Clustering statistics and dynamics

CLUSTERING STATISTICS & DYNAMICS R. Juszkiewicz1 , F. R. Bouchet2 Copernicus Center, Warsaw, Poland. Institut d’Astrophysique de Paris, France. 1 arXiv:astro-ph/9602134v1 26 Feb 1996 2 Abstract Since the appearance of the classical paper of Lifshitz almost half a century ago, linear stability analysis of cosmological models is textbook knowledge. Until recently, however, little was known about the behavior of higher than linear order terms in the perturbative expansion. These terms become important in the weakly nonlinear regime of gravitational clustering, when the rms mass density contrast is only slightly smaller than unity. In the past, theorists showed little interest in studying this regime, and for a good reason: only a decade ago, it would have been an academic excercise – at scales large enough to probe the weakly nonlinear regime, all measures of clustering were dominated by noise. This is no longer the case with present data. The purpose of this talk is to provide a brief summary of recent advances in weakly nonlinear perturbation theory. We present analytical perturbative results together with results of N-body experiments, conducted to test their accuracy. We compare perturbative predictions with measurements from galaxy surveys. Such comparisons can be used to test the gravitational instability theory and to constrain possible deviations from Gaussian statistics in the initial mass distribution; they can be also used to study the nature of physical processes that govern galaxy formation (“biasing”). We also show how future studies of velocity field statistics can provide a new way to determine the density parameter, Ω. To appear in: Clustering in the Universe, Proc. XXX Moriond Meeting, Les Arcs, 1995, Eds. S. Maurogordato et al. Editions Frontieres, Paris 1 Perturbation theory Tests of theories for the origin of the large scale structure of the universe ultimately depend on a comparison of model predictions with measurable quantities, derived from observations. The statistical measures we will discuss here are the low order N-point correlation functions. They have two clear advantages. First, they can be estimated from galaxy surveys with a reasonable degree of precision and reproducibility (see, e.g. [45, 46, 20, 51], and references therein). Second, correlation functions can be relatively easily related to dynamics (e.g. [45]). Their evolution can be studied by taking moments of the hydrodynamical equations of motion of an expanding self-gravitating pressureless fluid with zero vorticity, which is a good approximation of the real universe after the hydrogen recombination. Low order correlation functions can also be measured from N-body experiments. 1.1 Gravitational instability The full description of gravitational instability is nonlinear. The density contrast δ(x, t) = ρ(x, t) −1, hρi (1) the peculiar velocity v and the gravitational potential φ are related by the Euler, Poisson and continuity equations. They can be combined into one expression for the density contrast, δ̈ + 2H δ̇ − 23 ΩH 2 δ = 3 ΩH 2 δ 2 2 + a−2 ∇δ · ∇φ + a−2 ∇α ∇β [(1 + δ)vα vβ ] . (2) Here x = {xα } and t are, respectively, the comoving spatial coordinates and the cosmological time, the dots represent time derivatives, ∇α = ∂/∂xα , a(t) is the scale factor, H = ȧ/a is the Hubble parameter, and Ω = 8πGhρi/3H 2 is the density parameter. The brackets h. . .i denote ensemble averaging and ρ is the mass density. Perturbation theory rests on a conjecture that when the deviations from homogeneity are small, the first few terms of the expansion δ = δ1 + δ2 + δ3 + . . . (3) provide a reasonable approximation of the exact solution of eq. (2). The first, linear term is the well known Lifshitz [39] solution of eq. (2) with the right-hand side set to zero, δ1 = D(t)ε(x) + decaying mode , (4) where D(t) is the standard growing mode (see, e.g., §11 in [45]) and ε is a random field with statistical properties defined by initial conditions. The term δ2 = O(ε2 ) is the solution of the same equation with quadratic nonlinearities included iteratively by using δ1 as source terms (as in [45], §18). For a vanishing cosmological constant (Λ = 0) and arbitrary Ω, the fastest growing mode in this solution is [35] δ2 = D 2 (t) h 2 (1 3 i + κ)ε2 + ∇ε · ∇Φ + ( 12 − κ)ταβ ταβ , ταβ = ( 31 δαβ ∇2 − ∇α ∇β )Φ ; Φ(x) = − R d3 x′ ε(x′ )/4π|x − x′ | . (5) (6) The parameter κ is a slowly varying function of time; its dependence on Ω is extremely weak, too. In the range 0.05 ≤ Ω ≤ 3, the Ω-dependence is well approximated by [8] κ [Ω(t)] ≈ 3 14 Ω−2/63 . The higher order terms can be constructed out of the linear solution in a similar way. 1 (7) 1.2 Dynamical evolution of correlations The main source of difficulties in studies of the dynamics of gravitational instability is the nonlinearity of the equations