An improved model of HII bubbles during the epoch of reionization

0.87 MB·

by  Aseem Paranjape

Mon.Not.R.Astron.Soc.000,000–000(0000) Printed22May2014 (MNLATEXstylefilev2.2)
Hii
An improved model of bubbles during the epoch of
reionization
Aseem Paranjape1(cid:63) and T. Roy Choudhury2†
1ETH Zurich, Department of Physics, Institute for Astronomy, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland
2National Centre for Radio Astrophysics, TIFR, Post Bag 3, Ganeshkhind, Pune 411007, India
22May2014
4
ABSTRACT
1
Thesizedistributionofionizedregionsduringtheepochofreionization–akeyingredientin
0
2 understanding the Hi power spectrum observable by 21cm experiments – can be modelled
analytically using the excursion set formalism of random walks in the smoothed initial
y
densityfield.Todate,suchcalculationshavebeenbasedonsimplifyingassumptionscarried
a
forward from the earliest excursion set models of two decades ago. In particular, these
M
models assume that the random walks have uncorrelated steps and that haloes can form at
arbitrarylocationsintheinitialdensityfield.Weextendthesecalculationsbyincorporating
1
recent technical developments that allow us to (a) include the effect of correlations in the
2
stepsofthewalksinducedbyarealisticsmoothingfilterand(b)moreimportantly,account

] for the fact that dark matter haloes preferentially form near peaks in the initial density. A
O comparison with previous calculations shows that including these features, particularly
C the peaks constraint on halo locations, has large effects on the size distribution of the Hii
bubbles surrounding these haloes. For example, when comparing models at the same value
.
h of the globally averaged ionized volume fraction, the typical bubble sizes predicted by our
p
model are more than a factor 2 larger than earlier calculations. Our results can potentially
o- have a significant impact on estimates of the observable Hi power spectrum.
r
t Key words: dark ages, reionization, first stars – intergalactic medium – cosmology:
s
theory – large-scale structure of Universe.
a
[

2
v
41 INTRODUCTION the mean transmittance of Lyman-α flux in the quasar ab-
9 sorption spectra becomes so low (Fan et al. 2006) that the
An outstanding problem in present day cosmology is to under-
9 constraints on the neutral hydrogen fraction and photoioniza-
7standthereionizationofneutralhydrogenthatoccursbetween tionrateturnouttobehighlymodel-dependent(Galleranietal.
.20(cid:38)z(cid:38)6 (Furlanetto et al. 2006b; Pritchard & Loeb 2012).
1 2006;Bolton&Haehnelt2007;Galleranietal.2008a,b;Bolton
Fromtheobservationalviewpoint,thestudyofthereionization
0 et al. 2011). Similarly, fluctuations in the CMBR polarization
epoch is extremely interesting because this is the last phase of
4 at large angular scales are expected to be cosmic variance
cosmic evolution that can be observed directly. Theoretically,
1 dominated, and hence it is unlikely that CMBR observations
:reionization is linked to the formation of the first galaxies and wouldbeabletoprovidemoreinformationonreionizationthan
vstars. It is the ionizing photons from these sources which are
i what is available at present (Holder et al. 2003; Mitra et al.
Xbelieved to be the drivers of the initial phase of reionization;
2012). A potential way to constrain the reionization history
observations of the reionization history could therefore yield
r at high redshifts is by detecting the 21cm signal from neutral
aimportantinformationaboutthenatureandevolutionofthese hydrogen (Hi) at those redshifts. A large effort is being di-
sources. These ionizing sources are expected to form ionized
rectedtowardsthisendusingradiointerferometricinstruments
regions (bubbles, in what follows) around themselves, which
such as GMRT1 (Paciga et al. 2011, 2013), LOFAR2 (Harker
then overlap and proceed to complete the process of reioniza-
etal.2010;Yatawattaetal.2013),MWA3 (Tingayetal.2013;
tion. According to current models, this process should have
Bowman et al. 2013), etc. In addition, one expects significant
occurred over the redshift range 20(cid:38)z(cid:38)6 (Wyithe & Loeb
breakthroughs from future radio telescopes such as the SKA4
2003; Choudhury & Ferrara 2005, 2006; Alvarez et al. 2006;
(Mellema et al. 2013).
Pritchard et al. 2010; Mitra et al. 2011, 2012).
Observationally, constraining reionization at z (cid:38) 6 has
been extremely challenging for various reasons. For example,
1 http://www.gmrt.ncra.tifr.res.in/
2 http://www.lofar.org/
(cid:63) Email:[email protected] 3 http://www.haystack.mit.edu/ast/arrays/mwa/
† E-mail:[email protected] 4 http://www.skatelescope.org/
(cid:13)c 0000RAS
2 Paranjape & Choudhury
Since the size distribution of the ionized bubbles depends dom walks; these walks are assumed to have steps that are
onthenatureofthesourcesfromwhichtheyform,andinturn uncorrelated with each other, which would be the case if the
determines the observable 21cm signal, building analytical smoothing filter for the density were sharp in Fourier space.
models of the evolution of this distribution can give us a This introduces some technical simplifications in the analysis,
handle on interpreting future observations and constraining whileusingamorerealisticfiltersuchasaTopHatinrealspace
astrophysical and cosmological models. This has been the leads to correlations in the steps which are difficult to treat in
motivation behind much recent analytical (Furlanetto et al. full generality. Additionally, the FZH04 calculation explicitly
2004;Zaldarriagaetal.2004;Furlanettoetal.2006a;McQuinn uses the Bond et al. (1991) expression for the mass fraction in
et al. 2006; Wyithe & Loeb 2007; Wyithe & Morales 2007) collapsed objects, which relies on the spherical collapse model
and numerical work on this subject (Iliev et al. 2006; Mellema and further treats all locations in the initial density field on
etal.2006;Zahnetal.2007;McQuinnetal.2007;Trac&Cen equalfooting,whereasoneexpectshaloestoformpreferentially
2007; Semelin et al. 2007; Shin et al. 2008; Lidz et al. 2008; near peaks in the initial density (Bardeen et al. 1986; Bond
Baek et al. 2009; Thomas et al. 2009; Choudhury et al. 2009). 1989) with a dynamical evolution that is better described by
Since the physics governing the formation of the first sources ellipsoidal collapse (Bond & Myers 1996; Sheth et al. 2001).
is poorly understood, theoretical models often contain a large Recentworkhasshownthatthereexistsimpleyetremark-
number of uncertain parameters (Pritchard et al. 2010; Mitra ablyaccurateapproximationsthataccountforthecorrelations
et al. 2011, 2012; Shull et al. 2012). In this regard, analytical in the steps of the random walks induced by a realistic filter
or semi-analytical models are therefore very useful in probing (Musso & Sheth 2012, 2013). These approximations make it
large regions of parameter space. relatively simple to additionally include the effect of centering
A particular class of models, which estimate the size dis- the walks at peaks in the initial density (Musso & Sheth 2012;
tributionofthesebubblesandtheir21cmsignals,arebasedon Paranjape & Sheth 2012) as well as including the effects of
the excursion set approach (Press & Schechter 1974; Epstein ellipsoidal collapse, leading to a modified approach of Excur-
1983; Peacock & Heavens 1990; Bond et al. 1991) which has sion Set Peaks (ESP) that agrees with halo abundances and
traditionally been applied to the problem of estimating the clustering measurements in N-body simulations at the ∼10%
mass function, growth and clustering of dark matter haloes level (Paranjape et al. 2013b). These improvements, which we
(Lacey & Cole 1993; Mo & White 1996; Bond & Myers 1996; describe in detail below, form the basis of the current work5.
