_{23} c^2_{13} \\left( c^2_{23} + s^2_{23} s^2_{13} \\right) \\sin ^2 \\Delta_{31} \n\\nonumber \\\\\n&+& 4 \\epsilon s^2_{23} c^2_{13} \n\\left(c^2_{12} c^2_{23} + s^2_{12} s^2_{23} s^2_{13} - 2 J_r \\cos \\delta \\right)\n\\Delta_{31} \\sin 2\\Delta_{31} \n\\nonumber \\\\\n&-& \n4 \\epsilon^2 s^2_{23} c^2_{13} \n\\left(c^2_{12} c^2_{23} + s^2_{12} s^2_{23} s^2_{13} - 2 J_r \\cos \\delta \\right) \\Delta_{31}^2 \\cos 2\\Delta_{31},\n\\label{Pmumu-vac2}\n\\end{eqnarray}\nwhere $J_r \\equiv c_{12} s_{12} c_{23} s_{23} s_{13}$. \n\nWe examine whether a simple ansatz for $\\Delta m^2_{\\alpha \\alpha}$ in (\\ref{P-alpha-alpha-vacuum-ansatz}), \n\\begin{eqnarray}\n\\Delta m^2_{\\alpha \\alpha} = \\Delta m^2_{31} - s_{\\alpha} \\Delta m^2_{21}, \n\\label{Dm2-eff-ansatz-vac}\n\\end{eqnarray}\ncan be matched to (\\ref{Pmumu-vac2}) to order $\\epsilon$.\nThe $\\nu_{\\alpha}$ survival probability $P(\\nu _\\alpha \\rightarrow \\nu _\\alpha) $ in (\\ref{P-alpha-alpha-vacuum-ansatz}) can be expanded to a power series of $\\Delta_{21}$ as \n\\begin{eqnarray}\n&& P(\\nu _\\alpha \\rightarrow \\nu _\\alpha) =\nC_{\\alpha \\alpha} - A_{\\alpha \\alpha} \\sin^2 \\Delta_{31} \n+ A_{\\alpha \\alpha} (s_{\\alpha} \\Delta_{21}) \\sin 2\\Delta_{31} \n- A_{\\alpha \\alpha} (s_{\\alpha} \\Delta_{21})^2 \\cos 2 \\Delta_{31}.  \n\\nonumber \\\\\n\\label{Pmumu-2flavor-2nd}\n\\end{eqnarray}\nThe equations (\\ref{Pmumu-vac2}) and (\\ref{Pmumu-2flavor-2nd}) matches (for $\\alpha=\\mu$) to order $\\Delta_{21}$ if  \n\\begin{eqnarray}\nC_{\\mu \\mu} &=& 1 - 4 \\epsilon^2 \\left(s^2_{12} c^2_{23} + c^2_{12} s^2_{23} s^2_{13} + 2 J_r \\cos \\delta \\right) \\left(c^2_{12} c^2_{23} + s^2_{12} s^2_{23} s^2_{13} - 2 J_r \\cos \\delta \\right) \\Delta_{31}^2\n\\nonumber \\\\\nA_{\\mu \\mu} &=& 4 s^2_{23} c^2_{13} \\left( c^2_{23} + s^2_{23} s^2_{13} \\right), \n\\nonumber \\\\\ns_{\\mu} A_{\\mu \\mu} &=& 4 s^2_{23} c^2_{13} \n\\left(c^2_{12} c^2_{23} + s^2_{12} s^2_{23} s^2_{13} - 2 J_r \\cos \\delta \\right).\n\\label{Aeff-vac}\n\\end{eqnarray}\nThat is, $P(\\nu _\\mu \\rightarrow \\nu _\\mu)$ in vacuum can be written in the effective two-flavor form. \n\n\nThen, dividing the last line by the second, we obtain \n\\begin{eqnarray}\ns_{\\mu} \n&=& c^2_{12} -\n\\frac{ \\cos 2\\theta_{12} \\tan^2 \\theta_{23} s^2_{13} + 2 \\frac{J_r}{c^2_{23}} \\cos \\delta \n}{\n1 + \\tan^2 \\theta_{23} s^2_{13} }.\n\\label{s-mu-vac}\n\\end{eqnarray}\nWe note that the matching between (\\ref{Pmumu-vac2}) and (\\ref{Pmumu-2flavor-2nd}) to order $\\Delta_{21}^2$ is not possible with the current ansatz (\\ref{P-alpha-alpha-vacuum-ansatz}) because the coefficient of $\\Delta_{21}^2 \\cos 2 \\Delta_{31}$ term must be $A_{\\mu \\mu} s_{\\mu}^2$, which does not mach with (\\ref{Pmumu-vac2}).\\footnote{\nIntroduction of the similar perturbative ansatz for $A_{\\mu \\mu}$ does not resolve this issue.\n}\nWith $s_{\\mu}$ in (\\ref{s-mu-vac}), the effective $\\Delta m^2_{\\mu \\mu} (= \\Delta m^2_{31} - s_{\\mu} \\Delta m^2_{21})$ in vacuum is given to order $\\epsilon s_{13}$ by the formula \n\\begin{eqnarray}\n\\Delta m^2_{\\mu \\mu} \n&=& s^2_{12} \\Delta m^2_{31} + c^2_{12} \\Delta m^2_{32} + 2 \\frac{J_r}{c^2_{23}} \\cos \\delta \\Delta m^2_{21}, \n\\label{Dm2-eff-mumu-vac}\n\\end{eqnarray}\nwhich reproduces the expression of $\\Delta m^2_{\\mu \\mu}$ in ref.~\\cite{Nunokawa:2005nx}. \n\nA similar treatment with ansatz (\\ref{P-alpha-alpha-vacuum-ansatz}) for $P(\\nu _{e} \\rightarrow \\nu _{e})$ in vacuum gives the effective $\\Delta m^2_{e e}$ without expanding by $s_{13}$ as \n\\begin{eqnarray}\n\\Delta m^2_{ee} &=& c^2_{12} \\Delta m^2_{31} + s^2_{12}  \\Delta m^2_{32},  \n\\label{Dm2-eff-ee-vac}\n\\end{eqnarray}\nagain reproducing the formula for $\\Delta m^2_{ee}$ in ref.~\\cite{Nunokawa:2005nx}. In the rest of this paper, we will refer eqs.~(\\ref{Dm2-eff-mumu-vac}) and (\\ref{Dm2-eff-ee-vac}) as the NPZ formula for effective $\\Delta m^2$. \n\n\\section{Effective two-flavor form of survival probability in matter }\n\\label{sec:P-alpha-alpha-matt}\n\nIn matter, we don't know apriori whether the effective two-flavor form of the survival probability makes sense. Therefore, it is not obvious at all if there is such a concept as effective $\\Delta m^2_{\\alpha \\alpha}(a) $ in matter. Fortunately, very recently, there was a progress in our understanding of this issue.\n\nThe authors of ref.~\\cite{Minakata:2015gra} have shown to all orders in matter effect (with uniform density) as well as in $\\theta_{13}$ that $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ can be written in an effective two-flavor form \n\\begin{eqnarray}\nP(\\nu_{e} \\rightarrow \\nu_{e}: a) &=& \n1 -  \\sin^2 2 \\tilde{\\phi}~\\sin^2 \\frac{ (\\lambda_{+} - \\lambda_{-} ) L }{4E} \n\\label{Pee-matter-SC}\n\\end{eqnarray}\nto first order in their expansion parameter $\\epsilon_r$, where $\\tilde{\\phi}$ is $\\theta_{13}$ in matter and $\\lambda_{\\pm}$ denote the eigenvalues of the states which participate the 1-3 level crossing.\\footnote{\nIn their framework, which is dubbed as the ``renormalized helio-perturbation theory'', they used a slightly different expansion parameter $\\epsilon_r \\equiv \\Delta m^2_{21} \/ \\Delta m^2_{ren}$, where $\\Delta m^2_{ren} \\equiv \\Delta m^2_{31} - s^2_{12} \\Delta m^2_{21}$, which is identical with $\\Delta m^2_{ee}$ in vacuum, eq.~(\\ref{Dm2-eff-ee-vac}). See ref.~\\cite{Minakata:2015gra} for the explicit definitions of $\\tilde{\\phi}$, $\\lambda_{\\pm}$ etc.\n}\nThis provides us an {\\em existence proof} of the concept of the effective two-flavor form of the survival probability in matter. \n\nFrom (\\ref{Pee-matter-SC}), $\\Delta m^2_{ee} (a)$ in matter is given by (see \\cite{Minakata:2015gra})\n\\begin{eqnarray}\n\\Delta m^2_{ee} (a) = \n\\vert \\lambda_{+} - \\lambda_{-} \\vert = \n\\sqrt{ \\left( \\Delta m^2_{ren} - a \\right)^2 + 4 s^2_{13} a \\Delta m^2_{ren} } \n\\label{Dm2ee-matter-SC}\n\\end{eqnarray}\nwhere $a$ denotes the Wolfenstein matter potential \\cite{Wolfenstein:1977ue} which in our convention depend on energy $E$ as \n\\begin{eqnarray}\na & = & 2\\sqrt{2} G_F N_e E \\approx 1.52 \\times 10^{-4} \\left( \\frac{Y_{e}~\\rho}{\\rm g.cm^{-3}} \\right) \\left( \\frac{E}{\\rm GeV} \\right) {\\rm eV}^2,  \n\\label{matt-potential}\n\\end{eqnarray}\nwhere $G_F$ denotes the Fermi constant, $N_e$ the number density of electrons, $Y_e$ the electron fraction and $\\rho$ is the density of matter. For simplicity and clarity we will work with the uniform matter density approximation in this paper. \n\nThen, the natural question is: Is the similar effective two-flavor form of the survival probability available in $\\nu_{\\mu}$ disappearance channel within the same framework? Unfortunately, the answer appears to be {\\em No}.\n\nOne of the charming features of the framework developed in ref.~\\cite{Minakata:2015gra} is that the oscillation probability takes the canonical form, the one with the same structure as in vacuum, of course with replacing the quantities $U_{\\alpha i}$ and $\\Delta_{ji}$ by the corresponding ones in matter. For the canonical or the vacuum-like structure of $P(\\nu _\\alpha \\rightarrow \\nu _\\alpha)$, look at the first line in eq.~(\\ref{P-alpha-alpha-vac}). Therefore, generally speaking, it contains the three terms with kinematic sine functions of the three differences of the eigenvalues, with exception of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ mentioned above. If one looks at the eigenvalue flow diagram as a function of the matter potential (figure 3 in \\cite{Minakata:2015gra}) one would be convinced that there is no good reason to expect that the effective two-flavor form of the survival probability holds, except for the asymptotic regions $a \\rightarrow \\pm \\infty$. After all, the system we are dealing with is the {\\em three-flavor neutrino mixing } so that we must expect generically the genuine three-flavor structure.  \n\nThen, the readers may ask: Is this the last word for answering the question ``Is there any sensible definition of effective two-flavor form of the survival probability in matter?''. Most probably the answer is {\\em No}. Nonetheless, we will show in the rest of this paper that circumventing the conclusion in the last paragraph faces immediate difficulties. In a nutshell, we will show that the effective two-flavor form of the survival probability is possible in matter to order $\\epsilon$ at least formally in both $\\nu_{e}$ and $\\nu_{\\mu}$ disappearance channels. But, we show that the effective $\\Delta m^2_{\\mu \\mu} (a)$ in matter cannot be regarded as a physically sensible quantity by being $L$ (baseline) dependent. On the other hand, $\\Delta m^2_{ee} (a)$ does not suffer from the same disease.   \n\nOur approach is that we limit ourselves to a simpler perturbative framework in which however the effect of matter to all orders is kept, because it is the key to the present discussion. It allows us to write the survival probability by simple analytic functions and the fact that each quantity has explicit form would allow us clearer understanding. \n\n\\subsection{Ansatz for the effective two-flavor form of survival probability in matter}\n\\label{sec:ansatz}\n\nWith the above explicit example of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ in mind, we examine the similar ansatz as in vacuum for the effective two-flavor form of survival probability in matter. We postulate the same form of ansatz as in vacuum, which is assumed to be valid up to order $\\epsilon$, but allowing more generic form of $\\Delta m^2_{\\alpha \\alpha}$ ($\\alpha= e, \\mu, \\tau$):\n\\begin{eqnarray} \nP(\\nu _\\alpha \\rightarrow \\nu _\\alpha: a) &=&\nC_{\\alpha \\alpha} (a) - A_{\\alpha \\alpha} (a) \\sin ^2 \\left(\\frac{\\Delta m^2_{\\alpha \\alpha} (a) L}{4E} \\right), \n\\label{P-alpha-alpha-matter} \n\\\\\n\\Delta m^2_{\\alpha \\alpha} (a) &=& \n\\Delta m^2_{\\alpha \\alpha} (a)^{(0)} + \\epsilon\n\\Delta m^2_{\\alpha \\alpha} (a)^{(1)}. \n\\label{Dm2-eff-alpha-alpha}\n\\end{eqnarray}\nThe lessons we learned in the case in vacuum suggest that the restriction to order $\\epsilon$ is necessary to keep the expression of the effective two-flavor probability sufficiently concise. Notice that the quantities $A_{\\alpha \\alpha}$, $C_{\\alpha \\alpha}$, and $\\Delta m^2_{\\alpha \\alpha}$ in eq.~(\\ref{P-alpha-alpha-matter}) depend not only on the mixing parameters but also on the matter potential $a$, as explicitly indicated in (\\ref{P-alpha-alpha-matter}).\n\nWe then follow the procedure in section~\\ref{sec:2flavor-vac} to determine the form of $\\Delta m^2_{\\alpha \\alpha} (a)^{(0)}$ and $\\Delta m^2_{\\alpha \\alpha} (a)^{(1)}$. As was done in the previous section we occasionally use a concise notation \n$\\Delta_{\\alpha \\alpha} (a) \\equiv \\frac{ \\Delta m^2_{\\alpha \\alpha} (a) L }{ 4E }$. The effective two-flavor form of $P(\\nu_{\\alpha} \\rightarrow \\nu_{\\alpha})$, eq.~(\\ref{P-alpha-alpha-matter}), can be expanded in terms of $\\epsilon$\n\\begin{eqnarray} \nP(\\nu_\\alpha \\rightarrow \\nu _\\alpha: a) &=&\nC_{\\alpha \\alpha} (a) - A_{\\alpha \\alpha} (a)\n\\left( \\sin^2 \\Delta_{\\alpha \\alpha}^{(0)} (a) + \\epsilon \\Delta_{\\alpha \\alpha}^{(1)} (a) \\sin 2 \\Delta_{\\alpha \\alpha}^{(0)} (a) \\right) \n+ \\mathcal{O} (\\epsilon^2).\n\\label{Palpha-alpha-2flavor3}\n\\end{eqnarray}\nTherefore, if the expressions of the survival probabilities take the form (\\ref{Palpha-alpha-2flavor3}), then they can be written as the effective two flavor forms which are valid to order $\\epsilon$.\n\n\\subsection{Perturbative framework to compute the oscillation probabilities}\n\\label{sec:framework}\n\nWe use the $\\sqrt{\\epsilon}$ perturbation theory formulated in ref.~\\cite{Asano:2011nj} to derive the suitable expressions of the survival probabilities in matter. In this framework the oscillation probabilities are computed to a certain desired order of the small expansion parameter $\\epsilon \\equiv \\Delta m^2_{21} \/ \\Delta m^2_{31} \\simeq 0.03$ assuming $s_{13} \\sim \\sqrt{\\epsilon} $. Notice that the measured value of $\\theta_{13}$ is $s_{13} = 0.147$, the central value of the largest statistics measurement \\cite{An:2016ses}, so that $s^2_{13} = 0.021 \\sim \\epsilon$. We use the survival probabilities computed to second order in $\\epsilon$, which means to order $\\epsilon s^2_{13}$ and $s^4_{13}$. Inclusion of these higher-order corrections implies to go beyond the Cervela et al. formula \\cite{Cervera:2000kp}. We will see that it is necessary to keep the former higher-order term to recover the NPZ formula in the vacuum limit.\n\nWhile we expand the oscillation probabilities in terms of $\\epsilon$ and $s_{13}$, we keep the matter effect to all orders. It is the key to our discussion, and furthermore keeping all-order effect of matter may widen the possibility of application of this discussion to various experimental setups of the long-baseline (LBL) accelerator neutrino experiments considered in the literature. The parameter which measures relative importance of the matter effect to the vacuum one is given by \n\\begin{eqnarray} \nr_{A} &\\equiv& \\frac{ a }{ \\Delta m^2_{31} } = \n\\frac{ 2\\sqrt{2} G_F Y_{e} \\rho} { \\Delta m^2_{31} m_{N} } E \n\\nonumber \\\\\n&=&\n0.89 \n\\left(\\frac{|\\Delta m^2_{31}|}{2.4 \\times 10^{-3}\\mbox{eV}^2}\\right)^{-1}\n\\left(\\frac{\\rho}{2.8 \\text{g\/cm}^3}\\right) \\left(\\frac{E}{10~\\mbox{GeV}}\\right), \n\\label{rA-def-value}\n\\end{eqnarray}\nwhere $m_{N}$ denotes the unified atomic mass unit, and we assume $Y_{e}=0.5$ in this paper. \n$r_{A}$ appears frequently in the expressions of the oscillation probabilities, as will be seen below. Notice that the ratio $r_{A}$ of matter to vacuum effects can be sizeable for neutrino energies of $\\sim 10$ GeV in the LBL experiments. It should also be noticed that $r_{A}$ depends linearly on neutrino energy $E$. In what follows, we use the formulas of the probabilities given in \\cite{Asano:2011nj} without explanation, leaving the derivation to the reference.\n\n\\section{Effective $\\Delta m^2_{ee}$ in matter }\n\\label{sec:Dm2-ee-matt}\n\nGiven the perturbative expressions of the survival probabilities we can derive the effective $\\Delta m^2_{\\alpha \\alpha} (a)$ in matter by using the matching condition with (\\ref{Palpha-alpha-2flavor3}), in the similar way as done in section~\\ref{sec:2flavor-vac}. In addition to the abbreviated notation $\\Delta_{j i} \\equiv \\frac{\\Delta m^2_{j i} L }{4E}$ introduced in (\\ref{Delta-ji-def}), we use the notation \n\\begin{eqnarray}\n\\Delta_{a} \\equiv \\frac{a L }{4E} = r_{A} \\Delta_{31}\n\\label{Delta-a-def}\n\\end{eqnarray}\nwith the matter potential $a$ defined in (\\ref{matt-potential}) to simplify the expressions of the oscillation probabilities. Notice that, unlike $r_{A}$, $\\Delta_{a}$ is energy independent, but $\\Delta_{a}$ depends on the baseline $L$, $\\Delta_{a} \\propto L$. \n\n\\subsection{Effective two-flavor form of $P(\\nu_{e} \\rightarrow \\nu_{e})$ and $\\Delta m^2_{ee}$ in matter}\n\\label{sec:Pee-matt}\n\nWe first discuss the $\\nu_{e}$ survival probability $P(\\nu_{e} \\rightarrow \\nu_{e})$ in matter. As we remarked in section~\\ref{sec:framework}\nwe need to go to second order in $\\epsilon$:\\footnote{\nHere is a comment on behaviour of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ in region of energy for $r_{A} \\simeq 1$. Though it may look like that $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ is singular in $r_{A} \\rightarrow 1$ limit, it is not true. The apparent singularity cancels. But, it is not the end of the story. Despite no singularity at $r_{A}=1$, the perturbative expressions of the oscillation probabilities in region of $r_{A}$ close to 1 display the problem of inaccuracy. The cause of the problem is due to the fact that we are expanding the probability by $s_{13}$, by which we miss the effect of resonance enhancement of flavor oscillation. If fact, one can observe the improvement of the accuracy at around $r_{A}$ close to 1 by including $s^4_{13}$ terms. See figure 3 of ref.~\\cite{Asano:2011nj}. \n}\n\\begin{eqnarray}\nP(\\nu_{e} \\rightarrow \\nu_{e}: a) &=& \n1 - 4 s^2_{13} \\frac{ 1 }{ (1 - r_{A})^2 } \n\\sin^2 \\left[ (1 - r_{A}) \\Delta_{31} \\right] \n\\nonumber \\\\\n&+& \n4 \\left[ \ns^4_{13} \\frac{ (1 + r_{A})^2 }{ (1 - r_{A})^4 } \n- 2 s^2_{12} s^2_{13} \\frac{ \\epsilon r_{A} }{ (1 - r_{A})^3 } \n\\right] \n\\sin^2 \\left[ (1 - r_{A}) \\Delta_{31} \\right] \n\\nonumber \\\\\n&-& \n4 \\left[ \n2 s^4_{13} \\frac{ r_{A} }{ (1 - r_{A})^3 } \n- s^2_{12} s^2_{13} \\frac{ \\epsilon }{ (1 - r_{A})^2 } \n\\right] \n\\Delta_{31} \\sin \\left[ 2 (1 - r_{A}) \\Delta_{31} \\right] \n\\nonumber \\\\\n&-&\n4 c^2_{12} s^2_{12} \n\\left( \\frac{ \\epsilon }{ r_{A} } \\right)^2 \n\\sin^2 \\Delta_{a}.\n\\label{Pee-matter}\n\\end{eqnarray}\nThe leading order depletion term in $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ in (\\ref{Pee-matter}) is of order $\\epsilon$, and the remaining terms (second to fourth lines) are of order $\\epsilon^2$. $\\bar{\\nu}_{e}$ survival probability can be discussed just by flipping the sign of the matter potential $a$.\n\n\nWe notice in eq.~(\\ref{Pee-matter}) that even in the two flavor limit, $\\epsilon \\rightarrow 0$ and $s_{13} \\rightarrow 0$, the effective $\\Delta m^2$ is modified from $\\Delta m^2_{31}$ to $\\Delta m^2_{ee} (a)^{(0)} = (1 - r_{A}) \\Delta m^2_{31}$ due to the strong, order unity, matter effect in the $\\nu_{e}$ channel. Notice that in view of eq.~(\\ref{rA-def-value}) the change can be sizeable at energies $E \\gsim$ a few GeV. Thus, the effective $\\Delta m^2_{ee} (a)$ in matter, and generically $\\Delta m^2_{\\alpha \\alpha} (a)$ as we will see later, inevitably become dynamical quantities, which depend on neutrino energy $E$.\n\nThe matching between (\\ref{Pee-matter}) and the two-flavor form in (\\ref{Palpha-alpha-2flavor3}) can be achieved as follows:  \n\\begin{eqnarray}\nC_{ee} (a) &=& 1 - 4 c^2_{12} s^2_{12} \n\\left( \\frac{ \\epsilon }{ r_{A} } \\right)^2 \n\\sin^2 \\Delta_{a}, \n\\nonumber \\\\\nA_{ee} (a) &=& \n\\frac{ 4 s^2_{13} }{ (1 - r_{A})^2 } \n\\left[ 1 \n+ 2 s^2_{12} \\frac{  \\epsilon r_{A} }{ (1 - r_{A}) } \n- s^2_{13} \\frac{ (1 + r_{A})^2 }{ (1 - r_{A})^2 } \n\\right], \n\\nonumber \\\\\n\\epsilon A_{ee} (a) \\Delta m^2_{ee} (a)^{(1)} &=& \n\\frac{ 4 s^2_{13} }{ (1 - r_{A})^2 } \n\\left[\n2 s^2_{13} \\frac{ r_{A} }{ (1 - r_{A}) } \n- s^2_{12}  \\epsilon\n\\right] \\Delta m^2_{31}.\n\\label{coeff-ee-matter}\n\\end{eqnarray}\nIt is remarkable to see that all the terms in (\\ref{Pee-matter}) including $\\mathcal{O} (\\epsilon^2)$ terms can be organized into the effective two-flavor form in (\\ref{P-alpha-alpha-matter}). Using the second and the fourth lines of (\\ref{coeff-ee-matter}) we obtain to first order in $\\epsilon$: \n\\begin{eqnarray}\n\\epsilon \\Delta m^2_{ee} (a)^{(1)}\n=\n\\frac{\\epsilon A_{ee} (a) \\Delta m^2_{ee} (a)^{(1)} }{ A_{ee} (a) } \n= \\left[\n2 s^2_{13} \\frac{ r_{A} }{ (1 - r_{A}) } \n- s^2_{12} \\epsilon\n\\right]\n\\Delta m^2_{31} \n\\label{Dm2-1st}\n\\end{eqnarray}\nwhere we have kept terms up to order $\\epsilon$ in the second line in (\\ref{Dm2-1st}). Thus, the effective $\\Delta m^2$ in matter in the $\\nu _e \\rightarrow \\nu _e$ channel is given as $\\Delta m^2_{ee} (a) = \\Delta m^2_{ee} \\vert^{(0)} + \\epsilon \\Delta m^2_{ee} (a)^{(1)}$, \n\\begin{eqnarray}\n\\Delta m^2_{ee} (a) &=& \n(1 - r_{A}) \\Delta m^2_{31} + \n\\left[ 2 s^2_{13} \\frac{ r_{A} }{ (1 - r_{A}) } - \\epsilon s^2_{12} \n\\right] \\Delta m^2_{31},\n\\nonumber \\\\ &=& \n(1 - r_{A}) \\Delta m^2_{ee} (0) +  \nr_{A} \\left[ \\frac{ 2 s^2_{13} }{ (1 - r_{A}) } - \\epsilon s^2_{12} \\right] \\Delta m^2_{31}, \n\\label{Dm2-eff-ee-matter}\n\\end{eqnarray}\nwhich obviously reduces to the NPZ formula $\\Delta m^2_{ee} \\vert_{ \\text {vac} } =\\Delta m^2_{ee} (0) = c^2_{12} \\Delta m^2_{31} + s^2_{12} \\Delta m^2_{32}$ in the vacuum limit. \n\nThus, we have learned that $\\nu_{e}$ (and $\\bar{\\nu}_{e}$) survival probability in matter can be casted into the effective two-flavor form (\\ref{P-alpha-alpha-matter}) in a way parallel to that in vacuum. But, the nature of the effective $\\Delta m^2_{ee} (a)$ is qualitatively changed in matter: It becomes a dynamical quantity which depends on energy, not just a combination of fundamental parameters as it is in vacuum. It is inevitable once we recognize that the leading-order effective $\\Delta m^2_{ee}$ in matter is given by $\\Delta m^2_{ee} (a)^{(0)} = (1 - r_{A}) \\Delta m^2_{31}$ in the two-flavor limit. \n\nOne may argue that the expression of $\\Delta m^2_{ee} (a)$ in eq.~(\\ref{Dm2-eff-ee-matter}) does not make sense because it is singular at $r_{A} \\rightarrow 1$ limit. It might sound a very relevant point because the survival probability itself is singularity free, as mentioned in the footnote 4. But, we argue that the singularity of $\\Delta m^2_{ee} (a)$ at $r_{A} =1$ is very likely to be superficial. Let us go back to $\\Delta m^2_{ee} (a)$ in eq.~(\\ref{Dm2ee-matter-SC}) which is obtained by using the renormalized helio-perturbation theory \\cite{Minakata:2015gra} with all order effect of $\\theta_{13}$. It is perfectly finite in the limit $r_{A} \\rightarrow 1$. One can easily show that by expanding $\\Delta m^2_{ee} (a)$ in (\\ref{Dm2ee-matter-SC}) by $s_{13}$ one reproduces the result in (\\ref{Dm2-eff-ee-matter}).\\footnote{\nThis exercise has first been suggested to the author by Stephen Parke. \n}\nIt means that the singularity in $\\Delta m^2_{ee} (a)$ at $r_{A} = 1$ is an artifact of the expansion around $s_{13}=0$. In fact, one can easily convince oneself that the expansion of the eigenvalues in terms of $s_{13}$ is actually an expansion in terms of $s^2_{13}\\frac{r_{A}}{ (1-r_{A})^2 }$. \n\n\\subsection{Energy dependence of $\\Delta m^2_{ee} (a)$} \n\\label{sec:E-dep-Dm2}\n\n\\begin{figure\n\\begin{center}\n\\vspace{3mm}\n\\includegraphics[width=0.48\\textwidth]{Dm2-ee-ratio-10.pdf}\n\\includegraphics[width=0.48\\textwidth]{Dm2-ee-ratio-lowE.pdf}\n\\end{center}\n\\caption{\nIn the left panel, plotted is the ratio $\\Delta m^2_{ee} (a) \/ \\Delta m^2_{ee} (0)$ as a function of $E$ in units of GeV. The right panel is to magnify the low energy region of $\\Delta m^2_{ee} (a) \/ \\Delta m^2_{ee} (0)$ for anti-neutrinos, \nshowing that the matter effect in $\\Delta m^2_{ee} (a)$ is tiny, at a level of $\\sim \\text{a few} \\times 10^{-4}$ in MeV energy region. The mixing parameters and the matter density that we used are: \n$\\Delta m^2_{31} = 2.4 \\times 10^{-3}$ eV$^2$, $\\Delta m^2_{21} = 7.5 \\times 10^{-5}$ eV$^2$, $\\sin^2 \\theta_{13} = 0.022$, $\\sin^2 \\theta_{12} = 0.30$, and $\\rho=2.8~\\text{g\/cm}^3$. \n}\n\\label{fig:Dm2-ratio-ee}\n\\end{figure}\n\nIn figure~\\ref{fig:Dm2-ratio-ee}, the ratio $\\Delta m^2_{ee} (a) \/ \\Delta m^2_{ee} (0)$ is plotted as a function of neutrino energy $E$ in units of GeV. The left panel of Fig.~\\ref{fig:Dm2-ratio-ee} indicates that $\\Delta m^2_{ee} (a)$ decreases linearly with $E$ in a good approximation, the behaviour due to the leading order term $\\Delta m^2_{ee} (a)^{(0)} = (1 - r_{A}) \\Delta m^2_{31}$. Our expression of $\\Delta m^2_{ee} (a)$ cannot be trusted beyond $E \\simeq 7$ GeV because the turn over behaviour seen in figure~\\ref{fig:Dm2-ratio-ee} starting at the energy signals approach to the resonance enhancement at $E \\simeq 11$ GeV. An estimation of the resonance width via the conventional way yields the results $\\pm 3.3$ GeV around the resonance, inside which our perturbation theory breaks down. The estimated width is consistent with what we see in figure~\\ref{fig:Dm2-ratio-ee}. The deviation from the linearity below that energy represents the effect of three-flavor correction, the second term in the last line in (\\ref{Dm2-eff-ee-matter}), and its smallness indicates that this effect is small, and it is nicely accommodated into the effective two-flavor $\\Delta m^2_{ee} (a)$. \n\nThe right panel in Fig.~\\ref{fig:Dm2-ratio-ee} shows $\\Delta m^2_{ee} (a)$ for the antineutrino channel at low energies relevant for reactor electron antineutrinos. We see that the matter effect is extremely small, $0.05\\%$ even at $E=6$ MeV, which justifies the commonly used vacuum approximation for $\\Delta m^2_{ee}$ for reactor neutrino analyses \\cite{An:2016ses,RENO:2015ksa}. \n\n\\subsection{Energy dependence of the minimum of $P(\\nu_{e} \\rightarrow \\nu_{e})$} \n\\label{sec:E-dep-Peemin}\n\nIn section~\\ref{sec:Pee-matt}, the formula for the effective $\\Delta m^2_{ee} (a)$ in matter was derived in an analytic way, eq.~(\\ref{Dm2-eff-ee-matter}). The question we want to address in this section is to what extent the energy dependent $\\Delta m^2_{ee} (a)$ is sufficient to describe the behaviour of $\\nu_{e}$ disappearance probability at around $E=E_{min}$, the highest-energy  minimum of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$. For this purpose, we construct a simple model of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ in which the matter (therefore energy) dependence exists only in $\\Delta m^2_{ee} (a)$: \n\n\\vspace{2mm}\n\\noindent\n{\\bf Simple model}: We ignore the energy dependence of $C_{ee} (a)$ and $A_{ee} (a)$ in the effective two-flavor form eq.~(\\ref{P-alpha-alpha-matter}) of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$, while keeping the energy dependence in $\\Delta m^2_{ee} (a)$.\n\n\\vspace{2mm}\n\n\\noindent\nThe spirit of the model is that the energy dependent $\\Delta m^2_{ee} (a)$ plays a dominant role in describing the behaviour of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ at around $E=E_{min}$. We want to test this simple model to know to what extent the spirit is shared by the actual $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ in matter. \n\n\\begin{figure\n\\begin{center}\n\\vspace{-4.4mm}\n\\includegraphics[width=0.48\\textwidth]{Prob_ee.pdf}\n\\includegraphics[width=0.48\\textwidth]{E_min_rho.pdf}\n\\end{center}\n\\vspace{-6mm}\n\\caption{ The left panel: The survival probability $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ is plotted as a function of neutrino energy $E$ obtained by numerically solving the neutrino evolution equation for various values of matter density between $\\rho = 0$ and $\\rho = 8$ $\\frac{ \\text{g} }{ \\text{cm}^3}$. The blue-solid and red-dashed lines are for the normal and inverted mass orderings, respectively. \nThe right panel: The highest-energy solution $E_{min}$ of the equation $\\frac{ d }{d E} P(\\nu_{e} \\rightarrow \\nu_{e}: a) = 0$ is plotted as a function of $\\rho$ in units of $\\frac{ \\text{g} }{ \\text{cm}^3}$. $E_{min}$ is with use of the same color line symbols as in the left panel. Also plotted are the solution of eq.~(\\ref{Emin-eq}) obtained in the simple model described in the text.\nThe mixing parameters used are: \n$\\Delta m^2_{ee} = 2.4 \\times 10^{-3}$ eV$^2$, $\\Delta m^2_{21} = 7.54 \\times 10^{-5}$ eV$^2$, $\\sin^2 \\theta_{12} = 0.31$, and $\\sin^2 2\\theta_{13} = 0.089$. \n}\n\\label{fig:Pee}\n\\end{figure}\n\nIn figure~\\ref{fig:Pee}, in the left panel, plotted is the survival probability $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$ as a function of neutrino energy $E$ obtained by solving exactly (within numerical precision) the neutrino evolution equation for various values of matter density between $\\rho = 0$ and $\\rho = 8$ $\\frac{ \\text{g} }{ \\text{cm}^3}$ to vary the strength of the matter effect.\nThe blue-solid and red-dashed lines are for the normal and inverted mass orderings, respectively. In mid between the blue and red colored lines there is a black solid line which corresponds to $P(\\nu_{e} \\rightarrow \\nu_{e})$ in vacuum. \nIn the right panel in figure~\\ref{fig:Pee}, plotted with the same line symbols as in the left panel is the highest-energy solution $E_{min}$ of the equation $\\frac{ d }{d E} P(\\nu_{e} \\rightarrow \\nu_{e}: a) = 0$ as a function of $\\rho$ in units of $\\frac{ \\text{g} }{ \\text{cm}^3}$. $E_{min}$ corresponds to so called the dip energy at the first minimum of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$. The thin blue-solid and red-dotted lines are the solution of $\\frac{ d }{d E} P(\\nu_{e} \\rightarrow \\nu_{e}: a) = 0$ of the simple model for the normal and inverted mass orderings, respectively. \n\nWe now try to understand qualitatively figure~\\ref{fig:Pee}, and compare $E_{min}$ predicted by the simple model to the one obtained by using the numerically computed survival probability. The solution of $\\frac{ d }{d E} P(\\nu_{e} \\rightarrow \\nu_{e}: a) = 0$ in the simple model is given by \n\\begin{eqnarray}\n\\frac{ \\Delta m^2_{ee} (E) L }{ 2 E } = \\pm \\pi, \n\\label{E-min}\n\\end{eqnarray}\nwhere the sign $\\pm$ corresponds to the normal and inverted mass orderings, respectively. \nTo simplify the expression we use the notations \n\\begin{eqnarray}\nr_{A} &\\equiv& \\pm A E, \n\\hspace{10mm}\nA \\equiv \\frac{ 2\\sqrt{2} G_F Y_{e} \\rho} { \\vert \\Delta m^2_{31} \\vert m_{N} }, \n\\hspace{10mm} \nE_{vom} \\equiv \\frac{ \\vert \\Delta m^2_{31} \\vert L }{ 2 \\pi }.\n\\label{notations}\n\\end{eqnarray}\nThen, by using (\\ref{Dm2-eff-ee-matter}) the $E_{min}$-determining equation (\\ref{E-min}) becomes \n\\begin{eqnarray}\n1 \\mp AE \n\\pm \\left[ 2 s^2_{13} \\frac{ AE }{ 1 \\mp AE } - \\epsilon s^2_{12} \\right] = \n\\frac{ E }{ E_{vom} }.