The Halo Occupation Distribution of HI from 21cm Intensity Mapping at Moderate Redshifts
aa r X i v : . [ a s t r o - ph . C O ] D ec Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 5 November 2018 (MN L A TEX style file v2.2)
The Halo Occupation Distribution of HI from 21cmIntensity Mapping at Moderate Redshifts
J. Stuart B. Wyithe , Michael J. I. Brown School of Physics, University of Melbourne, Parkville, Victoria, Australia School of Physics, Monash University, Clayton, Victoria 3800, AustraliaEmail: [email protected]
ABSTRACT
The spatial clustering properties of HI galaxies can be studied using the formalism ofthe halo occupation distribution (HOD). The resulting parameter constraints describeproperties like gas richness verses environment. Unfortunately, clustering studies basedon individual HI galaxies will be restricted to the local Universe for the foreseeablefuture, even with the deepest HI surveys. Here we discuss how clustering studies ofthe HI HOD could be extended to moderate redshift, through observations of fluctua-tions in the combined 21cm intensity of unresolved galaxies. In particular we make ananalytic estimate for the clustering of HI in the HOD. Our joint goals are to estimate i ) the amplitude of the signal, and ii ) the sensitivity of telescopes like the AustralianSKA Pathfinder to HOD parameters. We find that the power spectrum of redshifted21cm intensity could be used to study the distribution of HI within dark matter halosat z & . HI ) at epochs between the local Universe, and redshifts probedby damped Ly α absorbers. Key words: cosmology: large scale structure, observations – galaxies: halos, statistics– radio lines: galaxies
The cosmic star-formation rate has declined by more thanan order of magnitude in the 8 billion years since z ∼ z . . . & c (cid:13) Wyithe & Brown bution (HOD; e.g. Peacock & Smith 2000; Seljak 2000; Scoc-cimarro et al. 2001; Berlind & Weinberg 2002; Zheng 2004).The HOD includes contributions to galaxy clustering frompairs of galaxies in distinct halos which describes the cluster-ing in the large scale limit, and from pairs of galaxies withina single halo which describes clustering in the small scalelimit. The latter contribution requires a parametrisation torelate the number and spatial distribution of galaxies withina dark matter halo of a particular mass. It is by constrainingthis parameterisation that observed clustering can be usedto understand how galaxies are distributed.By comparison with the massive optical redshift sur-veys, the largest survey of HI selected galaxies contains only ∼ ∼
10% of the HIPASS sample. HI satellitegalaxies are therefore less significant in number and in termsof their contribution to clustering statistics than are satel-lites in optically selected galaxy redshift surveys.These results from HOD modeling of HI galaxy clus-tering therefore quantify the extent to which environmentgoverns the HI content of galaxies in the local Universe andconfirms previous evidence that HI galaxies are relativelyrare in overdense environments (Waugh et al. 2002; Corteset al. 2008). Wyithe et al. (2009) found a minimum halomass for HIPASS galaxies at the peak of the redshift distri-bution of M ∼ M ⊙ (throughout this paper we refer tothe halo mass as M and the HI mass as M HI ), and showedthat less than 10% of baryons in HIPASS galaxies are in theform of HI. Their analysis also revealed that the fingers-of-god in the redshift space correlation function are sensitiveto the typical halo mass in which satellite galaxies reside,and indicated that the HI rich satellites required to producethe measured 1-halo term must be preferentially in grouprather than cluster mass halos.As described above, the clustering of HI galaxies can bestudied at z = 0 using HIPASS. However in the future withthe advent of the Square Kilometer Array (SKA) and itspathfinders the volume and redshift range over which clus-tering of HI galaxies can be studied will greatly increase. Onthe other hand, these studies will still be limited to moder-ate redshifts of z . . z ∼ . z ∼ .