Sheth1998;Monaco1999;Shethetal.2001;Zhang&Hui2006; Our main idea in this paper is to investigate the effects
Maggiore&Riotto2010;Paranjapeetal.2012;Musso&Sheth these improvements have on the size distribution of the Hii
2012; Achitouv et al. 2013). In this context, this approach es- bubbles. The FZH04 model, with its assumptions of sharp-k
sentially maps a counting problem in the evolved dark matter filtering and spherical collapse at randomly chosen locations,
densityfieldto thestatisticsof randomwalksin the smoothed formedthebasisforgeneratingsemi-numericalionizationmaps
initial density field as a function of smoothing scale, through in a three-dimensional box (Mesinger & Furlanetto 2007; Geil
some simplifying assumption on the dynamical evolution of &Wyithe2008;Mesingeretal.2011),withlaterimprovements
gravitational nonlinearities such as spherical (Gunn & Gott in the semi-numerical calculations that included sharp-k fil-
1972; Press & Schechter 1974) or ellipsoidal (Bond & Myers ters consistently throughout the calculations (e.g., the FFRT
1996; Sheth et al. 2001) collapse. The latter leads to a density scheme of Zahn et al. 2011). These semi-numerical estimates
thresholdfortherandomwalks,andthemassfractioninhaloes of the ionization maps were found to match the results of ra-
ofmassmisequatedtothe‘first-crossingdistribution’,i.e.,the diative transfer simulations to some degree of accuracy (Zahn
fraction of walks that first cross this threshold at Lagrangian et al. 2007, 2011; Mesinger et al. 2011). It is almost a decade
smoothingscaleR ∝m1/3 asthescaleisdecreasedfromlarge since the first simple analytical treatments were introduced,
L
values. Since the initial density field is expected to be close to which were useful in building intuition. Since there have been
Gaussian, this allows for simple approximate solutions for the technical improvements in modelling dark matter on the ana-
abundance and clustering of halos. lyticalsideasmentionedabove,webelieveitisworthrepeating
Furlanetto, Zaldarriaga, & Hernquist (2004, hereafter, the calculation to see whether the improvements transfer to
FZH04) first pointed out that, by treating collapsed dark mat- the bubble distributions too. We focus on analytical results
terhaloesasthehostsofthesourcesofreionization,essentially here and leave a detailed comparison with numerical models
the same counting problem applies to the growth of ionized to future work.
Hii bubbles during reionization as well, and can be used to The plan of the paper is as follows. In section 2 we de-
estimate the size distribution of these bubbles under some scribe the analytical excursion set formalism, starting with
simplifying assumptions. (We describe their model in detail
below.) Under the excursion set ansatz, similarly to the case
of haloes, the comoving number density dn/dlnR0 of ionized 5 Anothercomplicationisthattheexcursionsetansatzformally
bubbles of Lagrangian volume V =4πR3/3 is mapped to the considersrandomwalkscenteredatafixedlocationforanensemble
0 0
first-crossing distribution f of a specific threshold or ‘barrier’ ofinitialconditionsandthereforepotentiallymissestheeffectsof
by random walks in the smoothed initial density δ as the correlationsbetween walkscenteredatdifferentlocations.E.g.,the
0
numerical algorithm of Bond & Myers (1996) explicitly accounts
smoothing scale R is decreased from large values:
0
forthesecorrelations,andtherehavebeenattemptstoaccountfor
(cid:12) (cid:12)
V0dldnnR0 =S0f(S0) (cid:12)(cid:12)(cid:12)ddllnnRS00(cid:12)(cid:12)(cid:12) , (1) t&heBsearekffaencats2a0n0a2ly;tSichaeltlyhf2o0r1s1h)a.rpR-ekcefinltterwinogrk(s(eHe,aeh.ng.,&ScPaanrnaanpjiaepcoe
where S = (cid:10)δ2(cid:11) is the variance of density fluctuations on 2014)hasshown,however,thattheESPframeworkofParanjape
0 0 et al. (2013b) approximates the effects of these ‘correlated walks’
scale R .
0 remarkablywellwhencomparedwithnumericalresultssimilarto
One set of assumptions inherent in the FZH04 analysis thoseofBond&Myers(1996).SincetheESPapproachisasimpler
derives from the original excursion set formalism presented and more realistic solution of this problem, we will not explore
by Bond et al. (1991) and pertains to the nature of the ran- alternateformalismshere.
(cid:13)c 0000RAS,MNRAS000,000–000
Modelling HII bubbles during reionization 3
a recapitulation of the FZH04 model in section 2.1, followed anditscomovingEulerianradiusisR=(1+z)R .Wehave
phys
by a description of the ESP framework and the associated assumed δ >0 since we will not consider situations in which
0
modifications to the bubble size distribution calculation in underdensevoidscontainenoughsourcestobefullyionized.By
section2.2.Theresultsofthecomparisonbetweenthetwoare the time this occurs, the ionized regions are expected to have
described in Section 3. We perform two kinds of comparisons, percolated through most of the volume in the Universe, with
onewhenassumingthesameastrophysicalionizationefficiency reionization progressing very rapidly. It is not clear whether
for both models (section 3.1) and another where we fix the the simplifying assumptions of the excursion set calculation
globally averaged ionized volume fraction to be the same in remain valid in this regime. We will return to this issue in
both models (section 3.2). We conclude in section 4. future work where we compare our analytical models with
numerical calculations.
The mass fraction contained in all ionizing sources inside
2 FORMALISM the volume V is
(cid:90) M
In this section we describe the formalism we use to estimate f (M,V)≡ dmf(m|M,V)
coll
the size distribution of ionized bubbles. We start with a sum- mmin
mary of the approach presented by FZH04, and then describe (cid:90) smin
= dsf(s|δ ,S )=f (δ ,S ), (6)
the modifications introduced by considering peaks theory and 0 0 coll 0 0
random walks with correlated steps. S0
where s ≡ σ2(m ). The mass fraction ionized by these
min 0 min
sources is therefore ζf , which means that a region can be
coll
2.1 FZH04 calculation with sharp-k walks treated as completely ionized when
The basic assumptions that go into the FZH04 calculation are ζf (M,V)(cid:62)1. (7)
coll
that (i) any halo above a minimum mass m is an ionizing
min
sourceand(ii)suchahaloofmassmionizesamassm =ζm This,however,doesnotaccountfortheeffectof“extra”ionizing
ion
with ζ >1 an efficiency factor related to various astrophysical photons from neighbouring regions. The idea here is that if a
quantities. In the following we will frequently use the relation regionhasacollapsefractionsuchthatζfcoll (cid:29)1,thensomeof
betweenmassscalem,LagrangianradiusR andthevariance the ionizing photons can “leak” into neighbouring regions. To
L
σ2ofthesmoothedlinearlyextrapolateddensitycontrastgiven include this effect, FZH04 proposed that an ionized region be
0
by assignedavolumeV providedthatV isthelargest volumefor
whichthecondition(7)issatisfied,whichpreventsovercounting
(cid:90)
σ2(m)≡(cid:10)δ2 (cid:11)= dlnk ∆2(k)W2(kR ), (2) ofoverlapping/nestedionizedbubbles6.Thiscountingproblem
0 RL L
is therefore very similar to the one of counting Lagrangian
where ∆2(k)≡k3P(k)/2π2 is the dimensionless matter power regions that will eventually form halos, and can be treated in
spectrum linearly extrapolated to present epoch, W(kR ) is the same excursion set language. There are some differences
L
thesmoothingfilterinFourierspace,andtheLagrangianradius between the two counting problems, however, which we will
is related to the mass through discuss below.
