### 2. GRAVITATIONAL INSTABILITY

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 g be the position, peculiar velocity and peculiar gravitational potential in comoving distance units, corresponding to ax, av, and a2 g in physical units, with a (t) the universal expansion factor. Let the mass-density fluctuation be ( - bar) / . 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):

dot + · v + · (v ) = 0, (1)

vdot + 2Hv + (v · )v = - g , (2)

2 g = (3/2)H2 , (3)

with H and 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, dotdot + 2H dot = (3/2)H2 . The growing mode of the solution, D (t), is irrotational and can be expressed in terms of f() H-1 Ddot / D 0.6 (see Peebles 1993, eq. 5.120). The linear relation between density and velocity is

= 0 -(Hf)-1 · v. (4)

The use of 0 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 | · v| obtains values 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 0 becomes a severe underestimate at large ||. This explains why Equation (4) is invalid in the non-linear epoch even where = 0 ; the requirements that d3 x = 0 by definition and · v d3 x = 0 by isotropy imply - · v > at || << 1. Fortunately, the small variance of · v given promises that some function of the velocity derivatives may be a good local approximation to .

 Figure 1. Quasi-linear velocity-to-density approximations. approx(v) - true. The mean and standard deviation are from large standard-CDM N-body simulations normalized to 8 = 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) (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)

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 c involves second- and third-order terms (mv2, mv3) as well. The relation (6) is not easily invertible to provide · v or v when is given, but a useful approximation derived from simulations is · v = -f / (1 + 0.18).

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)

c2 = -f-1 · v + (4 / 7)f-2 m v2 ,

m v2 = i j > ii vi ðj vj - ðj vi ði vj). (7)

The factor 4 / 7 replaces 1 in the second-order term of 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 · v given , with mv2 replaced by an analogous expression mg2, involving the gravitational acceleration g.

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

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

+ a3 f-3[( · v)3 - µ3] + . . . (8)

Because the first two terms vanish when integrated over a large volume, the moments µn <( · 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 0.3 and a3 -0.1, tested for values 0.1-1 and smoothing radii 5-12 h-1Mpc at 8 = 1 (8 is the rms of unsmoothed mass- in top-hat spheres of radius 8 h-1Mpc). The structure of Equation (8) makes it robust to uncertain features such as , 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 , 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). c, of scatter ~ 0.1, is an excellent approximation for 1 but it is a slight overestimate at the negative tail. c2 and b do better at the negative tail, but they are severe underestimates in the positive tail. 3( · v) is an excellent robust fit over the whole quasi-linear regime. c3 is constructed from the three terms in the expansion of 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 ).