Evolution of galaxy clustering

J. S. Bagla*Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA

Accepted 1998 April 30. Received 1998 April 29; in original form 1997 November 13

A B S T R A C TWe study the evolution of the correlation function of dark matter haloes in the CDM class ofmodels. We show that the halo correlation function does not evolve in proportion with thecorrelation function of the underlying mass distribution. The earliest haloes to collapse, whichcorrespond to rare peaks in the density field, cluster very strongly. The amplitude of the halocorrelation function decreases from its initial, large, value. This decrease continues until theaverage peaks have collapsed, after which the amplitude grows slowly. This behaviour isshown to be generic and the epoch of minimum amplitude depends only on the rmsfluctuations in mass at the relevant scale and, to a much smaller extent, on the slope of thepower spectrum at that scale. We discuss the relevance of this result for interpretation ofobservations of galaxy and quasar clustering.

Key words: galaxies: formation – cosmology: theory – early Universe – large-scale structureof Universe.

1 I N T RO D U C T I O N

It is believed that structures like galaxies and clusters of galaxiesformed by gravitational amplification of small perturbations. Thisimplies that clustering in the mass distribution in the Universealways increases with time. In this model, galaxies form in highlyover-dense haloes of dark matter. Evolution of galaxy clusteringand its relation to the clustering in the underlying mass distributionis an important question that needs to be addressed before we cancorrectly interpret the observations of galaxy clustering. In thispaper we discuss one approach to this problem and comment on therelevance of our results for interpretation of observations of galaxyclustering and its evolution.

Clustering properties of haloes have been studied by manyauthors (see, for example, Gelb & Bertschinger 1994b). Thesestudies show that there is no simple, scale-independent relationbetween the correlation functions for the mass distribution andhaloes. The evolution of clustering properties of haloes has beenstudied by Brainerd & Villumsen (1994). They computed the halo–halo correlation function and found that it evolves very slowly inmost cases. In some cases they found a decreasing phase before theamplitude of the halo correlation function starts growing. Recently,some authors have studied the effects of a decreasing correlationfunction, and an epoch of minimum amplitude of clustering, on theangular correlation function of galaxies (Ogawa, Roukema &Yamashita 1997).

In this paper we argue that the halo correlation function, forhaloes above a given mass, has a generic behaviour: its amplitude isvery large at early times, and decreases very rapidly. The amplitudereaches a minima and then increases slowly.

The paper is organized as follows: Section 2 outlines a model forevolution of halo clustering. In Section 3 we describe the numericalsimulations used for testing the model. This section also containsdetails of our approach for identifying haloes and calculation of thehalo correlation function. Results are presented in Sections 4 and 5we discuss the generalization to galaxy clustering and the relevanceof our results for interpretation of observations. We summarize themain conclusions in Section 6.

2 ‘ T H E H A L O G R A I L’

In most models of structure formation, the initial density field isassumed to be a Gaussian random field. Gravitational instabilityleads to amplification of density perturbations, and over-denseregions collapse to form virialized haloes. If the threshold densitycontrast for formation of haloes is much larger than the rmsdispersion in density contrast, as is the case at early times, only afew rare peaks collapse into haloes. These peaks are expected tocluster strongly (Bardeen et al. 1986) compared to the almostsmooth mass distribution, and are poor tracers of the underlyingmass distribution. On the other hand, if the threshold is less than, orcomparable to, the rms dispersion, almost all peaks collapse to formhaloes and the halo number density traces the mass distribution atscales (much) larger than the typical inter-halo separation. Thus weexpect the haloes to become ‘better’ tracers of the mass distributionwith time. In the following discussion, we quantify these statementsand present a simple model for evolution of the halo correlationfunction.