The FZH04 calculation is based on the original treatment
m=(4π/3)ρ¯R3, (3)
L by Bond et al. (1991) which studies random walks in the
with ρ¯the present mean density of the Universe. overdensity δ smoothed using a filter that is sharp in k-space7.
Let f(m|M,V)dm be the mass fraction in haloes having Further assuming the mass-independent critical overdensity or
mass within m and m+dm embedded in a region of Eulerian “barrier” δ (z) predicted by spherical collapse, this leads to a
c
volume V which encloses a total mass M =(4π/3)ρ¯R03. Then, closed form expression for the mass fraction given by
denotingS0 =σ02(M)ands=σ02(m),wecanchangevariables (cid:90) smin
to write f (M,V)= dsf(δ (z)−δ ;s−S )
coll,sh−k c 0 0
S0
f(m|M,V)dm=f(s|δ0,S0)ds. (4) (cid:32) (cid:33)
δ (z)−δ
=erfc c 0 , (8)
The quantity f(s|δ ,S ) is typically calculated using a condi- (cid:112)
0 0 2(s −S )
min 0
tionalfirstcrossingdistributionofachosenbarrier,byrandom
walks in the density contrast that are constrained to pass where f(δc(z),s(m)) gives the uncondtional mass fraction in
through δ on scale S . halosofmassmatredshiftz,andthefirstequalityfollowsfrom
0 0
In writing equation (4), we have implicitly assumed that thefactthatsharp-k filteredwalkshaveuncorrelatedsteps,so
there is a one-to-one mapping between Lagrangian regions
of scale S that have a linear density contrast δ , and Eu-
0 0 6 Of course this is only approximate since real bubbles are not
lerian regions of (comoving) volume V = 4πR3/3 that en-
expectedtobespherical.Additionally,theassumptionofadetermin-
close mass M (and therefore have a nonlinear density contrast
isticvalueforζ isalsoknowntobeanoversimplification.Sinceour
δNL =M/(ρ¯V)−1).Thisistrue,e.g.,ifoneassumesspherical goalinthispaperisonlytocomparerelativedifferencesbetweentwo
evolution of the volume (Sheth 1998). In this case, assuming analyticalmodels,wewillretainthesesimplifyingassumptionsand
anEinstein-deSitterbackground(whichisanexcellentapprox- leaveamorerealisticcomparisonwithnumericalresultstofuture
imation at the redshifts of interest), the comoving Lagrangian work.
radiusR ofthesphereisrelatedtoitsphysicalEulerianradius 7 NotethatinusualapplicationsoftheBondetal.(1991)formalism
0
R at redshift z through (Gunn & Gott 1972) oneassumesasharp-k filterwhencalculatingthemassfractionbut
phys areal-spaceTopHatfilterwhencalculatingthevarianceσ2(m).The
0
Rphys = 3 (1−cosθ); 1 = 3·62/3 (θ−sinθ)2/3, sameassumptionisimpliedintheoriginalFZH04calculations.In
R 10δ 1+z 20δ ourlatercalculationwewillself-consistentlyusereal-spaceTopHat
0 0 0
(5) filteringforbothδ andσ2(m).
0
(cid:13)c 0000RAS,MNRAS000,000–000
4 Paranjape & Choudhury
that the conditional mass fraction is simply the unconditional Secondly, the traditional excursion set formalism is based on
one evaluated with a simple “shift of origin” (Lacey & Cole counting randomly placed regions, while halos preferentially
1993).Thisexactresultwillnolongerbetruewhenconsidering form near initial density peaks: this has been the guiding prin-
realistic filters below. ciple behind several works in the past (Bardeen et al. 1986;
An implicit assumption here is that at very large scales Appel & Jones 1990; Bond & Myers 1996; Manrique et al.
where S → 0 and δ → 0, one ends up with the condition 1998;Hanami2001)andhasrecentlybeenexplicitlyverifiedin
0 √ 0
erfc(cid:0)δ (z)/ 2s (cid:1)<ζ−1.Guaranteeingthisimpliesarestric- simulations(Ludlow&Porciani2011;Hahn&Paranjape2014).
c min
tion on allowed (ζ,m ) values. A similar restriction will also Ignoring the fact that the locations of collapse are special and
min
apply in our later calculation including the peaks constraint. notarbitrarilyplacedisknowntoleadtosystematicallywrong
Sincetheerrorfunctionisamonotonicfunctionofitsargument, predictionsforthefinalhalomassfunction(Shethetal.2001).
the condition (7) becomes an inequality on δ , Toimproveuponthiscalculation,wemustthereforeaccountfor
0
√ thesetwoeffects:bothfirstpassageproblems(forhalosaswell
δ (cid:62)δ (z)− 2K(ζ)(s −S )1/2 ≡B (S ;ζ,z), (9)
0 c min 0 FZH 0 as ionized regions) must now be solved with correlated steps
whereK(ζ)=erfc−1(ζ−1)=erf−1(1−ζ−1).Hencetheproblem due to a nontrivial filter W(kRL), and the conditional mass
oflocatingionizedbubblesisreducedtosolvingafirstpassage fractionfcoll mustbecalculatedwiththeadditionalconstraint
problem with the moving barrier B (S ;ζ,z). FZH04 did that the walks describing halos be centered on peaks8.
FZH 0
this by using the same sharp-k formalism applied to a linear Recentwork(Musso&Sheth2012,MS12)hasshownthat
approximation to this barrier, the effects of correlations arising from realistic filtering such
as the TopHat or Gaussian can be accurately described using
BFZHlin(S0;ζ,z)=B0+B1S0 (10) a simple approximation. Essentially, this involves recognizing
√ √ that the filter induces strong correlations between the steps of
where B = δ (z) − K(ζ) 2s and B = K(ζ)/ 2s .
0 c min 1 min the walks (Paranjape et al. 2012). This means that the first-
With this approximation, the solution for the first crossing
crossing condition that solves the cloud-in-cloud problem and,
distribution is given by (Sheth 1998)
in principle, corresponds to an infinite number of constraints,
fFZH(S0;ζ)= (cid:112)B0 e−B02/2S0e−B0B1e−B12S0/2. (11) can be replaced with the simpler up-crossing condition (Bond
2πS3 1989; Bond et al. 1991) which requires exactly two constraints
0
and can be solved analytically (see also Musso & Sheth 2013).
