Submitted to the ApJ
Preprint typeset using LATEX style emulateapj v. 6/22/04
EFFICIENT SIMULATIONS OF EARLY STRUCTURE FORMATION AND REIONIZATION
Andrei Mesinger & Steven Furlanetto
Yale Center for Astronomy and Astrophysics, Yale University, New Haven, CT 06520
Submitted to the ApJ
ABSTRACT
Detailed theoretical studies of the high-redshift universe, and especially reionization, are generally
forced to rely on time-consuming N-body codes and/or approximate radiative transfer algorithms. We
present a method to construct semi-numerical “simulations”, which can efficiently generate realizations
of halo distributions and ionization maps at high redshifts. Our procedure combines an excursion-
set approach with first-order Lagrangian perturbation theory and operates directly on the linear
density and velocity fields. As such, the achievable dynamic range with our algorithm surpasses
the current practical limit of N-body codes by orders of magnitude. This is particularly significant in
studies of reionization, where the dynamic range is the principal limiting factor because ionized regions
reach scales of tens of comoving Mpc. We test our halo-finding and ionization-mapping algorithms
separately against N-body simulations with radiative transfer and obtain excellent agreement. We
compute the size distributions of ionized and neutral regions in our maps. We find even larger ionized
bubbles than do purely analytic models at the same volume-weighted mean hydrogen neutral fraction,
x̄HI, especially early in reionization. We also generate maps and power spectra of 21-cm brightness
temperature fluctuations, which for the first time include corrections due to gas bulk velocities. We
find that velocities widen the tails of the temperature distributions and increase small-scale power,
though these effects quickly diminish as reionization progresses. We also include some preliminary
results from a simulation run with the largest dynamic range to date: a 250 Mpc box that resolves
halos with massesM ≥ 2.2×108M⊙. We show that accurately modeling the late stages of reionization,
x̄HI ∼< 0.5, requires such large scales. The speed and dynamic range provided by our semi-numerical
approach will be extremely useful in the modeling of early structure formation and reionization.
Subject headings: cosmology: theory – early Universe – galaxies: formation – high-redshift – evolution
1. INTRODUCTION
Accurately modeling the formation of bound struc-
tures is invaluable for understanding any process in
the early universe. Reionization, the epoch when
radiation from early generations of astrophysical objects
managed to ionize the intergalactic medium (IGM), is
particularly sensitive to the distribution of collapsed
structure. Current observations paint a complex pic-
ture of the reionization epoch (Mesinger & Haiman
2004; Wyithe & Loeb 2004; Fan et al. 2006;
Mesinger & Haiman 2006; Malhotra & Rhoads 2004;
Furlanetto et al. 2006c; Malhotra & Rhoads 2006;
Page et al. 2006; Kashikawa et al. 2006; Totani et al.
2006). The next generation of instruments (James
Webb Space Telescope; 21-cm instruments such as the
Low Frequency Array and the Mileura Widefield Array
Low-Frequency Demonstrator; CMB polarization mea-
surements with Planck, etc.), could potentially shed light
on this poorly understood milestone. Unfortunately,
we still do not have accurate models of reionization
with which to interpret these upcoming (and current)
observations.
The main difficulty lies in the enormous dynamic range
required. Ionized regions are expected to reach charac-
teristic sizes of tens of comoving Mpc (Furlanetto et al.
2004c; Furlanetto & Oh 2005), which is over seven or-
ders of magnitude in mass larger than the pertinent cool-
ing mass, corresponding to gas with a temperature of
T ∼ 104 K (e.g. Efstathiou 1992; Thoul & Weinberg
1996; Gnedin 2000b; Shapiro et al. 1994). The required
dynamic range is even larger if smaller “minihalos” be-
low this cooling threshold are important during reion-
ization. Because of the steep mass dependence of halo
abundances, halos with masses close to the cooling mass
could dominate the photon budget. Hence modeling
reionization requires simulation box sizes of hundreds
of megaparsecs on a side, with extremely high resolu-
tion. Attempts to overcome these obstacles have gener-
ally followed the same fundamental and well-trod path
(e.g. Gnedin 2000a; Razoumov et al. 2002; Ciardi et al.
2003; Sokasian et al. 2003; Iliev et al. 2006b; Zahn et al.
2007; Trac & Cen 2006): (1) N-body codes are run to
generate halo distributions; (2) a simple prescription is
used to relate the halo mass to an ionizing efficiency; (3)
approximate methods (generally so-called ray-tracing al-
gorithms) are used to model radiative transfer (RT) on
large scales.
Even with modest halo resolution
(Springel & Hernquist 2003) of tens of dark matter
particles per halo, such schemes are computationally
limited to box sizes of tens of megaparcecs, if they
wish to resolve the likely cooling mass. McQuinn et al.
(2006a) extended the mass resolution of their sim-
ulations by using a merger tree scheme to populate
sub-grid scales with unresolved halos in a stochastic
manner. Such hybrid schemes are useful for extending
the dynamic range, but merger trees require a number
of corrections to achieve consistent mass functions (see,
e.g., Sheth & Pitman 1997; Benson et al. 2005, and Fig.
1 in McQuinn et al. 2006a) and to track individual halos
with redshift. Moreover, although they are perfectly
adequate for many purposes (including studying the
http://arxiv.org/abs/0704.0946v1
large-scale features of reionization), they prevent one
from taking full advantage of the simulation.
Aside from dynamic range, the other main limiting fac-
tor in all of the above numerical approaches is speed.
Even if the relevant scales can be resolved with N-body
codes, such as may be the case in the early phase of reion-
ization or with hybrid stochastic schemes. The codes
themselves generally take days to run on large super-
computing clusters, with the approximate RT algorithms
consuming a few additional days. The computational
cost of each simulation makes it difficult to explore the
full range of parameter space for reionization, which is
particularly large because we know so little about high-
redshift galaxies.
The computational cost becomes truly prohibitive if
hydrodynamics is included: the largest such simulation
of reionization performed to date spanned only 10h−1
Mpc (Sokasian et al. 2003). Including self-consistent de-
scriptions of galaxy formation – even at the approximate
level currently implemented in lower-redshift cosmolog-
ical simulations (e.g., Springel & Hernquist 2003) – re-
quires hydrodynamics, so N-body simulations of reion-
ization are limited to semi-analytic prescriptions for star
formation, feedback, etc. It is therefore worthwhile to
explore even simpler schemes.
The purpose of this paper is to introduce approximate
but efficient methods for generating halo distributions at
high redshifts as well as for generating the associated ion-
ization maps. We apply an excursion-set approach (e.g.
Bond et al. 1991; Lacey & Cole 1993) to the filtering of
a realization of the linear density field and then adjust
halo locations with first-order perturbation theory. We
can thus generate halo distributions at any given red-
shift, without explicitly including information from any
higher redshifts. This scheme is an updated form of
the “peak-patch” formalism developed and validated by
Bond & Myers (1996a,b), although it was conceived and
implemented completely independently. We then apply
a similar technique to obtain the ionization field from the
halo field. This part is similar to the schemes described
in Zahn et al. (2005, 2007), except applied to our effi-
ciently built halo distributions. As such, our methods
allow us to make general predictions about non-linear
processes, such as structure formation and reionization,
without making use of time-guzzling cosmological sim-
ulations. The speed of our approach also allows us to
explore a larger dynamic range than is possible with cur-
rent cosmological simulations while preserving detailed
spatial information (at least in a statistical sense), un-
like purely analytic models.
This paper is organized as follows. In § 2, we introduce
and test the components of our halo finding algorithm.
In § 3, we introduce and test our HII bubble finding
algorithm. In § 4, we use our semi-numerical scheme
to generate maps and power spectra of expected 21-cm
brightness temperature fluctuations throughout reioniza-
tion. In § 5, we summarize our key findings and present
our conclusions.
Unless stated otherwise, we quote all quantities in co-
moving units. We adopt the background cosmological
parameters (ΩΛ, ΩM, Ωb, n, σ8, H0) = (0.76, 0.24,
0.0407, 1, 0.76, 72 km s−1 Mpc−1), consistent with the
three–year results of the WMAP satellite (Spergel et al.
2006).
2. SEMI-NUMERICAL SIMULATIONS OF HALO
PROPERTIES
In brief, our algorithm generates a linear density field
and identifies halos within it. Because only linear evo-
lution is required, the algorithm is fast and flexible. We
generate 3D Monte-Carlo realizations of the linear den-
sity field on a box with sides of length L = 100 Mpc and
N = 12003 grid cells. As such, we are able to take advan-
tage of many pre-existing tools operating on the linear
density field alone. Our method consists of the following
principal steps:
1. creating the linear density and velocity fields
2. filtering halos from the linear density field using
the excursion-set formalism
3. adjusting halo locations using their linear-order
displacements
Step (1) only needs to be done once for each realization,
since it is independent of redshift. As mentioned above,
steps (2) and (3) need only be performed on redshifts of
interest, i.e. since our output at redshift z is independent
of any outputs at higher redshifts, there is no need for
our code to “run down” to z, as is the case for N-body
codes.
Our algorithm is an updated and simplified version of
the “peak-patch” algorithm of Bond & Myers (1996a);
we refer the interested reader there for more detailed
explanations of some steps. A simpler version has also
been used by Scannapieco et al. (2002) to study metal
enrichment at high redshifts.
We perform our semi-numerical simulations on a sin-
gle desktop Mac Pro with two dual-core 3.00 GHz Quad
Xeon processors and 16 GB of RAM. ForN = 12003, step
(1) takes ∼ 1 hour. For a given redshift in our range of
interest, specifically for z = 8.75, steps (2) and (3) take
∼ 2.5 hours. To achieve comparable halo mass resolution
(including halos with M ∼> 10
7M⊙) with a minimum of
∼ 500 particles per halo (Springel & Hernquist 2003), N-
body codes would require a prohibitively large number
of particles, N ∼ 1012! Below we describe in detail the
components of our model.
2.1. The Linear Density Field
Our linear density field is generated in much the same
way as it is for N-body codes. We briefly outline the
procedure here.
The density field of the universe, δ(x) ≡ ρ(x)/ρ̄ − 1,
in the linear regime1 is well-represented as a Gaussian
random field, whose statistical properties are fully de-
fined by its power spectrum, σ2(k) ≡ 〈|δ(k)|2〉. Here,
δ(k) is the Fourier transform of δ(x), and the standard
assumption of isotropy implies σ2(k) = σ2(k) while ho-
mogeneity implies that there are no density fluctuations
with wavelengths larger than the box size L = V 1/3. We
use the following, standard (e.g. Bagla & Padmanabhan
1997; Sirko 2005) Fourier transform conventions:
δ(k) =
δ(x)e−ik·x , (1)
1 In linear theory, density perturbations evolve in redshift as δ(z)
= δ(0)D(z), where D(z) is the linear growth factor normalized
so that D(0) = 1 (e.g., Liddle et al. 1996). Unless the redshift
dependence is noted explicitly, from this point forward we will work
with quantities linearly-extrapolated to z = 0.
with the inverse transform being
δ(x) =
δ(k)eik·x . (2)
The discrete simulation box only permits a finite set
of wavenumbers: k = ∆k(i, j, k), where ∆k = 2π/L and
i, j, k are integers in the range (-
N/2]. For each
independent wavenumber,2 we assign
δ(k) =
σ2(k)
(ak + ibk) , (3)
where ak and bk are drawn from a zero-mean Gaussian
distribution with unit variance. We use the power spec-
trum from Eisenstein & Hu (1999). Then the real-space
density field, δ(x), is obtained by performing an inverse
Fourier transform on δ(k).
2.2. The Linear Velocity Field
We construct a linear velocity field corresponding to
our linear density field using the standard Zel’Dovich
approximation (c.f. Zel’Dovich 1970; Efstathiou et al.
1985; Sirko 2005):
x1=x+ ψ(x) , (4)
v≡ ẋ1 = ψ̇(x) , (5)
δ(x)=−∇·[(1 + δ(x))ψ(x)]
≈−∇ · ψ(x) , (6)
where x and x1 denote initial (Lagrangian) and updated
(Eulerian) coordinates, respectively, ψ(x) is the displace-
ment vector, and the last equation follows from the conti-
nuity criterion, with the final approximation using linear-
ity, δ(x) ≪ 1. We note again that all units are comoving,
unless stated otherwise. From the above, one can relate
the velocity mode in our simulation at redshift z to the
linear density field:
v(k, z) =
Ḋ(z)δ(k) , (7)
where for computational convenience differentiation is
performed in k-space.
Another convenient property of this first-order
Zel’Dovich approximation is that the velocity field can
be decomposed into purely spatial, vx(x), and purely
temporal, vz(z), components:
v(x, z) = vz(z)vx(x) , (8)
where vz(z) = Ḋ(z) and vx(x) is the inverse Fourier
transform of ikδ(k)/k2. This is computationally conve-
nient, as we only need to compute the vx(x) field once
in order to be able to scale it for all redshifts, and it
also allows us to write a simple, exact expression for the
integrated linear displacement field, Ψ. When eq. (8) is
integrated from some large initial z0 [D(z0) ≪ D(z)], the
total displacement is just
Ψ(x, z)= [D(z)−D(z0)]vx(x)
≈D(z)vx(x) (9)
2 Since δ(x) is real-valued, only half of the k-modes defined
above are independent. The other half are determined by the usual
Hermitian constraints for real-valued functions (see for example
Hockney & Eastwood 1988; Bagla & Padmanabhan 1997).
We make use of this displacement field to adjust the halo
locations obtained by our filtering procedure (see § 2.4),
as well as to adjust the linear density field for our 21-cm
temperature maps (see § 4).
In principle, one could obtain non-linear velocities by
mapping the linear overdensity to a corresponding non-
linear overdensity obtained from a spherical collapse
model (Mo & White 1996), and then taking the time
derivative of the non-linear overdensity. However, due
to the large spread in the dynamical times of the non-
linear density field, accurately capturing the time evolu-
tion is non-trivial. Furthermore, although the non-linear
density field implicitly captures the velocities of collaps-
ing gas, mapping each pixel’s linear density to its non-
linear counterpart independently of other nearby pixels
does not properly preserve correlations on larger scales.
Hence, we choose to use the linear density field directly
in estimating velocities. For the purposes of studying the
ionization field, we are further justified in this procedure
because our final ionization maps are smoothed on large
scales, on which most pixels are still in the linear regime
at the high redshifts of interests. It is possible to include
higher-order contributions to the Zel’Dovich approxima-
tion where necessary (e.g., Scoccimarro & Sheth 2002).
