Annu. Rev. Astron. Astrophys. 1994. 32: 371-418
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Having assumed GI, the LSS can be traced backward in time to recover the initial fluctuations and constrain statistics which characterize them as a random field, e.g. the power spectrum (PS), and the probability distribution functions (PDFs). ``Initial'' here may refer either to the linear regime at z ~ 103 after the onset of the self-gravitating matter era, or to the origin of fluctuations in the early universe before being filtered on sub-horizon scales during the plasma-radiation era. The PS is filtered on scales leq 100 h-1Mpc by DM-dominated processes, but its shape on scales geq 10 h-1Mpc is not affected much by recent non-linear effects (because the rapid density evolution in superclusters roughly balances the slow evolution in voids at the same wavelength). The shape of the one-point PDF, on the other hand, is expected to survive the plasma era unchanged but it develops strong skewness even in the mildly-non-linear regime. Thus, the present day PS can be used as is to constrain the origin of fluctuations (large scale) and the nature of the DM (small scale), while the PDF needs to be traced back to the linear regime first. (Review: Efstathiou 1990.)

7.1 Power Spectrum - Dark Matter

The competing LSS formation scenarios are reviewed in Peebles (1993, Section 25). If the DM is all baryonic, then by nucleosynthesis constraints (see Kolb & Turner 1990, Section 4) the universe must be of low density, Omega < 0.2, and a viable model for LSS is the Primordial Isocurvature Baryonic model (PIB) with several free parameters, typically of large relative power on large scales. With Omega ~ 1 the non-baryonic DM constituents are either ``hot'' or ``cold'', and the main competing models are CDM, HDM, and MDM - a 7:3 mixture of the two (e.g. Blumenthal et al. 1988; Davis et al. 1992; Klypin et al. 1993). The main difference in the DM effect on the PS arises from free-streaming damping of the ``hot'' component of fluctuations on galactic scales.

BULK VELOCITY. A simple and robust statistic related to the PS is the amplitude V of the vector average of the smoothed velocity field, v, over a volume defined by a normalized window function WR(r) of a characteristic scale R (e.g. top-hat),

V identinteg d3 x WR(x) v(x),

< V2 > = f2 / 2 pi2 integ0infty dk P(k) tilde WR2(k). (22)

< V2 > is predicted for a linear model with a density spectrum P (k), where tilde WR2(k), the Fourier transform of WR (r), emphasizes waves geq R. The bulk velocity is obtained from the observed radial velocities by minimizing Equation (11). The report by Dressler et al. (1987) for the 7S Es sampled within ~ 60 h-1Mpc was V = 599 ± 104 towards (l, b) = (312°, +6°), which was interpreted prematurely as being in severe excess of the predictions of common theories. However, this measurement cannot be directly compared to the predictions for a top-hat sphere because the effective window is much smaller due to the nonuniform sampling (SG) and weighting (Kaiser 1988). The SG bias can be crudely corrected by volume weighting as in POTENT (Section 4.2), at the expense of large noise. Courteau et al. (1993) find for the tentative Mark III data: V40 = 335 ± 38 (295°, +35°) and V60 = 360 ± 40 (279°, +11°), where VR refers to a top-hat sphere of radius R h-1Mpc. Alternatively, V can be computed from the POTENT v field by simple vector averaging from the grid. Figure 7 shows two results, one minimizing the SG bias by Vi weighting, and the other reducing the random errors by weighting propto sigmai-2 (Section 4.2). V60 is found to be in the range 270-360 km s-1 (296°,+11°) - smaller than previous estimates. The additional random error from Monte-Carlo noise simulations is typically 15%, not including cosmic scatter due to the fact that only one sphere has been sampled.

Figure 7. The bulk velocity in a top-hat sphere of radius Reff about the LG as recovered by POTENT from the Mark III data. Shown are |V| (filled dots), Vx (triangles), Vy (squares), and Vz (hexagons). Distances and velocities are in km s-1. The two results shown reflect the systematic uncertainty. The 1 sigma uncertainty due to random distance errors is appeq 15% (Dekel et al. 1994).