7, as this is the redshift atwhich ASKAP no longer has the sensitivity to study indi-vidual galaxies. Our goal is not to provide a detailed methodfor extracting detailed HOD parameters from an observedpower spectrum of redshifted 21cm fluctuations. This wouldrequire calibration against N-body simulations, which is pre-mature at this time. Rather, we present an analytic modelfor the 21cm power spectrum in the HOD, and investigatewhich of its properties could be constrained by observationsusing a telescope like ASKAPThe paper is organised as follows. We begin by sum-marising the formalism for the HOD model, and introduceHOD modeling of 21cm intensity fluctuations in §
2. Wediscuss the potential sensitivity of ASKAP to these fluctu-ations in §
3. We then present our forecast constraints onHOD parameters in § §
5. We summarise our findings in §
6. Inour numerical examples, we adopt the standard set of cos-mological parameters (Komatsu et al. 2009), with values ofΩ m = 0 .
24, Ω b = 0 .
04 and Ω Q = 0 .
76 for the matter, baryon,and dark energy fractional density respectively, h = 0 .
73, forthe dimensionless Hubble constant, and σ = 0 .
81 for thevariance of the linear density field within regions of radius8 h − Mpc. c (cid:13) , 000–000 he HOD of Neutral Hydrogen We begin by reviewing the halo occupation distributionformalism for galaxies (e.g. Peacock & Smith 2000; Seljak2000; Scoccimarro et al. 2001; Berlind & Weinberg 2002;Zheng 2004) which we describe only briefly, referring thereader to the above papers for details. The technique of sur-face brightness mapping will not allow resolution of individ-ual galaxies, but rather the measurement of fluctuations inthe surface brightness of unresulved galaxies. However weutilise a halo model formalism where galaxies are traced,rather than a form where the density field is a continuousfunction. This is because the HI is found in discrete galaxies,and treating individual galaxies allows us to explicitely cal-culate the HI mass weighted galaxy bias. The HOD model isconstructed around the following simple assumptions. First,one assumes that there is either zero or one central galaxythat resides at the centre of each halo. Satellite galaxies arethen assumed to follow the dark matter distribution withinthe halos. The mean number of satellites is typically as-sumed to follow a power-law function of halo mass, whilethe number of satellites within individual halos follows aPoisson (or some other) probability distribution. The two-point correlation function on a scale r can be decomposedinto one-halo ( ξ ) and two-halo ( ξ ) terms ξ ( r ) = [1 + ξ ( r )] + ξ h ( r ) , (1)corresponding to contributions to the correlation functionfrom galaxy pairs which reside in the same halo and in twodifferent halos respectively (Zheng 2004).The 2-halo term can be computed as the halo correla-tion function weighted by the distribution and occupationnumber of galaxies within each halo. The 2-halo term of thegalaxy power spectrum (PS) is P ( k ) = P m ( k ) » n g Z M max dM dndM h N i M b ( M ) y g ( k, M ) – , (2)where P m is the mass PS and y g is the normalised Fouriertransform of the galaxy distribution, which is assumed to fol-low a Navarro, Frenk & White (1997; NFW) profile (see e.g.Seljak 2000; Zheng 2004). Here ¯ n g is the mean number den-sity of galaxies. We assume the Sheth Tormen (1999) massfunction dn/dM using parameters from Jenkins et al. (2001)throughout this paper. To compute the halo bias b ( M ) weuse the Sheth, Mo and Tormen (2001) fitting formula. Thequantity M max is taken to be the mass of a halo with separa-tion 2 r . The 2-halo term for the correlation function followsfrom ξ h ( r ) = 12 π Z ∞ P ( k ) k sin krkr dk. (3)In real space the 1-halo term can be computed using(Berlind & Weinberg 2002)1 + ξ h ( r ) = 12 πr ¯ n × Z ∞ dM dndM h N ( N − i M R vir ( M ) F ′ „ r R vir « , (4)where h N ( N − i M is the average number of galaxy pairswithin halos of mass M . The distribution of multiple galaxieswithin a single halo is described by the function F ′ ( x ), whichis the differential probability that galaxy pairs are separated by a dimensionless distance x ≡ r/R vir . As is common in theliterature, we assume that there is always a galaxy located atthe center of the halo, and others are regarded as satellitegalaxies. The contribution to F ′ is therefore divided intopairs of galaxies that do, and do not involve a central galaxy,and is computed assuming that satellite galaxies follow thenumber-density distribution of an NFW profile. With thisassumption, the term in the integrand of equation (4) reads h N ( N − i M F ′ ( x )= h N − i M F ′ cs ( x ) + h ( N − N − i M F ′ ss ( x ) . (5)where F ′ ( x ) is the pair-number-weighted average of thecentral-satellite pair distribution F ′ cs ( x ) and the satellite-satellite pair distribution F ′ ss ( x ) (see, e.g., Berlind & Wein-berg 2002; Yang et al. 2003; Zheng 2004), The HOD method of estimating and modeling the cluster-ing of HI galaxies will not work at redshifts beyond z ∼ . , LOFAR , PAPER , 21CMA ) and moreambitious designs are being planned (SKA ). During theepoch of reionization, the PS of 21cm brightness fluctuationsis shaped mainly by the topology of ionized regions. How-ever the situation is expected to be simpler following reion-ization of the intergalactic medium (IGM; z .
