The Origin of Cosmic Structure

The origin of cosmic structure

C.S. Frenk University of Durham, Durham, UK

Abstract A timely combination of new theoretical ideas and observational discoveries has brought about significant advances in our understanding of cosmic evolution. In the current paradigm, our Universe has a flat geometry, is undergoing accelerated expansion, and is gravitationally dominated by elementary particles that make up cold dark matter. Within this framework, it is possible to calculate the emergence of galaxies and other structures from small quantum fluctuations imprinted during an epoch of inflationary expansion shortly after the Big Bang. I review the basic concepts that underlie our understanding of the formation of cosmic structure and some of the tools required to calculate it. Although many questions remain unresolved, a coherent picture for the formation of cosmic structure in now beginning to emerge.

1 Introduction The origin of structure in the Universe is a central problem in physics. Its solution will not only inform our understanding of the processes by which matter became organized into galaxies and clusters, but it will also help uncover the identity of dark matter, offer insights into events that happened in the early stages of the Big Bang, and provide a useful check on the values of the fundamental cosmological parameters estimated by other means. The problem of the origin of cosmic structure is well posed because the initial conditions—small perturbations in the density and velocity ﬁeld of matter—are, in principle, known from Big Bang theory and observations of the early Universe, and the basic physical principles involved are understood. The behaviour of dark matter is governed primarily by gravity, while the formation of the visible parts of galaxies involves gas dynamics and radiative processes of various kinds. The early stages of the evolution of these perturbations can be followed by applying linear theory, but the later stages require large computer simulations. With these two methods, it is possible to follow the development of structure from primordial perturbations to the point where the model can be compared with observations. Over the past few years, huge progress has been made in quantifying observationally the properties of galaxies not only in the nearby Universe, but also in the very distant Universe. Since the clustering pattern of galaxies is rich in information about physics and cosmology, much effort is invested in map- ping the distribution of galaxies at different epochs. Two large surveys, the US-based Sloan Digital Sky Survey [1] and the Anglo-Australian ‘2-degree Field Galaxy Redshift Survey’ (2dFGRS) [2], are revolu- tionizing our view of the nearby Universe with order of magnitude increases in the amount of available data. Similarly, new data collected in the past decade or so have opened up the high redshift Universe1 to detailed statistical study [3]. The advent of large computers, particularly parallel supercomputers, and the development of efﬁ- cient algorithms have enabled the accuracy and realism of simulations to keep pace with observational progress. With the wealth of data now available, simulations are essential for the interpretation of astronomical data and to link the data to physical and cosmological theory.

1 In cosmology, distances to galaxies are estimated from the redshift of their spectral lines; higher redshifts correspond to more distant galaxies and thus to earlier epochs.

239 C.S. FRENK

2 Building a model To build a model of large-scale structure, four key factors need to be speciﬁed: (i) the content of the Universe, (ii) the initial conditions, (iii) the growth mechanism, and (iv) the values of the fundamental cosmological parameters. Each of these is discussed in turn.

2.1 The content of the Universe

Densities are usually expressed in terms of the cosmological density parameter, Ω = ρρcrit, where the critical density, ρcrit, is the value that makes the geometry of the Universe ﬂat. The main constituents of the Universe and their contribution to Ω are listed in Table 1.

Table 1: The content of the Universe Component Contribution to Ω CMB radiation Ω = 47 10 5 r × − Massless neutrinos Ω = 3 10 5 ν × − 2 mν Massive neutrinos Ω = 6 10 h i ν × − 1 eV Baryons Ω = 0037 0009 b (of which stars ) Ω = (00023 00023) 00003 s − Dark matter ΩDM 026 004 ' Dark energy Ω 070 004 Λ '

The main contribution to the extragalactic radiation field today is the Cosmic Microwave Back- ground (CMB), the redshifted radiation remaining from the Big Bang. These photons have been propa- gating freely since the epoch of ‘recombination’, approximately 300,000 years after the Big Bang. The CMB provides a direct observational window to the conditions that prevailed in the early Universe. The Big Bang also produced neutrinos that today have an abundance comparable to that of photons. We do not yet know for certain the mass of the neutrino (if any), but even for the largest masses that seem plau- sible at present ( 01 eV), neutrinos make a negligible contribution to the total mass budget (although ∼ they could be as important as baryons). The abundance of baryons, dark matter, and dark energy can be inferred, as discussed below, by combining data on inhomogeneities measured in the CMB with data on galaxy clustering [4]. Independent estimates of Ωb can be obtained with reasonable precision by comparing the abundance of deuterium predicted by Big Bang theory with observations of the absorption lines produced by intergalactic gas clouds at high redshift, seen along the line-of-sight to quasars [5]. Baryons, the overwhelming majority of which are not in stars today, are also dynamically unimportant (except, perhaps, in the cores of galaxies). Dark matter makes up most of the matter content of the Universe today. To the now firm dynamical evidence for its existence in galaxy haloes, even more direct evidence has been added by the phenomenon of gravitational lensing that has now been detected around galaxy haloes (see, for example, Refs. [6, 7]), in galaxy clusters (see Ref. [8]), and in the general mass field (see Ref. [9] and references therein). The distribution of dark matter in rich clusters can be reconstructed in fair detail from the weak lensing of distant background galaxies, in what amounts virtually to the imaging of cluster dark matter. Various dynamical tests give values of ΩDM 03, consistent with the CMB estimates and also with ' other, independent determinations such as those based on the baryon fraction in clusters [10, 11] and on the evolution in the abundance of galaxy clusters [12, 13]. Since ΩDM is much larger than Ωb, it follows that dark matter cannot be made of baryons. The most popular candidate for dark matter is a hypothetical elementary particle, such as those predicted by supersymmetric theories of particle physics. These particles are generically referred to as cold dark matter. Hot dark matter is also possible: for example, if the neutrino had a mass of 5 eV. However, early cosmological simulations show that the galaxy distribu- ∼ tion in a Universe dominated by hot dark matter would not resemble that observed in our Universe [14].

240 THE ORIGIN OF COSMIC STRUCTURE

A recent addition to the cosmic budget is dark energy, evidence for which was first provided by studies of type Ia supernovae2 [17, 18]. These presumed ‘standard candles’ can now be observed at redshifts between 0.5 and 1 and beyond. Those at z (0.3–1) are fainter than would be expected if universal ∼ expansion were decelerating, indicating that expansion is, in fact, accelerating. Within standard Fried- mann cosmology, there is only one agent that can produce an accelerating expansion. This is nowadays known as dark energy, a generalization of the cosmological constant first introduced by Einstein, which could, in principle, vary with time. The supernova evidence is consistent with the value Ω 07. Fur- Λ ' ther, independent evidence for dark energy is provided by a recent joint analysis of CMB data (see next section) and the 2dFGRS [19]. Amazingly, when the components are added together, the data are consistent with a flat Universe, Ωtot = 1.