\n\\label{Emin-eq}\n\\end{eqnarray}\nIt is a quadratic equation for $E$ with an obvious solution that is not written here. The solution of (\\ref{Emin-eq}) is plotted by the thin-solid and dotted lines in the right panel of figure~\\ref{fig:Pee}. \n\nThe qualitative behaviour of the solution of (\\ref{Emin-eq}) can be understood by a perturbative solution of (\\ref{Emin-eq}) with the small parameters $\\epsilon$ and $s^2_{13} \\sim \\epsilon$. To first order in $\\epsilon$ it reads \n\\begin{eqnarray}\nE_{min} = \n\\frac{ E_{vom} }{ 1 \\pm A E_{vom}  }\n\\left[ 1 \\pm \\left( 2 s^2_{13} A E_{vom}  - \\epsilon s^2_{12} \\right) \\right]. \n\\label{Emin-sol}\n\\end{eqnarray}\nNoticing the value of $A$, \n\\begin{eqnarray} \nA &=& \n\\frac{ 2\\sqrt{2} G_F Y_{e} \\rho} { \\vert \\Delta m^2_{31} \\vert m_{N} } = \n0.032 \n\\left(\\frac{|\\Delta m^2_{31}|}{2.4 \\times 10^{-3}\\mbox{eV}^2}\\right)^{-1}\n\\left(\\frac{\\rho}{1 \\text{g\/cm}^3}\\right) \\mbox{GeV}^{-1}, \n\\label{A-def-value}\n\\end{eqnarray}\nwhich is small for $\\rho \\lsim 3~\\text{g\/cm}^3$, an approximately linear $\\rho$ dependence $E_{min} \\approx E_{vom} \\left( 1 \\mp A E_{vom} \\right)$ is expected. But, in region of $\\rho \\gsim 6~\\text{g\/cm}^3$ a visible nonlinearity is expected in particular in the case of inverted mass ordering. They are in good agreement with the simple model prediction plotted by the thin-solid and dotted lines in the right panel of figure~\\ref{fig:Pee}. It confirms that the perturbative solution (\\ref{Emin-sol}) captures the main feature of the simple model. We note, however, that the agreement between the simple model prediction and the numerically computed $E_{min}$ (blue-solid and red-dashed lines) is rather poor as seen in the same figure. \n\nThus, despite qualitative consistency exists to certain extent, we see that the simple model fails to explain the quantitative features of $\\rho$ dependence of the first minimum of $P(\\nu_{e} \\rightarrow \\nu_{e}: a)$. It indicates that the energy dependent $\\Delta m^2_{ee} (a)$ is not sufficient to describe the behaviour of $\\nu_{e}$ disappearance probability at around its first minimum. That is, the matter effect brings the energy dependences into the coefficients $C_{ee}$ and $A_{ee}$ in (\\ref{P-alpha-alpha-matter}) as strongly as to modify $\\Delta m^2_{ee} (a)$. Therefore, though the perfectly consistent effective two-flavor approximation exists for $\\nu_{e}$ survival probability in matter, its quantitative behaviour at around the highest-energy minimum cannot be described solely by the energy-dependent $\\Delta m^2_{ee} (a)$. This is in contrast to the situation in vacuum that introduction of $\\Delta m^2_{ee}$ allows to describe the result of precision measurement of $P(\\nu_{e} \\rightarrow \\nu_{e})$ in reactor experiments very well \\cite{An:2016ses,RENO:2015ksa}.\n\n\\section{Effective $\\Delta m^2_{\\mu \\mu}$ in matter }\n\\label{sec:Dm2-mumu-matt}\n\n\\subsection{Effective two-flavor form of $P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu})$ and $\\Delta m^2_{\\mu \\mu}$ in matter}\n\\label{sec:Pmumu-matt}\n\nWe now discuss $\\Delta m^2_{\\mu \\mu}$ in matter. Here, we need the survival probability $P(\\nu _\\mu \\rightarrow \\nu _\\mu; a)$ only up to second order in $s_{13}$ and first order in $\\epsilon \\equiv \\frac{\\Delta m^2_{21}}{\\Delta m^2_{31}}$, because the leading order term is of order unity. These terms were calculated previously by many authors, see e.g., \\cite{Cervera:2000kp,Akhmedov:2004ny,Minakata:2004pg}. It can be written as the effective two flavor form (\\ref{Palpha-alpha-2flavor3}) with the coefficients \n\\begin{eqnarray}\nC_{\\mu \\mu} (a) &=& \n1 - 4 \\biggl[ \ns^2_{23} s^2_{13}  \\left(\\frac{1}{1 - r_{A}}\\right)^2  - 2 \\epsilon J_r \\cos \\delta \\frac{1}{r_{A} (1 - r_{A})} \\biggr] \n\\sin ^2 \\Delta_{a}, \n\\nonumber \\\\\nA_{\\mu \\mu} (a) &=& \n4 \n\\left[\nc^2_{23} s^2_{23} - s^2_{23} s^2_{13} \\left(\\frac{1}{1 - r_{A}}\\right)^2 \n\\left( \\cos 2\\theta_{23} + 2 s^2_{23} \\sin ^2 \\Delta_{a} \\right) \n \\right. \n  \\nonumber \\\\ \n  && \\left. \n   \\hspace{18mm} \n+ 2 \\epsilon J_r \\cos \\delta \\frac{1}{r_{A} (1 - r_{A})} \n\\left( \n\\cos 2\\theta_{23} r_{A}^2 + 2 s^2_{23} \\sin^2\\Delta_{a} \n\\right) \n\\right], \n\\nonumber \\\\\n\\epsilon \\Delta m^2_{\\mu \\mu} (a)^{(1)} A_{\\mu \\mu} (a) &=& \n2 s^2_{23} \n\\left[\n2 c^2_{23} \\biggl\\{ \ns^2_{13} \\left(\\frac{r_{A}}{1 - r_{A}} \\right) \n- \\epsilon c^2_{12}  \n\\biggr\\}\n\\right. \n\\nonumber \\\\ \n&& \\left. \n\\hspace{-6mm} \n- \\biggl\\{ s^2_{23} s^2_{13} \n\\left(\\frac{1}{1 - r_{A}}\\right)^2  \n- 2 \\epsilon J_r \\cos \\delta \\frac{1}{r_{A} (1 - r_{A})} \n\\biggr\\} \\frac{ \\sin 2\\Delta_{a} }{ \\Delta_{31} }\n\\right] \n\\Delta m^2_{31}, \n\\label{matching2}\n\\end{eqnarray}\nwhere $J_r \\equiv c_{12} s_{12} c_{23} s_{23} s_{13}$. \n\nUsing the last two equations in (\\ref{matching2}), the first order correction term in the effective $\\Delta m^2_{\\mu \\mu} (a)$ can be calculated, to order $s^2_{13} \\sim \\epsilon$ and $\\epsilon s_{13}$, as \n\\begin{eqnarray}\n&& \\epsilon \\Delta m^2_{\\mu \\mu} (a)^{(1)} = \n\\frac{ \\epsilon \\Delta m^2_{\\mu \\mu} (a)^{(1)} A_{\\mu \\mu} (a) }{ A_{\\mu \\mu} (a) }\n\\nonumber \\\\ \n&=&\n\\left[\n- \\epsilon c^2_{12} \n+ s^2_{13} \\left(\\frac{r_{A}}{1 - r_{A}} \\right) \n- \\biggl\\{ \\frac{ 1 }{ 2 } s^2_{13} \\tan^2 \\theta_{23} \n\\frac{1}{( 1 - r_{A} )^2 } \n- \\epsilon \\frac{ J_r \\cos \\delta }{ c^2_{23}  } \\frac{1}{r_{A} (1 - r_{A})} \n\\biggr\\} \\frac{ \\sin 2\\Delta_{a} }{ \\Delta_{31} } \n\\right] \\Delta m^2_{31}.\n\\nonumber \\\\ \n\\label{Dm2-mumu-1st-again}\n\\end{eqnarray}\nThen, finally, $\\Delta m^2_{\\mu \\mu} (a)= \\Delta m^2_{\\mu \\mu} (a)^{(0)} + \\epsilon \\Delta m^2_{\\mu \\mu} (a)^{(1)}$ can be obtained as \n\\begin{eqnarray}\n&& \\Delta m^2_{\\mu \\mu} (a) = \ns^2_{12} \\Delta m^2_{31} + c^2_{12} \\Delta m^2_{32} \n\\nonumber \\\\\n&+& \\left[\ns^2_{13} \\left\\{ \\frac{r_{A}}{1 - r_{A}} \n- \\tan^2 \\theta_{23}\n\\left(\\frac{1}{1 - r_{A}}\\right)^2 \\frac{ \\sin 2\\Delta_{a} }{ 2 \\Delta_{31} } \n\\right\\} \n+ 2 \\frac{ \\epsilon J_r \\cos \\delta }{ c^2_{23}  } \\frac{1}{ (1 - r_{A})} \n\\frac{ \\sin 2\\Delta_{a} }{ 2 \\Delta_{a} } \n\\right] \\Delta m^2_{31}.\n\\nonumber \\\\\n\\label{Dm2-eff-mumu-matt}\n\\end{eqnarray}\n\nIn the vacuum limit, noticing that $\\epsilon \\Delta m^2_{\\mu \\mu} (a)^{(1)} \\rightarrow \n\\left( - c^2_{12} + 2 \\frac{ J_r }{ c^2_{23}  } \\cos \\delta \\right) \\Delta m^2_{21}$ as $a \\rightarrow 0$, we obtain \n\\begin{eqnarray}\n\\Delta m^2_{\\mu \\mu} (0) \n&=& s^2_{12} \\Delta m^2_{31} + c^2_{12} \\Delta m^2_{32} \n+ 2 \\frac{J_r}{c^2_{23}} \\cos \\delta~\\Delta m^2_{21}, \n\\label{Dm2-eff-mumu-vac2}\n\\end{eqnarray}\nwhich again reproduces the NPZ formula for $\\Delta m^2_{\\mu \\mu}$ in vacuum. \n\nNow, we have to address the conceptual issue about the result of $\\Delta m^2_{\\mu \\mu} (a)$ in (\\ref{Dm2-eff-mumu-matt}). Though it depends on energy through $r_{A} \\propto E$ we do not think it a problem. See the discussion in the previous section. However, there is a problem of $L$-dependence of $\\Delta m^2_{\\mu \\mu} (a)$. \nNotice that $\\Delta_{a} \\equiv aL\/ 4E$ is $L$-dependent and is $E$ independent. Therefore, the last two terms of $\\Delta m^2_{\\mu \\mu} (a)$ in (\\ref{Dm2-eff-mumu-matt}) have a peculiar dependence on baseline length $L$.\\footnote{\nIn short baseline, or in low-density medium, $\\Delta_{a} \\ll 1$, the $L$ dependence in (\\ref{Dm2-eff-mumu-matt}) goes away because \n\\begin{eqnarray}\n\\frac{ \\sin 2\\Delta_{a} }{ 2 \\Delta_{31} } \\approx \n\\frac{ \\Delta_{a} }{ \\Delta_{31} } = r_{A}, \n\\hspace{10mm}\n\\frac{ \\sin 2\\Delta_{a} }{ 2 \\Delta_{a} } \\approx 1.\n\\end{eqnarray}\nBut, this is just very special cases of possible experimental setups. \n}\nBecause of the $L$-dependence of $\\Delta m^2_{\\mu \\mu} (a)$ in (\\ref{Dm2-eff-mumu-matt}), unfortunately, we cannot consider it as the sensible quantity as the effective parameter which describes the physics of $\\nu_{\\mu}$ survival probability in matter.\\footnote{\nSome examples of $L$-dependent (actually $L\/E$-dependent) effective $\\Delta m^2_{ee}$ in vacuum are discussed recently with the critical comments \\cite{Parke:2016joa}.\n}\n\nPutting aside the problem of $L$-dependence of $\\Delta m^2_{\\mu \\mu} (a)$, we examine its matter potential dependence by examining energy dependence of $\\Delta m^2_{\\mu \\mu} (a) \/ \\Delta m^2_{\\mu \\mu} (0)$. From figure~\\ref{fig:Dm2-ratio-mumu}, one can see that the matter effect correction to $\\Delta m^2_{\\mu \\mu}$ is only a few \\% in the ``safe'' region $E \\lsim 7$ GeV.\n\nAs will be commented at the end of appendix~\\ref{sec:hesitation-theorem} the $\\nu_{\\tau}$ appearance probability $P(\\nu_{\\tau} \\rightarrow \\nu_{\\tau} )$ can be obtained from $P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu} )$ by the transformation $c_{23} \\rightarrow - s_{23}, s_{23} \\rightarrow c_{23}$. Therefore, $\\Delta m^2_{\\tau \\tau} (a)$ can be obtained by the same transformation from $\\Delta m^2_{\\mu \\mu} (a)$. \n\n\n\\begin{figure\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Dm2-mumu-ratio-10.pdf}\n\\end{center}\n\\caption{\nThe ratio \n$\\Delta m^2_{\\mu \\mu} (a) \/ \\Delta m^2_{\\mu \\mu} (0)$ is plotted as a function of $E$ in units of GeV. We take $L=1000$ km. The mixing parameters and the matter density used are the same as in figure~\\ref{fig:Dm2-ratio-ee}.\n}\n\\label{fig:Dm2-ratio-mumu}\n\\end{figure}\n\n\\subsection{Matter potential dependence of $\\Delta m^2_{ee}$ and $\\Delta m^2_{\\mu \\mu}$} \n\nThe matter potential dependence of the effective $\\Delta m^2$ is very different between $\\Delta m^2_{ee} (a)$ and $\\Delta m^2_{\\mu \\mu} (a)$, as shown in the previous sections. In contrast to the strong matter dependence of $\\Delta m^2_{ee} (a)$, $\\Delta m^2_{\\mu \\mu} (a)$ shows only a weak dependence on the matter potential $a$. \n\nTo understand the difference, in particular, the weak matter effect in $P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu} )$, we derive in appendix~\\ref{sec:hesitation-theorem} a general theorem about the matter potential dependence of the various oscillation probabilities, which may be called as the ``matter hesitation theorem''. It states that the matter potential dependent terms in the oscillation probabilities $P(\\nu_{\\alpha} \\rightarrow \\nu_{\\beta})$ ($\\alpha, \\beta = e, \\alpha, \\tau$) receive the suppression factors of at least $s^2_{13}$, or $\\epsilon s_{13}$, or $\\epsilon^2$, where $\\epsilon \\equiv \\Delta m^2_{21} \/ \\Delta m^2_{31}$ as defined in (\\ref{epsilon-def}). That is, the matter effect hesitates to come in before computation reaches to these orders. Given the small values of the parameters, $s^2_{13} \\simeq 0.02$, or $\\epsilon s_{13}  \\simeq 4.5 \\times 10^{-3}$, or $\\epsilon^2 \\simeq 10^{-3}$, the theorem strongly constrains the matter potential dependence of the oscillation probabilities. Our discussion simply generalizes the similar one given in ref.~\\cite{Kikuchi:2008vq}. \n\nLet us apply the matter hesitation theorem to $P(\\nu_{\\mu} \\rightarrow \\nu_{\\mu} )$, whose expression is given (though in a decomposed way) in (\\ref{Palpha-alpha-2flavor3}) with (\\ref{matching2}). It reveals the feature of large vacuum term corrected by the suppressed matter effect terms, as dictated by the theorem. Then, we immediately understand the reason why the matter effect dependent terms in $\\Delta m^2_{\\mu \\mu} (a)$, the second line in (\\ref{Dm2-eff-mumu-matt}), are suppressed with the factors either $s^2_{13}$ or $\\epsilon s_{13}$, explaining its smallness and the weak energy dependence of $\\Delta m^2_{\\mu \\mu} (a)$. \n\nThen, a question might arises: Given the universal (channel independent) suppression of the matter effect why it can produce a strong modification to $\\Delta m^2_{ee}$ in vacuum? Look at first (\\ref{Pee-matter}) to notice that all the terms in $1 - P(\\nu_{e} \\rightarrow \\nu_{e} )$ is matter dependent, and they are all equally suppressed by $s^2_{13}$ or by smaller factors. Therefore, the theorem itself is of course valid. But, since all the terms are universally suppressed by small factors, the suppression itself does not tell us how strongly the matter potential affects $1 - P(\\nu_{e} \\rightarrow \\nu_{e} )$. It turned out that the matter effect significantly modifies $1 - P(\\nu_{e} \\rightarrow \\nu_{e} )$, as we have leaned in section~\\ref{sec:Dm2-ee-matt}. The feature stems from the structure of matter Hamiltonian $\\propto \\text{diag} [a, 0, 0]$, which allows $\\nu_{e}$ to communicate directly with the matter potential. Even after including the three flavor effect, this feature dominates. \n\n\\section{Effective two-flavor approximation of appearance probability in vacuum }\n\\label{sec:P-beta-alpha-vac}\n\nIn this paper, so far, we have discussed the validity of the concept of effective two-flavor form of the disappearance probability, and the associated effective $\\Delta m^2$ in vacuum and in matter. Do these concepts have validities also for the appearance probability? Since we have questioned the validity of the notion of effective $\\Delta m^2$ in matter our discussion in this section primarily deal with the possible validity of effective appearance $\\Delta m^2$ in vacuum. \n\nThe appearance probability $P(\\nu_{\\beta} \\rightarrow \\nu_{\\alpha})$ ($\\beta \\neq \\alpha$) in vacuum can be written to order $\\epsilon$ in the form \n\\begin{eqnarray} \nP(\\nu_{\\beta} \\rightarrow \\nu_{\\alpha}) =\n4 A_{31}^{\\beta \\alpha} \\sin^2 \\Delta_{31} +\n4 A_{32}^{\\beta \\alpha} \\sin^2 \\Delta_{32} +\n8 J_{r} c^2_{13} \\sin \\delta \n\\sin \\Delta_{21} \\sin \\Delta_{31} \\sin \\Delta_{32} \n\\label{P-beta-alpha-vac}\n\\end{eqnarray}\nwhere the sign of CP-odd term in (\\ref{P-beta-alpha-vac}) is normalized for $\\beta=e$ and $\\alpha=\\mu$. The coefficients $A_{31}^{\\beta \\alpha}$ etc are given in table~\\ref{tab:coefficient-A}.\n\\begin{table}[h!]\n\\begin{center}\n\\caption{\nThe coefficients $A_{31}^{e \\mu}$ etc. used in eq.~(\\ref{P-beta-alpha-vac}) are tabulated. The similar expressions for other channel, e.g., $A_{31}^{e \\tau}$ can be obtained by the appropriate transformation from $A_{31}^{e \\mu}$. See e.g., ref.~\\cite{Kikuchi:2008vq}.\n}\n\\label{tab:coefficient-A} \n\\begin{tabular}{c|c}\n\\hline \n\\hline \n$A_{31}^{e \\mu}$ & \n$c^2_{12} s^2_{23} c^2_{13} s^2_{13} + c^2_{13} J_r \\cos \\delta$ \\\\\n\\hline \n$A_{32}^{e \\mu}$ & \n$s^2_{12} s^2_{23} c^2_{13} s^2_{13} - c^2_{13} J_r \\cos \\delta$ \\\\\n\\hline \n$A_{31}^{\\mu \\tau}$ & \n$c^2_{23} s^2_{23} c^2_{13} \\left( s^2_{12} - c^2_{12} s^2_{13} \\right) - \\cos 2 \\theta_{23} c^2_{13} J_r \\cos \\delta$ \\\\\n\\hline \n$A_{32}^{\\mu \\tau}$ & \n$c^2_{23} s^2_{23} c^2_{13} \\left( c^2_{12} - s^2_{12} s^2_{13} \\right) + \\cos 2 \\theta_{23} c^2_{13} J_r \\cos \\delta$ \\\\\n\\hline \\hline \n$B_{e \\mu}$ & \n$s^2_{23} c^2_{13} s^2_{13} + 2 c^2_{13} J_r \\sin \\delta \\Delta_{21}$ \\\\\n\\hline\n$B_{\\mu \\tau}$ & \n$c^2_{23} s^2_{23} c^4_{13} + 2 c^2_{13} J_r \\sin \\delta \\Delta_{21}$ \\\\\n\\hline \\hline \n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn complete analogy to the case of survival probability we define the effective $\\Delta m^2$ for appearance channel, $\\Delta m^2_{\\beta \\alpha} \\equiv \\Delta m^2_{\\beta \\alpha} (0)$, removing ``$(0)$'' (which signals that it is in vacuum) since all the effective $\\Delta m^2$ in this section are in vacuum, as \n\\begin{eqnarray} \n\\Delta m^2_{31} = \\Delta m^2_{\\beta \\alpha} + s_{\\beta \\alpha} \\Delta m^2_{21}. \n\\label{Dm2-beta-alpha}\n\\end{eqnarray}\nThe effective two-flavor form \n\\begin{eqnarray} \nP(\\nu_{\\beta} \\rightarrow \\nu_{\\alpha}) = \n4 B_{\\beta \\alpha} \\sin^2 \\Delta_{\\beta \\alpha}, \n\\label{effective-2flavor-app}\n\\end{eqnarray}\nwhere $\\Delta_{\\beta \\alpha} \\equiv \\frac{\\Delta m^2_{\\beta \\alpha} L}{4E}$, is obtained by requiring that the order $\\epsilon$ terms that arise from the first two terms in (\\ref{P-beta-alpha-vac}) cancel out. Notice that to order $\\epsilon$ the CP-odd term in (\\ref{P-beta-alpha-vac}) merely renormalizes the coefficient of the effective two-flavor form. The cancellation condition determines $s_{\\beta \\alpha}$ as \n\\begin{eqnarray}\ns_{e \\mu} \n&=& s^2_{12} -\n\\frac{ J_r \\cos \\delta \n}{\ns^2_{23} s^2_{13} }, \n\\nonumber \\\\\ns_{\\mu \\tau} \n&=& c^2_{12} + \n\\cos 2 \\theta_{12} \\tan^2 \\theta_{13} + \n\\frac{ \\cos 2\\theta_{23} }{ c^2_{23} s^2_{23} c^2_{13} }\nJ_r \\cos \\delta. \n\\label{s-beta-alpha-vac}\n\\end{eqnarray}\nThe resultant coefficients $B_{\\beta \\alpha}$ for the two-flavor form (\\ref{effective-2flavor-app}) are also tabulated in table~\\ref{tab:coefficient-A}. \nNotice that $s_{e \\mu}$ cannot be expanded in terms of $s_{13}$, because $P(\\nu_{e} \\rightarrow \\nu_{\\mu}) =0$ at $s_{13}=0$. The second term of $s_{e \\mu}$ signals discrepancy between disappearance $\\Delta m^2_{ee}$ and appearance $\\Delta m^2_{e \\mu}$ in vacuum. Similarly, the difference between $s_{\\mu \\tau}$ in (\\ref{s-beta-alpha-vac}) and $s_{\\mu}$ in (\\ref{s-mu-vac}) indicate the discrepancy between disappearance and appearance effective $\\Delta m^2$. If expanded in terms of $s_{13}$ and keeping to order $\\epsilon s_{13}$, $s_{\\mu}$ in (\\ref{s-mu-vac}) and $s_{\\mu \\tau}$ in (\\ref{s-beta-alpha-vac}) are given by \n$s_{\\mu} = c^2_{12} - \\frac{ 2 }{ c^2_{23} } J_r \\cos \\delta$ and $s_{\\mu \\tau} \n= c^2_{12} + \\frac{ \\cos 2\\theta_{23} }{ c^2_{23} s^2_{23} } J_r \\cos \\delta$, respectively. They lead to the effective $\\Delta m^2$ in disappearance and appearance channels as (without expanding by $s_{13}$ in the $\\nu_e$ channel)\n\\begin{eqnarray}\n\\Delta m^2_{e e} &=& \n\\Delta m^2_{31} - s^2_{12} \\Delta m^2_{21}, \n\\nonumber \\\\\n\\Delta m^2_{e \\mu} &=& \n\\Delta m^2_{31} - \\left( s^2_{12} - \\frac{ J_r \\cos \\delta }{ s^2_{23} s^2_{13} } \\right) \\Delta m^2_{21}, \n\\nonumber \\\\\n\\Delta m^2_{\\mu \\mu} &=& \n\\Delta m^2_{31} - \\left( c^2_{12} - \\frac{ 2 }{ c^2_{23} } J_r \\cos \\delta \\right) \\Delta m^2_{21}, \n\\nonumber \\\\\n\\Delta m^2_{\\mu \\tau} &=& \n\\Delta m^2_{31} -  \n\\left( c^2_{12} + \\frac{ \\cos 2\\theta_{23} }{ c^2_{23} s^2_{23} } J_r \\cos \\delta \\right) \\Delta m^2_{21}.  \n\\label{Dm2-vac-summary}\n\\end{eqnarray}\n\nTo summarize the results of discussion in this section, we have shown that the effective two-flavor form of appearance probabilities in vacuum can be defined  with suitably defined effective $\\Delta m^2$ in parallel to those in disappearance channels. However, the notable feature is that the appearance effective $\\Delta m^2$ is different from the corresponding disappearance effective $\\Delta m^2$ by an amount of order $\\epsilon$ which is proportional to $J_r \\cos \\delta$. \n\nWhat is the meaning of this result? Is it natural to expect that the difference is only the term proportional to $J_r \\cos \\delta$? The effective $\\Delta m^2$ is defined in such a way that it absorbs certain effects which come from the genuine three-flavor properties of the oscillation probability, thereby making it the ``two-flavor'' form. The $\\delta$ dependence, not only $\\sin \\delta$ but also $\\cos \\delta$, is one of the most familiar examples of such three-flavor effect \\cite{Asano:2011nj}. The relative importance of $\\cos \\delta$ term is different between the probabilities in the appearance and disappearance channels, and it is reflected to the difference the effective $\\Delta m^2$. Thus, the feature we see in (\\ref{Dm2-vac-summary}) is perfectly natural. The fact that the difference between the appearance and disappearance effective $\\Delta m^2$ consists only of $\\cos \\delta$ term is due to our restriction to first order in $s_{13}$.\n\nOne may ask if the similar discussion can go through for the effective two-flavor form of appearance probabilities in matter. The answer to this question is far from obvious to the present author. Even in the simpler case of $\\nu_e$ related channels in which $P(\\nu_e \\rightarrow \\nu_e)$ has the two-flavor form (see eq.~(\\ref{Pee-matter-SC})) it is unlikely that $P(\\nu_e \\rightarrow \\nu_\\mu)$ can be written as the similar two-flavor form under the framework of $\\epsilon$ perturbation theory. If one looks at eq.~(3.14) in \\cite{Minakata:2015gra}, $P(\\nu_e \\rightarrow \\nu_\\mu)$ has a structure similar to (\\ref{P-beta-alpha-vac}), but all the eigenvalue differences are of order unity. For more about this point see the discussion in the next section. \n\n\\section{Conclusion and Discussion}\n\\label{sec:conclusion}\n\nIn this paper, we have discussed a question of whether the effective two-flavor approximation of neutrino survival probabilities is viable in matter. We gave an affirmative answer using the perturbative treatment of the oscillation probabilities to order $\\epsilon^2$ (to order $\\epsilon$ in $\\nu_{\\mu}$ channel) with the small expansion parameters $\\epsilon = \\Delta m^2_{21} \/ \\Delta m^2_{31}$ assuming $s_{13} \\sim \\sqrt{\\epsilon}$. It allows us to define the effective $\\Delta m^2_{\\alpha \\alpha} (a)$ ($\\alpha = e, \\mu, \\tau$) in matter in an analogous fashion as in vacuum. However, the resultant expression of $\\Delta m^2_{\\alpha \\alpha} (a)$ poses the problem. \n\nIn neutrino oscillation in vacuum the oscillation probability is a function of $L\/E$. However, the effective $\\Delta m^2_{\\alpha \\alpha}$ ($\\alpha = e, \\mu, \\tau$) is defined such that it depends neither on $E$, nor $L$. It is a combination of the fundamental parameters in nature. \nIn matter, however, $\\Delta m^2_{\\alpha \\alpha} (a)$ becomes $E$ dependent, which may be permissible because it comes from the Wolfenstein matter potential $a \\propto E$. In fact, in $\\nu_{e}$ disappearance channel, we have a sensible definition of $\\Delta m^2_{ee} (a)$, eq.~(\\ref{Dm2ee-matter-SC}), in leading order in the renormalized helio-perturbation theory. However, in the $\\nu_{\\mu}$ disappearance channel, we have observed that $\\Delta m^2_{\\mu \\mu} (a)$ (and $\\Delta m^2_{\\tau \\tau} (a)$) is $L$ dependent, although we did the same construction of the effective two-flavor form of the survival probability as in the $\\nu_{e}$ channel. It casts doubt on whether it is a sensible quantity to define. Certainly, it is an effective quantity which results when we seek the two-flavor description of the three-flavor oscillation probabilities in our way. Nonetheless, the basic three-flavor nature of the phenomena seems to prevent such two-flavor description in the $\\nu_{\\mu}$ channel. Thus, the effective $\\Delta m^2$ in matter does not appear to have any fundamental physical significance. \n\nOne may ask: Is $L$ dependence of $\\Delta m^2_{\\mu \\mu} (a)$ an artifact of the perturbative treatment of $s_{13}$ dependence? We strongly suspect that the answer is {\\em No}, though it is very difficult to give an unambiguous proof of this statement at this stage. A circumstantial evidence for the above answer is that we have used the same method of formulating the effective two-lavor approximation in both $\\nu_{e}$ and $\\nu_{\\mu}$ channels. In contrast to $\\Delta m^2_{\\mu \\mu} (a)$, $\\Delta m^2_{ee} (a)$ does not have problem of $L$ dependence. It should also be emphasized that the perturbative expression of $\\Delta m^2_{ee} (a)$ can be obtained from the ``non-perturbative'' expression (\\ref{Dm2ee-matter-SC}) derived by using the renormalized helio-perturbation theory. Notice that the expression (\\ref{Dm2ee-matter-SC}) is free from any ``singularity'' at $r_{A}=1$. Therefore, we have no reason to doubt validity of our method used to formulate the effective two-lavor approximation, which treats both the $\\nu_{e}$ and $\\nu_{\\mu}$ channels in an equal footing. \n\nPutting aside the above conceptual issue, we have examined the matter effect dependences of $\\Delta m^2_{ee} (a)$ and $\\Delta m^2_{\\mu \\mu} (a)$. In fact, they are very different. It produces a strong linear energy dependence for $\\Delta m^2_{ee} (a)$, whereas $\\Delta m^2_{\\mu \\mu} (a)$ only has a weak energy dependence with magnitude of a few \\% level. We expect that the effect of deviation of $\\Delta m^2_{ee} (a)$ from the vacuum expression can be observed in a possible future super-LBL experiments, such as neutrino factory, with $\\nu_{e}$ detection capability. \n\nWe have also examined the question of whether the similar effective two-flavor form of appearance probability exists with the ``appearance effective $\\Delta m^2$''. We have shown that in vacuum it does under the same framework of expanding to order $\\epsilon$. We have observed that the effective $\\Delta m^2$ in disappearance and appearance channels in vacuum differ by the terms proportional to $\\epsilon J_r \\cos \\delta$. In matter, the effective two-flavor form is very unlikely to exist in the current framework. \n\nA remaining question would be: What is the meaning of finding, or not finding, the effective two-flavor description of the neutrino oscillation probability in vacuum and in matter? In vacuum we have shown that to order $\\epsilon$ such description is tenable in both the appearance and the disappearance channels. It is not too surprising because we restrict ourselves into the particular kinematical region at around the first oscillation maximum, and are expanding by $\\epsilon$ to first order, whose vanishing limit implies the two-flavor oscillation. What may be worth remarking is that the effective two-flavor description does not appear to work  in matter under the same approximation as used in vacuum. Nothing magical happens here. Due to the eigenvalue flow as a function of the matter potential all the three eigenvalue differences becomes order unity, and the $\\epsilon \\rightarrow 0$ limit does not render the system the two-flavor one. \n\nFinally, in an effort to understand the reasons why the matter potential dependence of $\\Delta m^2_{\\mu \\mu} (a)$ is so weak, we have derived a general theorem which states that the matter potential dependent terms in the oscillation probability are suppressed by a factor of one of $s^2_{13}$, or $\\epsilon s_{13}$, or $\\epsilon^2$. See appendix~\\ref{sec:hesitation-theorem}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn Gan \\& Xia~\\cite{GX15}, Stein's method for the compound Poisson (and the special cases of Poisson and negative binomial) distribution conditioned upon being greater than $m$ was formulated. This work was initially motivated for the modelling of extreme events such as earthquakes, where the incidence of one extreme event would often lead to many more, hence an understanding of the conditional distribution is often of significant importance. In this paper, we will aim to extend conditional random variable approximation to conditional random process approximation.\n\nTo give some motivation for the following work, we introduce the example of Hawkes processes, a class of processes commonly used in financial literature. For a recent survey see Laub, Taimre \\& Pollett~\\cite{LTP2015}. A Hawkes point process is a point process in time where the incidence rate is in some sense `excited' by other recent points in the process. Intuitively, this is a natural choice of model for earthquakes as given one earthquake has occured, numerous aftershocks typically occur. If we were to approximate such a process with a simple distribution like Poisson, given that there should be a large mass at 0, this approximation would likely be inaccurate. In contrast to such an approach, if we were instead to focus on approximating the distribution given we know that one incident has occurred, we may be able to achieve an accurate approximation. In the case of a Hawkes process, if we know the base\/unexcited incidence rate, the probability of zero events occurring is known, and hence understanding the conditional distribution would lead directly to understanding the complete distribution.\n\nCompound Poisson point process approximation theory using Stein's method was first developed by Barbour \\& M{\\aa}nsson~\\cite{BM02}. However, there are many technical difficulties with the approach and the utility of the results are somewhat limited due to the great generality of the compound Poisson distribution.\nIn contrast to this, Poisson point process approximation via Stein's method has been far more successful, and as a result, we shall therefore focus on formulating conditional Poisson point process approximation theory.\n\nIn this paper we will let $\\G$ denote a locally compact complete separable metric space, and $\\mathcal{H}$ denote the space of all locally finite point measures on $\\G$.\n\nStein's method for Poisson point process approximation was initiated by Barbour~\\cite{barbour88} and Barbour \\& Brown~\\cite{BB92} as a generalisation of the Stein-Chen method by Chen~\\cite{chen75}, and was later refined by Brown, Weinberg \\& Xia~\\cite{BWX00}, Xia~\\cite{xia05} and Xia \\& Schuhmacher~\\cite{SX08}. The general approach is to utilise a generator for which the associated stationary distribution is a Poisson point process with intensity measure $\\Lb$, denoted by $\\Po(\\Lb)$, and then to use suitable techniques involving couplings to find the relevant bounds. For a given configuration $\\xi \\in \\mathcal{H}$, define the operator $\\cA$ on a suitably rich family of functions $h$\n\\[ \\mathcal{A} h(\\xi) = \\int_\\G \\left[h(\\xi+\\d_\\a) - h(\\xi)\\right]\\Lb(\\d_\\a) + \\int_\\G \\left[ h(\\xi-\\d_\\a) - h(\\d_\\a)\\right]\\xi(d\\a). \\]\n\nThe operator $\\cA$ is the generator of a spatial immigration-death process with immigration rate $\\Lb$ on $\\Gamma$, unit per capita death rate and the associated stationary distribution to the generator $\\mathcal{A}$ is a Poisson process with mean measure $\\bm\\L$. We now define $h_f(\\xi)$ to be the solution to the following (if it exists),\n\\[ \\cA h_f(\\xi) = f(\\xi) - \\Po(\\bm{\\L})(f),\\]\nfor all $f$ from a suitable family of functions $\\mathcal{F}$. Then given a point process $\\Xi$, the aim is to use properties of the function $h_f$ to estimate $|\\E f(\\Xi) - \\Po(\\bm\\L)(f)|$ by finding a bound for $|\\cA h_f(\\Xi)|$.\n\nWe will use the term \\emph{conditional} to mean conditional upon at least $m$ atoms in the entire space $\\Gamma$. $\\Xi$ follows the distribution of a \\emph{conditional Poisson point process}, if it has the distribution of a Poisson point process conditional on having at least $m$ atoms in $\\Gamma$. Its distribution will be denoted by $\\Po^\\resm(\\Lb)$. Note in general we will use the notation $\\vphantom{a}^{(m)}$ to denote conditional upon $m$ atoms.\n\n\nWe will now associate a conditional Poisson point process with the limiting distribution of a spatial immigration-death process, in a similar fashion to how it is defined for conditional random variable approximation in Gan \\& Xia~\\cite{GX15}. Given the process is in configuration $\\xi$, with $|\\xi| \\geq m$, then the process will stay at this configuration for an exponentially distributed time with mean $\\frac{1}{|\\xi|\\ind_{|\\xi|>m} + \\L}$, where $\\L = \\Lb(\\G)$. With probability $\\frac{\\L}{|\\xi|\\ind_{|\\xi|>m} + \\L}$, the process will then add a new point into the system at a point following distribution $\\Lb\/\\L$ , or with probability $\\frac{|\\xi|\\ind_{|\\xi|>m} }{|\\xi|\\ind_{|\\xi|>m} + \\L}$, one of the existing points chosen uniformly at random will be removed. The generator of such a process is\n\\begin{align}\n\\cA^\\resm h(\\xi) = \\int_\\G \\left[h(\\xi+\\d_\\a) - h(\\xi)\\right]\\Lb(\\d_\\a) + \\int_\\G \\left[ h(\\xi-\\d_\\a) - h(\\d_\\a)\\right]\\xi(d\\a) \\cdot \\ind_{|\\xi| > m}. \\label{generator}\n\\end{align}\n\n\\begin{lemma}\nThe unique stationary distribution for the generator $\\cA^\\resm$ is $\\Po^\\resm(\\Lb)$.