6) – whenonly dense pockets of self-shielded hydrogen, such as dampedLy α absorbers (DLA) and Lyman-limit systems (LLS) sur-vive (Wyithe & Loeb 2008; Chang et al. 2007; Pritchard &Loeb 2008). These DLA systems are thought to be the highredshift equivalents of HI rich galaxies in the local Universe(Zwaan et al. 2005b). We do not expect 21cm self absorp-tion to impact the level of 21cm emission. This conclusion http://astro.berkeley.edu/ ∼ dbacker/EoR/ http://web.phys.cmu.edu/ ∼ past/ (cid:13) , 000–000 Wyithe & Brown is based on 21cm absorption studies towards damped Ly α systems at a range of redshifts between z ∼ z ∼ . α systems have a spin temperature that is large relativeto the temperature of the cosmic microwave background ra-diation, and will therefore have a level of emission that isindependent of the kinetic gas temperature (e.g. Kanekar &Chengalur 2003). Thus the intensity of 21cm emission canbe directly related to the column density of HI. As mentioned above, low spatial resolution observationscould be used to detect surface brightness fluctuations in21cm emission from the cumulative sum of HI galaxies,rather than from individual sources of emission. Here the PSis a more natural observable than the correlation function,since a radio interferometer records visibilities that directlysample the PS. In the linear regime the 21cm PS followsdirectly from the PS of fluctuations in mass P m ( k ) (Wyithe& Loeb 2009) P HI ( k ) ≈ T x h b i M P m ( k ) , (6)where T b = 23 . z ) / mK is the brightness tem-perature contrast between the mean IGM and the CMBat redshift z , and h b i M is the HI mass weighted halo bias.Note that we have used the subscript HI rather than themore usual in order to reduce confusion with the sub-scripts for the 1-halo and 2-halo PS terms. The fractionof hydrogen that is neutral is described by the parameter x HI ≡ Ω HI / (0 . b ). We assume x HI = 0 .
01 (correspond-ing to Ω HI ∼ × − , Zwaan et al. 2005a) throughoutthis paper. The resulting PS is plotted in the right panel ofFigure 1 (dashed line). The constant T b hides the implicitassumptions that the 21cm emission from the galaxies is notself absorbed, and that the spin temperature is much largerthan the temperature of the cosmic microwave background.On small scales a model is needed to relate HI mass to halomass. To achieve this we modify the HOD formalism as out-lined below. Since surface brightness fluctuations depend on the totalHI mass within a halo rather than on number counts ofindividual galaxies, the number of galaxies, and the numberof galaxy pairs per halo in the HOD formalism need to beweighted by the HI mass per galaxy. In analogy with theHOD formalism, we distribute this mass between centraland satellite galaxies. We define h M HI , c i M and h M HI , s i M tobe the mean HI mass of central galaxies and of the combinedsatellite galaxies within a halo of mass M respectively.To compute the 2-halo PS, we replace h N i M in equa-tion (2) with the mean value of the total HI mass in a haloof mass M , i.e. h M HI i M = h M HI , c i M + h M HI , s i M , yielding P , gg ( k ) = T x P m ( k ) × » ρ HI Z M max dM dndM h M HI i M b ( M ) y g ( k, M ) – , (7) where, ¯ ρ HI is the mean density of HI contributed by all galax-ies in the IGM. The 2-halo term ξ h, HI ( r ) follows from sub-stitution into equation (3).To compute the 1-halo term we again weight the numberof galaxies by their HI mass. In difference from the calcula-tion of the 1-halo term for galaxy clustering, the distribu-tion of satellite masses will be important in addition to thenumber. This aspect of the HOD modeling will require sim-ulation for a proper treatment (e.g. Bagla & Khandai 2009).However for the purposes of our analysis it is sufficient toassume that most of the satellite HI for a halo mass M iscontained within satellites of similar mass (as would be thecase for a steep power-law mass function with a lower cutofffor example). We therefore further define h m HI , s i M to be themean HI mass of satellite galaxies within a halo of mass M .The coefficients in equation (5) are then modified to yield h ( N − h m HI , s i M × h M HI , c i M i = h M HI , c i M h M HI , s i M (8)and h ( N − h m HI , s i M × ( N − h m HI , s i M i h M HI , s i M h ( N − N − i M = h ( N − i M , and have assumed that h M s i M = h ( N − i M × h m s i M . The modified