The Lagrangian bubble size distribution then follows from
This also allows the peaks constraint (Bardeen et al. 1986)
setting f(S )→f (S ;ζ) in equation (1).
0 FZH 0 to be included in a straightforward way (MS12; Paranjape &
A useful quantity is the fraction of Lagrangian volume
Sheth2012)andleadstoananalyticalprescription(Excursion
(i.e., the mass fraction) filled by the ionized bubbles; in this
SetPeaks;ESP)forthehalomassfunctionandhalobiaswhich
model this is given by
describes the results of N-body simulations to within ∼10%
(cid:90) ∞ (cid:18) dn (cid:19) (Paranjape et al. 2013b).
Q = dlnR V
0 0 0dlnR In Appendix A1 we describe these calculations in some
R0,min 0 detail. The conditional mass fraction in the ESP case works
1(cid:20) (cid:18)ν2 −b(cid:19) (cid:18)ν2 +b(cid:19)(cid:21)
= e−2berfc √0,min +erfc √0,min , (12) out to
2 2 ν0,min 2 ν0,min (cid:90) smin
f (M,V)= dsf (s|δ ,S ). (13)
where R0,min = (3ζmmin/4πρ¯)1/3 is the smallest possible coll S0 ESP 0 0
Lagrangian bubble size, and in the second line we defined
where f (s|δ ,S ) is given by equation (A7). The condition
ESP 0 0
b=B B and ν =B /σ (ζm ). Notice that, contrary
0 1 0,min 0 0 min for identifying ionized regions (7) can be now generalized to a
to what would be expected based on photon number con-
condition
servation arguments (Furlanetto et al. 2004), in general this
√
expression for Q0 is not equal to ζerfc(cid:0)δc(z)/ 2smin(cid:1). This is δ0 (cid:62)BHII(S0;ζ,z). (14)
a generic feature of excursion set calculations and is shared by
For each S , the barrier B (S ;ζ,z) is the value of δ where
the peaks-based approach we describe below. Although this 0 HII 0 0
ζf (δ ,S )=1, which must be solved for numerically since
issueofnon-conservationofphotonnumberhasbeendiscussed coll 0 0
there is no closed form expression for the conditional mass
in the past (e.g., Zahn et al. 2007), we believe it deserves a
fraction f (s|δ ,S ). This is the most time-consuming part
more thorough investigation which we leave to future work. ESP 0 0
of the calculation.
To summarize, adopting the traditional sharp-k excursion
Knowing the barrier B (S ;ζ,z), the first passage prob-
setanalysishasthefollowingeffects.Firstly,thisfixesthecon- HII 0
lem (14) can be solved using the MS12 approximation:
ditional mass fraction f as a complementary error function,
coll
and the form of its argument in terms of δ0 and S0. Next, this f (S ;ζ)=(cid:90) ∞ dv(v−B(cid:48) )p(B ,v;S ), (15)
fixes the shape of the barrier B (S ;ζ). And finally, it fixes HII 0 HII HII 0
FZH 0 B(cid:48)
thesolutiontothefirstpassageprobleminthepresenceofthis HII
barrier. where v ≡ dδ0/dS0, BH(cid:48) II ≡ dBHII/dS0 and p(BHII,v;S0) is
a bivariate Gaussian with a covariance matrix fixed by the
correlation between the density contrast δ and its derivative
0
2.2 Accounting for peaks and correlated steps v at scale S . This expression can be written in closed form
0
(equation 5 of MS12) and, in the present case, is valid for
The original excursion set formulation of Bond et al. (1991)
0<S <σ2(ζm ).
has two shortcomings. Firstly, it uses sharp-k filtering when 0 0 min
studyingrandomwalksinδ,whichleadstosometechnicalsim-
plifications(thewalkshaveuncorrelatedratherthancorrelated 8 Theionizedregionsneedn’tbecenteredondensitypeaks,sothe
steps)butislessrealisticthanusingreal-spaceTopHatfiltering. peaksconstraintwillnotenterthere.
(cid:13)c 0000RAS,MNRAS000,000–000
Modelling HII bubbles during reionization 5
TheapproximationsinherentintheMS12(andESP)ana-
lytical formulae will break down if the value of δ approaches Name Ωm σ8 ns h Ωb
0
that of the halo barrier (see Figure 3 of Musso et al. 2012). Planck13 0.315 0.829 0.960 0.673 0.0487
This will never happen if B (S ;ζ,z)<B (S ;z) which WMAP1 0.30 0.90 1.00 0.70 0.046
HII 0 halo 0
is the case here. WMAP3 0.25 0.80 1.00 0.70 0.040
Table 1.Cosmologicalmodelsandparametersusedinthepaper.
2.3 From Lagrangian to Eulerian coordinates
So far we have not explicitly discussed how to convert the other cosmologies, one we refer to as WMAP1 (Spergel et al.
Lagrangianresult(15)intothedistributionofEulerianbubble 2003) and the other as WMAP3 (Spergel et al. 2007). One
sizes implied by our notation fcoll(M,V). Making the relation can see from the Table that the primary difference between
explicit requires an extra step, namely, we assume the regions Planck13 and WMAP1 is in the value of σ while WMAP3
8
tobesphericalandthenuseequation(5)replacingδ0 withthe and Planck13 differ mainly in the values of Ωm and Ωb. (The
value of the barrier BHII(S0;ζ,z). WMAP1 parameter values are the same as those used by
The Lagrangian calculation gives us a comoving number FZH04.) We use Camb (Lewis, Challinor, & Lasenby 2000)9
density dn/dlnR0 of bubbles of size R0 or volume V0 given to generate all transfer functions. The value of mmin, which
by equation (1) with f(S0)→fHII(S0;ζ). The Lagrangian to appearsinexpressionsforthecollapsedfraction,ischosensuch
Eulerian mapping preserves the comoving number density of that only atomically cooled haloes are allowed to form stars
the bubbles, since the comoving volume of the Universe (or (we set the associated virial temperature to T =104K.).
vir
simulation box) remains constant and there is a one-to-one Now, assuming we fix the cosmology to one of the above,
mapping between the Lagrangian and Eulerian radius of each it is possible to compare bubble size distributions at a given z
bubble. So we have in two different ways: (i) we can fix the value of the efficiency
V ddlnnR = VV0 (cid:18)ddllnnRR0(cid:19)−1S0fHII(S0;ζ) (cid:12)(cid:12)(cid:12)(cid:12)ddllnnRS00(cid:12)(cid:12)(cid:12)(cid:12) . (16) pthaeragmiveetnerreζdswhhifitc,howri(lili)gwiveecraisnentoormdiaffleizreentthevacluurevsesoftoQthaet
same value of Q by using a different value of ζ and then
The Jacobian dlnS0/dlnR0 is standard, and the ratio V/V0 compare. The first approach is conceptually straightforward
follows from equation (5). All we need then is the Jacobian as it captures the difference in size distributions when the
dlnR/dlnR0 which is given by astrophysical properties of the sources remain the same. The
(cid:12) (cid:12) (cid:18) (cid:19) advantageofthesecondmethodisthatitnormalizesallmodels
dlnR =1−(cid:12)(cid:12)dlnS0(cid:12)(cid:12)dlnBHII 1− 3θ(θ−sinθ) . (17) to the same value of the global ionization fraction, and hence
dlnR0 (cid:12)dlnR0(cid:12) dlnS0 2(1−cosθ)2
the differences would be only in the topology of the ionized
Converting from Lagrangian to Eulerian coordinates can po- regions.Wewillpresentresultsusingboththeapproaches,first
tentially lead to quantitative differences in the calculations at when ζ is kept constant, and next with Q kept fixed.
the end since bubbles are defined by overdense regions and
hence their expansion could be different from the background.