MACH NUMBER. The bulk velocity is robust but it relates to the normalization of the PS, not predicted from first principles by any of the competing theories but rather normalized by some other uncertain observation, e.g. the CMB fluctuations or the galaxy distribution with an unknown biasing factor. A statistic which measures the shape of the PS free of its normalization is the cosmic Mach number (Ostriker & Suto 1990), defined as M ident V / S, where S is the rms deviation of the local velocity from the bulk velocity,

S2 ident integ d3 x WR(x) [v(x)-v]2,

< S2 > = f2 / 2pi2 integ0infty dk P (k) [1 - tildeWR2(k)]. (23)

M(R, Rs) measures the ratio of power on large scales gtapprox R to power on small scales gtapprox Rs. Strauss et al. (1993) derived M = 1.0 for the local Ss (Aaronson et al. 1982a) with R ~ 20 h-1Mpc and Rs -> 0, and found ~ 5% of their CDM simulations to have M as large - a marginal rejection or consistency depending on taste. On larger scales, using POTENT with R = 60 h-1Mpc (top-hat) and Rs = 12 h-1Mpc (Gaussian), the tentative Mark III data yield M ~ 1, which roughly coincides with the rms expected from CDM but has only ~ 5% probability to be that low for a PIB spectrum (Omega = 0.1, h = 1, fully ionized, normalized to sigma8 = 1) (Kolatt et al., in preparation).

POWER SPECTRUM. The velocity field by POTENT enables a preliminary determination of the mass PS itself in the range 10-100 h-1Mpc (Kolatt & Dekel 1994b). It is best determined by the potential field, which is smoother and less sensitive to non-linear effects than its derivatives v and delta. The PS was compared with the predictions of theoretical models which were N-body simulated, ``observed'', and fed into POTENT before the PS was computed using the same procedure applied to the observed data, thus eliminating the effect of systematic errors and estimating the random errors. The preliminary results indicate that the shape of the PS in the limited range sampled resembles a CDM-like PS with a shape parameter Gamma ~ 0.5 (referring in CDM to Omega h), with the power index bending toward n appeq 0 by 100 h-1Mpc. The normalization for the mass PS, also obtained by Seljak & Bertschinger (1994), is f sigma8 = 1.3 ± 0.3. This is not too sensitive to the PS shape, but still note that it mostly determined by data at ~ 30 h-1Mpc. With sigma8opt approx 1 for optical galaxies, this implies betaopt approx 1.3 ± 0.3 (compare Section 8). If Omega = 1, the quadrupole in delta T / T by COBE corresponds to sigma8 = 1.0, 0.6, 0.5 with ~ 45% error (including cosmic scatter) for CDM, MDM and HDM spectra (yet to be calculated for an open universe), i.e., CDM is fine while HDM and MDM are ~ 2sigma low. In comparison, the PS of the different luminous objects are all well described by a CDM-like PS with Gamma approx 0.25, with the relative bias factors for Abell clusters, optical galaxies and IRAS galaxies in the ratios 4.5:1.3:1 (± 6% rms, e.g. Peacock & Dodds 1994). If indeed sigma8opt ~ 1 for optical galaxies, then IRAS galaxies are slightly anti-biased and betaI approx 1.3.betaopt.

A DISCREPANCY ON A VERY LARGE SCALE? The galaxy velocity field in the local ~ 60 h-1Mpc which has been studied quite accurately seems to be in general agreement with other observations and with the predictions of the common GI scenarios, but there is a disturbing hint for a possible discrepancy with our basic hypotheses on larger scales. Lauer and Postman (1993, LP) measured the bulk flow of the volume-limited system of 119 Abell/ACO clusters within z < 150 h-1Mpc, estimating distances from luminosity versus slope of the surface-brightness profile in brightest cluster galaxies (BCG) with claimed 16% error. The LG motion with respect to this system is found to be 561 ± 284 km s-1 towards (l, b) = (220°,-28°), roughly 80° off the direction of the LG-CMB velocity (276°,+30°). The inferred bulk velocity of the cluster system (of effective radius ~ 80-110 h-1Mpc) relative to the CMB is V = 689 ± 178 km s-1 towards (343°,+52°) (± 23°), in the general direction of the LG motion, the galaxy bulk flow, the GA and the background Shapley concentration. Taken at face value this is a high velocity over a large scale; the rms bulk velocity within R = 100 h-1Mpc as predicted by the common theories normalized to COBE is below 200 km s-1. However, the errors are large too.