2.2 Initial conditions The idea that galaxies and other cosmic structures are the result of the slow amplification by gravity of small primordial perturbations present in the mass density at early times goes back at least to the 1940s [20]. However, it was not until the early 1980s that a physical mechanism capable of producing small perturbations was identified. This is the mechanism of inflation, an idea attributed to Guth [21] that changed the face of modern cosmology. Inflation is produced by the dominant presence of a quantum scalar field, which rolls slowly from a false to the true vacuum, maintaining an approximately constant energy density and causing the early Universe to expand exponentially for a brief period of time. Quan- tum fluctuations in the inflation field are blown up to macroscopic scales and become established as genuine adiabatic ripples in the energy density. Simple models of inflation predict the general properties of the resulting fluctuation field: it has Gaussian distributed amplitudes and a near scale-invariant power spectrum [22]. In 1992, after three decades of ever more sensitive searches, evidence for the presence of small fluctuations in the early Universe was finally obtained. Since the matter and radiation fields were coupled prior to recombination, fluctuations in mass density are reflected in the temperature of the radiation. Temperature fluctuations in the CMB were discovered by the COBE satellite [23] and are now being measured with increasing accuracy, particularly by detectors deployed in long-flight balloons [24–26]. The spectrum of temperature fluctuation is just what inflation predicts: it is scale invariant on large scales and shows a series of ‘Doppler’ or ‘acoustic’ peaks, which are the result of coherent acoustic oscillations experienced by the photon–baryon fluid before recombination. The characteristics of these peaks depend on the values of the cosmological parameters. For example, the location of the first peak is primarily determined by the large-scale geometry of the Universe and thus by the value of Ω. Current data imply a flat geometry. The spectrum of primordial fluctuations generated by, for example, inflation evolves with time in a manner that depends on the content of the Universe and the values of the cosmological parameters. The dark matter acts as a sort of filter, inhibiting the growth of certain wavelengths and promoting the growth of others. Following the classical work of Bardeen et al. [27], transfer functions for different kinds of dark matter (and different types of primordial fluctuation fields, including non-Gaussian cases) have been computed. In Gaussian models, the product (in Fourier space) of the primordial spectrum and the transfer function, together with the growing mode of the associated velocity field, provides the initial conditions for the formation of cosmic structure. 2The possibility that dark energy could be the dynamically dominant component had been anticipated by theorists from studies of the cosmic large-scale structure (see, for example, Ref. [15]), and was considered in the first simulations of structure formation in cold dark matter universes [16].

241 C.S. FRENK

2.3 Growth mechanism Primordial fluctuations grow by gravitational instability: overdense fluctuations expand linearly, at a re- tarded rate relative to the Universe as a whole, until eventually they reach a maximum size and collapse non-linearly to form an equilibrium (or ‘virialized’) object whose radius is approximately half the physical size of the perturbation at maximum expansion. The theory of fluctuation growth is lucidly explained by Peebles [28]. Although gravitational instability is now widely accepted as the primary growth mechanism re- sponsible for the formation of structure, it is only very recently that firm empirical evidence for this process has been found. Gravitational instability causes the inflow of material around overdense regions. From the perspective of a distant observer, this flow gives rise to a characteristic infall pattern, which, in principle, is measurable in a galaxy redshift survey by comparing the two-point galaxy correlation function along and perpendicular to the line-of-sight. In this space, the infall pattern resembles a butter- fly [29]. This pattern was clearly seen for the first time in the 2dFGRS3 [30].

2.4 Cosmological parameters After decades of debate, the values of the fundamental cosmological parameters are finally being measured with some precision. The main reason for this is the accurate measurement of the acoustic peaks in the CMB temperature anisotropy spectrum, whose location, height, and shape depend on the values of the cosmological parameters. Parameter degeneracies do exist but some of these can be broken using other data, for example, the distant type Ia supernovae or the 2dFGRS [19]. The CMB data alone do not constrain the Hubble constant, but these data combined with the 2dFGRS give a value, in units of 100 km s 1 Mpc 1, of h = 068 007, which agrees with results from the HST key project [31] and − − other methods. In addition to h and the other parameters listed in Table 1, the other important number in studies of large-scale structure is the amplitude of fluctuations in primordial density, which is usually parametrized by the quantity σ8 (the linearly extrapolated value of the top-hat filtered fluctuation am- 1 plitude on the fiducial scale of 8h− Mpc). The best estimate of this quantity comes from the observed 06 abundance of rich galaxy clusters, which gives σ8Ω = 05, with an uncertainty of about 10% [32–34].

3 The Friedmann model This section provides a very brief review of the main equations that describe the evolution of the mean properties of the Universe. The co-moving separation, x, is related to physical separation through

r = a(t)x (1) The Friedmann equations are a˙ 2 kc2 8π Λc2 + = G ρ + (2) a a2 3 3 and a¨ 4πG 3P Λc2 = H˙ + H2 = ρ + + (3) a − 3 c2 3 where a is the expansion factor of the Universe, k the curvature, ρ the mass–energy density, Λ the cosmological constant, and H = a˙a the Hubble constant.

3 Strictly speaking, the ‘butterﬂy’ pattern does not prove the existence of infall since the continuity equation would ensure a similar pattern even if velocities were induced by non-gravitational processes. However, it can be shown that such velocities, if present, would rapidly decay.

242 THE ORIGIN OF COSMIC STRUCTURE

The ﬁrst law of thermodynamics implies that (P being the pressure):

dρ P + 3H ρ + = 0 (4) dt c2 It is easy to show that for non-relativistic matter ρ a 3, whereas for radiation ρ a 4. g ∝ − r ∝ − 2 The cosmological density parameter is deﬁned as Ω = ρρcrit, where ρcrit = 3H 8πG. Using the subscripts m, γ and Λ to denote the contributions of matter, radiation, and cosmological constant respectively, for a ﬂat Universe we have

Ωm + Ωγ + ΩΛ = 1 (5)

4 The growth of density fluctuations—linear theory 4.1 Linear fluctuations To study the evolution of linear perturbations, we write the density and velocity fields, ρ(r t) and u(r t), as:

ρ(r t) = ρ(t)(1 + δ(r t)) (6) u = H(t)r + v(r t) (7) where v is the peculiar velocity. For linear ﬂuctuations, δ 1 and v 1.

