Nonlinear clustering during the cosmic Dark Ages and its effect on the 21-cm background from minihalos

10 Pages·0.47 MB·English

by  Ilian T. Iliev

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed2February2008 (MNLATEXstylefilev2.2)
Nonlinear clustering during the cosmic Dark Ages and its
effect on the 21-cm background from minihalos
Ilian T. Iliev,1 Evan Scannapieco,1 Hugo Martel,2 and Paul R. Shapiro2
1 Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy
2 Department of Astronomy, University of Texas, Austin, TX 78712-1083
3
0
0
2 2February2008

n
a
J ABSTRACT

8
Hydrogen atoms inside virialized minihalos (with Tvir 6 104K) generate a radia-
2
tionbackgroundfromredshifted21-cmlineemissionwhoseangularfluctuationsreflect
v
clustering during before and during reionization. We have shown elsewhere that this
6
1 emission may be detectable with the planned Low Frequency Array (LOFAR) and
2 Square Kilometer Array (SKA) in a flat Cold Dark Matter Universe with a cosmo-
9 logicalconstant (ΛCDM). This is a direct probe of structure during the “Dark Ages”
0 at redshifts z &6 and down to smaller scales than have previously been constrained.
2
In our original calculation, we used a standard approximation known as the “linear
0
/ bias”[e.g. Mo & White (1996)].Here we improve upon that treatmentby considering
h
the effect of nonlinear clustering. To accomplish this, we develop a new analytical
p
methodforcalculatingthenonlinearEulerianbiasofhalos,whichshouldbeusefulfor
-
o otherapplicationsaswell.Predictionsofthismethodarecomparedwiththeresultsof
r ΛCDMN-bodysimulations,showingsignificantlybetteragreementthanthestandard
t
s linearbiasapproximation.Whenappliedtothe21-cmbackgroundfromminihalos,our
a
: formalismpredicts fluctuations that differ from our originalpredictions by up to 30%
v at low frequencies (high-z)and small scales.However,within the range of frequencies
i
X and angular scales at which the signal could be observable by LOFAR and SKA as
r currentlyplanned, the differences are smalland our originalpredictions proverobust.
a
Our results indicate that while a smaller frequency bandwidth of observationleads to
ahighersignalthatismoresensitivetononlineareffects,thiseffectiscounteractedby
theloweredsensitivityoftheradioarrays.Wecalculatethebestfrequencybandwidth
forthese observationsto be ∆νobs ∼2MHz.Finally we combineoursimulations with
ourpreviouscalculationsofthe21-cmemissionfromindividualminihalostoconstruct
illustrative radio maps at z =9.
Key words: cosmology:theory—diffuseradiation—intergalacticmedium—large-
scale structure of universe — galaxies: formation — radio lines: galaxies
1 INTRODUCTION
Despite striking progress over the past decade, cosmologists have still failed to see into the “Dark Ages” of cosmic time.
Although the details of this epoch between recombination at redshift z ∼ 103 and reionization at z & 6 are crucial for
understanding issues ranging from early structure formation to the process of reionization itself, no direct observations of
anykindhavebeenmadein thisredshift range. Recently,we madethefirst proposal for such direct observations(Iliev et al.
2002, from hereafter Paper I) based on collisional excitation of the hydrogen 21-cm line in the warm, dense, neutral gas in
virializedminihalos(haloswithvirialtemperatureT 6104K),thefirstbaryonicstructurestoemergeinthestandardCDM
vir
universe.Weshowedthatcollisionalexcitationissufficienttoincreasethespintemperatureofhydrogenatomsinsideminihalos
abovethetemperaturethecosmicmicrowavebackground(CMB).Minihalosshouldgenerallyappearinemissionwithrespect
to the CMB, producing a background “21-cm forest” of redshifted emission lines, well-separated in frequency. Unlike all
previousworks(Hogan & Rees1979;Scott & Rees1990;Subramanian & Padmanabhan1993;Madau, Meiksen & Rees1997;
2 I. T. Iliev et al.
Shaveret al.1999;Tozzi et al.2000),thismechanismdoesnotrequiresourcesofLyαradiationfor“pumping”the21-cmline
and decoupling it from the CMB.
InPaperIwecalculatedthe21-cmemission propertiesofindividualminihalosindetail,alongwiththetotalbackground
and its large-scale fluctuations due to clustering. Although individual lines and the overall background are too weak to be
readily detected, the background fluctuations should be measurable on ∼10′−100′ scales with the currently-planned radio
arraysLOFARand SKA.Wedemonstrated that suchobservations can beusedtoprobethedetails ofreionization aswell as
measure thepower spectrum of densityfluctuations at far smaller scales than havebeen constrained previously.