\n\\end{lemma}\n\\begin{proof} We can apply Theorem~7.1 from Preston~\\cite{preston75} to show the existence and uniqueness of the stationary distribution for $\\cA^\\resm$. It remains to show that if $\\Xi \\sim \\Po^\\resm (\\Lb)$ then $\\E \\cA^\\resm h(\\Xi)= 0$, which can be verified via a direct calculation.\n\\end{proof}\nIt should be noted that we can also censor the immigration rate at a level $n>m$ and we would then have a Poisson point process conditioned on having a number of atoms between $m$ and $n$ as the stationary distribution. In line with the original motivation, for this paper we will just focus on conditioning from below.\n\nAs per usual in Stein's method, for any bounded function $f$, we set up a Stein equation, and hope to solve for a $h^\\resm_f(\\xi)$ that satisfies\n\\begin{align}\n \\cA^\\resm h_f^\\resm(\\xi) = f(\\xi) - \\Po^\\resm(\\Lb)(f),\\label{CPPsteineq}\n\\end{align}\nwhere $\\Po^\\resm(\\Lb)(f) = \\E f(Z^\\resm)$ and $Z^\\resm \\sim \\Po^\\resm(\\Lb)$.\n\\begin{lemma}\nFor all bounded functions $f$,\n\\begin{align*}\nh_f^\\resm(\\xi)= -\\int_0^\\infty \\E \\left[ f(\\zx - \\poml(f)\\right]dt,\n\\end{align*}\nwhere $\\zxt\\dot$ is a spatial immigration-death point process with generator $\\cA^\\resm$ and $Z_\\xi^\\resm(0) = \\xi$, is well defined and is the solution to the Stein equation \\Ref{CPPsteineq}.\n\\end{lemma}\nOur metric of choice to evaluate the distance between two point processes will be the $\\overline{d}_2$ metric first introduced in Xia~\\cite{xiathesis} and systematically studied in Schuhmacher \\& Xia~\\cite{SX08}. Important to note is that the metric $\\bar{d_2}$ encapsulates convergence in distribution of point processes, see Proposition~2.3 of Schuhmacher \\& Xia~\\cite{SX08}. \n\n\\begin{definition}\nFor $\\xi = \\sum_{i=1}^n \\d_{x_i}$, $\\eta = \\sum_{i=1}^m \\d_{y_i} \\in \\cH$ with $n \\geq m$, and $d_0\\leq 1$ as the metric on $\\Gamma$, the metric $\\bar{d}_1$ is defined by\n\\[ \\overline{d}_1(\\xi, \\eta) = \\frac{1}{n} \\left( \\min_{\\pi \\in \\Pi_n} \\sum_{i=1}^m d_0(x_{\\pi(i)}, y_i) + (n-m) \\right),\\]\nwhere $\\Pi_n$ is the set of all permutations of $\\{1, \\ldots, n\\}.$\n\\end{definition}\n\\begin{definition}\nLet $\\cF_{\\bar{d}} = \\{ f : \\cH \\rightarrow [0,1]: |f(\\xi) - f(\\eta)| \\leq \\overline{d}_1(\\xi,\\eta)\\ \\forall\\ \\xi,\\eta \\in \\cH\\}$. Then for any two point distributions $P$, $Q$ on $\\cH$, define\n\\[\\overline{d}_2(P,Q) = \\sup_{f \\in \\cF_{\\bar{d}}} \\left| \\int f dP  - \\int f dQ \\right|.\\]\n\\end{definition}\n\nIn this paper, we will assume that $\\Lb$ is \\emph{diffuse}, that is, it has no atoms in $\\G$. If $\\Lb$ is not diffuse, we can approximate it by a diffuse measure accurately by \\emph{lifting} the process as described in Chen \\& Xia~\\cite{CX04}. Furthermore for any configuration $\\xi$ we will also without loss of generality assume that the points in $\\xi$ are all distinct. Similarly to how we can assume $\\Lb$ is diffuse, for any non-distinct points in a configuration $\\xi$ we can `shift' points by a small amount and then take limits due to the continuity of the metric $\\bar{d}_1$.\n\nTo succesfully apply Stein's method, we typically require over the family of functions $f \\in \\cF_{\\bar{d}}$, bounds for:\n\\begin{align}\n\\| \\D h^\\resm \\| &:= \\sup_{\\xi} \\|\\D h_f^\\resm(\\xi;\\a)\\| :=\\sup_{\\xi} \\sup_{f,\\a} | h_f^\\resm(\\xi + \\d_\\a) - h_f^\\resm(\\xi)|,\\label{D1}\\\\\n\\| \\D^2 h^\\resm \\| &:= \\sup_{\\xi} \\|\\D^2 h_f^\\resm(\\xi;\\a,\\b)\\|\\notag\\\\\n&:=\\sup_{\\xi}\\sup_{f,\\a,\\b} |  h_f^\\resm(\\xi + \\d_\\a + \\d_\\b) - h_f^\\resm(\\xi+ \\d_\\a)  - h_f^\\resm(\\xi + \\d_\\b) + h_f^\\resm(\\xi)|.\\label{D2}\n\\end{align}\n\\begin{theorem}\\label{firstdiff}\nDefine\n\\[K_1 :=  \\min \\left( \\frac{1}{m}, \\frac{0.95 + \\log^+\\L}{\\L} \\right).\\]\nFor $m \\geq 1$,\n\\[\\|\\D h^\\resm\\| \\leq \\frac{1}{\\L} + (m+1)K_1,\\]\nand if $\\L > m+2$,\n\\[\\|\\D h^\\resm\\| \\leq\\frac{1}{\\L(\\L - m)} + \\frac{\\L}{\\L - m} K_1,\\]\n\\end{theorem}\n\\begin{theorem}\\label{seconddiff}\nDefine\n\\[ K_2 :=\\frac{2\\log \\L}{\\L} \\ind_{\\L \\geq 1.76} + \\frac{1}{m+1} \\ind_{\\L < 1.76}. \\]\nFor $m \\geq 1$,\n\\begin{align*}\n \\|\\D^2 h^\\resm\\| \\leq &\\min\\Big\\{\\frac{2}{\\L} + 2(m+1)K_1,\\\\\n&\\frac{(4m+3)(m+3)}{(m+3)(2m+2)\\lambda + 2\\lambda^2} + \\frac{4m(m+1)(m+3)}{(m+3)(2m+2) + 2\\lambda} K_1+K_2\\Big\\},\n\\end{align*}\nand if $\\L > m+2$,\n\\begin{align*}\n\\|\\D^2 h^\\resm\\| \\leq \\min\\left\\{ \\frac{2}{\\L(\\L - m)} + \\frac{2\\L}{\\L - m}K_1,\\frac{3\\L + m}{\\L(\\L-m)(\\L+m)} + \\frac{4\\L m}{(\\L-m)(\\L+m)}K_1 + K_2\\right\\}.\n\\end{align*}\n\\end{theorem}\n\nThis paper will be set out in the following manner. Section 2 will focus on proving the above bounds for the Stein factors, this will also include a short diversion into some non-uniform bounds for the first and second difference of $h$, and section 3 will include a short application. \n\n\\section{Bounds for the Stein factors}\n\\subsection{Bounds for the first difference of $h$}\nWhen deriving bounds for unconditional Poisson point process approximations, the canonical technique to calculate bounds for \\Ref{D1} and \\Ref{D2} is to couple the additional points at $\\a$ and $\\b$ independently to $Z_\\xi \\dot$. More precisely, we can set\n\\begin{align*}\nZ_{\\xi + \\a}(t) &= Z_{\\xi}(t) + \\d_\\a \\ind_{\\tau_\\a > t},\\\\\nZ_{\\xi + \\a + \\b}(t) &= Z_{\\xi}(t) + \\d_\\a \\ind_{\\tau_\\a > t} + \\d_\\b \\ind_{\\tau_\\b > t},\n\\end{align*}\nwhere $\\tau_\\a, \\tau_\\b$ are independent exponential random variables with mean 1, and are independent of $Z_\\xi \\dot$. As a result, the bounds for \\Ref{D1} and \\Ref{D2} will generally depend upon how long it takes for the particles at $\\a$ and $\\b$ to die. In our conditional setting, one would hope we would be able to similarly `separate' the extra point at $\\a$ from $\\xi$, and run a (not necessarily independent) pure death process for the point at $\\a$. In the case of conditional Poisson approximation, this approach works, but it does not for point processes. The problem is created by the location of deaths, where as in random variable approximation we essentially only care about the total number. The following reveals the problems that keeping track of particle locations generates.\n\nSuppose $m=1$, we would like to define a coupling such that:\n\\begin{align} Z_{\\xi + \\d_\\a}^\\resone(t) \\stackrel{d}{=} Z_\\xi^\\resone(t) + \\d_\\a \\ind_{\\tau_\\a>t},\\label{approx}\\end{align}\nfor some stopping time $\\tau_\\a$. However, such a coupling does not exist because of the following reason. $Z_{\\xi + \\d_\\a}^\\resone(\\cdot) = \\d_\\a$ is a configuration that can be achieved with a positive probability, but it is not a possible realisation for $Z_\\xi^\\resone(\\cdot) + \\d_\\a \\ind_{\\tau_\\a>\\cdot}$, as $|Z_\\xi^\\resone(\\cdot)| \\geq 1$. To deal with the lack of a direct coupling, we shall formulate an `approximate' coupling that enables us to use the formulation of the right hand side of \\Ref{approx} in the following manner\n\\begin{align}\n\\left|\\D h^\\resm_f(\\xi;\\a)\\right| &= \\left| \\int_0^\\infty \\left[ \\E f(\\zxa) - \\E f(\\zx) \\right] dt \\right|\\notag\\\\\n\t&\\leq \\left| \\int_0^\\infty \\left[ \\E f(\\zxa) - \\E f(\\zx + \\d_\\a \\ind_{\\tau_a>t}) \\right] dt \\right|\\notag\\\\\n\t&\\phantom{V} +  \\left| \\int_0^\\infty \\left[ \\E f(\\zx + \\d_\\a \\ind_{\\tau_a>t}) - \\E f(\\zx) \\right] dt \\right|.\\label{split}\n\\end{align}\nAn important question is, how do we best define $\\tau_\\a$ so as to minimise these two integrals in \\Ref{split}? Examining the first integral of \\Ref{split}, due to our choice of metric, it would be best if we defined $\\tau_\\a$ such that the two processes $\\zxat \\dot$ and $\\zxt \\dot + \\d_\\a \\ind_{\\tau_\\a > \\dot}$ both have the same number of particles at all times. To this end, we can define $\\tau_\\a$ such that \n\\begin{align}\n\\Pr\\left(\\tau_\\a > t\\left| |Z_\\xi^\\resm\\dot|\\right)\\right. = \\exp\\left\\{-\\int_0^t (1 + m \\ind_{|Z_{\\xi}^\\resm(s)| = m}) ds\\right\\}. \\label{taua}\n\\end{align} In essence, given $\\zxat \\dot$ and $\\zxt \\dot + \\d_\\a \\ind_{\\tau_\\a > \\dot}$ have more than $m+1$ particles, then they can be coupled exactly. If the number of particles reaches $m+1$ and the point at $\\a$ is still alive, then in $\\zxt \\dot + \\d_\\a \\ind_{\\tau_\\a > \\cdot}$, all the death rates are forced to the single point at $\\a$.\n\nThe decomposition of \\Ref{split} may initially seem to be somewhat arbitrary, however there is an interpretation for this decomposition if we think about $|\\D h_f^\\resm(\\xi;\\a)|$ in the following manner. Essentially what needs to be considered is how long it takes for $\\zxat \\dot$ and $\\zxt \\dot$ to coalesce, and the aim is to find a coupling so this occurs as quickly as possible. Naively, one might think that we are simply waiting for the point at $\\a$ to die, and until it dies, $\\zxat \\dot$ will have exactly one more point than $\\zxt\\dot$. \nAs long as $|\\zxat\\dot| > m+1$, we can couple of the two processes exactly. The problem occurs if we reach a state where $|\\zxat\\dot| = m+1$ and the point at $\\a$ is still alive. At this time, $\\zxat\\dot$ would still have per capita death rate, but as $\\zxt\\dot$ would only have $m$ particles, it would therefore have a net death rate of 0. From here it is possible that a particle that is not $\\a$ will die from $\\zxat\\dot$, and hence we need to account for this. \n\nThe decomposition in \\Ref{split} is designed to address this issue exactly. Notice that only one of the integrals in \\Ref{split} is ever non-zero at any given time. The term in the second integral is going to be $0$ for $t > \\tau_\\a$. Similarly, the term in the first integral is $0$ until $t > \\tau_\\a$, and then from this point onwards, it may be non-zero. In this manner, we can see that the second integral essentially takes care of $|\\D h_f^\\resm(\\xi;\\a)|$ until an additional death occurs in $\\zxat\\dot$ compared to $\\zxt\\dot$, and the first integral accounts for the chance that the additional death may not be the point at $\\a$. Before we prove Theorem~\\ref{firstdiff} we will need a number of lemmas.\n\n\\begin{lemma}\\label{zmz0}\nThere exists a coupling such that\n\\[ | Z_\\xi^\\resm \\dot| \\geq |Z_\\xi\\dot| \\geq |Z_0\\dot|,\\]\nwhere $Z_\\xi \\dot:= Z_\\xi^{(0)}\\dot$, and $Z_0\\dot$ is a process that follows generator $\\mathcal{A}^{(0)}$ with $Z_\\xi(0) = \\emptyset.$\n\\end{lemma}\n\\begin{proof}\nFor $|Z_\\xi\\dot|$, the birth and death rates are, $\\forall i \\in \\{ 0,1,\\ldots \\}$,\n\\begin{align*}\n\\a_i = \\L ,\\ \\ \n\\b_i = i.\n\\end{align*}\nFor $|\\zxt\\dot|$, the birth and death rates are, $\\forall i \\in \\{ 0,1,\\ldots \\}$,\n\\begin{align*}\n\\a_i = \\L, \\ \\ \n\\b_i = \\begin{cases}\n0 & i = m,\\\\\ni & \\text{otherwise.}\n\\end{cases}\n\\end{align*}\n\nFrom the above, it is clear that the birth rates of both processes match, and the death rates for $|Z_\\xi\\dot|$ are greater than those for $|\\zxt\\dot|$. Intuitively, as $|\\zxt\\dot|$ has strictly lower death rates, it should stochastically dominate $|Z_\\xi\\dot|$. To rigorously show the required result, we can use the coupling from Lindvall~(p.~163)~\\cite{lindvall92}.\n\nFor the second inequality, it is sufficient to note that we can use a coupling to define\n\\begin{align*}\nZ_\\xi(t) = Z_0(t) + D_\\xi(t),\n\\end{align*}\nwhere $D_\\xi(t)$ is a pure-death process with unit per capita death rate independent of $Z_0(t)$. For details of this coupling, see Proposition~3.5 in Xia~\\cite{xia05}. \n\\end{proof}\n\n\n\\begin{lemma}\\label{alphadeath}\nIf $|\\xi| = k\\geq m$, $\\xi(\\{\\a\\})=0$ and $\\tau_{k} = \\inf \\{ t : |\\zxa| = k\\}$, then\n\\[p_{\\L,k} := \\Pr\\left(Z_{\\xi + \\d_\\a}^\\resm(\\tau_{k}) (\\{\\a\\}) = 1\\right) \\leq \\min \\left(\\frac{k}{\\L}, \\frac{k}{k+1}\\right).\\]\n\\end{lemma}\n\\begin{proof}\nFirst, we note that as the process has more than $m$ particles for all $t < \\tau_k$, each particle can be treated independently over this time period. As a result, it suffices to consider $S$, the number of original points from the configuration $\\xi + \\d_\\a$ that still remain in the system at time $\\tau_k$. Due to the independence, given $S$ we then know that each original point is equally likely to survive with probability $\\frac{S}{k+1}$, and we can use this fact to calculate $p_{\\L,k}$. \n\nIf we let $N = k-S$, be the number of new points in the system at time $\\tau_{k}$ that have originated from immigration, it suffices to study $N$, and furthermore $N$ has the same distribution as that of $|Z_0(\\tau_{k})|$, where $Z_0\\dot$ is the process that tracks immigrants and their deaths from the process $\\zxat\\dot$. Therefore conditioning upon $|Z_0(\\tau_{k})|$,\n\\begin{align*}\np_{\\L,k} &= \\sum_{i=0}^{m}\\Pr\\left( Z_{\\xi + \\d_\\a}^\\resm(\\tau_{k}) (\\{\\a\\}) = 1 \\Big{|} |Z_0(\\tau_{k})| = i\\right) \\Pr(| Z_0(\\tau_{k})| = i).\n\\end{align*}\nIf at time $\\tau_k$ there are $i$ particles in our system that arrived by immigration, and $k$ particles in total at time $\\tau_{k}$, then there must be $k-i$ surviving original points in the system, and therefore\n\\begin{align*}\np_{\\L,k} &= \\sum_{i=0}^k \\left( \\frac{k - i}{k+1} \\right)\\cdot \\Pr( |Z_0(\\tau_{k})| = i)\\\\\n\t&=1 - \\frac{1}{k+1} \\E (|Z_0(\\tau_{k})|+1).\n\\end{align*}\nBrown \\& Xia~\\cite{BX01} (Eq.~5.17), showed that $$\\E | Z_0(\\tau_{k})| = -1 + (k+1) \\left( \\frac{\\bar{F}(k-1)}{\\bar{F}(k)} - \\frac{k}{\\L} \\right),$$\nwhere $\\bar{F}(i) = \\sum_{j=i}^\\infty \\Po(\\L)(\\{j\\})$. Therefore\n\\begin{align*}\np_{\\L,k} &= 1 - \\left( \\frac{\\bar{F}(k-1)}{\\bar{F}(k)} - \\frac{k}{\\L} \\right) \\leq \\frac{k}{\\L}.\n\\end{align*}\nThe above bound is redundant if $\\L < k$. To achieve the $\\L$-independent bound, we consider the last transition of the process at time $\\tau_{k}$. Assuming the point at $\\a$ is alive at this time, the probability of the point at $\\a$ surviving the final death event is $\\frac{k}{k+1}$. Therefore if $\\L > k+1$, we use $\\frac{k}{\\L}$ as our bound, otherwise we can use $\\frac{k}{k+1}$. This bound can not be improved in general as for $m=0$ and $k=0$ or $\\L = 0$, equality holds.\n\\end{proof}\n\n\\begin{lemma}\\label{lemmaI1}\nFor $m \\geq 1$, define\n\\begin{align*}\nI_1(\\xi) := \\int_0^\\infty \\E \\left[ f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) - f(\\zx) \\right] dt,\n\\end{align*}\nwhere $U$ is chosen uniformly at random from $\\xi$. Let $|I_1| := \\sup_{\\xi : |\\xi|\\geq m} | I_1(\\xi)|$, then\n\\begin{align*}\n|I_1| &\\leq (m+1) \\left[ \\frac{1}{\\L m} + K_1 \\right].\n\\end{align*}\nFurthermore, if $\\L > m+1$,\n\\begin{align*}\n|I_1| &\\leq \\frac{1}{m(\\L-m)} + \\frac{\\L}{\\L-m}\\cdot K_1.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nIf we pair the death times of $U$ in $\\zxt\\dot$ with $\\a$ in $\\zxaut\\dot$, then both $\\zxaut\\dot$ and $\\zxt\\dot$ can be coupled in such a way that their birth times and death times are matched. Note that we get a complete coupling, i.e. the two processes become identical, when the particle at $\\a$ and the corresponding $U$ leave the systems. Noting that the time of death for the particle at $\\a$, $T_\\a$, satisfies $\\Pr\\left(T_\\a > t \\big| |\\zxu|\\right) = \\exp \\{ -\\int_0^t (1 - \\ind_{|\\zxut(s)| = m-1} )ds \\}$, it can be seen that as the death rate for the particle at $\\a$ `turns off' when there are $m$ particles in the system, if $\\xi$ is larger, then the point at $\\alpha$ is more likely to die sooner. Therefore it suffices to consider the worst case scenario $|\\xi| = m$. Define\n\\begin{align*}\nS &:= \\inf\\{ t : |Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)| = m+1\\}\\sim \\exp(\\L),\\\\\nT &:= \\inf\\{ t : |Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)| = m, t > S\\}.\n\\end{align*}\nUsing the strong Markov property,\n\\begin{align}\n&\\left| \\int_0^\\infty  \\left[ \\E f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) - f(\\zx) \\right] dt \\right| \\notag\\\\\n&= \\left| \\E \\int_0^{T_\\a}  \\left[  f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) - f(\\zx) \\right] dt \\right| \\notag\\\\\n\t&\\leq \\E \\int_0^{S}\\left| f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t))-  f(\\zx)\\right|dt\\notag\\\\\n\t&\\ \\ +\\E \\int_{S}^{T} \\left|  f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) -  f(\\zx) \\right| \\ind_{T_\\a > t}\\ dt  + \\Pr(T_\\a > T) |I_1|.\\label{thing}\n\\end{align}\nFor the first integral of \\Ref{thing}, recalling the choice of $\\cF_{\\bar{d}}$ for our metric,\n\\begin{align*}\n&\\E \\int_0^{S} \\left| f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t))-  f(\\zx) \\right| dt \\leq \\E \\int_0^{S} \\frac{1}{m} dt= \\frac{1}{m} \\E S = \\frac{1}{\\L m}.\n\\end{align*}\nFor the second integral in \\Ref{thing},\n\\begin{align*}\n&\\E \\int_{S}^{T} \\left|  f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) -  f(\\zx) \\right| \\ind_{T_\\a > t}\\ dt\\\\\n\t& = \\E \\int_{S}^{T} \\left|  f(Z_{\\xi - \\d_U}^{(m-1)}(t) + \\d_\\a) -  f(Z_{\\xi - \\d_U}^{(m-1)}(t) + \\d_U) \\right| \\ind_{T_\\a > t}\\ dt\\\\\n\t&\\leq \\E \\int_{S}^T \\frac{1}{|Z_{\\xi - \\d_U}^{(m-1)}(t)|+1} \\cdot e^{-(t-S)} dt \\\\\n\t&\\leq \\E \\int_{S}^\\infty \\frac{1}{|Z_{\\xi - \\d_U}^{(m-1)}(t)|+1} \\cdot e^{-(t-S)} dt \\\\\n\t&\\leq \\int_0^\\infty e^{-t} \\E \\Bigg{[}\\frac{1}{|Z_{Z_{\\xi - \\d_U}^{(m-1)}(S)}^{(m-1)*}(t)|+1}\\Bigg{]}dt \\leq K_1,\n\\end{align*}\nwhere we have used the strong Markov property, $Z_\\xi^{(m-1)*}\\dot$ is an independent process that follow generator $\\cA^{(m-1)}$, Lemma~\\ref{zmz0} in the last inequality, and that the death rate for the particle $\\a$ is 1 in the time interval $[S,T]$. Noting that that we have $\\Pr(T_\\a > T) =p_{\\L,m}$, using Lemma~\\ref{alphadeath} in the following equation yields the result.\n\\[ |I_1| \\leq \\frac{1}{\\lambda m} + K_1 + p_{\\lambda,m}|I_1|.\\]\n\\end{proof}\n\\begin{lemma}\\label{lemmaI2}\nFor $m \\geq 1$, define\n\\[I_2(\\xi) := \\int_0^\\infty \\left[ \\E f(\\zxa) - \\E f(\\zx + \\d_\\a \\ind_{\\tau_\\a>t})\\right]dt,\\]\nwhere $\\tau_\\a$ is defined as in \\Ref{taua}. Let $|I_2| := \\sup_{\\xi : |\\xi| \\geq m} |I_2(\\xi)|$, then\n\\begin{align*}\n\t|I_2| \\leq \\frac{1}{\\L} + mK_1.\n\\end{align*}\nIf $\\L > m+2$, then\n\\begin{align*}\n|I_2| &\\leq \\frac{1}{\\L(\\L - m)} + \\frac{m}{\\L - m} K_1.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nAs long as $|\\zxat\\dot| > m+1$ and $|\\zxt\\dot + \\d_\\a \\ind_{\\tau_\\a>\\cdot}| > m+1$, the death rates for both processes at all points are identical, and we can couple the two processes identically. A problem occurs if we reach the state where there are only $m+1$ points in both systems, and the point at $\\a$ is still alive. In $\\zxat\\dot$, each particle has per capita death rate, but in $Z_{\\xi + \\d_\\a}^\\resm\\dot + \\d_\\a \\ind_{\\tau_\\a>\\cdot}$, only the point at $\\a$ has death rate $m+1$. This scenario is where the two processes could diverge. \n\n\n\nSimilarly to Lemma~\\ref{lemmaI1} it suffices to consider the case $|\\xi| = m$. We decompose $|I_2(\\xi)|$ by considering the first transition of both processes. Our final coupling time will be when the point at $\\a$ is dead in both processes. We can define a coupling such that\n\\begin{itemize}\n\\item With probability $\\frac{1}{m+1+\\L}$ the first transition will be a death and $\\a$ will be the point chosen to die from both $\\zxat\\dot$ and $Z_{\\xi}^\\resm\\dot + \\d_\\a \\ind_{\\tau_\\a>\\cdot}$.\n\\item With probability $\\frac{m}{m+1+\\L}$ the first transition will be a death, and $\\a$ in $Z_{\\xi}^\\resm\\dot + \\d_\\a \\ind_{\\tau_\\a>\\cdot}$ will die, but a uniformly selected point $U$ from $\\xi$ of $\\zxat\\dot$ will die.\n\\item With probability $\\frac{\\L}{m+1+\\L}$ the first transition will be an immigration. \n\\end{itemize}\nIf the first transition is immigration, we can couple the points of two processes exactly until the return to $m+1$ particles. If the point at $\\a$ dies before the return to $m+1$ particles, then the coupling is complete, if not, then we essentially return to the initial starting state.  Therefore,\n\\begin{align}\n|I_2(\\xi)| &\\leq \\frac{m}{m+1+\\L} \\left| \\int_0^\\infty \\left[ \\E f(Z_{\\xi + \\d_\\a - \\d_U}^\\resm(t)) - f(\\zx) \\right] dt \\right| + \\frac{\\L}{m+1+\\L} \\cdot p_{\\L,m+1} |I_2|,\\label{I2bit}\n\\end{align}\nwhere $p_{\\L,m+1}$ is defined as in Lemma~\\ref{alphadeath}. Using Lemma~\\ref{alphadeath} and noting that the integral in \\Ref{I2bit} can be bounded by $|I_1|$ in Lemma~\\ref{lemmaI1}, some rearrangement yields the lemma.\n\\end{proof}\nWe are now ready to complete the proof of Theorem \\ref{firstdiff}.\n\\begin{proof}[Proof of Theorem \\ref{firstdiff}]\nLemma \\ref{lemmaI2} accounts for the first half of \\Ref{split}. For the second integral of \\Ref{split}, \n\\begin{align*}\n&\\left|\\int_0^\\infty \\left[ \\E f(\\zx + \\d_\\a \\ind_{\\tau_\\a>t}) - \\E f(\\zx) \\right] dt\\right|\\\\\n&= \\left|\\int_0^\\infty \\E \\left[ f(\\zx + \\d_\\a) - f(\\zx) \\middle| \\tau_\\a > t \\right] \\Pr(\\tau_\\a > t) dt\\right|\\\\\n&\\leq \\int_0^\\infty\\E \\left[ \\frac{1}{|Z_\\xi^{(m)}(t)|+1} \\right] e^{-t} dt \\leq K_1\n\\end{align*}\nwhere $\\tau_\\a$ is defined as in \\Ref{taua}. \nThe first bound in $K_1$ is true as $|Z_\\xi^{(m-1)}(t) + 1| \\geq m$. For the other bound, using Lemma~\\ref{zmz0},\n\\begin{align*}\n\\E \\left[ \\frac{1}{|Z_\\xi^{(m-1)}(t)| + 1} \\right] \\leq \\E \\left[ \\frac{1}{|Z_0(t)| + 1} \\right],\n\\end{align*}\nand the bound for $\\int_0^\\infty e^{-t} \\E  \\left[ \\frac{1}{|Z_0(t)| + 1} \\right] dt$ can be found in Schuhmacher \\& Xia~\\cite{SX08}. Putting everything together we achieve the final bound.\n\\end{proof}\n\\begin{remark}\\label{sameasuncond}\nIn the proof of Lemma~\\ref{lemmaI2}, if $m=0$ and $|\\xi| = 0$, notice that the probability of the first transition being a death, but $\\a$ not dying from $Z_{\\xi + \\d_\\a}^{(0)}$ is 0. Therefore, the processes never diverge and hence $|I_2| = 0$. Furthermore, this implies $\\|\\D h^{(0)}\\| \\leq K_1$ (albeit with the small modification of using $1$ for the constant term instead of $\\frac{1}{m}$), consistent with the unconditional bounds of Schuhmacher \\& Xia~\\cite{SX08}.\n\\end{remark}\n\n\\begin{remark}\\label{sameasuncond2}\nIt is worth comparing the above bound to bounds in the unconditional case. Proposition~4.1 from Schuhmacher \\& Xia~\\cite{SX08} gives\n\\begin{align}\n\\|\\D h^{(0)} \\| \\leq \\min\\left(1, \\frac{0.95 + \\log^+\\L}{\\L}\\right). \\label{SXfirstdiff}\n\\end{align}\nTherefore our bound in the conditional case is generally slightly worse than the bounds in the unconditional case. However, for large $\\L$, the bound is asymptotically the same as $K_1$, so it appears that the additional term is not too bad as long as $\\L$ is of a reasonable size. \n\\end{remark}\n\nWhen $\\L$ is small, the bounds in Theorem~\\ref{firstdiff} are large and therefore may not be useful in applications. In the unconditional scenario, when $\\L$ is small, the constant bound of $1$ is used. An important question is, does there exist a $\\L$-independent bound for $\\|\\Delta h^\\resm\\|$ like in the unconditional case? The answer to this appears to be no. Consider\n\\begin{align*}\n\\left| \\int_0^\\infty \\E \\left[ f(\\zxa) - f(\\zx) \\right] dt \\right|,\n\\end{align*}\nwith a starting configuration such that $\\left|\\xi\\right| = m$ and $\\L$ is very small. The first transition for $|\\zxat\\dot|$ is going to be a death with probability $\\frac{m+1}{m+1+\\L}$, or an immigration with probability $\\frac{\\L}{m+1+\\L}$. Given that $\\L$ is small, this implies that the first transition is almost certainly going to be a death, and with probability $\\frac{m}{m+1}$ the particle chosen for death is not going to be $\\a$. Meanwhile $\\zxt\\dot$ is going to be unchanged with high probability as the only possible transition is an immigration step upwards which occurs with rate $\\L$. We are therefore likely to reach a state where the processes are differing by a single pair of particles, but the expected time until the next transition is exactly the expected time until the next immigration, $\\frac{1}{\\L}$. Hence, it appears there will unavoidably be a $\\L$-dependent component in the bound. This problem does not occur in the unconditional case as the death rate for the point at $\\a$ is always 1, it does not `turn off' as it does in our conditional scenario.\n\n\n\n\\subsection{Bounds for the second difference of $h$}\\label{CPPPseconddiff}\nWe now seek to extend our approximate coupling idea to calculate bounds for $\\|\\D^2h^\\resm\\|$? Again, we would like to use the canonical couplings of the form\n\\begin{align*}\n\\zxab &\\stackrel{d}{=} \\zx + \\d_\\a \\ind_{\\tau_\\a > t} + \\d_\\b \\ind_{\\tau_\\b > t},\\\\\n\\zxa &\\stackrel{d}{=} \\zx + \\d_\\a \\ind_{\\tau_\\a > t},\\\\\n\\zxb &\\stackrel{d}{=} \\zx + \\d_\\b \\ind_{\\tau_\\b > t},\n\\end{align*}\nfor some waiting times $\\tau_\\a$ and $\\tau_\\b$, and again the conditioning induces complications. In the unconditional case, we can define $\\tau_\\a$ and $\\tau_\\b$ as two independent exponential random variables with rate 1. For exactly the same reasons outlined in the previous subsection, there do not exist any waiting times $\\tau_a, \\tau_b$ that satisfy the coupling above. As a result, we instead approach the problem by again using `approximate' couplings from before to enable us to maintain some semblance of independence.\n\nGiven the filtration of $|\\zxt\\dot|$, we choose to define $\\tau_\\a$ and $\\tau_\\b$ such that $\\tau_\\a$ and $\\tau_\\b$ are conditionally independent copies of the waiting time with distribution as in \\Ref{taua} . This approach presents not only similar complications as discussed earlier for the first difference, but also gives different net death rates for $\\zxabt\\dot$ when compared to $\\zxt\\dot + \\d_\\a \\ind_{\\tau_\\a > \\cdot} + \\d_\\b \\ind_{\\tau_\\b > \\cdot}$. Consider\n\\begin{align}\n&\\int_0^\\infty \\E \\left[ f(\\zxab) - f(\\zxa) - f(\\zxb) + f(\\zx) \\right] dt\\notag\\\\\n\t&= \\int_0^\\infty \\E \\left[ f(\\zxab) - f(\\zx + \\d_\\a \\ind_{\\t_\\a > t} + \\d_\\b \\ind_{\\t_\\b > t}) \\right.\\notag\\\\\n\t&-  f(\\zxa) + f(\\zx + \\d_\\a \\ind_{\\t_\\a > t}) - f(\\zxb) + f(\\zx + \\d_\\b \\ind_{\\t_\\b > t}) \\Big] dt\\notag\\\\\n\t&+ \\int_0^\\infty \\E \\left[ f(\\zx + \\d_\\a \\ind_{\\t_\\a > t} + \\d_\\b \\ind_{\\t_\\b > t}) - f(\\zx + \\d_\\a \\ind_{\\t_\\a > t})\\right.\\notag\\\\\n\t&\\phantom{VVVVV}- \\left. f(\\zx + \\d_\\b \\ind_{\\t_\\b > t}) + f(\\zx) \\right] \\label{fourparts}dt.\n\\end{align}\nIn \\Ref{fourparts}, the absolute value of last integral can be shown to be bounded by existing unconditional bounds, similar to the way the second half of \\Ref{split} was bounded by its unconditional equivalent. Therefore, the main work is to find a bound for the first integral of \\Ref{fourparts}.\n\n\\begin{lemma}\\label{lemmaI3}\nFor $m \\geq 1$, define,\n\\begin{align}\nI_3(\\xi) :=& \\int_0^\\infty \\E \\left[ f(\\zxab) - f(\\zx + \\d_\\a \\ind_{\\t_\\a > t} + \\d_\\b \\ind_{\\t_\\b > t}) \\right.\\notag\\\\\n\t&-  f(\\zxa) + f(\\zx + \\d_\\a \\ind_{\\t_\\a > t}) \\notag\\\\\n\t&- f(\\zxb) + f(\\zx + \\d_\\b \\ind_{\\t_\\b > t}) \\Big] dt\\label{six}\n\\end{align}\nwhere $\\tau_\\a$ and $\\tau_\\b$ are defined as in~\\Ref{taua}. Let $|I_3| := \\sup_{\\xi:|\\xi| \\geq m} |I_3(\\xi)|$, then\n\\begin{align*}\n|I_3| &\\leq \\frac{(4m+3)(m+3)}{(m+3)(2m+2)\\lambda + 2\\lambda^2} + \\frac{4m(m+1)(m+3)}{(m+3)(2m+2) + 2\\lambda} K_1.\n\\end{align*}\nFurthermore, if $\\L > m+2$,\n\\begin{align*}\n|I_3| & \\leq \\frac{3\\L + m}{\\L(\\L-m)(\\L+m)} + \\frac{4\\L m}{(\\L-m)(\\L+m)}K_1.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nTo begin, note as before, it suffices to assume that $|\\xi|=m$ as the worst case scenario. We will decompose $|I_3|$ by conditioning upon the first transition of the processes and assume $|\\xi| = m$.\n\nThe major complication is that the net death rates of the first transition are not the same for the six processes in \\Ref{six}. For the six processes the net death rates are $m+2,\\ 2m+2,\\ m+1,\\ m+1,\\ m+1,\\ m+1$ in the order given by \\Ref{six}. To deal with this, we define the following coupling for the first transition time $T$. The first transition will occur after an exponential time with rate parameter $\\L + 2m + 2$. We need to carefully take note of which particles die in each of the six processes given in the order of \\Ref{six}.\n\\vspace{-0.2cm}\n\nCase 1: With probability $\\frac{m}{\\L + 2m + 2}$,\n$U$ dies, $\\b$ dies,\n$U$ dies, $\\a$ dies,\n$U$ dies,  $\\b$ dies, where $U$ is chosen uniformly from the points in $\\xi$.\n\nCase 2: With probability $\\frac{1}{\\L + 2m + 2}$,\n$\\a$ dies,  $\\a$ dies,\n$\\a$ dies,  $\\a$ dies,\n$\\b$ dies,  nothing dies.\n\nCase 3: With probability $\\frac{1}{\\L + 2m + 2}$,\n$\\b$ dies, $\\b$ dies,\nnothing dies, nothing dies,\nnothing dies, $\\b$ dies.\n\nCase 4: With probability $\\frac{m}{\\L + 2m + 2}$,\nnothing dies, $\\a$ dies,\nnothing dies, nothing dies,\nnothing dies, nothing dies.\n\nAnd finally with probability $\\frac{\\L}{\\L + 2m + 2}$ an immigration occurs at the same location for all the processes. If the first transition is an immigration step, then we can couple each successive pair of processes (in the order of \\Ref{six}) exactly, until the process $\\zxabt\\dot$ returns to a state with $m+2$ particles. Furthermore, for points that exist in more than one pair of processes, they can also be coupled across the pairs. For example, the point at $\\a$ exists in the first, second, third and fourth processes, so we couple the death time of $\\a$ to be the same for all four processes. When $|\\zxabt\\dot|$ first returns to a state with $m+2$ particles we need only check if the points at $\\a$ and $\\b$ have died or not. If one of them has died, then the integrand in \\Ref{six} becomes zero immediately. \n\nSimple calculations show that case 2 and case 3 cancel each other out exactly, upon some further simplification, and noting that the integrand of $I_3$ contributes nothing until the first transition each pair of processes have the same configurations,\n\\begin{align*}\n|I _3| \\leq &\\frac{m}{\\L + 2m + 2} \\left[3 |I_1| + \\| \\Delta h^\\resm \\|\\right] +  \\frac{\\L}{\\L + 2m + 2} p^{(2)}_{\\L, m+2} |I_3|,\n\\end{align*}\nwhere $p^{(2)}_{\\L, m+2}$ represents the probability that given the first transition was an immigration step, the points $\\a$ and $\\b$ are both still alive upon the first return to $m+2$ particles for the process $\\zxabt\\dot$. Noting that $p^{(2)}_{\\L, m+2} \\leq p_{\\L, m+2}$ from Lemma~\\ref{alphadeath} as the event that both the points at $\\a$ and $\\b$ survive is a subset of the event that the point at $\\a$, survives $p^{(2)}_{\\L, m+2} \\leq \\min\\{ \\frac{m+1}{m+3},\\frac{m+2}{\\L}\\}$ (the $\\frac{m+1}{m+3}$ comes from the fact that both points need to survive at least once death event), the bounds in Lemmas \\ref{lemmaI1} and \\ref{firstdiff} give the final result.