of motion. Indeed, let us write eq. (2) for δ(x, t) and δ ′ = δ(x′ , t), then multiply the first equation by δ ′ , and the second by δ. The result of adding these two equations and averaging is ξ¨ + 2H ξ˙ − 3ΩH 2 ξ + 2a−2 ∇2 ξv = 3ΩH 2 hδ 2 δ ′ i + . . . , (8) where ξ = hδδ ′ i and ξv = hv · v′ i are the two-point density and velocity correlation functions, while the right-hand side of the equation involves 3- and 4-point correlations, which we left out except for one term given as an example. The full expression is not the point here; the important message is that we have an infinite hierarchy of equations. Mathematically, this is similar to the BBGKY hierarchy: in order to solve for an N-point function, we need to know (N + 1)- and (N +2)-point correlations. Perturbation theory may help, but not without further assumptions, necessary to close the infinite hierarchy of correlation functions. Here we will assume that ε(x) is a random Gaussian field. This approach, also known as the random phase approximation, can be justified theoretically for a wide class of models of the early universe [46]. Independently of whether we take “inflationary” arguments seriously or not, the random phase assumption is observationally testable. As we will show below, comparing model predictions with observations can help us decide whether the early universe was Gaussian or not. The greatest advantage of our assumption is simplicity. Indeed: all odd-order correlation functions of ε vanish, and all even-order moments can be reduced to the two-point function. For example, for the first four moments we have h 1 i = h 123 i = 0, h 12 i = ξL (|x1 − x2 |, t) D −2 , and h 1234 i = D −4 ξL (|x1 − x2 |, t) ξL (|x3 − x4 |, t) + cycl. (two terms) , (9) where h 12 . . . N i ≡ h ε(x1)ε(x2 ) . . . ε(xN ) i, while ξL in eq. (9) is h δ1 (x1 , t) δ1 (x2 , t) i, the linear 2-point correlation function. Using these relations and the perturbative expansion, we can express the 3- and 4-point functions on the right-hand side of eq. (8) in terms of ξL . This will close the hierarchy, allowing us to calculate weakly nonlinear corrections for the 2-point function ξ. The same procedure can be applied for higher order correlation functions. This simple idea is at the root of all perturbative results, discussed below. Perturbative calculations are usually easier after applying the Fourier transform. The Fourier transform of ξ is the power spectrum, P (k, t), where k is the comoving wavenumber. The linear term in the perturbative expansion for P (k, t) is PL (k, t) = Z ξL (r, t) exp(ik · r) d3 k . (10) The basic properties of linear evolution, as well as the effect of nonlinear interactions can be deduced from the functional form of δ1 and δ2 . Indeed, according to eq. (4), all fluctuations evolve without changing the shape of their density profiles, so that we can expect that the linear power spectrum evolves without changing its slope (the growth rate is the same for all scales). This is indeed the case. If PL (k, t) ∝ k n at some initial time t0 , than at later times PL (k, t) = PL (k, t0 ) D 2 (t)/D2 (t0 ) ∝ k n . A second characteristic property of a linear field is that its distribution function remains Gaussian; only its width, hδ12 i grows in time as D 2 (t). Neither of these properties remain valid when nonlinear interactions become important. The nonlinear terms, describing tidal interactions in eq. (5) redistribute power across the spectrum, and growth rates may be enhanced at some wavenumbers and suppressed at others: the logarithmic slope n(k, t) = d(log P )/d(log k) is no longer constant, it can now change with 2 time. Moreover, the nonlinear evolution introduces deviations from Gaussian statistics. The lowest-order moment which can detect such deviations is the skewness, hδ 3 i. The Gaussian distribution is symmetric about δ = 0, and its skewness is zero. According to eq. (5), however, the fluctuations with positive initial density contrast δ1 grow more rapidly than those with δ1 < 0. Hence, nonlinear interactions introduce asymmetry in the distribution, and we can expect that with time, gravity will induce a non-zero skewness in the distribution. The earliest (to our knowledge) analytic results of weakly nonlinear calculations were obtained by Zeldovich et al. and Peebles [56, 19, 45]. They considered power-law initial conditions, PL ∝ k n , and showed that unless n < 4, the nonlinear terms become larger than linear at large scales, and the perturbation theory breaks down. A somewhat more systematic approach, going beyond scaling relations was introduced in the eighties by Juszkiewicz et al. [33, 34] and Vishniac [53], who used Eulerian perturbation theory and the random phase approximation to study the first non-trivial, weakly-nonlinear corrections for second moments – P (k, t) and ξ(r, t). Peebles [45] used the same