Consider the distribution of haloes of mass M, or larger, beforetypical haloes of this mass have collapsed. To quantify this, we firstdefine a bias parameter nðM; zÞ ¼ dc=jðM; zÞ, where jðMÞ is the

Mon. Not. R. Astron. Soc. 299, 417–424 (1998)

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* E-mail: jasjeet@ast.cam.ac.uk

(linearly extrapolated) rms dispersion in the density contrast at massscale M and dc is the linearly extrapolated density contrast at whichhaloes are expected to collapse and virialize. We use the valuedc ¼ 1:68, obtained from the spherical top hat collapse model(Gunn & Gott 1972). Most haloes of mass M collapse whennðMÞ < 1. At early times, n q 1, and the number density ofcollapsed haloes is very small (compared to, say, %=M, where % isthe background density at that epoch). We can write the linearcorrelation function of these rare peaks, at scales whereyðM; rÞ=yðM; 0Þ p 1, as (Bardeen et al. 1986)

yHðM; rÞ ¼ exp n2 yðM; rÞyðM; 0Þ

� �¹ 1 ð1Þ

Here yðM; rÞ is the correlation function of the smoothed densityfield (smoothed at mass scale M) evaluated at scale r, and yH is thehalo correlation function. The halo correlation has an exponentiallylarge amplitude over the range of scales wheren2½yðM; rÞ=yðM; 0Þÿ > 1. At very large scales, the halo correlationfunction is related to the mass correlation as yHðM; rÞ.n2½yðM; rÞ=yðM; 0Þÿ. It follows that at early times, characterizedby n2 q 1, the halo correlation function has a much largeramplitude than the mass correlation function at all scales.

Time evolution of the linear correlation function for haloes, asdescribed in equation (1), is controlled by the function n – the ratioyðM; rÞ=yðM; 0Þ being a function of scale only.

nðM; zÞ ¼dc

jðM; zÞ¼

Dþðz*Þ

DþðzÞ¼

1 þ z1 þ z*

: ð2Þ

Here Dþ is the growing mode for density perturbations in lineartheory and z* is fixed by requiring nðM; z*Þ ¼ 1. The last equality isvalid only in the Einstein–de Sitter Universe as in that caseDþðzÞ ~ ð1 þ zÞ¹1. Dþ is a monotonically increasing function oftime, implying that n is a monotonically decreasing function.Therefore, at early times, the amplitude of correlation function ofhaloes is a rapidly decreasing function of time. In terms ofcorrelation bias,1 defined as

b2ðr; zÞ ¼yHðr; zÞyðr; zÞ

; ð3Þ

we can say that at early times, bias b increases more rapidly thanð1 þ zÞ with redshift.

Equation (1) gives only the linearly extrapolated correlationfunction for haloes. However, the qualitative behaviour, beingexponentially strong, should survive the non-linear evolution.

At later epochs, when nðMÞ < 1, equation (1) is no longer valid.By this time, most haloes of mass M have collapsed and the halodistribution begins to trace the underlying mass distribution. There-fore, further evolution of the halo distribution must reflect thegrowth of density perturbations. There is another way of explainingthis for hierarchical models: low-mass haloes merge and give rise tomore massive haloes. As gravity brings haloes closer for merger, thehalo correlation function must increase. However, the rate of growthof correlation function will be slow as anti-biased haloes continueto collapse for some time. In other words, the correlation bias willchange with redshift at a rate lower than 1 þ z.

Thus, there are two effects that contribute to the evolution ofcorrelation function of haloes: the intrinsic correlation function ofhaloes that have collapsed, and, gravitational instability. Theseeffects act in opposite directions. At early times the rapidly

changing intrinsic correlation function of the few haloes thathave collapsed dominates. At late times, gravitational clusteringtakes over, leading to growth of the correlation function. At someintermediate epoch, these effects must cancel each other leading toa stationary amplitude of the halo correlation function. This is likelyto happen when n < 1.

3 N U M E R I C A L S I M U L AT I O N S

We used a set of simulations of the standard CDM model, normal-ized so that the linearly extrapolated rms fluctuations in density,smoothed with a spherical top hat window function at the scale of8 h¹1Mpc, is j8 ¼ 0:6. We used h ¼ 0:5. All simulations werecarried out with a PM (particle–mesh) code and 1283 particles in abox with the same number of cells.

We used the friends-of-friends (FOF) algorithm with a linkinglength of 0:3 grid length to identify dense haloes. This is larger thanthe ‘traditional’ value of 0:2 and has been chosen so that the massfunction matches the Press–Schechter (Press & Schechter 1975)mass function with dc ¼ 1:68. We need to do this to compensate forthe lower resolution of a PM code. For the purpose of computing thecorrelation function, we used haloes with 10 or more particles. Thiswas done to avoid noise arising from erroneous detections ofsmaller groups.