The data have been subjected by LP to careful statistical tests. With a few tens of points contributing to the velocity in each direction, the shot-noise can clearly contribute a false signal of several hundred km s-1. For assessing the theoretical implications, the actual observing scheme of LP was applied to N-body simulations of several competing scenarios (Strauss et al. 1994). Clusters were placed at the 119 highest density peaks of appropriate mass, and Galactic extinction and observational errors were modeled. As noted by LP, the fact that the measured velocity vector lies away from the ZOA increases the statistical significance of this measurement. Taking the error ellipsoid into account, the probability by means of chi2 of the LP result in the model universes simulated is 2.6-5.8%. Feldman & Watkins (1994), using a different technique, find the probabilities to be 6-10%. The LP bulk velocity is thus a ~ 2sigma deviation from several common theories. This is probably not enough for a serious falsification of these models, but it is certainly an intriguing result which motivates more accurate investigations. The BCG method is limited to one measurement per cluster, so an obvious strategy to reduce the errors would be to collect many TF and Dn - sigma distances per cluster.

A true ~ 700 km s-1 velocity at R ~ 100 h-1Mpc would be in serious conflict with GI. First, if this velocity is typical then it predicts larger CMB fluctuations than observed on ~ 2°. Second, the gravitational acceleration on the LP sphere as estimated from the spatial distribution of clusters on even larger scales (Section 8.1, Scaramella et al. 1991) predicts a flow of only ~ 200 km s-1. The way to interpret the LP result in the context of the conventional theories is either as a ~ 2-2.5sigma statistical fluke, or as a biased result due to a yet-unresolved systematic error in the BCG method or in the sample, e.g. a significant velocity biasing of the clusters (which is contrary to the naive expectations from GI, however).

7.2 Back in Time

The forward integration of the GI equations by analytic approximations or by N-body simulations cannot be simply reversed despite the time reversiblity of gravity. It is especially hopeless in collapsed systems where memory has been erased, but the case is problematic even for linear systems. When attempting backwards integration, the decaying modes (e.g. Peebles 1993, Section 5), having left no detectable trace at present, t0, would amplify noise into dominant spurious fluctuations at early times. This procedure has a negligible probability of recovering the very special initial state of almost uniform density and tiny velocities which we assume for the real universe at ti. This is a problem of mixed boundary conditions: some of the six phase-space variables per particle are given at ti and some at t0. This problem can be solved either by eliminating the decaying modes or by applying the principle of least action.

ZEL'DOVICH TIME MACHINES. If the velocity field is irrotational, the Euler equation (2) can be replaced by the Bernoulli equation for the potentials, dotPhiv - (del Phiv)2/2 = -2H Phiv + Phig. The Zel'dovich approximation, restricted to the growing mode, requires that each side vanish: one side relates Phiv and Phig linearly and the other is the ``Zel'dovich-Bernoulli'' (ZB) equation (Nusser & Dekel 1992), dotphiv - (Ddot / 2) (del phiv)2 = 0, with the potentials in units of a2Ddot. The ZB equation can be easily integrated backwards with a guaranteed uniform solution at ti. phiv at t0 is extractable from observations of velocities (Section 4) or galaxy density (Section 5), and the initial v and delta can be derived from the initial phiv using linear theory. While the ZB approximation conserves momentum (like delta0) one can alternatively satisfy continuity under the Zel'dovich approximation (like deltac), and obtain a second-order equation for phig which is somewhat more accurate (Gramman 1993b, Equation 2.24, 2.25). The recovered initial deltai has deeper valleys and shallower hills compared to naive recovery using linear theory, e.g. the GA is less eccentric than assumed by Bertschinger & Juszkiewicz (1988).

RECOVERING THE IPDF. An important issue is whether or not the initial fluctuations were Gaussian. A Gaussian field is characterized by the joint PDFs of all order being generalized Gaussians (cf., Bardeen et al. 1986), and in particular the one-point probability of delta is P(delta)propto exp[-delta2 / (2sigma2)]. Common Inflation predicts Gaussian fluctuations but non-Gaussian fluctuations are allowed by certain versions of Inflation (cf., Kofman et al. 1990) and by models where the perturbations are seeded by cosmic strings, textures, or explosions (see Peebles 1993, Section 16). The present density PDF develops a log-normal shape due to non-linear effects (Coles & Jones 1991; Kofman et al. 1994): the tails become positively skewed because peaks collapse to large densities while the density in voids cannot become negative, and the middle develops negative skewness as density hills contract and valleys expand. On the other hand, the PDF of present-day velocity components is insensitive to quasi-linear effects (Kofman et al. 1994).