\n\\end{proof}\nWe now have everything we need to complete the proof of Theorem~\\ref{seconddiff}.\n\\begin{proof}[Proof of Theorem~\\ref{seconddiff}]\nThe first set of bounds are simply twice the bounds for the first difference given in Theorem~\\ref{firstdiff}. \n\nFor the remaining bounds, recall \\Ref{fourparts}. \nLemma~\\ref{lemmaI3} gives a a bound for the first integral. We now need only examine the last integral. \n\\begin{align}\n&\\left|\\int_0^\\infty \\E \\left[ f(\\zx + \\d_\\a \\ind_{\\t_\\a > t} + \\d_\\b \\ind_{\\t_\\b > t}) - f(\\zx + \\d_\\a \\ind_{\\t_\\a > t})\\right.\\right.\\notag \\\\\n\t&\\left.\\phantom{VVVVV}- \\left. f(\\zx + \\d_\\b \\ind_{\\t_\\b > t}) + f(\\zx) \\right]dt\\right|\\notag\\\\\n\t&=\\left| \\int_0^\\infty \\E \\left[ f(\\zx + \\d_\\a + \\d_\\b) - f(\\zx + \\d_\\a)\\right. \\right.\\notag\\\\\n\t&\\left. \\left. \\phantom{VVVVV}- f(\\zx + \\d_\\b)+ f(\\zx)\\middle| \\tau_\\a > t, \\tau_\\b > t \\right]\\Pr(\\tau_\\a > t, \\tau_\\b > t) dt\\right|\\notag\\\\\n\t&\\leq \\int_0^\\infty e^{-2t} \\E \\left[ \\frac{1}{|\\zx| + 2} + \\frac{1}{|\\zx|+1} \\right] dt\\label{constantone}\\\\\n\t&\\leq \\int_0^\\infty e^{-2t} \\E \\left[ \\frac{1}{|Z_\\xi(t)| + 2} + \\frac{1}{|Z_\\xi(t)| + 1} \\right] \\label{parttwo}dt,\n\\end{align}\nwhere in the first inequality we have used that the conditional independence of $\\tau_\\a$ and $\\tau_\\b$ given the natural filtration of $\\zxt\\dot$,\n\\[ \\Pr\\left(\\tau_\\a>t, \\tau_\\b > t \\left| |\\zxt\\dot|\\right)\\right. = \\exp \\left\\{-\\int_0^t (2 + 2 m \\ind_{|\\zxt(s)| = m}) ds \\right\\} \\leq e^{-2t},\\]\nand for the second inequality we have used Lemma~\\ref{zmz0}.\n\nWe can bound \\Ref{constantone} by\n\\begin{align*}\n\\int_0^\\infty& e^{-2t} \\E \\left[ \\frac{1}{|\\zx| + 2} + \\frac{1}{|\\zx|+1} \\right]dt \\\\\n\t&\\leq \\int_0^\\infty e^{-2t}\\left[ \\frac{1}{m+2} + \\frac{1}{m+1}\\right] dt\\\\\n\t&= \\frac{2m+3}{(m+1)(m+2)} \\cdot \\frac{1}{2} < \\frac{1}{m+1}.\n\\end{align*}\nSchuhmacher \\& Xia~\\cite{SX08} bound the quantity in \\Ref{parttwo} with ${\\frac{2\\log \\L}{\\L}}$ when ${\\L\\geq 1.76}$. Recalling our definition\n\\[ K_2 := \\frac{2\\log \\L}{\\L} \\ind_{\\L \\geq 1.76} + \\frac{1}{m+1} \\ind_{\\L < 1.76}. \\]\nWe therefore have,\n\\begin{align}\n\\|\\D^2 h^\\resm\\| \\leq |I_3| + K_2,\\label{D2parts}\n\\end{align}\nand the bound now follows by using the bound from Lemma~\\ref{lemmaI3}.\n\\end{proof}\n\nAn interesting question is whether there are equivalent statements to Remarks~\\ref{sameasuncond} and \\ref{sameasuncond2} for the second difference of $h$. \n\nIf $m=0$, then $\\tau_\\a$ and $\\tau_\\b$ are independent exponential random variables, and hence it is easily seen that $|I_3|=0$. Therefore, this approach is consistent with the unconditional bounds of Schuhmacher \\& Xia~\\cite{SX08}. \n\nThe answer to the question of whether there exists a $\\L$-independent bound for $\\|\\D h^\\resm \\|$ is the same as for the first difference. Using the same arguments as in Remark~\\ref{sameasuncond2} we can see that there appears to always be a need for a $\\L$-dependent component.\n\n\\subsection{Non-uniform bounds for Stein factors}\nThe bounds in Theorems~\\ref{firstdiff} and \\ref{seconddiff} are both uniform in the choice of $\\xi$ and are of order $\\frac{\\log(\\L)}{\\L}$, where this term comes from the unconditional bounds derived in Schuhmacher \\& Xia~\\cite{SX08}. Using a counterexample based on Brown \\& Xia~\\cite{BX95}, Schuhmacher \\& Xia~\\cite{SX08} show that the logarithmic terms in these bounds can not be removed if we want to use uniform Stein factors. In Brown, Weinberg \\& Xia~\\cite{BWX00}, the logarithmic terms are removed from Stein factors in the $d_2$ metric by allowing the Stein factors to rely upon the configurations involved, and alternate upper bounds can be derived. This argument is later simplified into a more elegant result in Brown \\& Xia~\\cite{BX01} and non-uniform bounds in the $\\bar{d}_2$ metric are also given in Schuhmacher \\& Xia~\\cite{SX08}. In this subsection we will also derive non-uniform bounds for our Stein factors. Fortunately, given the lemmas we have already proven, the results are not difficult to arrive at.\n\nFirstly, we will require the following non-uniform unconditional bounds.\n\\begin{lemma}[Schuhmacher \\& Xia~\\cite{SX08}]\\label{L1L2}\n\\begin{align*}\n\\int_0^\\infty e^{-t} \\E \\left[ \\frac{1}{|Z_\\xi(t)|+1} \\right]dt \\leq L_1 := \\frac{1- e^{-(|\\xi| \\wedge \\L)}}{|\\xi| \\wedge \\L}.\n\\end{align*}\n\\begin{align*}\n\\int_0^\\infty e^{-2t} \\E \\left[ \\frac{1}{|Z_\\xi(t)| + 2} + \\frac{1}{|Z_\\xi(t)| + 1} \\right] dt \\leq L_2 := \\min\\left\\{ \\frac{1}{|\\xi| \\wedge \\L}, \\frac{1.09}{|\\xi| + 1} + \\frac{1}{\\L} \\right\\}.\n\\end{align*}\n\\end{lemma}\n\\begin{corollary}\nFor $m \\geq 1$,\n\\begin{align*}\n\\|\\D h^\\resm_f(\\xi;\\a)\\| \\leq \\frac{m+1}{|\\xi| + 1} \\left( \\frac{1}{\\L} + m L_1 \\right) + L_1,\n\\end{align*}\nand if $\\L > m+2$\n\\begin{align*}\n\\|\\D h^\\resm_f(\\xi;\\a)\\| \\leq \\frac{m+1}{|\\xi|+1} \\left( \\frac{1}{\\L (\\L - m)} + \\frac{m}{\\L-m} L_1 \\right) + L_1.\n\\end{align*}\nFor the second difference,\n\\begin{align*}\n\\|\\D^2 & h_f^\\resm(\\xi;\\a,\\b)\\| \\leq \\min\\Bigg\\{ \\frac{2m+2}{|\\xi| + 1} \\left( \\frac{1}{\\L} + m L_1 \\right) + 2L_1,\\\\\n\t&\\frac{(m+2)(m+1)}{(|\\xi|+2)(|\\xi|+1)} \\left[\\frac{(4m+3)(m+3)}{(m+3)(2m+2) + 2\\lambda} + \\frac{4m(m+1)(m+3)}{(m+3)(2m+2)\\lambda + 2\\lambda^2} L_1\\right] + L_2\\Bigg\\}.\n\\end{align*}\nFurthermore, if $\\L > m+2$,\n\\begin{align*}\n\\|\\D^2 h_f^\\resm(\\xi;\\a,\\b)\\| &\\leq \\min\\Bigg\\{ \\frac{2m+2}{|\\xi|+1} \\left( \\frac{1}{\\L (\\L - m)} + \\frac{m}{\\L-m} L_1 \\right) + 2L_1,\\\\\n&\\ \\ \\ \\ \\frac{(m+2)(m+1)}{(|\\xi|+2)(|\\xi|+1)} \\left[ \\frac{3\\L + m}{\\L(\\L-m)(\\L+m)} + \\frac{4\\L m}{(\\L-m)(\\L+m)}L_1 \\right]+ L_2\\Bigg\\}.\n\\end{align*}\n\\end{corollary}\n\n\\begin{proof}\nWe again use our approximate couplings as before, and define $\\tau_\\a$ as in \\Ref{taua}. Then\n\\begin{align*}\n|\\D h^\\resm_f(\\xi;\\a)| &\\leq \\left| \\int_0^\\infty \\E \\left[ f(\\zxa) - f(\\zx + \\d_\\a \\ind_{\\tau_\\a > t}) \\right] dt\\right|\\\\\n\t& \\ \\ \\ + \\left| \\int_0^\\infty \\E \\left[ f(\\zx + \\d_\\a \\ind_{\\tau_\\a > t}) - f(\\zx) \\right] dt \\right|.\n\\end{align*}\nFollowing the same argument as in the proof of Theorem~\\ref{firstdiff}, the second integral can be bound by $L_1$. \n\nFor the first integral, recall that in the proof of Lemma~\\ref{lemmaI1}, we used the `worst case scenario' of $|\\xi| = m$. To achieve a non-uniform bound, we simply relax that restriction. Consider $|I_2(\\xi)|$ where $|\\xi|=k > m$. The processes only diverge in the manner described in the proof of Lemma~\\ref{lemmaI2} if the particle at $\\a$ is alive upon reaching a state where there are $m+1$ particles in the system. Therefore\n\\begin{align*}\n|I_2(\\xi)| &= \\left| \\int_0^\\infty \\E \\left[ f(\\zxa) - f(\\zx + \\d_\\a \\ind_{\\tau_\\a > t}) \\right] dt \\right|\\\\\n\t&\\leq p_{\\L,k} \\cdot p_{\\L,k-1} \\cdot \\ldots \\cdot p_{\\L,m+1} |I_2|\\\\\n\t&\\leq \\frac{k}{k+1} \\cdot \\frac{k-1}{k} \\cdot \\ldots \\cdot \\frac{m+1}{m+2} |I_2|\\\\\n\t&= \\frac{m+1}{k+1} |I_2|\n\\end{align*}\nNote that we can use exactly the same proof for bounding $|I_2|$ as in Lemma~\\ref{lemmaI2} but using our non-uniform bound $L_1$ instead of $K_1$. \nFor $\\| \\D^2 h_f^\\resm(\\xi;\\a,\\b)\\|$, the first bounds are simply twice the first difference for our non-uniform bounds. For the second bounds, similarly to the first difference we note that as in \\Ref{D2parts},\n\\begin{align*}\n\\|\\D^2h_f^\\resm(\\xi;\\a,\\b)\\| \\leq |I_3(\\xi)| + L_2.\n\\end{align*}\nWe can then use the same argument that the processes only diverge if both particles at $\\a$ and $\\b$ are alive upon the first time the process $\\zxabt\\dot$ reaches a state with $m+2$ particles. Therefore if $|\\xi| = k>m$,\n\\begin{align*}\n|I_3(\\xi)| &\\leq p_{\\L,k+1}^{(2)}\\cdot p_{\\L,k}^{(2)} \\cdot \\ldots \\cdot p_{\\L,m+2}^{(2)}|I_3|\\\\\n\t&\\leq \\frac{k}{k+2} \\cdot \\frac{k-1}{k+1} \\cdot \\ldots \\cdot \\frac{m+1}{m+3} |I_3|\\\\\n\t&= \\frac{(m+2)(m+1)}{(k+2)(k+1)}|I_3|.\n\\end{align*}\nAfter substituting $L_1$ and $L_2$ for $K_1$ and $K_2$ in the bounds from Theorem~\\ref{seconddiff} we then get the final results.\n\\end{proof}\n\\begin{remark}\nNote that we use the somewhat crude bound of $p_{\\L,k} \\cdot p_{\\L,k-1} \\cdot \\ldots \\cdot p_{\\L,m+1} \\leq \\frac{m+1}{k+1}$ in the bound for $|I_1(\\xi)|$ (and the similar bound for $|I_3(\\xi)|$) purely for simplicity. Using Lemma~\\ref{alphadeath} appropriately, a sharper $\\L$-dependent bound could also be devised.\n\\end{remark}\n\\subsection{Conditional Bernoulli process approximation}\nIn this section we give a simple example of conditional Poisson point process approximation using Stein's method.\n\nLet $\\G= [0,1]$ with metric $d_0(x,y) = |x-y|$, and let $X_1, \\ldots, X_n$ be i.i.d. Bernoulli random variables with common parameter $p$. $\\Xi = \\sum_{i=1}^n X_i \\d_{i\/n}$ defines a \\emph{Bernoulli process}.\n\nLet $T_1, \\ldots, T_n$ be i.i.d.\\ uniform random variables on $[0,1]$ which are independent of the $X_i$'s. $W = \\sum_{i=1}^n X_i \\d_{T_i}$ defines a \\emph{binomial process}.\n\nWe seek to approximate a conditional Bernoulli process $\\Xi^\\resone$ with a conditional Poisson process $\\Po^\\resone(\\Lb)$, conditioning upon at least $1$ atom. To achieve this, we will first compare the conditional Bernoulli process $\\Xi^\\resone$ with the conditional binomial process $W^\\resone$, and then compare the conditional binomial process with the conditional Poisson process $\\Po^\\resone(\\Lb)$. \n\\begin{theorem}\nUsing the setup above, if we set $\\Lb(dx) = np\\ dx$,\n\\begin{align}\n\\bar{d}_2(\\cL(\\Xi^\\resone), \\Po^\\resone(\\Lb)) &\\leq \\frac{1}{1-(1-p)^n} \\cdot \\Bigg[ \\left( \\frac{1}{2n} + \\frac{p}{2} \\right) \\wedge \\frac{1}{\\sqrt{3np}} \\notag\\\\\n\t&\\ \\ \\ + \\frac{p + 2p(0.95 + \\log^+(np))}{(0.5 \\vee \\sqrt{(n-1)p(1-p)})}\\Bigg],\\label{bound1}\n\\end{align}\nand if $\\L > 3$,\n\\begin{align}\n\\bar{d}_2(\\cL(\\Xi^\\resone), \\Po^\\resone(\\Lb)) &\\leq \\frac{1}{1-(1-p)^n} \\cdot \\Bigg[ \\left( \\frac{1}{2n} + \\frac{p}{2} \\right) \\wedge \\frac{1}{\\sqrt{3np}} \\notag\\\\\n\t&\\ \\ \\ + \\frac{p + np^2(0.95 + \\log(np))}{(np-1)\\cdot (0.5 \\vee \\sqrt{(n-1)p(1-p)})}\\Bigg].\\label{bound2}\n\\end{align}\n\\end{theorem}\n\\begin{proof}\nXia \\& Zhang~\\cite{XZ08} have shown that,\n\\[d_2(\\cL(\\Xi), \\cL(W)) \\leq \\left(\\frac{1}{2n} + \\frac{p}{2} \\right) \\wedge \\frac{1}{\\sqrt{3np}}.\\]\nNoting that $|\\Xi| = |W|$, $\\Pr(|\\Xi| \\geq 1) = \\Pr(|W| \\geq 1)$, and $d_2$ and $\\bar{d}_2$ are the same given both configurations have the same number of particles, hence\n\\[\\bar{d}_2(\\cL(\\Xi^\\resone), \\cL(W^\\resone)) \\leq \\frac{1}{1-(1-p)^n} \\left[\\left(\\frac{1}{2n} + \\frac{p}{2} \\right) \\wedge \\frac{1}{\\sqrt{3np}}\\right].\\]\nWe now need to find a bound for $\\bar{d}_2(\\cL(W^\\resone), \\Po^\\resone(\\Lb))$, where we set $\\Lb(dx) = np\\ dx$. To this end, we need to find a bound for $|\\E \\cA^\\resone h(W^\\resone)|$, where\n\\begin{align}\n\\E \\cA^\\resone h(W^\\resone) &= \\E \\left\\{\\int_0^1 \\left[ h(W^\\resone + \\d_\\a) - h(W^\\resone)\\right] \\Lb(d\\a)\\right\\}\\notag\\\\\n\t&\\ \\ - \\E \\left\\{\\int_0^1 \\left[ h(W^\\resone) - h(W^\\resone - \\d_\\a) \\right] W^\\resone(d\\a) \\ind_{|W^\\resone| \\geq 2}\\right\\}.\\label{CPPex1}\n\\end{align}\nWe re-write the first term in \\Ref{CPPex1} as\n\\begin{align}\n&\\E \\left\\{\\int_0^1 \\left[ h(W^\\resone + \\d_\\a) - h(W^\\resone)\\right] \\Lb(d\\a)\\right\\}\\notag\\\\\n &= \\frac{p}{\\Pr(|W| \\geq 1)} \\sum_{i=1}^n  \\E \\left\\{ \\left[ h(W + \\d_{S_i}) - h(W) \\right] \\ind_{|W| \\geq 1} \\right\\}\\label{CPPex2},\n\\end{align}\nwhere the $S_i$ are i.i.d.\\ uniform random variables on $[0,1]$. For the second term of \\Ref{CPPex1},\n\\begin{align}\n&\\E \\left\\{ \\int_0^1 \\left[ h(W^\\resone) - h(W^\\resone - \\d_\\a) \\right] W^\\resone(d\\a) \\ind_{|W^\\resone| \\geq 2}\\right\\}\\notag\\\\\n\t&\\ \\  =\\frac{1}{\\Pr(|W| \\geq 1)} \\sum_{i=1}^n \\E \\left\\{ \\left[ h(W) - h(W-\\d_{T_i}) \\right] X_i \\ind_{|W| \\geq 2} \\right\\}\\notag\\\\\n\t&\\ \\ = \\frac{p}{\\Pr(|W| \\geq 1)} \\sum_{i=1}^n \\E \\left\\{ \\left[ h(W^i + \\d_{T_i}) - h(W^i) \\right] \\ind_{|W^i| \\geq 1} \\right\\}\\label{CPPex3},\n\\end{align}\nwhere $W^i = W - X_i\\d_{T_i}$, and the last equality is true as $W$ and $W - \\d_{T_i}$ are only different if $X_i=1$ which occurs with probability $p$. Note that since $W^i$ and $T_i$ are independent, we can replace $T_i$ with $S_i$ without changing the value of the expectation. We can now combine~\\Ref{CPPex1}, \\Ref{CPPex2} and \\Ref{CPPex3} to give\n\\begin{align}\n\\left|\\E\\cA^\\resone h(W^\\resone)\\right|  &= \\frac{p}{\\Pr(|W| \\geq 1)} \\Bigg|\\sum_{i=1}^n \\E \\Big[ (h(W + \\d_{S_i}) - h(W))\\ind_{|W| \\geq 1}\\notag\\\\\n\t&\\ \\ \\ \\  - (h(W^i + \\d_{S_i}) - h(W^i)) \\ind_{|W^i| \\geq 1} \\Big] \\Bigg|\\label{CPPex4}\\\\\n\t&=\\frac{p^2}{\\Pr(|W| \\geq 1)} \\Bigg|\\sum_{i=1}^n \\E \\Big[ (h(W^i + \\d_{T_i} + \\d_{S_i}) - h(W^i + \\d_{T_i}))\\notag\\\\\n\t&\\ \\ \\ \\  - (h(W^i + \\d_{S_i}) - h(W^i)) \\ind_{|W^i| \\geq 1} \\Big] \\Bigg|\\label{tada}\\\\\n\t&\\leq \\frac{np^2}{\\Pr(|W| \\geq 1)} \\| \\D^2 h^\\resone \\|,\\notag\n\\end{align}\nwhere in the last equality we used the fact that the summand is non-zero only if $X_1 = 1$, which occurs with probability $p$, and the indicator variable in \\Ref{tada} can be dropped off by noting that we can arbitrarily set $h(\\emptyset) = \\E h(\\d_{S_i})$.\n\nAs an alternative to this bound, we can utilise the approach in Section~4.2 of Schuhmacher \\& Xia~\\cite{SX08}, and use bounds of the first difference of $h$. Following similar arguments to before, using \\Ref{CPPex4} we can show that\n\\begin{align*}\n\\left|\\E\\cA^\\resone h(W^\\resone)\\right|  &= \\frac{np}{\\Pr(|W| \\geq 1)} \\Bigg| \\E \\Big[ (h(W + \\d_{S_0}) - h(W))\\ind_{|W| \\geq 1}\\\\\n\t&\\ \\ \\ \\  - (h(W^1 + \\d_{S_0}) - h(W^1)) \\ind_{|W^1| \\geq 1} \\Big] \\Bigg|,\n\\end{align*}\nwhere $S_0$ independent of $W$ and is uniform on $[0,1]$. Define\n\\[ g(i) = \\E\\big[ h(W + \\d_{S_0}) - h(W) \\big| |W| = i\\big] = \\E \\left[ h\\left(\\sum_{j=1}^i \\d_{T_j}+ \\d_{S_0} \\right) - h\\left(\\sum_{j=1}^i \\d_{T_j} \\right)\\right].\\]\nThen\n\\begin{align}\n|\\E \\cA^\\resone h(W^\\resone)| &= \\frac{np}{\\Pr(|W| \\geq 1)} \\left|\\E [ g(|W|) - g(|W^1|) ] \\right|\\notag\\\\\n\t&= \\frac{np^2}{\\Pr(|W| \\geq 1)} \\left|\\E [g(|W^1| + 1) - g(|W^1|) ] \\right|\\notag\\\\\n\t&\\leq \\frac{2np^2}{\\Pr(|W| \\geq 1)}\\| g \\| d_{TV}(\\cL(|W^1|+1), \\cL(|W^1|)),\\label{CPPex5}\n\\end{align}\nas similarly to earlier $|W| = |W^1+1|$ with probability $p$, and noting that for any two non-negative integer random variables $Z_1, Z_2$ (see Appendix A.1 of Barbour, Holst \\& Janson~\\cite{BHJ}),\n\\[ d_{TV}(\\cL(Z_1), \\cL(Z_2)) = \\frac{1}{2} \\sup_{f: \\Z_+ \\to [-1,1]} | \\E f(Z_1) - \\E f(Z_2)|.\\]\nFurthermore, from Lemma 1 of Barbour \\& Jensen~\\cite{BJ89},\n\\begin{align}d_{TV}(\\cL(|W^1|+1), \\cL(|W^1|)) \\leq 1 \\wedge \\frac{1}{2\\sqrt{(n-1)p(1-p)}}.\\label{CPPex6}\\end{align}\nNoting that $\\|g\\| \\leq \\|\\D h^\\resone \\|$, \\Ref{CPPsteineq}, \\Ref{CPPex5} and \\Ref{CPPex6} imply\n\\begin{align*}\n\\bar{d}_2(\\cL(W^\\resone), \\Po^\\resone(\\Lb)) &= \\sup_f| \\E \\cA h_f(W)|\\\\\n\t &\\leq \\frac{\\| \\D h^\\resone\\| \\cdot np^2}{(1-(1-p)^n) \\cdot (0.5 \\vee \\sqrt{(n-1)p(1-p)})}.\n\\end{align*}\nHence if $\\L = np > 3$,\n\\begin{align*}\n\\bar{d}_2(\\cL(W^\\resone), \\Po^\\resone(\\Lb)) \\leq \\frac{p + np^2(0.95 + \\log(np))}{(1-(1-p)^n) (np-1)\\cdot (0.5 \\vee \\sqrt{(n-1)p(1-p)})},\n\\end{align*}\notherwise,\n\\begin{align*}\n\\bar{d}_2(\\cL(W^\\resone), \\Po^\\resone(\\Lb)) \\leq \\frac{p + 2p(0.95 + \\log^+(np))}{(1-(1-p)^n) \\cdot (0.5 \\vee \\sqrt{(n-1)p(1-p)})}.\n\\end{align*}\n\\end{proof}\nWe assess these bounds under two scenarios. First, when $n$ is fixed and $p \\to 0$. In this scenario we would use \\Ref{bound1} as our bound. The bound appears to not be particularly good due to the term $\\left( \\frac{1}{2n} + \\frac{p}{2} \\right) \\wedge \\frac{1}{\\sqrt{3np}}$. This is because this bound was derived by Xia \\& Zhang~\\cite{XZ08} with the intention to be used for large $n$. There exists a possibility that the bound \\Ref{bound1} can be improved for this scenario by improving the unconditional bound for $d_2(\\cL(\\Xi), \\cL(W))$.\n\nIn the case where $p$ is fixed and $n \\to \\infty$, we use the bound in \\Ref{bound2}, and this bound appears to be quite good. The bound is of order \n\\[\\frac{\\sqrt{p}(0.95 + \\log(np))}{\\sqrt{n(1-p)}},\\]\nwhich is asymptotically equivalent to the unconditional bounds. Considering Remark~\\ref{sameasuncond2}, this is what we would expect.\n\\begin{remark}\nIt would undoubtedly be nice to have an example of a Hawkes point process. However the inherent `independent increment' nature of the Poisson process would make this unsuitable for any non-trivial Hawkes point processes, as a Hawkes point process usually will contain clustering. A more natural choice of point process for approximation would be a conditional compound Poisson point process. This paper is intended as a first step into understanding how one can manipulate generators to approximate conditional point processes. It remains to be seen if it can be generalised to a wider class of point processes by applying different conditions to the immigration or death processes. \n\\end{remark}\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction \\label{sec1}}\n\nThe causes of the world's worst industrial disaster at Bhopal on 2--3 December 1984 are still being debated in the international media, more than 25 years after the event. Was it caused by neglect, parsimony, or procrastination by Union Carbide on training, safety and maintenance? Corruption, sabotage and cover-up? Inadequate government regulation? Any or all of the above may well have helped set up the worst possible scenario --- for it could not have been any worse --- but they are  contributing factors rather than causes. (A brief account of the disaster is given in the Appendix.) \n\nThe primary \\textit{cause} of the thermal runaway that led to the venting of a lethal mist of methyl isocyanate  (MIC) over Bhopal city was physicochemical.  In this work I present a stability analysis of a simple dynamical model for the MIC-H$_2$O reacting system, revealing oscillatory thermal misbehaviour that cannot be predicted using classical ignition theory.  A similar instability is shown to govern the explosive thermal decomposition of the organic hydroperoxide triacetone triperoxide, in liquid solution. The provenance of oscillatory thermal instability on the nanoscale is elucidated, and shown to lie in the ability of the reactant molecules to store energy in the internal molecular motions --- in other words, the heat capacity. \n\nDespite the enormity of the Bhopal disaster little or no research has been published that elucidates the fundamental physico-chemical cause of thermal runaway in liquid reactive systems such as MIC hydrolysis. In terms of achieving the Millenium Development Goals (MDG) it seems rather important that thermoreactive processes in liquids are thoroughly investigated and the knowledge disseminated widely.  Given the the horrific legacy of the disaster, the long term adverse health effects in children and adverse reproductive effects in women of MIC exposure that have been well-documented \\citep{Mishra:2009},  such knowledge is relevant to the MDG of Child Health and Maternal Health.  More generally people have a right to expect that thermally unstable and hazardous liquids are stored safely. \n\nThe~Millennium~Development Goals~Report~\\citep{mdg:2010} highlights the challenges posed by conflicts and armed violence to human security and MDG achievements. A new and growing threat to people's security is the use of liquid peroxide explosives by terrorists. The ingredients for making such bombs are cheap and widely available and they cannot be detected by metal detectors and nitrogenous explosives detectors, or distinguished from hand lotion by x-ray machines. Liquid peroxide-based explosives were used in the suicide attacks on the London transit system in 2005, which killed 56 people, and  the terrorists convicted of the foiled 2006 transatlantic aircraft conspiracy had plotted to blow up a number of planes using liquid peroxide explosives. (Many more accounts of  peroxide misuse incidents are easily found on the web.) To this day there are severe restrictions on carrying liquids through security barriers at most airports. It seems grimly inevitable that the use of liquid peroxide explosives as mass murder weapons will increase. Knowledge of their fundamental mechanism of action may help to counter their use. \n\n\n\n\nThe open literature on theoretical and experimental validations of thermal  runaway criteria and parametric sensitivity  in batch reactors and storage tanks was summarized by  \\citet{Velo:1996}. In defining critical conditions they, along with  other authors cited therein, begin with the assumption that storage tanks can be modelled as well-stirred batch reactors with linear thermal coupling to the environment. \nHowever batch reactors have no non-trivial steady states, and there is no general theory for determining whether a thermal excursion will grow or decay.  It is  shown in this work that a simple model with nonequilibrium steady states that is also spatially homogeneous  --- the continuous-flow stirred tank reactor (CSTR) paradigm --- can provide great insight into thermoreactive instabilities in liquid systems, and provide fundamental causative information that cannot easily be extracted from detailed numerical simulations that include convective motions. \n\nIn section \\ref{sec2} I describe the chemical reactions and provide the relevant data, taken from the literature, for the physical properties of the reactants and thermodynamic and kinetic parameters. The CSTR paradigm is described in section \\ref{sec3} and the equations are given using dimensionally consistent units and in terms of dimensionless variables and parameter groups. In section \\ref{sec4} the results of numerical stability analyses of the equations are given, where numerical values of the parameters for MIC hydrolysis and for triacetone triperoxide thermal decomposition in solution were used in the equations. Some points regarding the applicability of the CSTR paradigm are discussed in section \\ref{sec5}, and the nanoscale aspects of oscillatory thermal runaway are elucidated through examining the behaviour in the limits of the two timescales of the relaxation oscillation. A summary of the conclusions is given in section \\ref{sec6}. \n\n\n\n\n\n\\section{ \\label{sec2}Chemistry and  data} \n\n\\subsection{MIC hydrolysis}\n\nIsocyanates hydrolyse exothermically to the corresponding amine and carbon dioxide.  In excess water isocyanates react exothermically with the hydrolysis product amine to form the disubstituted urea \\citep{Saunders:1948,Dsilva:1986}. With MIC the product is N,N-dimethyl urea and the reaction sequence is as follows:\n \\begin{align} \n \\text{CH}_3\\text{NCO}_\\text{(l)}  + \\text{H}_2\\text{O}_\\text{(l)}  &\\overset{k_1(T)}{\\longrightarrow}  \\text{CH}_3\\text{NH}_{2\\text{(aq)}}  + \\text{CO}_{2\\text{(aq)}}  \\tag{R1}\\\\\n \\text{CH}_3\\text{NCO}_\\text{(l)} + \\text{CH}_3\\text{NH}_{2\\text{(aq)}}  &\\overset{k_2(T)}{\\longrightarrow}  \\text{CH}_3\\text{NHCONHCH}_{3\\text{(aq)}} .\\tag{R2}\n \\end{align} \n For reaction R2 $\\Delta H_2 (298\\,\\text{K})= -174.6$\\,kJ\/mol and for the sequence overall $\\Delta H_\\text{tot}(298\\,\\text{K})=-255$\\,kJ\/mol \\citep{Lide:2009}. \n \n\n\n\nA chemical analysis of the residue in the MIC storage tank (Tank 610) at the Union Carbide plant at Bhopal, sampled seventeen days after the event, found a variety of MIC condensation products, mainly the cyclic trimer \\citep{Dsilva:1986}.  However, experiments  indicated that these condensations must have been initiated at temperatures and pressures well above the normal boiling point of MIC, so for modelling the initial thermal runaway these reactions need not be considered. No kinetic data are available for reaction R2 so only reaction R1 is used in the model. It will be seen from the results that reaction R1 alone is sufficient to induce thermal runaway. Relevant physicochemical data are given in table \\ref{table1}. \n\\begin{table}\\caption{\\label{table1}Physical, kinetic, and thermochemical parameters for MIC hydrolysis.}\n{\\begin{tabular}{p{0.55\\columnwidth}p{0.15\\columnwidth}p{0.2\\columnwidth}}\n\\hline\\hline\nMolecular weight MIC &57.051&\\\\\nSpecific heat capacity $C_p^\\circ$(298) MIC &1959\\,J\/(kg\\,K)& \\citet{Perry:2008}\\\\\nSpecific heat capacity $C_p^\\circ$(298) H$_2$O&4181\\,J\/(kg\\,K)&\\\\\nBoiling point  MIC at 1\\,atm& 38.3{\\degr}C& \\citet{Lide:2009}\\\\\nDensity  MIC  at 25{\\degr}C & 0.9588\\,g\/cm$^3$&\\citet{Lide:2009} \\\\\nR1 reaction enthalpy& 80.4\\,kJ\/mol&\\citet{Lide:2009}$^\\dag$\\\\\nR1 activation energy&65.4\\,kJ\/mol&\\citet{Castro:1985}\\\\\nR1 pseudo first order frequency factor & 4.13e08\/s&\\citet{Castro:1985}\\\\\n\\hline\\hline\n\\end{tabular}}\n$^\\dag$From standard enthalpies of formation at 298\\,K. \n\\end{table}\n\n\n\n\\subsection{Thermal decomposition of triacetone triperoxide (TATP)}\nTriacetone triperoxide, a cycle trimer,  is an explosive made by mixing acetone and hydrogen peroxide, both of which substances are legal, cheap and readily available over the counter. Pure TATP is a white crystalline powder that is soluble in organic solvents. The thermal decomposition of TATP does not involve combustion; the main reaction products are acetone,  some carbon dioxide, and ozone \\citep{Eyler:2000,Oxley:2002}. Its high explosive power is in part due to the large entropy increase of the formation of four gas molecules from one condensed-phase molecule \\citep{Dubnikova:2005}. Relevant parameters for the thermal decomposition of TATP in toluene are given in table \\ref{tatp}. \n\n\\begin{table}\\caption{\\label{tatp}Physical, kinetic, and thermochemical parameters for thermal decomposition of TATP in toluene.}\n{\\begin{tabular}{p{0.55\\columnwidth}p{0.15\\columnwidth}p{0.2\\columnwidth}}\n\\hline\\hline\nMolecular weight  &222.2356\\,g\/mol&\\\\\nSpecific heat capacity $C_p^\\circ$(298) toluene &1698.25\\,J\/(kg\\,K)& \\citet{Perry:2008}\\\\\nBoiling point toluene  at 1\\,atm&110.8 {\\degr}C& \\\\\nDensity of toluene at 298\\,K&866.9\\,kg\/m$^3$&\\\\\nReaction enthalpy & 330--420\\,kJ\/mol$^\\dag$&\\cite{Dubnikova:2005}\\\\\nActivation energy&178.52\\,kJ\/mol&\\citet{Eyler:2000}\\\\\nFrequency factor & 9.57e16\/s&\\citet{Eyler:2000}\\\\\nFeed concentration of TATP& 2\\,mol\/kg&\\\\\n\\hline\\hline\n\\end{tabular}}\n$^\\dag$Depending on reaction products. \n\\end{table}\n\n\\section{The CSTR paradigm\\label{sec3}}\n The spatially homogeneous flow reactor, or reacting mass or volume, in which a reactant undergoes a first order, exothermic conversion is a simple but elucidatory model for thermoreactive systems when it is appropriate to ignore convection, because as a dynamical system it has non-trivial steady states that can be analysed for stability. The dynamical mass and enthalpy equations may be written as\n\\begin{align}\nM\\frac{\\text{d}c}{\\text{d}t} =  &Mze^{-E\/RT}c + F(c_f-c) \\label{e1}\\\\\nMC_r\\frac{\\text{d}T}{\\text{d}t} &=  (-\\Delta H)Mze^{-E\/RT}c +F(C_fT_a-C_rT) -L(T-T_a)  \\label{e2}.\n\\end{align}\nThe symbols and quantities are defined in table \\ref{table2}. \nFor numerical and comparative reasons  it is more convenient to work with the following dimensionless system corresponding to equations  (\\ref{e1}--\\ref{e2}):\n\\begin{align}\n\\frac{\\text{d}x}{\\text{d}\\tau}&=-xe^{-1\/u}+f(1-x)\\label{e4}\\\\\n\\varepsilon\\frac{\\text{d}u}{\\text{d}\\tau}&=xe^{-1\/u} +\\varepsilon f(\\gamma u_a- u) - \\ell(u-u_a),\\label{e5}\n\\end{align}\nwhere the dimensionless groups are defined in table \\ref{table2}.\nNumerical analysis of equations (\\ref{e4}--\\ref{e5}) was carried out using values of the dimensionless groups calculated from the data in tables \\ref{table1} and table \\ref{tatp} and assigned values of the inverse residence time $f$, heat loss coefficient $\\ell$, and inflow concentration $c_f$.  \n\n\\begin{table}[h]\\caption{\\label{table2}Quantities,  definitions,   and units.}\n{\\vbox{\\begin{tabular}{p{0.04\\columnwidth}p{0.5\\columnwidth}p{0.04\\columnwidth}p{0.1\\columnwidth}}\n\\hline\\hline\n$A$& reaction frequency&&s$^{-1}$\\\\\n$c$ & $c(t)$, concentration of reactant & &mol\/kg\\\\\n$c$ & inflow reactant concentration & &mol\/kg\\\\\n$C_r$&specific heat capacity of reaction mixture& &J\/kg\\,K\\\\\n$C_f$&specific heat capacity of the inflow stream& &J\/kg\\,K\\\\\n$E$ & activation energy &&J\/mol\\\\\n$F$ & flow through rate  & &kg\/s\\\\\n$\\Delta H$& reaction enthalpy& &J\/mol\\\\\n$M$ &mass of reaction mixture & &kg\\\\\n$R$ &gas constant & 8.314&J\/mol\\,K\\\\\n$t$ & time & &s\\\\\n$T$& $T(t)$ reaction temperature &&K\\\\\n$T_a$ & ambient temperature &&K\\\\\n$L$ & heat loss coefficient &&W\/K\\\\ \n\\hline\n\\end{tabular}\n\\begin{tabular}{p{0.04\\columnwidth}p{0.4\\columnwidth}p{0.04\\columnwidth}p{0.2\\columnwidth}}\n$\\varepsilon$&${C}_rE\/c_f(-\\Delta H)R$& $\\tau$& $ tA$\\\\\n$f$&$ F\/MA$&$u$&$ RT\/E$\\\\\n$\\gamma$&$C_f\/C_r$&$u_a$&$ RT_a\/E$\\\\\n$\\ell$&$ LE\/c_fMA(-\\Delta H)R$&$x$ & $ c\/c_f$\\\\ \n\\hline\\hline\n\\end{tabular}}\n}\n\\end{table}\n\n\\section{Results} \\label{sec4}\n\\subsection{MIC hydrolysis}\n\nFrom equations \\ref{e4} and \\ref{e5} in the steady state we can define the rate of reactive heat generation as the nonlinear term\n$$\nr_g\\equiv fe^{-1\/u}\/(e^{-1\/u}+f),\n$$\nand the rate of non-reactive cooling as the linear term\n$$\nr_c\\equiv -u(\\varepsilon f+\\ell)+u_a(\\varepsilon\\gamma f+\\ell).