In principle, we should identify haloes with mass in a given rangeand compute the correlation function for these at different epochs.However, the following reasons force us to use a different strategy.

(i) The FOF algorithm is known to link dynamically distincthaloes (Gelb & Bertschinger 1994a). (Smaller linking lengths tendto dissolve some haloes.) This leads to an incorrect estimate of themass of haloes. We will also underestimate the number of close pairsof haloes, and therefore, the amplitude of the correlation function.

(ii) Finite mass resolution in numerical simulations leads to theover-merging problem (Moore, Katz & Lake 1996). The over-merging problem in collisionless simulations makes it difficult toestimate the number of haloes of a given mass that may havesurvived inside bigger haloes.

These problems are particularly serious at late times when thenumber of haloes is large and there are many cluster sized haloespresent in the system. To circumvent these problems we adopt adifferent approach. We do not consider haloes as being one uniteach. Instead, we compute the correlation function of particlescontained in these haloes. This strategy is also appropriate forstudying the evolution of galaxy correlation function, as galaxies areknown to survive inside groups and clusters of galaxies. Although ourmethod ignores the large range in masses of galaxies by using thisparticular method, we are able to include the contribution of verymassive haloes that are likely to contain many galaxies.

4 R E S U LT S

To show that haloes cluster very strongly at early times, we haveplotted the haloes in Fig. 1, along with a subset of all the N-bodyparticles, in a slice from a CDM simulation with a box size of76:8 h¹1Mpc. We have shown the same slice at four redshifts; z ¼ 3,2, 1 and 0. The thickness of the slice is 3 h¹1Mpc (comoving). Theparticles in haloes (Mhalo $ 1:2 1012 M() are shown as thick dotsand the N-body particles are shown as thin dots. A box on thebottom-left corner shows the projected area per halo, if the haloeswere distributed uniformly then there should be one halo in a regionof this size on an average. It is clear from this figure that the halo

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1 Note that this bias is different from n defined earlier. This degeneracy innomenclature is unfortunate and we will use the term correlation bias for thisone to avoid confusion.

distribution is very non-uniform at early times even though theunderlying mass distribution is fairly homogeneous. It is alsoobvious that the halo distribution develops along the ‘skeleton’ ofthe late time mass distribution. Halos form along filaments andpancakes, or their intersections. Other simulations also show asimilar pattern of evolution. (To see the variation with linkinglength used for locating haloes, see the plots in Brainerd &Villumsen 1994.)

We describe the clustering properties using the averaged two-point correlation function yðrÞ, described as

yðrÞ ¼3r3

�r0

x2yðxÞdx ¼3J3ðrÞ

r3 ð4Þ

where yðxÞ is the two-point correlation function. The averagedcorrelation function is a measure of the excess neighbours around atypical point in a sphere of radius r, whereas the two-pointcorrelation function is a measure of excess neighbours in a sphericalshell at r.

We have plotted the averaged correlation function for the massdistribution and the halo distribution in Fig. 2 for simulations likethose shown in Fig. 1. The error bars denote the 2j dispersionaround the mean evaluated using three realizations of the CDMpower spectrum. The left panel shows the correlation function formass and the right panel shows the correlation function for masscontained within haloes (Mhalo $ 1:2 × 1012 M(). These figuresshow that the clustering in mass increases monotonically, asexpected. These figures also confirm that the amplitude of the

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Figure 1. This figure shows the distribution of mass and haloes (Mhalo $ 1:2 1012 M() in a slice from a CDM simulation. The thickness of the slice is 3 h¹1Mpc.The halo particles are marked as thick points. To highlight the highly non-uniform distribution of haloes, we have drawn a box in the lower left corner of eachframe that shows the region that should contain one halo if these were distributed uniformly. The upper-left panel shows the slice at z ¼ 3, the upper-right panelshows the same at z ¼ 2, the lower-left column shows the same slice at z ¼ 1 and the lower-right panel is for z ¼ 0.

halo correlation function is large at early times and it decreases veryrapidly. The amplitude of the halo correlation function reaches aminima at n < 1:0 and then increases again. To show this point moreclearly, we have plotted the averaged halo correlation function attwo scales as a function of redshift in Fig. 3. Filled squares show theaveraged correlation function at 15 h¹1Mpc and stars show theaveraged correlation function at 5 h¹1Mpc. A dashed line andarrows mark the epoch when n ¼ 1 for these haloes.