2.3. Halo Filtering
In standard Press-Schechter theory (PS; see e.g.,
Press & Schechter 1974; Bond et al. 1991; Lacey & Cole
1993), the halo mass function can be written as
∂n(> M, z)
δc(z)
σ2(M)
∂σ(M)
c (z)
2σ2(M)
where n(> M, z) is the mean number density of halos
with total mass greater than M , ρ̄ = ΩMρcrit is the
mean background matter density, δc(z) ∼ 1.68/D(z) is
the scale-free critical over–density evaluated in the case
of spherically symmetric collapse (Peebles 1980), and
σ2(M) =
σ2(k)W 2(k,M) , (11)
is the squared r.m.s. fluctuation in the mass enclosed
within a region described by the filter function,W (k,M),
normalized to integrate to unity.
Although the PS mass function in eq. (10) is in fair
agreement with simulations, especially for halos near the
characteristic mass, at low redshifts it underestimates
the number of high–mass halos and overestimates the
number of low–mass halos when compared with large nu-
merical simulations (e.g. Jenkins et al. 2001). A mod-
ified expression shown to fit low-redshift simulation re-
sults more accurately (to within ∼ 10%) was obtained
by Sheth & Tormen (1999):
∂n(> M, z)
= − ρ̄
∂[ln σ(M)]
ν̂e−ν̂
where ν̂ ≡
aδc(z)/σ(M), and a, p, and A are fitting
parameters. Sheth et al. (2001) derive this form of the
mass function by including shear and ellipticity in model-
ing non–linear collapse, effectively changing the scale-free
critical over–density δc(z), into a function of filter scale,
δc(M, z) =
aδc(z)
1 + b
σ2(M)
aδ2c (z)
. (13)
Here b and c are additional fitting parameters (a is
the same as in eq. 12). For the constants above, we
adopt the recent values obtained by Jenkins et al. (2001),
who studied a large range in redshift and mass: a =
0.73, A = 0.353, p = 0.175, b = 0.34, c = 0.81. We
note, however, that the situation at high redshifts is
less clear: studies disagree on the relative accuracy of
the Press & Schechter (1974) and Jenkins et al. (2001)
forms (Reed et al. 2003; Iliev et al. 2006b; Zahn et al.
2007). Our algorithm can be trivially modified to ac-
commodate other choices for the mass function; fortu-
nately, for the purposes of the ionization maps (see §3),
the choice of mass function makes very little difference
because all have a similar dependence on the local den-
sity (Furlanetto et al. 2006a).
The mass functions in equations (10) and (12) can be
obtained by the standard excursion set random walk pro-
cedure. The approach is to smooth the density field
around a point, x, on successively smaller scales start-
ing with M → ∞ [where σ2(M) → 0] and to identify
the point as belonging to the halo with the largest M
such that δ(x,M) > δc(M, z). If W
2(k,M) is chosen to
have a sharp cut-off, this procedure amounts to a random
walk of δ(x,M) along the mass axis, since the change in
δ(x,M) as the scale is shrunk is independent of δ(x,M)
for a top-hat filter in k-space (see eq. 11).
We perform this procedure on our realization of the
linear density field by filtering the field using a real-
space top-hat filter3, starting on scales comparable to
the box size and going down to grid cell scales, in log-
arithmic steps of width ∆M/M = 1.2.4 At each filter
scale, we use the scale-dependent barrier in eq. (13)
to mark a collapsed halo if δ(x,M) > δc(M, z). Filter
scales large enough that collapsed structure is extremely
unlikely, δc(M, z) > 7σ(M), are skipped (Mesinger et al.
2005). Since this procedure treats each cell as the center
of a spherical filter, neighboring pixels are not properly
placed in the same halo. Because of this, we discount
halos which overlap with previously marked halos.
As mentioned above, this algorithm is similar
to the “peak-patch” approach first introduced by
Bond & Myers (1996a). The primary differences are:
3 There is a slight swindle in the current application of this for-
malism. The filter function is assumed to be a top-hat in k-space
in order to facilitate the analytic random walk approach described
above. However, when the power spectrum is normalized to ob-
servations [i.e. σ(R = 8h−1Mpc] = σ8), the filter that is used to
define the mass M corresponding to R is a top-hat filter in real
space. Nevertheless, it has been shown that the mass function is
not very sensitive to this filter choice (Bond et al. 1991).
4 We note that Mesinger et al. (2005) required a much smaller
step size at these redshifts, ∆M/M ∼ 0.1, in order to produce ac-
curate mass functions using 1D Monte-Carlo random walks. How-
ever, here we find that we can reproduce accurate mass functions
with a larger step size, since in our 3D realization of the density
field, “overstepping” δc(M, z) due to a large filter step size can be
compensated with a small offset in the filter center, i.e. by cen-
tering the filter in a neighboring cell. This is the case since over-
stepping δc(M, z) means that some dense matter between the two
filter scales was “missed”. In a 1D Monte-Carlo random walk this
matter is unrecoverable; however, in a 3D realization of the density
field, the missed matter will be picked up by a filter centered on a
neighboring cell.
(1) we use the Jenkins et al. (2001) barrier to identify
halos (rather than calculating the strain tensor to ac-
count for ellipsoidal collapse), (2) we do not separately
identify peaks in the density field (this step is not re-
quired given modern computing power), and (3) we use
the “full exclusion” criterion for preventing halo overlap.
Bond & Myers (1996b) found that a “binary exclusion”
method in which pairs of overlapping halos are compared
and eliminated was somewhat more accurate. However,
at the high redshifts of interest to us, halo overlap is rare,
and we are primarily interested in the large-scale prop-
erties of the halo field, which are relatively insensitive to
the details of the overlap criterion.
We also note that our halo finder is similar
in spirit to the PTHalos algorithm introduced by
Scoccimarro & Sheth (2002) to generate mock galaxy
surveys at low redshifts. There are two key differences.
First, at present we use only first-order perturbation the-
ory to displace the particles.5 This limits us to higher
redshifts, where velocities are smaller. However, our al-
gorithm does not require particles in order to resolve ha-
los and hence can accommodate a considerably larger
dynamic range than PTHalos.
Mass functions resulting from this procedure are shown
as points in Figure 1, with error bars indicating 1-σ Pois-
son uncertainties and bin widths spanning our mass filter
steps. Dotted red curves denote PS mass functions gen-
erated by eq. (10); short–dashed blue curves denote ex-
tended PS conditional mass functions generated by eq.
(10) but also taking into account the absence of den-
sity modes longer than the box size; long–dashed green
curves denote mass functions generated using the Sheth-
Tormen correction in eq. (12). The upper (lower) set
of curves and points correspond to redshifts of z=6.5
(z=10). The dotted and short–dashed curves overlap at
these redshifts due to our large box size (L = 100 Mpc),
so we are immune to the finite box effects pointed out by
Barkana & Loeb (2004).
Fig. 1 shows that we obtain accurate mass functions
for M ∼> 10
8M⊙. Our procedure seems to underpre-
dict the abundance of halos with masses approaching
the cell size, Mcell ∼ 107M⊙. However, as the Jeans
mass corresponding to a gas temperature of ∼ 104 K
is MJ(z ∼ 8) ∼ 108M⊙, in subsequent calculations, we
only use halos with masses greater than Mmin = 10
Using thisMmin, we match the collapse fraction obtained
by integrating eq. (12) to better than ∼ 10%.
This mass cutoff corresponds to the minimum tem-
perature required for efficient atomic hydrogen cooling
and would be the pertinent mass scale if: (1) the H2
cooling channel is suppressed, e.g. due to a perva-
sive Lyman-Werner (LW) background, and if (2) photo-
ionization feedback is ineffective at suppressing gas cool-
ing and collapse onto higher mass halos. While feed-
back at high redshifts remains poorly-constrained, both
of these assumptions seem reasonable during the mid-
dle stages of reionization on which we focus. A dis-
sociating LW background is likely to have established
5 We note that a similar scheme to ours has been independently
created by O. Zahn (private communication). This scheme uses
a simple Press-Schechter barrier but adjusts halo locations follow-
ing second-order Lagrangian perturbation theory. However, he has
found that the second-order corrections make very little difference
to the map.
Fig. 1.— Mass functions generated from our halo filtering pro-
cedure discussed in §2.3 are shown as points. Dotted red curves
denote PS mass functions generated by eq. (10); short–dashed
blue curves denote extended PS conditional mass functions gener-
ated by eq. (10) but also taking into account the absence of density
modes longer than the box size; long–dashed green curves denote
mass functions generated using the Sheth-Tormen correction in eq.
(12). The upper (lower) set of curves and points correspond to
redshifts of z=6.5 (z=10).
itself well before the universe is significantly ionized
(Haiman et al. 1997). Model-dependent empirical evi-
dence supporting the suppression of star formation in
smaller mass halos, M ∼< Mmin, can also be gleaned
from WMAP data (Haiman & Bryan 2006). Further-
more, although early work suggested that an ionizing
background could partially suppress star formation in
halos with virial temperatures of Tvir ∼< 3.6 × 10
(M ∼< 2×10
9M⊙) (Thoul & Weinberg 1996), more recent
studies (Kitayama & Ikeuchi 2000; Dijkstra et al. 2004)
find that at high redshifts (z ∼> 3), self-shielding and the
increased cooling efficiency could be strong countering
effects for halos with virial temperatures Tvir > 10
We postpone a more detailed analysis of the reionization
footprint left by photo-ionization feedback to a future
work.
2.4. Adjusting Halo Locations
Once the halo field is obtained, we use the displace-
ment field obtained through eq. (9) to adjust the halo
locations at each redshift. This corrects for the enhanced
halo bias in Eulerian space with respect to our filtering,
which is done in Lagrangian space (i.e. using the initial
locations at large z). For computational convenience, we
smooth the 12003 velocity field onto a coarser-grained
2003 grid before adjusting halo locations. The choice of
resolution, where each cell is (100 Mpc)/200 = 0.5 Mpc
on a side, is somewhat arbitrary here, and we have veri-
fied that our halo and 21-cm power spectra are unaffected
Fig. 2.— Halo power spectra at z = 8.7, with L = 20 h−1Mpc
and cosmological parameters taken from McQuinn et al. (2006a).
The solid red curve is the halo power spectrum from an N-body
simulation obtained from McQuinn et al. (2006a) (c.f. the bottom
panel of their Fig. 2). The short-dashed green and the long-dashed
violet curves are obtained from our filtering procedure with and
without the halo location adjustments, respectively.
by this choice. We also note that in linear theory, the
mean velocity dispersion inside a (0.5 Mpc)3 sphere with
mean density at z = 10 is a factor of ∼10 lower than the
r.m.s. bulk velocity of such regions, so smoothing over
smaller scale velocities appears reasonable. Furthermore,
we keep in mind that our “endproducts” in this work are
ionization and 21-cm temperature fluctuation maps, for
which such “low-resolution” is more than adequate (com-
pare, e.g., to N-body simulations of reionization, which
typically have similar cell sizes for the radiative transfer
component).
In Figure 2, we plot the halo power spectrum, de-
fined as ∆hh(k, z) = k
3/(2π2V ) 〈|δhh(k, z)|2〉k, where
δhh(x, z) ≡ Mcoll(x, z)/〈Mcoll(z)〉 − 1 is the collapsed
mass field.6 The solid red curve is the halo power spec-
trum from a 20 h−1 Mpc N-body simulation at z = 8.7
obtained from McQuinn et al. (2006a) (c.f. the bottom
panel of their Figure 2). The short-dashed green and the
long-dashed violet curves are obtained from our filtering
procedure (matching the assumed cosmology) with and
without the halo location adjustments, respectively. We
note that ignoring the cumulative motions of halos re-
sults in an underestimate of the power of long-wavelength
modes of the halo field by a factor of ∼ 2 in this case.
The average Eulerian bias of these halos is ∼ 2, about
half of which comes from the correction from Lagrangian
to Eulerian coordinates.
After the halo locations are adjusted according to lin-
ear theory, our halo power spectrum agrees almost per-
6 We use the collapsed mass field, rather than the individual
galaxies, because we calculate the power from the smoothed cells.
fectly with the simulation. By design our procedure in-
cludes Poisson fluctuations in the halo number counts,
which dominate the power spectrum at k ∼> 5 h/Mpc
and are lost in purely analytic estimates (McQuinn et al.
2006a). We also note that both the halo mass func-
tions and power spectra are statistical tests and hence
the agreement shown here does not imply that our halo
field has a one-to-one mapping with an N-body halo
field sourced by identical initial conditions. Indeed,
Gelb & Bertschinger (1994) showed that those particles
located nearest initial linear density peaks are not nec-
essarily incorporated into massive galaxies. The “peak
particle” algorithm is less robust than our smoothing
technique, but we still do not expect to recover halo
masses or locations precisely. We plan on doing a “one-
on-one” comparison between halo fields obtained from
our halo finder to those obtained from N-body codes
in a future work. However, it is certainly encouraging
that the very similar “peak-patch” group finding formal-
ism of Bond & Myers (1996a) did very well when com-
pared “one-on-one” to N-body codes at large mass scales
(Bond & Myers 1996b).
In Figure 3 we show slices through the halo field from
our simulation box at z = 8.25, generated by the above
procedure, again with (right panel) and without (left
panel) the halo location adjustments. In the figure, the
halo field is mapped to a lower resolution 4003 grid for
viewing purposes. Each slice is 100 Mpc on a side and
0.25 Mpc deep. Collapsed halos are shown in blue. Vi-
sually, it is obvious that peculiar motions increase halo
clustering.
3. GENERATING THE IONIZATION FIELD
Once the halo field is generated as described above,
we can perform a similar filtering procedure (also us-
ing the excursion-set formalism) to obtain the ionization
field (similar methods have been discussed by Zahn et al.
2005, 2007). The time required for this final step is a
function of x̄HI, with large x̄HI requiring less time than
small x̄HI. Specifically, at x̄HI ∼ 0.5 this step takes ∼ 15
minutes to generate a 2003 ionization box on our work-
station.
