Annu. Rev. Astron. Astrophys. 1992. 30: 499-542 Copyright © 1992 by Annual Reviews. All rights reserved

3.5 Growth of Linear Perturbations

In all the homogeneous and isotropic cosmologies, linear cold matter perturbations / grow at a rate that does not depend on their comoving spatial scale (e.g. Peebles 1980). An explicit expression for the amplitude of a growing perturbation (Heath 1977) is

28.

where a' is the dummy integration variable, and da / d is to be viewed as a known function of a or a', in our case given explicitly by Equation 9. Equation 28 is normalized so that the fiducial case of _M = 1, = 0 gives the familiar scaling (a) = a, with coefficient unity.

Different values of _M, lead to different linear growth factors from early times (a 0) to the present (a = 1, da / d = 1). Denoting the ratio of the current linear amplitude to the fiducial case by ₀(_M, ) we have

29.

(The remarkable approximation formula - good to a few percent in regions of plausible _M, - follows from Lahav et al 1991 and Lightman & Schechter 1990.) Figure 7 shows numerical values for ₀ (_M, ) for the region in the (, _tot) plane previously seen in Figures 1 and 4. One sees that as _M is reduced from unity, both along the line = 0 and along the line _tot = 1, the growth of perturbations is suppressed, but somewhat less suppressed in the _tot = 1 case. The reason is that, for fixed _M, linear growth effectively stopped at a redshift (1 + z) = _M^-1 in the open case (when the universe became curvature dominated), but, more recently, at (1 + z) = _M^-1/3 in the flat case (when the universe became dominated).

Figure 7. Growth factor for linear perturbations, as contours in the (M, tot) plane, normalized to unity for the case M = 1, = 0. There is relatively less suppression of growth as M is decreased along the line tot = 1 than along the line = 0; but for credible values of M the difference is not a large factor. Perturbation growth approaches at the ``loiter line,'' but for credible M it occurs at too small a redshift to explain quasars (see Figure 1).

To the right of the line _tot = 1 in Figure 7, one sees values of ₀ (_M, ) that are greater than 1, in fact approaching infinity at the loitering cosmology line (cf Figure 1 and discussion above). Loitering cosmologies allow the arbitrarily large growth of linear perturbations, since the perturbations continue to grow during the (arbitrarily long) loiter time.

Related to the growth of linear perturbations is the relation between peculiar velocity v and peculiar acceleration g, or (as a special case) the radial infall velocity v_rad around a spherical perturbation of radius . These quantities depend not directly on Equation 28, but on its logarithmic derivative, the exponent in the momentary power law relating to a,

30.

the relation being

31.

where <> is the overdensity averaged over the interior of the sphere of radius (Peebles 1980, Section 14). One can calculate accurately by taking the derivative of Equation 28, using Equation 9, and solving the resulting integral numerically. Lahav et al (1991), however, give an approximation formula valid for all redshifts z,

32.

Figure 8 plots f(z) for our standard models A-E. One sees that, at small redshifts, peculiar velocities depend almost entirely on _M and are quite insensitive to . This is because they are driven by the matter perturbations in primarily the most recent Hubble time. Looking back to redshifts z 1, however, the peculiar velocities do start depending on , allowing in principle for observational tests (but see Lahav et al 1991 for caveats).

Figure 8. Peculiar velocities around fixed-density condensations as a function of redshift. for models B-E, relative to model A. For z 1, peculiar velocities, and other related dynamical effects, are extremely insensitive to the value of .