The fact that bias is high at early times is an ‘expected’ result thatis reflected in all the models for evolution of bias (see below).However, what we are pointing out here with our simple model, thatis justified by Fig. 3, is that the bias evolves faster than 1 þ z atepochs where n q 1 and slower than 1 þ z at epochs where n p 1.This result is contained in the expression for bias derived by Mo &White (1996) but not in the model by Fry (1996).

In simulations of the CDM model with different box size, we findthat the amplitude of halo correlation function reaches its minimumat lower n, i.e. at a later epoch than n ¼ 1, for more negative indices.We find the range nmin ¼ 0:96 (neff ¼ ¹2:4) to nmin ¼ 1:1 (neff ¼

¹1:9). Here, the index neff is defined as

neff ¼ ¹3 þ∂ ln j2 rð Þ

∂ ln rð5Þ

where the derivative is evaluated at the mass scale of haloes. We arestudying the variation of nmin with the index for power-law models.

The shape of both the mass correlation and halo correlationchanges with time. The halo correlation function becomes less steepwith time, whereas the mass correlation function becomes steeper.If we compare the shape and amplitude of the halo and masscorrelation function at z ¼ 0 in Fig. 2 then it is apparent that the halocorrelation function has a higher amplitude and it is steeper than the

mass correlation function. To quantify this, we have plotted thecorrelation bias as a function of scale at four redshifts in Fig. 4. Thecorrelation bias is defined here using the averaged correlationfunction in the manner of equation (3). The left panel shows thesefor ‘mass-weighted’ halo correlation function, i.e. correlationfunction of mass in haloes. The right panel shows bias computedfrom unweighted halo–halo correlation function. These differ atsmall scales where the effect of over merging is important. At latetimes, the lower curve shows strong suppression of bias at smallscales owing to this effect. However, at early times, this effect isrelatively unimportant and the unweighted halo correlation functiongives a better estimate of bias. At late times bias computed from thecorrelation function of mass in haloes (left panel) should resemblethe galaxy correlation function. The curve corresponding to z ¼ 0shows that bias is a function of scale and it increases at small scales.At large scales it approaches an asymptotic value and does notchange much beyond 10 h¹1Mpc.

The qualitative behaviour of the correlation bias as a function ofscale does not evolve very strongly. However, at early times theincrease in bias at smaller scales is more prominent than at latetimes. This is a reflection of the change in shape of the matter andhalo correlation functions. The scale dependence of correlation biashas important implications for inversion of galaxy power spectrumto obtain the initial power spectrum (see Section 5.1).

Fig. 3 shows the variation of correlation function for haloes attwo comoving scales (5 and 15 h¹1Mpc) as a function of redshift.To compare with theoretical models, we first describe the figure interms of correlation bias. At early times bias varies at a rate fasterthan 1 þ z, the rate of variation is close to 1 þ z at around n ¼ 1and then it slows down at lower redshifts. In other words, biasvaries very rapidly at early times but at late times the variation isvery slow.

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Figure 2. This figure shows the results from simulations of CDM spectrum, like the one shown in Fig. 1. The averaged correlation function y is shown as afunction of scale r (h¹1Mpc). The left panel shows the evolution of clustering in the total mass distribution. As redshift decreases, y increases monotonically. Wehave shown the points for z ¼ 3 (*), z ¼ 2 (.), z ¼ 1 (×), z ¼ 0:5 (A), z ¼ 0:25 (K) and z ¼ 0 (⊕ ). The right panel shows the correlation function of haloes withMhalo $ 1:2 1012 M(. The amplitude of the halo correlation function decreases rapidly at early times, reaches a minima around z ¼ 1 and then increases again.The error bars show the 6 1j error computed from three realizations of the same power spectrum.