\n$$\nFrom classical ignition theory the reacting mixture self-heats if $r_g$ exceeds $r_l$. Thermal runaway occurs if $r_g$ exceeds $r_l$ beyond a system-specific threshold; for the hydrolysis of MIC this is taken as the boiling point of MIC. These rates were computed using data from table \\ref{table1} and  plotted in figure~\\ref{figure1}, where the temperature is labeled in dimensional units. \n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.6]{figure1.pdf}}\n  \\caption[]{\\label{figure1}Rates of reactive heat generation $r_g$ (red) and heat loss $r_l$ (blue) versus  $T$ from equations (\\ref{e4}--\\ref{e5}). $u_a=0.0379$ (corresponding to $T_a=292\\,K $), $f=1.7$, $\\ell=700$, $\\varepsilon=10$. }\n\\end{figure}\nWe see that the system self-heats until the reaction temperature $T$ reaches the steady state temperature of $\\sim$305\\,K at which the heating and cooling rates are balanced. Since the boiling point of MIC is 312\\,K, according to this diagram  the Bhopal disaster did not happen.  On the basis of this diagram we would not expect a thermal runaway to develop, even when the ambient temperature is allowed to drift slowly up to 292\\,K. \n\nHowever thermal balance diagrams such as that in figure \\ref{figure1} can be dangerously misleading because they infer stability rather than assess stability rigorously, although such diagrams are often used in chemical reactor engineering. The steady states, periodic solutions, and stability analysis of equations (\\ref{e4}--\\ref{e5}) were computed numerically \\citep{Doedel} and yielded a dramatically different picture of the the thermal stability of MIC hydrolysis. Figure \\ref{figure2} shows a bifurcation diagram in which the steady state temperature  and the temperature amplitude envelope of periodic solutions are plotted as a function of $T_a$. \n\\begin{figure}[ht]\n\\centerline{\n\\includegraphics[scale=0.9]{figure2.pdf} }\n  \\caption[]{\\label{figure2} Bifurcation diagram. Stable steady states are plotted with solid blue line, unstable steady states with dashed red line, and the amplitude envelope of periodic solutions is marked with thin dotted magenta line. $H_1$ and $H_2$ label the Hopf bifurcation and the large  \\textbf{\\ding{81}} marks the change in stability of the limit cycles. $f=1.7$, $\\ell=700$, $\\varepsilon=10$.}\n\\end{figure}\n\nThe steady state is stable at $T_a\\approx 286\\,K$, the temperature at which the tank of MIC had been held for several months. As $T_a$ is quasistatically increased the reaction temperature $T$ increases slowly, but at $T_a=290.15\\,K$ the stability analysis flags an abrupt change in the nature of the solutions. At this point the steady state solutions become unstable at a Hopf bifurcation and the hard onset of a high amplitude thermal oscillation ensues. Clearly, at $T_a=292\\,K $ we have catastrophic thermal runaway, contrary to the  prediction given by figure \\ref{figure1}. (In the resulting superheated fluid the exothermic condensation reactions would increase the temperature even further.) \n\nThis is quite different from classical ignition of a thermoreactive system, which occurs at a steady-state turning point. The dynamics of oscillatory thermal runaway can be understood by studying the close-up of the region around the Hopf bifurcation $H_1$ shown in figure  \\ref{figure3}. \n\\begin{figure}[t]\n\\centerline{\n\\includegraphics[scale=0.7]{figure3.pdf}}\n  \\caption[]{\\label{figure3} Close-up of the region around the Hopf bifurcation $H_1$ in figure \\ref{figure2}. }\n\\end{figure}\n$H_1$ is subcritical and the emergent limit cycle is \\textbf{un}stable. The amplitude envelope of the unstable limit cycles is marked with a thin dotted line; they grow as $T_a$ is \\textbf{de}creased. At the turning point \\raisebox{-1ex}{\\Large\\textbf{*}} of the periodic solution branch  the limit cycles become stable. Thermal runaway \\textit{may} occur if there are significant perturbations while $T_a$ is within the regime \\raisebox{-1ex}{\\Large\\textbf{*}}--$H_1$, and it \\textit{must} occur when $T_a$ drifts above $H_1$.  In principle the rapid thermal excursion takes the system to the amplitude maximum of the stable limit cycle.  In reality the reactant and products have vaporised, the pressure has soared,  the peak temperature is far above the boiling point of MIC, and the system must vent or explode. But it must be emphasized that the thermal runaway is due to oscillatory instability rather than classical ignition at a turning point. \n\n\nThe presence of oscillatory instability is all-pervasive and dominant in this system. This can be appreciated by inspection of figure \\ref{figure4}, \n\\begin{figure}[h]\n\\centerline{\n\\includegraphics[scale=0.7]{figure4.pdf}}\n  \\caption[]{\\label{figure4} The locus of Hopf bifurcations is marked with a solid line, the locus of steady-state turning points is marked with dashed line.  $\\ell=700$, $\\varepsilon=10$. }\n  \\end{figure}\n  a plot of the loci of the steady state turning points and the Hopf bifurcations of equations (\\ref{e4}--\\ref{e5}) over the two parameters $u_a$ and the inverse residence time $f$. \nIn the filled region thermal runaway will always be oscillatory. The bistable regime, indicated by the dashed line, occurs at very high flow rates (short residence times). However, classical thermal runaway at a steady state turning point does not occur because the oscillatory instability is still present and dominant.\n\nTwo computed time series for $F=0.0016$\\,kg\/s are compared in figure \\ref{figure5}. \n\\begin{figure}[h]\n\\centerline{\n\\includegraphics[scale=.4]{cstr3ts1}\n}\n\\caption{\\label{figure5} Computed time series for MIC hydrolysis with $F=0.0016$\\,kg\/s,  $L=560$\\,W\/K. Upper plot: $T_c=308.4$\\,K, lower plot: $T_c=308.5$\\,K}\n\\end{figure}\nIn the upper plot $T_a=308.4$\\,K and the oscillations decay to a stable steady state. However, the onset of thermal instability is violent: in the lower plot $T_a=308.5$\\,K and the transient does not decay but swings into sustained high amplitude relaxation oscillations with a period of about 166\\,s. \n\n\nThe behaviour of the system under a slow upwards drift of the ambient temperature can be simulated easily; a time series with $\\d T_a\/\\d t=0.02^\\circ{C}$\/s is shown in figure \\ref{figure6}, \n\\begin{figure}[h]\n\\centerline{\n\\includegraphics[scale=.4]{cstr3ts3}\n}\n\\caption{\\label{figure6} Computed time series for MIC hydrolysis with drift in thermostat temperature of 0.02\\degr{C}\/s.  $F=0.0016$\\,kg\/s,  $L=560$\\,W\/K. }\n\\end{figure}\nwhich confirms the abrupt onset of the instability. Of note is the  decay in amplitude of the oscillations as the thermostat temperature \\textit{increases}; physically this occurs because the reactant is consumed faster than it is supplied as the temperature increases. \n\n\\subsection{Triacetone triperoxide}\nThe steady states, periodic solutions, and stability analysis of equations (\\ref{e4}--\\ref{e5}) were computed using the data for TATP thermal decomposition in table \\ref{tatp} and results are shown in figures  \\ref{tatp-2p} and \\ref{tatp-ts1}.  \n\\begin{figure}[h]\n\\centerline{\n\\includegraphics[scale=0.35]{tatp-2p}}\n  \\caption{\\label{tatp-2p}  The locus of Hopf bifurcations for the triacetone triperoxide system. }\n\\end{figure}\n\nIn figure \\ref{tatp-2p} the loci of the Hopf bifurcations are plotted over $T_a$ and $F$, and a point within the oscillatory region was selected to compute the time series in figure\\ref{tatp-ts1}. The system exhibits violent relaxation oscillations, suggesting that explosive thermal decomposition of TATP is initiated at the onset of this oscillatory behaviour rather than by classical ignition. \n\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.5]{tatp-ts1}}\n  \\caption{\\label{tatp-ts1} Strong relaxation oscillations in the TATP thermal decomposition. The time series was computed for $T_a=460$\\,K, $F=5$e-04\\,kg\/s. }\n\\end{figure}\n\\clearpage\n\n\\section{Discussion \\label{sec5}}\n\nThe tendency for oscillatory thermal runaway may be typical of exothermically reactive organic liquids. In the work of \\citet{Ball:1995a} the hydration of 2,3-epoxy-1-propanol in a CSTR was found to exhibit similar non-classical thermal misbehaviour. Here it is shown that  the thermal runaway that led to the Bhopal disaster may have been initiated at an oscillatory instability, and that liquid peroxide explosions may be initiated at an oscillatory instability rather than by classical thermal ignition. Similar results have been obtained using parameters for the decomposition of cumene hydroperoxide  in equations (\\ref{e4}--\\ref{e5}) \\citep{Ball:2010}. \n\nIs it realistic to model a reacting volume inside a storage tank --- or, for that matter, a peroxide bomb --- as a well-stirred flow reactor? Yes, on a timescale over which the reacting volume remains relatively constant and gradientless relative to the much faster rate of reaction. \nFor the purposes of this analysis in which the focus is on the dynamics we can circumscribe a reacting volume in which the spatial gradients are insignificant in comparison to the time evolution, and therefore can be neglected.  In this case the CSTR paradigm is appropriate. If this approximation does not hold, then we are free to reduce the circumscribed volume until it does. There is nothing particularly artificial or manipulative in doing this; it is just a simplest case scenario for which the powerful tools of stability and bifurcation theory can be applied. Much of the heat transfer would be convective rather than conductive, and on convective timescales the approximation does not hold --- but that is for a separate study. \n\n\\subsection{Nanoscale aspects of oscillatory thermal instability} \nSince this is a book about all things nano, it is pertinent to elucidate the mechanism of oscillatory thermal instability on the nanoscale. But before we zoom in to nano spatial scales we need to discuss the characteristics of relaxation oscillators in general and examine the two time scales of the relaxation oscillation solutions of equations \\ref{e4} and \\ref{e5}. \n\nRelaxation oscillators are often and easily implemented in electrical and electronic circuits,   but any dissipative dynamical system with nonequilibrium steady states  and  two or more dynamical state variables has the potential to exhibit relaxation oscillations. In general they are not smoothly sinusoidal in form. Instead, relaxation oscillators are characterised by energy dissipation and energy accumulation occurring on different timescales. There may be relatively slow dissipation until the system reaches a threshold state at which the internal energy rapidly and nonlinearly increases, or rapid dissipation   followed by slow but accelerating increase  in internal energy. This latter two-time dynamics occurs with the thermochemical oscillator in  equations \\ref{e4} and \\ref{e5}. \n\nThe two time scales in each period evident in figure \\ref{figure5} are shown in figure~\\ref{ro}. On the `fast' time scale, left, during the few seconds between the temperature maximum and about 990\\,s the concentration remains close to zero. On the `slow' time scale, right, the system barely heats at all over the long interval, although evidently some reaction takes place because reactant accumulates nonlinearly, reaches a maximum, then declines slowly before the exponential spike.  \n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=.25]{relax-oscillation}}\n  \\caption{\\label{ro} Left: a `fast' or dissipative time interval. Right: a 'slow' time interval.}\n\\end{figure}\n\n\n We can extract these two time scales approximately  from equations \\ref{e4} and \\ref{e5}. Define `fast' or `stretched' or dissipative  time  as $\\tau^\\prime\\equiv \\tau\/\\varepsilon $ and recast the equations with  $\\tau^\\prime$ as the independent variable:\n\\begin{align}\n\\frac{\\d x}{\\d \\tau^\\prime} &= \\varepsilon\\left(-xe^{-1\/u} + f\\left(1-x\\right)\\right) \\label{e6}\\\\ \n\\frac{\\d u}{\\d  \\tau^\\prime}&= xe^{-1\/u} +\\varepsilon f(\\gamma u_a- u) - \\ell(u-u_a) .\\label{e7}\n\\end{align} \nFor $\\varepsilon$ sufficiently small we have $\\d x\/\\d  \\tau^\\prime \\approx 0$ and $x\\approx x_0 \\approx 0$ since the reactant is almost fully depleted, and $\\d u\/\\d \\tau^\\prime\\approx -\\ell(u-u_a)$, so that $u$ decays as $u\\approx \\exp(-\\ell\\tau^\\prime)(u_0 - u_a) + u_a = \\exp(-\\ell\\tau^\\prime)(u_\\text{max} - u_a) + u_a$ since the temperature amplitude is at a maximum when $x=0$ at $\\tau^\\prime_\\text{max}=\\tau^\\prime_0\\equiv 0$.  \nAs $\\tau^\\prime$ becomes `large' $u\\rightarrow u_a$; in  figure \\ref{ro} left this occurs around $\\sim$1000\\,s.  However  reactant  is gradually accumulating and the system evolves in  `slow' time. \n\n\nOver much of the `slow' time interval the system behaviour can be understood by dividing equation \\ref{e5} through by $\\varepsilon$.  At low temperatures and with large $\\varepsilon$ we then have \n$\\d u\/\\d \\tau\\approx -f(u-u_a)$, which gives $u\\approx \\exp(-f\\tau)(u_0 - u_a) + u_a$, and since $u_0\\equiv u_a $ on this time scale we have $u\\approx u_a$. In figure \\ref{ro}  this approximation holds over much of the `slow' time interval:  the internal energy of the system increases but the temperature remains almost constant, although slowly increasing. \n During this time reactant accumulates, but reaction does take place: $x$ evolves as $x\\approx f(1-\\exp(-b\\tau)\/b$, where $b=\\exp(-1\/u_a)$.  \n\nIf reaction takes place in `slow' time but the system is barely heating up, where does the heat of reaction go? Now we focus on  the nanoscale. \n\nThe answer is that the heat is stored in the internal motions of the reactant mixture molecules. Referring to the definition of the dimensionless group $\\varepsilon$ in table \\ref{table2} we see that $\\varepsilon$ is large if the specific heat capacity $C_r$ of the reaction mixture is large. The specific heat capacity of a substance is a function of the number of degrees of freedom of motion that are available to its constituent particles. A monoatomic perfect gas has only the kinetic energy of each atom so translational motion in three dimensions is the only motion it can undergo. The equipartition theorem tells us that the three translational contributions to the constant volume molar heat capacity $C_v^\\text{tr}=(3\/2)R=12.47$\\,J\/(mol\\,K), thus the kinetic energy contribution does manifest as temperature change. \n\nPolyatomic molecules have many additional degrees of freedom though, and physical properties of  liquids are governed much more by the potential energy of the system than the  kinetic energy.  Potential energy is stored in intramolecular rotational and vibrational degrees of freedom and in the vibrational intermolecular force potential. The mean translational energy of each liquid molecule is the same as that for gases, $(3\/2)kT$, where $k$ is Boltzmann's constant, and since $C_p=C_v + R$ we can take the molecular translational contribution to the molar heat capacity of a liquid as\n$$\nC_p^\\text{tr}=20.78\\,\\text{J\\,mol}^{-1}\\text{K}^{-1}.\n$$\nSince the molar heat capacity of MIC is $111.76\\,\\text{J\\,mol}^{-1}\\text{K}^{-1}$ and that of water is $75.29\\,\\text{J\\,mol}^{-1}\\text{K}^{-1}$, on a molar basis the MIC-H$_2$O liquid system can store a large amount of heat in the internal rotational modes and in the vibrational intermolecular potentials. (The internal vibrational modes are not excited at this `slow' temperature.) \n\nThe specific heat capacity of MIC, as opposed to the molar heat capacity,  is not particularly large, but that of water is anomalously high due to strong intermolecular forces. For the reaction mixture it is it is large enough to depress the first and third terms on the right hand side of equation \\ref{e5} relative to the second term at low temperature. The rapid temperature spike and transition to `fast' time occurs because the activation energy is high enough to make the reaction very temperature-sensitive. In other words, the Arrhenius term in equation \\ref{e5} can take over very rapidly  once the reaction zone reaches a certain temperature. At the temperature peak the system enters the 'fast' dissipative time interval, since the reactant is fully depleted. The specific heat capacity cannot affect the dynamics on this time scale. \n\nThis analysis suggests that containment systems with large heat capacity for thermoreactive liquids may not suppress the oscillatory instability, although they would certainly lengthen the `slow' timescale. This may or may not be a good thing. A lengthy `slow' interval may provide enough time before the thermal runaway to deactivate or quench the system in other ways. On the other hand, it might give a false sense of security: \\textit{`Nothing has happened over the last four hours, we might as well leave it.'} \n\n\n\\section{Conclusions\\label{sec6}}\n\n\\begin{enumerate}\n\\item The CSTR paradigm was applied to investigate the thermal stability of MIC hydrolysis and the thermal decomposition of triacetone triperoxide in solution. \n\\item Stability analyses of the steady state solutions of the dynamical model found that in both cases thermal runaway occurs due to the hard onset of a thermal oscillation at a subcritical Hopf bifurcation. Classical thermal ignition at a steady state turning point does not occur in these systems and over the thermal regime of interest they are dominated by oscillatory instability. \n\\item This non-classical oscillatory thermal misbehaviour may be generic in liquid thermoreactive systems where the specific heat capacity and activation energy are high. \nThese results provide new information  about the cause of thermal runaway that may inform improved designs of storage systems for  thermally unstable liquids and  better management of organic peroxide based explosives. \n\\end{enumerate}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nPlasmonic metal \\glspl{np} are fundamental components in several emerging technologies, including sensing \\cite{NugDarCus19, DarKhaTom21}, light-harvesting \\cite{GenAbdBer21}, solar-to-chemical energy conversion \\cite{AslRaoCha18, LiCheRic21, DuCTagWel18} and catalysis \\cite{ZhoLouBao21, DuCTagWel20, HouCheXin20, YamKuwMor21}.\nThe properties that set these materials apart for these applications are their high surface-to-volume ratios and high optical absorption cross sections at visible frequencies \\cite{Boh83, LanKasZor07}, the latter being due to the presence of a \\gls{lsp} resonance \\cite{KreVol95}.\nIn particular plasmonically-driven catalysis is an active research field, addressing important chemical reactions such as ethylene epoxidation, CO oxidation or \\ce{NH3} oxidation that are catalyzed by illuminating \\glspl{np}, e.g., of the noble metals Ag \\cite{ChrXinLin11, YamKuwMor21}, Au \\cite{DuCTagWel18, LiCheRic21, SahYanMas22} or Cu \\cite{DuCTagWel20, HouCheXin20}.\n\nThe \\gls{lsp}, which is a collective electronic excitation, is excited by absorption of light and decays within tens of femtoseconds \\cite{BerMusNea15, RosKuiPus17, ZhoSweZha18, AslRaoCha18, RosErhKui20, KumRosKui19, KumRosMar19, RosErhKui20, VilLeiMar22} into a highly non-thermal (usually referred to as ``hot'') distribution of electrons and holes \\cite{BroHalNor15, GonMun15, RomHesLis19, Khu19, Khu20, HatMenZhe21, HawSilBer21}.\nChemical reactions can then be catalyzed by \\glspl{hc} transiently populating orbitals of nearby molecules \\cite{LinAslBoe15, AslRaoCha18}, which can, e.g., lower reaction barriers \\cite{ZhoSweZha18}.\nTwo variants of this process can be distinguished.\nIn the \\textit{direct} \\gls{hc} transfer process \\cite{KhuPetEic21} (also known as chemical interface damping) the \\gls{lsp} decays into an electron-hole pair, where one of the carriers is localized on the reactant molecule and the other on the \\gls{np}.\nIn the \\textit{indirect} \\gls{hc} transfer process both carriers are generated in the \\gls{np}, and at a later time scattered into molecular orbitals.\nThe efficiency and importance of these processes as well as their competition with thermal effects, is still a matter of intense debate \\cite{DubUnSiv20, Jai20, ZhoSweZha18, SivBarUn19, ZhoSweRob19}, highlighting the importance of more detailed atomistic studies.\nThe direct \\gls{hc} transfer process is promising in terms of efficiency and selectivity \\cite{ChrXinLin11, LinAslBoe15}, and has been studied experimentally \\cite{SeeTheLou19}, in theory \\cite{KhuPetEic21}, and by computational \\textit{ab-initio} models \\cite{MaGao19, KumRosMar19, KumRosKui19}.\nTypically the focus lies on understanding \\gls{hc} generation at surfaces \\cite{SunNarJer14, SeeTheLou19}, but there has not yet been a detailed account of the dependence of \\gls{hc} transfer on molecular position and orientation, and whether there are handles for tuning \\gls{hc} devices to particular molecules in all probable states of thermal motion.\nYet these aspects are crucial for direct \\gls{hc} transfer processes, which exhibit an intricate dependence on the hybridization of molecular and surface states as also shown in this work.\n\nIn this work, we study plasmon decay and carrier generation across a \\gls{np}-molecule junction, which is the initial step in the direct \\gls{hc} transfer process.\nWe consider plasmonic Ag, Au, and Cu \\glspl{np} in combination with a CO molecule, whose excitations are energetically much higher than the plasmon resonance of any of the \\glspl{np} considered here.\nIn a \\gls{rttddft} \\cite{YabBer96} framework, we drive the system with an ultra-fast laser pulse to induce a plasmon.\nWe simulate the electron dynamics in the system until the plasmon has decayed, and then analyze the distribution of carriers over the ground state \\gls{ks} states.\nTo this end, we employ and extend the analysis methods previously developed by us \\cite{KumRosKui19, KumRosMar19, RosErhKui20}.\n\nWe consider a wide range of geometrical configurations.\nSpecifically, we treat (111) on-top, (111) \\gls{fcc}, (100) hollow and corner sites of molecular adsorption on the \\gls{np} and various distances between the molecule and \\gls{np}.\nWe map out the \\gls{hc} transfer efficiency as a function of the \\gls{np}-molecule geometry (adsorption site, distance, molecular bond length), excitation energy, and material (Ag, Au, Cu).\n\nThe article is structured as follows.\nWe present and discuss our results for Ag \\gls{np}+CO systems driven by a laser pulse tuned to the \\gls{lsp} resonance in \\autoref{subsec:geometry-dependence}, \\autoref{subsec:level-alignment} and \\autoref{subsec:site-distribution}.\nIn \\autoref{subsec:pulse-dependence} we quantify the dependence of carrier generation on the pulse frequency, including Ag, Au, and Cu \\glspl{np}.\nFinally, in \\autoref{sec:conclusions} we discuss the implications of our findings.\nDetails concerning our methodology, the systems under study as well as the parameters used in computations are provided in \\autoref{sec:methods}.\n\n\n\\section{Results and discussion}\n\\label{sec:results}\n\n\\subsection{Adsorption geometry-dependent carrier generation in Ag}\n\\label{subsec:geometry-dependence}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics{Fig1.pdf}\n    \\caption{\n        Geometry dependence of \\gls{hc} generation in \\ce{Ag201} \\gls{np}+CO.\n        (a) Optical spectrum of the bare \\gls{np}.\n        The frequency \\SI{3.8}{\\electronvolt} of the driving laser is marked by an arrow above the spectrum.\n        (b) Model of the \\gls{np} with the axes along which the \\gls{np}-molecule distance is varied.\n        (c-d) Fractions of generated electrons (c) and holes (d) on the molecule (\\autoref{eq:region-electron-generation}) after \\SI{30}{\\femto\\second} and binding energies (\\autoref{eq:binding-energy}) as a function of distance and site.\n    }\n    \\label{fig:geometry-dependence}\n\\end{figure*}\n\nWe consider the CO molecule at a range of distances from the \\ce{Ag201} \\gls{np} (111) on-top, (111) \\gls{fcc}, (100) hollow and corner sites.\nThe plasmon (\\SI{3.8}{\\electronvolt}, \\autoref{fig:geometry-dependence}a) and first optical excitation of CO (\\SI{14.5}{\\electronvolt}, \\autoref{sfig:CO}) are not resonant and the optical response of the combined \\gls{np}+CO system is not strongly dependent on geometry (\\autoref{sfig:lspr}).\n\nThe bond length of CO, which is \\SI{1.144}{\\angstrom} in the free molecule, increases when adsorbed to the \\gls{np} (corner: \\SI{1.150}{\\angstrom}, (111) on-top: \\SI{1.151}{\\angstrom}, (111) \\gls{fcc}: \\SI{1.170}{\\angstrom} and (100) hollow \\SI{1.184}{\\angstrom}).\nHigher coordination numbers for the C atom thus result in larger bond lengths.\nThis is in agreement with previous studies on the extended (111) Ag surface \\cite{GajEicHaf04}, and can be understood by considering that as C shares more electron density in bonds with metal atoms, the CO bond is weakened.\n\nAs we vary the \\gls{np}-CO distance (\\autoref{fig:geometry-dependence}b), we observe minima in the binding energy curves around \\SI{250}{\\milli\\electronvolt}-\\SI{500}{\\milli\\electronvolt} at \\SI{1.5}{\\angstrom}-\\SI{2.3}{\\angstrom} (\\autoref{fig:geometry-dependence}c).\nHere, we define the binding energy as\n\\begin{align}\n    \\label{eq:binding-energy}\n    E^\\text{(site)}_\\text{bind}(d) = E^\\text{(site)}(d) - E_\\text{NP}^\\text{(site)} - E_\\text{mol}^\\text{(site)},\n\\end{align}\nwhere $E_\\text{NP}^\\text{(site)}$ and $E_\\text{mol}^\\text{(site)}$ are the energies of the \\gls{np} and molecule, respectively, taken from two separate calculations, representing infinite separation.\nNote that in this definition (\\autoref{eq:binding-energy}), we take $E_\\text{NP}^\\text{(site)}$ and $E_\\text{mol}^\\text{(site)}$ as the energies of the \\gls{np} and molecule as in the relaxed configuration for the specific site, emphasized by the superscript (site).\nAllowing the molecule to relax at each distance effectively widens the adsorption curve but does not affect the main conclusions drawn here (\\autoref{sfig:binding_relax}, \\autoref{sfig:hc-generation}).\nFixing the bond length reduces, however, the degrees of freedom and simplifies the following discussion, whence we adopt this constraint here.\n\nWe drive the Ag \\gls{np}+CO system with a Gaussian laser pulse tuned to the \\gls{lsp} frequency $\\hbar \\omega = \\SI{3.8}{\\electronvolt}$.\nWithin the first tens of femtoseconds a plasmon forms in the \\gls{np} and decays into resonant excitations, for which the electron-hole energy difference equals $\\hbar\\omega$.\nThe plasmon formation and decay process in similar systems has previously been studied in detail by our group \\cite{RosKuiPus17, RosErhKui20} and is not covered here.\n\nWe then measure the fraction of generated electrons in the molecule (\\autoref{eq:region-electron-generation}) after plasmon decay (\\autoref{fig:geometry-dependence}c).\nWhile intuition would suggest this quantity to decrease monotonically with decreasing wave function overlap at increasing distances, we find the fraction of generated \\glspl{hc} to be of similar magnitude measuring between 0.5 and \\SI{2}{\\%} over a wide range of distances with several site specific features.\nA smooth decay to zero only occurs beyond 4 to \\SI{5}{\\angstrom}.\nBelow this threshold several of the sites feature one or two peaks, including near \\SI{2.1}{\\angstrom} and \\SI{3.3}{\\angstrom} for the (111) on-top site, \\SI{2.7}{\\angstrom} for the (100) hollow site and \\SI{2.7}{\\angstrom} and \\SI{3.9}{\\angstrom} for the corner site.\nOnly the (111) \\gls{fcc} site appears relatively feature-less.\n\nBy contrast, the binding energies depend smoothly on distance and approach zero already at 3 to \\SI{4}{\\angstrom}.\nThe landscape of electron generation on the molecule thus extends further than the features in the potential energy surface and is more sensitive to the underlying shifts in eigenenergies and wave function overlaps.\nOur findings imply that across-interface \\gls{hc} generation can be effective even at quite long distances (up to \\SI{5}{\\angstrom}) from the \\gls{np}, and does not require molecular adsorption.\n\nThe fraction of holes generated on the molecule (\\autoref{eq:region-hole-generation}), on the other hand, decays smoothly with distance (\\autoref{fig:geometry-dependence}d) reaching a maximum of \\SI{0.2}{\\%} to \\SI{0.8}{\\%}.\n\n\\subsection{Origin of the \\texorpdfstring{\\gls{hc}}{HC} generation distance dependence}\n\\label{subsec:level-alignment}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{Fig2.pdf}\n    \\caption{\n        Level alignment between molecular and \\gls{np} \\gls{pdos} for (111) on-top (a-c) and corner (d-f) sites.\n        (a, d) Molecular \\gls{pdos} as a function of distance.\n        As the molecule approaches the \\gls{np} the \\gls{lumo} shifts to lower energies, eventually splitting into several branches.\n        The \\gls{pdos} for the \\gls{np} and molecule at far separation are indicated above the plot, where the \\gls{np} \\gls{pdos} has been shifted by the pulse frequency.\n        Shaded regions correspond to (a selection of) large values in the shifted \\gls{np} \\gls{pdos}.\n        (b, e) Electron distribution as a function of distance. (c, f) The fraction of electrons generated in the molecule.\n    }\n    \\label{fig:level-alignment}\n\\end{figure}\n\nTo explain the rich behavior of the across-interface electron generation as a function of distance we study the molecular \\gls{pdos}  (\\autoref{fig:level-alignment}a,c) and the energy distribution of \\glspl{hc} generated on the molecule (\\autoref{eq:region-energetic-electron-generation}, \\autoref{fig:level-alignment}b,d).\nWe note that as the molecule only contains a small fraction of the electrons in the system, the \\gls{np} \\gls{pdos} makes up the most part of the total \\gls{dos} and is practically independent of distance.\nThe molecular \\gls{pdos}, however, is strongly site and distance-dependent.