approach to calculate the gravitationally induced skewness, while Fry, in a truly seminal paper, showed how to calculate a reduced correlation function of arbitrary order N, using the perturbative expansion, truncated at the (N − 1) -st order [23]. Unfortunately, N-body simulations, available in the early eighties were too crude to study the range of validity of perturbative methods. Moreover, the galaxy surveys at scales, large enough to probe the weakly nonlinear regime of clustering, were dominated by noise. This made further progress in analytic calculations difficult and the field became dormant for several years. With time, the dynamical range of the simulations improved by more than an order of magnitude [55]. Simultaneously, the length scales in galaxy surveys, at which correlations can be accurately measured, increased by an order of magnitude [20], making weakly nonlinear theory useful in their analysis. As a result of these developments, the study of the evolution of the mass distribution in the expanding universe has become a rich topic for analytic perturbative methods. The revival of enthusiasm for analytic calculations produced many new results. It is our purpose to briefly summarize some of these new results here. For a more detailed approach, see e.g. the recent lecture notes of Bouchet [6] and the review by Sahni & Coles [48]. 1.3 Spatial smoothing The last issue we need to discuss before moving on to the next section is spatial smoothing. It is useful to employ a field δ̄, such that δ(x, t) is replaced by δ̄1 + δ̄2 + . . ., with δ̄N (x, t) = Z δN (x′ , t) w(|x − x′ | /R) d3x′ , (11) where w(x/R) is a spherically symmetric window function. R is the characteristic comoving smoothing length: w decreases rapidly for x/R ≫ 1. Its Fourier representation acts as a low pass filter, suppressing the fuctuations with wavenumbers k ≫ 1/R. The usual normalization R condition requires d3 r w(r) = 1. Smoothing over a ball of radius R is also called “top hat” smoothing; another popular filter is a fuzzy ball with a Gaussian profile of width R. Note that the effects of smoothing and gravitational evolution are interchangeable only for δ1 ; for terms of second or higher order, these two effects do not commute (this is evident from eq. (5); see also [30]). For a power-law initial spectrum, the smoothed linear variance is hδ̄12 i = σL2 (R, t) = Z w(x1 /R)w(x2 /R)ξL (|x1 − x2 |) d3 x1 d3 x2 ∝ R−(n+3) , (12) so the condition n > −3 ensures that the clustering process is hierarchical, and proceeds from small to large scales. (this constraint, together with the Zeldovich-Peebles condition n < 4, 3 broadly determines the range of “reasonable” values of n; a mix of observational arguments and theoretical prejudce shrinks this range to −2 < ∼n < ∼1, see e.g. [46]). The smoothing procedure is needed for at least two reasons. First, at any stage of the evolution, in order to be able to use the perturbation theory, we need to smooth-out the strong nonlinearities. In all hierarchical models, this can be done by any a low pass filter of width R > Rnl , where Rnl (t) is the current nonlinear scale, defined by the condition σL2 (Rnl , t) = 1. The second reason for introducing the smoothing is practical: all observable quantities, derived from galaxy surveys or numerical experiments, involve spatial smoothing. Hereafter, we will deal with smoothed fields almost without exceptions (which will be explicitly stated), so to simplify notation, we will omit the bars over δ and δN . Generally, we will also assume Gaussian initial conditions (again, all exceptions will be properly described). 2 New results for the second moment; the previrialization conjecture The perturbative series (3) can be used to expand the second moment of the density field in powers of the linear variance, σL2 . Squaring δ and averaging, we get σ 2 ≡ hδ 2 i = hδ12 i + hδ22 i + 2hδ1 δ3 i + . . . = σL2 + (I22 + I13 ) σL4 + O(σL6 ). (13) The second and third term in this expansion are the lowest order nonlinear corrections to linear theory. These terms are of order σL4 because O(σL3 ) terms vanish under the random phase assumption. The quantities I22 and I13 can be determined from perturbation theory. They were recently calculated for power-law initial conditions by Scoccimarro & Frieman [50] and by Lokas et al. [41]. Scoccimarro & Frieman explored the evolution of an unsmoothed density field. Lokas et al. studied spatially smoothed fields, and found that nonlinear tidal interactions redistribute the power in the spectrum in such a way, that for n > −1, the correction I22 + I13 is negative. The growth rate of density fluctuations is inhibited: σ is smaller than the linear prediction σL . The magnitude of this suppression increases with n, or with the relative amount of small scale power in the initial spectrum. For n < −1, there is an opposite