As with the FZH04 calculation, the Lagrangian vol- 3.1 Models with the same ζ
ume fraction in ionised bubbles is given by Q =
(cid:82)∞ dlnR (V dn/dlnR ) with R =(3ζm /4π0ρ¯)1/3; In this section, we compare the predictions of different models
R0,min 0 0 0 0,min min forthesamevalueofζ,whichwillbeconstantatζ =40inour
this needs a numerical calculation in the ESP case. The corre-
fiducial model. We first compare the barriers, ionized volume
sponding Eulerian volume fraction is
fractions and bubble size distributions predicted by the ESP
(cid:90) ∞ (cid:18) dn (cid:19) and FZH04 models, and then study the effect of varying the
Q= dlnR V
dlnR cosmological parameters and form of ζ within the ESP model.
Rmin
(cid:90) ∞ (cid:18) dn (cid:19)(cid:18)V (cid:19)
= dlnR V (18)
R0,min 0 0dlnR0 V0 3.1.1 Comparison of barriers
where Rmin is the Eulerian size corresponding to R0,min. Be- Figure 1 shows the behaviour of the barrier BHII for differ-
low we will study the normalized bubble size distributions ent cases. As we will see later, many of the results can be
Q−01V0dn/dlnR0 and Q−1Vdn/dlnR. understood from the heights and shapes of these barriers. We
show the Hii barrier for the FZH04 (dashed black) and ESP
(fine-dashed red) models for two different redshifts z=12,16
3 RESULTS forζ =40.Physically,ionizedbubblesformhierarchicallywith
smallerbubblesformingfirstandthengrowingandmergingto
We present the main results of our calculation in this section.
formlargerbubbles.Theexcursionsetapproachverynaturally
WewillconsiderGaussianinitialconditionsandaflatLambda-
accounts for this feature, which is very similar (and, in fact,
cold dark matter (LCDM) cosmology throughout, with three
related)tothehierarchicalgrowthofdarkmatterhalos(Bond
different sets of parameters which are summarized in Table 1.
et al. 1991; Lacey & Cole 1993). The barriers for both the
TheparametersΩ andΩ arethepresenttotalandbaryonic
m b models are higher at larger redshifts implying that relatively
matter fractions, respectively, the Hubble constant is H =
0 fewerrandomwalkswouldcrossthethreshold,withthetypical
100hkm/s/Mpc, σ is the r.m.s. of linear density fluctuations
8 first crossing occurring at larger S or smaller bubble size;
0
smoothed in spheres of size 8h−1Mpc, and n is the scalar
s in other words, fewer bubbles with smaller sizes will form at
spectral index. Our fiducial cosmology, which we refer to as
Planck13,iscompatiblewiththerecentresultsfromthePlanck
Collaboration et al. (2013). Additionally, we will consider two 9 http://lambda.gsfc.nasa.gov/toolbox/tb camb form.cfm
(cid:13)c 0000RAS,MNRAS000,000–000
6 Paranjape & Choudhury
Figure 1. The ionization barrier for the FZH04 (dashed black) Figure 2. The mass fraction sfhalo(s) in halos of mass m where
and ESP (fine-dashed red) cases for two representative redshifts s = σ2(m), as a function of δ2(z)/s at different redshifts. The
0 c
z = 12,16. The cosmology chosen is Planck13 and the value of originalexcursionsetpredictionofBondetal.(1991,dottedblack,
ζ =40. Also shown are the linear approximations to the barriers markedBCEK91)isuniversalandthereforeidenticalatallredshifts.
(dottedblackforFZH04andsolidredforESP).Seetextfordetails. TheESPpredictionofParanjapeetal.(2013b,solid,fromtopto
bottom z = 0,6,9,12) is non-universal and predicts lower mass
fractionsneartheminimummassathigherredshifts.Thisexplains
thedifferenceinoverallbarrierheightsoftheFZH04andESPmodels
earlier times. Other than sharing this generic feature related
asdiscussedinthetext.
to hierarchical growth, the barriers for the ESP and FZH04
models are qualitatively quite different from each other. We
first discuss the nature and consequences of these differences
Section2.Sincethemassfunctionissteeplyfalling,thecollapse
and then provide a physical justification of their origin.
fraction is dominated by halo masses close to the minimum
First, at fixed redshift, we see that the ESP barrier is
mass m at any redshift. Figure 2 shows the unconditional
higherthanthecorrespondingFZH04one.Asdiscussedabove, min
halo mass fractions sf (s) as a function of ν2 ≡δ2(z)/s at
thisimpliesfewerbubbleswithsmallersizesintheESPmodel halo c
different redshifts as predicted in the original excursion set
as compared to FZH04. One can thus expect that the total
calculation of Bond et al. (1991, dotted black line) and in
fractionalvolumecontainedwithinionizedregionsQwouldbe
ESP (solid lines). The prediction for the former is universal
smaller in the ESP case as compared to FZH04. √
IftheonlydifferencebetweentheESPandFZH04barriers (sfhalo(s)=νe−ν2/2/ 2π) and therefore independent of red-
were a constant vertical shift, then one would also expect that shift in this plot, while in the latter case (equation A3 using
thatthebubblesizedistributionshowsasmallercharacteristic sf(s) = νf(ν)/2) the mass fraction decreases at higher red-
radius in ESP as compared to FZH04. However, we see from shifts (the curves from top to bottom show z=0,6,9,12). All
Figure 1 that the slope of the ESP barrier at each redshift is thez>0ESPcurvesstartat10 ν =νmin =δc(z)/σ0(mmin(z)).
also larger than the corresponding FZH04 one. An increase in The difference in the height of the barriers at a given redshift
barrier slope tends to decrease the typical first crossing S by isthereforelargelydrivenbythefactthatathighredshiftsthe
0
makingitmoredifficultforwalkstocrossatlargeS ,andhence ESP calculation predicts fewer collapsed objects near the min-
0
increases the characteristic bubble size. This is most easily imum mass than the original excursion set approach, making
seen in equation (11) for the case of walks with uncorrelated it more difficult to ionize the Universe.
stepsinthepresenceofalinearlyincreasingbarrier:increasing The non-universality in the ESP halo mass fraction arises
the slope B1 shifts the exponential cutoff due to the term fromthemassdependenceofthequantitiesV∗andγ thatenter
e−B12S0/2 tosmallervaluesofS0 andhencelargerbubblesizes. thecalculation(seeAppendixA1).Thiseffectwasdiscussedby
(It also decreases the normalisation due to the term e−B0B1, Paranjapeetal.(2013b)andwasshowntoagreewellwiththe
non-universality in low redshift halo mass functions calibrated
but this can be countered by an appropriate decrease of B .)