4.2 The rate of growth equation For an ideal, non-relativistic ﬂuid, the equations of conservation of mass and momentum are:

∂ρ + (ρu) = 0 (8) ∂t ∇ · ∂u P + u u = ∇ Φ (9) ∂t · ∇ − ρ − ∇ where Φ is the gravitational potential, which, for a self-gravitating ﬂuid, is given by Poisson’s equation

3P 2Φ = 4πG(ρ + tot ) Λ (10) ∇ tot c2 − where the sufﬁx tot indicates that all matter contributes to the gravitational potential, while the conservation equations apply separately to the baryonic ﬂuid and also to any component of non-baryonic dark matter. The linearized form of Eq. (8) is

δ˙ + Hv δ + v = 0 (11) · ∇ ∇ · and the linearized form of Eq. (9) is

∂v δρ + Hv + Hv v = ∇ φ (12) ∂t · ∇ − ρ − ∇ where φ is the perturbative part of Φ:

3δP 2φ = 4πG δρ + tot (13) ∇ tot c2

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In co-moving coordinates, these equations become

2 2 ∂ δ ∂δ r δP 2 x +2H = ∇ + r φ (14) ∂t2 | ∂t x ρ ∇ where 3δP 2φ = 4πG(δρ + tot ) (15) ∇r tot c2

4.3 Solution for cold matter (dust) For cold matter (dust), the growth equation becomes

3 δ¨ + 2Hδ˙ H2Ω δ = 0 (16) − 2 m For Ωm = 1, this has the solution

23 1 δ = A(x)t + B(x)t− (17) where the ﬁrst term corresponds to a growing mode and the second to a decaying mode. For a Λ-dominated Universe, Ω 1, the solution is Λ '

2Ht δ = A(x) + B(x)e− (18) so that the growing mode is no longer growing, but is frozen at a constant value.

In the general case, Ωm + ΩΛ = 1, a feel for the solution may be obtained by writing the solution as

δ = δ0(x)ag(a Ωm0) (19) where the correction factor g is constant at early times and scales as 1a at late times. For Ωm = 13, g g = 125 as a 0. Thus, fluctuations in the matter distribution have grown by less than a factor of → i → 800 since recombination. Since the cosmic microwave background radiation indicates that fluctuations 4 in the baryon distribution at recombination had amplitude of less than 10− , the existence of non-linear structures today (δ > 1) implies that the growth of fluctuations must have been driven by non-baryonic dark matter that is not relativistic at the epoch of recombination, trec, and does not couple to the radiation. After trec, the baryons decouple and are free to fall into the dark matter potential wells.

4.4 The Jeans mass The Jeans mass is derived by comparing the relative strength of gravitational and pressure forces acting on a perturbation. If, for an adiabatic perturbation, we write

ik x δP δρ e · (20) ∝ ∝ then Eq. (16) gives,

k2c2 δ¨ + 2Hδ˙ = 4πGρ s δ (21) − a2 Fluctuations can only grow if their wavelength is greater than the Jeans length, λJ, which in physical units is π 12 λ = c (22) J s Gρ

244 THE ORIGIN OF COSMIC STRUCTURE corresponding to masses greater than the Jeans mass

4π λ 3 M = ρ J (23) J 3 m 2

Note that the Jeans mass is deﬁned for any mass component of density ρm (e.g. baryons, cold dark matter, etc.), but the density that comes into the Jeans length, ρ, includes all gravitating components.

4.5 The Me´sza´ros effect A perturbation in a collisionless, non-relativistic matter component (e.g., cold dark matter) when the Universe is dominated by a relativistic component, e.g., radiation, experiences reduced growth. To see this, write Eq. (21) with δP = 0 and δρnrρnr:

δ¨ + 2Hδ˙ 4πGρ δ = 0 (24) − nr changing variables to ρ a y = nr = (25) ρr aeq 4 2 where aeq is the expansion factor at the epoch of matter–radiation equality (zeq = 43 10 Ω h ), and × 0 using the Friedmann equation with k = 0, we obtain

d2δ 2 + 3y dδ 3y + = 0 (26) d2y 2y(1 + y) dy − 2y(1 + y) The growing solution is 3 δ 1 + y (27) + ∝ 2 so that before zeq (y < 1), the growing mode is practically frozen. (The total growth in the interval t = 0 teq is only 5/2.) After zeq, the solution rapidly matches the growth rate in a matter-dominated − Einstein–de Sitter model, δ t23. ∝

4.6 Evolution of superhorizon perturbations The Hubble radius,

c R = (28) H H describes the typical size over which physical processes can operate coherently. Throughout most of the Universe’s evolution, a(t) tn with n < 1. Thus, H t 1 and R t, so that R grows faster ∝ ∝ − H ∝ H than proper length (r a tn). There is therefore a time, t = t (λ) when the proper radius of a ∝ ∝ enter perturbation equals RH. For t < tenter, the wavelength is larger than the Hubble radius. In analogy with the Jeans mass, deﬁne the horizon mass as

1 M = πρR3 (29) H 6 H

To calculate the evolution of perturbations with λ > RH, we need GR. However, we can take a short-cut that gives the correct answer. Regard the perturbation as a slightly closed (positively curved) Universe superposed on a ﬂat Universe with the same expansion rate. This leads to

ρ ρ ka2 δ 1 − 2 = (30) ≡ ρ0 8πGρ03

245 C.S. FRENK and ﬂuctuations outside the horizon grow as

a 2 a2 for t < t δ − eq (31) ∝ ρ ∝ a for t > teq .

5 Evolution of individual perturbations 5.1 Adiabatic and isothermal perturbations There are two basic types of perturbation: adiabatic or curvature (affecting both matter and radiation) and isothermal or entropic (affecting only matter). These are related by

δρm δT 3δρr 3 δm = 3 = δr (32) ≡ ρ T 4ρr ≡ 4 During the radiation era, isothermal ﬂuctuations in the baryons do not grow because of the strong frictional drag between matter and the uniform radiation ﬁeld. After recombination, perturbations in matter evolve in the same way regardless of whether they are initially adiabatic or isothermal. If perturbations are produced by ‘microscopic’ physics, then we expect them to be adiabatic.

5.2 Perturbations in dark matter Consider the evolution of a perturbation in dark matter, with proper wavelength λ a, which enters the ∝ horizon at a = aenter. Dark matter particles can be relativistic at early times; their velocity dispersion v c for a < a . For a > a , the velocity decays as v a 1 (because dark matter behaves like an ' nr nr ∝ − adiabatically expanding ﬂuid). For dark matter v plays the role of the speed of sound. Thus, the Jeans length, Eq. (22), is

a2 for a < a v nr a for a < a < a (33) λJ 12 nr eq ∝ ρ ∝  12  a for aeq < a . There are three stages in the evolution of a mode that enters the horizon between anr and aeq (these are the most relevant):

Stage A: a < aenter

Here λ > λH, so perturbation grows. From Eq. (31),

δ a2 (34) ∝

Stage B: aenter < a < aeq

Now λ < λH and λ > λJ so pressure cannot stop collapse. However, the perturbation cannot grow because of the Me´sza´ros effect, so δ = const (35)

Stage C: aeq < a

Still λ < λH and λ > λJ but the perturbation can now grow, so

δ a (36) ∝ We can also express the evolution in terms of the Jeans mass

2 a for a < anr 3 MJ ρDMλJ const for anr < a < aeq (37) ∝ ∝  32  a− for aeq < a , 

246 THE ORIGIN OF COSMIC STRUCTURE which can be compared with the horizon mass

2 a for a < anr 3 3 MH ρDMdH a for anr < a < aeq (38) ∝ ∝  32  a for aeq < a .