In the current paper we extend these results to smaller scales. Our previous calculations showed that the fluctuation
signalincreases asthebeam sizeandfrequencybandwidthdecrease, whichcorresponds tosampling the21-cmemission from
minihalos within smaller volumes. As these volumes correspond in turn to length scales that are more nonlinear, nonlinear
effects havethegreatest impact on theangular scales and frequency bandwidthsat which thesignal is thestrongest.
This issue is of particular importance as our previous investigations relied on the standard, simplified nonlinear bias of
Mo & White (1996), which breaks down at many of the relevant redshifts and length scales. For example, an rms density
fluctuationinaColdDarkMatteruniversewithacosmologicalconstant(ΛCDM)atz =6(8),isstronglynonlinear(σ(M)>1)
for M 6 109M⊙ (6.2×107M⊙), which roughly corresponds to the region sampled by beam sizes 9” (3”), for frequency
bandwidths12kHz(3kHz),respectively.Nonlineareffectscaninfluencethepredictedbackgroundfluctuationsonevenlarger
scales, up to few comoving Mpc (corresponding to few hundredkHzfrequency bandwidthsand 1-10 arc min beams), even if
these scales are still not strongly nonlinear. Additionally, rare halos are always more strongly clustered than the underlying
density distribution (i.e. thebias is >1), again bringing nonlinear issues to thefore.
Inordertoaddresstheseimportantissues wehavecarried out aseriesofhigh-redshift N-bodysimulations ofsmall scale
structureformation,whichweusetoconstruct21-cmlineradiomapsthatillustratetheexpectedfluctuationsintheemission.
Asnocurrentsimulations areabletospanthefulldynamicrangerelevantto21-cm emission, however,weextendourresults
by developing a new formalism for calculating the nonlinear Eulerian bias of halos, which is based on the Lagrangian bias
formalism of Scannapieco & Barkana (2002, hereafter SB02). We describe this approach in detail in this paper, verify it by
comparingitwiththeresultsofourN-bodysimulations,andapplyittotocalculateimprovedpredictionsforthefluctuations
in the 21-cm emission. These are then compared to the predictions given in Paper I, quantifying the impact of nonlinear
effects on minihalo emission.
Thestructureofthisworkisasfollows. In§2wedescribeaset ofsimulations ofminihaloemission at highredshift,and
usethesetoconstructsimulatedmapsatsmallangularscales.In§3wepresentourimprovedcalculation oftheEulerianbias
andverifyitbycomparison withthenumericalsimulationspresentedin§2.In§4wemodifythecalculation oftheradiation
backgroundfromminihalostoincorporatethecontributionduetononlinearclusteringofsources,accordingtotheformalism
described in § 3, and describe theresults of ournonlinear formalism. Conclusions are given in § 5.
2 NUMERICAL SIMULATIONS AND SIMULATED 21-CM RADIO MAPS
Wesimulatedtheformationofminihalosinthreecubiccomputationalvolumesofcomovingsize1Mpc,0.5Mpc,and0.25Mpc,
respectively. We used a standard Particle-Particle/Particle-Mesh (P3M) algorithm (Hockney & Eastwood 1981), with 1283
equal-massparticles,a2563 PMgrid,andasofteninglengthof0.3gridspacing.Hereandthroughoutthispaperweconsidera
flatΛCDMmodelwithdensityparameterΩ =0.3,cosmological constantλ =0.7,Hubbleconstant H =70kms−1Mpc−1,
0 0 0
baryondensityparameterΩ =0.02h−2 (whereh=H /100kms−1Mpc−1),andnotilt.Theinitialconditionswheregenerated
b 0
using the transfer function of Bardeen et al. (1986) with the normalization of Bunn & White (1997) (for details, see Martel
& Matzner 2000, §2). All simulations started at redshift z =50 and terminated at redshift z =9. To identify minihalos, we
used astandard friends-of-friends algorithm with a linkinglength equal to0.25 timesthemean particle spacing. Werejected
halos composed of 20 particles or less. Table 1 lists the comoving size L of the box, the comoving value of the softening
box
length η, the total mass M inside thebox, and the particle mass M .