The first model we consider assumes that all the haloes areassumed to form at the same redshift z*. These haloes form withsome initial local bias2 b*. The model then evolves the clustering ofthese haloes by solving the continuity equation. For comparison, wehave plotted (solid lines) the prediction of this model through thelow redshift points. The model predicts

b ¼z* þ b*

1 þ z*þ z

b* ¹ 11 þ z*

¼ 1 þ ð1 þ zÞðb0 ¹ 1Þ: ð6Þ

b0 is the bias the z ¼ 0. As is clear from this equation, an unbiasedset of tracers always remain unbiased. At early times, the combinedassumption of conservation of the numbers of haloes and a localbias is not valid as the rate of formation of haloes is high and theseare not fair tracers of the mass distribution – this leads to the strongdisagreement of predictions with the simulation data at early times.

Mo & White (1996) have developed a model for biasing in whichthe correlation bias is assumed to be independent of scale for haloesof any given mass. They predict variation of the form

b ¼ 1 þ1dc

n2 ¹ 1ÿ �

¼ 1 þ1dc

1 þ z1 þ z*

� �2

¹1

� �ð7Þ

where the second equality holds only for an Einstein–deSitterUniverse.

Matarrese et al. (1997) have generalized the model of Mo &White (1996) by combining the bias for different mass scales. They

use many different ansatz for variation of ‘visibility’ of haloes andcompute the correlation bias in each case. In general, they get thefollowing form for correlation bias

bðzÞ ¼ c1 þ c2ð1 þ zÞb ð8Þ

where some of the parameters can be fixed/related by choosing anappropriate ansatz. Using the coefficients for the appropriateMmin ð¼ MhaloÞ and accounting for the fact that we are using adifferent normalization, we have computed the predicted amplitudeof correlation function for comparison with the values obtainedfrom simulations. The predictions of their model are shown asdotted lines in Fig. 3. It is clear that although the match is good athigh redshifts, this model does poorly at low redshifts. The mainreason being that the parent model, Mo & White (1996), deals withthe unweighted halo correlation function. Therefore any compar-ison with a weighted halo correlation function will not work.However, as we argued in Section 3, the mass weighted correlationfunction is a better candidate for the galaxy correlation function.Unweighted halo correlation function predicts an anti-bias at latetimes. This is never seen in the weighted correlation function andhence the large mismatch.

Some authors have constructed analytical models to understandthe evolution of bias (Catelan et al. 1997 and the references citedtherein) field where the concept is generalized from statistical bias(only a function of epoch) to a bias that depends both on positionand epoch. In these models the mapping from the initial halodistribution to the final one is performed using perturbative orapproximate methods.

5 D I S C U S S I O N

In Section 4 we described evolution of the correlation function formass contained in haloes of mass greater than a given cut-off. Theseresults, when applied to galaxies, have many important implica-tions. In Section 5.1, we will discuss the calculation of the initialpower spectrum from the observed galaxy correlation function inview of the results presented in Section 4. In Section 5.2 we turn tothe question of evolution of galaxy/quasar clustering and its relationwith the evolution of halo clustering. Lastly, we outline someimplications of these results for evolution of the intergalacticmedium and galaxy formation models in Section 5.3.

5.1 Galaxy correlation function and the initial powerspectrum

In this section, we assume that the halo distribution and galaxydistribution are the same at the present epoch. This is a reasonableassumption for studying galaxy clustering at a given epoch, as longas the mass of haloes is not too different from the mass of typicalgalaxies studied in surveys.

The shapes of mass and galaxy correlation functions are different,even at late times (Figs 2, 3 and 4). These differences introduce errorsin calculation of the initial power spectrum from observations ofgalaxy clustering using scaling relations (Peacock & Dodds 1996).

Fig. 5 shows the non-linear index nnl as a function of the linearindex nlin of the averaged correlation function. We define nnl as inequation (5) except that j2 is replaced by y. This relation betweenthe indices is obtained by using the power law fit (Bagla &Padmanabhan 1997) in the quasi-linear regime (1 # y # 200)3 to

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2 Local bias is defined as the ratio between density contrasts in galaxynumber and the underlying mass distribution. The model by Fry (1996)assumes that the local bias is independent of position. The assumption oflocal bias implies a correlation bias that is independent of scale.