From these scalings and the value of MJ(zeq), we can compute MJ. For example, MJ = 32 14 2 2 32 × 10 (Ωh )− (aaeq)− M for a > aeq.

5.3 Adiabatic and isothermal perturbations

The epoch of baryon-radiation equality (ρb = ρr) is given by

Ωb 4 2 1 + zbr = = 39 10 (Ωbh ) (39) Ωr ×

2 Current estimates of Ωbh (002 0001) indicate that zbr < zrec. For a < arec, the adiabatic speed of sound for the baryons is

12 12 12 (a) ∂P c ∂ρm − c 3 ρm − cz = = 1 + = 1 + (40) ∂ρ √3 ∂ρr √3 4 ρr S S Thus, we have the following three regimes:

c for a < a √3 br 12 (a) c 1+z c =  for abr < a < arec S  √3 1+zbr  12  γkT for a < a . mp rec   The speed of sound plummets from 108 m/s before recombination to 5 105 m/s just after recom- ∼ ∼ × bination, as the radiation pressure ceases to act on the gas and the baryons experience only gas pressure. Following the same reasoning as for dark matter, we ﬁnd

3 a for a < aeq 32 3 a for aeq < a < abr MJb ρbλJ  (41) ∝ ∝  const for abr < a < arec  const for arec < a .  Thus, for example,  3 14 Ωb 2 2 a MJb = 32 10 (Ωh )− M for a < aeq (42) × Ω a eq and 32 3 Ωb 2 12 a − MJb = 25 10 (Ωh )− M for arec < a (43) × Ω a rec For an adiabatic perturbation with λ > λJb(aeq)

a2 for a < a ∂δ enter δb const for aenter < a < arec (44) ≡ ρ b ∝   a for arec < a . 

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Adiabatic perturbations with λ < λJb(aeq) during the epoch aenter < a < arec oscillate as acoustic (a) waves with the speed of sound cS and constant amplitude. Photons are able to diffuse from ﬂuctuations with M < MD(t) << MJb where

32 154 7 Ωb 2 54 1 + z − MD 8 10 (Ωh )− M (45) ' × Ω 1 + z eq so no baryon ﬂuctuations below this mass survive.

Perturbations in baryons can only grow after arec, but perturbations in dark matter can grow from 2 aeq. During this time, perturbations in dark matter grow by a factor (arecaeq) (TeqTrec) 21Ωh . ' ' When baryons decouple at recombination, they quickly ‘fall into’ the potential wells created by dark matter. Thus, δb grows rapidly for a short time after arec until it catches up with δDM. Thereafter both grow as a.

We have assumed Ω = 1. If Ω = 1, only the period a > arec is affected. As we saw in Section 2.3, 6 when Ω = 1, ﬂuctuations eventually stop growing when the Universe begins to expand too fast. Growth 6 stops when Ω z 1, i.e. for z 1Ω . 0 ' ≤ 0

6 Power spectrum of density fluctuations In this section, we collect the results of the preceding section to arrive at a simple statistical description of the density and velocity fluctuation fields.

6.1 Mathematical background The density ﬁeld, δ(x t), can be written as a sum of plan waves in Fourier space:

1 ik x 3 δ(x t) = δ (k t)e · d k (2π)3 k Z

ik x 3 δk(k t) = δ(x t)e− · d x (46) Z One integral may be seen to be the inverse of the other by the identity

ik x 3 3 3 f(k)e · d xd k = (2π) f(0) (47) Z Z The fact that δ(x t) averages to zero, δ(x t) = 0, implies that δ (0) = 0. h i k The term δk is usually assumed to be a ‘Gaussian random field’, which means that the waves have random phases. This means that the field can be specified entirely by its variance or power spectrum

P (k) δ (k) 2 (48) ≡ | k | For an isotropic distribution, the power spectrum, averaged over all possible realizations, must be independent of direction, P (k) = P (k). Often, P (k) is approximated as h i P (k) kn (49) ∝ In practice, P (k) is not a power law, i.e. n varies with scale. We deﬁne δρ δM 1 = = δ(x)d3x (50) ρ M V Z

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If δk is Gaussian, so is δMM. The average density ﬂuctuation (averaged over the whole of space) is zero δM 1 δM 3 1 3 = d x = δ(x + x0)d xd3x0 (51) M V M V V u ZVu u ZVu ZV where the outer integral ranges over all space (of nominal volume Vu) and the inner integral ranges over the volume V , centred on x0. The mean squared value of δMM is

δM 2 1 σ2(M) = = P (k)W (k; V )d3k (52) M (2π)3V * + u Z where W is the window function

2 1 ik x 3 W (k; V ) = e · d x (53) V ZV A useful relation when manipulating Fourier integrals is

0 i(k k ) x 3 3 e − · d x = (2π) δD(k k0) (54) − Z where δD is the Dirac delta function. For a spherical volume of radius x it can be shown that 9 W = [sin(kx) kx cos(kx)]2 (55) (kx)6 −

The exact form of the window function is not crucial. For a ﬁnite volume, W always tends to one at small k (large scales) and tends to zero at large k (small scales). Often W is approximated by a sharp cut-off in k-space 1 for kx 1 W (kx) = ≤ (56) 0 for kx > 1 , since M x3, provided n > 3. We will express the present-day value of σ, assuming linear evolution, ∝ − as

α M − 3 + n σ α = (57) 0 ≡ M 6 0 This gives the relative amplitude of ﬂuctuations on different scales according to linear theory. Of course, σ scales with time in the same way as δ, i.e. as in Eq. (19):

σ = σ0ga (58)

6.2 Dark matter power spectra Density perturbations are thought to be generated by quantum fluctuations during inflation. These are adiabatic, Gaussian perturbations. As the Universe inflates, the perturbation is stretched beyond the horizon. Once inflation ends, however, perturbations eventually re-enter the horizon. According to self- similarity arguments, when they do so they all have the same amplitude, independent of wavelength (or mass). Smaller fluctuations re-enter the horizon first.