tot part
Oncethehalosareidentified,wecanusetheirpropertiesanddistributiontocomputethecorrespondingradiomaps.We
assign to each halo a 21-cm flux based on its mass and redshift of formation by modeling thehalos as Truncated Isothermal
Spheres (Shapiro, Iliev & Raga 1999; Iliev & Shapiro 2001). The 21-cm fluxes from individual minihalos are obtained by
solving the radiative transfer equation self-consistently through each halo to obtain the line-integrated flux, as described in
Paper I. Let us consider a cylindrical volume with radius R = ∆θ (1+z)D (z)/2, and length L , where D (z) is
beam A box A
the angular diameter distance, ∆θ is the beam size of the observation. Then the beam-averaged differential antenna
beam
temperatureδT is calculated using equation 6 of Paper I, which can be discretized as follows:
b
δT (x)= c(1+z)4 1 (∆ν ) (δT ) A e−(x−xi)2/2R2, (1)
b 2ν H(z) πR2L eff i b,ν0 i i
0 box
i
X
where H(z) is the Hubble constant at redshift z, ν is the rest-frame line frequency, πR2L is the comoving volume of
0 box
Nonlinear clustering of minihalos 3
Table 1.ParametersoftheN-bodysimulations
Lbox (Mpc) η (kpc) Mtot(M⊙) Mpart(M⊙)
1.0 1.172 4.079×1010 1.945×104
0.5 0.586 5.099×109 2.431×103
0.25 0.293 6.374×108 3.039×102
the beam (∆ν ) , (δT ) , and A are the effective line width, the line-center differential brightness temperature, and the
eff i b,ν0 i i
geometriccross-sectionofhaloi,respectively,andweuseGaussianfilterinordertoensurethatthecontributionofaparticular
minihalo to a particular pixelvaries smoothly with thelocation of theminihalo.
The resulting maps are shown in Figure 1. Due to the small box sizes that are required to resolve the minihalos, the
beamsusedtoproducethemapsarealsoverysmall,rangingfrom4”to0.25”.Thefluxesfromsuchsmallbeamsizesarewell
below thesensitivity limits of thecurrently planned radio arrays LOFARand SKA.Additionally,thelarger-box (1 Mpcand
0.5 Mpc) simulations donot havesufficient massresolution toresolve thesmallest halos, while thelarger-mass minihalos are
notpresentinthesmallerboxsimulations. Thereforethefluxlevelsaresomewhat underestimatedinallcases, andshouldbe
considered only as illustrative of fluctuationson very small scales.
3 NONLINEAR CLUSTERING REGIME AND COMPARISON TO SIMULATIONS
3.1 An analytic approach to nonlinear clustering
As our simulations are unable to span a sufficient range of scales to both resolve all minihalos and reproduce the physical
scales accessible to observations, we adopt instead an alternate technique to estimate the full nonlinear signal. We apply
an approximate formalism that is an extension of the one developed in SB02. A more complete discussion of this method,
refinement of the basic formalism, and detailed comparisons with simulations will be presented in a separate paper. In this
section we summarize the basic features of this approach and show that it is sufficient for our purposes in this investigation.
For an alternative approach see Catelan, Matarese, & Porciani (1998).
In SB02 the authors extended the Press-Schechter formalism as reinterpreted by Bond et al. (1991) to calculate the
numberdensityofcollapsed objects intworegions ofspaceinitially separated byafixedcomovingdistance.Inthisapproach
virialized halos are associated with linear density peaks that fall above a critical value usually taken to be δ =1.686. Here
c
the linear overdensity field is defined as δ(x)D−1(z) ≡ ρ(x,z)/ρ¯(z)−1, where ρ(x) is the linear mass density as a function
of position and and ρ¯is the mean mass density. Note that here the linear overdensity field δ(x) is described in terms of its
value extrapolated to z =0, and its evolution with time is subsumed by the “linear growth factor” D(z)=δ(0)/δ(z). Other
choices for δ and its evolution have been explored in the literature, by fitting to simulations (Sheth,Mo, & Tormen 2001;
c
Jenkinset al. 2001)orincorporatingtheZel’dovichoradhesionapproximations(Lee & Shandarin1999;Menci2001).While
thesemethodsimprovetheaccuracyofthePress-Schechtertechnique,theyprovidenoinformationaboutnonlinearclustering.
Herewe strive to develop a formalism that is applicable to nonlinear clustering regardless of therecipe chosen for δ .
c
Once δ determined, the presence of a halo can be associated with a random walk procedure. At a given redshift we
c
consider the smoothed density in a region around a point in space. We begin by averaging over a large mass scale M, or
equivalently, by including only small comoving wavenumbers k. We then lower M, adding k modes, and adjusting δ(M)
accordingly. This amounts to arandom walk in which the“time variable” is thevariance associated with thefilter mass and
the “spatial variable” is the overdensity itself. This walk continues until the averaged overdensity crosses δ D−1(z) and we
c
assume that thepoint belongs to a halo with a mass M corresponding to this filterscale.
Theproblemofhalocollapseattwopositionsinspacecanbesimilarlyassociatedwithtworandomwalksinthepresence
of a barrier of height δ D−1(z). These random walks are extremely correlated at large scales, and become less and less so as
c
themodesofsmallerandsmallerscalesareadded.InSB02thisprocesswasapproximatedbyacompletelycorrelatedrandom
walkdowntoanoverallvarianceS =ξ,whereξ isthecorrelation functionthataccountsfortheexcessprobabilityoffinding
a second halo at a fixed distance from a given halo. This correlated random walk was then followed by uncorrelated walks
down to the finalvariances associated with themasses of theobjects, S =σ2(M ) and σ2(M ).