effect – the fluctuation growth rate is enhanced (σ > σL ). The transition occurs at n = −1, when the correction is close to zero and σ ≈ σL . Lokas et al. compared these perturbative results with a series of N-body simulations and found a good agreement in a wide dynamical range, up to σ ≈ 1. These results are in qualitative agreement with the so-called previrialization conjecture of Davis & Peebles [18]. According to Davis & Peebles, tidally induced nonradial motions of small scale inhomogeneities within collapsing mass concentrations should resist gravity and slow down the collapse. Until recently the size of the previrialization effect was still under discussion, with some N-body studies reproducing the effect and others claiming its absence. For example, Peebles [47] found strong support for the previrialization effect, while Evrard & Crone [21] reached opposite conclusions. This apparent controversy can be resolved, using the perturbative results we just discussed. Indeed, Peebles in his simulations assumed n = 0, while Evrard & Crone assumed n = −1: a spectral index, for which nonlinear effects are minimal. Jain & Bertschinger [32] used perturbation theory and N-body experiments to investigate the nonlinear transfer of power in a CDM spectrum. Their results are consistent with Lokas et al. and Scoccimarro & Frieman. Finally, we must mention the important work of Makino et al. [42], who succeeded in reducing the dimensionality of the mode coupling integrals of Vishniac and Juszkiewicz et al. [53, 34]. This significantly simplified the derivation of all of the new results described in this section. 4 3 The third moment and deviations from Gaussian behavior As we saw in Section 1, nonlinear gravitational evolution drives the field δ away from the initial Gaussian distribution. Deviations from Gaussianity can be measured by cumulants, also called “reduced” or “connected” moments. A cumulant of order N is MN = ∂ N ln hexδ i/∂xN , evaluated at x = 0 . (14) For the first four cumulants, this expression generates the mean, M1 = hδi = 0, the variance, M2 = hδ 2 i = σ 2 , the skewness, M3 = hδ 3 i, and the kurtosis, M4 = hδ 4 i − 3σ 4 . For a zero-mean Gaussian distribution, all cumulants vanish except M2 . The skewness measures the asymmetry of the distribution, while the kurtosis measures the flattening of the tails relative to a Gaussian. Using perturbation theory, Fry [23] showed that in an Einstein-de Sitter universe, MN scale like σ 2N −2 . Hence, to lowest non-vanishing order in perturbation theory, the ratio SN = MN / σ 2N −2 (15) is time-independent. Juszkiewicz et al. [36] have recently used this property and the Edgeworth series to describe the weakly nonlinear evolution of the probability distribution function (PDF) of the density field. The Edgeworth expansion is a standard statistical tool, used to represent quasi-gaussian probability distribution functions (PDFs). Juszkiewicz et al. obtained an expansion in powers σ, with the first few terms, given by p(ν) = h 1+ 1 3! S3 H3 (ν)σ + n 1 4! S4 H4 (ν) + 10 6! o i S32 H6 (ν) σ 2 g(ν) + O (g(ν)σ 4 ) , (16) where ν = δ/σ, g = (2π)−1/2 exp(−ν 2 /2), and HJ is a Hermite polynomial of degree J. A similar expansion was obtained independently by Bernardeau & Kofman [4]. The Edgeworth expansion is particuraly useful when only a few low-order reduced moments are known; it can be also used to estimate low-order SN -s from the shape of empirically measured PDF near the peak (in realistic situations, the tails, and/or high-N moments are poorly known anyway). An alternative, Laplace-transform representation for the PDF, which does require the knowledge of all SN -s to N = ∞, was proposed by Bernardeau [5]. 3.1 Skewness induced by gravity The acual values of the parameters SN can be also calculated perturbatively. The results generally depend on the shape of the initial power spectrum and the shape of the window function. Let us begin with the skewness, responsible for the lowest order nonlinear correction to the Gaussian PDF in eq. (16). Upon raising both sides of equation (3) to the third power and averaging, we get hδ 3 i = S3 σ 4 = hδ13 i + h3δ12 δ2 i + . . . (17) In models, which satisfy the random phase assumption, the lowest order non-vanishing term in this expansion is O(σ)4 . The scaling with σ is completely different in models with nongaussian initial conditions and an intrinsic skewness hδ13 i = 6 0: the lowest order term is ∝ σ 3 . The skewness parameter, S3 in eq. (17) was first calculated by Peebles [45], assuming Ω = 1 and no smoothing. The latter constraint prevents any comparison of this result with observations or N-body simulations. To deal with this problem, we calculated S3 for a smoothed density field with a power-law initial spectrum and for two different kinds of spatial filters: a gaussian and a top hat [35]. The result