\nAt long distances the \\gls{pdos} is comprised of a single \\gls{lumo} level at \\SI{2.8}{\\electronvolt}, which shifts to lower energies with decreasing distance, eventually splitting into several branches.\nMost of these branches are above the Fermi level (\\autoref{sfig:pdos}) and thus represent unoccupied hybridized states.\n\nThe hot electron distribution in the molecule clearly mirrors the shape of the \\gls{pdos} as electrons are generated in unoccupied molecular levels.\nThe distribution of electrons in each \\gls{pdos} branch is, however, not simply proportional to the corresponding \\gls{pdos} weight.\nAs the transitions ($\\varepsilon_i \\rightarrow \\varepsilon_a$) induced by the pulse are resonant with the pulse frequency ($\\varepsilon_a - \\varepsilon_i = \\hbar \\omega_\\text{pulse} = \\SI{3.8}{} \\pm \\SI{0.37}{\\electronvolt}$, the value after $\\pm$ denotes the half-width at half-maximum of the Gaussian pulse in frequency space), the electron distribution is determined by the energetic alignment of the molecular \\gls{pdos} with the \\gls{np} \\gls{pdos}.\nIt is worth noting that various states in the \\gls{np} couple with differing strength to the molecular states; the alignment with particular peaks in the \\gls{np} \\gls{pdos} is therefore important (\\autoref{sfig:hcdist-map})\nThe across-interface electron generation is thus enhanced at energies $\\varepsilon$ where (1) the molecular \\gls{pdos} is large at $\\varepsilon$, (2) the \\gls{np} \\gls{dos} is large at $\\varepsilon - \\hbar\\omega$ and (3) the transition dipole moment between the corresponding \\gls{np} and molecular states is sizable.\nIt is important to emphasize that it is the combination of these effects that dictates the response.\nIt is, however, usually much simpler to obtain the \\glspl{pdos} than the transition dipole matrix elements, and we may assess the basic possibility for \\gls{hc} transfer already on this basis.\nThis is apparent, e.g., in the decomposition of the electrons distribution in terms of the underlying single-particle excitations (\\autoref{sfig:hcdist-map}).\nThe variation of the \\gls{hc} transfer probability with distance is the result of transitions from two occupied states \\SI{0.95}{} and \\SI{1.20}{\\electronvolt} below the Fermi energy and one unoccupied state \\SI{2.46}{\\electronvolt}.\nThe transitions involving these three states at \\SI{3.41}{} and \\SI{3.66}{\\electronvolt} are slightly different from the excitation frequency of \\SI{3.8}{\\electronvolt} considered in this section but are still activated due to the finite linewidth of the excitation pulse.\nIn fact, in \\autoref{subsec:pulse-dependence} below, we will find that a slight reduction in the excitation frequency leads to a notable increase in the \\gls{hc} transfer probability.\n\nIn a similar manner we can understand the across-interface generation of holes, with the rule that an occupied state $\\varepsilon$ in the molecule must align with a peak in the \\gls{np} \\gls{pdos} at $\\varepsilon + \\hbar\\omega$.\nAs the CO \\gls{homo} level is at \\SI{-4.8}{\\electronvolt} in the free molecule (long distance limit), hole generation is not possible with the pulse frequency \\SI{3.8}{\\electronvolt}.\nTransfer is only possible at close distances where hybridized branches of the \\gls{homo} and \\gls{lumo} appear in the region $-\\SI{3.8}{\\electronvolt} < \\varepsilon < \\SI{0}{\\electronvolt}$, beginning at distances around \\SI{2.5}{\\angstrom} (\\autoref{sfig:pdos}).\n\nThe energetic level alignment is thus a good descriptor in predicting across-interface \\gls{hc} generation.\nOther factors, such as the amount of wave function overlap and the orbital momentum character of states, play a role, i.e., in determining the coupling strength between occupied and unoccupied states, but the energetic level alignment is sufficient to qualitatively explain the distance and site dependence.\nIn the case of \\ce{Ag201} with \\SI{3.8}{\\electronvolt} pulse frequency the constructive contributions to the \\gls{lsp} stem from delocalized sp-band states \\cite{RosKuiPus17} that also have a larger spatial extent.\nThis provides a rationale for the rather long-ranged effect that we observe here.\nBased on this insight, one could expect the effect to be shorter ranged in non-noble metals such as Pd and Pt, for which the Fermi level lies inside the d-band.\nA deeper investigation of this question is, however, beyond the scope of the present study.\nFinally, we note that for larger \\glspl{np}, where the \\gls{dos} between d-band onset and Fermi level is more smeared out, we expect the energetic level alignment to be less noticeable.\n\n\\subsection{Comparison of across-interface electron generation to surface electron distribution}\n\\label{subsec:site-distribution}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics{Fig3.pdf}\n    \\caption{\n        Energy distribution of electrons generated on the molecule and on the \\gls{np} corner site that the molecule approaches.\n        While the former varies non-monotonically, the latter is practically unchanged with distance.\n    }\n    \\label{fig:site-distribution}\n\\end{figure}\n\nTo further emphasize that the energy distribution of electrons generated on the molecule depends on energetic level alignment, we compute the energy distribution of electrons on the nearest metal atom (the integral of \\autoref{eq:region-energetic-electron-generation} in the metal-atom Voronoi cell) as a function of distance (\\autoref{fig:site-distribution}).\nWe focus specifically on the corner site, where the distance dependence of the electron distribution on the molecule exhibits two clear maxima.\nWe find that the electron distribution on the adsorption site is practically distance-independent.\n\nOnly at the smallest considered distances does the electron distribution on the molecule resemble the electron distribution at the adsorption site on the metal.\nHence it is not enough to know the electron distribution on surfaces and surface sites for a bare \\gls{np} to predict across-interface electron generation in combined \\gls{np} + molecule systems.\nEquipped with distributions for the bare \\gls{np} alone one misses for example that about 2 times more electrons are generated at \\SI{2.5}{\\angstrom} for the corner site, than at \\SI{1.9}{\\angstrom}.\nIn other words, the surface electron distribution is an insufficient predictor for across-interface electron generation.\n\n\\subsection{Adsorption geometry and pulse frequency-dependent \\texorpdfstring{\\gls{hc}}{HC} generation}\n\\label{subsec:pulse-dependence}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics{Fig4.pdf}\n    \\caption{\n        Geometry dependence of electron generation in \\ce{Ag201}, \\ce{Au201} and \\ce{Cu201} \\glspl{np}+CO.\n        (a) Optical spectra of the bare \\glspl{np}.\n        (b-e) Fractions of generated electrons on the molecule (\\autoref{eq:region-electron-generation}) after plasmon decay as a function of distance and site with pulse frequency \\SI{3.8}{\\electronvolt} (Ag), \\SI{2.5}{\\electronvolt} (Au) and \\SI{2.7}{\\electronvolt} (Cu).\n    }\n    \\label{fig:element-dependence}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics{Fig5.pdf}\n    \\caption{\n        Pulse-frequency dependence of the electron generation in CO.\n        Fraction of generated electrons (a-c) and holes (d-f) on the molecule as a function of distance and pulse frequency for the (111) on-top site in Ag (a, d), Au (b, e) and Cu (c, f).\n        For reference, the crosses in the figure mark the distance corresponding to the adsorption minimum, and the pulse frequency used in \\autoref{fig:element-dependence}.\n    }\n    \\label{fig:pulse-dependence}\n\\end{figure*}\n\nWe now extend our study to also include Au and Cu.\nThe s-electrons have nearly identical \\glspl{dos} in the \\ce{Ag201}, \\ce{Au201} and \\ce{Cu201} \\glspl{np} but the d-band onsets differ (Ag: \\SI{3.7}{\\electronvolt}, Au: \\SI{2.1}{\\electronvolt}, Cu: \\SI{2.3}{\\electronvolt} below the Fermi level; \\autoref{sfig:NP-dos}).\nAs a consequence of the earlier d-band onset \\ce{Au201} and \\ce{Cu201} lack the well defined \\gls{lsp} peak of \\ce{Ag201}\\cite{CazDolRub00} (\\autoref{fig:element-dependence}a).\nThe binding energy curves are also similar to Ag, with the main difference that the molecule binds more strongly and closer to Cu (\\autoref{sfig:binding}).\n\nWe drive the \\gls{np}+CO with a Gaussian laser pulse (Ag: frequency \\SI{3.8}{\\electronvolt}, Au: \\SI{2.5}{\\electronvolt}, Cu: \\SI{2.7}{\\electronvolt}) and measure the fraction of electrons generated on the molecule (\\autoref{fig:element-dependence}b-e).\nWe observe similar trends in Ag and Cu, both exhibiting peaks near \\SI{2.1}{\\angstrom} for the (111) on-top site, \\SI{2.7}{\\angstrom} for the (100) hollow site and \\SI{2.7}{\\angstrom} and \\SI{3.9}{\\angstrom} for the corner site.\nOnly the \\SI{3.3}{\\angstrom} peak in the (111) on-top site of Ag lacks a counterpart in Cu.\nIn contrast to Ag and Cu, the Au \\gls{np} shows smooth trends, without pronounced peaks, of decreasing electron generation on the molecule with increasing distance.\n\nThe similarity in electron generation for Ag and Cu can be explained by a similar distance dependence of the molecular orbital hybridization (\\autoref{sfig:pdos}).\nWe note that while the resonance condition is not the same for Ag and Cu ($\\hbar\\omega=\\SI{3.8}{\\electronvolt}$ and \\SI{2.7}{\\electronvolt}, respectively), the similar energy-spacing between hybridized molecular levels is enough to yield similar electron generation curves.\nThe \\SI{3.3}{\\angstrom} peak is the only clear feature that is missing in Cu, the reason being that the molecular orbital is too far from the Fermi level (\\SI{2.8}{\\electronvolt}, to be compared to $\\hbar\\omega=\\SI{2.7}{\\electronvolt}$).\nThe hybridization behavior in Au differs from the behavior in Ag and Cu.\nAt long distances the CO \\gls{lumo} is further from the Fermi level in Au than in Ag or Cu (due to Au having a higher work function, and us considering different bond length of the molecule for each metal), preventing electron generation.\nAs the distance decreases the orbital hybridizes more strongly, splitting into more branches.\nAs a consequence, at small distances there are more \\gls{pdos} branches in which electron generation occurs, leading to a smoother distance dependence.\n\nBased on our observations we should expect the pulse frequency $\\hbar\\omega$ to act as a handle for tuning the electron generation through the approximate (barring electron-hole coupling) resonance condition $\\varepsilon_a - \\varepsilon_i = \\hbar\\omega$.\nIndeed, the electron generation depends non-monotonically on both distance and pulse frequency (\\autoref{fig:pulse-dependence}a-c).\nFor example, by lowering the pulse frequency we can get rid of the dip in electron generation at \\SI{2.9}{\\angstrom} for the Ag (111) on-top site; using $\\hbar\\omega =\\SI{3.1}{\\electronvolt}$ the feature at the same distance instead becomes a maximum.\nChoosing the pulse frequency appropriately the fraction of electrons generated on the molecule can be as high as \\SI{8.9}{\\%} for Ag (distance \\SI{2.7}{\\angstrom}, $\\hbar\\omega =\\SI{3.1}{\\electronvolt}$),  \\SI{1.1}{\\%} for Au (distance \\SI{1.9}{\\angstrom}, $\\hbar\\omega =\\SI{2.2}{\\electronvolt}$), and  \\SI{2.3}{\\%} for Cu (distance \\SI{2.1}{\\angstrom}, $\\hbar\\omega =\\SI{2.2}{\\electronvolt}$).\nWhile the electron generation in the Ag and Cu systems shows a complex dependence on pulse frequency and distance, in the case of Au, it is almost monotonic in both dimensions.\nHowever, in both endpoints of the considered pulse frequencies the behavior for Au becomes more interesting; at low frequencies the distance dependence becomes very sharp, as fewer \\gls{pdos} branches fall into the relevant energy range, and at high frequencies there is electron generation at long distances, due to the free-molecule \\gls{lumo} falling into the relevant energy range.\n\nIn the case of Ag (\\autoref{fig:pulse-dependence}d) and Cu (\\autoref{fig:pulse-dependence}f), holes are generated at small distances (where hybridized branches of the \\gls{lumo} orbital are below the Fermi energy; \\autoref{sfig:pdos}) with a weak dependence on pulse frequency.\nFor Au (\\autoref{fig:pulse-dependence}e) hole generation becomes relatively strong at intermediate distances (\\SI{3}{} - \\SI{5}{\\angstrom}) and large pulse frequencies.\nThis behavior originates from the \\gls{homo} orbital, which in the long-distance limit resides \\SI{3.9}{\\electronvolt} below the Fermi energy, but shifts to lower energies with decreasing distance (i.e., out of the range $\\varepsilon_a - \\varepsilon_i = \\hbar\\omega, \\varepsilon_a > 0$, thus limiting hole generation).\n\nThe frequency of the exciting light is thus an excellent handle for tuning the fraction of carriers generated on the molecule, which is especially interesting in applications where selectivity is important.\nIn molecules with several orbitals close enough to the Fermi energy to be optically accessible, the generation of electrons in one orbital could be favored over the other.\nIt is, however, important at this stage to remember that changing the pulse frequency also changes the total optical absorption and thus the total number of generated carriers.\nWe therefore also consider the total, pulse-frequency and distance-dependent, amount of electrons generated on the molecule (\\autoref{sfig:pulse-dependence}, that is, contrary to before, \\emph{without} expressing it as a fraction of electrons generated in the entire \\gls{np}+CO system).\nIn particular for the Ag \\gls{np}, which has an exceptionally sharp absorption spectrum, the pulse dependence is affected, with a maximum in total electron generation on the molecule occurring using a pulse frequency of \\SI{3.6}{\\electronvolt} (to be compared to a maximum in the fraction of electrons generated on the molecule at \\SI{3.1}{\\electronvolt}).\nAs a final note, we point out that it should be possible to simultaneously tune the pulse frequency to the desired resonance condition $\\varepsilon_a - \\varepsilon_i = \\hbar\\omega$, and the \\gls{lsp} resonance of the \\gls{np} (thus the optical absorption) to the pulse frequency, by taking advantage of the fact that the \\gls{lsp} is more sensitive the size and shape of the \\gls{np} than, e.g., the \\gls{dos}.\n\n\\section{Conclusions and outlook}\n\\label{sec:conclusions}\n\nIn this study we have investigated the geometry dependence of \\gls{hc} generation across noble metal-molecule interfaces due to plasmon absorption and decay.\nWe have found that typically up to 0.5 -- \\SI{3}{\\%} of all electrons generated in the system end up on the molecule after plasmon decay, even up to distances of \\SI{5}{\\angstrom}, which is well outside the region of chemisorption.\nBy tuning the excitation frequency, we are able to achieve up to \\SI{8.9}{\\%} electrons generated in the molecule, at the expense of lower absolute amount of electrons generated.\nThese findings suggest that direct \\gls{hc} transfer is a relevant process in plasmon decay, and that the process does not require chemisorption of molecules to the absorbing medium.\n\nWe have also shown that the fraction of generated electrons on the molecule depends on the geometry of the molecule and \\gls{np} in rather intricate fashion.\nThis geometry dependence can be understood in terms of the energy landscape of hybridized molecular orbitals; as an orbital shifts so it is resonant with certain peaks in the \\gls{np} \\gls{pdos} an increase in the probability for \\gls{hc} transfer can be expected.\nThe distance-dependent behavior of the hybridization differs enough for the various sites so that also the carrier generation differs.\nFor larger \\glspl{np}, these effects could be less important, as the \\gls{dos} would be smoother.\n\nIn \\gls{hc} transfer processes the rate of charge carrier generation across the metal-molecule interface is in competition with various loss channels, such as the rates of reemission, and scattering and subsequent thermalization of excited carriers with phonons, surfaces, and other carriers \\cite{BerMusNea15, Khu19}.\nAs these loss channels are currently beyond the reach of our calculations, we have to view our results as an upper bound on the efficiency of \\gls{hc} transfer, i.e., out of all photon absorption events that occur, up to 0.5 - \\SI{3}{\\%} (or \\SI{8.9}{\\%} when tuning the excitation frequency) result in electron transfer to the molecule.\nIt is worthwhile to point out that \\gls{hc} generation is a quantized process, where it is very unlikely that there is at one time more than one excited plasmon at a time, under illumination conditions that are realistic for energy-harvesting applications \\cite{Khu19, Khu20}.\nEach plasmon decays into one electron-hole pair where one carrier can be either in the molecule or in the metal, and we should thus consider our computed fractions as probabilities.\n\nWhile the \\gls{hc} transfer landscape has a detailed structure, it is important to remember that in reality both the molecule and the \\gls{np} are subjected to thermal motion.\nSince it is the weakest interaction in the combined system, one can expect the relative motion of \\gls{np} and molecule to have the most pronounced thermal effect on level alignment, effectively broadening the peaks in the \\gls{hc} transfer probability curves.\nConsidering $kT \\approx \\SI{25}{\\milli\\electronvolt}$ at room temperature and the calculated binding energy curves (\\autoref{fig:geometry-dependence}c), we should expect the molecule to move much less than \\SI{1}{\\angstrom} along the distance axis, but in reality the molecule is free to move laterally as well as rotate.\nSampling the landscape of \\gls{hc} transfer in the full space of molecular movement is a high-dimensional problem that can be addressed in future work.\nFor the purpose of a quantitative comparison to experimental realizations, it is crucial to consider that while metal states are accurately described with our level of theory \\cite{KuiOjaEnk10}, molecular states are not.\nWe should thus expect transfer maxima to occur for different geometrical configurations, due to slightly shifted (hybridized) molecular orbitals; however, our general conclusions are still valid.\n\nThe importance of ground state hybridization for \\gls{hc} transfer implies that theoretical modeling should not be restricted to considering bare metal surfaces, without taking interactions with molecules into account.\nThe distance-dependent hybridization of molecular orbitals should be explicitly taken into account for meaningful predictions.\nOur results suggest that since already the ground state of the hybridized system is a good descriptor for prediction of \\gls{hc} generation on molecules, rapid screening of candidates of good systems can be performed, without conducting expensive real-time simulations.\n\nWe close by commenting on the possibilities for tuning \\gls{hc} transfer suggested by the results of this study.\n\\Gls{hc} devices can be designed by tuning the resonance condition to achieve a desired purpose; handles for tuning to a certain molecular orbital are the \\gls{np} \\gls{dos}, surface substitutions that affect the hybridization of the orbital, and the frequency of the incoming light.\nAs we have demonstrated, the tuning of the latter influences the absorption cross section so that there is a trade-off between high fraction of carriers transferred and high amount of carriers generated in total.\nIt is possible, however, to shift the absorption maximum with a rather small impact on level alignment, by modifying the \\gls{np} shape and size, so that there is a maximum in both absorption and transfer.\nIn this way one ought to be able to maximize \\gls{hc} transfer.\nFurthermore, the sharp \\gls{lsp} resonances of Ag \\glspl{np} could possibly be utilized in the design of highly selective catalysts that work with broadband (solar) light; if the \\gls{np} \\gls{pdos} consists of one particularly strong peak, and that peak is resonant to one specific molecular orbital with the  frequency of the \\gls{lsp} resonance, then transfer to that specific orbital will be preferred over transfer to other orbitals.\n\n\n\\section*{Data Availability}\nThe data generated in this study are openly available via Zenodo at \\url{https:\/\/doi.org\/10.5281\/zenodo.6524101}, Ref.~\\onlinecite{FojRosErh22Data}.\n\n\n\n\\section*{Software used}\nThe VASP\\cite{KreHaf93, KreFur96, KreFur96a, KreJou99} suite with the \\gls{paw}\\cite{Blo94} method and the vdW-df-cx\\cite{BerHyl14, KliBowMic09, KliBowMic11, RomSol09} \\gls{xc}-functional was used for the total energy calculations and structure relaxations.\nThe GPAW{} package \\cite{MorHanJac05, EnkRosMor10} with \\gls{lcao} basis sets \\cite{LarVanMor09} and the \\gls{lcao}-\\gls{rttddft} implementation \\cite{KuiSakRos15} was used for the \\gls{rttddft} calculations.\nThe \\textsc{ase} library \\cite{LarMorBlo17} was used for constructing and manipulating atomic structures.\nThe NumPy \\cite{HarMilvan20}, SciPy \\cite{VirGomOli20} and Matplotlib \\cite{Hun07} Python packages and the VMD software \\cite{HumDalSch96, Sto98} were used for processing and plotting data.\nThe Snakemake \\cite{MolJabLet21} package was used for managing the calculation workflow.\n\n\n\n\\section*{Acknowledgments}\nWe gratefully acknowledge helpful discussions with Mikael Kuisma.\nThis research has been funded by the Knut and Alice Wallenberg Foundation (2015.0055, 2019.0140; J.F., P.E.), the Swedish Foundation for Strategic Research Materials framework (RMA15-0052; J.F., P.E.), the Swedish Research Council (2015-04153, 2020-04935; J.F., P.E.), the European Union's Horizon 2020 research and innovation programme under the Marie Sk{\\l}odowska-Curie grant agreement No~838996 (T.P.R.), and by the Academy of Finland under grant No~332429 (T.P.R.).\nThe computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at NSC, C3SE and PDC partially funded by the Swedish Research Council through grant agreement no. 2018-05973 as well as by the CSC -- IT Center for Science, Finland, and by the Aalto Science-IT project, Aalto University School of Science.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section[]{Introduction} \nIf we assume that the stellar halo is in an approximately steady state, we can characterise it with distribution functions (DFs) $f(\\vJ)$ that depend only on the constants of stellar motion $J_i$ \\citep{jeans+16}. Using actions as the constants of motion has several clear advantages. First, action coordinates can be complemented by canonically conjugate variables, the angles, to obtain a complete coordinate system for phase space. Second, it is straightforward to add DFs for the thin and thick discs, the bulge and the dark halo to the DF of the stellar halo to build up a complete Galaxy model \\citep{piffl+15}. Third, actions are adiabatic invariants and therefore can be used to examine phenomena such as adiabatic contraction \\citep[e.g.][]{piffl+15}. Finally, the actions $J_r$, $J_{\\phi}$ and $J_z$ quantify excursions of an orbit in the radial, azimuthal and vertical directions, and thus are a natural set of labels for categorizing orbits.\n\n\\cite{bell+08} established that much of the halo is comprised of substructures, thought to be relics of disrupted satellites and globular clusters. These clumps disperse in positions and velocities but their ages and metallicities remain bunched \\citep{bell+10}. We expect each clump of halo stars sharing the same metallicity and age to have its own DF $f(\\mathbf{J},\\theta,[\\mathrm{Fe\/H}],\\tau)$. Over time as the clump phase mixes, its DF becomes a function of actions, metallicity, and age only, i.e. an extended distribution function (EDF) of the type introduced by \\cite{sanders+15} for the Milky Way disc(s), and developed by \\cite{das+16} for the Milky Way halo K giants. The EDF of the entire halo is simply the sum of these individual EDFs. The ages and metallicities of stars are thus associated with separations in action space that may manifest as gradients in real and velocity space.\n\nSeveral methods have been explored in the literature for determining the distribution of ages of halo stars. These include estimating the main-sequence turn-off temperature and combining it with metallicities and isochrones, finding $10$--$12\\,$Gyr \\citep{jofre+11}, $10.5 \\pm 1.5\\,$Gyr \\citep{guo+16}, and a minimum age of $8\\,$Gyr \\citep{hawkins+14}. \\cite{kali+12} find an age of $11.4\\pm 0.7\\,$Gyr for local field halo white dwarf stars, by finding a simple relation between their current and progenitor masses using stellar evolution models. There is some evidence in the literature for an old inner halo with a small dispersion in ages compared to a younger outer halo with a larger dispersion in ages \\citep{marquez+94}. \\cite{preston+91} and \\cite{santucci+15} analyse blue horizontal branch (BHB) stars and find that the mean unreddened colour $B-V$ increases outwards to 40 kpc. Interpreting this as an age gradient amounts to a spread of roughly $2$--$2.5\\,$Gyr in age, with the oldest stars concentrated in the central 15 kpc of the Galaxy.\n\nIn this work, we revisit the case for a stellar population gradient in the halo using a spectroscopic sample of BHB stars, within the context of EDFs. Following the work of \\cite{das+16}, where only a very weak metallicity gradient was found in the K giants, we attribute any gradient in the stellar population to ages. We introduce the EDFs, the initial mass function, and the gravitational potential in Section ~\\ref{sec:halomod}. We consider four cases; the first two fit the position and metallicity observables and the second two fit position, velocity, and metallicity observables. In Section ~\\ref{sec:data}, we introduce the spectroscopic sample of BHBs and the methods used to select them. In Section ~\\ref{sec:fitdata}, the method used to explore the posterior distribution of the observations given the stellar halo model, is described. The observables are a single realization of the convolution of the EDF with the selection function (SF) imposed in selecting the sample. The method of deriving the SF is discussed here. Section ~\\ref{sec:results} presents the fits to the observables and properties of our EDFs. Section ~\\ref{sec:discuss} compares this work to the literature and assembles an interpretation of the results. We conclude in Section ~\\ref{sec:conc}.\n\\section[]{Stellar halo models}\\label{sec:halomod}\nAn extended distribution function (EDF) gives the probability density of\nstars in the space specified by the phase-space coordinates\n$(\\mathbf{x},\\mathbf{v})$ and the variables that characterise stellar properties, such as\nmass $m$, age $\\tau$, and chemistry ([Fe\/H], $[\\alpha\/\\mathrm{Fe}],\\ldots$). Below the mass at which the stellar lifetime becomes equal to $\\tau$, we assume that the halo's EDF is proportional to the Kroupa IMF \\citep{kroupa+93},   \n\\begin{equation}\n\t\\epsilon(m) = \n\t\\begin{cases}\n\t\t0.035m^{-1.5} &\\mathrm{if} \\, 0.08 \\leq m < 0.5\\\\\n\t\t0.019m^{-2.2} &\\mathrm{if} \\, 0.5 \\leq m < 1.0\\\\\n\t\t0.019m^{-2.7} &\\mathrm{if} \\, m \\geq 1.0\\,.\n\t\\end{cases}\n\\end{equation}\nwhere $m$ is in solar masses. The EDF vanishes at higher masses. Thus the only explicit dependencies of the EDF are on phase-space coordinates, [Fe\/H], and age, and the EDF can be considered either a function $f(\\mathbf{x},\\mathbf{v},\\mathrm{[Fe\/H]}, \\tau)$ or a function $f(\\mathbf{J},\\mathrm{[Fe\/H]}, \\tau)$. For actions we use $J_r, J_\\phi\\equiv L_z$ and $J_z$.  Where required, we use the St\\\"{a}ckel Fudge \\citep{binney+12} to convert $(\\vx,\\vv)$ to $\\vJ$. We take the gravitational potential to be axisymmetric, and thus cylindrical polar coordinates $(R,\\phi,z,v_R,v_{\\phi},v_z)$ are a natural choice. \n\nBelow we introduce the EDFs and the gravitational potential. The former includes two `pseudo' EDFs that depend only on positions and metallicities. We include these both as an intermediate step to constructing the full phase-space EDFs and as a means to comparing with past works that only fit density profiles. They are equivalent to full phase-space EDFs integrated over velocities.\n\\subsection[]{Separable EDF of density and metallicity (constant axis ratio)}\\label{ssec:mod1}\nThe EDF is specified as\n\\begin{equation}\\label{eq:mod1}\n{\\mathrm{d}N\\over\\mathrm{d}^3\\mathbf{x} \\,\\mathrm{d}\\mathrm{[Fe\/H]}}\n= f(\\mathbf{x},G) = Af_{\\mathrm{bpl}}(R,z)f_{\\mathrm{m}}(G), \n\\end{equation}\nwhere $A$ is a normalization constant enforcing a total probability of one and \n\\begin{equation}\\label{eq:G}\nG = -\\ln ([\\mathrm{Fe\/H}]_{\\mathrm{max}} - \\mathrm{[Fe\/H]}).\n\\end{equation}\n\n\\noindent The function $f_\\mathrm{bpl}$ is the density profile and is specified by a broken power law and a constant axis ratio \\citep{deason+12a}\n\\begin{equation}\nf_{\\mathrm{bpl}}(R,z)= \n\t\\begin{cases}\n    \t\\left(\\frac{R^2+z^2\/q^2}{r_{\\mathrm{b}}^2}\\right)^{-\\alpha_{\\mathrm{in}}}\/2,& \\text{if } R^2+z^2\/q^2 \\leq r_{\\mathrm{b}}^2\\\\\n    \t\\left(\\frac{R^2+z^2\/q^2}{r_{\\mathrm{b}}^2}\\right)^{-\\alpha_{\\mathrm{out}}\/2}, & \\text{otherwise}\n\t\\end{cases}\n\\end{equation}\nWe consider the metallicity to be a lognormal in $\\mathrm{[Fe\/H]}$ \\citep{das+16}\n\\begin{equation}\\label{eq:fm}\n\tf_{\\mathrm{m}}(G) = {\\rm e}^{G}\\frac{{\\rm e}^{-\\frac{G^2}{2\\sigma^2}}}{\\sigma\\sqrt{2\\pi}},\n\\end{equation}\nwhere \n\\begin{equation}\n\t\\sigma^2 = -\\ln(\\mathrm{[Fe\/H]}_{\\mathrm{max}} - \\mathrm{[Fe\/H]}_{\\mathrm{peak}}).\n\\end{equation}\nThus, the distribution is specified by a maximum metallicity, and a metallicity at which the distribution peaks. The difference between these metallicities cannot be greater than 1. A glance at the distribution of metallicities in the observations (Fig.~\\ref{fig:obshist}) suggests this is suitable.\n\nModel 1 is thus specified by the parameter set\n\\begin{equation}\n\tM_1(q,\\alpha_{\\mathrm{in}},\\alpha_{\\mathrm{out}},r_{\\mathrm{b}},\\mathrm{[Fe\/H]}_{\\mathrm{max}},\\mathrm{[Fe\/H]}_{\\mathrm{peak}})\n\\end{equation}\n\\subsection[]{Separable EDF of density and metallicity (variable axis ratio)}\\label{ssec:mod2}\nThe EDF is specified as\n\\begin{equation}\\label{eq:mod2}\n{\\mathrm{d}N\\over\\mathrm{d}^3\\mathbf{x} \\,\\mathrm{d}\\mathrm{[Fe\/H]}}\n= f(\\mathbf{x},G) = Af_{\\mathrm{spl}}(R,z)f_{\\mathrm{m}}(G), \n\\end{equation}\nwhere $A$ is a normalization constant enforcing a total probability of one. $G$ and $f_m$ are defined by Equations ~\\eqref{eq:G} and ~\\eqref{eq:fm}, respectively. $f_\\mathrm{spl}$ is the density profile and is specified by a single power law with a variable axis ratio \\citep{xue+15}\n\\begin{equation}\nf_{\\mathrm{spl}}(R,z) = (R^2+z^2\/q(r)^2)^{-\\alpha\/2},\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{split}\nr &= (R^2+z^2)^{-1\/2}\\\\\nq(r) &= q_{\\infty} - (q_{\\infty} - q_0)\\exp\\left(1 - \\frac{\\sqrt{r^2+r_0^2}}{r_0}\\right).