Figure 3. This plot shows the averaged halo correlation function at twoscales as a function of redshift. Filled squares show the averaged correlationfunction at 15 h¹1Mpc and stars show the averaged correlation function at5 h¹1Mpc. Error bars depict the 61j dispersion around the mean computedfrom three realizations of the power spectrum. A dashed line and arrowsmark the epoch when n ¼ 1 for haloes of mass Mhalo $ 1:2 × 1012 M(. It isclear that the minimum amplitude is reached near that epoch. Thick linesmark the predicted evolution of halo correlation function in the model by Fry(1996). Dotted lines mark the evolution of bias in the model of Mo & White(1996) where we have used the expression for effective bias from Matarreseet al. (1997).

3 For a power-law correlation function, y ¼ 3y=ð3 ¹ gÞ. Thus for g ¼ 1:8, assuggested by observations, this relation between the indices can be used upto about 8 h¹1Mpc.

the scaling relation between the linear and the non-linearcorrelation function (Hamilton et al. 1991). This figure shows thatthis relation flattens out for indices above nnl ¼ ¹1. Two reasonscontribute to this flattening.

(1) The index nlin ¼ ¹1 is a ‘critical index’ in the sense thatpower spectra with an index close to nlin ¼ ¹1 change shape so thatnnl is closer to this critical index in the quasi-linear regime (Bagla &Padmanabhan 1997).

(2) Gravitational clustering of hierarchical models (n > ¹3)restricts the non-linear indices below nnl # 3=y (Bagla & Padma-nabhan 1997). In other words, the non-linear index of the powerspectrum must be in the range ¹3 # nnl # 0, irrespective of theinitial index.

Gravitational instability acts to decrease (but not to erase) thedifferences between different initial conditions. Therefore, anyuncertainties in the non-linear mass correlation function translateinto much larger uncertainties in the initial power spectrum. Thedifference in the shape/slope and the amplitude of the galaxycorrelation function and the mass correlation function is one suchuncertainty.

In order to assess this amplification of uncertainty in a quantita-tive manner, we have mapped a narrow range of non-linear indices,nnl ¼ ¹1:2 6 0:1 (g ¼ 1:8 6 0:1), to the corresponding linearindices. The permitted range of linear indices is much larger(nlin . ¹1:5 6 0:2; g . 1:5 6 0:2). Therefore, even a small errorin the non-linear index4 of the power spectrum can lead to a muchlarger error in the linear power spectrum. In addition to theuncertainty in slope, uncertainty in amplitude also contributes tothe error.

The above discussion shows that inverting the galaxy correlationfunction to determine the initial power spectrum leads to amplifica-tion of uncertainties. This is especially true of scales in the non-linear regime where, in general, correlation bias depends on scale.At larger scales, bias is expected to be scale independent and themain source of error will be the uncertain difference in amplitude.

5.2 Halos and galaxies

Discussion in this paper has assumed, so far, that galaxy clusteringand halo clustering are the same thing. Although this is a reasonableassumption if we are interested in only one epoch with an appro-priate choice of minimum halo mass, the same is not true for therelative evolution of the two distributions. Here, we list someplausible alternatives and outline the evolution of galaxy correla-tion function for each one.

At any given epoch, there are more low mass galaxies than highmass galaxies. This suggests that the galaxy correlation function isdominated by the smallest galaxies.5 In such a case, the galaxycorrelation function must evolve in a manner similar to the halocorrelation function and its amplitude must go through a minimum atsome epoch. For low mass galaxies, this minimum could be at z q 0.

The assumption outlined above applies in a much better way toquasars as these cannot form in very low mass haloes and theredshift at which the amplitude of quasar correlation reaches theminima may be low enough to be observable. If the minimum halomass associated with quasars is greater than 1011 M( then theobserved quasar correlation function should have a higher ampli-tude at z > 2 than at z < 1. Considering the small probability of agiven halo hosting a quasar at a given time, the more appropriatemeasure for clustering of quasars is provided by the unweightedhalo correlation function.

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5 There is, of course, the problem of identifying the same population ofgalaxies at different redshifts.