1 2
This procedure lends itself directly to calculating the overall increase in numberdensity due to infall. The fact that the
overdensity on large scales will contract the distances between halos provides an additional contribution at the length scale
x such that σ2(x)=ξ, the scale down to which the evolution of the two points is completely correlated. Let δ be the linear
0
overdensity at redshift zero at that scale. The probability distribution of δ , Q is then simply given by a Gaussian, with a
0 0
barrier imposed at ν(z)≡δ D−1(z):
c
Q (ν,δ ,ξ)=[G(δ ,ξ)−G(2ν−δ ,ξ)]θ(ν−δ ), (2)
0 0 0 0 0
4 I. T. Iliev et al.
Figure 1. (upper panels) Differential brightness temperature radio map of the 21-cm emission from minihalos at z =9, produced as
describedinthetextfromΛCDMN-bodysimulationof:(upperleftpanel)boxsize1Mpcandbeamsize∆θbeam=4′′,and(upperright
panel) box size 0.5 Mpc and beam size ∆θbeam =2′′. (lower left panel) box size 0.25 Mpc and ∆θbeam =1′′, and (lower right panel)
boxsize0.25Mpcand∆θbeam=0.25′′.
Nonlinear clustering of minihalos 5
where G(δ,ξ)≡(2πξ)−1/2e−δ2/2ξ, and θ is theHeaviside step function.
Wenowconsiderf (ν(z),σ2,σ2,δ ,ξ)dσ2dσ2,thejoint probabilityofhavingpointAinahalowith masscorresponding
2 1 2 0 1 2
totherangeσ2 toσ2+dσ2 andpointB inahalowithmasscorrespondingtotherangeσ2 toσ2+dσ2,whoserandomwalks
1 1 1 2 2 2
pass though a value of δ at the σ2(M)=ξ scale. In this case we obtain an expression analogous to equation (37) of SB02
0
∂ ∂ ν(z) ∞ ν(z) ∞
f (ν(z),δ ,σ2,σ2,ξ(r)) = 2 dδ − dδ 2 dδ − dδ Q (ν,δ ,δ ,σ2,σ2,ξ(r)), (3)
2 0 1 2 ∂σ2∂σ2 1 1 2 2 12 1 2 1 2
1 2 (cid:20) Z−∞ Z−∞ (cid:21)(cid:20) Z−∞ Z−∞ (cid:21)
wherenowQ ≡G(δ −δ ,σ2−ξ)G(δ −δ ,σ2−ξ),andthereaderis referred to SB02for amore detailed derivation of f
12 1 0 1 2 0 2 2
from the underlyingprobability distribution.
Averaging equation (3) over the probability distribution for δ as given in equation (2) and carrying out the partial
0
derivativesweobtaintheEulerian“bivariate”numberdensity,thejointprobabilityofhavingpointAlieinahalointhemass
rangeM toM +dM andpointB atacomovingdistancer lieinahalointhemassrangeM toM +dM ataredshiftz:
1 1 1 2 2 2
d2n2 ρ¯ dσ2 ρ¯ dσ2 ∂ξ ∂ξ
12,E (r,z)= 1 2 1− 1− f (ν(z),σ2,σ2,ξ(r)), (4)
dM dM M dM M dM ∂σ2 ∂σ2 2,E 1 2
1 2 1 (cid:12) 1(cid:12) 2 (cid:12) 2(cid:12)(cid:18) 1(cid:19)(cid:18) 2(cid:19)
(cid:12) (cid:12) (cid:12) (cid:12)
where (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
ν
f (ν,σ2,σ2,ξ)≡ dδ Q (δ ,ξ)g(D−1δ )f(ν−δ ,ξ−σ2)g(D−1δ )f(ν−δ ,ξ−σ2), (5)
2,E 1 2 0 0 0 0 0 1 0 0 2
Z−∞
and f(ν,σ2) ≡ (2π)−1/2(ν/σ3)e−ν2/2σ2. Here g(D−1δ ) is ρ/ρ¯ at the scale at which the points A and B are completely
0
correlated. This can be thought of as the contraction of a large spherical perturbation that surrounds both points and
contains a total mass M of material, where σ2(M)=ξ. This contribution is then just (1+D−1δ ) in the linear regime, and
0
for our purposes herewe assume g(D−1δ )=(1+D−1δ ) for all values as this reproduces well theresults from simulations.