\n\\end{split}\n\\end{equation}\nModel 2 is thus specified by the parameter set\n\\begin{equation}\n\t\\\n    M_2(q_{\\infty},q_0,\\alpha,r_0,\\mathrm{[Fe\/H]}_{\\mathrm{max}},\\mathrm{[Fe\/H]}_{\\mathrm{peak}})\n\\end{equation}\n\\subsection[]{Separable EDF of phase space and metallicity}\\label{ssec:mod3}\n\nThe EDF is specified as\n\\begin{equation}\\label{eq:mod3}\n\t{\\mathrm{d}N\\over\\mathrm{d}^3\\mathbf{x}\\,\\mathrm{d}^3\\mathbf{v}\\,\\mathrm{d}\\mathrm{[Fe\/H]}} \\ = f(\\mathbf{J},G) = Af_{\\mathrm{ps}}(\\mathbf{J})f_{\\mathrm{m}}(G),\n\\end{equation}\nwhere $A$ is a normalization constant enforcing a total probability of one. $f_{\\mathrm{m}}$ is given by Equation \\eqref{eq:fm}, and $f_{\\mathrm{ps}}$ is from \\cite{posti+15}\n\\begin{equation}\\label{eq:postiDF}\n\t\\begin{split}\n\t\tf_{\\mathrm{ps}}(\\mathbf{J})  &=  \\frac{[1+J_0\/h(\\mathbf{J})]^{\\beta_{\\mathrm{in}}}}{[1+g(\\mathbf{J})\/J_0]^{\\beta_{\\mathrm{out}}}}\\\\\n\t\th(\\mathbf{J}) \t\t\t\t&= a_rJ_r + a_{\\phi}|J_{\\phi}| + a_zJ_z\\\\\n\t\tg(\\mathbf{J})  \t\t\t\t&= b_rJ_r + b_{\\phi}|J_{\\phi}| + b_zJ_z.\n\t\\end{split}\n\\end{equation}\n\n\\noindent For $|\\vJ|\\gg J_0$, $f_{\\rm ps}$ is dominated by $g(\\vJ)$, and for $|\\vJ|\\ll J_0$, $f_{\\rm ps}$ is\ndominated by $h(\\mathbf{J})$. Both $g(\\vJ)$ and $h(\\vJ)$\nare homogeneous functions of the actions of degree one.\n\nThe parameters $(a_r,a_{\\phi},a_z,b_r,b_{\\phi},b_z)$ control the shape of the density and velocity ellipsoids. Rescaling the $a_i$ and $b_i$ by the same factor has no effect on the model if accompanied by a rescaling of $J_0$. This degeneracy is eliminated by imposing the conditions $\\sum_i a_i=\\sum_i b_i=3$. \n\nModel 3 is thus specified by \n\\begin{equation}\nM_3(\\beta_{\\mathrm{in}},\\beta_{\\mathrm{out}},J_0,a_r,a_{\\phi},b_r,b_{\\phi},\\mathrm{[Fe\/H]}_{\\mathrm{max}},\\mathrm{[Fe\/H]}_{\\mathrm{peak}}).\n\\end{equation}\n\\subsection[]{Correlated EDF of phase space, metallicity, and age, with rotation}\\label{ssec:mod4}\nThe EDF is specified as\n\\begin{equation}\\label{eq:mod4}\n\t\\begin{split}\n\t{\\mathrm{d}N\\over\\mathrm{d}^3\\mathbf{x}\\,\\mathrm{d}^3\\mathbf{v}\\,\\mathrm{d}\\mathrm{[Fe\/H]}\\,\\mathrm{d}\\tau} &= f(\\mathbf{J},G,\\tau) \\\\\n    \t\t\t\t\t\t\t  &= Af_{\\mathrm{psr}}(\\mathbf{J})f_\\mathrm{m}(G)f_{\\mathrm{psa}}(\\mathbf{J},\\tau),\n\t\\end{split}\n\\end{equation}\nwhere $A$ and $f_{\\mathrm{m}}$ are as above. $f_{\\mathrm{psr}}$ is given by\n\\begin{equation}\\label{eq:psr}\n\t\\begin{split}\n\t\tf_{\\mathrm{psr}}(\\mathbf{J}) &= R(J_{\\phi})f_{\\mathrm{ps}}(\\mathbf{J})\\\\\n\t    R(J_{\\phi}) & = 1 + x\\tanh\\left(\\frac{J_{\\phi}}{J_0}\\right),\t\n    \\end{split}\n\\end{equation}\nwhere $f_{\\mathrm{ps}}$ is given by Equation \\eqref{eq:postiDF}. The prefactor $R(J_{\\phi})$ splits the phase-space DF into even and odd components, introducing the possibility for rotation. $x$ governs the strength of the rotation. $f_{\\mathrm{psa}}$ is given by\n\\begin{equation}\\label{eq:psa}\n\tf_{\\mathrm{psa}} = \\delta\\left(\\tau - \\left[a_{\\tau} + b_{\\tau}\\ln{\\frac{J_t}{J_0}}\\right]\\right), \n\\end{equation}\nwhere $J_{\\mathrm{t}}$ is the total action\n\\begin{equation}\nJ_{\\mathrm{t}} = \\sqrt{J_r^2+J_{\\phi}^2 + J_z^2}.\n\\end{equation}\n\\noindent This implies a single age at each total action, which is a guide to apocentric radius. $b_{\\tau}$ encodes the dependence on actions, and $a_{\\tau}$ is the age for which the total action is equal to the transition action $J_0$. Increasing $|b_{\\tau}|$ increases the age gradient within the halo, with $b_{\\tau} < 0$ implying that mean age decreases with radius. Increasing $a_{\\tau}$ makes the halo older at every radius.\n\nModel 4 is specified by\n\\begin{equation}\nM_4(\\beta_{\\mathrm{in}},\\beta_{\\mathrm{out}},J_0,a_r,a_{\\phi},b_r,b_{\\phi},\nx,\\mathrm{[Fe\/H]}_{\\textrm{max}},\\mathrm{[Fe\/H]}_{\\textrm{peak}},a_{\\tau},b_{\\tau}).\n\\end{equation}\n\\begin{table}\n \\centering\n  \\caption{Parameters of the Galactic potential.\\label{tab:potpars}}\n  \\begin{tabular}{lll}\n  \t\\hline\n  \tComponent     \t\t&Parameter     \t\t\t\t\t &Value\\\\\n  \t\\hline\n  \tThin          \t\t&$R_\\mathrm{d}$ (kpc)     \t\t\t\t &2.682\\\\\n  \t\t\t\t\t    &$z_\\mathrm{d}$ (kpc)     \t\t\t\t &0.196\\\\\n  \t\t\t\t\t    &$\\Sigma_\\mathrm{d} (M_{\\odot}$kpc$^{-2}$)  &5.707$\\times10^8$\\\\\n  \t\\hline\n  \tThick       \t\t\t&$R_\\mathrm{d}$ (kpc)     \t\t\t\t &2.682\\\\\n  \t\t\t\t\t    &$z_\\mathrm{d}$ (kpc)     \t\t\t\t &0.701\\\\\n  \t\t\t\t\t    &$\\Sigma_\\mathrm{d} (M_{\\odot}$kpc$^{-2}$)  &2.510$\\times10^8$\\\\\n  \t\\hline\n  \tGas         \t\t\t&$R_\\mathrm{d}$ (kpc)     \t\t\t\t &5.365\\\\\n  \t\t\t\t\t    &$z_\\mathrm{d}$ (kpc)     \t\t\t\t &0.040\\\\\n  \t\t\t\t\t    &$\\Sigma_\\mathrm{d} (M_{\\odot}$kpc$^{-2}$)  &9.451$\\times10^7$\\\\\n  \t\t\t\t\t    &$R_{hole}$ (kpc)\t\t\t\t &4.000\\\\\n  \t\\hline\n  \tBulge\t  \t\t\t&$\\rho_0 (M_{\\odot}$kpc$^{-3}$)    &9.490$\\times10^{10}$\\\\\n  \t\t\t\t\t\t&$q$\t\t\t\t\t\t\t\t &0.500\\\\\n  \t\t\t\t\t\t&$\\gamma$\t\t\t\t\t\t &0.000\\\\\n  \t\t\t\t\t\t&$\\delta$\t\t\t\t\t\t &1.800\\\\\n  \t\t\t\t\t\t&$r_0$ (kpc)\t\t\t\t\t\t &0.075\\\\\n  \t\t\t\t\t\t&$r_\\mathrm{t}$ (kpc)\t\t\t\t\t     &2.100\\\\\n  \t\\hline\n  \tDark halo    &$\\rho_0 (M_{\\odot}$kpc$^{-3}$)    &1.815$\\times10^7$\\\\\n  \t\t\t\t\t\t&$q$\t\t\t\t\t\t\t\t &1.000\\\\\n  \t\t\t\t\t\t&$\\gamma$\t\t\t\t\t\t &1.000\\\\\n  \t\t\t\t\t\t&$\\delta$\t\t\t\t\t\t &3.000\\\\\n  \t\t\t\t\t\t&$r_0$ (kpc)\t\t\t\t\t\t &14.434\\\\\n  \t\t\t\t\t\t&$r_\\mathrm{t}$ (kpc)\n\t\t\t\t\t\t&$\\infty$\\\\\n  \t\\hline\n  \\end{tabular}\n\\end{table}\n\\subsection[]{The gravitational potential}\nAs in previous works, we use the composite potential proposed by \\cite{dehnen+98}, generated by thin and thick stellar discs, a gas disc, and two spheroids representing the bulge and the dark halo. The densities of the discs are\ngiven by\n\\begin{equation}\n\t\\rho_\\mathrm{d}(R,z) = \\frac{\\Sigma_0}{2z_\\mathrm{d}}\\exp\\left[-\\left(\\frac{R}{R_\\mathrm{d}} + \\frac{|z|}{z_\\mathrm{d}} + \\frac{R_{\\mathrm{hole}}}{R}\\right) \\right],\n\\end{equation}\nwhere $R_\\mathrm{d}$ is the scale length, $z_\\mathrm{d}$, is the scale height, and\n$R_{\\mathrm{hole}}$ controls the size of the hole at the centre of the disc,\nwhich is only non-zero for the gas disc. The densities of the bulge and dark\nhalo are given by\n\\begin{equation}\n\t\\rho(R,z) = \\rho_0\\frac{(1+m)^{(\\gamma-\\delta)}}{m^{\\gamma}}\\,\\exp\\left[-(mr_0\/r_{\\mathrm{t}})^2\\right],\n\\end{equation}\nwhere \n\\begin{equation}\n\tm(R,z) = \\sqrt{(R\/r_0)^2 + (z\/qr_0)^2}. \n\\end{equation}\n$\\rho_0$ sets the density scale, $r_0$ is a scale radius, and the parameter $q$ is the axis ratio of the isodensity surfaces. The exponents $\\gamma$ and $\\delta$ control the inner and outer slopes of the radial density profile, and $r_\\mathrm{t}$ is a truncation radius. \n\nThe adopted parameter values are taken from \\cite{piffl+14} and given in Table~\\ref{tab:potpars}. They specify a spherical NFW halo that is not truncated\n($r_\\mathrm{t}=\\infty$). The stellar halo contributes only\nnegligible mass, and thus can be considered included in the contributions of the bulge and dark halo. \n\n\\section[]{Observational constraints}\\label{sec:data}\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[scale=0.46]{segue2}\n\t\\caption{One-dimensional distributions for the SEGUE-II BHBs in Galactic coordinates, for sky positions, apparent magnitudes, line-of-sight velocities, proper motions, and metallicities.\\label{fig:obshist}}\n\\end{figure*}\nHere, we introduce the observations that we will use to constrain parameters\nof the EDFs. We adopt a left-handed coordinate system in which positive $v_R$ is away from the Galactic centre and positive $v_{\\phi}$ is in the direction of Galactic rotation. To convert from Galactocentric coordinates to heliocentric coordinates we assume that the Sun is located at $(R_0,z_0) =\n(8.3,0.014)\\,{\\rm kpc}$ \\citep{schonrich+12,binney+97}, that the local standard of\nrest (LSR) has an azimuthal velocity of $238\\,\\mathrm{km\\,s}^{-1}$, and that the velocity of\nthe Sun relative to the LSR is $(v_R,v_{\\phi},v_z) = (-14.0, 12.24,\n7.25)\\,\\mathrm{km\\,s}^{-1}$ \\citep{schonrich+12}.\n\n\\subsection[]{BHB sample}\nWe use the BHB sample of \\cite\n{xue+11}, which is based on Data Release 10 of the Sloan Digital Sky Survey (SDSS) and in particular the Sloan Extension for Galactic Understanding and Exploration surveys (SEGUE) within it. The sample consists of equatorial coordinates $(\\alpha,\\delta)$, apparent magnitudes, colours, line-of-sight velocities $v_{||}$, and spectroscopic metallicities $\\mathrm{[Fe\/H]}$. We supplement the catalogue values with proper motions ($\\mu_l^* = \\dot{l}\\cos b$ and $\\mu_b = \\dot{b}$)  downloaded from SkyServer's CasJobs\\footnote{\\url{http:\/\/skyserver.sdss.org\/CasJobs\/}}, by cross matching within 15 arcsec. We include proper motion measurements to use all available data, but note that the uncertainties are often $> 100\\%$ and so we do not expect much, if any, extra constraining power to come from their inclusion. We constrain the sample to SEGUE-II stars only, as the selection criteria are not fully known for SDSS Legacy and SEGUE-I stars. SEGUE II focuses on distant stars and therefore only uses `faint' plates. SEGUE I used both `bright' and `faint' plates to cover a larger range in apparent magnitudes. We remove stars on cluster and test\nplates and apply the cut $\\mathrm{[Fe\/H]}\\le-1.4$ to reduce the contamination from\ndisc stars \\citep{schonrich+12}. We exclude stars on plates that intersect the two polygons given by \\cite{fermani+13a} as containing the Sagittarius stream. We use the relation of \\cite{fermani+13a} to relate apparent magnitudes, colours, and metallicity to a\nheliocentric distance\\footnote{We shorten `heliocentric distance' to\ndistance for the remainder of the paper.} $s$ for the BHBs. Thus our observables are defined by the vector\n\\begin{equation}\\label{eqn:obs}\n\t\\mathbf{u} = (l,b,s,v_{||},\\mu_l^*,\\mu_b,\\mathrm{[Fe\/H]}).\n\\end{equation}\n\\begin{table*}\n \\centering\n  \\caption{Combined selection criteria in terms of apparent magnitude, colour indices, and metallicity. $n^*$ is the number of stars after applying the combined selection criteria, and $u$, $g$, $r$, and $i$ refer to apparent magnitudes in Sloan's $ugriz$ colour-magnitude system. \\label{tab:selfunc}}\n  \\begin{tabular}{lllll}\n  \t\\hline\n\tProgramme \t&$n^*$    &Apparent  \t&Colour\t\t\t\t     &Metallicity\\\\\t\n\t\t \t\t&\t \t&magnitude \t\t&\t\t\t\t\t     &\\\\\t\n\t\\hline\t\t\n\tSEGUE II \t&701\t &$15.5<g<19.0$\t&$0.8<(u-g)<1.5$\t\t &$\\text{[Fe\/H]}<-1.4$\\\\\n\t \t\t\t& \t\t &$r>12.5$\t\t&$-0.4<(g-r)<0.0$\t\t &\\\\\n\t\\hline\t\n\t\\end{tabular}\n\\end{table*}\n\\subsection[]{Selection criteria}\nOur selection function is the overlap between the original spectroscopic\ntargeting criteria, criteria imposed by \\cite{xue+11}, and further criteria applied by us. The selection on sky positions is given by the coverage of the spectroscopic plates on the sky, and on each plate by the completeness of the spectroscopic sample, i.e. the number of stars observed compared to the potential number of targets. The remaining combined selection criteria relate to apparent magnitudes and various colour indices. These are summarised in Table~\\ref{tab:selfunc}. The phase-space, colour, and metallicity distributions for the sample are shown in Fig.~\\ref{fig:obshist}. \n\\section[]{Fitting the data}\\label{sec:fitdata}\nWe construct the likelihood of models by the method of \\cite{mcmillan+13} that  \\cite{das+16} used to fit an EDF to halo K giants. The form and contributing terms of the likelihood are described in detail here. \n\\subsection[]{The likelihood from Bayes' law}\nThe total likelihood $\\mathcal{L}$ of a model $M$ is given by the product over all stars $i$ of the individual likelihoods $\\mathcal{L}^i=P(\\mathbf{u}^i|SM)$ of\nmeasuring the star's catalogued coordinates $\\mathbf{u}^i$ given the model $M$ and that it is in the survey $S$. By Bayes' law this is\n\\begin{equation}\t\n\t\t\\mathcal{L}^i = \\frac{P(S|\\mathbf{u}^i)P(\\mathbf{u}^i|M)}{P(S|M)}.\n\\end{equation}\n\\noindent $P(S|\\mathbf{u}^i)$ is the probability that the star is in the survey given the observables i.e. the `selection\nfunction' (Section ~\\ref{ssec:selfunc}). $P(\\mathbf{u}^i|M)$ is the EDF convolved with the error distribution of the observables (Section ~\\ref{ssec:errconv}). $P(S|M)$ is \nthe probability that a randomly chosen star in the model enters the catalogue (Section ~\\ref{ssec:normfactor}). The total log-likelihood is\n\\begin{multline}\\label{eqn:logL}\n\t\\log \\mathcal{L} = \\sum_{i=k}^{n_*} \\log \\mathcal{L}^i = \n\t \\sum_{i=k}^{n_*}\\log\\left(P(S|\\mathbf{u}^i)\\right) \n+\\\\\n\\hskip2cm \\sum_{i=k}^{n_*}\\log\\left(P(\\mathbf{u}^i|M)\\right) - n_*\\log\\left(P(S|M)\\right),\n\\end{multline}\nwhere $n_*$ is the number of stars. \n\\subsection[]{Selection function}\\label{ssec:selfunc}\nThere is no selection on line-of-sight velocities or proper motions. Moreover, we assume that the selection function is separable as\n\\begin{equation}\n\t\\begin{split}\n\t\tP(S|\\mathbf{u}^i) =& \\,p(S|l,b,s,\\mathrm{[Fe\/H]})\\\\\n\t   \t\t\t      \t  =& \\,p(S|l,b)\\,p(S|s,\\mathrm{[Fe\/H]}).\n\t\\end{split}\n\\end{equation}\nThe selections on sky positions $p(S|l,b)$ and distance\/metallicity $p(S|s,\\mathrm{[Fe\/H]})$ are described below.\n\\subsubsection[]{Selection on sky positions}\\label{sssec:skypossf}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{lb_selfunc}\n \\caption{Selection function as a function of sky positions,  $p(S|l,b)$.\\label{fig:lbsfmaps}}\n\\end{figure}\nThe selection on $l$ and $b$, $p(S|l,b)$, depends on the coordinates of the SEGUE-II plates and the completeness fraction, i.e. the fraction of photometrically identified targets for which spectra were obtained. The completeness fraction depends strongly on $|b|$ because close to the plane available targets are numerous so an individual star has a low probability of being allocated a fibre. For coordinates within $1.49\\ensuremath{^\\circ}$ of the centre of a plate, the selection function equals the completeness fraction for that plate. Thus\n\\begin{equation}\n\tP(S|l,b) = \\frac{N_{\\mathrm{spec,plate}}}{N_{\\mathrm{phot,plate}}},\n\\end{equation}\nwhere $N_{\\mathrm{spec,plate}}$ is the number of BHBs on the plate that make the spectroscopic sample and $N_{\\mathrm{phot,plate}}$ is the number of BHBs in the photometric sample within the same patch in the sky. We evaluate these fractions by searching for BHBs in the spectroscopic and photometric samples in the regions covered by each of the plates, using SkyServer's CasJobs. In Fig.~\\ref{fig:lbsfmaps} the colour scale shows $p(S|l,b)$. The dependence of $p(S|l,b)$ on $b$ is evident.\n\\begin{figure*}\n\\centering\n\t\\subfloat[]{\n\t\t\\includegraphics[scale=0.4]{specphot_limrange_19}\n\t}\n\t\\subfloat[]{\n\t\t\\includegraphics[scale=0.4]{CDF_gmag}\n}\n\\caption{Probability (a) and cumulative (b) $g$-band apparent magnitude distributions for the photometric and spectroscopic samples.\\label{fig:magdist}}\n\\end{figure*}\n\\begin{figure}\n\\centering\nq\\includegraphics[scale=0.5]{singleagechem_selfunc}\n \\caption{Selection function as a function of distance and metallicity, assuming a single age of $9$ (top), $11$ (middle), and $13$ (bottom) Gyr, with locations of observed stars superimposed.\\label{fig:chemsfmaps_singleage}}\n\\end{figure}\n\\subsubsection[]{Selection on distance and metallicity}\nThe selection probability in terms of distance and metallicity depends on the assumed IMF, the survey selection with respect to apparent magnitudes and colours, and the isochrones used to relate apparent magnitudes and colours to intrinsic properties. We assume the survey selection on colour to be uniform over the ranges in Table~\\ref{tab:selfunc}. The signal-to-noise ratio however decays with apparent magnitude, and therefore the survey selection is not uniform within the imposed apparent magnitude range. Fig.~\\ref{fig:magdist} shows the $g$-band apparent magnitude distribution for our sample of SEGUE-II stars and for stars in the SDSS photometric sample found in Section ~\\ref{sssec:skypossf} to lie within the plate dimensions and the selection criteria of Table~\\ref{tab:selfunc}. The left panel shows the distributions normalized to unity for $g$-band apparent magnitudes between $15.5$ and $19.0$, just within the range imposed by the SEGUE-II targetting criteria for BHBs ($15.5<g<20.3$). The right panel shows the associated cumulative distribution function. The plots show that the single faint SEGUE-II plates do an excellent job of capturing the apparent magnitude distribution out to a $g$-band apparent magnitude of $19.0$. Therefore we take the selection on apparent magnitude to be uniform within the ranges specified in Table~\\ref{tab:selfunc} and zero otherwise.\n\nWe use the $\\alpha$-enhanced B.A.S.T.I. isochrones for a mass-loss parameter $\\eta=0.4$ \\citep{pietrin+06} to relate apparent magnitudes to intrinsic stellar properties in the sample. We tabulate on a grid in $s$ and [Fe\/H] the probability that a star randomly chosen at birth passes the magnitude and colour cuts\n\\begin{equation}\np(S|\\tau,s,\\mathrm{[Fe\/H]}) = \\int \\mathrm{d} m \\, \\epsilon(m) p(S|m,\\tau,s,\\mathrm{[Fe\/H]}).\n\\end{equation}\nFor a given value of $s$, we integrate each isochrone (specified by [Fe\/H] and $\\tau$) over the mass distribution, by adding a star's IMF value to $p(S|s,\\mathrm{[Fe\/H]},\\tau)$ only if it passes the magnitude and colour cuts. We repeat this for a grid of distances, and create an interpolant of $p(S|s,\\mathrm{[Fe\/H]},\\tau)$ that gives the selection function probability for any distance, metallicity, and age within the specified domain. \n\nThe colour scale in Fig.~\\ref{fig:chemsfmaps_singleage} shows $p(S|s,\\mathrm{[Fe\/H]},\\tau)$ for $\\tau = 9, \\, 11$, and $13$ Gyr. The top plot shows that the BHBs cannot be described by a single age of $9\\,$Gyr, as stars of metallicities higher than $-2.3$ would then not be seen. However the stars could be described by a single age of $11$ or $12\\,$Gyr, i.e. all observed stars lie within the boundaries of non-negligible probabilities predicted for these single ages. We consider two possible age distributions for the BHBs. The first is a delta function, centred on $11\\,$Gyr \\citep{jofre+11,das+16}, and the second is a function of the actions, as prescribed by Equation \\eqref{eq:psa}. \n\nFor the first case, a glance at Fig.~\\ref{fig:chemsfmaps_singleage} shows that $p(S|s,\\mathrm{[Fe\/H]},\\tau=11)$ is approximately constant within well-defined boundaries. Therefore to avoid the time-consuming engagement with the isochrones, we assume a constant probability within a box, so \n\\begin{equation}\\label{eqn:chemsfp_const}\np(S|s,\\mathrm{[Fe\/H]}) =\n\t\\begin{cases}\n    \t\\multirow{2}{*}{1}   & \\mathrm{if}\\,10\\, \\mathrm{kpc}\\leq s\\leq 50\\, \\mathrm{kpc}\\\\\n                              & \\mathrm{and}\\,-3.6\\leq\\mathrm{[Fe\/H]}\\leq -1.4\\\\\n\t\t0 &\\, \\mathrm{otherwise.}\\\\\n\t\\end{cases}\n\\end{equation}\nWe adopt this age distribution for the models specified in Sections \\ref{ssec:mod1}, \\ref{ssec:mod2}, and \\ref{ssec:mod3} ($M_1$, $M_2$, and $M_3$). For the model described in Section \\ref{ssec:mod4} ($M_4$), we assume the age to depend on actions within the same distance and metallicity limits as specified in Equation ~\\eqref{eqn:chemsfp_const}.\n\n\\subsection[]{Convolution of the EDF with the error distribution}\\label{ssec:errconv}\nWe neglect errors in sky coordinates and adopt independent Gaussian error\ndistributions for $v_\\parallel$, $\\mu_l^*$, $\\mu_b$, [Fe\/H], and $\\log s$.\nThus the multi-variate error distribution is\n\\begin{equation}\\label{eqn:convint}\n\tC^{(7)}(\\mathbf{u}^i,\\mathbf{u}^{\\prime},\\mathbf{p}^i) \\equiv\n\t\\prod_{l=1}^2\\delta(u_l^i-u_l')\\prod_{l=3}^{7}C(u_l^i,u_l^{\\prime},p_l^i),\n\\end{equation}\nwhere true values of observables are indicated by primes and\n\\begin{equation}\nC(u^i,u',p^i)\\equiv{\\exp\\left[{-(u^i-u')^2\/(2p^{i2})}\\right]\\over\\sqrt{2\\pi} p^i}.\n\\end{equation}\nThe\nconvolution of the EDF with the error distribution of a given star is\n\\begin{multline}\n\tP(\\mathbf{u}^i|M) = \\\\\n\t\\int \\mathrm{d}^7\\mathbf{u}^{\\prime} \\, C^7(\\mathbf{u}^i,\\mathbf{u}^{\\prime},\\mathbf{p}^i) \\, f(\\mathbf{\\mathbf{x},\\mathbf{v},\\mathrm{[Fe\/H]}})) \\left|\\frac{\\partial(\\mathbf{x},\\mathbf{v},\\mathrm{[Fe\/H]})}{\\partial(\\mathbf{u}^{\\prime})}\\right| .\n\\end{multline}\nThe Jacobian determinant here is proportional to $s^4\\cos b$\n\\citep{mcmillan+12}. The\nintegral is calculated using a fixed Monte Carlo sample of 2000 points per star to eliminate Poisson noise in likelihood evaluations \\citep{mcmillan+13}.\n\\begin{table}\n \\centering\n  \\caption{Median and 68$\\%$ confidence intervals for $M_1$, $M_2$, and $M_4$, and MAP estimates for $M_3$. A `*' indicates that the parameter is either set prior to the runs, or fixed by other parameters. \\label{tab:bestfitpars}}\n\\begin{tabular}{lllll}\n  \t\\hline\n  \tParameter\t\t\t\t&$M_1$\t \t\t\t\t\t&$M_2$\t\t\t\t\t\t&$M_3$\t\t\t&$M_4$\\\\\n    \\hline\n    $q$\t\t\t\t\t\t&$0.72_{-0.030}^{+0.031}$\t&-\t\t\t\t\t\t\t&-\t\t\t\t&-\\\\\n    $\\alpha_{\\mathrm{in}}$\t&$3.61_{-0.16}^{+0.15}$\t\t&-\t\t\t\t\t\t\t&-\t\t\t\t&-\\\\\n    $\\alpha_{\\mathrm{out}}$\t&$4.75_{-0.28}^{+0.30}$\t\t&-\t\t\t\t\t\t\t&-\t\t\t\t&-\\\\\n\t$r_{\\mathrm{b}}$\/kpc\t&$29.87_{-3.55}^{2.80}$\t    &-\t\t\t\t\t\t\t&-\t\t\t\t&-\\\\ \n    \\hline\n    $q_{0}$\t\t\t\t\t&-\t\t\t\t\t\t\t&$0.39_{-0.09}^{+0.08}$\t\t&-\t\t\t\t&-\\\\\n    $q_{\\infty}$\t\t\t&-\t\t\t\t\t\t\t&$0.81_{-0.050}^{+0.055}$\t&-\t\t\t\t&-\\\\\n    $r_0$\/kpc\t\t\t\t&-\t\t\t\t\t\t\t&$7.32_{-1.73}^{+1.88}$\t\t&-\t\t\t\t&-\\\\\n    $\\alpha$\t\t\t\t&-\t\t\t\t\t\t\t&$4.65_{-0.23}^{+0.25}$\t\t&-\t\t\t\t&-\\\\\n    \\hline\n    $\\beta_{\\mathrm{in}}$\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$2.05$\t\t\t&$2.17_{-0.52}^{+0.44}$\\\\\n    $\\beta_{\\mathrm{out}}$\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$4.72$\t\t\t&$4.65_{-0.31}^{+0.33}$\\\\\t    $J_0$     \t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$1635$\t\t\t&$1635*$\\\\\n    $a_r$\t\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$1.31$\t\t\t&$0.70_{-0.24}^{+0.27}$\\\\\n    $a_{\\phi}$\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$0.93$\t\t\t&$0.88_{-0.22}^{+0.25}$\\\\\n    $a_z$\t\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$0.78*$\t\t&$1.42*$\\\\\n    $b_r$\t\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$1.11$\t\t\t&$1.33_{-0.15}^{+0.17}$\\\\\n    $b_{\\phi}$\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$0.60$\t\t\t&$0.54_{-0.09}^{+0.10}$\\\\\n    $b_z$\t\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$1.29*$\t\t&$1.13*$\\\\\n    $x$\t\t\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&$0.34$\t\t\t&$0.07_{-0.16}^{+0.19}$\\\\\n    \\hline\n    $\\mathrm{[Fe\/H]}_{\\mathrm{max}}$   &$-0.83_{-0.017}^{+0.018}$\t&$-0.83_{-0.017}^{+0.018}$\t&$-0.83_{-0.017}^{+0.018}$\t\t&$-0.80_{-0.030}^{+0.053}$\\\\\n    $\\mathrm{[Fe\/H]}_{\\mathrm{peak}}$\t&$-1.77_{-0.018}^{+0.020}$\t&$-1.77_{-0.018}^{+0.020}$\t&$-1.77_{-0.018}^{+0.020}$\t\t&$-1.73_{-0.035}^{+0.059}$\\\\\n    \\hline\n \t$a_{\\tau}$\t\t\t\t&-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&-\t\t\t\t&$12.00_{-0.29}^{+0.37}$\\\\\n    $b_{\\tau}$\t\t\t    &-\t\t\t\t \t\t\t&-\t\t\t\t\t\t\t&-\t\t\t\t&$-0.69_{-0.14}^{+0.24}$\\\\\n\\hline\n  \\end{tabular}\n\\end{table}\n\\subsection[]{The normalization factor}\\label{ssec:normfactor}\nThe normalization of $\\mathcal{L}$ is given by\n\\begin{multline}\nP(S|M) = \\\\\n\\int\\mathrm{d}^7\\mathbf{u}^{\\prime}\\,P(S|\\mathbf{u}^{\\prime})\\,f(\\mathbf{x},\\mathbf{v},\\mathrm{[Fe\/H]})\\left|\\frac{\\partial(\\mathbf{x},\\mathbf{v},\\mathrm{[Fe\/H]}}{\\partial(\\mathbf{u}^{\\prime})}\\right| .\n\\end{multline}\nWe approximate the integrals over sky coordinates by sums of the remaining five-dimensional integrals evaluated at the centre of\neach SEGUE-II plate \\citep{das+16}. Since $P(S|\\mathbf{u}^{\\prime})$ is multiplied by  $n_*$ in Equation (\\ref{eqn:logL}), the integrals must be calculated to a high degree of accuracy \\citep{mcmillan+12}. We calculate them to an accuracy of 0.1\\% using a Python wrapper for the cubature method\\footnote{This wrapper can be downloaded from \\url{https:\/\/github.com\/saullocastro\/cubature}.}. The procedure is parallelised over 16 cores using the distributed memory tool in the \\texttt{multiprocessing} package of Python.\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[scale=0.4]{emcee_bplfehdf_constdistfehsf_triangle}\n\t\\caption{{\\tt emcee} results for fitting the DF $M_1$. The plots along the diagonal illustrate 1-D probability distributions for each of the model parameters. The remaining plots show joint probability distributions between all pairs of parameters. The contours show the 1-$\\sigma$ and 2-$\\sigma$ confidence levels. \\label{fig:emcee_m1}}\n\\end{figure*}\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[scale=0.4]{emcee_splpdf_constdistfehsf_triangle}\n\t\\caption{{\\tt emcee} results for fitting the DF $M_2$. The plots along the diagonal illustrate 1-D probability distributions for each of the model parameters. The remaining plots show joint probability distributions between all pairs of parameters. The contours show the 1-$\\sigma$ and 2-$\\sigma$ confidence levels.\\label{fig:emcee_m2}}\n\\end{figure*}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[scale=0.25]{emcee_bplfehdf_constdistfehsf_triangle_metallicity}\n\t\\caption{{\\tt emcee} results for the metallicity DF of $M_1$, $M_2$, and $M_3$. The plots along the diagonal illustrate 1-D probability distributions for each of the model parameters. The remaining plot shows the joint probability distribution between the two parameters of the metallicity DF. The contours show the 1-$\\sigma$ and 2-$\\sigma$ confidence levels.\\label{fig:emcee_feh}}\n\\end{figure}\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[scale=0.55]{emcee_Jfehrotdf_varydistfehsf_agerestrict_020816_triangle}\n\t\\caption{{\\tt emcee} results for fitting the EDF $M_4$. The plots along the diagonal are 1-D probability distributions for each of the model parameters. The remaining plots show joint probability distributions between all pairs of parameters. The contours show the 1-$\\sigma$ and 2-$\\sigma$ confidence levels.\\label{fig:emcee_m4}}\n\\end{figure*}\n\\subsection[]{Exploring the posterior distribution}\nWe explore the posterior distribution using two algorithms. The first method is the Nelder-Mead \\texttt{amoeba} method \\citep{nelder+65}, a downhill-simplex optimization routine for locating an extremum of a multi-dimensional function when the derivatives of that function are not known. This algorithm has been shown to work well in many non-linear optimization problems, although it suffers from the common issue that one may only discover a local maximum, as opposed to a global one (if it exists). The second algorithm we use is the affine-invariant Monte Carlo Markov Chain approach implemented in the \\texttt{emcee} package for Python \\citep{foreman+13}. The approach uses an interacting ensemble of `walkers' and has been shown to be more effective at dealing with narrow degeneracies than simpler approaches, such as basic Metropolis-Hastings sampling. We now outline our priors and method of\nsampling for each of our four models:\n\\begin{itemize}\n\\item{\\textbf{$M_1$:} Spatial DF and metallicity DF with constant axis ratio (Equation ~\\ref{eq:mod1}) and box-uniform distance-metallicity selection function (Equation ~\\ref{eqn:chemsfp_const}). We fit all parameters using \\texttt{emcee} with 40 walkers and 10,000 steps each. The priors are set as uniform within the following ranges:\n\n\\begin{enumerate}\n\t\\item{$0<q<1$}\n    \\item{$1<\\alpha_{\\mathrm{in}}<\\alpha_{\\mathrm{out}}<7$}\n    \\item{$15<r_{\\mathrm{b}}<40$ kpc}\n    \\item{$0<\\mathrm{[Fe\/H]}_{\\mathrm{max}}-\\mathrm{[Fe\/H]}_{\\mathrm{peak}}<1$}\n\\end{enumerate}}\n\\item{\\textbf{$M_2$:} Spatial DF and metallicity DF with variable axis ratio (Equation ~\\ref{eq:mod2}) and box-uniform distance-metallicity selection function (Equation ~\\ref{eqn:chemsfp_const}). We fit the spatial DF parameters only using \\texttt{emcee} with 40 walkers and 10,000 steps each. The metallicity DF parameters are independent and therefore the same as found for $M_1$. The priors are uniform within the following ranges:\n\\begin{enumerate}\n    \\item{$q_0>0$}\n    \\item{$q_{\\infty}>0$}\n    \\item{$0<r_0<50$ kpc}    \t\n    \\item{$0<\\alpha<10$}\n\\end{enumerate}}\n\\item{\\textbf{$M_3$:} Action-based DF and metallicity DF (Equation ~\\ref{eq:mod3}) with box-uniform distance-metallicity selection function (Equation ~\\ref{eqn:chemsfp_const}). We first fit [$\\beta_{\\mathrm{in}}$, $\\beta_{\\mathrm{out}}$, $J_0$] using the \\texttt{amoeba} algorithm, with the flattening\/anisotropy and rotation parameters fixed at those for an isotropic and round halo, and the metallicity parameters fixed at those found for $M_1$. We then fit [$a_r$,$a_{\\phi}$,$b_r$,$b_{\\phi}$] using the \\texttt{amoeba} algorithm, with the density parameters fixed from the first stage, and metallicity parameters fixed from $M_1$.