Figure 4. This figure shows correlation bias as a function of scale for four redshifts. This is shown for haloes of mass Mhalo $ 1:2 × 1012 M( at z ¼ 3, 2, 1 and 0.Bias decreases monotonically with redshift so the points for z ¼ 3 are the ones at the top and the points for z ¼ 0 are at the bottom. Error bars depict the 61j

dispersion around the mean computed from three realizations of the power spectrum. The left panel shows these for mass-weighted halo correlation function. Theright panel shows bias computed from unweighted halo–halo correlation function. These differ at small scales where the effect of over merging is important. Atlate times, the lower curve shows strong suppression of bias at small scales owing to this effect.

4 If we assume the difference between halo clustering and mass clustering tobe the only source of error, then we can estimate its magnitude fromsimulations. The curves for z ¼ 0 in Fig. 2 show this difference to beDn . 0:2 for Mhalo $ 1:2 × 1012 M(, twice the uncertainty of 0:1 we used inour example.

Recent estimates of the quasar correlation function, though basedon small data sets, show that clustering of quasars is indeed strongerat higher redshifts (Kundic 1997; La Franca, Andreani & Cristiani1997). Authors of the second paper claim that the observedevolution of quasar clustering cannot be explained in all modelsof AGN activity. Our analysis suggests that this need not be true.Quasars must show decreasing clustering amplitude if the AGNactivity is not correlated with the large-scale environment.

The assumption of a mass threshold also applies very well togroups and clusters of galaxies. Therefore, the cluster correlationfunction should show a higher amplitude at higher redshifts.

We can use a different ansatz: we can assume that the brightestgalaxies reside in haloes that have formed relatively recently. Wecan identify the mass scale M*ðzÞ by assuming that these correspondto nðM*Þ . 1. For hierarchical models, we know thatjðM1; zÞ > jðM2 > M1; zÞ, implying that nðM2; zÞ > nðM1Þ. As ahigh n corresponds to a high correlation bias, it is clear that thehalo correlation function with threshold M1 will have a loweramplitude than the halo correlation function with threshold M2.This indicates that the correlation function of M* haloes at z1 willhave a much lower amplitude than its equivalent at z2. Therefore weexpect rapid evolution of the galaxy correlation function, muchfaster than the linear growth rate, if galaxies correspond to M*

haloes. This may be a good model for evolution of clustering of faintgalaxies (Brainerd, Smail & Mould 1995).

The above discussion shows that the evolution of galaxy cluster-ing is likely to depend strongly on our choice of relation betweenhaloes and galaxies at different redshifts. If we identify similarobjects at different redshifts, then these will show either a decreas-ing correlation function or nearly constant clustering in comovingco-ordinates. On the other hand, if we are working with the mostprominent/numerous objects at each redshift then we should seerapid evolution of clustering. Gastrophysical effects, in general,make the relation of haloes and galaxies a little fuzzy, and this willresult in an uncertain linear combination of the two types ofevolution of clustering.

Considering the uncertain relation of the relative evolution ofgalaxy/halo and mass clustering, the rate of growth of the correla-tion function at large scales should not be interpreted as the lineargrowth rate of density perturbations. Direct determination of

cosmological parameters by assuming the two rates to be identicalcan lead to wrong results.

5.3 Galaxy clustering and reionization of the IGM

Observations show that the intergalactic medium (IGM) is ionizedat the highest redshifts we can probe using known quasars andgalaxies. Scenarios for reionization of the IGM fall in two basiccategories: ionization by quasars, which have a low number densityand hence the IGM has a very patchy structure at early times.Ionization by proto-galaxies, dwarf galaxies or star clusters that aredistributed uniformly, leading to a quick and homogeneousreionization.

We have shown in this paper that haloes, irrespective of theirmass, cluster very strongly at early times. Therefore, the reioniza-tion of the IGM will be patchy at a scale much larger than n¹1=3,where n is the number density of the sources of ionization. The non-uniformity of haloes at early times is illustrated in Fig. 1.