0 0
Finally we definethe Eulerian bias as
d2n2 dn dn −1
b2ξD−2 ≡ 12,E 1,E 2,E −1 (6)
E dM dM dM dM
1 2 1 2
(cid:16) (cid:17)
wherewedividebydn /dM,asinglepointnumberdensitythataccountsfortheselfcorrelationsbetweenthecollapsedpeaks
E
andtheoverdensesphere,whichareover-countedinequation(4).Tocomputethisprobabilityweagain carryoutanaverage
over the probability distribution (2), but in this case we consider f (ν,δ ,σ2)dσ2 = f(ν−δ ,σ2−ξ)dσ2, the probability of
1 0 1 1 0 1
havinga single point in ahalo with amass corresponding totherangeσ2 toσ2+dσ2 whose random walk passes throughδ
1 1 1 0
at theσ2(M)=ξ scale. This gives dn /dM =(ρ¯/M)|dσ2/dM|(1+∂ξ/∂σ2)f (ν,σ2) where
E E
ν
f (ν,σ2,ξ)≡ dδ Q (δ ,ν,ξ)g(δ D−1)f(ν−δ ,ξ−σ2). (7)
E 0 0 0 0 0
Z−∞
Combining these expressions yields our final expression for thebias
f (ν,σ2,σ2,ξ)
b2ξD−2 = 2,E 1 2 −1. (8)
E f (ν,σ2,ξ)f (ν ,σ2,ξ)
E 1 E 2 2
Note that at large distances we can work to order δ2 to determinethe asymptotic limit as ξ/σ2−→0. In thislimit
0
ν2 1
g(δ D−1)f(ν−δ ,σ2−ξ)−→ 1−D−1δ − +1 +O(δ2) f(ν,σ2−ξ), (9)
0 0 0 δ σ2 δ 0
c c
(cid:20) (cid:18) (cid:19) (cid:21)
and the only surviving term in equation (8) is the cross term between terms of order δ . All other terms cancel out between
0
thenumerator and thedenominator, giving
ν/σ2−1 ν/σ2−1
b2 = 1+ 1 1+ 2 , (10)
E δ δ
c c
(cid:18) (cid:19)(cid:18) (cid:19)
Thusour formalism reproduces theusual bias as in Mo & White(1996) at large distance, as was used in Paper I.
Finally, we definethe flux-averagedcorrelation between objects as
dM dM F(M )(dn/dM )F(M )(dn/dM )b2ξD−2
¯b2ξD−2 = 1 2 1 1 2 2 E , (11)
E 2
R R dMF(M)(dn/dM)
where F(M) is the line-inte(cid:2)gRrated fluxof a miniha(cid:3)lo of mass M.
3.2 Nonlinear properties and comparisons with simulations
In Fig. 2 we plot b2ξD−2 as calculated from equation (8) and the standard expression, equation (10), using the fit to the
E
density fluctuation power spectrum given by Eisenstein & Hu (1999). In each panel we show the behavior of 107M⊙ and
6 I. T. Iliev et al.
Figure 2. Comparisonof standard and nonlinear bias. In all panels, the solidcurves give b2D−2ξ as calculated inequation (8) while
E
the dotted linesgive the standard bias expressions as used inPaper I. For each case, the upper curves assume afixed mass of107M⊙,
correspondingtoasphericalperturbationwithacomovingradiusofapproximately38kpc,whilethelowercurvesassume afixedmass
of106M⊙,correspondingtoacomovingradiusof18kpc.
106M⊙ objects at four different redshifts, which span the detectable range. At large distances, the clustering expressions
approacheachotherasymptotically,asexpectedfromequation(10).Ontheotherhand,thesolidlinesshootupdramatically
atthesmallestdistances.Thisisbecausetheseparationbetweenthehalosiscomparabletotheirradii,andthusthelikelihood
offindingasecondcollapsedhaloatthesamepointbecomesinfiniteasr−→0.Atintermediatedistancesthebehaviorismore
complex. For relatively rare objects, corresponding to high redshifts, the nonlinear values consistently exceed the standard
ones.Forexample,atadistanceof 0.1comovingMpc, thenonlinear107M⊙ valueisalmost twicethat ofthestandard result
at z=20, while it is only about 70% of the standard result at z=6.
To compare this behavior to numerical results, we calculated the correlations between halos for each of our simulations.