Let us compute the shape of the power spectrum at some time, say trec. Consider a ﬂuctuation that enters the horizon in the matter-dominated regime. Since σ δ, from Eq. (19) σH = σreca g , where ∝ H i the subscript H refers to horizon crossing. Thus,

249 C.S. FRENK

σ σ t 23 σ = H = H 0 (59) rec g a g t i H i H where σH = const (i.e. independent of mass). During this time, the horizon mass scales as

3 MH ρ (ctH) tH (60) ∝ H ∝ 2 because ρ t in the matter-dominated regime. Since at aenter, M MH, ∝ − ' 23 23 σrec t− M − (61) ∝ H ∝ Comparing with Eq. (17), 2 α = or n = 1 (62) 3 This is called the Harrison–Zeldovich spectrum, P (k) k. ∝ Density ﬂuctuations cannot grow during the radiation-dominated era so the power spectrum is maintained at the value at horizon crossing. Thus σ const and, from Eq. (17), α 0, so n 3. It → → → − is usual to write P (k) = T 2(k)k (63) where T (k) is the transfer function. For Cold Dark Matter (CDM), a ﬁtting formula is

ln(1 + 234q) 2 3 4 14 T (k) = [1 + 389q + (161q) + (546q) + (671q) ]− (64) CDM 234q where k 1 q = h− Mpc (65) Γ and the shape parameter Γ = Ωm0h (66) Hot Dark Matter (HDM) corresponds to the case where the dark matter sparticles are still relativistic at matter–radiation equality. In this case, the dark matter particles free-stream out of the perturbations as soon as they come into the horizon so ﬂuctuations are erased until such time as the particles cease to be relativistic. The cut-off wavelength is

1 λFS m− (67) ∝ X where mX is the mass of the HDM particle. A ﬁtting formula for HDM is

39q 21q2 ΓHDM(k) e− − (68) ' The power spectrum (or, more precisely, k3 δ 2) for CDM and HDM is shown in Fig. 1. Using the | k| program at the URL http://www/physics.nyu.edu/matiasz/CMBFAST/cmbfast.html accurate transfer functions may be obtained.

7 Cosmic microwave background The discovery of fluctuations in the CMB by COBE in 1992 [23] is one of the most important discoveries in twentieth-century science. These fluctuations are a manifestation of the density fluctuations that give rise to galaxies. They connect the physics of the very early Universe to the Universe of galaxies and give us information about events during inflation and about the nature of dark matter. In this section, we will explain the origin and significance of the recent data obtained by the WMAP satellite.

250 THE ORIGIN OF COSMIC STRUCTURE

Fig. 1: The linear power spectra for cold dark matter and hot dark matter

7.1 Angular power spectrum The CMB temperature distribution on the celestial sphere can be expanded in spherical harmonics:

m=+l T (θ φ) T0 ∆T (θ φ) ∞ − = almYlm(θ φ) (69) T0 ≡ T l=0 m= l X X− where T0 is the mean CMB temperature. This is analogous to the Fourier expansion of δ. The l = 0 mode (monopole) gives the mean temperature; l = 1 (dipole) is mostly due to the motion of our galaxy ( 600 km/s) produced by local mass ﬂuctuations (Coma cluster, Great Attractor, ' etc.); l = 2 (quadrupole) and higher modes are produced by intrinsic anisotropies produced either at trec (primary) or between trec and t0 (secondary). The multipole l is related to the angle on the sky by

600 θ (70) ' l for l > 0. The angular power spectrum (analogous to P (k)) is

C < a 2 > (71) l ≡ | lm| The autocorrelation function is deﬁned as δT δT C(θ) =< (nˆ ) (nˆ ) > (72) T 1 T 2 where cos θ = nˆ nˆ and the average is over the ensemble of possible skies. Since we have only one 1 · 2 sky, we use the ergodic hypothesis: we can replace an average over realizations with an average over

251 C.S. FRENK different patches of a single realization. (But this is not possible on large scales, where measurements are limited by cosmic variance.) Using the addition theorem for spherical harmonics, it is easy to show that

1 ∞ C(θ) = (2l + 1)C P (cos θ) (73) 4π l l Xl=2 where Pl is the Legendre polynomial.

7.2 Primary anisotropies There are three basic effects occurring at the surface of last scattering.

(i) Gravitational (Sachs–Wolfe) perturbations: photons from high-density regions at last scattering have to climb out of potential wells and are thus redshifted. (ii) Intrinsic (adiabatic) perturbations: in high-baryon-density regions, the coupling of matter and radiation also compresses the radiation, giving rise to high T . (iii) Velocity (Doppler) perturbations: the velocity of the plasma at recombination produces Doppler shifts in frequency and thus in temperature.

A full treatment requires the solution of the Boltzmann equation. (This is what CMBFAST does.)

7.2.1 The Sachs–Wolfe effect This is the dominant contribution to ∆TT on large scales. Density fluctuations generate fluctuations in the gravitational potential, δφ. Photons emitted from a potential well need to climb out and are subject to (a) gravitational redshift and (b) time dilation (we see photons at different times or values of a). Since T φ, (a) ' ⇒ ∆T δφ = (74) T c2 Since, the time dilation δtt δφc2, (b) ⇒ ∆T δa 2 δt 2 δφ = = = (75) T a −3 t −3 c2 The net effect is ∆T 1 δφ = (76) T 3 c2 7.2.2 Intrinsic perturbations For adiabatic fluctuations, Eq. (32) ⇒ ∆T 1 δρ = (77) T 3 ρ

7.2.3 Doppler perturbations Density ﬂuctuations induce streaming motions. This produces a ∆T , because some electrons are moving toward the observer when they last scatter the radiation and others are moving away. From the continuity equation ∆T v δρ λ (78) T ' c ' ρ ct

252 THE ORIGIN OF COSMIC STRUCTURE

The Sachs–Wolfe effect dominates for scales 1 Gpc (i.e. larger than horizon at trec); Doppler ≥ effects take over for scales 1 Gpc, but are soon dominated by adiabatic effects on the smallest scales. ≤ The horizon scale at trec corresponds to l 100–200. On these scales an acoustic peak is produced. ' This was first detected from balloon-borne measurements [24] and has now been confirmed by WMAP data [4,35]. The peak results form the acoustic oscillations of sub-Jeans mass fluctuations in the photon– baryon fluid prior to trec. These standing waves have a phase relation between density and velocity. When the Universe recombines, waves interfere constructively, giving rise to the peak and its harmonics. The physical length-scale at which the first peak occurs corresponds to the size of the sound horizon, RSH c trec at the surface of last scattering. This is approximately fixed in all cosmological ' s models. However, the angular scale subtended by RSH depends on cosmology via the angular diameter– distance relation. For a flat Universe, the peak occurs at l 200. For a positively curved Universe, the ' angle subtended is larger, so the peak appears at smaller l. For a negatively curved Universe, the angle subtended is smaller, so the peak appears at larger l. Thus, the position of the peak gives an estimate of Ωtot. The WMAP data give Ωtot = 100 002. The properties of the various peaks depend on other cosmological parameters in a complicated fashion. The Cls measured by WMAP can be used to estimate the values of these parameters. Remarkably, the WMAP data agree very well with predictions made from CDM theory in the early 1980s!