0
by Tinker et al. (2008). The ESP mass function has not yet
A qualitatively similar change in characteristic size occurs for
been tested at high redshifts; in future work we will perform
walkswithcorrelatedsteps(seeequation15).Thiseffectwould
such a test to assess the level of accuracy of this predicted
therefore compete with the one due to the relative difference
non-universalityduringtheepochofreionization.Insection3.2
in barrier height in determining the shape of the bubble size
below we will study the case when we normalise both models
distribution.
The reason for the difference in overall height of the ESP
barriers as compared to FZH04 can be understood as follows.
Recall that the collapse fraction fcoll(M,V) is an integral over 10 Most of the redshift dependence of νmin arises from δc(z) ∝
themassfractionf(m|M,V)inhalosofmassmasdiscussedin D(0)/D(z)whereD(z)isthelineartheorygrowthfactor.
(cid:13)c 0000RAS,MNRAS000,000–000
Modelling HII bubbles during reionization 7
to have the same global ionized volume fraction, in which case
the overall height of the barriers becomes irrelevant.
The reason for the difference in slopes of the ESP and
FZH04 barriers, on the other hand, is related to the peaks
constraint in ESP which is absent in the FZH04 model. To
understand why, let us make some drastic approximations
which still retain the qualitative features of the problem. (For
technical details of the full calculation see Appendix A.) To
startwith,sinceweareinterestedinafixedvalueofthecollapse
fraction (equal to 1/ζ), let us assume that we only need to
study the single scale s = s(m ) where the characteristic
∗ ∗
mass m , as discussed above, depends on redshift and the
∗
values of δ and S . The overall relative behaviour of the two
0 0
models then follows if we understand the behaviour in each
model of the typical walk in δ as the smoothing scale S
0 0
changes, such that δ approaches the halo barrier for S →s
0 0 ∗
and δ →0 for S →0. (For the purposes of this qualitative
0 0
argument we can safely assume that the halo barrier is simply
δ (z).)Thistypicalwalkiscloselyrelatedtothedensity profile
c
surrounding the locations of patches in the initial density that
are destined to form halos. (To see this, note that the profile
gives the typical density as a function of distance from the
chosen location, while the walk gives the density integrated Figure3.TheevolutionofthevolumefractionQinionizedregions
in concentric spheres centered at this location.) Finally, the fordifferentmodelsusedinthepaper.Seetextfordetails.
key point is that density profiles around peaks of δ are steeper
than those around randomly chosen locations. This is easy to
understand on physical grounds, since in order for (cid:126)x to be a 3.1.2 Comparison of Q(z)
peak, the density in all directions around(cid:126)x must decrease by
definition, while this is not required for an arbitrary point (for WenextshowinFigure3theevolutionofQasafunctionofz
a detailed calculation, see equations 7.8-7.11 and Figure 8 of fordifferentbarriers.Unlessstatedotherwise,allcurvesinthis
Bardeenetal.1986).Consequently,thetypicalwalkofinterest plotusethePlanck13cosmologyandζ =40.Thedottedblack
to us (which is a proxy for the Hii barrier in this case) is a line shows the Lagrangian ionized fraction from the original
steeper function of S when constrained to be centered on FZH04 calculation (equation 12). If we use the same linear
0
peaks as compared to random locations11. barrier BFZHlin (equation 10) as FZH04 but the MS12 approx-
Figure 1 also shows linear approximations to the barriers imation with correlated steps for the bubble distribution, i.e.,
for the two models. For the FZH04 case, the linear fit (dotted use the barrier (10) in equation (15), we obtain a Lagrangian
black) is the one in equation (10), which is the same as used Q(z) as shown by the dashed black curve (marked “FZH04 +
byFZH04andessentiallycorrespondstoastraightlinewitha corr. steps”). The effect of correlated steps reduces the value
slopeequaltotheslopeofthebarrieratS =0.Itisclearthat of Q by about a factor 2 at z(cid:38)12. The solid blue line shows
0
the fit is quite good for small S and is reasonable for higher the Lagrangian prediction for Q(z) in the ESP case. Clearly,
0
valuesofS .Ontheotherhand,thelinearapproximationused the effect of including ESP is much more drastic than that of
0
for the ESP case (solid red) is actually a least-squares fit to accounting for correlated steps alone as it reduces the value of
the full calculation. As one can see, the fit is excellent for a Q by nearly a factor of 10 at z =16 and about a factor 5 at
wide range of S values. We have checked that qualitatively z=12 compared to the FZH04 case. This was anticipated in
0
similar results are obtained when using different values of the our discussion of the barrier shapes above.
cosmological parameters or ionization efficiency. Hence from AllthethreecasesdiscussedsofarareforLagrangianbub-
now on, we will work with the linear fits for the both FZH04 blesizes.WhenweconverttoEuleriansizesintheESPmodel,
and ESP cases. thevalueofQ(long-dashedbluecurve)decreasesfurther.This
isaconsequenceofthefactthatbubblesformpreferentiallyin
mildly overdense regions and hence their evolved sizes would
be smaller than the corresponding Lagrangian ones. While the
differenceswiththeoriginalFZH04arenowconsiderablylarger
11 Essentiallythesamereasoningexplainswhytheso-calledpeak- at all redshifts, the difference with respect to the Lagrangian
background split halo bias in the ESP model is higher at large ESP calculation is not that dramatic (which was also antici-
masses than the corresponding quantity associated with, e.g. the pated by FZH04). Overall, then, the ESP Eulerian model at
Sheth & Tormen (1999) or Tinker et al. (2008) mass functions, fixed ζ predicts a substantially different reionization history
a point that was discussed by Paranjape et al. (2013b). These than the FZH04 model.
arguments rest on the excursion set ansatz which says that the
For later comparison, we also show Q(z) for the ESP Eu-
number of halos in a given region at late times can be predicted
lerian case for the WMAP1 (dot-dashed red) and WMAP3
bystudyingtheenvironmentsofindividualpoints(here,peaks)in
(fine-dashed yellow) cosmologies. The WMAP1 curve is sub-
theinitialconditions.Inthepresentcaseoneisfurtheridentifying
someoftheseenvironments(asdeterminedbytheHiibarrier)as stantiallyhigherthanthecorrespondingPlanck13curve,which
beingionizedbubbles,whichexplainswhythenumberofbubblesis is a consequence of the higher value of σ8 in WMAP1. The
sensitivetothepeaksconstrainteventhoughthebubblesthemselves fact that the Planck13 and WMAP3 curves are very close to
arenotnecessarilycenteredonpeaks. each other means that changing the values of Ωm and Ωb has
(cid:13)c 0000RAS,MNRAS000,000–000
8 Paranjape & Choudhury
cosmologicalparametersonthebubbledistribution.Wealready
saw the effect on Q(z) in Figure 3. The left panel of Figure 5
shows the normalised Eulerian bubble distributions for the
Planck13 (solid red), WMAP1 (dashed black) and WMAP3
(dot-dashed blue) cosmologies, at z =12,16. These Eulerian
distributions are very similar to the corresponding Lagrangian
ones apart from a small shift towards smaller radii in the
former (e.g., compare the solid red curves in the left panel of
Figure5withthesolidredcurvesinFigure4).Thedifferences
are non-negligible mainly at high redshift; this is because at
high-z the peak of the distribution is close to or smaller than
R , so that a small shift in sizes leads to relatively large
0,min
shiftsinQandtheshapeofthedistribution.Atlowerredshifts
thecharacteristicbubblesizeissignificantlylargerthanR
0,min
andsmallshiftshaveamuchsmallerimpact.Thisalsoexplains
the relative difference between the solid and fine-dashed red
curves of Figure 3.