There are two main differences between the halo fil-
tering and the HII bubble filtering procedures: (1) HII
bubbles are allowed to overlap, and (2) the excursion
set barrier (the criterion for ionization) becomes, as per
Furlanetto et al. (2004a):
fcoll(x1,M, z) ≥ ζ−1 , (14)
where ζ is some efficiency parameter and fcoll(x1,M, z)
is the fraction of mass residing in collapsed halos inside
a sphere of mass M = 4/3πR3ρ̄[1 + 〈δnl(x1, z)〉R], with
mean physical overdensity 〈δnl(x1, z)〉R, centered on Eu-
lerian coordinate x1, at redshift z.
Equation (14) is only an approximate model and makes
several simplifying assumptions about reionization. In
particular, it assumes a constant ionizing efficiency per
halo and ignores spatially-dependent recombinations and
radiative feedback effects. It can easily be modified to
include these effects (e.g., Furlanetto et al. 2004b, 2006a;
Furlanetto & Oh 2005), and we plan to do so in future
work. Here we present the simplest case in order to best
match current RT numerical simulations.
This prescription models the ionization field as a two-
phase medium, containing fully-ionized regions (which
we refer to as HII bubbles) and fully-neutral regions.
This is obviously much less information than can be
gleaned from a full RT simulation, which precisely tracks
the ionized fraction. However, HII bubbles are typi-
cally highly-ionized during reionization, and for many
purposes (such as for 21 cm maps) this two-phase ap-
proximation is perfectly adequate.
In order to “find” the HII bubbles at each redshift we
smooth the halo field onto a 2003 grid. Then we filter
the halo field using a real-space top-hat filter, starting on
scales comparable to the box size and decreasing to grid
cell scales in logarithmic steps of width ∆M/M = 0.33.
At each filter scale, we use the criterion in eq. (14)
to check whether the region is ionized. If so, we flag
all pixels inside that region as ionized. We do this for
all pixels and scales, regardless of whether the resulting
bubble would overlap with other bubbles. Note, there-
fore, that the nominal ionizing efficiency ζ that we use
as an input parameter does not equal (1 − x̄HI)/fcoll.
They typically differ by . 30%, with ζfcoll < 1 − x̄HI
early in reionization and ζfcoll > 1− x̄HI late in reioniza-
tion). Unfortunately, we thus cannot use our algorithm
to self-consistently predict the time evolution of the ion-
ized fraction (rather, that must be prescribed from some
other model). Of course, the same is true for N-body
simulations, because the evolution of the ionized fraction
depends on the evolving ionization efficiency of galaxies
and cannot be self-consistently included in any present-
day simulation.
In order to obtain the density field used in eq. (14),
δnl(x1, z), we use the Zel’Dovich approximation on our
linear density field, δ(x), in much the same manner as
we did to adjust our halo field in § 2.4. Starting at some
arbitrarily large initial redshift (we use z0 = 50), we
discretize our high-resolution 12003 field into “particles”
whose mass equals that in each grid cell. We then use
the displacement field (eq. 9) to move the particles to
new locations at each redshift. This resulting mass field
is then smoothed onto our lower resolution 2003 box to
obtain δnl(x1, z). We then recalculate the velocity field
(§ 2.2) using the new densities.
Zahn et al. (2007) showed that a very similar HII bub-
ble filtering procedure performed on an N-body halo field
was able to reproduce the ionization topology obtained
through a ray-tracing RT algorithm fairly well. Their
algorithm differs from ours in two ways. First, they used
a slightly different barrier definition; however, this dif-
ference has only a small impact on the ionization topol-
ogy.7 More importantly, for each filter scale at each pixel,
Zahn et al. (2007) flag only the center pixel as ionized if
the barrier is crossed, whereas we flag the entire filtered
sphere.
In order to test our bubble filtering algorithm, we ex-
ecute it on the same N-body halo field at z = 6.89 as
was used to generate the bottom panels of Fig. 3 in
Zahn et al. (2007). We compare analogous ionization
maps created using various algorithms in Figure 4. All
7 Specifically, in order to match the physics of their simulations
better, they required
dt fcoll > ζ
−1. However, the density mod-
ulation ends up nearly identical to our model, so the topology is
almost unchanged.
Fig. 3.— Slices through the halo field from our simulation box at z = 8.25. The halo field is generated on a 12003 grid and then mapped
to a 4003 grid for viewing purposes. Each slice is 100 Mpc on a side and 0.25 Mpc deep. Collapsed halos are shown in blue. The left panel
shows the halo field directly filtered in Lagrangian space; the right panel maps the field to Eulerian space according to linear theory (see
§ 2.4 and eq. 9). The right panel corresponds to the bottom-left (x̄HI = 0.53) ionization field in Figure 5.
slices are 93.7 Mpc on a side and 0.37 Mpc deep, with ζ
adjusted so that the mean neutral fraction in the box is
x̄HI = 0.49. Ionized regions are shown as white. The left-
most and right-most panels are taken from Zahn et al.
(2007). The left-most panel was created by perform-
ing their bubble filtering procedure directly on the lin-
ear density field (without explicitly identifying halos).
The second panel was created by performing their bub-
ble filtering procedure on their N-body halo field, but
with the slightly different barrier definition in eq. (14).
The third panel was created by performing our bubble
filtering procedure on the same N-body halo field, but ig-
noring density fluctuations outside of halos (i.e. setting
〈δnl(x1, z)〉R = 0), which we have verified give nearly
identical bubble maps as our full procedure (so long as
x̄HI is fixed). The right-most panel was created using
an approximate RT algorithm (Abel & Wandelt 2002;
Sokasian et al. 2001, 2003) on the same halo field.
It is immediately obvious from Fig. 4 that all of the
approximate maps (first three panels) reproduce the RT
map (right-most panel) fairly well. Even the HII bub-
ble filtering performed directly on the linear density field
(left-most panel) performs well, which is encouraging, as
that is the starting point for our semi-numerical proce-
dure and we only improve on this scheme.
Figure 4 shows that our HII bubble filtering algorithm
is an excellent approximation to RT. The similar algo-
rithm proposed by Zahn et al. (2007) also performs well.
In comparison, our algorithm produces somewhat more
“bubbly” maps but appears to better capture the connec-
tivity of HII regions. Both are an obvious improvement
on directly filtering the linear density field.
Of course, in our full algorithm we identify halos from
the linear density field (rather than from simulations), so
our method consumes comparable processing time to the
one used to generate the leftmost panel in Figure 4, once
the halos have been identified. Moreover, we are able
to capture the “stochastic” component of the halo bias
that causes the relatively large differences between the
leftmost panel and the full RT simulation. That is, the
algorithm used to generate the leftmost panel uses the
large-scale linear density field to predict the distribution
of halos (Zahn et al. 2005, 2007). In reality, the rela-
tion is not deterministic because of random fluctuations
in the small-scale modes comprising each region. This
leads to nearly Poisson scatter in the halo number densi-
ties (Sheth & Lemson 1999; Casas-Miranda et al. 2002)
that can substantially modify the bubble size distribution
whenever sources are rare, particularly early in reion-
ization (Furlanetto et al. 2006a). By directly sampling
the small-scale modes to build the halo distribution, we
better recover this scatter (at least statistically, as illus-
trated by Fig. 2). Another way to include this scatter
is by directly sampling halos from an N-body simulation
(as in Zahn et al. 2007, or the second panel of Fig. 4),
although that obviously requires much more computing
power.
3.1. Ionization Maps
Now that we have demonstrated in turn the success of
our halo and bubble filtering procedures, we present the
resulting ionization maps when the two are combined.
In Figure 5, we show 100 Mpc × 100 Mpc × 0.5 Mpc
slices through our 2003 ionization field at z = 10, 9,
8.25, 7.25 (left to right across rows). With the assump-
tion of ζ = 15.1, these redshifts correspond to x̄HI = 0.89,
0.74, 0.53, 0.18, respectively. As has been pointed out by
Furlanetto et al. (2004c), the neutral fraction is the more
relevant descriptor; bubble morphologies at a constant
x̄HI vary little with redshift (see also McQuinn et al.
Fig. 4.— Slices from the ionization field at z = 6.89 created using different algorithms. All slices are 93.7 Mpc on a side and 0.37 Mpc
deep, with the mean neutral fraction in the box being x̄HI = 0.49. Ionized regions are shown as white. The left-most panel was created
by performing the bubble filtering procedure of Zahn et al. (2007) directly on the linear density field. The second panel was created by
performing their bubble filtering procedure on their N-body halo field, but with the slightly different barrier definition in eq. (14). The
third panel was created by performing our bubble filtering procedure described in § 3 on the same N-body halo field. The right-most panel
(from Zahn et al. 2007) was created using an approximate RT algorithm on the same halo field.
2006a). The bottom-left panel corresponds to the halo
field in the top-right panel of Fig. 3, generated on a
high-resolution 12003 grid.
To quantify the ionization topology resulting from our
method, we calculate the size distributions of both the
ionized and neutral regions. We randomly choose a pixel
of the desired phase (neutral or ionized), and record the
distance from that pixel to a phase transition along a
randomly chosen direction. We repeat this Monte Carlo
procedure 107 times. Volume-weighted probability dis-
tribution functions (PDFs) produced thusly are shown
by the solid curves in Figure 6 for ionized regions (top
panel) and neutral regions (bottom panel). Curves corre-
spond to (z, x̄HI) = (10, 0.89), (9.25, 0.79), (8.50, 0.61),
(8.00, 0.45), (7.50, 0.27), (7.00, 0.10), from left to right
in the top panel, respectively (or from right to left in
the bottom panel). All curves are normalized so that the
probability density integrates to unity.
It is useful to compare these distributions to the an-
alytic bubble mass function of Furlanetto et al. (2004c);
although this analytic approach is motivated by the same
excursion set barriers as our semi-numerical approach, it
does not account for the full geometry of sources. We
compute the probability distribution from the analytic
model by assuming purely spherical bubbles and convolv-
ing with the volume-weighted distance to the sphere’s
edge:
p(r) dr =
2πr2 dr
(1− x̄HI)
dRnb(R)
, (15)
where nb(R) is the comoving number density of bub-
bles with radii between R and R + dR (taken from
Furlanetto et al. 2004c).
Several points are evident from Figures 5 and
6. As expected (e.g., Furlanetto et al. 2004c, 2006a;
McQuinn et al. 2006a), there is a well-defined bubble
scale at each neutral fraction, despite some scatter in
the sizes. This scale also gets more pronounced (i.e. the
PDF peaks more) as reionization progresses; this is a
result of the changing shape of the underlying matter
power spectrum (Furlanetto et al. 2006a).
Also, the purely analytic estimates underpredict the
size distributions at all values of the neutral fraction,
though they do become increasingly accurate as the neu-
tral fraction decreases. This trend is perhaps counterin-
tuitive, as the analytic model, which rests on the assump-
tion of spherical bubbles, should perform best when the
bubbles are isolated, as one would expect at earlier times,
i.e. high neutral fractions. However, looking at the top-
left panel of Fig. 5, the typical bubbles filling most of
the ionized volume overlap due to the strong clustering
of early sources and bubbles. This results in many “over-
lapping pairs of spheres” at early times, resulting from
merging HII bubbles sourced by clustered sources. Thus
the spherical bubble-based analytic model underpredicts
the true size distribution, using our “mean free path” def-
inition of bubble sizes above. This effect was not noted
by previous studies (Zahn et al. 2007), because they used
a different definition of bubble sizes, based on spherical
filters used to flag regions in which x̄HI < 0.1. As time
progresses and the universe becomes more ionized, this
“overlapping pair of spheres” effect becomes less and less
dominant (see Fig. 5), and the analytic model becomes
increasingly more accurate.
Finally, the size distributions of neutral regions pre-
sented in the bottom panel of Fig. 6 are a new result and
potentially important for the 21-cm signal (which origi-
nates in neutral hydrogen, of course). In the later stages
of reionization, when the topology has transformed to
isolated neutral islands in a sea of ionized gas, this fig-
ure pinpoints the typical sizes of “mostly neutral” pixels
that continue to emit strongly. In contrast to the ionized
regions, the neutral regions (defined in this way) do not
grow substantially during reionization. From x̄HI = 0.89
to x̄HI = 0.1, the peak of the distribution shifts only by a
factor of ∼ 6, whereas the peak of the ionized region dis-
tribution shifts by a factor of ∼40 over the same range.
The reason for this is also evident in Figure 5: even when
the universe is mostly neutral, space is dotted with is-
lands of ionized gas, such that our “mean free path”–type
size distributions never become too large. The converse
does not hold true for ionized regions. However, a slight
parallel for ionized islands in a mostly neutral IGM, could
be found in Lyman limit systems (LLS) inside larger HII
regions (e.g. Barkana & Loeb 2002; Shapiro et al. 2004;
Miralda-Escudé et al. 2000), though it is not clear how
prevalent such neutral clumps are at high redshifts.
Throughout this paper, we have used a L = 100
Mpc “simulation” box. This size facilitates compar-
ison of our results with those from recent hybrid N-
Fig. 5.— Slices through the 2003 ionization field at z = 10, 9, 8.25, 7.25 (left to right across rows). With the assumption of ζ = 15.1,
these redshifts correspond to x̄HI = 0.89, 0.74, 0.53, 0.18, respectively. All slices are 100 Mpc on a side and 0.5 Mpc deep. The bottom-left
panel corresponds to the halo field in the top-right panel of Fig. 3, generated on a high-resolution 12003 grid.
body works (Zahn et al. 2007; McQuinn et al. 2006a;
Iliev et al. 2006a; Trac & Cen 2006). However, the speed
of our semi-numerical approach allows us to explore
larger cosmological scales while still consistently resolv-
ing the small halos that could dominate the photon bud-
get during reionization. As mentioned previously, exist-
ing N-body codes must resort to merger-tree methods
to populate their distribution of small-mass halos, even
for box sizes ∼< 100 Mpc (McQuinn et al. 2006a). In
this spirit, we present some preliminary results from a
N = 15003, L = 250 Mpc simulation, capable of directly
resolving halos with masses M ∼> 2.2× 10
8M⊙, with re-
sulting mass functions accurate to better than a factor
of two even at the smallest scale. This resolution pushes
the RAM limit of our machine and so each redshift can
take several hours to complete.8
In Figure 7, we compare size distributions of ionized
8 We note here that our halo-finding algorithm requires signifi-
cantly higher resolution than does predicting the ionization field
directly from the linear density field smoothed on larger scales
(Zahn et al. 2005, 2007). The latter method can be extended to
even larger boxes, though at the price of a somewhat less accurate
ionization map (compare the left and right panels in Fig. 4).