8 Cosmological simulations The linear power spectra of Fig. 1 may be regarded as the ‘initial conditions’ from which structure will form by the gravitational amplification of small fluctuations. Today, structures on scales of galaxy clusters and below are strongly non-linear. To follow the evolution of fluctuations into the non-linear regime in full generality requires numerical simulation. Operationally, the problem of the cosmic large-scale structure can be divided into two parts: understanding the clustering evolution of dark matter and understanding the gaseous and radiative processes that lead to the formation of galaxies. Specialized simulation techniques have been developed to tackle both aspects. The evolution of dark matter is most often calculated using N-body techniques, implemented through a variety of efficient algorithms, such as P3M (particle–particle/particle–mesh [36]), AP3M (the adaptive mesh version of P3M [37]), and hierarchical trees [38,39]. Gaseous and radiative processes are followed by combining a hydrodynamics code with an N-body code. Numerical hydrodynamic techniques used in cosmology include Eulerian methods [40], Lagrangian codes based on Smooth Particle Hydrodynamics (SPH) [41], and hybrid codes [42,43]. These techniques have different strengths and weaknesses, but, where a simple comparison has been made, they all give similar results for the simplest cosmological problems [44]. There has been rapid growth in the size and power of cosmological simulations in the two and a half decades since this technique was introduced to the subject by Peebles [45]. One way to measure this growth is by the number of particles employed in the simulations. The size of the largest simulations has grown exponentially, in a manner reminiscent of the well-known ‘Moore’s law’, which describes the increase in CPU speed with time, except that the advent of massively parallel supercomputers led to a sudden order-of-magnitude jump in size toward the end of the last decade. The largest simulation carried out to date is the 10-billion particle ‘Millennium’ N-body simulation performed by the Virgo Consortium, an international collaboration of researchers in Canada, Germany, and the UK.

8.1 Large-scale structure With large N-body simulations, it is possible to characterize the statistical properties of dark matter distribution with very high accuracy. For example, Fig. 2 shows the 2-point correlation function, ξ(r), of dark matter (a measure of its clustering strength) in the ‘ΛCDM’ cosmology, a ﬂat cold dark matter model in which Ωdm = 03, ΩΛ = 07 and h = 07 [46]. The statistical error bars in this estimate are actually smaller than the thickness of the line. Similarly, higher order clustering statistics, topological measures,

253 C.S. FRENK

Fig. 2: Two-point correlation functions. The dotted line shows the dark matter ξdm(r) [46]. The solid line shows the galaxy predictions of Benson et al. [47], with Poisson errors indicated by the dashed lines. The points with error bars show the observed galaxy ξgal(r) [48]. The galaxy data are discussed in the text. (Adapted from Ref. [49].) the mass function and clustering of dark matter haloes, and the time evolution of these quantities can all be determined very precisely from these simulations [50, 51]. In a sense, the problem of the large-scale distribution of dark matter in the ΛCDM model can be regarded as largely solved. In contrast to the clustering of dark matter, the process of galaxy formation is still poorly understood. How then can dark matter simulations be compared with observational data that, for the most part, refer to galaxies? On large scales a very important simplification applies: for Gaussian theories like CDM, it can be shown that if galaxy formation is a local process, that is, if it depends only upon local physical conditions (density, temperature, etc.), then, on scales much larger than those associated with individual galaxies, the galaxies must trace mass, i.e. ξgal(r) ξDM(r) [52]. It suffices therefore ∝ to identify a random subset of the dark matter particles in the simulation to obtain an accurate prediction of the properties of galaxy clustering on large scales. This idea (complemented on small scales by an empirical prescription in the manner described by Cole et al. [53]) has been used to construct the mock versions of a region of the APM galaxy survey and of a slice of the 2dFGRS displayed in Fig. 3, which also shows the real data for comparison in each case. By eye, at least, it is very difficult to distinguish the mock from the real data. A quantitative comparison between simulations and the real world is displayed in Fig. 4. The symbols show the estimate of the power spectrum in the 2dFGRS survey [54]. This is the raw power spectrum convolved with the survey window function and can be compared directly with the line showing the theoretical prediction obtained from the mock catalogues, which have exactly the same window function. The agreement between the data and the ΛCDM model is remarkably good.

8.2 Galaxy formation Understanding galaxy formation is a much more difﬁcult problem than understanding the evolution of dark matter distribution. In CDM theory, galaxies form when gas, initially well mixed with dark matter,

254 THE ORIGIN OF COSMIC STRUCTURE

Fig. 3: A 1◦ thick slice through the 2dF galaxy redshift survey. The radial coordinate is redshift and the angular coordinate is right ascension. The top-left panel is the real data and the other two panels are mock catalogue constructed from the Hubble volume simulations [51].

Fig. 4: The power spectrum of the 2dFGRS (symbols) compared with the power spectrum predicted in the ΛCDM model (line). Both power spectra are convolved with the 2dFGRS window function. The model predictions come from dark matter simulations and assume that, on large scales, the distribution of galaxies traces the distribution of mass. (Adapted from Ref. [54].)

255 C.S. FRENK cools and condenses into emerging dark matter haloes. In addition to gravity, a non-exhaustive list of the processes that now need to be taken into account includes the shock heating and cooling of gas into dark haloes, the formation of stars from cold gas and the evolution of the resulting stellar population, the feedback processes generated by the ejection of mass and energy from evolving stars, the production and mixing of heavy elements, the extinction and re-radiation of stellar light by dust particles, and the formation of black holes at the centres of galaxies and the influence of the associated quasar emission. These processes span an enormous range of length and mass scales. For example, the parsec scale relevant to star formation is a factor of 108 smaller than the scale of a galaxy supercluster. The best that can be done given current computing techniques is to model the evolution of dark matter and gas in a cosmological volume with resolution comparable to a single galaxy. Subgalactic scales must then be regarded as ‘subgrid’ scales and followed by means of phenomenological models based either on our current physical understanding or on observations. In the approach known as ‘semi- analytic’ modelling [55], even gas dynamics is treated phenomenologically using a simple, spherically symmetric model to describe the accretion and cooling of gas into dark matter haloes. It turns out that this simple model works surprisingly well if judged by the good agreement with results of full N-body/gas- dynamical simulations [47, 56, 57]. The main difficulty encountered in cosmological gas-dynamical simulations arises from the need to suppress a cooling instability present in hierarchical clustering models like CDM. The building blocks of galaxies are small clumps that condense at early times. The gas that cools within them has very high density, reflecting the mean density of the Universe at that epoch. Since the cooling rate is proportional to the square of the gas density, in the absence of heat sources, most of the gas would cool in the highest levels of the mass hierarchy, leaving no gas to power star formation today or even to provide the hot, X-ray emitting plasma detected in galaxy clusters. Known heat sources are photo-ionization by early generations of stars and quasars, as well as the injection of energy from supernovae and active galactic nuclei. These processes, which undoubtedly happened in our Universe, belong to the realm of subgrid physics that cosmological simulations cannot resolve. Different treatments of this ‘feedback’ result in different amounts of cool gas and can lead to very different predictions for the properties of the galaxy population. This is a fundamental problem that afflicts cosmological simulations even when they are complemented by semi-analytic techniques. In this case, the resolution of the calculation can be extended to arbitrarily small mass haloes, perhaps allowing a more realistic treatment of feedback. Although they are less general than full gas-dynamical simulations, simulations in which the evolution of gas is treated semi-analytically make experimentation with different prescriptions relatively simple and efficient [58–60].