Recall that the main difference between Planck13 and
WMAP1isinthevalueofσ whilethemaindifferencebetween
8
Planck13 and WMAP3 is in the value of Ω and Ω . As with
m b
Q(z) in Figure 3, the bubble distributions for Planck13 and
WMAP3 are almost identical, indicating that the effect of
Figure 4.NormalisedLagrangiansizedistributionsofionizedbub- varying the density parameters Ω and Ω is negligible. On
m b
blesforthreecases:originalFZH04calculation(dottedblack),FZH04 the other hand, the effect of varying σ is quite drastic as can
8
linear barrier with the MS12 approximation for correlated steps be seen by comparing the curves for Planck13 and WMAP1.
(dashed blue) and the ESP prediction (solid red). The curves are Increasing σ leads to higher Q and larger bubble sizes.
8
shown for two different redshifts with ζ = 40 for the Planck13
Inourfiducialbasemodel,wehavetakenζ tobeindepen-
cosmology.
dent of the halo mass m. The parameter ζ is a combination of
various astrophysical quantities like the initial mass function
a much smaller effect on the reionization history, a feature we of stars within galaxies, the stellar spectra, the fraction of
will see again when discussing the bubble size distributions baryonic mass converted to stars and the escape fraction. The
below. values and evolutionary properties of most of these quantities
are unknown. Hence it is natural to investigate when ζ is al-
lowedtobeafunctionofm.Thisisstraightforwardinpractice,
3.1.3 Comparison of bubble size distributions since the only change required in the formalism is that the
condition (7) which sets the barrier upon using equation (13)
Our next goal is to see the effect of correlated steps and ESP
must be replaced with the condition
on the bubble size distributions. We plot the normalized La-
grangian size distribution Q−1Vdn/dlnR for different cases (cid:90) smin
dsf (s|δ ,S )ζ(m(s))(cid:62)1.
in Figure 4 for two redshifts z = 12,16. As before, we com- ESP 0 0
S0
pare results from the original FZH04 calculation (dotted black
Since the integral over the conditional ESP mass fraction
curves),theFZH04linearbarrierwithcorrelatedsteps(dashed
was numerical to begin with, this introduces no additional
blue curves), and the Lagrangian ESP prediction (solid red
complication in our calculations.
curves). All curves use the Planck13 cosmology with ζ =40.
We choose ζ to have a mass-dependent form ζ(m) =
Onecanseethattheeffectofcorrelatedstepsisnoticeable
40(cid:0)m/108.1h−1M (cid:1)α.Wechoosetwovaluesofα=0(fiducial
at relatively lower redshifts and almost neligible at z=16. At (cid:12)
case) and 2/3. The case α=2/3 is motivated by observations
z=12, we see that the effect is to decrease the location of the
of low-mass galaxies in the nearby universe (Kauffmann et al.
maximum and lengthen the tail at lower radii. The effect of
2003) which suggest that the efficiency of the conversion of
including ESP in the formalism is more prominent. At z=12
baryons into stars in galaxies with masses < 3×1010M
itshiftsthedistributiontolowerbubblesizescomparedtothe (cid:12)
increases with halo mass, perhaps as a result of supernova
originalFZH04,whilethisisreversedatz=16.Thediscussion
feedback processes.. The comparison between the two cases
in section 3.1.1 shows that the overall effect of ESP is the
is shown in the right hand panel of Figure 5 (solid red for
combination of two competing effects, namely a decrease in
α=0anddot-dashedblueforα=2/3;thesolidredcurvesfor
typical size due to the larger overall height of the ESP barrier
each z in the two panels of the Figure are identical). We can
andanincreaseduetoitssteeperslopeascomparedtoFZH04.
see that when we take the large mass haloes to be relatively
Combined with the large relative difference in the predictions
more efficient than the small ones, the peak of the bubble
forQ(z)seeninFigure3,thisshowsthatthenumberofbubbles
size distribution shifts to larger radii. This is qualitatively
dn/dlnR of a given radius R would be very different for the
consistent with the results of Furlanetto et al. (2006a).
two models.
3.1.4 Effect of cosmology and form of ζ 3.2 Models with the same Q
Having understood the main effects of including ESP in the So far we have compared different models keeping the value
formalism, we now proceed to see the effect of varying the of ζ fixed. We found that the improvements in the excursion
(cid:13)c 0000RAS,MNRAS000,000–000
Modelling HII bubbles during reionization 9
Figure 5. Left panel:NormalisedEulerianbubblesizedistributionattwodifferentredshiftswithζ=40forthreecosmologies;Planck13
(solidred),WMAP1(dashedblack)andWMAP3(dot-dashedblue).Right panel:ThesameforthePlanck13cosmologybutforζ(m)∝mα
fortwochoicesα=0(solidred)andα=2/3(dot-dashedblue).Notethatweusethesameefficiencyforbothmodels.Thesolidredcurvesat
eachredshiftareidenticalacrossthetwopanels.
setformalismcouldsignificantlyaffecttheheightandshapeof corresponding FZH04 ones (compare Figure 1). As discussed
the ionization barrier, and consequently the global ionization earlier, this relative steepness is the physical consequence of
fractionQaswellasthebubblesizedistribution.However,one the peaks constraint in ESP, which is absent in the FZH04
can argue that the value of ζ is highly uncertain, and hence modelandisagenuinedifferencebetweenthetwomodelsthat
any change arising from the mass functions in two models is evidently not degenerate with the physics of reionization
can possibly be compensated by changing the value of ζ. In as captured by a constant value of ζ. (Of course, it might be
that case, one may never be able to distinguish between the possible to compensate for the nature of the peaks constraint
two models observationally (unless one has an independent by allowing a different mass dependence of ζ in the FZH04
constraint on ζ, either theoretically or from observations). For model compared to ESP; we do not explore this here.)