is particularly simple in the latter case, S3 = 34 7 − (n + 3) + O(σ 2 ) , 5 (18) where the allowed range of power indexes is −3 ≤ n < 1 (outside of this range S3 diverges). Note that here, as well as in Section 2, we quote only results for the Einstein-de Sitter case. All these results, however, are also true for a much wider class of models with arbitrary Ω and Λ (this is an approximate statement, but the approximation is good enough for all practical purposes). For example, for 0 ≤ Λ/3H 2 ≤ 1 − Ω and 0.05 ≤ Ω ≤ 3 , the term ‘34/7’ in eq. (18) should be replaced by 4 + 4κ(Ω) = 34 7 + 6 7 (Ω−0.03 − 1) . (19) The corrections, introduced by deviations from the Einstein-de Sitter model were first derived by Bouchet et al. [8] for arbitrary Ω, and by Bernardeau [2] and Bouchet et al. for arbitrary Λ [10]. These results were recently confirmed by Catelan et al. [14], who used a significantly different method of derivation. For our purposes here, the most important implication of all this is that Gaussian initial conditions and weakly nonlinear gravitational instability generate the scaling relation MN ∝ σ 2N −2 (σ < 1) , (20) and SN , the ratio of the above moments, up to terms of order σ 2 in perturbation theory is time-, Ω- and Λ-independent; it is determined only by the shape of the initial power spectrum. Bernardeau [2] generalized eq. (18) for a smoothing scale-dependent slope, n = n(R). He showed, that this expression for S3 (R) remains valid when n is replaced by the effective local slope, n(R) = − d(log σL2 (R))/d(log R) − 3 . (21) As we have seen, to lowest non-vanishing order in perturbation theory, S3 is time independent. We can expect that S3 (R) will be scale-independent as well in the range of smoothing scales, for which n(R) is constant. A completely different picture arises when the primeval PDF is significantly skew. The lowest order non-vanishing contribution to skewness gives S3 (R) ∝ 1/σL ∝ R−(n+3)/2 D(t)−1 , (22) so in all hierarchical (n > −3) models we can expect a strong scale- and time-dependence, with S3 “blowing up” at large scales and at high redshift. As pointed out by several authors, this property can be used to distinguish observationally Gaussian initial conditions from their non-gaussian alternatives [17, 35, 49]. Chodorowski & Bouchet [16] find that the distinction can be made even sharper by studying the scaling properties of S4 (see also Fry & Scherrer [25]). Using methods similar to those discussed above, Catelan & Moscardini [15], Lokas et al. [40], and Bernardeau [2] calculated the parameter S4 for power-law spectra and top hat as well as Gaussian windows. The last two papers also provide closed-form expressions for S4 = S4 (n). Bernardeau [5] solved the problem of calculating SN -s of arbitrarily high order for top hat smoothing. Most of the above perturbative results for S3 and S4 were tested against N-body simulations [1, 7, 35, 36, 38, 40, 54], showing remarkable agreement up to σ ≈ 1. On a comoving scale R, the Edgeworth expansion agrees with the N-body simulations within ∼ 1/σ(R) standard deviations from zero [36]. The Laplace transform perturbative PDF agrees with the N-body PDF even when σ is slightly greater than unity [5]. 3.2 Velocity statistics Unlike the reduced moments of the density field, the cumulants of the expansion scalar, θ = ∇ · v/H are very sensitive to the value of Ω. For example, the skewness parameter is given by 6 the expression [3] hθ3 i / hθ2i2 = − Ω−0.6 h 26 7 − (n + 3) i . (23) In principle, this can provide a new method of determining Ω from the velocity data alone (rather than from the comparison with the density field, as in more conventional methods). The existing velocity surveys, however, are by far too shallow to allow accurate measurements of the variance and skewness of θ. We have to wait for better data. Weakly nonlinear theory can also be used to study the dynamics of relative motions in pairs of galaxies, which induce the anisotropy of the redshift space galaxy correlations [37]. This can improve the accuracy of Ω determinations from presently available redshift surveys. 3.3 Lagrangian methods So far, we have focused on Eulerian perturbation theory. An interesting alternative is the Lagrangian approach. It follows particle trajectories rather than the evolution of δ(x, t). The local density is δ = 1/J − 1, where J is the Jacobian of the transformation from initial particle positions to positions at time t. The famous Zeldovich approximation [57] is a linear order Lagrange solution. In its original form, it can not be used for statistical studies because it predicts infinite cumulants for δ. As pointed out by Grinstein & Wise [29], the problem can be at least partially fixed by expanding 1/J − 1 in powers of spatial derivatives of the linear order Eulerian velocity field. The cumulants, generated by such an