The galaxy autocorrelation function, ξgal(r), calculated by applying the semi-analytic model of Cole et al. [60] to a large N-body simulation is plotted in Fig. 2 above. On large scales, it follows ξDM(r) quite closely, but on small scales it dips below the mass autocorrelation function. This small-scale ‘antibias’ has also been seen in N-body/gas-dynamical simulations of the ΛCDM cosmology [61–63]. The galaxy autocorrelation function in the simulations of Benson et al. [47] agrees remarkably well with the observational data. This is a genuine success of the theory. The differences between the small-scale clustering of galaxies and dark matter result from the interplay between the clustering of dark matter haloes and the occupation statistics of galaxies in haloes, which, in turn, are determined by the physics of galaxy formation. This conclusion, discussed in detail by Benson et al. [47], has led to the development of an analytic formulation known as the ‘halo model’ [64–66].

9 Conclusions Unlike most computational problems in many areas of science, the cosmological problem is blessed with known, well-specified initial conditions. Within a general class of models, it is possible to calculate the properties of primordial perturbations in the cosmic energy density generated by quantum processes during an early inflationary epoch. In a wide family of inflationary models, these perturbations are

256 THE ORIGIN OF COSMIC STRUCTURE adiabatic, scale invariant, and have Gaussian-distributed Fourier amplitudes. The model also requires an assumption about the nature of dark matter and the possibilities have now been narrowed down to non-baryonic candidates, of which cold dark matter particles seem the most promising. An empirical test of the initial conditions for the formation of structure predicted by the model is provided by cosmic microwave background radiation. The tiny temperature fluctuations it exhibits have exactly the properties expected in the model. Furthermore, CMB data can be used to fix some of the key model parameters, such as Ω and Ωb, and the data can be combined with other recent datasets, such as the 2dFGRS, to allow the determination of many of the remaining parameters, e.g. Ωm, ΩΛ, and h. It is this specificity of the cosmological problem that has turned simulations into the primary tool connecting cosmological theory to astronomical observations. In addition to well-specified initial conditions, the cosmological dark matter problem has the ad- vantage that the only important physical interaction is gravity. The problem can thus be posed as a gravitational N-body problem and approached using the many sophisticated techniques developed over the past two decades to tackle this problem. Although on small scales there remain a number of unresolved issues, it is fair to say that on scales larger than a few megaparsecs the distribution of dark matter in CDM models is essentially understood. The inner structure of dark matter haloes, on the other hand, is still a matter for debate and the mass function of dark matter haloes has only been reliably established by simulations down to masses of order 1011 M . Resolving these outstanding issues is certainly within reach, but will require carefully designed simula tions and large amounts of computing power. The frontier of the subject at present lies in simulations of the formation, evolution and structure of galaxies. This problem first requires a treatment of gas dynamics in a cosmological context— a number of techniques, relying on direct simulations or on semi-analytical approximations, are being explored. Realistic models of galaxy formation, however, will require much more than a correct treatment of cooling gas. Such models will have to include a plethora of astrophysical phenomena such as star formation, feedback, metal enrichment, etc. The huge disparity between the submegaparsec scales on which these processes operate and the gigaparsec scale of the large-scale structure makes it impossible to contemplate a comprehensive ab initio calculation. The way forward is clearly through a hybrid approach, combining direct simulation of the processes operating on a limited range of scales with a phenomenological treatment of the others. There is currently a great deal of activity in the phenomenology of galaxy formation. In spite of the uncertainties that remain, all the indications are that our Universe is well described by a model in which

(i) the overall geometry is flat; (ii) the dominant dynamical components are cold dark matter ( 30%) and dark energy ( 70%), ∼ ∼ with baryons playing a supporting role ( 4%); ∼ (iii) the initial conditions are quantum fluctuations in the primordial energy density generated during inflation; and (iv) the structure has grown primarily as a result of the gravitational instability experienced by mass fluctuations in an expanding Universe.

A sceptic is entitled to feel that the current paradigm is odd, to say the least. Not only is there a need to invoke vast amounts of as yet undetected non-baryonic cold dark matter, but there is also a need to account for the dominant presence of a dark energy whose very existence is a mystery within conven- tional models of fundamental physics. Odd as it may seem, however, this model accounts remarkably well for a large and diverse collection of empirical facts that span 13 billion years of evolution.

Acknowledgements I am grateful to my collaborators for their contribution to the work reviewed here, especially Carlton Baugh, Andrew Benson, Shaun Cole, Adrian Jenkins, Cedric Lacey, John Peacock, and Simon White.