Fig. 6.— Size distributions (see definition in text) of ionized
(top panel) and neutral (bottom panel) regions. Curves correspond
to (z, x̄HI) = (10, 0.89), (9.25, 0.79), (8.50, 0.61), (8.00, 0.45),
(7.50, 0.27), (7.00, 0.10), from left to right in the top panel, re-
spectively (or from right to left in the bottom panel). Solid curves
are produced from our simulation while dotted curves correspond
to the analytic mass function. All curves are normalized so that
the probability distribution integrates to unity.
(top panel) and neutral (bottom panel) regions from our
two different simulation boxes. Curves correspond to (z,
x̄HI) = (9.00, 0.80), (8.00, 0.56), (7.00, 0.21), from left
to right in the top panel, respectively (or from right to
left in the bottom panel, respectively). Solid curves are
generated from our fiducial, N = 12003, L = 100 Mpc,
simulation while dashed curves are generated from our
larger simulation with N = 15003, L = 250 Mpc. The
cell size in all ionization maps is 0.5 Mpc on a side, with
the efficiency parameter, ζ, adjusted to obtain matching
values of x̄HI, and we set the minimum halo mass to
Mmin = 2.2× 108M⊙ even in the higher resolution runs
for easier comparison.
As reionization progresses, an increasing number of
large HII regions are “missed” by the L = 100 Mpc sim-
ulation. Interestingly, the analogous trend in the neutral
region size distributions (bottom panel) is weaker. This is
most likely because the “ionized island” effect limits the
size distributions of neutral regions as described above.
4. 21-CM TEMPERATURE FLUCTUATIONS
A natural application of our “simulation” technique is
to predict 21-cm brightness temperatures during reion-
ization. The offset of the 21-cm brightness temperature
from the CMB temperature, Tγ , along a line of sight
(LOS) at observed frequency ν, can be written as (e.g.
Furlanetto et al. 2006b):
δTb(ν)=
TS − Tγ
1 + z
(1− e−τν0 ) (16)
Fig. 7.— Size distributions of ionized (top panel) and neutral
(bottom panel) regions from different simulation boxes. Curves
correspond to (z, x̄HI) = (9.00, 0.80), (8.00, 0.56), (7.00, 0.21),
from left to right in the top panel, respectively (or from right to
left in the bottom panel). Solid curves are generated from our
fiducial, N = 12003, L = 100 Mpc, simulation while dashed curves
are generated from a larger simulation with N = 15003, L = 250
Mpc. The cell size in all ionization maps is 0.5 Mpc on a side,
with the efficiency parameter, ζ, adjusted to get matching values
of x̄HI and the minimum halo mass set to Mmin = 2.2 × 10
for comparison purposes.
≈ 9(1 + z)1/2xHI(1 + δnl)
dvr/dr +H
where TS is the gas spin temperature, τν0 is the opti-
cal depth at the 21-cm frequency ν0, δnl is the physical
overdensity (see discussion under eq. 14), H is the Hub-
ble parameter, dvr/dr is the comoving gradient of the
line of sight component of the comoving velocity, and all
quantities are evaluated at redshift z = ν0/ν − 1. The
final approximation makes the standard assumption that
TS ≫ Tγ for all redshifts of interest during reionization
(e.g. Furlanetto 2006) and also that dvr/dr ≪ H . We
verify in our simulation that dvr/dr < H for all neutral
pixels.
Maps of δTb(x, ν) generated in this manner are shown
in Figure 8. All slices are 100 Mpc on a side, 0.5 Mpc
deep, and correspond to (z, x̄HI) = (9.00, 0.74), (8.25,
0.53), (7.50, 0.27), from left to right. The top panels take
into account the velocity correction term in eq. (16),
while the bottom panels ignore it.
As seen in Fig. 8, velocities typically increase the con-
trast in temperature maps, making hot spots hotter and
cool spots cooler. We also see that temperature hot
spots, which correspond to dense pixels, tend to clus-
ter around the edges of HII bubbles, especially smaller
bubbles. This occurs because HII bubbles correlate with
peaks of the density field and long-wavelength biases in
the density field can extend beyond the edge of the ion-
ized region. This enhanced contrast might be useful in
Fig. 8.— Brightness temperature of 21-cm radiation relative to the CMB temperature. All slices are 100 Mpc on a side, 0.5 Mpc deep,
and correspond to (z, x̄HI) = (9.00, 0.74), (8.25, 0.53), (7.50, 0.27), left to right. Top panels include the velocity correction term in eq.
(16), while the bottom panels do not. For animated versions of these pictures, see http://pantheon.yale.edu/∼am834/Sim.
the detection of the boundaries of ionized regions with fu-
ture 21-cm experiments. As reionization progresses most
hot spots become swallowed up by HII bubbles, and the
effects of velocities diminish.
In Figure 9 we plot the dimensionless 21-
cm power spectrum, defined as ∆221(k, z) =
k3/(2π2V ) 〈|δ21(k, z)|2〉k, where δ21(x, z) ≡
δTb(x, z)/ ¯δTb(z) − 1. Solid blue curves take into
account gas velocities, while dashed red curves do not.
Curves correspond to (x̄HI, z) = (0.79, 9.25), (0.61,
8.50), (0.45, 8.00), (0.27, 7.50), (0.10, 7.00), bottom to
top. Error bars on the bottom dashed curve denote 1-σ
Poisson uncertainties; fractional errors in a given bin
are the same for all curves. As reionization progresses,
small-scale power is traded for large-scale power, and the
curves become flatter. Note that, with our dimensionless
definition of the power spectrum, curves with smaller
x̄HI have larger values of ∆
21(k, z). This is because the
mean brightness temperature offset drops quite rapidly
as reionization progresses, since ¯δTb(z) ∝ x̄HI, but the
scatter remains significant (see Fig. 6) and thus the
fractional perturbation, δ21(x, z), increases throughout
reionization.
Finally, in Figure 10 we plot dimensional power spec-
tra, ¯δTb(z)
2∆221(k, z). The curves correspond to (x̄HI, z)
= (0.80, 9.00), (0.56, 8.00), (0.21, 7.00), top to bottom at
large k, respectively. The dotted green curves are gen-
erated from a large, high-resolution “simulation”, with
Fig. 9.— Dimensionless 21-cm power spectra for (x̄HI, z) =
(0.79, 9.25), (0.61, 8.50), (0.45, 8.00), (0.27, 7.50), (0.10, 7.00),
bottom to top. Solid blue curves take into account gas velocities,
while dashed red curves do not.
http://pantheon.yale.edu/~am834/Sim
Fig. 10.— Dimensional 21-cm power spectra. The curves corre-
spond to (x̄HI, z) = (0.80, 9.00), (0.56, 8.00), (0.21, 7.00), top to
bottom at large k, respectively. The dotted green curves are gen-
erated from a large, high-resolution “simulation”, with N = 15003
and L = 250 Mpc, with no velocity contribution to the power spec-
tra. Solid blue curves and dashed red curves are generated with
our fiducial N = 12003 and L = 100 Mpc simulation, with and
without the velocity contribution, respectively.
N = 15003 and L = 250 Mpc, with no velocity contribu-
tion to the power spectra. Solid blue curves and dashed
red curves are generated with our fiducial N = 12003
and L = 100 Mpc simulation, with and without the ve-
locity contribution, respectively. The cell size in all δTb
maps is 0.5 Mpc on a side, with the efficiency parame-
ter, ζ, adjusted to achieve matching values of x̄HI and
the minimum halo mass set to Mmin = 2.2× 108M⊙ for
comparison purposes.
As seen in Figures 9 and 10, velocities make a mod-
est contribution to the 21 cm power spectrum, boosting
power on small scales early in reionization. Note that
the apparent slight decrease in power at small k when
velocities are included is well within the errors from av-
eraging over the few modes available to us on the largest
scales (e.g., see Poisson error bars on the bottom dashed
curve in Fig. 9). While the maximum δTb value in our
simulation box increases by a factor of a few when ve-
locities are included, most of the pixels are only slightly
affected. When the power spectrum is plotted in a di-
mensional version, ¯δTb(z)
2∆221(k, z), small-scale power is
boosted by ∼ 40% at (x̄HI, z) = (0.80, 9.00), with this en-
hancement monotonically decreasing as reionization pro-
gresses. Linear theory predicts that velocities enhance
the density power spectrum by a factor of 1.87 when
x̄HI = 1 (Kaiser 1987). In fact we do recover this en-
hancement for a fully neutral IGM; however, as predicted
by analytic models (McQuinn et al. 2006b), the ionized
bubbles rapidly remove most of this amplification.
Figure 10 also confirms the inferences drawn from Fig.
7, primarily that larger box sizes are needed to capture
the ionization topology at the end stages of reionization.
Comparing the dashed red to the dotted green curves in
Fig. 10, we note that our fiducial L = 100 Mpc simula-
tions are accurate for scales smaller than k ∼> 0.2 Mpc
(or λ . 30 Mpc). As reionization progresses, larger scales
lose power more rapidly than in the L = 250 Mpc simu-
lation. This is again evidence that very large scale simu-
lations are needed to model the middle and late stages of
reionization. Thus the speed and high resolution of our
semi-numeric approach will be extremely useful in future
modeling of reionization.
5. CONCLUSIONS
We introduce a method to construct semi-numeric sim-
ulations that can efficiently generate realizations of halo
distributions and ionization maps at high redshifts. Our
procedure combines an excursion-set approach with first-
order Lagrangian perturbation theory and operates di-
rectly on the linear density and velocity fields. As such,
our algorithm can exceed the dynamic range of exist-
ing N-body codes by orders of magnitude. As this is
the main limiting factor in simulating the ionized bubble
topology throughout reionization, when ionized regions
reach scales of tens of comoving Mpc, this will be partic-
ularly useful in such studies. Moreover, the efficiency of
the algorithm will allow us to explore the large parame-
ter space required by the many uncertainties associated
with high-redshift galaxy formation.
We find that our halo finding algorithm compares well
with N-body simulations on the statistical level, yield-
ing both accurate mass functions and power spectra. We
have not yet compared our halo distribution with sim-
ulations on a point-by-point basis, but we do not ex-
pect perfect agreement because of the vagaries of the ex-
cursion set approach. However, it is encouraging that
a very similar algorithm independently developed by
Bond & Myers (1996a) fares quite well in a comparison
of high-mass halos.
Our HII bubble finding algorithm captures the bubble
topology quite well, as compared to ionization maps from
ray-tracing RT algorithms at an identical x̄HI. Our al-
gorithm is similar to other codes, although we build the
ionization map from our excursion set halo field rather
than directly from the linear density field or from halos
found in an N-body simulation (Zahn et al. 2005, 2007).
Compared to codes built only from the linear density
field, we can better track the “stochastic” component of
the bias, though at the cost of somewhat more compu-
tation and a harder limit on resolution. On the other
hand, our scheme is much faster than using an N-body
code and offers superior dynamic range.
We create ionization maps using a simple efficiency pa-
rameter and compute the size distributions of ionized
and neutral regions. Our size distributions are gener-
ally shifted to larger scales when compared with purely
analytic models (Furlanetto et al. 2004c) at the same
mean neutral fraction. The discrepancy lies in the fact
that, at their core, the purely analytic models are based
on ensemble-averaged distributions of isolated spheres.
Hence they do not capture overlapping bubble shapes,
which are most important at large x̄HI (when the bubbles
are small and random fluctuations in the source densities,
as well as clustering, are most important).
In this paper, we have confined ourselves to a sim-
ple ionization criterion (essentially photon counting;
Furlanetto et al. 2004c). However, our algorithm can
easily accommodate more sophisticated prescriptions, so
long as they can be expressed either with the excursion
set formalism (Furlanetto & Oh 2005; Furlanetto et al.
2006a) or built from the halo field (in a similar way to
semi-analytic models of galaxy formation embedded in
numerical simulations).
We also use our procedure to generate maps and power
spectra of the 21-cm brightness temperature fluctuations
during reionization. We note that temperature hot spots
generally cluster around HII bubbles, especially in the
early phases of reionization. Because HII bubbles cor-
relate with peaks of the density field, long-wavelength
biases in the density field can extend beyond the edge of
the ionized region, with the resulting overdensities ap-
pearing as hot spots. This effect might be useful for
detecting the boundaries of ionized regions with future
21-cm experiments. We study the imprint of gas bulk ve-
locities on 21-cmmaps and power spectra, an effect which
was not included in previous studies. We find that ve-
locities do not have a major impact during reionization,
although they do increase the contrast in temperature
maps, making some hot spots hotter and some cool spots
cooler. Velocities also increase small-scale power, though
the effect decreases with decreasing x̄HI.
We also include some preliminary results from a sim-
ulation run with the largest dynamical range to date:
a 250 Mpc box which resolves halos with masses M ∼>
2.2 × 108M⊙. This simulation run confirms that ex-
tremely large scales are required to model the late stages
of reionization, x̄HI ∼< 0.5, when the typical scale of ion-
ized bubbles becomes several tens of Mpc.
The speed and dynamic range provided by our semi-
numeric approach will be extremely useful in the mod-
eling of early structure formation and reionization. Our
ionization maps can be efficiently folded into analyses of
current and upcoming high-redshift observations, espe-
cially 21-cm surveys.
We thank Greg Bryan for many helpful conversations
concerning the inner workings of cosmological simula-
tions and the generation of initial conditions. We also
thank Oliver Zahn for permitting the use of the halo field
from his simulation output as well as for several interest-
ing discussions. We thank Mathew McQuinn for provid-
ing the halo power spectra from his simulation as well
as for associated helpful comments. We thank Zoltan
Haiman, Greg Bryan, Oliver Zahn and Mathew McQuinn
for insightful comments on a draft version of this paper.
This research was supported by NSF-AST-0607470.
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  We present a method to construct semi-numerical ``simulations'', which can
efficiently generate realizations of halo distributions and ionization maps at
high redshifts. Our procedure combines an excursion-set approach with
first-order Lagrangian perturbation theory and operates directly on the linear
density and velocity fields. As such, the achievable dynamic range with our
algorithm surpasses the current practical limit of N-body codes by orders of
magnitude. This is particularly significant in studies of reionization, where
the dynamic range is the principal limiting factor. We test our halo-finding
and HII bubble-finding algorithms independently against N-body simulations with
radiative transfer and obtain excellent agreement. We compute the size
distributions of ionized and neutral regions in our maps. We find even larger
ionized bubbles than do purely analytic models at the same volume-weighted mean
hydrogen neutral fraction. We also generate maps and power spectra of 21-cm
brightness temperature fluctuations, which for the first time include
corrections due to gas bulk velocities. We find that velocities widen the tails
of the temperature distributions and increase small-scale power, though these
effects quickly diminish as reionization progresses. We also include some
preliminary results from a simulation run with the largest dynamic range to
date: a 250 Mpc box that resolves halos with masses M >~ 2.2 x10^8 M_sun. We
show that accurately modeling the late stages of reionization requires such
large scales. The speed and dynamic range provided by our semi-numerical
approach will be extremely useful in the modeling of early structure formation
and reionization.