expression for the 1-halo termtherefore becomes T x + ξ h, HI ( r ) = T x πr ¯ ρ Z ∞ dndM » h M HI , c i M h M HI , s i M F ′ cs „ r R vir « + h M HI , s i M F ′ ss „ r R vir «– R vir ( M ) dM. (10)The correlation function follows from ξ HI ( r ) = [ T x + ξ , HI ( r )] + ξ , HI ( r ).In order to evaluate this expression the HI mass occu-pation of a halo of mass M must be parameterised, and isobviously quite uncertain. For illustration, we choose the fol-lowing polynomial form, with a minimum halo mass ( M min )and characteristic scale ( M ) where satellites contribute HImass that is comparable to the central galaxy, h M HI , c i M ∝ M γ c if M > M min = 0 otherwise . (11)and h M HI , s i M = M HI , c „ MM « γ s if M > M min = 0 otherwise . (12)The average HI mass within a halo of mass M > M min istherefore h M HI i M = h M HI , c i M + h M HI , s i M ∝ M γ c » „ MM « γ s – (13)Note that the constant of proportionality in equations (11)and (13) is not specified but cancels with the same factor Note that the correlation function has dimensions of mK, andtherefore does not have the usual interpretation of probabilityabove random for finding a galaxy pair of separation r .c (cid:13) , 000–000 he HOD of Neutral Hydrogen in ρ HI in equations (7) and (10). From experience of thegalaxy HOD there will be degeneracy between the parame-ters M and γ s . We therefore make the simplification of set-ting γ c = γ s ≡ γ in our parameterisation for the remainderof this paper. The left panel of Figure 1 shows the real-spacecorrelation function at z = 0 . γ = 0 . M min = 10 M ⊙ and M = 10 M ⊙ . Thismodel serves as our fiducial case throughout this paper, andis motivated by the parameters derived from estimates forHIPASS galaxies (Wyithe et al. 2009). In particular we notethe value of γ = 0 . γ ∼ . Since a radio interferometer directly measures the 3 di-mensional distribution of 21cm intensity it is more pow-erful to work in redshift space, where line-of-sight infall(Kaiser 1987) can be used to break the degeneracy betweenneutral fraction and galaxy bias (Wyithe 2008). In additionto gravitational infall the shape of the redshift space PSwill be complicated by peculiar motions of galaxies withingroups or clusters, which produce the so-called fingers-of-god in the redshift space correlation function. In the case of21cm fluctuations, the internal velocities of HI in galaxieswill also contribute to the fingers-of-god. In this paper weuse the combination of the real-space HOD 21cm PS P R , HI ( k ) = 4 π Z drξ HI ( r ) sin krkr r dr, (14)and the dispersion model to estimate the redshift space PSincluding these effects. The dispersion model is written P z , HI ( k perp , k los ) = P R , HI ( k )(1 + βµ ) (1 + k σ k µ / − (15)where µ is the cosine of the angle between the line-of-sightand the unit-vector corresponding to the direction of a par-ticular mode, k perp = kµ , k los = k p − µ , β = Ω . / h b i M and h b i M is the average HI mass weighted halo bias. Thequantity σ k is a constant which describes a “typical” ve-locity dispersion for galaxies and parametrises the promi-nence of the fingers-of-god. Simulations indicate a value of σ k ∼ / (1 + z )km s − (Lahav & Suto 2004). We note thatthe redshift space PS could have been generated through a2-d Fourier transform of the redshift space correlation func-tion computed within the HOD model using the formalismdescribed in Tinker (2007), allowing additional constraintson HOD parameters to be placed based on the prominenceof the fingers-of-god as in Wyithe et al. (2009). In particularthe assumption of γ < σ k .The left panel of Figure 2 shows the resulting redshiftspace PS for the fiducial model. The large scale motionsinduced by infall into overdense regions can be seen as an extension of the PS along the line-of-sight at small k , whilethe fingers of god are manifest as a compression at large k .The right panel of Figure 1 shows the corresponding spher-ically averaged redshift space PS (solid line) P sphz , HI ( k ) = Z P z , HI ( k perp , k los ) dk perp dk los . (16)For comparison, the dotted lines in the right hand panel ofFigure 2 show the 1-halo and 2-halo contributions to thespherically averaged redshift space PS.The spherically averaged redshift space PS can be com-pared with the linear real-space 21cm PS estimated basedon the neutral fraction x HI = 0 .
