The only non-standard term involves the factor $\frac{\mathrm{d} q}{\mathrm{d} \eta }$. As usual, the perturbation Ψ of the unperturbed phase-space distribution f0 = 1/(exp (ε/T0) − 1) is introduced as
(36)
\begin{eqnarray*}
f = f_0(\epsilon) \left(1 + \Psi \right)\, ,
\end{eqnarray*}
where the co-moving energy ε reads
(37)
\begin{eqnarray*}
\epsilon \equiv \left(q^2 + a^2 \frac{m_{V_\pm }^2}{{\cal S}^2\, T_0^2}\right)^{1/2}\, .
\end{eqnarray*}
The scaling function $\mathcal {S}$ is defined in equation (5). Since the term $\frac{\mathrm{d} q}{\mathrm{d} \eta }$ is determined by the geodesic equation for a massive point particle, we may write
(38)
\begin{eqnarray*}
\frac{\mathrm{d} q}{\mathrm{d} \eta } \frac{\partial f}{\partial q} &amp;=&amp; \left(q \dot{\phi } - \epsilon n_i \partial _i \psi - \frac{a^2 m_{V_\pm } \dot{m}_{V_\pm }}{q} \right)\nonumber\\
&amp;&amp; \times \,\left(\frac{\partial f_0}{\partial q} (1 + \Psi) + f_0 \frac{\partial \Psi }{\partial q}\right).
\end{eqnarray*}
The use of the geodesic equation for a quasi-particle must be questioned, if this particle associates with pure quantum fluctuations (Hofmann 2017). If at all, temperature fluctuations in the V± sector can thus only be coherently propagated via the low-frequency regime of γ fluctuations in terms of classical electromagnetic waves (Hofmann 2016b). To do this, a residual interaction between V± and γ is required. Albeit when such an interaction occurs (Hofmann 2016a), its efficiency in conveying the coherent propagation of V± temperature fluctuations must be questioned, especially at high temperatures (Falquez et al. (2010). To ignore the V± Boltzmann equations and associated source terms in linearized Einstein equations thus is a physically motivated option. On the other hand, considering the evolution of V± temperature fluctuations via the coherently propagating low-frequency sector in γ implies that ε = q in the V± geodesic equation. At the same time, $m_{V_\pm }\gt 0$ is required in f0. Since the structure of temperature fluctuations is mainly imprinted before and during recombination, setting $m_{V_\pm }=0$ in the geodesic equation does not influence the prediction for the power spectra in practice. Note that due to equation (37) an explicit dependence of f0 on η needs to be considered via a = a(η). Transforming equation (35) into k space and otherwise following the standard procedure of linear perturbation theory (Ma & Bertschinger 1995), one arrives at the following hierarchy:
(39)
\begin{eqnarray*}
\dot{\Psi }_0 = - k\Psi _1 - \frac{\mathrm{d} \ln f_0}{\mathrm{d} \ln q}\dot{\phi } - \frac{1+\Psi _0}{f_0} \frac{\partial f_0}{\partial \eta }\, ,
\end{eqnarray*}
(40)
\begin{eqnarray*}
\dot{\Psi }_1 = \frac{k}{3 }(\Psi _0 - 2 \Psi _2) - \frac{1}{3} \frac{\mathrm{d} \ln f_0}{\mathrm{d} \ln q} k \psi - \frac{\Psi _1}{f_0} \frac{\partial f_0}{\partial \eta },
\end{eqnarray*}
(41)
\begin{eqnarray*}
\dot{\Psi _l} = \frac{k}{(2l+1)}\left[l \Psi _{l-1}-(l+1)\Psi _{l+1} \right] -\frac{\Psi _l}{f_0} \frac{\partial f_0}{\partial \eta },
\end{eqnarray*}
where the $\Psi _l(\vec{k},q,\eta)$ are the expansion coefficients for $\Psi (\vec{k},\hat{n},q,\eta)$ into Legendre polynomials.
Citation: Hofmann 2017
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