Submitted to the ApJ
Preprint typeset using LATEX style emulateapj v. 6/22/04
EFFICIENT SIMULATIONS OF EARLY STRUCTURE FORMATION AND REIONIZATION
Andrei Mesinger & Steven Furlanetto
Yale Center for Astronomy and Astrophysics, Yale University, New Haven, CT 06520
Submitted to the ApJ
ABSTRACT
Detailed theoretical studies of the high-redshift universe, and especially reionization, are generally
forced to rely on time-consuming N-body codes and/or approximate radiative transfer algorithms. We
present a method to construct semi-numerical “simulations”, which can efficiently generate realizations
of halo distributions and ionization maps at high redshifts. Our procedure combines an excursion-
set approach with first-order Lagrangian perturbation theory and operates directly on the linear
density and velocity fields. As such, the achievable dynamic range with our algorithm surpasses
the current practical limit of N-body codes by orders of magnitude. This is particularly significant in
studies of reionization, where the dynamic range is the principal limiting factor because ionized regions
reach scales of tens of comoving Mpc. We test our halo-finding and ionization-mapping algorithms
separately against N-body simulations with radiative transfer and obtain excellent agreement. We
compute the size distributions of ionized and neutral regions in our maps. We find even larger ionized
bubbles than do purely analytic models at the same volume-weighted mean hydrogen neutral fraction,
x̄HI, especially early in reionization. We also generate maps and power spectra of 21-cm brightness
temperature fluctuations, which for the first time include corrections due to gas bulk velocities. We
find that velocities widen the tails of the temperature distributions and increase small-scale power,
though these effects quickly diminish as reionization progresses. We also include some preliminary
results from a simulation run with the largest dynamic range to date: a 250 Mpc box that resolves
halos with massesM ≥ 2.2×108M⊙. We show that accurately modeling the late stages of reionization,
x̄HI ∼< 0.5, requires such large scales. The speed and dynamic range provided by our semi-numerical
approach will be extremely useful in the modeling of early structure formation and reionization.
Subject headings: cosmology: theory – early Universe – galaxies: formation – high-redshift – evolution
1. INTRODUCTION
Accurately modeling the formation of bound struc-
tures is invaluable for understanding any process in
the early universe. Reionization, the epoch when
radiation from early generations of astrophysical objects
managed to ionize the intergalactic medium (IGM), is
particularly sensitive to the distribution of collapsed
structure. Current observations paint a complex pic-
ture of the reionization epoch (Mesinger & Haiman
2004; Wyithe & Loeb 2004; Fan et al. 2006;
Mesinger & Haiman 2006; Malhotra & Rhoads 2004;
Furlanetto et al. 2006c; Malhotra & Rhoads 2006;
Page et al. 2006; Kashikawa et al. 2006; Totani et al.
2006). The next generation of instruments (James
Webb Space Telescope; 21-cm instruments such as the
Low Frequency Array and the Mileura Widefield Array
Low-Frequency Demonstrator; CMB polarization mea-
surements with Planck, etc.), could potentially shed light
on this poorly understood milestone. Unfortunately,
we still do not have accurate models of reionization
with which to interpret these upcoming (and current)
observations.
The main difficulty lies in the enormous dynamic range
required. Ionized regions are expected to reach charac-
teristic sizes of tens of comoving Mpc (Furlanetto et al.
2004c; Furlanetto & Oh 2005), which is over seven or-
ders of magnitude in mass larger than the pertinent cool-
ing mass, corresponding to gas with a temperature of
T ∼ 104 K (e.g. Efstathiou 1992; Thoul & Weinberg
1996; Gnedin 2000b; Shapiro et al. 1994). The required
dynamic range is even larger if smaller “minihalos” be-
low this cooling threshold are important during reion-
ization. Because of the steep mass dependence of halo
abundances, halos with masses close to the cooling mass
could dominate the photon budget. Hence modeling
reionization requires simulation box sizes of hundreds
of megaparsecs on a side, with extremely high resolu-
tion. Attempts to overcome these obstacles have gener-
ally followed the same fundamental and well-trod path
(e.g. Gnedin 2000a; Razoumov et al. 2002; Ciardi et al.
2003; Sokasian et al. 2003; Iliev et al. 2006b; Zahn et al.
2007; Trac & Cen 2006): (1) N-body codes are run to
generate halo distributions; (2) a simple prescription is
used to relate the halo mass to an ionizing efficiency; (3)
approximate methods (generally so-called ray-tracing al-
gorithms) are used to model radiative transfer (RT) on
large scales.
Even with modest halo resolution
(Springel & Hernquist 2003) of tens of dark matter
particles per halo, such schemes are computationally
limited to box sizes of tens of megaparcecs, if they
wish to resolve the likely cooling mass. McQuinn et al.
(2006a) extended the mass resolution of their sim-
ulations by using a merger tree scheme to populate
sub-grid scales with unresolved halos in a stochastic
manner. Such hybrid schemes are useful for extending
the dynamic range, but merger trees require a number
of corrections to achieve consistent mass functions (see,
e.g., Sheth & Pitman 1997; Benson et al. 2005, and Fig.
1 in McQuinn et al. 2006a) and to track individual halos
with redshift. Moreover, although they are perfectly
adequate for many purposes (including studying the
http://arxiv.org/abs/0704.0946v1
large-scale features of reionization), they prevent one
from taking full advantage of the simulation.
Aside from dynamic range, the other main limiting fac-
tor in all of the above numerical approaches is speed.
Even if the relevant scales can be resolved with N-body
codes, such as may be the case in the early phase of reion-
ization or with hybrid stochastic schemes. The codes
themselves generally take days to run on large super-
computing clusters, with the approximate RT algorithms
consuming a few additional days. The computational
cost of each simulation makes it difficult to explore the
full range of parameter space for reionization, which is
particularly large because we know so little about high-
redshift galaxies.
The computational cost becomes truly prohibitive if
hydrodynamics is included: the largest such simulation
of reionization performed to date spanned only 10h−1
Mpc (Sokasian et al. 2003). Including self-consistent de-
scriptions of galaxy formation – even at the approximate
level currently implemented in lower-redshift cosmolog-
ical simulations (e.g., Springel & Hernquist 2003) – re-
quires hydrodynamics, so N-body simulations of reion-
ization are limited to semi-analytic prescriptions for star
formation, feedback, etc. It is therefore worthwhile to
explore even simpler schemes.
The purpose of this paper is to introduce approximate
but efficient methods for generating halo distributions at
high redshifts as well as for generating the associated ion-
ization maps. We apply an excursion-set approach (e.g.
Bond et al. 1991; Lacey & Cole 1993) to the filtering of
a realization of the linear density field and then adjust
halo locations with first-order perturbation theory. We
can thus generate halo distributions at any given red-
shift, without explicitly including information from any
higher redshifts. This scheme is an updated form of
the “peak-patch” formalism developed and validated by
Bond & Myers (1996a,b), although it was conceived and
implemented completely independently. We then apply
a similar technique to obtain the ionization field from the
halo field. This part is similar to the schemes described
in Zahn et al. (2005, 2007), except applied to our effi-
ciently built halo distributions. As such, our methods
allow us to make general predictions about non-linear
processes, such as structure formation and reionization,
without making use of time-guzzling cosmological sim-
ulations. The speed of our approach also allows us to
explore a larger dynamic range than is possible with cur-
rent cosmological simulations while preserving detailed
spatial information (at least in a statistical sense), un-
like purely analytic models.
This paper is organized as follows. In § 2, we introduce
and test the components of our halo finding algorithm.
In § 3, we introduce and test our HII bubble finding
algorithm. In § 4, we use our semi-numerical scheme
to generate maps and power spectra of expected 21-cm
brightness temperature fluctuations throughout reioniza-
tion. In § 5, we summarize our key findings and present
our conclusions.
Unless stated otherwise, we quote all quantities in co-
moving units. We adopt the background cosmological
parameters (ΩΛ, ΩM, Ωb, n, σ8, H0) = (0.76, 0.24,
0.0407, 1, 0.76, 72 km s−1 Mpc−1), consistent with the
three–year results of the WMAP satellite (Spergel et al.
2006).
2. SEMI-NUMERICAL SIMULATIONS OF HALO
PROPERTIES
In brief, our algorithm generates a linear density field
and identifies halos within it. Because only linear evo-
lution is required, the algorithm is fast and flexible. We
generate 3D Monte-Carlo realizations of the linear den-
sity field on a box with sides of length L = 100 Mpc and
N = 12003 grid cells. As such, we are able to take advan-
tage of many pre-existing tools operating on the linear
density field alone. Our method consists of the following
principal steps:
1. creating the linear density and velocity fields
2. filtering halos from the linear density field using
the excursion-set formalism
3. adjusting halo locations using their linear-order
displacements
Step (1) only needs to be done once for each realization,
since it is independent of redshift. As mentioned above,
steps (2) and (3) need only be performed on redshifts of
interest, i.e. since our output at redshift z is independent
of any outputs at higher redshifts, there is no need for
our code to “run down” to z, as is the case for N-body
codes.
Our algorithm is an updated and simplified version of
the “peak-patch” algorithm of Bond & Myers (1996a);
we refer the interested reader there for more detailed
explanations of some steps. A simpler version has also
been used by Scannapieco et al. (2002) to study metal
enrichment at high redshifts.
We perform our semi-numerical simulations on a sin-
gle desktop Mac Pro with two dual-core 3.00 GHz Quad
Xeon processors and 16 GB of RAM. ForN = 12003, step
(1) takes ∼ 1 hour. For a given redshift in our range of
interest, specifically for z = 8.75, steps (2) and (3) take
∼ 2.5 hours. To achieve comparable halo mass resolution
(including halos with M ∼> 10
7M⊙) with a minimum of
∼ 500 particles per halo (Springel & Hernquist 2003), N-
body codes would require a prohibitively large number
of particles, N ∼ 1012! Below we describe in detail the
components of our model.
2.1. The Linear Density Field
Our linear density field is generated in much the same
way as it is for N-body codes. We briefly outline the
procedure here.
The density field of the universe, δ(x) ≡ ρ(x)/ρ̄ − 1,
in the linear regime1 is well-represented as a Gaussian
random field, whose statistical properties are fully de-
fined by its power spectrum, σ2(k) ≡ 〈|δ(k)|2〉. Here,
δ(k) is the Fourier transform of δ(x), and the standard
assumption of isotropy implies σ2(k) = σ2(k) while ho-
mogeneity implies that there are no density fluctuations
with wavelengths larger than the box size L = V 1/3. We
use the following, standard (e.g. Bagla & Padmanabhan
1997; Sirko 2005) Fourier transform conventions:
δ(k) =
δ(x)e−ik·x , (1)
1 In linear theory, density perturbations evolve in redshift as δ(z)
= δ(0)D(z), where D(z) is the linear growth factor normalized
so that D(0) = 1 (e.g., Liddle et al. 1996). Unless the redshift
dependence is noted explicitly, from this point forward we will work
with quantities linearly-extrapolated to z = 0.
with the inverse transform being
δ(x) =
δ(k)eik·x . (2)
The discrete simulation box only permits a finite set
of wavenumbers: k = ∆k(i, j, k), where ∆k = 2π/L and
i, j, k are integers in the range (-
N/2]. For each
independent wavenumber,2 we assign
δ(k) =
σ2(k)
(ak + ibk) , (3)
where ak and bk are drawn from a zero-mean Gaussian
distribution with unit variance. We use the power spec-
trum from Eisenstein & Hu (1999). Then the real-space
density field, δ(x), is obtained by performing an inverse
Fourier transform on δ(k).
2.2. The Linear Velocity Field
We construct a linear velocity field corresponding to
our linear density field using the standard Zel’Dovich
approximation (c.f. Zel’Dovich 1970; Efstathiou et al.
1985; Sirko 2005):
x1=x+ ψ(x) , (4)
v≡ ẋ1 = ψ̇(x) , (5)
δ(x)=−∇·[(1 + δ(x))ψ(x)]
≈−∇ · ψ(x) , (6)
where x and x1 denote initial (Lagrangian) and updated
(Eulerian) coordinates, respectively, ψ(x) is the displace-
ment vector, and the last equation follows from the conti-
nuity criterion, with the final approximation using linear-
ity, δ(x) ≪ 1. We note again that all units are comoving,
unless stated otherwise. From the above, one can relate
the velocity mode in our simulation at redshift z to the
linear density field:
v(k, z) =
Ḋ(z)δ(k) , (7)
where for computational convenience differentiation is
performed in k-space.
Another convenient property of this first-order
Zel’Dovich approximation is that the velocity field can
be decomposed into purely spatial, vx(x), and purely
temporal, vz(z), components:
v(x, z) = vz(z)vx(x) , (8)
where vz(z) = Ḋ(z) and vx(x) is the inverse Fourier
transform of ikδ(k)/k2. This is computationally conve-
nient, as we only need to compute the vx(x) field once
in order to be able to scale it for all redshifts, and it
also allows us to write a simple, exact expression for the
integrated linear displacement field, Ψ. When eq. (8) is
integrated from some large initial z0 [D(z0) ≪ D(z)], the
total displacement is just
Ψ(x, z)= [D(z)−D(z0)]vx(x)
≈D(z)vx(x) (9)
2 Since δ(x) is real-valued, only half of the k-modes defined
above are independent. The other half are determined by the usual
Hermitian constraints for real-valued functions (see for example
Hockney & Eastwood 1988; Bagla & Padmanabhan 1997).
We make use of this displacement field to adjust the halo
locations obtained by our filtering procedure (see § 2.4),
as well as to adjust the linear density field for our 21-cm
temperature maps (see § 4).
