Annu. Rev. Astron. Astrophys. 1994. 32:
371-418 Copyright © 1994 by . All rights reserved |
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 a^{2 }_{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):
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^{-1}Mpc = 100 km s^{-1}).
In the linear approximation, the GI equations can be combined into a time evolution equation, dotdot + 2H dot = (3/2)H^{2} . 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
The use of _{0} is limited to the small dynamical range between a few tens of megaparsecs and the ~ 100 h^{-1}Mpc extent of the current samples. In contrast, the sampling of galaxies enables reliable dynamical analysis with smoothing as small as ~ 10 h^{-1}Mpc, 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 d^{3} x = 0 by definition and · v d^{3} 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^{-1}Mpc (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,
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)
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 (m_{v2}, m_{v3}) 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)
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 m_{v2} replaced by an analogous expression m_{g2}, 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)
Because the first two terms vanish when integrated over
a large volume, the moments
Figure 1 demonstrates the accuracy of the explicit quasi-linear approximations using CDM N-body simulations and 12 h^{-1}Mpc 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 ).