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References [1] D.G. York et al. (The SDSS Collaboration), Astron. J. 120 (2000) 1579. [2] M. Colless et al. (The 2dFGRS Team), Mon. Not. R. Astron. Soc. 328 (2001) 1039. [3] C.C. Steidel, M. Giavalisco, M. Pettini, M. Dickinson and K.L. Adelberger, Astrophys. J. 462 (1996) L17. [4] D.N. Spergel et al., Astrophys. J. Suppl. 148 (2003). [5] D. Tytler, J.M. O’Meara, N. Suzuki and D. Lubin, Phys. Scr. 85 (2000) 12. [6] P. Fischer et al., Astron. J. 120 (2000) 1198. [7] G .Wilson, N. Kaiser, G. Luppino and L.L. Cowie, Astrophys. J. 555 (2001) 572. [8] D. Clowe, G.A. Luppino, N. Kaiser and I.M. Gioia, Astrophys. J. 539 (2000) 540. [9] L. Van Waerbeke et al., Astron. Astrophys. 374 (2001) 757. [10] S.D.M. White, J.F. Navarro, A.E. Evrard and C.S. Frenk, Nature 366 (1993) 429. [11] A.E. Evrard, Mon. Not. R. Astron. Soc. 292 (1997) 289. [12] V.R. Eke, S. Cole, C.S. Frenk and J.P. Henry, Mon. Not. R. Astron. Soc. 298 (1998) 1145. [13] S. Borgani et al., Astrophys. J. 561 (2001) 13. [14] S.D.M. White, C.S. Frenk and M. Davis, Astrophys. J. 274 (1983) L1. [15] G. Efstathiou, W.J. Sutherland and S.J. Maddox, Nature 348 (1990) 705. [16] M. Davis, G. Efstathiou, C.S. Frenk and S.D.M. White, Astrophys. J. 292 (1985) 371. [17] A. Riess et al., Astron. J. 116 (1998) 1009. [18] S. Perlmutter et al. (The Supernova Cosmology Project), Astrophys. J. 517 (1999) 565. [19] G. Efstathiou et al. (The 2dFGRS Team), Mon. Not. R. Astron. Soc. 330 (2002) 29. [20] E.M. Lifshitz, J. Phys. 10 (1946) 116. [21] A.H. Guth, Phys. Rev. D23 (1981) 347. [22] A. Starobinskii, Phys. Lett. 117B (1982) 175. [23] G. Smoot et al., Astrophys. J. 396 (1992) 1. [24] P. de Bernardis et al., Nature 404 (2000) 995. [25] S. Hanany et al., Astrophys. J. Lett. 545 (2000) 5. [26] E.M. Leitch et al., Astrophys. J. 568 (2002) 28L. [27] J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, Astrophys. J. 304 (1986) 15. [28] P.J.E. Peebles, The Large Scale Structure of the Universe (Princeton University Press, Princeton, 1980). [29] N. Kaiser, Mon. Not. R. Astron. Soc. 227 (1987) 1. [30] J. Peacock et al., (The 2dFGRS Team), Nature 410 (2001) 169. [31] W.L. Freedman et al., Astrophys. J. 553 (2001) 47. [32] V.R. Eke, S. Cole and C.S. Frenk, Mon. Not. R. Astron. Soc. 282 (1996) 263. [33] P.T.P. Viana and A. Liddle, Mon. Not. R. Astron. Soc. 281 (1996) 323. [34] E. Pierpaoli, D. Scott and M. White, Mon. Not. R. Astron. Soc. 325 (2001) 77. [35] C.L. Bennett et al., Astrophys. J. 148 (2003) 1. [36] G. Efstathiou, M. Davis, C.S. Frenk and S.D.M. White, Astrophys. J. Suppl. 57 (1985) 241. [37] H.M.P. Couchman, P.A. Thomas and F.R. Pearce, Astrophys. J. 452 (1995) 797. [38] J. Barnes and P. Hut, Nature 324 (1986) 446. [39] V. Springel, N. Yoshida and S.D.M. White, New Astronomy 6 (2001) 79. [40] R. Cen, Astrophys. J. Suppl. 78 (1992) 341. [41] R.A. Gingold and J.J. Monaghan, Mon. Not. R. Astron. Soc. 181 (1977) 375.

258 THE ORIGIN OF COSMIC STRUCTURE

[42] N.Y. Gnedin, Astrophys. J. Suppl. 97 (1995) 231. [43] U-L. Pen, Astrophys. J. Suppl. 115 (1998) 19. [44] C.S. Frenk et al., Astrophys. J. 525 (1999) 554. [45] P.J.E. Peebles, Astron. J. 75 (1970) 13. [46] A. Jenkins, C.S. Frenk, F.R. Pearce, P. Thomas, J. Colberg, S.D.M. White, H. Couchman, J. Pea- cock, G. Efstathiou and A. Nelson, Astrophys. J. 499 (1998) 20. [47] A.J. Benson, S. Cole, C.S. Frenk, C.M. Baugh and C.G. Lacey, Mon. Not. R. Astron. Soc. 311 (2000) 79. [48] C.M. Baugh, Mon. Not. R. Astron. Soc. 280 (1996) 26. [49] A.J. Benson, C.S. Frenk, C.M. Baugh, S. Cole and C.G. Lacey, Mon. Not. R. Astron. Soc. 327 (2001) 1041. [50] A. Jenkins, C.S. Frenk, S.D.M. White, J.M. Colberg, S. Cole, A.E. Evrard and N. Yoshida Mon. Not. R. Astron. Soc. 321 (2001) 372. [51] A.E. Evrard, T. MacFarland, H.M.P. Couchman, J.M. Colberg, N. Yoshida, S.D.M. White, A. Jenk- ins, C.S. Frenk, F.R. Pearce, G. Efstathiou, J. Peacock and P. Thomas (The Virgo Consortium), Astrophys. J. 573 (2002) 7. [52] P. Coles, Mon. Not. R. Astron. Soc. 262 (1993) 1065. [53] S. Cole, S. Hatton, D.H. Weinberg and C.S. Frenk, Mon. Not. R. Astron. Soc. 300 (1998) 945. [54] W. Percival et al. (The 2dFGRS Team), Mon. Not. R. Astron. Soc. 327 (2001) 1297. [55] S.D.M. White and C.S. Frenk, Astrophys. J. 379 (1991) 52. [56] A.J. Benson, F.R. Pearce, C.S. Frenk, C.M. Baugh and A.R. Jenkins, Mon. Not. R. Astron. Soc. 320 (2001) 261. [57] N. Yoshida, F. Stoehr,V. Springel and S.D.M. White, Mon. Not. R. Astron. Soc. 335 (2002) 762. [58] G. Kauffmann, S.D.M. White and B. Guiderdoni, Mon. Not. R. Astron. Soc. 264 (1993) 201. [59] R.S. Somerville and J.R. Primack, Mon. Not. R. Astron. Soc. 310 (1999) 1087. [60] S. Cole, C.G. Lacey, C.M. Baugh, and C.S. Frenk, Mon. Not. R. Astron. Soc. 319 (2000) 168. [61] F.R. Pearce, A.R. Jenkins, C.S. Frenk, J. Colberg, S.D.M. White, P. Thomas, H.M.P. Couchman, J.A. Peacock and G. Efstathiou (The Virgo Consortium), Astrophys. J. 521 (1999) 992. [62] F.R. Pearce, A.R. Jenkins, C.S. Frenk, S.D.M. White, P. Thomas, H.M.P. Couchman, J.A. Peacock and G. Efstathiou (The Virgo Consortium), Mon. Not. R. Astron. Soc. 326 (2001) 649. [63] R. Dave, L. Hernquist, N. Katz and D. Weinberg, in Proc. Rencontres Internationales de l’IGRAP, ‘Clustering at High Redshift’, Marseille, 1999, astro-ph/9910221. [64] U. Seljak, Mon. Not. R. Astron. Soc. 318 (2000) 203. [65] J.A. Peacock and R. Smith, Mon. Not. R. Astron. Soc. 318 (2000) 1144. [66] A. Berlind and D. Weinberg, Astrophys. J. 575 (2002) 587.

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