In principle, one could obtain non-linear velocities by
mapping the linear overdensity to a corresponding non-
linear overdensity obtained from a spherical collapse
model (Mo & White 1996), and then taking the time
derivative of the non-linear overdensity. However, due
to the large spread in the dynamical times of the non-
linear density field, accurately capturing the time evolu-
tion is non-trivial. Furthermore, although the non-linear
density field implicitly captures the velocities of collaps-
ing gas, mapping each pixel’s linear density to its non-
linear counterpart independently of other nearby pixels
does not properly preserve correlations on larger scales.
Hence, we choose to use the linear density field directly
in estimating velocities. For the purposes of studying the
ionization field, we are further justified in this procedure
because our final ionization maps are smoothed on large
scales, on which most pixels are still in the linear regime
at the high redshifts of interests. It is possible to include
higher-order contributions to the Zel’Dovich approxima-
tion where necessary (e.g., Scoccimarro & Sheth 2002).
2.3. Halo Filtering
In standard Press-Schechter theory (PS; see e.g.,
Press & Schechter 1974; Bond et al. 1991; Lacey & Cole
1993), the halo mass function can be written as
∂n(> M, z)
δc(z)
σ2(M)
∂σ(M)
c (z)
2σ2(M)
where n(> M, z) is the mean number density of halos
with total mass greater than M , ρ̄ = ΩMρcrit is the
mean background matter density, δc(z) ∼ 1.68/D(z) is
the scale-free critical over–density evaluated in the case
of spherically symmetric collapse (Peebles 1980), and
σ2(M) =
σ2(k)W 2(k,M) , (11)
is the squared r.m.s. fluctuation in the mass enclosed
within a region described by the filter function,W (k,M),
normalized to integrate to unity.
Although the PS mass function in eq. (10) is in fair
agreement with simulations, especially for halos near the
characteristic mass, at low redshifts it underestimates
the number of high–mass halos and overestimates the
number of low–mass halos when compared with large nu-
merical simulations (e.g. Jenkins et al. 2001). A mod-
ified expression shown to fit low-redshift simulation re-
sults more accurately (to within ∼ 10%) was obtained
by Sheth & Tormen (1999):
∂n(> M, z)
= − ρ̄
∂[ln σ(M)]
ν̂e−ν̂
where ν̂ ≡
aδc(z)/σ(M), and a, p, and A are fitting
parameters. Sheth et al. (2001) derive this form of the
mass function by including shear and ellipticity in model-
ing non–linear collapse, effectively changing the scale-free
critical over–density δc(z), into a function of filter scale,
δc(M, z) =
aδc(z)
1 + b
σ2(M)
aδ2c (z)
. (13)
Here b and c are additional fitting parameters (a is
the same as in eq. 12). For the constants above, we
adopt the recent values obtained by Jenkins et al. (2001),
who studied a large range in redshift and mass: a =
0.73, A = 0.353, p = 0.175, b = 0.34, c = 0.81. We
note, however, that the situation at high redshifts is
less clear: studies disagree on the relative accuracy of
the Press & Schechter (1974) and Jenkins et al. (2001)
forms (Reed et al. 2003; Iliev et al. 2006b; Zahn et al.
2007). Our algorithm can be trivially modified to ac-
commodate other choices for the mass function; fortu-
nately, for the purposes of the ionization maps (see §3),
the choice of mass function makes very little difference
because all have a similar dependence on the local den-
sity (Furlanetto et al. 2006a).
The mass functions in equations (10) and (12) can be
obtained by the standard excursion set random walk pro-
cedure. The approach is to smooth the density field
around a point, x, on successively smaller scales start-
ing with M → ∞ [where σ2(M) → 0] and to identify
the point as belonging to the halo with the largest M
such that δ(x,M) > δc(M, z). If W
2(k,M) is chosen to
have a sharp cut-off, this procedure amounts to a random
walk of δ(x,M) along the mass axis, since the change in
δ(x,M) as the scale is shrunk is independent of δ(x,M)
for a top-hat filter in k-space (see eq. 11).
We perform this procedure on our realization of the
linear density field by filtering the field using a real-
space top-hat filter3, starting on scales comparable to
the box size and going down to grid cell scales, in log-
arithmic steps of width ∆M/M = 1.2.4 At each filter
scale, we use the scale-dependent barrier in eq. (13)
to mark a collapsed halo if δ(x,M) > δc(M, z). Filter
scales large enough that collapsed structure is extremely
unlikely, δc(M, z) > 7σ(M), are skipped (Mesinger et al.
2005). Since this procedure treats each cell as the center
of a spherical filter, neighboring pixels are not properly
placed in the same halo. Because of this, we discount
halos which overlap with previously marked halos.
As mentioned above, this algorithm is similar
to the “peak-patch” approach first introduced by
Bond & Myers (1996a). The primary differences are:
3 There is a slight swindle in the current application of this for-
malism. The filter function is assumed to be a top-hat in k-space
in order to facilitate the analytic random walk approach described
above. However, when the power spectrum is normalized to ob-
servations [i.e. σ(R = 8h−1Mpc] = σ8), the filter that is used to
define the mass M corresponding to R is a top-hat filter in real
space. Nevertheless, it has been shown that the mass function is
not very sensitive to this filter choice (Bond et al. 1991).
4 We note that Mesinger et al. (2005) required a much smaller
step size at these redshifts, ∆M/M ∼ 0.1, in order to produce ac-
curate mass functions using 1D Monte-Carlo random walks. How-
ever, here we find that we can reproduce accurate mass functions
with a larger step size, since in our 3D realization of the density
field, “overstepping” δc(M, z) due to a large filter step size can be
compensated with a small offset in the filter center, i.e. by cen-
tering the filter in a neighboring cell. This is the case since over-
stepping δc(M, z) means that some dense matter between the two
filter scales was “missed”. In a 1D Monte-Carlo random walk this
matter is unrecoverable; however, in a 3D realization of the density
field, the missed matter will be picked up by a filter centered on a
neighboring cell.
(1) we use the Jenkins et al. (2001) barrier to identify
halos (rather than calculating the strain tensor to ac-
count for ellipsoidal collapse), (2) we do not separately
identify peaks in the density field (this step is not re-
quired given modern computing power), and (3) we use
the “full exclusion” criterion for preventing halo overlap.
Bond & Myers (1996b) found that a “binary exclusion”
method in which pairs of overlapping halos are compared
and eliminated was somewhat more accurate. However,
at the high redshifts of interest to us, halo overlap is rare,
and we are primarily interested in the large-scale prop-
erties of the halo field, which are relatively insensitive to
the details of the overlap criterion.
We also note that our halo finder is similar
in spirit to the PTHalos algorithm introduced by
Scoccimarro & Sheth (2002) to generate mock galaxy
surveys at low redshifts. There are two key differences.
First, at present we use only first-order perturbation the-
ory to displace the particles.5 This limits us to higher
redshifts, where velocities are smaller. However, our al-
gorithm does not require particles in order to resolve ha-
los and hence can accommodate a considerably larger
dynamic range than PTHalos.
Mass functions resulting from this procedure are shown
as points in Figure 1, with error bars indicating 1-σ Pois-
son uncertainties and bin widths spanning our mass filter
steps. Dotted red curves denote PS mass functions gen-
erated by eq. (10); short–dashed blue curves denote ex-
tended PS conditional mass functions generated by eq.
(10) but also taking into account the absence of den-
sity modes longer than the box size; long–dashed green
curves denote mass functions generated using the Sheth-
Tormen correction in eq. (12). The upper (lower) set
of curves and points correspond to redshifts of z=6.5
(z=10). The dotted and short–dashed curves overlap at
these redshifts due to our large box size (L = 100 Mpc),
so we are immune to the finite box effects pointed out by
Barkana & Loeb (2004).
Fig. 1 shows that we obtain accurate mass functions
for M ∼> 10
8M⊙. Our procedure seems to underpre-
dict the abundance of halos with masses approaching
the cell size, Mcell ∼ 107M⊙. However, as the Jeans
mass corresponding to a gas temperature of ∼ 104 K
is MJ(z ∼ 8) ∼ 108M⊙, in subsequent calculations, we
only use halos with masses greater than Mmin = 10
Using thisMmin, we match the collapse fraction obtained
by integrating eq. (12) to better than ∼ 10%.
This mass cutoff corresponds to the minimum tem-
perature required for efficient atomic hydrogen cooling
and would be the pertinent mass scale if: (1) the H2
cooling channel is suppressed, e.g. due to a perva-
sive Lyman-Werner (LW) background, and if (2) photo-
ionization feedback is ineffective at suppressing gas cool-
ing and collapse onto higher mass halos. While feed-
back at high redshifts remains poorly-constrained, both
of these assumptions seem reasonable during the mid-
dle stages of reionization on which we focus. A dis-
sociating LW background is likely to have established
5 We note that a similar scheme to ours has been independently
created by O. Zahn (private communication). This scheme uses
a simple Press-Schechter barrier but adjusts halo locations follow-
ing second-order Lagrangian perturbation theory. However, he has
found that the second-order corrections make very little difference
to the map.
Fig. 1.— Mass functions generated from our halo filtering pro-
cedure discussed in §2.3 are shown as points. Dotted red curves
denote PS mass functions generated by eq. (10); short–dashed
blue curves denote extended PS conditional mass functions gener-
ated by eq. (10) but also taking into account the absence of density
modes longer than the box size; long–dashed green curves denote
mass functions generated using the Sheth-Tormen correction in eq.
(12). The upper (lower) set of curves and points correspond to
redshifts of z=6.5 (z=10).
itself well before the universe is significantly ionized
(Haiman et al. 1997). Model-dependent empirical evi-
dence supporting the suppression of star formation in
smaller mass halos, M ∼< Mmin, can also be gleaned
from WMAP data (Haiman & Bryan 2006). Further-
more, although early work suggested that an ionizing
background could partially suppress star formation in
halos with virial temperatures of Tvir ∼< 3.6 × 10
(M ∼< 2×10
9M⊙) (Thoul & Weinberg 1996), more recent
studies (Kitayama & Ikeuchi 2000; Dijkstra et al. 2004)
find that at high redshifts (z ∼> 3), self-shielding and the
increased cooling efficiency could be strong countering
effects for halos with virial temperatures Tvir > 10
We postpone a more detailed analysis of the reionization
footprint left by photo-ionization feedback to a future
work.
2.4. Adjusting Halo Locations
Once the halo field is obtained, we use the displace-
ment field obtained through eq. (9) to adjust the halo
locations at each redshift. This corrects for the enhanced
halo bias in Eulerian space with respect to our filtering,
which is done in Lagrangian space (i.e. using the initial
locations at large z). For computational convenience, we
smooth the 12003 velocity field onto a coarser-grained
2003 grid before adjusting halo locations. The choice of
resolution, where each cell is (100 Mpc)/200 = 0.5 Mpc
on a side, is somewhat arbitrary here, and we have veri-
fied that our halo and 21-cm power spectra are unaffected
Fig. 2.— Halo power spectra at z = 8.7, with L = 20 h−1Mpc
and cosmological parameters taken from McQuinn et al. (2006a).
The solid red curve is the halo power spectrum from an N-body
simulation obtained from McQuinn et al. (2006a) (c.f. the bottom
panel of their Fig. 2). The short-dashed green and the long-dashed
violet curves are obtained from our filtering procedure with and
without the halo location adjustments, respectively.
by this choice. We also note that in linear theory, the
mean velocity dispersion inside a (0.5 Mpc)3 sphere with
mean density at z = 10 is a factor of ∼10 lower than the
r.m.s. bulk velocity of such regions, so smoothing over
smaller scale velocities appears reasonable. Furthermore,
we keep in mind that our “endproducts” in this work are
ionization and 21-cm temperature fluctuation maps, for
which such “low-resolution” is more than adequate (com-
pare, e.g., to N-body simulations of reionization, which
typically have similar cell sizes for the radiative transfer
component).
In Figure 2, we plot the halo power spectrum, de-
fined as ∆hh(k, z) = k
3/(2π2V ) 〈|δhh(k, z)|2〉k, where
δhh(x, z) ≡ Mcoll(x, z)/〈Mcoll(z)〉 − 1 is the collapsed
mass field.6 The solid red curve is the halo power spec-
trum from a 20 h−1 Mpc N-body simulation at z = 8.7
obtained from McQuinn et al. (2006a) (c.f. the bottom
panel of their Figure 2). The short-dashed green and the
long-dashed violet curves are obtained from our filtering
procedure (matching the assumed cosmology) with and
without the halo location adjustments, respectively. We
note that ignoring the cumulative motions of halos re-
sults in an underestimate of the power of long-wavelength
modes of the halo field by a factor of ∼ 2 in this case.
The average Eulerian bias of these halos is ∼ 2, about
half of which comes from the correction from Lagrangian
to Eulerian coordinates.
After the halo locations are adjusted according to lin-
ear theory, our halo power spectrum agrees almost per-
6 We use the collapsed mass field, rather than the individual
galaxies, because we calculate the power from the smoothed cells.
fectly with the simulation. By design our procedure in-
cludes Poisson fluctuations in the halo number counts,
which dominate the power spectrum at k ∼> 5 h/Mpc
and are lost in purely analytic estimates (McQuinn et al.
2006a). We also note that both the halo mass func-
tions and power spectra are statistical tests and hence
the agreement shown here does not imply that our halo
field has a one-to-one mapping with an N-body halo
field sourced by identical initial conditions. Indeed,
Gelb & Bertschinger (1994) showed that those particles
located nearest initial linear density peaks are not nec-
essarily incorporated into massive galaxies. The “peak
particle” algorithm is less robust than our smoothing
technique, but we still do not expect to recover halo
masses or locations precisely. We plan on doing a “one-
on-one” comparison between halo fields obtained from
our halo finder to those obtained from N-body codes
in a future work. However, it is certainly encouraging
that the very similar “peak-patch” group finding formal-
ism of Bond & Myers (1996a) did very well when com-
pared “one-on-one” to N-body codes at large mass scales
(Bond & Myers 1996b).
In Figure 3 we show slices through the halo field from
our simulation box at z = 8.25, generated by the above
procedure, again with (right panel) and without (left
panel) the halo location adjustments. In the figure, the
halo field is mapped to a lower resolution 4003 grid for
viewing purposes. Each slice is 100 Mpc on a side and
0.25 Mpc deep. Collapsed halos are shown in blue. Vi-
sually, it is obvious that peculiar motions increase halo
clustering.