The observed PDFs today agree with N-body simulations of Gaussian initial conditions (Bouchet et al. 1993), but they have only limited discriminatory power against initial non-Gaussianities; the development of a density PDF with a general log-normal shape may occur even in certain cases of non-Gaussian initial fluctuations (e.g. Weinberg & Cole 1992), and the velocity PDF becomes Gaussian under general conditions due to the central limit theorem whenever the velocity is generated by several independent density structures. A more effective strategy seems to be to take advantage of the full dynamical fields at t0, trace them back in time, and use the linear fields to discriminate between theories. The Eulerian Zel'dovich approximation can be used to directly recover the initial PDF (IPDF) as follows (Nusser & Dekel 1993). The tensor ð vi / ð xj derived from v(x) is transformed to Lagrangian variables q(x). The corresponding eigenvalues µi ident ð vi / ð xi and lambdai ident ð vi / ð qi are related via the key relation lambdai = µi / (1 - f-1µi). In the Zel'dovich approximation v propto Ddot so the Lagrangian derivatives lambdai are traced back in time by simple scaling propto Ddot-1. The initial densities at q(x) can then be computed using linear theory, deltain propto - (lambda1 + lambda2 + lambda3), and the IPDF is computed by bin counting of deltain values across the Eulerian grid, weighted by the present densities at the grid points.

A key feature of the recovered IPDF is that it is sensitive to the assumed value of Omega when the input data is velocities (Section 8.4), and is Omega-independent if the input is density. Thus it can be used to robustly recover the IPDF from the density field of the 1.2 Jy IRAS survey (Nusser et al. 1994). The IPDF so determined is insensitive to galaxy biasing in the range 0.5 leq b leq 2, at least for the power-law biasing relation assumed. Errors were evaluated using mock IRAS-like catalogs, and the IPDF was found to be consistent with Gaussian, e.g. the initial skewness S and kurtosis K are limited at the 3sigma level to -0.65 < S < 0.36 and -0.82 < K < 0.62 - useful for evaluating specific non-Gaussian models. The first-year COBE measurements are consistent with Gaussian but the noise limits their discriminatory power to strongly non-Gaussian models (Smoot et al. 1994).

GAUSSIANIZATION. A simple method for recovering the initial fluctuations from the galaxy distribution under the assumption that they were Gaussian is based on the assertion that both gravitational evolution and biased galaxy formation tend to preserve the rank order of density in cells. The non-Gaussian distribution of galaxies in cells simply needs to be ``Gaussianized'' in a rank-preserving way (Weinberg 1991). The initial conditions can then be evolved forward using an N-body code and compared with the observed galaxy distribution in z-space until convergence. A self-consistent solution for a redshift survey in the PP region was found if bopt ~ 2, while it was impossible to match the structure on small and large scales simultaneously with GI, Gaussian fluctuations and no biasing. This result derived independently of Omega implies a high Omega once betaopt is determined by another method (Section 8), but note that the limited surveyed region may be an ``unfair'' sample.

LEAST ACTION. The general GI problem with mixed boundary conditions lends itself naturally to an application of Hamilton's action principle (Peebles 1989; 1990; 1993). The comoving orbit xi (t) of each mass point mi is parametrized in a way that satisfies the boundary conditions xi (t0) = xi 0 and lim t->0 a2 dotxi = 0, and the action

S = integ0t0 L dt = integ0t0 dt sumi [(1/2)mi a2 dotxi2 - mi Phig (xi)] (24)

is minimized to determine the free parameters. The orbits can be xi (t) = xi (t0) + Sigman fn (t) Ci, n, where fn (t) each satisfy the boundary conditions, and Ci, n are the parameters. The problem can be solved to any desired accuracy by increasing n, and a proper choice of fn's helps the series to converge rapidly to the desired solution. A generalization of the Zel'dovich approximation of the sort fn (t) = [D (t) - D(t0)]n is particularly efficient (Giavalisco et al. 1993). In a preliminary application of this scheme to a redshift sample (Shaya et al. 1994) the galaxies are assumed to trace mass and be self-gravitating. The complication caused by observing redshifts is solved by an iterative procedure: a tentative guess is made for the xi at t0 based on TF distances to a sub-sample of galaxies and a crude flow model, and the least-action solution provides peculiar velocities, i.e., redshifts, whose deviations from the observed redshifts are used to correct the xi for the next iteration, until convergence. After the original study of the history of the Local Group (Peebles 1989), the method was applied to a sample based on 500 groups within 30 h-1Mpc of the LG from the Nearby Galaxy Catalog. The results indicate that the unknown tidal forces from outside the sampled volume have a significant effect on the recovered fields and should be incorporated in future applications. The solution obtained depends on the assumed Omega, so a comparison with independent TF distances can in principle constrain Omega (Section 8.2).

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