expansion are finite. Unfortunately, however, they disagree with the proper Eulerian perturbation theory and with all N-body experiments [15, 35, 36]. The Zeldovich approximation fails particularly badly when applied to the velocity divergence, θ [3, 36]. Two lessons follow from this example. The first is that dynamical estimates of Ω, derived from reconstruction schemes for the initial PDF of δ or θ, which rely entirely on the Zeldovich approximation (see e.g. [44]), should be treated with caution. The second lesson is positive: the Zeldovich approach is more powerful than it seems. For example, under this approximation, κ = 0 in eq. (19) and therefore for a top hat S3 = 4 − (n + 3). Depending on n, this causes about 20% to 50% underestimate of the true S3 , derived from second order Eulerian theory. Note however, that we used first order Lagrangian theory to derive the Zeldovich prediction for S3 . To make a fair comparison, we should confront this result with its first order Eulerian counterpart: the linear prediction S3 = 0, which is much further away from the true value! What we have just seen, is an illustration of a more general phenomenon: at any fixed order in perturbation theory, the Lagrangian approach is likely to remain valid for stronger density contrasts than the Eulerian approach. This happens because the requirement of the smallness of the (Lagrangian) particle displacements is less restrictive than the requirement of the smallness of |δ|. This idea, which must have motivated Zeldovich, remains valid at higher orders in perturbation theory. The idea to extend the Lagrangian perturbation theory beyond the linear order was introduced by Moutarde et al. [43] and generalized by Bouchet et al. [8] and Buchert & Ehlers [13]. Lagrangian coordinates have been used to derive second- and third-order perturbative solutions in the Einstein-de Sitter case [12, 13], as well as in more general cosmological models [8, 10], and to study distortions of correlations in redshift space [10, 31]. We are convinced that the full potential of this approach to perturbation theory is far from being exhausted. 3.4 Convergence studies As we have seen, perturbative expansions for hδ 2 i and hδ 3 i, truncated at the lowest-order or next to the lowest-order non-vanishing contribution, are in remarkable agreement with N-body 7 simulations in a wide dynamical range, up to σ ≈ 1. The precise range of validity of perturbation theory is still under investigation. Apart from comparisons with N-body experiments, the convergence can be also studied within the perturbation theory itself, by comparing contributions coming from terms of increasing order. Such studies may answer the question: why does perturbation theory work so well? Scoccimarro & Frieman [50] obtained closed-form expressions for I2 = I22 + I13 in the case of no smoothing (R = 0). Although individually, I22 and I13 suffer from “infrared” (k → 0) divergencies, their sum is finite and approximately n-independent; for −2 ≤ n ≤ 2 , the correction term is [41, 50] I2 ≈ 1.82 . (24) The divergent terms in I22 and I13 cancel each other. The probable source of this phenomenon is the Galilean invariance of the equations of motion [50]. In order to make a quantitative comparison with N-body results possible, Lokas et al. [40] included the effects of smoothing in their calculations. Introduction of a finite smoothing length (R 6= 0) generates “ultraviolet” (k → ∞) divergences at n > −1; I2 diverges unless PL (k) has a shortwave cutoff at k = kc . For kc R ≫ 1, the correction term is [41] I2 ≈ −(0.6kc R)n+1 . (25) We can expect that perturbation theory breaks down when σL reaches the value, at which the first two terms in the expansion (13) become comparable, |I2 |σL4 = σL2 . Therefore, equation (25) suggests that for a given kc R, the maximal value of σL , determining the range of validity of perturbation theory, should gradually decrease with increasing n. Comparisons with N-body experiments show that this is indeed the case [40]. Note that a shortwave cutoff naturally appears in N-body experiments; kc−1 corresponds to the mean interparticle separation, or the Nyquist scale. Scoccimarro & Frieman [50] have also calculated the O(σ 2 ) term in the expansion for S3 for the unsmoothed density field. They found that O(σL2 ) terms in the expansions for S3 do not “overtake” the lowest-order non-vanishing terms until σL ≈ 0.5, in qualitative agreement with the results of direct N-body tests of perturbation theory, discussed above. A quantitative comparison, however, will be possible only after smoothing effects are included (Scoccimarro & Frieman, in preparation). 4 Observations Before comparing theoretical predictions to observations, let us discuss two possible sources of confusion. 