3. GENERATING THE IONIZATION FIELD
Once the halo field is generated as described above,
we can perform a similar filtering procedure (also us-
ing the excursion-set formalism) to obtain the ionization
field (similar methods have been discussed by Zahn et al.
2005, 2007). The time required for this final step is a
function of x̄HI, with large x̄HI requiring less time than
small x̄HI. Specifically, at x̄HI ∼ 0.5 this step takes ∼ 15
minutes to generate a 2003 ionization box on our work-
station.
There are two main differences between the halo fil-
tering and the HII bubble filtering procedures: (1) HII
bubbles are allowed to overlap, and (2) the excursion
set barrier (the criterion for ionization) becomes, as per
Furlanetto et al. (2004a):
fcoll(x1,M, z) ≥ ζ−1 , (14)
where ζ is some efficiency parameter and fcoll(x1,M, z)
is the fraction of mass residing in collapsed halos inside
a sphere of mass M = 4/3πR3ρ̄[1 + 〈δnl(x1, z)〉R], with
mean physical overdensity 〈δnl(x1, z)〉R, centered on Eu-
lerian coordinate x1, at redshift z.
Equation (14) is only an approximate model and makes
several simplifying assumptions about reionization. In
particular, it assumes a constant ionizing efficiency per
halo and ignores spatially-dependent recombinations and
radiative feedback effects. It can easily be modified to
include these effects (e.g., Furlanetto et al. 2004b, 2006a;
Furlanetto & Oh 2005), and we plan to do so in future
work. Here we present the simplest case in order to best
match current RT numerical simulations.
This prescription models the ionization field as a two-
phase medium, containing fully-ionized regions (which
we refer to as HII bubbles) and fully-neutral regions.
This is obviously much less information than can be
gleaned from a full RT simulation, which precisely tracks
the ionized fraction. However, HII bubbles are typi-
cally highly-ionized during reionization, and for many
purposes (such as for 21 cm maps) this two-phase ap-
proximation is perfectly adequate.
In order to “find” the HII bubbles at each redshift we
smooth the halo field onto a 2003 grid. Then we filter
the halo field using a real-space top-hat filter, starting on
scales comparable to the box size and decreasing to grid
cell scales in logarithmic steps of width ∆M/M = 0.33.
At each filter scale, we use the criterion in eq. (14)
to check whether the region is ionized. If so, we flag
all pixels inside that region as ionized. We do this for
all pixels and scales, regardless of whether the resulting
bubble would overlap with other bubbles. Note, there-
fore, that the nominal ionizing efficiency ζ that we use
as an input parameter does not equal (1 − x̄HI)/fcoll.
They typically differ by . 30%, with ζfcoll < 1 − x̄HI
early in reionization and ζfcoll > 1− x̄HI late in reioniza-
tion). Unfortunately, we thus cannot use our algorithm
to self-consistently predict the time evolution of the ion-
ized fraction (rather, that must be prescribed from some
other model). Of course, the same is true for N-body
simulations, because the evolution of the ionized fraction
depends on the evolving ionization efficiency of galaxies
and cannot be self-consistently included in any present-
day simulation.
In order to obtain the density field used in eq. (14),
δnl(x1, z), we use the Zel’Dovich approximation on our
linear density field, δ(x), in much the same manner as
we did to adjust our halo field in § 2.4. Starting at some
arbitrarily large initial redshift (we use z0 = 50), we
discretize our high-resolution 12003 field into “particles”
whose mass equals that in each grid cell. We then use
the displacement field (eq. 9) to move the particles to
new locations at each redshift. This resulting mass field
is then smoothed onto our lower resolution 2003 box to
obtain δnl(x1, z). We then recalculate the velocity field
(§ 2.2) using the new densities.
Zahn et al. (2007) showed that a very similar HII bub-
ble filtering procedure performed on an N-body halo field
was able to reproduce the ionization topology obtained
through a ray-tracing RT algorithm fairly well. Their
algorithm differs from ours in two ways. First, they used
a slightly different barrier definition; however, this dif-
ference has only a small impact on the ionization topol-
ogy.7 More importantly, for each filter scale at each pixel,
Zahn et al. (2007) flag only the center pixel as ionized if
the barrier is crossed, whereas we flag the entire filtered
sphere.
In order to test our bubble filtering algorithm, we ex-
ecute it on the same N-body halo field at z = 6.89 as
was used to generate the bottom panels of Fig. 3 in
Zahn et al. (2007). We compare analogous ionization
maps created using various algorithms in Figure 4. All
7 Specifically, in order to match the physics of their simulations
better, they required
dt fcoll > ζ
−1. However, the density mod-
ulation ends up nearly identical to our model, so the topology is
almost unchanged.
Fig. 3.— Slices through the halo field from our simulation box at z = 8.25. The halo field is generated on a 12003 grid and then mapped
to a 4003 grid for viewing purposes. Each slice is 100 Mpc on a side and 0.25 Mpc deep. Collapsed halos are shown in blue. The left panel
shows the halo field directly filtered in Lagrangian space; the right panel maps the field to Eulerian space according to linear theory (see
§ 2.4 and eq. 9). The right panel corresponds to the bottom-left (x̄HI = 0.53) ionization field in Figure 5.
slices are 93.7 Mpc on a side and 0.37 Mpc deep, with ζ
adjusted so that the mean neutral fraction in the box is
x̄HI = 0.49. Ionized regions are shown as white. The left-
most and right-most panels are taken from Zahn et al.
(2007). The left-most panel was created by perform-
ing their bubble filtering procedure directly on the lin-
ear density field (without explicitly identifying halos).
The second panel was created by performing their bub-
ble filtering procedure on their N-body halo field, but
with the slightly different barrier definition in eq. (14).
The third panel was created by performing our bubble
filtering procedure on the same N-body halo field, but ig-
noring density fluctuations outside of halos (i.e. setting
〈δnl(x1, z)〉R = 0), which we have verified give nearly
identical bubble maps as our full procedure (so long as
x̄HI is fixed). The right-most panel was created using
an approximate RT algorithm (Abel & Wandelt 2002;
Sokasian et al. 2001, 2003) on the same halo field.
It is immediately obvious from Fig. 4 that all of the
approximate maps (first three panels) reproduce the RT
map (right-most panel) fairly well. Even the HII bub-
ble filtering performed directly on the linear density field
(left-most panel) performs well, which is encouraging, as
that is the starting point for our semi-numerical proce-
dure and we only improve on this scheme.
Figure 4 shows that our HII bubble filtering algorithm
is an excellent approximation to RT. The similar algo-
rithm proposed by Zahn et al. (2007) also performs well.
In comparison, our algorithm produces somewhat more
“bubbly” maps but appears to better capture the connec-
tivity of HII regions. Both are an obvious improvement
on directly filtering the linear density field.
Of course, in our full algorithm we identify halos from
the linear density field (rather than from simulations), so
our method consumes comparable processing time to the
one used to generate the leftmost panel in Figure 4, once
the halos have been identified. Moreover, we are able
to capture the “stochastic” component of the halo bias
that causes the relatively large differences between the
leftmost panel and the full RT simulation. That is, the
algorithm used to generate the leftmost panel uses the
large-scale linear density field to predict the distribution
of halos (Zahn et al. 2005, 2007). In reality, the rela-
tion is not deterministic because of random fluctuations
in the small-scale modes comprising each region. This
leads to nearly Poisson scatter in the halo number densi-
ties (Sheth & Lemson 1999; Casas-Miranda et al. 2002)
that can substantially modify the bubble size distribution
whenever sources are rare, particularly early in reion-
ization (Furlanetto et al. 2006a). By directly sampling
the small-scale modes to build the halo distribution, we
better recover this scatter (at least statistically, as illus-
trated by Fig. 2). Another way to include this scatter
is by directly sampling halos from an N-body simulation
(as in Zahn et al. 2007, or the second panel of Fig. 4),
although that obviously requires much more computing
power.
3.1. Ionization Maps
Now that we have demonstrated in turn the success of
our halo and bubble filtering procedures, we present the
resulting ionization maps when the two are combined.
In Figure 5, we show 100 Mpc × 100 Mpc × 0.5 Mpc
slices through our 2003 ionization field at z = 10, 9,
8.25, 7.25 (left to right across rows). With the assump-
tion of ζ = 15.1, these redshifts correspond to x̄HI = 0.89,
0.74, 0.53, 0.18, respectively. As has been pointed out by
Furlanetto et al. (2004c), the neutral fraction is the more
relevant descriptor; bubble morphologies at a constant
x̄HI vary little with redshift (see also McQuinn et al.
Fig. 4.— Slices from the ionization field at z = 6.89 created using different algorithms. All slices are 93.7 Mpc on a side and 0.37 Mpc
deep, with the mean neutral fraction in the box being x̄HI = 0.49. Ionized regions are shown as white. The left-most panel was created
by performing the bubble filtering procedure of Zahn et al. (2007) directly on the linear density field. The second panel was created by
performing their bubble filtering procedure on their N-body halo field, but with the slightly different barrier definition in eq. (14). The
third panel was created by performing our bubble filtering procedure described in § 3 on the same N-body halo field. The right-most panel
(from Zahn et al. 2007) was created using an approximate RT algorithm on the same halo field.
2006a). The bottom-left panel corresponds to the halo
field in the top-right panel of Fig. 3, generated on a
high-resolution 12003 grid.
To quantify the ionization topology resulting from our
method, we calculate the size distributions of both the
ionized and neutral regions. We randomly choose a pixel
of the desired phase (neutral or ionized), and record the
distance from that pixel to a phase transition along a
randomly chosen direction. We repeat this Monte Carlo
procedure 107 times. Volume-weighted probability dis-
tribution functions (PDFs) produced thusly are shown
by the solid curves in Figure 6 for ionized regions (top
panel) and neutral regions (bottom panel). Curves corre-
spond to (z, x̄HI) = (10, 0.89), (9.25, 0.79), (8.50, 0.61),
(8.00, 0.45), (7.50, 0.27), (7.00, 0.10), from left to right
in the top panel, respectively (or from right to left in
the bottom panel). All curves are normalized so that the
probability density integrates to unity.
It is useful to compare these distributions to the an-
alytic bubble mass function of Furlanetto et al. (2004c);
although this analytic approach is motivated by the same
excursion set barriers as our semi-numerical approach, it
does not account for the full geometry of sources. We
compute the probability distribution from the analytic
model by assuming purely spherical bubbles and convolv-
ing with the volume-weighted distance to the sphere’s
edge:
p(r) dr =
2πr2 dr
(1− x̄HI)
dRnb(R)
, (15)
where nb(R) is the comoving number density of bub-
bles with radii between R and R + dR (taken from
Furlanetto et al. 2004c).
Several points are evident from Figures 5 and
6. As expected (e.g., Furlanetto et al. 2004c, 2006a;
McQuinn et al. 2006a), there is a well-defined bubble
scale at each neutral fraction, despite some scatter in
the sizes. This scale also gets more pronounced (i.e. the
PDF peaks more) as reionization progresses; this is a
result of the changing shape of the underlying matter
power spectrum (Furlanetto et al. 2006a).
Also, the purely analytic estimates underpredict the
size distributions at all values of the neutral fraction,
though they do become increasingly accurate as the neu-
tral fraction decreases. This trend is perhaps counterin-
tuitive, as the analytic model, which rests on the assump-
tion of spherical bubbles, should perform best when the
bubbles are isolated, as one would expect at earlier times,
i.e. high neutral fractions. However, looking at the top-
left panel of Fig. 5, the typical bubbles filling most of
the ionized volume overlap due to the strong clustering
of early sources and bubbles. This results in many “over-
lapping pairs of spheres” at early times, resulting from
merging HII bubbles sourced by clustered sources. Thus
the spherical bubble-based analytic model underpredicts
the true size distribution, using our “mean free path” def-
inition of bubble sizes above. This effect was not noted
by previous studies (Zahn et al. 2007), because they used
a different definition of bubble sizes, based on spherical
filters used to flag regions in which x̄HI < 0.1. As time
progresses and the universe becomes more ionized, this
“overlapping pair of spheres” effect becomes less and less
dominant (see Fig. 5), and the analytic model becomes
increasingly more accurate.
Finally, the size distributions of neutral regions pre-
sented in the bottom panel of Fig. 6 are a new result and
potentially important for the 21-cm signal (which origi-
nates in neutral hydrogen, of course). In the later stages
of reionization, when the topology has transformed to
isolated neutral islands in a sea of ionized gas, this fig-
ure pinpoints the typical sizes of “mostly neutral” pixels
that continue to emit strongly. In contrast to the ionized
regions, the neutral regions (defined in this way) do not
grow substantially during reionization. From x̄HI = 0.89
to x̄HI = 0.1, the peak of the distribution shifts only by a
factor of ∼ 6, whereas the peak of the ionized region dis-
tribution shifts by a factor of ∼40 over the same range.
The reason for this is also evident in Figure 5: even when
the universe is mostly neutral, space is dotted with is-
lands of ionized gas, such that our “mean free path”–type
size distributions never become too large. The converse
does not hold true for ionized regions. However, a slight
parallel for ionized islands in a mostly neutral IGM, could
be found in Lyman limit systems (LLS) inside larger HII
regions (e.g. Barkana & Loeb 2002; Shapiro et al. 2004;
Miralda-Escudé et al. 2000), though it is not clear how
prevalent such neutral clumps are at high redshifts.
Throughout this paper, we have used a L = 100
Mpc “simulation” box. This size facilitates compar-
ison of our results with those from recent hybrid N-
Fig. 5.— Slices through the 2003 ionization field at z = 10, 9, 8.25, 7.25 (left to right across rows). With the assumption of ζ = 15.1,
these redshifts correspond to x̄HI = 0.89, 0.74, 0.53, 0.18, respectively. All slices are 100 Mpc on a side and 0.5 Mpc deep. The bottom-left
panel corresponds to the halo field in the top-right panel of Fig. 3, generated on a high-resolution 12003 grid.
body works (Zahn et al. 2007; McQuinn et al. 2006a;
Iliev et al. 2006a; Trac & Cen 2006). However, the speed
of our semi-numerical approach allows us to explore
larger cosmological scales while still consistently resolv-
ing the small halos that could dominate the photon bud-
get during reionization. As mentioned previously, exist-
ing N-body codes must resort to merger-tree methods
to populate their distribution of small-mass halos, even
for box sizes ∼< 100 Mpc (McQuinn et al. 2006a). In
this spirit, we present some preliminary results from a
N = 15003, L = 250 Mpc simulation, capable of directly
resolving halos with masses M ∼> 2.2× 10
8M⊙, with re-
sulting mass functions accurate to better than a factor
of two even at the smallest scale. This resolution pushes
the RAM limit of our machine and so each redshift can
take several hours to complete.8
In Figure 7, we compare size distributions of ionized
8 We note here that our halo-finding algorithm requires signifi-
cantly higher resolution than does predicting the ionization field
directly from the linear density field smoothed on larger scales
(Zahn et al. 2005, 2007). The latter method can be extended to
even larger boxes, though at the price of a somewhat less accurate
ionization map (compare the left and right panels in Fig. 4).