Inthiscaseb2ξD−2 isdirectlycomparable tothehalocorrelation function ξ (r),theexcessprobabilityof findingaparticle
E i,j
within a mass bin m and a particle within a mass bin m separated by a distance r. This expression is symmetric, so that
i j
ξ (r)=ξ (r).WedefinehN (r)iasthemeannumberofparticlesinamassbinm ,located withinaradiusr ofaparticle
i,j j,i i,j j
in amass binm ,averaged overall particles in amass bin M .This isis obtained bydividingthetotal numberof pairs with
i i
separation r′<r, bythe numberN of i particles,
i
4πN r
hN (r)i= j r3+3 ξ (r′)r′2dr′ , (12)
i,j 3V i,j
box (cid:20) Z0 (cid:21)
whereN isthenumberof particles ofmass m and V is thevolume.Finally,wedifferentiateequation(12)toget ξ (r).
j j box i,j
Usingthesedefinitionsweexaminedthedistributionofhalosineachofoursimulationsatz =15andz =9.Ineachcase
wedividedthehalosintomassbinscontainingroughly thesamenumber(∼600) ofobjects andcompared theircorrelations
to those given by equation (8).In the left panels of Fig. 3 we compare themass averaged correlation between halos,
−1
dn dn dn dn
ξ¯ = b2ξD−2 , (13)
i,j dM dM E dM dM
Zmi i Zmj j ! Zmi i Zmj j!
to the numerical value, ξ , for objects in three mass bins in our 1 Mpc3 simulation at z =15 (Fig. 3). For all mass ranges,
i,j
there is a good match between the numerical ξ and our analytic expression, with both values equaling or exceeding the
i,i
linear predictions at most distances. The only case in which the predictions of the simulations fall below equation (10), is
at large distances, where the separations between halos begin to approach the simulation box size, and some damping is to
be expected. Comparisons with our smaller box simulations yielded comparable results, but with larger damping at these
distances, indicating that the analytic and simulated correlations are a good match at all reliable separations. On the other
hand, the cross-correlation functions are a poorer match, particularly if the mass bins are very different. For all correlations
however, our nonlinear estimate falls between the standard and simulated results, indicating that the correlations between
halos exceed those predicted by equation (10), and that ourapproach gives a conservative estimate of this excess.
IntherightpanelsofFig.3,werepeatthiscomparison withour1Mpc3 simulationatz=9.Inthiscase,wedividedour
halosintofivemassbins,threeofwhichareshowninthisfigure.Inthiscasethereisgoodagreementbetweenthecorrelations
and cross-correlations obtained from the simulations and our nonlinear estimates, for all mass scales considered. In fact the
analyticalestimatesareconsistentwiththeresultsofthesimulationsforalldistancesthataresmallrelativetothebox,apart
from the dip at small distances in the m m case, which is most likely due to small-number statistics. Note that unlike the
1 1
Nonlinear clustering of minihalos 7
Figure 3. Comparisonbetweennonlinearanalyticexpressionsforthebiasandsimulationsattwodifferentredshifts.Intheleftpanels
z = 15 and the halos are divided into three mass bins: 4.1×105M⊙ 6m1 65.5×105M⊙, 5.6×105M⊙ 6m2 61.0×106M⊙, and
1.1×106M⊙ 6m3 67.7×107M⊙. In the right panels z = 9 and halos are divided into five mass bins, three of which are shown:
4.1×105M⊙6m164.7×105M⊙,6.2×105M⊙6m368.2×105M⊙,and1.3×106M⊙6m563.4×106M⊙.Inbothcases,thepoints
arethecorrelationfunctionascalculatedfromthesimulations,whilethesolidlinesgiveouranalyticestimatesasgivenbyequation(8).
The error bars are 2-σ estimates of the statistical noise given by 2ξi,i(r)/ Ni,i(<r). Finally, the dashed lines give correlations as
estimatedfromthestandardapproach, equation(10).
p
higher redshift case, our nonlinear estimates now fall below the standard analytic approach at larger distances, as we saw in
thelower redshift cases in Fig. 2.
4 FLUCTUATIONS OF THE 21-CM EMISSION FROM MINIHALOS: EFFECTS OF THE
NONLINEAR BIAS
The amplitude of q-σ (i.e. q times the rms value) angular fluctuations in the differential brightness temperature δT (or,
b
equivalently,of theflux)in thelinear regime are given by
hδT2i1/2 =qb(z)σ δT (14)
b p b
(Paper I),where b(z) is themean flux-weightedbias, and
8D−2(z) ∞ 1 sin2(kLx/2)J2[kR(1−x2)1/2] P(k)
σ2= dk dx 1 (1+fx2)2 , (15)
p π2L2R2 x2(1−x2) k2
Z0 Z0
where P(k) is the linear power spectrum at z = 0, and the factor (1+fx2)2, where f ≈ [Ω(z)]0.6 is the correction to
the cylinder length for the departure from Hubble expansion due to peculiar velocities (Kaiser 1987). In order to apply the
formalismdevelopedin§3,weusethatthepowerspectrumistheFouriertransformofthecorrelationfunctionξ(r),obtaining
32D−2(z) ∞
σ2= drr2ξ(r)f(r,R,L), (16)
p πL2R2
Z0
where
1 ∞ sin(kr)sin2(kLx/2)J2[kR(1−x2)1/2]
f(r,R,L)≡ dx(1+fx2)2 dk 1 . (17)
kr x2 k2(1−x2)
Z0 Z0
8 I. T. Iliev et al.
Figure4.Predicted3-σdifferentialantennatemperaturefluctuationsatz=7andz=8.5vs.observerfrequencybandwidth∆νobs for
standardΛCDMmodelusingMo&Whitebias(long-dashedcurves)atangularscales(a)(left)∆θ=9′,and(b)(right)∆θ=25′.Also
indicatedisthepredictedsensitivityofLOFARandSKAradioarraysforintegrationtimes100h(dashed)and1000h(dotted)(forright
panelssensitivitycurvesforLOFARandSKAareidentical)assumingrmssensitivity∝ν−2.4 (seehttp://www.lofar.org/science).