\nThe priors are uniform within the following ranges:\n\\begin{enumerate}\n\t\\item{$0 < \\beta_{\\mathrm{in}} < 3$}\n    \\item{$\\beta_{\\mathrm{out}}>3$}\n    \\item{$a_{\\mathrm{r}}>0.3$}\n    \\item{$a_{\\phi}>0.3$}\n    \\item{$a_{\\mathrm{r}}+a_{\\phi}<2.7$}\n    \\item{$b_{\\mathrm{r}}>0.3$}\n    \\item{$b_{\\phi}>0.3$}\n    \\item{$b_{\\mathrm{r}}+b_{\\phi}<2.7$}\n\\end{enumerate}}\n\\item{\\textbf{$M_4$:} Action-based DF, metallicity DF, and age DF (Equation ~\\ref{eq:mod4}) with age-varying distance-metallicity selection function. We first fit [$x$, $a_r$, $a_{\\phi}$, $b_r$, $b_{\\phi}$] using the \\texttt{amoeba} algorithm, with the density and metallicity parameters fixed at those found for $M_3$. Then we fit [$a_{\\tau}$, $b_{\\tau}$] using the \\texttt{amoeba} algorithm, with the density, flattening\/anisotropy, rotation, and metallicity parameters fixed at those found for the first stage. We then fit all parameters using the \\texttt{amoeba} algorithm, and again using \\texttt{emcee} with 22 walkers and 500 steps each. We require fewer steps than for $M_1$ and $M_2$  as we start very close to the MAP estimate. However, it cannot be guaranteed that the chains have converged - the limit is simply a function of computational resources (22 walkers and 500 steps take about three weeks to run across 16 cores). The priors are as for $M_2$ with the following set to be uniform in the given ranges:\n\\begin{enumerate}\n\t\\item{$a_{\\tau}<14.5$}\n\\end{enumerate}}\n\\end{itemize}\nThe high-level implementation is in Python and uses the \\underline{A}ction-based G\\underline{A}laxy \\underline{M}odelling \\underline{A}rchitecture (\\texttt{AGAMA}\\footnote{\\texttt{AGAMA} can be downloaded from\n \\url{https:\/\/github.com\/GalacticDynamics-Oxford\/AGAMA}}). This is a galaxy-modelling library in C++ consisting of several layers that together provide a complete package for constructing galaxy models. The innermost layer provides a range of mathematical tools that include integration, interpolation, multi-dimensional samplers, and units. The central layers provide classes for gravitational potentials, action finders converting phase-space coordinates to actions, and DFs. The outermost layer comprises Python wrappers for several of the C++ classes, and an additional suite of Python routines for fitting DFs, self-consistent modelling, generating mock catalogues, isochrone interpolation, and selection functions. A more detailed description of the library will be given elsewhere (Vasiliev et al. in prep).\n\\section[]{Results}\\label{sec:results}\nHere we discuss the favoured parameters determined for our four models, and their uncertainties where derived. We assess the quality of the fit of the models to the observables and investigate the moments associated with $M_4$.\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.9]{model_data_comp_1d}\n \\caption{In colour: one-dimensional distributions of mock phase-space\nobservables (cyan - $M_1$, orange - $M_2$ red - $M_3$, green - $M_4$). The error bars show Poisson errors.\nGrey regions: 2000 Monte Carlo resamplings of the data from the error\ndistributions. Progressively lighter shades of grey indicate $1\\sigma$ to $3\\sigma$ and 100\\% regions for the resampled observables.\\label{fig:datafit_1d}}\n\\end{figure*}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.9]{model_data_comp_feh_1d}\n \\caption{In colour and grey: as Fig.~\\ref{fig:datafit_1d} but\n for metallicities.\\label{fig:fehfit_1d}}\n\\end{figure}\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.63]{model_data_comp_2d_Jfehrotdf_varydistfehsf}\n \\caption{Colour-filled contours: two-dimensional\ndistributions of measured observables. Black contours: distributions of the mock observables for $\\mathrm{M}_4$.\n\\label{fig:datafit_2d}}\n\\end{figure*}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.55]{density}\n\t\\caption{Real-space density: (1) contour map of $M_4$ (top panel), (2) radial density profiles of models $M_1$ (cyan), $M_2$ (orange), and $M_4$ (green) (second panel), (3) logarithmic radial density gradient (third panel), and (4) axis ratio for models $M_1$, $M_2$, and $M_4$ with the same colour coding. \\label{fig:model_densitymoment}}\n\\end{figure}\n\\subsection[]{The best-fit parameters}\nTable~\\ref{tab:bestfitpars} gives the median of the recovered parameters for $M_1$, $M_2$, and $M_4$ and the MAP estimates for the parameters after the multi-stage fit for $M_3$. From $M_1$, an axis ratio $q\\sim0.7$ is recovered, with a halo that transitions from a power law index of $\\alpha_{\\mathrm{in}}\\sim-3.6$ to $\\alpha_{\\mathrm{out}}\\sim-4.8$ at a break radius $r_{\\mathrm{b}}\\sim30$ kpc. In the case of $M_2$, the axis ratio varies from $q_0\\sim0.4$ to $q_{\\infty}\\sim0.8$, transitioning at a radius $r_0\\sim7$ kpc. The power-law index of the slope is $\\alpha\\sim-4.7$. Both density profiles would yield a divergent mass if extended towards the centre, and are therefore not physical there. The metallicity DF has a cut-off at $\\mathrm{[Fe\/H]}_{\\mathrm{max}} = -0.83$ dex and peaks at $\\mathrm{[Fe\/H]}_{\\mathrm{peak}} = -1.77$ dex.\n\n\nFor models $M_3$ and $M_4$, the slope in actions steepens smoothly from $\\beta_{\\mathrm{in}}\\sim-2.1$ to $-2.2$ for actions below $\\sim1600$ kpc km s$^{-1}$ to a slope of $\\beta_{\\mathrm{out}}\\sim-4.7$ at larger actions. Allowing a distribution of ages as in $M_4$ results in a minor change in the inner and outer halo power-law indices. Since any rotation within the halo is weak ($x\\sim0$), introducing the part in the EDF odd in $J_{\\phi}$ probably has little effect on the optimal values of the weights on the actions $a_r, b_r$  and so on. However running an \\texttt{emcee} chain, which ensures the parameter space is fully explored, finds different action weights between $M_3$ and $M_4$. In the latter, the isodensity ellipsoids are flattened at low and high actions ($a_{\\phi}\\ll a_z$ and $b_{\\phi}\\ll b_z$) and the velocity ellipsoids are elongated in the radial direction in the inner halo ($a_r\\ll a_z$). The age model predicts a mean age of $\\sim 12.0\\,$Gyr, with a negative dependence on actions, i.e. ages decrease outwards. \n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.6]{dispmoments}\n \\caption{Velocity dispersions predicted by the\n best-fitting EDF. Going from the left to right: spherical radial\nvelocity dispersion, spherical angular velocity dispersion, and spherical azimuthal velocity dispersion.\n\\label{fig:model_dispmoments}}\n\\end{figure*}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.7]{beta}\n \\caption{Spherical anisotropy parameter predicted by the best-fitting EDF plotted as a map in $R$ and $z$ (top panel) and against spherical radius for a range of colatitudes $\\theta$ (bottom panel).\n\\label{fig:model_beta}}\n\\end{figure}\n\\subsection[]{Uncertainties on the recovered parameters} \nTable~\\ref{tab:bestfitpars} gives the 68\\% confidence intervals for parameters of $M_1$, $M_2$, $M_3$ (metallicity DF parameters only) and $M_4$. Figs.~\\ref{fig:emcee_m1}, ~\\ref{fig:emcee_m2}, ~\\ref{fig:emcee_feh}, and ~\\ref{fig:emcee_m4} present the 1-D and joint probability distributions from the {\\tt emcee} runs. The metallicity DF is independent of the spatial DF in $M_1$, $M_2$ and of the phase-space DF in $M_3$. Its parameters are thus shown separately.\n\nThe uncertainties on the parameters of $M_1$ and $M_2$ are of the order of $\\sim5$--$10\\%$ for the spatial DF, except for the inner axis ratio and flattening transition radius of $M_2$, which are more uncertain ($\\sim 20\\%$). There are positive correlations between the outer slope and between the break radius of $M_1$, and the outer axis ratio, flattening transition radius, and slope of $M_2$.\n\nFor models $M_1$, $M_2$, and $M_3$, the uncertainties on the maximum and peak metallicities are smaller, at the level of $\\sim1$--$2\\%$, showing that they are well constrained. Fig.~\\ref{fig:emcee_feh} shows a correlation between the two metallicity parameters that arises from the difference between them being limited to unity.\n\nThe uncertainties on the parameters of $M_4$ vary greatly. Those on the parameters of the metallicity DF are greater than the uncertainties in $M_1$ and $M_2$ because of the flexibility introduced with the age model, which modifies the distance-metallicity selection function. The uncertainties on the action weights vary between $\\sim 15$ to $35 \\%$. The uncertainty on the rotation parameter is high and translates to a 68\\% confidence interval $\\sim[-10, 30]$ kms$^{-1}$ for the rotation speed.\n\nThe mean age in the case of no dependence on actions has an uncertainty $\\sim0.33\\,$Gyr, primarily towards higher ages, because the distance-metallicity selection function varies much less there. The dependence on age is quite uncertain, again partly because of the insensitivity of the distance-metallicity selection function to the oldest ages. \n\nThe 1-D marginalized probability distributions in Fig.~\\ref{fig:emcee_m4} are unimodal, but often skewed. Asymmetrical distributions include the mean age ($a_{\\tau}$) in the case of no dependence on actions, and the dependence on actions ($b_{\\tau}$), both of which have an extended tail towards higher values. The inner halo power-law index ($\\beta_{\\mathrm{in}}$) has an asymmetrical distribution with an extended tail towards lower values. Several correlations exist between parameters. There is a weak, negative correlation between the power-law indices of the inner and outer halo ($\\beta_{\\mathrm{in}}$ and $\\beta_{\\mathrm{out}}$), i.e. a lower value of the index in the inner halo can be partly compensated by a larger value of the index in the outer halo. There is also a correlation between the weight on the radial action in the outer halo ($b_r$) and the weight on the angular action there ($b_{\\phi}$), and to a lesser extent in the inner halo. This highlights a connection between the flattening and anisotropy, i.e. more flattened systems tend to have a higher degree of radial anisotropy. There is also the correlation between the parameters of the metallicity DF. Other correlations may also exist but are obscured by the lack of sufficient resolution in the \\texttt{emcee} runs.  \n\\subsection[]{Fits to the observables}\\label{ssec:fits}\nWe assess the quality of the model fits by generating mock catalogues at the measured sky positions, using an adaptive sampling-rejection method from \\texttt{AGAMA}. The product of the SF and EDF is sampled in $\\log (s\/\\mathrm{kpc}) $, $\\log \\mathrm{[Fe\/H]}$, and for $M_3$ and $M_4$ in $\\tanh (v_r\/\\mathrm{km\\,s}^{-1})$, $\\tanh(\\mu_l^*\/\\mathrm{mas})$, and $\\tanh(\\mu_b\/\\mathrm{mas})$. In these coordinates the value of the EDF varies less strongly. Each proposed sample has noise added to it according to the errors measured on the observables at those sky positions. The parameters used to generate the mock samples in each case correspond to the MAP estimates.\n\nThe coloured points (cyan - $M_1$, orange - $M_2$, red - $M_3$, and green - $M_4$) joined by lines in Figs.~\\ref{fig:datafit_1d}\nand \\ref{fig:fehfit_1d} show histograms of the mock catalogues. The region\ncovered by analogous histograms of 2000 resamplings of the error\ndistributions of the measured observables are shown by the grey regions. In plots for observables with small errors, such as $l$, $b$, $s$, and $v_\\parallel$, the grey regions form fairly well defined curves. In plots for observables with large errors, namely $(\\mu_l^*,\\mu_b,\\mathrm{[Fe\/H]})$, the grey regions fill out a region of significant width. The coloured curves generally overlap with this region, indicating that the EDF and SF are together\ndoing a good job at describing the location of the observables. \n\nFig.~\\ref{fig:datafit_2d} compares histograms for the joint distribution of\npairs of phase-space observables $(l,b)$, $(s,b)$, $(\\mathrm{[Fe\/H]},b)$, etc. in the case of parameters corresponding to the MAP estimate obtained for $M_4$. The colour-filled contours show\ndistributions of mock observables, while the black contours show the\ndistributions of measured observables. In general there is good agreement\nbetween the mock and observational distributions.\n\\subsection[]{Moments of the recovered parameters of $M_4$}\\label{ssec:moments}\nWe now describe the model $M_4$ with the parameters corresponding to the MAP estimate.\n\\subsubsection[]{Density of stars in real space}\nFig.~\\ref{fig:model_densitymoment} shows the shape of the density distribution. The colour scale in the top panel shows $\\rho(R,z)$. Flattening of the contours is evident. By fitting ellipses to the isodensity curves, we obtain the radial density profile shown by the green curve in the second panel. The steepening of the density profile with increasing radius is shown by the green curve in the third panel, which gives the logarithmic radial density gradient. The slope steepens smoothly from $-2.2$ at $\\sim 2$ kpc to $\\sim -4$ in the outer halo. The green curve in the bottom panel shows that the halo is flattened ($q\\simeq0.6$ to $0.8$) throughout. Radial profiles of the logarithmic density, logarithmic density gradient, and axis ratio are shown also for $M_1$ (cyan) and $M_2$ (orange), each plotted against elliptical radius. The density profile of $M_1$ is steeper in the inner and outermost parts than that of $M_4$, while that of $M_2$ is steeper throughout. The axis ratio of $M_1$ is very similar to that of $M_4$. The axis ratio of $M_2$ is significantly lower than that of $M_1$ and $M_4$ in the inner halo and moderately higher in the outer halo.\n\\subsubsection[]{The velocity ellipsoid}\n\nFig.~\\ref{fig:model_dispmoments} shows the velocity dispersions of $M_4$. $\\sigma_r$ generally dominates at high $z$ throughout, implying radial anisotropy there. At low $z$, $\\sigma_{\\phi}$ dominates, implying tangential anisotropy. The contours of constant $\\sigma_r$ and $\\sigma_{\\theta}$ are elongated in the $z$ direction, while $\\sigma_{\\phi}$ contours are elongated in the $R$ direction. Fig.~\\ref{fig:model_beta} shows the spherical anisotropy parameter\n\\begin{equation}\n\\beta_{\\rm s}= 1 - \\frac{\\sigma_{\\theta}^2 + \\sigma_{\\phi}^2}{2\\sigma_r^2}.\n\\end{equation}\nThe degree of radial anisotropy increases from a tangential bias in the equatorial plane to $\\sim0.3$ at the highest point along the $z$-axis. The lower panel of Fig.~\\ref{fig:model_beta} shows radial profiles of the spherical anisotropy parameter against spherical radius along a range of polar angles, $\\theta$, where $\\theta = 0$ is along the $z$ axis and $\\theta = 90^\\mathrm{o}$ is in the equatorial plane. In the equatorial plane, the orbits vary from isotropic to mildly tangential. Nearer the $z$ axis, the profiles become more radially anisotropic.\n\n\\subsubsection[]{The distribution of ages}\nThe top panel of Fig.~\\ref{fig:model_age} shows the distribution of mean ages. Contours of constant mean age are flattened, approximately as the contours of constant density. The bottom panel shows the age map inferred from the actions of the stars in the SEGUE-II sample, convolved with their error distributions. In general stars at lower $R$ and $z$ have higher ages, but a high velocity implies large actions and therefore a relatively young age. Therefore the gradient in age with actions manifests as a small negative age gradient with radius $\\sim-0.03\\,$Gyr kpc$^{-1}$. The plot also suggests that we are biased towards observing younger stars.\n\\section[]{Discussion}\\label{sec:discuss}\nHere we focus our discussion on the results of fitting the full phase-space EDFs, how they compare to the literature, and highlight uncertainties that may impact these results.\n\\subsection[]{Our perspective on the stellar halo}\n\\subsubsection[]{Distribution of stars in action space}\nTraditionally metallicity and age gradients have been examined as a function of radius. However, stars are better characterised by their actions, which do not vary along orbits. A clear separation in action space becomes `smeared' in real space, and a glance at the picture in action space can be enlightening. We find that the EDF declines more rapidly with actions in the outer halo (slope $\\sim-5$) than in the inner halo (slope $\\sim-2$). The weights on the actions in the EDF suggest a flattened stellar system, which is tangential to isotropic in the equatorial plane, becoming more radially anisotropic as $z$ increases. The part of the EDF odd in $J_{\\phi}$ is negligible. We find the ages of the stars to be well predicted by a log-linear dependence on the total action, with higher ages at smaller actions. The gradient is thus negative ($-0.69\\,\\mathrm{Gyr}\\,\\mathrm{dex}^{-1}$). A single-age model is, however, also able to reproduce the observations.\n\n\\subsubsection[]{Distribution of stars in real space}\nThe slopes in action space translate to a density profile in real space that steepens with radius from a slope of $\\sim-2$ at $\\sim 2$ kpc  to $\\sim-4$ by 30 kpc. The gradient in ages with actions translates to a real-space gradient $\\sim -0.03\\,$Gyr kpc$^{-1}$, subject to a significant degree of uncertainty.\n\nThe weights on actions determine the shape of the density and velocity ellipsoids. The halo's axis ratio is roughly constant at $\\sim0.7$, very similar to what is implied by the broken power-law model. The orbital structure varies from mildly tangential to moderately radially anisotropic throughout, becoming more radial as you move from the equatorial plane towards the $z$ axis.\n\nThe negligible part of the EDF that is odd in $J_{\\phi}$ generates a very small level of rotation with a large uncertainty (our 68\\% confidence interval extends between $-10$ to $30$ km s$^{-1}$).\n\nFinally, we did not probe the distribution of metallicities in action space directly. It is however linked to the EDF through the selection function in distance and metallicity. The metallicity DF is well described by a single lognormal distribution with a peak at $\\sim-1.8$ dex and with a maximum at $\\sim-0.8$ dex.\n\\subsubsection[]{Is there a difference between the inner and outer halo?}\nThere is compelling evidence for differences between the inner and outer halo in action space, primarily due to the difference in the inner and outer slopes in actions. This manifests in real space as a steepening of the density profile with radius, a non-negligible negative age gradient with radius, and a variation in the anisotropy with radius.\n\nThere have been several claims of two populations in the halo \\citep[e.g.][]{carollo+07,beers+12,deason+13,hattori+13}. Although our sample is too small to rule out such a dichotomy, our models, which predict smooth transitions between the inner and outer halo, are sufficient to reproduce the current data.\n\\subsection[]{Comparison with previous work}\n\\subsubsection[]{Radial density profile}\nSeveral authors have determined density profiles for halo BHBs, finding a power-law index of $\\sim-2.5$ to$-3.5$ \\citep[e.g.][]{preston+91,kinman+94,sluis+98,dePropis+10}. Double power-law profiles have also been fitted to BHBs \\citep{deason+11a}, finding a power-law index of $-2.3$ in the inner halo, $-4.6$ in the outer halo, and a break radius of $27$ kpc. The first two power-law indices are very similar to our findings. Similar density profiles have been recovered for other stellar populations probing deep into the halo, such as RR Lyrae \\citep{watkins+09,sesar+13}, subdwarfs \\citep{smith+09b} and K giants \\citep{xue+15,das+16}. An axis ratio ranging from $0.5$--$0.6$ has been found in BHBs \\citep[e.g.][]{sluis+98,deason+11a} and $0.6$--$1.0$ in other stellar populations \\citep[e.g.][]{carollo+07,watkins+09,sesar+13,xue+15,smith+09b}.\n\\subsubsection[]{The velocity ellipsoid}\n\\cite{deason+11b} and \\cite{hattori+13} found the BHB stars in the Milky Way halo to exhibit a dichotomy between a prograde-rotating, comparatively metal-rich component ($\\mathrm{[Fe\/H]}>\u22122$) and a retrograde-rotating, comparatively metal-poor ($\\mathrm{[Fe\/H]}<\u22122$) component. \\cite{deason+11b} attribute the prograde metal-rich population to the accretion of a massive satellite ($\\sim10^9$M$_{\\odot}$) and the metal-poor population to the primordial stellar halo. The net retrograde rotation might then reflect an underestimate in the adopted LSR circular velocity. \\cite{fermani+13b} however  remeasured the rotation of the Milky Way stellar halo on two samples of BHB halo stars from SDSS with four different methods, and found a weakly prograde or non-rotating halo in all cases. They attributed the rotation gradient across metallicity measured by \\cite{deason+11b} on a similar sample of BHB stars to the inclusion of regions in the apparent magnitude-surface gravity plane known to be contaminated by substructures. \\cite{sirko+04} did not find any rotation in their sample of BHBs and \\cite{smith+09b} do not find any rotation in their sample of subdwarfs.\n\nSeveral authors \\citep{deason+12a,kafle+12,williams+15b,cunningham+16} found a radially biased velocity ellipsoid overall but some find a region around $\\sim20$ kpc with tangential anisotropy. From a similar sample of BHBs, \\cite{sirko+04} found isotropy and \\cite{hattori+13} found the metal-rich component to exhibit mild radial anisotropy, and the metal-poor component to exhibit tangential anisotropy. Analysis of other halo tracer populations have also arrived at a mixture of conclusions. \\cite{dehnen+06} found radial anisotropy in a mixture of globular clusters, horizontal-branch and red-giant stars, and dwarf spheroidal satellites. \\cite{smith+09a} found the velocity ellipsoid of SDSS subdwarfs to be radially biased. \\cite{carollo+10} found inner-halo metal-richer stars on radially anisotropic orbits, and outer-halo stars to be on less eccentric orbits. \n\nThe diversity in conclusions regarding halo anisotropy may be a result of several things. Spectroscopic surveys can have a strong selection bias on metallicity (not so much in BHBs, but in K giants, see \\cite{das+16}), and if there is a correlation between metallicity and dynamics, then a lack of treatment of such a bias can lead to erroneous conclusions about anisotropy. It is also true that the proper motions currently available are highly uncertain. Furthermore, it should be emphasised that our models are designed to fit the smooth, phase-mixed halo, and this may be why we do not reproduce the `dip' in anisotropy found by several authors at $20$ kpc. On a related note, we have attempted to remove substructures where possible - the exact samples used by various authors differ slightly in their observed velocity distributions and thus derive different anisotropy profiles.\n\nOur model qualitatively agrees with simulations at high $z$ (i.e. radial bias  increases outward), though with a lesser degree of radial bias throughout. In such simulated haloes, the primary mechanism for growth since $z\\sim2$ is thought to be accretion onto the halo through minor mergers \\citep{bullock+05,abadi+06}, and accreting objects have rather radial orbits.\n\\subsubsection[]{The distribution of chemical properties}\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.6]{age}\n \\caption{Age map predicted from the total action moment (top panel) and for the stars in the SEGUE-II sample (bottom panel).\n\\label{fig:model_age}}\n\\end{figure}\n\\cite{carollo+07} and \\cite{beers+12} found evidence for two metallicity components in a mixed sample of stellar types. \\cite{an+15} analysed a sample of main-sequence halo stars and found two components peaking at [Fe\/H]$\\sim-1.7$ and $\\sim-2.3$. \\cite{xue+15} and \\cite{das+16} reached similar conclusions about SDSS K giants.\n\nWe find one lognormal component is sufficient to describe the metallicity DF of the BHBs. The discrepancy may arise from a difference between the types of systems thought to contribute to halo stars. There is an age-metallicity bimodality in the Milky Way globular cluster system \\cite[e.g.][]{leaman+13}. \\cite{fiorentino+15} analysed the periods and luminosity amplitudes of field RR Lyrae stars and found that dwarf spheroidals lacked high-amplitude short-period variable stars; whereas these are found in globular clusters and massive dwarf irregulars such as the Sagittarius stream. \n\n \\cite{preston+91} detected a colour gradient in BHBs out to $\\sim12$ kpc, which \\cite{santucci+15} consolidated with a larger sample, extending out to $\\sim40$ kpc. They claimed that the \ngradient is independent of metallicity and therefore indicate a gradient in age. More massive systems penetrate deeper into the gravitational potential. We also expect these systems to have the oldest components because they would have grown over a longer timescale. From a similar argument however, we would also expect chemical evolution to have occurred more rapidly in these systems, therefore producing a metallicity gradient. It is unclear why the difference in the make-up of the inner and outer halos would manifest solely as an age gradient rather than a metallicity gradient. The metallicity gradient could just be small.\n\\subsection[]{Uncertainties in the analysis}\nThe errors quoted in this work represent statistical uncertainties only, rather than systematic errors, which are more difficult to characterise. We discuss possible sources of systematic errors below.\n\\subsubsection[]{Impact of resonances and chaos}\nWe have assumed a fully-integrable potential in which the number of isolating integrals of motion is equal to the number of degrees of freedom. In such cases a transformation from phase-space coordinates to angle-action coordinates can be done globally. Real potentials however permit families of resonantly trapped orbits. These resonant orbits possess actions, but require a different transformation. In non-integrable potentials, some fraction of the orbits will be chaotic. Chaotic orbits are not bound to the surface of a torus and instead fill the spaces between tori. Chaotic orbits have no orbital actions. \\cite{binney16} concluded that the net impact of resonant trapping on the dynamics of halo stars is likely to be small. \n\\subsubsection[]{A fixed potential and parametrised EDF}\nAn EDF can perfectly model the data only in the true potential. Therefore any imperfections in our choice of potential will both bias our EDF away from the true EDF and give rise to discrepancies between our best model and the data. Our success in reproducing the data suggests that our chosen potential is not seriously in error.  \nThe supposition of a particular functional form for the EDF can bias the results by restricting the set of possible solutions, despite allowing a range of density and anisotropy profiles. Our ansatz regarding the dependence of stellar ages on actions represents just one, physically-motivated, possibility that is simple to calculate. An age gradient is not forced by the EDF however; if none were needed by the data, age would have been found to be independent of actions.\n\\subsubsection[]{Stellar population assumptions}\nOur evaluation of the distance-metallicity selection function depended on  relations from isochrones between the age, mass, and metallicity of a star and its luminosity in various wavebands. Systematic errors arising from faulty isochrones are difficult to assess. The ability of our EDF to produce a similar density profile for the BHBs to that in the literature suggests that our selection function is not significantly in error.\n\\subsubsection[]{Impact of substructure}\nWe have masked the Sagittarius stream in this analysis, but we do not know how other, unmasked substructures may impact our assessment of the halo's structure. Our ability to sufficiently reproduce the phase-space observables after excluding the stream implies that either we are predominantly probing the smooth stellar halo with the data or the current data are too sparse to resolve the halo's substructure. \n\\section[]{Conclusions and further work}\\label{sec:conc}\nWe probed the chemodynamical structure of Milky Way halo BHBs by combining spatial and action-based EDFs that describe the locations of stars in phase space, metallicity, and age. The analysis allows a more natural description of the ages of BHBs in action space in which their separation is clearer than in real and velocity space. The specification of an EDF enables the incorporation of a realistic selection function that takes into account restrictions on sky positions, apparent magnitudes, and colours. In general, our models reproduce the observations well. This may be an argument that there is enough phase-mixed debris for action space to be smoothly populated (at least for relatively tightly bound orbits). i.e., there may be a part of the inner halo that will always be well represented by smooth models. Alternatively it may be because the data for BHBs are not yet rich enough to resolve most halo substructures. \n\nThe EDF of the BHBs is steeper at larger actions than at smaller actions. Older stars are found at smaller actions and younger stars at larger actions. The spatial distribution of the stars is similarly well reproduced by a broken power law with a constant axis ratio, a single power law with a variable axis ratio, and a gradually steepening power law with a variable axis ratio. Fitting positions and velocities simultaneously yields a density profile that steepens smoothly from $\\sim-2$ at $\\sim 2$ kpc to $\\sim-4$ in the outer halo. The halo is moderately flattened with an axis ratio $\\sim 0.7$ throughout. The overall metallicity distribution is well described by a single lognormal component that has a maximum metallicity at $\\sim-0.8$ dex and a peak at $\\sim-1.8$ dex. Our full phase-space EDF also allowed rotation - this could be at a level of $-10\\,$km s$^{-1}$ to $30\\,$km s$^{-1}$ at most but the median result favours no rotation. The stellar velocity ellipsoid varies from tangential bias in the equatorial plane to radially elongated at high $z$. Allowing a dependence of stellar ages on actions leads to an age gradient $\\sim-0.03$ kpc$^{-1}$, with moderate uncertainty. However, an EDF assuming approximately a single age of $11\\,$Gyr is also able to fit the observables well.\n\nThere are several possible directions for further work. The EDF could be applied to detect substructures in a richer sample of halo stars in the phase-space-metallicity domain. The EDF could be changed to make the transition between the inner and outer asymptotic slopes of the density profile sharper. The EDF could be further elaborated to include a dependence on [$\\alpha$\/Fe]. \n\n\\section*{Acknowledgements}\nThe research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7\/2007-2013)\/ERC grant agreement no.\\ 321067. PD thanks GitHub for providing free private repositories for educational use. AW acknowledges the support of STFC. PD is also grateful for fruitful discussions with members of the Oxford Galactic Dynamics group.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}