Annu. Rev. Astron. Astrophys. 1994. 32: 371-418
Copyright © 1994 by . All rights reserved

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This section provides a brief account of the standard theory of GI, and of the linear and quasi-linear approximations which serve the analysis of motions. Let x, v, and phig be the position, peculiar velocity and peculiar gravitational potential in comoving distance units, corresponding to ax, av, and a2 phig in physical units, with a (t) the universal expansion factor. Let the mass-density fluctuation be delta ident (rho - rhobar) / rho. The equations governing the evolution of fluctuations of a pressureless gravitating fluid in a standard cosmological background during the matter era are the Continuity equation, the Euler equation of motion, and the Poisson field equation (e.g. Peebles 1980; 1993):

deltadot + del · v + del · (v delta) = 0, (1)

vdot + 2Hv + (v · del)v = -del phig , (2)

del2 phig = (3/2)H2 Omega delta, (3)

with H and Omega varying in time. The dynamics do not depend on the value of the Hubble constant H; it is set to unity by measuring distances in km s-1 (1h-1Mpc = 100 km s-1).

In the linear approximation, the GI equations can be combined into a time evolution equation, deltadotdot + 2H deltadot = (3/2)H2Omega delta. The growing mode of the solution, D (t), is irrotational and can be expressed in terms of f(Omega) ident H-1 Ddot / D approx Omega0.6 (see Peebles 1993, eq. 5.120). The linear relation between density and velocity is

delta = delta0 ident -(Hf)-1 del · v. (4)

The use of delta0 is limited to the small dynamical range between a few tens of megaparsecs and the ~ 100 h-1Mpc extent of the current samples. In contrast, the sampling of galaxies enables reliable dynamical analysis with smoothing as small as ~ 10 h-1Mpc, where |del · v| obtains values geq 1 so quasi-linear effects play a role. Unlike the strong non-linear effects in virialized systems which erase any memory of the initial conditions, mild non-linear effects carry crucial information about the formation of LSS, and should therefore be treated carefully. Figure 1 shows that delta0 becomes a severe underestimate at large |delta|. This explains why Equation (4) is invalid in the non-linear epoch even where delta = 0 ; the requirements that integ delta d3 x = 0 by definition and integ del · v d3 x = 0 by isotropy imply -del · v > delta at |delta| << 1. Fortunately, the small variance of del · v given delta promises that some function of the velocity derivatives may be a good local approximation to delta.

Figure 1. Quasi-linear velocity-to-density approximations. Delta ident deltaapprox(v) - deltatrue. The mean and standard deviation are from large standard-CDM N-body simulations normalized to sigma8 = 1, Gaussian-smoothed with radius 12 h-1Mpc (see Mancinelli et al. 1994). Note the factor of 5 difference in scale between the axes.

A basis for useful quasi-linear relations is provided by the Zel'dovich (1970) approximation. The displacements of particles from their initial, Lagrangian positions q to their Eulerian positions x at time t are assumed to have a universal time dependence,

x(q, t) - q = D (t) psi (q) = f-1 v(q, t). (5)

For the purpose of approximating GI, the Lagrangian Zel'dovich approximation can be interpreted in Eulerian space, q(x) = x - f-1 v(x), provided that the flow is laminar (i.e., that multi-streams are appropriately smoothed over). The solution of the continuity equation then yields (Nusser et al. 1991)

delta c(x) = || I - f-1 ð v / ð x|| - 1, (6)

where the bars denote the Jacobian determinant, and I is the unit matrix. The Zel'dovich displacement is first order in f-1 and v, so delta c involves second- and third-order terms (mv2, mv3) as well. The relation (6) is not easily invertible to provide del · v or v when delta is given, but a useful approximation derived from simulations is del · v = -f delta / (1 + 0.18delta).

A modified approximation, which is derived by adding a second-order term to the Zel'dovich displacement (Moutarde et al. 1991) and truncating all the expressions at second order while solving the continuity equation, is (Gramman 1993a)

delta c2 = -f-1 del · v + (4 / 7)f-2 m v2 ,

m v2 = sumi sumj > ii vi ðj vj - ðj vi ði vj). (7)

The factor 4 / 7 replaces 1 in the second-order term of delta c. While terms are kept to second-order, it is still not an exact solution to the second-order equations of GI. This relation can be inverted in second-order to provide del · v given delta, with mv2 replaced by an analogous expression mg2, involving the gravitational acceleration g.

Since the variance of delta given del · v is small, one expects that a non-linear function of del · v which properly corrects for the systematic deviation can be a good quasi-linear approximation to delta (and vice versa). Assuming Gaussian initial fluctuations, Bernardeau (1992) found a solution in the limit of vanishing variance: deltab = [1 - (2 / 3)f-1del · v]3/2 - 1, which is easily invertible. A polynomial expansion with non-vanishing variance should have the form (Zehavi & Dekel, in preparation)

deltan (del · v) = -f-1 del · v + a2 f-2[(del · v)2 - µ2]

+ a3 f-3[(del · v)3 - µ3] + . . . (8)

Because the first two terms vanish when integrated over a large volume, the moments µn ident <(del · v)n> must be subtracted off to make the nth-order term vanish as well. The coefficients can be crudely approximated analytically (e.g. Bernardeau 1992) or, using CDM simulations and Gaussian smoothing, the best coefficients are a2 approx 0.3 and a3 approx -0.1, tested for Omega values 0.1-1 and smoothing radii 5-12 h-1Mpc at sigma8 = 1 (sigma8 is the rms of unsmoothed mass-delta in top-hat spheres of radius 8 h-1Mpc). The structure of Equation (8) makes it robust to uncertain features such as Omega, the shape of the fluctuation power spectrum, and the degree of non-linearity as determined by the fluctuation amplitude and the smoothing. Such robustness is crucial when using a quasilinear approximation for determining Omega, for example (Section 8).

Figure 1 demonstrates the accuracy of the explicit quasi-linear approximations using CDM N-body simulations and 12 h-1Mpc smoothing (Mancinelli et al. 1994). delta c, of scatter ~ 0.1, is an excellent approximation for delta leq 1 but it is a slight overestimate at the negative tail. delta c2 and delta b do better at the negative tail, but they are severe underestimates in the positive tail. delta3(del · v) is an excellent robust fit over the whole quasi-linear regime. delta c3 is constructed from the three terms in the expansion of delta c in powers of f-1 but with the numerical coefficients adjusted to achieve best fit in the simulation (-1.05, 0.9, 1.5 replacing unity, independent of Omega).

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