Fig. 6.— Size distributions (see definition in text) of ionized
(top panel) and neutral (bottom panel) regions. Curves correspond
to (z, x̄HI) = (10, 0.89), (9.25, 0.79), (8.50, 0.61), (8.00, 0.45),
(7.50, 0.27), (7.00, 0.10), from left to right in the top panel, re-
spectively (or from right to left in the bottom panel). Solid curves
are produced from our simulation while dotted curves correspond
to the analytic mass function. All curves are normalized so that
the probability distribution integrates to unity.
(top panel) and neutral (bottom panel) regions from our
two different simulation boxes. Curves correspond to (z,
x̄HI) = (9.00, 0.80), (8.00, 0.56), (7.00, 0.21), from left
to right in the top panel, respectively (or from right to
left in the bottom panel, respectively). Solid curves are
generated from our fiducial, N = 12003, L = 100 Mpc,
simulation while dashed curves are generated from our
larger simulation with N = 15003, L = 250 Mpc. The
cell size in all ionization maps is 0.5 Mpc on a side, with
the efficiency parameter, ζ, adjusted to obtain matching
values of x̄HI, and we set the minimum halo mass to
Mmin = 2.2× 108M⊙ even in the higher resolution runs
for easier comparison.
As reionization progresses, an increasing number of
large HII regions are “missed” by the L = 100 Mpc sim-
ulation. Interestingly, the analogous trend in the neutral
region size distributions (bottom panel) is weaker. This is
most likely because the “ionized island” effect limits the
size distributions of neutral regions as described above.
4. 21-CM TEMPERATURE FLUCTUATIONS
A natural application of our “simulation” technique is
to predict 21-cm brightness temperatures during reion-
ization. The offset of the 21-cm brightness temperature
from the CMB temperature, Tγ , along a line of sight
(LOS) at observed frequency ν, can be written as (e.g.
Furlanetto et al. 2006b):
δTb(ν)=
TS − Tγ
1 + z
(1− e−τν0 ) (16)
Fig. 7.— Size distributions of ionized (top panel) and neutral
(bottom panel) regions from different simulation boxes. Curves
correspond to (z, x̄HI) = (9.00, 0.80), (8.00, 0.56), (7.00, 0.21),
from left to right in the top panel, respectively (or from right to
left in the bottom panel). Solid curves are generated from our
fiducial, N = 12003, L = 100 Mpc, simulation while dashed curves
are generated from a larger simulation with N = 15003, L = 250
Mpc. The cell size in all ionization maps is 0.5 Mpc on a side,
with the efficiency parameter, ζ, adjusted to get matching values
of x̄HI and the minimum halo mass set to Mmin = 2.2 × 10
for comparison purposes.
≈ 9(1 + z)1/2xHI(1 + δnl)
dvr/dr +H
where TS is the gas spin temperature, τν0 is the opti-
cal depth at the 21-cm frequency ν0, δnl is the physical
overdensity (see discussion under eq. 14), H is the Hub-
ble parameter, dvr/dr is the comoving gradient of the
line of sight component of the comoving velocity, and all
quantities are evaluated at redshift z = ν0/ν − 1. The
final approximation makes the standard assumption that
TS ≫ Tγ for all redshifts of interest during reionization
(e.g. Furlanetto 2006) and also that dvr/dr ≪ H . We
verify in our simulation that dvr/dr < H for all neutral
pixels.
Maps of δTb(x, ν) generated in this manner are shown
in Figure 8. All slices are 100 Mpc on a side, 0.5 Mpc
deep, and correspond to (z, x̄HI) = (9.00, 0.74), (8.25,
0.53), (7.50, 0.27), from left to right. The top panels take
into account the velocity correction term in eq. (16),
while the bottom panels ignore it.
As seen in Fig. 8, velocities typically increase the con-
trast in temperature maps, making hot spots hotter and
cool spots cooler. We also see that temperature hot
spots, which correspond to dense pixels, tend to clus-
ter around the edges of HII bubbles, especially smaller
bubbles. This occurs because HII bubbles correlate with
peaks of the density field and long-wavelength biases in
the density field can extend beyond the edge of the ion-
ized region. This enhanced contrast might be useful in
Fig. 8.— Brightness temperature of 21-cm radiation relative to the CMB temperature. All slices are 100 Mpc on a side, 0.5 Mpc deep,
and correspond to (z, x̄HI) = (9.00, 0.74), (8.25, 0.53), (7.50, 0.27), left to right. Top panels include the velocity correction term in eq.
(16), while the bottom panels do not. For animated versions of these pictures, see http://pantheon.yale.edu/∼am834/Sim.
the detection of the boundaries of ionized regions with fu-
ture 21-cm experiments. As reionization progresses most
hot spots become swallowed up by HII bubbles, and the
effects of velocities diminish.
In Figure 9 we plot the dimensionless 21-
cm power spectrum, defined as ∆221(k, z) =
k3/(2π2V ) 〈|δ21(k, z)|2〉k, where δ21(x, z) ≡
δTb(x, z)/ ¯δTb(z) − 1. Solid blue curves take into
account gas velocities, while dashed red curves do not.
Curves correspond to (x̄HI, z) = (0.79, 9.25), (0.61,
8.50), (0.45, 8.00), (0.27, 7.50), (0.10, 7.00), bottom to
top. Error bars on the bottom dashed curve denote 1-σ
Poisson uncertainties; fractional errors in a given bin
are the same for all curves. As reionization progresses,
small-scale power is traded for large-scale power, and the
curves become flatter. Note that, with our dimensionless
definition of the power spectrum, curves with smaller
x̄HI have larger values of ∆
21(k, z). This is because the
mean brightness temperature offset drops quite rapidly
as reionization progresses, since ¯δTb(z) ∝ x̄HI, but the
scatter remains significant (see Fig. 6) and thus the
fractional perturbation, δ21(x, z), increases throughout
reionization.
Finally, in Figure 10 we plot dimensional power spec-
tra, ¯δTb(z)
2∆221(k, z). The curves correspond to (x̄HI, z)
= (0.80, 9.00), (0.56, 8.00), (0.21, 7.00), top to bottom at
large k, respectively. The dotted green curves are gen-
erated from a large, high-resolution “simulation”, with
Fig. 9.— Dimensionless 21-cm power spectra for (x̄HI, z) =
(0.79, 9.25), (0.61, 8.50), (0.45, 8.00), (0.27, 7.50), (0.10, 7.00),
bottom to top. Solid blue curves take into account gas velocities,
while dashed red curves do not.
http://pantheon.yale.edu/~am834/Sim
Fig. 10.— Dimensional 21-cm power spectra. The curves corre-
spond to (x̄HI, z) = (0.80, 9.00), (0.56, 8.00), (0.21, 7.00), top to
bottom at large k, respectively. The dotted green curves are gen-
erated from a large, high-resolution “simulation”, with N = 15003
and L = 250 Mpc, with no velocity contribution to the power spec-
tra. Solid blue curves and dashed red curves are generated with
our fiducial N = 12003 and L = 100 Mpc simulation, with and
without the velocity contribution, respectively.
N = 15003 and L = 250 Mpc, with no velocity contribu-
tion to the power spectra. Solid blue curves and dashed
red curves are generated with our fiducial N = 12003
and L = 100 Mpc simulation, with and without the ve-
locity contribution, respectively. The cell size in all δTb
maps is 0.5 Mpc on a side, with the efficiency parame-
ter, ζ, adjusted to achieve matching values of x̄HI and
the minimum halo mass set to Mmin = 2.2× 108M⊙ for
comparison purposes.
As seen in Figures 9 and 10, velocities make a mod-
est contribution to the 21 cm power spectrum, boosting
power on small scales early in reionization. Note that
the apparent slight decrease in power at small k when
velocities are included is well within the errors from av-
eraging over the few modes available to us on the largest
scales (e.g., see Poisson error bars on the bottom dashed
curve in Fig. 9). While the maximum δTb value in our
simulation box increases by a factor of a few when ve-
locities are included, most of the pixels are only slightly
affected. When the power spectrum is plotted in a di-
mensional version, ¯δTb(z)
2∆221(k, z), small-scale power is
boosted by ∼ 40% at (x̄HI, z) = (0.80, 9.00), with this en-
hancement monotonically decreasing as reionization pro-
gresses. Linear theory predicts that velocities enhance
the density power spectrum by a factor of 1.87 when
x̄HI = 1 (Kaiser 1987). In fact we do recover this en-
hancement for a fully neutral IGM; however, as predicted
by analytic models (McQuinn et al. 2006b), the ionized
bubbles rapidly remove most of this amplification.
Figure 10 also confirms the inferences drawn from Fig.
7, primarily that larger box sizes are needed to capture
the ionization topology at the end stages of reionization.
Comparing the dashed red to the dotted green curves in
Fig. 10, we note that our fiducial L = 100 Mpc simula-
tions are accurate for scales smaller than k ∼> 0.2 Mpc
(or λ . 30 Mpc). As reionization progresses, larger scales
lose power more rapidly than in the L = 250 Mpc simu-
lation. This is again evidence that very large scale simu-
lations are needed to model the middle and late stages of
reionization. Thus the speed and high resolution of our
semi-numeric approach will be extremely useful in future
modeling of reionization.
5. CONCLUSIONS
We introduce a method to construct semi-numeric sim-
ulations that can efficiently generate realizations of halo
distributions and ionization maps at high redshifts. Our
procedure combines an excursion-set approach with first-
order Lagrangian perturbation theory and operates di-
rectly on the linear density and velocity fields. As such,
our algorithm can exceed the dynamic range of exist-
ing N-body codes by orders of magnitude. As this is
the main limiting factor in simulating the ionized bubble
topology throughout reionization, when ionized regions
reach scales of tens of comoving Mpc, this will be partic-
ularly useful in such studies. Moreover, the efficiency of
the algorithm will allow us to explore the large parame-
ter space required by the many uncertainties associated
with high-redshift galaxy formation.
We find that our halo finding algorithm compares well
with N-body simulations on the statistical level, yield-
ing both accurate mass functions and power spectra. We
have not yet compared our halo distribution with sim-
ulations on a point-by-point basis, but we do not ex-
pect perfect agreement because of the vagaries of the ex-
cursion set approach. However, it is encouraging that
a very similar algorithm independently developed by
Bond & Myers (1996a) fares quite well in a comparison
of high-mass halos.
Our HII bubble finding algorithm captures the bubble
topology quite well, as compared to ionization maps from
ray-tracing RT algorithms at an identical x̄HI. Our al-
gorithm is similar to other codes, although we build the
ionization map from our excursion set halo field rather
than directly from the linear density field or from halos
found in an N-body simulation (Zahn et al. 2005, 2007).
Compared to codes built only from the linear density
field, we can better track the “stochastic” component of
the bias, though at the cost of somewhat more compu-
tation and a harder limit on resolution. On the other
hand, our scheme is much faster than using an N-body
code and offers superior dynamic range.
We create ionization maps using a simple efficiency pa-
rameter and compute the size distributions of ionized
and neutral regions. Our size distributions are gener-
ally shifted to larger scales when compared with purely
analytic models (Furlanetto et al. 2004c) at the same
mean neutral fraction. The discrepancy lies in the fact
that, at their core, the purely analytic models are based
on ensemble-averaged distributions of isolated spheres.
Hence they do not capture overlapping bubble shapes,
which are most important at large x̄HI (when the bubbles
are small and random fluctuations in the source densities,
as well as clustering, are most important).
In this paper, we have confined ourselves to a sim-
ple ionization criterion (essentially photon counting;
Furlanetto et al. 2004c). However, our algorithm can
easily accommodate more sophisticated prescriptions, so
long as they can be expressed either with the excursion
set formalism (Furlanetto & Oh 2005; Furlanetto et al.
2006a) or built from the halo field (in a similar way to
semi-analytic models of galaxy formation embedded in
numerical simulations).
We also use our procedure to generate maps and power
spectra of the 21-cm brightness temperature fluctuations
during reionization. We note that temperature hot spots
generally cluster around HII bubbles, especially in the
early phases of reionization. Because HII bubbles cor-
relate with peaks of the density field, long-wavelength
biases in the density field can extend beyond the edge of
the ionized region, with the resulting overdensities ap-
pearing as hot spots. This effect might be useful for
detecting the boundaries of ionized regions with future
21-cm experiments. We study the imprint of gas bulk ve-
locities on 21-cmmaps and power spectra, an effect which
was not included in previous studies. We find that ve-
locities do not have a major impact during reionization,
although they do increase the contrast in temperature
maps, making some hot spots hotter and some cool spots
cooler. Velocities also increase small-scale power, though
the effect decreases with decreasing x̄HI.
We also include some preliminary results from a sim-
ulation run with the largest dynamical range to date:
a 250 Mpc box which resolves halos with masses M ∼>
2.2 × 108M⊙. This simulation run confirms that ex-
tremely large scales are required to model the late stages
of reionization, x̄HI ∼< 0.5, when the typical scale of ion-
ized bubbles becomes several tens of Mpc.
The speed and dynamic range provided by our semi-
numeric approach will be extremely useful in the mod-
eling of early structure formation and reionization. Our
ionization maps can be efficiently folded into analyses of
current and upcoming high-redshift observations, espe-
cially 21-cm surveys.
We thank Greg Bryan for many helpful conversations
concerning the inner workings of cosmological simula-
tions and the generation of initial conditions. We also
thank Oliver Zahn for permitting the use of the halo field
from his simulation output as well as for several interest-
ing discussions. We thank Mathew McQuinn for provid-
ing the halo power spectra from his simulation as well
as for associated helpful comments. We thank Zoltan
Haiman, Greg Bryan, Oliver Zahn and Mathew McQuinn
for insightful comments on a draft version of this paper.
This research was supported by NSF-AST-0607470.
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