It is thought that the earliest clusters of stars formed when low-mass haloes Mhalo . 106 M( collapse after cooling by H2 linecooling (Tegmark et al. 1997). If these clusters of stars wereresponsible for reionization of IGM, then, as these have a verylarge number density, n¹1=3 < 10 kpcproper, the ionization structureof the IGM will be fairly homogeneous. However, ultravioletradiation from the first clusters leads to dissociation of H2 mole-cules and hence the Jeans mass increases by a considerable amount(Haiman, Rees & Loeb 1997). The second generation of collapsedobject are like dwarf galaxies Mhalo . 108¹9 M( and these have aslightly lower number density, n¹1=3 < 0:1 Mpcproper. A visualcomparison with Fig. 1 suggests that some parts of the Universewill be at a much greater distance from the nearest source ofionizing radiation. Therefore, one may expect patchiness at scalesof 1 Mpcproper, or about 10 Mpccomov. The early stages of reioniza-tion from such sources may be observed using the 21-cm tomo-graphy type of observations, though at a scale smaller than thatsuggested for quasars (Madau, Meiksin & Rees 1997). On the otherhand, if quasars are responsible for ionizing the IGM then the scaleof patchiness will be much bigger than that expected from thenumber density.

The relative distribution of haloes and the underlying mass inFig. 1 suggests that haloes form preferentially along filaments/pancakes or their intersections. Denser filaments and pancakes can‘resist’ being ionized much more efficiently than the under denseregions. Photo-ionization in the under-dense region will increasethe Jeans mass and suppress collapse of low-mass haloes(Efstathiou 1992) at late times. This may be the reason for theapparent lack of dwarf galaxies in voids. This effect will not be asprominent in the filaments as the recombination time is smaller,and, ionisation proceeds much more slowly along over-denseregions than it does in the under-dense regions.

6 C O N C L U S I O N S

In Section 5, we have highlighted some implications of the modeland simulation results presented in this paper. Our conclusions aresummarized below.

(i) The halo correlation function for haloes with mass greaterthan (Mhalo ¼ 1:2 × 1012 M() at z ¼ 3 is higher than the correlationfunction of haloes above the same threshold today. Therefore, it isonly to be expected that the galaxy correlation function at highredshifts should have a large amplitude (Steidel et al 1998; Bagla1998).

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Figure 5. This figure shows the non-linear index of the power spectrum as afunction of the linear index in the quasi-linear regime (1 # y # 200). Thisrelation is derived using a power-law fit to the relation between the linear andevolved correlation (Hamilton et al. 1991). We have shown a horizontalshaded region to mark the region g ¼ 1:8 6 0:1. This translates into a linearindex of glin . 1:5 6 0:2, marked by the vertical shaded region. Thisdemonstrates the amplification of uncertainty in the index of powerspectrum in the inversion process.

(ii) The halo correlation function (and the correlation function ofmass contained in haloes) has a different shape, as compared to thenon-linear correlation function of mass. Therefore, inverting thegalaxy correlation function to obtain the linear correlation function,or the power spectrum, for mass can lead to wrong answers at smallscales (l # 8 h¹1Mpc).

(iii) The growth rate of correlation function should not be used,directly, to compute omega/lambda by assuming linear growth orstable clustering – we have shown that the rate of evolution of halocorrelation function has little in common with the matter correlationfunction, and does not depend in any obvious way on cosmologicalparameters.

(iv) Quasar correlation function, as it corresponds to high-masshaloes, should show a decreasing phase at z > 1.

(v) Cluster correlation function should also show a decreasingphase, if we can define a sample of clusters at higher redshifts(z . 0:5) with the same mass threshold.

(vi) The rapid evolution of clustering of faint galaxies can beunderstood if these correspond to M* galaxies at high redshifts.

(vii) Earliest structures to form in the Universe will clusterstrongly, therefore the sources that reionize the Universe willoccur in groups that are further apart than expected from theirnumber density and a nearly uniform distribution. This implies thatthe reionization will be very patchy and it may be possible toobserve the patchiness using the redshifted 21-cm radiation(Madau, Meiksin & Rees 1997).

AC K N OW L E D G E M E N T S

I would like to thank Martin Rees, K. Subramanian, Ofer Lahav andShiv Sethi for many useful discussions. I acknowledge the supportof PPARC fellowship at the Institute of Astronomy.

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