1. Biasing. If galaxies form with different efficiency in different environments, then the galaxy distribution may paint a biased picture of the underlying mass distribution. Let us suppose that the smoothed galaxy field δg is a local, but not necessarily linear function of the smoothed mass density field: δg (x) = f [δ(x)], where f is an arbitrary function. Then, using a Taylor expansion for |δ| ≪ 1, it is possible to prove, as Fry & Gaztañaga [24] did, that local bias preserves the form of the scaling relations (20), while changing the constants SN . For example, S3 changes to [24, 36] S3g ≡ hδg3 i / hδ22i = S3 /b1 + 3b2 /b21 , (26) where bJ ≡ dJ f /dδ J , evaluated at δ = 0 (note that the linear bias model with all bJ = 0 for J > 1 must break down when b > 1 and δ < −1/b, since it predicts negative δg ). 8 2. Redshift distortions. Redshift surveys use radial velocities of galaxies instead of their true radial coordinates. As a result, galaxy peculiar motions distort the true spatial distribution and the two-point correlation function, ξ (cf. §20 in [46] and references therein). Bouchet et al. [10] and Hivon et al. [31] have used weakly nonlinear perturbation theory and Nbody simulations to study the effects of redshift space distortions on the three-point correlation function, ζ, and on S3 . They showed that although ζ, like ξ, is affected by the redshift space distortion, all appreciable, Ω-dependent effects cancel out for the moment ratio S3 . In the weakly-nonlinear domain, when σ < 1, and for spectral slopes −3 ≤ n ≤ −1, the differences between redshift-space S3 and its true value are smaller than 10%, whatever the value of Ω. Hence, the comparison between theoretical predictions and observations is not obscured by redshift space distortions provided we use the moment ratio S3 rather than reduced moments themselves. The parameters S3 and S4 have been estimated from the CfA and SSRS surveys [26], the IRAS 1.2 Jy sample [9], as well as for vaious angular catalogues [27, 52]. All these measurements show beautiful agreement with the predicted scaling relation (20). The dependence (26) of S3g on the shape of the bias function may explain why the value measured by Bouchet et al. [9] IRAS from the top-hat smoothed, IRAS redshift survey is rather low, S3g ∼ 1.5 . The galaxies observable in the infrared are underrepresented in rich clusters relative to optical galaxies, so the “bias function” that applies to IRAS galaxies may have negative curvature ( b2 < 0 ), pushing S3g below the value of S3 for the mass. If cluster avoidance is indeed the cause of IRAS the low value of S3g , then one would expect that optically selected surveys would yield higher values. The values of S3g , obtained recently by Gaztañaga [27] from the APM survey suggest that this is exactly the case. The value of S3g , deprojected from the two-dimensional APM data onto three-dimensional space is 3.16 ± 0.14 , assuming a top-hat filter with a radius R > 7 h−1 Mpc (here h is the Hubble constant in units 100 km s−1 Mpc−1 ; the latter condition ensures σg < 1 ). Substituting n = −1, appropriate for the slope, seen in the APM survey on these scales, into equation (18) yields S3 = 2.9 . Hence, for sufficiently large scales, the AP M APM results are in perfect agreement with second order perturbation theory: S3g = S3 . This agreement provides support for three theoretical assumptions: first, that gravitational instability is responsible for the growth of clustering; second, that the initial conditions were M ≈ 1 and Gausssian; and third, that the APM galaxies trace the mass, i.e., in eq. (26) bAP 1 AP M b2 ≈ 0 (of course, other combinations of b1 and b2 also fit the APM data, but they require a coincidental compensation of the linear, 1/b1 factor by the b2 term arising from nonlinear bias). This is bad news non-local biasing models like the model of Bower et al. [11], invoked in a desperate attempt to explain the mismatch of the observed power spectrum with standard CDM. Indeed, as demonstrated by Frieman & Gaztañaga [22], such models break the scaling relation (20) and are incompatible with observations. Most recently, Gaztañaga & Mähönen [28] have used the APM data to severely constrain the so-called global texture model. The source of problems for this model is its prediction that SJ parameters must ‘blow up’ at large scales, like S3 in our example in eq. (22). There is no trace of such behaviour in the APM galaxy distribution. Since the already available data can be used to constrain non-local biasing schemes and deviations from Gaussian distribution, we can expect that in the new generations of catalogues (like the SDSS and the 2dF, described in these proceedings) will finally tell us whether all we are looking at is just a Gaussian random process modified by plain gravity. Acknowledgements. We are grateful to the Organizers, Chantal Balkowski, Sophie Maurogordato, and J. Trân Thanh Vân, for making this