01 and the linear mass PS(dashed line, equation 6). On large (linear) scales the spher-ically averaged PS is larger in redshift space than in realspace. This is analogous to the excess power seen in red-shift space clustering of galaxy surveys (Kaiser 1987), andis due to an increase in the 21cm optical depth owing tovelocity compression (Barkana & Loeb 2005) towards highdensity regions. On small scales there is excess power abovethe linear theory expectation owing to the inclusion of thenon-linear 1-halo term. The 21cm PS shows a steepeningat large k owing to the mass weighting in the 1-halo term.This steepening is also seen in the simulations of Bagla &Khandai (2009). Figure 3 illustrates the sensitivity of the clustering and the21cm PS to variations in the HOD parameters. The solidlines repeat the fiducial model from Figure 1. For compar-ison, the dotted, dashed and dot-dashed lines show varia-tions on this model, with γ = 0 . M min = 10 M ⊙ and M = 10 M ⊙ respectively (with the remaining parametersset to their fiducial values in each case). On the largest scalesthe clustering amplitude is most sensitive to M min (dashedlines), which enables the typical host halo mass of HI to bemeasured from the PS amplitude (Wyithe 2008). Loweringthe value of M min (while keeping M fixed) implies a smallerfraction of HI in satellites, and hence a relative decrease ofpower on small scales. A smaller value of γ also leads to a rel-ative decrease of power on small scales because the flatterpower-law preferentially places mass in the more commonlow mass halos (with M < M ), and so lowers the fractionof HI in satellites (dotted lines). Conversely, a smaller valueof M leads to a larger fraction of HI in satellite systems, andhence an increase of small scale power (dot-dashed lines).The variation in shape and amplitude of the 21cm PSimplies that parameter values for a particular HOD modelcould be constrained if the PS were measured with sufficientsignal-to-noise. In the remainder of this paper we thereforefirst discuss the sensitivity of a radio interferometer to the21cm PS, and then estimate the corresponding constraintson the 5 parameters in our HOD model that could be placedusing observations of 21cm intensity fluctuations. In this paper we estimate the ability of a telescope likeASKAP to measure the clustering of 21cm intensity fluc-tuations, and hence to estimate HOD parameters, and the c (cid:13) , 000–000 Wyithe & Brown
Figure 1.
An example of the correlation function and PS of intensity fluctuations for a hypothetical HOD model at z = 0 . Left-handpanel: the model correlation function.
Right-hand panel:
The corresponding spherically averaged redshift space PS (solid line). In eachcase the 1-halo and 2-halo terms are plotted as dotted curves. For comparison we plot the spherically averaged sensitivity within pixelsof width ∆ k = k/
10 for an a radio interferometer resembling the design of ASKAP (thick gray line). We also show the real-space 21cmPS assuming a linear mass-density PS (dashed line). For calculation of observational noise an integration of 3000 hours was assumed,with a multiple primary beam total field of view corresponding to 30(1 + z ) square degrees (see text for details). The cutoff at largescales is due to foreground removal within a finite frequency band-pass. Figure 2.