Figure 5. Predicted 3-σ differential antenna temperature fluctuations at z = 8.5 and z = 20 vs. angular scale ∆θbeam for standard
ΛCDM model for observer frequency bandwidth ∆νobs = 2 MHz : using Mo & White bias (long-dashed curves) and current results
(solidcurves). Alsoindicated is the predicted sensitivity of LOFAR and SKA integration times of 100 h (short-dashed lines) and 1000
h(dotted lines),withcompactsubaperture(horizontal lines)andextended configurationneededtoachievehigherresolution(diagonal
lines)(seehttp://www.lofar.org/scienceandPaperIfordetails).
Using equations (14) and (16) themean squared angular fluctuations become
32 2 ∞
hδT2i= drr2 D−2b2(z)ξ(r)(δT )2 f(r,R,L). (18)
b πL2R2 b
(cid:16) (cid:17) Z0 (cid:2) (cid:3)
Weapplytheformalismdevelopedin§3bysimplyreplacing D−2b2(z)ξ(r)(δT )2 withthecorrespondingquantityD−2b2ξ(r)(δT )2
b E b
forthemeanEulerianflux-weightedbiascalculatedin§3.Asweshowed,atlargedistancesthetwoexpressionsareequivalent,
(cid:2) (cid:3)
while at smaller distances thenew formalism betterreproduces thenumerical results.
We start with deriving the optimal frequency bandwidth for observing the 21-cm emission from minihalos. As long as
∆ν and the beam size ∆θ are large enough to provide a fair sample of the halo distribution, the mean 21-cm flux is
obs beam
Nonlinear clustering of minihalos 9
Figure6.(a)(leftpanels)Predicted3-σdifferentialantennatemperaturefluctuationsatz=8.5andz=20atsmallangularscalesvs.
observer frequency bandwidth ∆νobs for standard ΛCDM model and for angular scale ∆θ = 1′, using Mo & White bias (long-dashed
curves)andcurrentresults(solidcurves).(b)(rightpanels)Predicted3-σdifferentialantennatemperaturefluctuationsat∆θbeam=25′
vs.redshiftzforΛCDMmodel,samenotationasina).Thepredictedsensitivityforintegrationtimes100h(dashed)and1000h(dotted)
of both LOFAR (“L”) and SKA (“S”), are plotted (for bottom panel, sensitivity curves for LOFAR and SKA are identical) assuming
rmssensitivity∝ν−2.4 (seehttp://www.lofar.org/science).
independentof ∆ν , while thefluctuations grow as ∆ν decreases. Howeverthesensitivity of theradio array deteriorates
obs obs
for smaller bandwidths as ∆ν−1/2 (e.g. Shaver et al. 1999; Tozzi et al. 2000). The results for 3−σ fluctuations at redshifts
obs
z = 7 and z = 8.5 vs. observed frequency bandwidth ∆ν for beam sizes ∆θ = 9′ and 25’ are shown in Fig. 4, along
obs beam
with the results obtained using the linear bias in order to facilitate comparison. As the bandwidth increases, the integration
time required to detect the signal decreases, reaching minimum at about ∆ν =2 MHz and remaining roughly unchanged
obs
thereafter. The fluctuations decrease with increasing bandwidth, however,hence the optimal bandwidth is about ∆ν ∼ 2
obs
MHz. We see that at large scales the current results are largely indistinguishable from the linear results, as expected, since
at such large scales the bias calculated in §3 reduces tothe linear expression in equation (10). For small valuesof ∆ν the
obs
differences between the two results grow to ∼10%. However, in theobservable range the two results hardly differ.
Illustrative results for 3−σ fluctuations at redshifts z = 8.5 and z = 20 vs. the beam size ∆θ for fixed observed
beam
frequencybandwidth∆ν =2MHzareshowninFig.5.TheapproximatesensitivitiesfortheLOFARandSKAradioarrays
obs
are also shown where appropriate (see Paper I for details), but from here on we concentrate on the differences between the
resultsfrom thetwoapproaches andtherobustnessofouroriginal predictionsratherthanobservability,which was discussed
in Paper I. At redshifts z . 15 using the linear bias gives an overestimate, although a small one compared to the various
uncertaintiesofthecalculation, whileathigherredshiftsusingthelinearbiasgivesasmallunderestimateofthefluctuations.