scientifically useful meeting happen; to Mother Nature for providing good skiing conditions, and last but not least, to Chantal and 9 Sophie as Editors, for their incredible patience with the authors, who broke several submission deadlines for this manuscript. This research was supported by the Polish Government KBN grants No. 2-P304-01607, 2-P304-01707, and by an NSF grant No. PHY94-07194. RJ thanks Alain Omont for his hospitality at Institut d’Astrophysique de Paris, CNRS, where part of the work reviewed here was done. References [1] Baugh, C.M., Gaztañaga, E., Efstathiou, G., 1994, Mon. Not. R. astr. Soc. 274, 1049. [2] Bernardeau, F., 1994, Ap. J. 433, 1. [3] Bernardeau, F., Juszkiewicz, R., Dekel, A., Bouchet, F.R., 1995, Mon. Not. R. astr. Soc. 274, 20. [4] Bernardeau, F., & Kofman, L., 1995, Ap. J. 443, 479. [5] Bernardeau, F., 1994, Astr. Astrophys. 291, 697. [6] Bouchet, F. R. 1995, in: Proc. of the “Enrico Fermi” School “Dark Matter in the Universe” held in Varenna (Societa Italiana di Fisica). [7] Bouchet, F.R., Hernquist, L., 1992, Ap. J. 400, 25. [8] Bouchet, F.R., Juszkiewicz, R., Colombi, S., Pellat, R., 1992, Ap. J. (Letters) 394, L5. [9] Bouchet, F.R., Strauss, M., Davis, M., Fisher, K.B., Yahil, A., & Huchra, J.P., 1993, Ap. J. 417, 36. [10] Bouchet, F.R., Colombi, S., Hivon, E., & Juszkiewicz, R., 1995, Astr. Astrophys. 296, 575. [11] Bower, R.G., Coles, P., Frenk, C.S., & White, S.D.M., 1993, Ap. J. 405, 403. [12] Buchert, T., 1994, Mon. Not. R. astr. Soc. 267, 811. [13] Buchert, T., & Ehlers, J., 1993, Mon. Not. R. astr. Soc. 264, 375. [14] Catelan, P., Lucchin, F., Mattarrese, S., Moscardini, L., 1995, Mon. Not. R. astr. Soc. 276, 39. [15] Catelan, P., & Moscardini, L., 1994, Ap. J. 426, 14. [16] Chodorowski, M., & Bouchet, F.R., 1996, Mon. Not. R. astr. Soc., in press. [17] Coles, P., Frenk, C.S., 1991, Mon. Not. R. astr. Soc. 253, 727. [18] Davis, M., & Peebles, P.J.E., 1977, Astrophys. J. Suppl. Ser. 34, 425. [19] Doroshkevich, A.G., Zeldovich, Ya. B., 1975 Astrophys. Space. Sci. 35, 55. [20] Efstathiou, G., 1996, in: Les Houches Lectures, Session LX, ed. R. Schaeffer, Elsevier Science Publishers (Netherlands). [21] Evrard, A.E., & Crone, M.M., 1992, Ap. J. (Letters) 394, L1. [22] Frieman, J., & Gaztañaga, E., Ap. J. 425, 392. [23] Fry, J.N., 1984, Ap. J. 279, 499. 10 [24] Fry, J.N., & Gaztañaga, E., 1993, Ap. J. 413, 447. [25] Fry, J.N., & Scherrer, R.J. 1994, Ap. J. 279, 36. [26] Gaztañaga, E., 1992, Ap. J. (Letters) 398, L17. [27] Gaztañaga, E., 1994, Mon. Not. R. astr. Soc. 268, 913. [28] Gaztañaga, E., & Mähönen, P., 1996, Ap. J. (Letters), in press [29] Grinstein, B., & Wise, M.B., 1987, Ap. J. 314, 448. [30] Goroff, M.H., Grinstein, B., Rey, S.-J., & Wise, M.B., 1986, Ap. J. 311, 6. [31] Hivon, E., Bouchet, F.R., Colombi, S., & Juszkiewicz, R., 1995, Astr. Astrophys. 298, 643. [32] Jain, B., Bertschinger, E., 1994, Ap. J. 431, 495. [33] Juszkiewicz, R., 1981, Mon. Not. R. astr. Soc. 197, 931. [34] Juszkiewicz, R., Sonoda, D.H., & Barrow, J.D., 1984, Mon. Not. R. astr. Soc. 209, 139. [35] Juszkiewicz, R., Bouchet, F.R., & Colombi, S., 1993, Ap. J. (Letters) 412, L9. [36] Juszkiewicz, R., Weinberg, D.H., Amsterdamski, P., Chodorowski, M., & Bouchet, F., 1995, Ap. J. 442, 39. [37] Juszkiewicz, R., Fisher, K.B., & Szapudi, I., 1996, Ap. J., in press. [38] Lahav, O., Itoh, M., Inagaki, S., & Suto, Y., 1993, Ap. J. 402, 387. [39] Lifshitz, E.M., 1946, J. Phys. U.S.S.R. 10, 116, (Zh.E.T.F. 16, 587). [40] Lokas, E.-L., Juszkiewicz, R., Weinberg, D.H., & Bouchet, F.R., 1995, Mon. Not. R. astr. Soc. 274, 730. [41] Lokas, E.-L., Juszkiewicz, R., Bouchet, F.R., & Hivon, E., 1996, Ap. J., in press. [42] Makino N., Sasaki M., Suto, Y., 1992, Phys. Rev. D46, 585. [43] Moutarde, F., Alimi, J.-M., Bouchet, F.R., & Pellat, R., 1991, Ap. J. 382, 377. [44] Nusser, A., & Dekel, A., 1993, Ap. J. 405, 437. [45] Peebles, P.J.E., 1980, The Large-Scale Structure of the Universe (Princeton University Press: Princeton). [46] Peebles, P.J.E., 1993, Principles of Physical Cosmology (Princeton University Press: Princeton). [47] Peebles, P.J.E., 1990, Ap. J. 365, 27. [48] Sahni, V., & Coles, P., 1995, Physics Reports, in press (preprint SISSA astro-ph/9505005). [49] Silk, J., & Juszkiewicz, R., 1991, Nature 353, 386. [50] Scoccimarro, R., & Frieman, J., 1995, Fermilab preprint. [51] Strauss, M.A., & Willick, J.A., 1995, Physics reports 261, 271. [52] Szapudi, I., Dalton, G., Efstathiou, G., & Szalay, A.S., 1995, preprint. 11 [53] Vishniac, E., 1983, Mon. Not. R. astr. Soc. 203, 345. [54] Weinberg, D.H., & Cole, S., 1992, Mon. Not. R. astr. Soc. 259, 652. [55] White, S.D.M., 1996, in: Les Houches Lectures, Session LX, ed. R. Schaeffer (Elsevier Science Publishers: Netherlands). [56] Zeldovich, Ya.B., 1965, Adv. Asron. Astrophys. 3, 241. [57] Zeldovich, Ya.B., 1970, Astr. Astrophys. 5, 84. 12