The redshift space PS of intensity fluctuations corresponding to the example in Figure 1.
Left-hand panel:
The redshift-space PS assuming a dispersion model with σ k = 650 / (1 + z )km s − . Right-hand panel:
Contours of the signal-to-noise (separated byfactors of √
10) within pixels of width ∆ k = k/
10. The thick contour corresponds to a signal-to-noise per pixel of unity. For calculationof observational noise an integration of 3000 hours was assumed, with a multiple primary beam total field of view corresponding to30(1 + z ) square degrees (see text for details). The cutoffs at large and small scales perpendicular to the line-of-sight are due to thelack of short and long baselines respectively. The cutoff at large scales along the line-of-sight is due to foreground removal within a finiteband-pass. total HI content of the Universe. The latter quantity, whichis not available from the clustering of resolved galaxies, couldbe used to bridge the gap in measurements of Ω HI (the cos-mic density of HI relative to the critical density) betweenthe local Universe where this quantity can be determinedfrom integration of the HI mass function, and z & α absorbers.To compute the sensitivity ∆ P HI ( k perp , k los ) of a radio-interferometer to the 21cm PS, we follow the procedureoutlined by McQuinn et al. (2006) and Bowman, Morales& Hewitt (2007) [see also Wyithe, Loeb & Geil (2008)].The important issues are discussed below, but the readeris referred to these papers for further details. The uncer-tainty comprises components due to the thermal noise, anddue to sample variance within the finite volume of the ob- servations. We also include a Poisson component due tothe finite sampling of each mode (Wyithe 2008), since thepost-reionization 21cm PS is generated by discrete clumpsrather than a diffuse IGM. We consider a telescope basedon ASKAP. This telescope is assumed to have 36 dish an-tenna with a density distributed as ρ ( r ) ∝ r − within adiameter of 2km. The antennae are each 12m in diameter,and being dishes are assumed to have physical and effec-tive collecting areas that are equal. We assume that fore-grounds can be removed over 80MHz bins, within a band-pass of 300MHz [based on removal within 1/4 of the avail-able bandpass (McQuinn et al. 2006)]. Foreground removaltherefore imposes a minimum on the wave-number acces-sible of k ∼ . z ) / . − Mpc − , although access tothe large scale modes is actually limited by the number ofshort baselines available. An important ingredient is the an- c (cid:13) , 000–000 he HOD of Neutral Hydrogen Figure 3.
Examples of correlation functions ( left panel ) and spherically averaged redshift space PS ( right panel ) of intensity fluctuationsfor different HOD models at z = 0 .
7. The solid lines repeat the fiducial model from Figure 1 (parameters listed in the top-right corner ofthe left panel). For comparison, the dotted, dashed and dot-dashed lines show variations on this model, with γ = 0 . M min = 10 M ⊙ and M = 10 M ⊙ respectively (with the remaining parameters set to their fiducial values in each case). For comparison with the PS,we also plot the spherically averaged sensitivity within pixels of width ∆ k = k/
10 for an a radio interferometer resembling the design ofASKAP (right panel, thick gray line). gular dependence of the number of modes accessible to thearray (McQuinn et al. 2006). ASKAP is designed to havemultiple primary beams facilitated by a focal plane phasedarray. We assume 30 fields are observed simultaneously for3000 hr each, yielding ∼ z ) square degrees [wherethe factor of (1 + z ) originates from the frequency depen-dence of the primary beam]. The signal-to-noise for obser-vation of the PS in the left panel of Figure 2 is shown inthe right panel of Figure 2. A telescope like ASKAP wouldbe most sensitive to modes of k perp ∼ . − − and k los ∼ . − . − . The spherically averaged signal-to-noise (within bins of ∆ k = k/
10) is shown in the right panelsof Figures 1 and 3 (grey curves). Comparison of the noisecurve with the variability of the PS amplitude and shapeamong different HOD models for the 21cm PS (Figure 3)indicates that a telescope like ASKAP would be sufficientlysensitive to generate constraints on the HOD. Moreover, thespatial scale on which the array would be most sensitivecorresponds to wave-numbers where we expect 1-halo and2-halo contributions to be comparable, indicating that suchobservations may constrain HOD model parameters. Thisstatement is quantified in the next section.