example,Furlanettoetal.(2006a)arguedthatthisisindeedthe Aspointedoutinsection3.1.1,asteeperbarrierwillfavour
case when one modifies the original FZH04 calculation to one larger bubble sizes, and we see from the right hand panel of
using the Sheth & Tormen (2002) conditional mass function, Figure 6 that this is indeed the case. This panel shows the
particularly for larger values of Q. The main reason for this is Lagrangian bubble size distributions for the linear barriers
that, although the collapsed mass fraction is different for the in the left panel, with the solid red and dotted black curves
twomassfunctions,thebubblesizedistributiondependsonthe showing the results for ESP and FZH04, respectively. We find
nature of the scale and density dependence of the conditional thatthebubblesizesaregenerallymuchlargerintheESPcase.
mass functions, which is quite similar for the cases analysed The differences between the two models are relatively less for
by Furlanetto et al. (2006a). larger Q; nevertheless, even at Q=0.75 the difference in the
In order to see if this is also the case when comparing value of R at the maximum of the distribution is more than a
ESP and FZH04, we compare these two models at a given factor of two. The FZH04 model predicts almost no bubbles
redshift z =8 for the Planck13 cosmology keeping the value with R(cid:38)40h−1 Mpc for Q=0.75, while the ESP calculation
of Q fixed. In practice, we do this by adjusting the value of shows bubbles as large as 60h−1 Mpc. This can potentially
ζ (we only consider constant ζ in this exercise) separately for have a significant impact on the Hi fluctuation signal at these
both models, until each model leads to the desired value of Q. scales.
We choose two values Q=0.25,0.75 and only display results Onewaytoverifythesedifferencesbetweenthetwomodels
for the Lagrangian calculation; the Eulerian calculation would would be to compare with semi-numeric and radiative transfer
lead to qualitatively similar conclusions. simulations. In fact, there is now substantial evidence showing
The ionization barriers are shown in the left hand panel that the semi-numerical models agree with the radiative trans-
ofFigure6;theformatofthispanelisthesameasinFigure1. fersimulationsquitewell(e.g.,Zahnetal.2007,2011;Mesinger
There are marked differences between the barriers for the etal.2011).Inaddition,Mesinger&Furlanetto(2007)showed
two cases even when we normalize the models to the same that the analytical calculations of FZH04 had a reasonable
Q. Although the differences are smaller for larger Q, which is match with their semi-numerical model, which gave rise to
consistent with the results of Furlanetto et al. (2006a), they the conventional wisdom that the analytical results of FZH04
are quite substantial even for Q=0.75. We see that while the agreewithsimulations.However,acarefullookrevealsthatthe
normalisationtothesameQasFZH04hasloweredtheoverall semi-numericalmodelofMesinger&Furlanetto(2007)isquite
heightoftheESPbarriers,theycontinuetobesteeper thanthe different from the semi-numerical models which were used for
(cid:13)c 0000RAS,MNRAS000,000–000
10 Paranjape & Choudhury
Figure 6.ComparisonoftheFZH04andESPcalculationsatredshiftz=8forthePlanck13cosmologyatthesamevalueofglobalionized
LagrangianvolumefractionQ.WeshowcomparisonsattwofixedvaluesQ=0.25,0.75.NotethatthescalesoftheaxesinthisFigureare
differentfromthepreviousFigures.Left panel:Ionizationbarriers.TheformatofthispanelisidenticaltoFigure1.TheESPbarriersare
systematicallysteeperthantheFZH04ones.Rightpanel:NormalisedLagrangianbubblesizedistributionscorrespondingtothelinearbarriers
oftheleftpanel.TheESPcalculation(solidred)predictssubstantiallylargerbubblesizesthanFZH04(dottedblack),evenatQ=0.75.
comparing with radiative transfer simulations. For example, form preferentially around density peaks. In particular, we
Zahnetal.(2007),whileidentifyingionizedcellsusingspheres haveusedtheExcursionSetPeaks(ESP)frameworkdescribed
ofvaryingsize,flagonlythecentralcellinasphericalregionas by Paranjape et al. (2013b). Our calculation of the bubble
ionized, whereas Mesinger & Furlanetto (2007) flag the whole sizedistributionfollowspreviousworkpresentedbyFurlanetto
sphere.The21cmFASTcodepresentedinMesingeretal.(2011) et al. (2004, FZH04) with the modifications mentioned above.
tooisdifferentforanumberofreasons(e.g.,theseauthorsuse We find that these modifications have a significant impact on
asharp-k filterforidentifyingionizedregionsandflagonlythe the size distribution of bubbles which we summarize here.
central cell as ionized, while Mesinger & Furlanetto 2007, use
a real-space spherical TopHat filter and flag the full spherical • Includingtheeffectofcorrelatedstepsintherandomwalks
region).Toourknowledge,theonlyworkwhichcomescloseto systematically lowers the ionized volume fraction (Figure 3)
comparingthesemi-numericalmodelofMesinger&Furlanetto as well as the characteristic bubble size, and enhances the low
(2007)withradiativetransfersimulationsisZahnetal.(2011). radius tail of the size distribution (Figure 4).
However, the density and halo fields in Zahn et al. (2011) are • The peaks constraint alters the conditional halo mass
obtained from N-body simulations, while they were derived function and leads to steeper excursion set barriers in our
using perturbation theory and the old excursion set formalism calculation as compared to FZH04 (Figure 1 and left panel of
in Mesinger & Furlanetto (2007). Hence, there does not exist Figure 6). This is true regardless of whether we compare the
any detailed comparison between the original semi-numeric models using the same astrophysics of reionization (fixed ζ) or
model of Mesinger & Furlanetto (2007) and numerical simu- keeping the ionized volume fraction the same (fixed Q). (See
lations, indirectly implying that the analytical calculations of section3.1.1foradetaileddiscussionoftherelativedifferences
FZH04 have never been carefully compared with the radiative in barrier shapes.)
transfersimulations.Wewillmakedetailedcomparisonsofour
• At fixed ζ, the barrier heights in ESP are systematically
analytical model with simulations in future work.
larger than in FZH04 (Figure 1), leading to a significantly
smaller ionizedvolumefraction(Figure3).Therelativetypical
bubble sizes are determined by the competition between the
relative heights and slopes of the barriers, and show a more
4 CONCLUSIONS
complex behaviour (Figure 4).
Inthispaper,wehaveuseanimprovedcalculationofthemass • At fixed Q, however, the primary difference between ESP
functionofcollapsedhaloesandappliedittocalculatethesize andFZH04isinthesteepnessofthebarriersmentionedabove,
distributionofionizedbubblesduringtheepochofreionization. and the ESP bubble size distribution in this case peaks at
Our mass function calculation incorporates recent develop- substantially larger sizes than the FZH04 one (Figure 6). Al-
ments in the excursion set approach (Musso & Sheth 2012; though the relative difference decreases with increasing values
Paranjape & Sheth 2012; Paranjape et al. 2013b) which allow of Q, we see differences in characteristic radius of a factor of 2
ustoimplementtheeffectofcorrelationsinthestepsintheran- even at values as large as Q=0.75. This is primarily due to
dom walks of the smoothed density induced by the smoothing thenatureofthepeaksconstraintintheconditionalhalomass
filter and, more importantly, account for the fact that haloes function which has been absent in all previous calculations,
(cid:13)c 0000RAS,MNRAS000,000–000