InFig.6(leftpanels)weplotthe3−σ fluctuationsatredshiftsz=8.5andz =20vs.theobserverfrequencybandwidth
∆ν for fixed beam size ∆θ =1′. Wehave chosen such a small beam size in order to investigate theupper limit of the
obs beam
deviation of the results due to the nonlinear clustering of halos. For small values of both ∆ν and ∆θ the differences
obs beam
areas large as 20-30%. In Fig. 6(right panels) weplot the3−σ fluctuationsfor beam sizes ∆θ =9′ and25′ vs. redshift
beam
z (i. e. the spectrum of fluctuations) for observer frequency bandwidth ∆ν = 2 MHz. Again, the same patterns emerge,
obs
where the linear bias approach overestimates the fluctuations at the lower end of the range of considered redshifts by up to
10 %, while underestimating thefluctuations at the higher redshifts, although by a smaller fraction.
5 CONCLUSIONS
Wehavedevelopedanewanalyticalmethodforestimatingthenonlinearbiasofhalosandusedittocalculatethefluctuations
of the 21-cm emission from the clustering of high-z minihalos. This method is likely to be useful in tackling a much wider
range of problems that lie beyond the capabilities of current numerical simulations, and will be refined and further verified
in a future publication. The minihalo bias predicted by our method at large scales reproduces the linear bias, as derived in
Mo & White (1996), and is much larger at small scales, confirming naive expectations. At intermediate scales, however, its
behaviorismorecomplexandbothmass-andredshift-dependent.Forveryrarehalosathigh-z(roughlyz>15),thestandard
10 I. T. Iliev et al.
linearbiasislowerthanournonlinearpredictionatalllengthscales.Atthelowerendoftheredshiftrangeweconsider(down
toz =6)thelinearbiasishigheratintermediatescales (fewhundredkpctofew Mpccomoving) andlower atsmaller scales.
We have also compared the predictions of our new method with the results of N-body numerical simulations, which we
used both toverify our approach and toproducesample radio maps.Dueto thelimited dynamical range of oursimulations,
however, these maps are only illustrative of thebehavior of thefluctuations on verysmall scales, below thesensitivity limits
of LOFAR and SKA. On the scales at which the simulations are reliable, we find excellent agreement between these results
andouranalytical approach.Themost significant discrepancies occurinthecalculation of cross-correlation functionsofvery
differentmassbinsathigherredshifts.However,thesedeparturesarerelativelymodest,andinallcasesourmethodreproduces
thesimulation results significantly better than thelinear bias.
Despite these differences, we find our original linear bias predictions for the fluctuations of the 21-cm emission from
minihalos to be robust at the scales and frequencies corresponding to observable signals. The prediction using the flux-
weightednonlinearbiasneverdepartsfromthelinearpredictionbymorethanfewpercentinthatrange,wellwithintheother
uncertaintiesof thecalculation. Thisrobustness ispartly accidental, however,andis duetothenonlinearbias varyingabove
andbelowthelinearonedependingonthelengthscale, leadingtopartialcancellation ofthedifferenceswhenthecorrelation
function is integrated over the length scales. For small observational bandwidths, ∆ν , there is less cancellation and the
obs
differences in the two predictions grow at all beam sizes, ∆θ . Similarly, if ∆θ is small, there is little cancellation,
beam beam
and thediscrepancies are larger at all bandwidths.In these cases thelinear bias gives an overestimate of the 21-cm emission
fluctuationsfromminihalosatthelowendoftheredshiftrange,andanunderestimateathigh-z,evenatlargevaluesof∆ν .
obs
However,whenboththebeamsizeandthefrequencybandwidtharelargethedifferencesinthetwoapproachesatsmallscales
are diluted, and the resulting 21-cm emission fluctuations become identical. Finally, we predict that the best observational
frequency bandwidth for improving the chances for detection is ∆ν ∼ 2 MHz. Thus it may be through a such frequency
obs
window that astronomers get theirfirst glimpses of the cosmological Dark Ages.
ACKNOWLEDGMENTS
Wewould liketo thankAndreaFerrara and RennanBarkana for usefulcomments and discussions. This work was supported
inpartbytheResearchandTrainingNetwork“ThePhysicsoftheIntergalacticMedium”setupbytheEuropeanCommunity
underthe contract HPRN-CT2000-00126 RG29185, NASA grants NAG5-10825 to PRS and NAG5-10826 to HM, and Texas
AdvancedResearchProgram3658-0624-1999toPRSandHM.EShasbeensupportedinpartbyanNSFMPS-DRFfellowship.
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