Based on our estimate of the sensitivity to the 21cm PS weforecast the ability of ASKAP to constrain the HI HOD.To begin, we assume the fiducial model P truez , HI ( k perp , k los ), asshown in Figure 2, and estimate the accuracy with whichthe parameters could be inferred. Our HOD model for the21cm PS has five parameters, M min , M , γ , x HI and σ k .For combinations of these parameters ( M min , M , γ , x HI , σ k ) that differ from the fiducial case (10 M ⊙ , 10 M ⊙ , 0.5,0.01, 650 / (1+ z )km/s), we compute a trial model for the realspace correlation function. We then use this to calculate thechi-squared of the difference between the fiducial and the trial models χ = (17) X k perp X k los „ P truez , HI − P z , HI ( k perp , k los | M min , M , γ, x HI , σ k , σ )∆ P z , HI ( k perp , k los ) « , and hence find the likelihood L ( M min , M , γ, x HI , σ k ) = Z dσ ′ dpdσ ′ exp „ − χ « . (18)The uncertainty introduced through imperfect knowledge ofthe PS amplitude (which is proportional to the normal-ization of the primordial PS, σ ) is degenerate with x H (Wyithe 2008). For this reason the uncertainty in σ hasbeen explicitly included in equation (18). We assume a Gaus-sian distribution dp/dσ for σ with σ = 0 . ± .
03 (Ko-matsu et al. 2009).Figure 4 shows an example of forecast constraints onHOD model parameters for a telescope like ASKAP, assum-ing a 3000hr integration of a single pointing [ ∼ z ) square degrees] centered on z = 0 .
7. Results are presentedin the upper panels of Figures 4 which shows contours ofthe likelihood in 2-d projections of this 5-parameter space.Here prior probabilities on log x HI , log M min , log M , γ and σ k are assumed to be constant. The contours are placed at60%, 30% and 10% of the peak likelihood. The lower panelsshow the marginalised likelihoods on the individual param-eters x HI , M min , M and γ .A deep integration of a single pointing for a telescopelike ASKAP would place some constraints on the minimummass (the projected uncertainty on M min is ∼ . ∼
20% constraint on γ ). In addition to these constraints onthe halo occupation distribution of HI, observations of the21cm PS would also provide a measurement of the globalneutral fraction (or equivalently Ω HI ), which would be con-strained with a relative uncertainty of 20% at z ∼ .
7. Thisindicates that 21cm intensity fluctuations could be used tomeasure the evolution of Ω HI from z ∼ z & . HI basedon the redshift space PS would be complimentary to thedetection of individual rare peaks, which could facilitating c (cid:13) , 000–000 Wyithe & Brown
Figure 4.
Example of forecast constraints on HOD model parameters from 21cm intensity fluctuations, assuming a 3000hr integrationof a 30(1 + z ) square degree field with an array based on the ASKAP design centered on z = 0 . upper panels show contours of the likelihood in 2-d projections of the 5-parameter space used for the HOD modeling of 21cm intensityfluctuations, while the lower panels show the marginalised likelihoods on individual parameters. Here prior probabilities on γ , log x HI ,log M min and log M are assumed to be constant. The contours are placed at 60%, 30% and 10% of the peak likelihood. The position ofthe dot indicates the peak likelihood in the 5-dimensional parameter space (i.e. the input model). a direct estimate of the cosmic HI mass density (Bagla &Khandai 2009). The combination of measurements for x HI and the HOD pa-rameters γ , M min and M indicates that the HI mass func-tion (summing both central and satellite galaxies) could beapproximated from the HOD using dndM HI = dndM dMdM HI . (19)where M HI = CM γ (1 + ( M/M ) γ ) and the constant C isevaluated from C = x HI . b / Ω M ) ρ m R ∞ M min M γ “ “ MM ” γ ” dndM dM . (20)The left panel of Figure 5 shows the mass function forthe fiducial model in Figure 1 (thick grey curve) as wellas ten HI mass functions computed assuming parametersdrawn at random from the joint probability distribution [ ∝ M − M − x − L ( M min , M , γ, x HI , σ k )], projections of whichare shown in Figure 4. While the values over which the HImass range extends in these realisations shows some vari-ability, the possibility of constraints on γ and x HI impliedby Figure 4 mean that the overall shape of the HI massfunction could be quite well constrained by observation ofa redshifted 21cm PS. In the central panel of Figure 5 weshow the corresponding the HI mass functions for centralgalaxies [obtained by instead substituting M HI = CM γ ]. Inthis case the range of realisations is much larger, which can be traced to the degeneracy between M min and M seen inFigure 4.In addition to the HI mass function, it would also bepossible to constrain the fraction of hydrogen within galaxiesthat is in atomic form. This number is given by f HI = x HI F col ( M min ) , (21)where F col ( M min ) is the fraction of dark-matter that is col-lapsed in halos more massive than M min . From the upperleft panel of Figure 4 we see that there is a degeneracy be-tween x HI and M min . Larger neutral fractions correspond tolower values of M min and hence larger collapsed fractions.As a result the ratio f HI would be very well constrained,as shown by the likelihood distribution in the right handpanel of Figure 5 (which is based on the distributions inthe left panel of Figure 5). The evolution of f , which canalso be measured locally from clustering with a value of f HI = 10 − . ± . (Wyithe, Brown, Zwaan & Meyer 2009)will provide an important ingredient for studies of the roleof HI in star formation. Due to the faintness of HI emission from individual galax-ies, even deep HI surveys will be limited to samples at rel-atively low redshift ( z . .
7) for the next decade. Howeverthese surveys will be able to detect fluctuations in 21cm in-tensity produced by the ensemble of galaxies out to higherredshifts, using observational techniques that are analogousto those being discussed with respect to the reionizationepoch at z & c (cid:13) , 000–000 he HOD of Neutral Hydrogen Figure 5.
Examples of the range for the total halo (left) and central galaxy (central panel) HI mass functions. In addition to the fiducialcase (thick lines, corresponding to the model in Figure 1), ten HI mass functions are shown in each case with parameters drawn fromthe probability distribution for HOD parameters in Figure 4 (dot-dashed lines). In the right panel we show forecast for correspondingconstraints on f HI . A 3000hr integration of a 30(1+ z ) square degree field with an array based on the ASKAP design centered on z = 0 . f HI was assumed to be constant. a result, studies of HI galaxy clustering could be extendedto redshifts beyond those where individual HI galaxies canbe identified through the use of 21cm intensity fluctuations.To investigate this possibility we have described an approx-imate model for the power spectrum of 21cm fluctuations,which is based on the halo occupation distribution formal-ism for galaxy clustering. Our goal for this paper has been touse this model to estimate the expected amplitude and fea-tures of the 21 cm power-spectrum, rather than to presenta detailed method for extracting the halo occupation of HIfrom an observed power-spectrum. This latter goal wouldrequire numerical simulations (e.g. Bagla & Khandai 2009).To frame our discussion we have made forecasts forASKAP, specifically with respect to the use of the 21cmpower spectrum as a probe of the occupation of HI in darkmatter halos. We have chosen z = 0 . ∼ α absorbers in the higher redshift Universe.The cosmic star-formation rate has declined by morethan an order of magnitude in the past 8 billion years (Lillyet al. 1996, Madau et al. 1996). Optical studies paint a some-what passive picture of galaxy formation, with the stellarmass density of galaxies gradually increasing and an increas-ing fraction of stellar mass mass ending up within red galax-ies that have negligible star-formation (e.g., Brown et al.2008). On the other hand, the combination of direct HI ob-servations at low redshift (Zwaan et al. 2005; Lah et al 2007)and damped Ly α absorbers in the spectra of high-redshiftQSOs (Prochaska et al. 2005) show that the neutral gas den- sity has remained remarkably constant over the age of theuniverse. The evolutionary and environmental relationshipsbetween the neutral gas which provides the fuel for star for-mation and the stars that form are central to understandingthese and related issues. The study of the halo occupationdistribution of HI based on 21cm fluctuations has the poten-tial to allow these studies to be made at redshifts beyondthose where individual galaxies can be observed in HI witheither existing or future radio telescopes. Acknowledgments.
The research was supported by theAustralian Research Council (JSBW).
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