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We shall now move on to the more realistic case of a multi-component universe consisting of radiation and collisionless dark matter. (For the moment we are ignoring the baryons, which we will study in Sec. 6). It is convenient to use y = a / aeq as independent variable rather than the time coordinate. The background expansion of the universe described by the function a(t) can be equivalently expressed (in terms of the conformal time eta) as

Equation 70 (70)

It is also useful to define a critical wave number kc by:

Equation 71 (71)

which essentially sets the comoving scale corresponding to matter-radiation equality. Note that 2x = kc eta and y approx kc eta in the radiation dominated phase while y = (1/4)(kc eta)2 in the matter dominated phase.

We now manipulate Eqs. (52), (55), (56), (57) governing the growth of perturbations by essentially eliminating the velocity. This leads to the three equations

Equation 72 (72)
Equation 73 (73)
Equation 74 (74)

for the three unknowns Phi, deltam, deltaR. Given suitable initial conditions we can solve these equations to determine the growth of perturbations. The initial conditions need to imposed very early on when the modes are much bigger than the Hubble radius which corresponds to the y << 1, k -> 0 limit. In this limit, the equations become:

Equation 75 (75)

We will take Phi(yi, k) = Phii(k) as given value, to be determined by the processes that generate the initial perturbations. First equation in Eq. (75) shows that we can take deltaR = -2Phii for yi -> 0. Further Eq. (53) shows that adiabaticity is respected at these scales and we can take deltam = (3/4) deltaR = -(3/2) Phii;. The exact equation Eq. (72) determines Phi' if (Phi, deltam, deltaR) are given. Finally we use the last two equations to set delta 'm = 3Phi ', delta 'R = 4Phi '. Thus we take the initial conditions at some y = yi << 1 to be:

Equation 76 (76)

with delta 'm(yi, k) = 3Phi '(yi, k); delta 'R(yi, k) = 4Phi '(yi, k).

Given these initial conditions, it is fairly easy to integrate the equations forward in time and the numerical results are shown in Figs 2, 3, 4, 5. (In the figures keq is taken to be aeqHeq.) To understand the nature of the evolution, it is, however, useful to try out a few analytic approximations to Eqs. (72) – (74) which is what we will do now.

Figure 2a Figure 2b

Figure 2. Left Panel:The evolution of gravitational potential Phi for 3 different modes. The wavenumber is indicated by the label and the epoch at which the mode enters the Hubble radius is indicated by a small arrow. The top most curve is for a mode which stays outside the Hubble radius for most of its evolution and is well described by Eq. (78). The other two modes show the decay of Phi after the mode has entered the Hubble radius in the radiation dominated epoch as described by Eq. (79). Right Panel: Evolution of entropy perturbation (see Eq. (87) for the definition). The entropy perturbation is essentially zero till the mode enters Hubble radius and grows afterwards tracking the dominant energy density perturbation.

4.1  . Evolution for lambda >> dH

Let us begin by considering very large wavelength modes corresponding to the keta -> 0 limit. In this case adiabaticity is respected and we can set deltaR approx (4/3)deltam. Then Eqs. (72), (73) become

Equation 77 (77)

Differentiating the first equation and using the second to eliminate deltam, we get a second order equation for Phi. Fortunately, this equation has an exact solution

Equation 78 (78)

[There is simple way of determining such an exact solution, which we will describe in Sec. 4.4.]. The initial condition on deltaR is chosen such that it goes to -2Phii initially. The solution shows that, as long as the mode is bigger than the Hubble radius, the potential changes very little; it is constant initially as well as in the final matter dominated phase. At late times (y >> 1) we see that Phi approx (9/10) Phii so that Phi decreases only by a factor (9/10) during the entire evolution if k -> 0 is a valid approximation.

4.2. Evolution for lambda << dH in the radiation dominated phase

When the mode enters Hubble radius in the radiation dominated phase, we can no longer ignore the pressure terms. The pressure makes radiation density contrast oscillate and the gravitational potential, driven by this, also oscillates with a decay in the overall amplitude. An approximate procedure to describe this phase is to solve the coupled deltaR - Phi system, ignoring deltam which is sub-dominant and then determine deltam using the form of Phi.

When deltam is ignored, the problem reduces to the one solved earlier in Eqs (64), (65) with w = 1/3 giving nu = 3. Since J3/2 can be expressed in terms of trigonometric functions, the solution given by Eq. (64) with nu = 3, simplifies to

Equation 79 (79)

Note that as y -> 0, we have Phi = Phii, Phi' = 0. This solution shows that once the mode enters the Hubble radius, the potential decays in an oscillatory manner. For ly >> 1, the potential becomes Phi approx -3Phii (ly)-2 cos(ly). In the same limit, we get from Eq. (65) that

Equation 80 (80)

(This is analogous to Eq. (68) for the radiation dominated case.) This oscillation is seen clearly in Fig 3. and Fig. 4 (left panel). The amplitude of oscillations is accurately captured by Eq. (80) for k = 100keq mode but not for k = keq; this is to be expected since the mode is not entering in the radiation dominated phase.

Figure 3

Figure 3. Evolution of deltaR for a mode with k = 100 keq. The mode remains frozen outside the Hubble radius at (k / keq)3/2(-deltaR) approx (k / keq)3/2 2Phi = 2 (in the normalisation used in Fig. 2 ) and oscillates when it enters the Hubble radius. The oscillations are well described by Eq. (80) with an amplitude of 6.

Figure 4a Figure 4b

Figure 4. Evolution of deltaR for two modes k = keq and k = 0.01 keq. The modes remain frozen outside the Hubble radius at (-deltaR) approx 2 and oscillates when it enters the Hubble radius. The mode in the right panel stays outside the Hubble radius for most part of its evolution and hence changes very little.

Figure 5

Figure 5. Evolution of |deltam| for 3 different modes. The modes are labelled by their wave numbers and the epochs at which they enter the Hubble radius are shown by small arrows. All the modes remain frozen when they are outside the Hubble radius and grow linearly in the matter dominated phase once they are inside the Hubble radius. The mode that enters the Hubble radius in the radiation dominated phase grows logarithmically until y = yeq. These features are well approximated by Eqs. (83), (85).

Let us next consider matter perturbations during this phase. They grow, driven by the gravitational potential determined above. When y << 1, Eq. (73) becomes:

Equation 81 (81)

The Phi is essentially determined by radiation and satisfies Eq. (61); using this, we can rewrite Eq. (81) as

Equation 82 (82)

The general solution to the homogeneous part of Eq. (82) (obtained by ignoring the right hand side) is (c1 + c2 lny); hence the general solution to this equation is

Equation 83 (83)

For y << 1 the growing mode varies as lny and dominates over the rest; hence we conclude that, matter, driven by Phi, grows logarithmically during the radiation dominated phase for modes which are inside the Hubble radius.

4.3. Evolution in the matter dominated phase

Finally let us consider the matter dominated phase, in which we can ignore the radiation and concentrate on Eq. (72) and Eq. (73). When y >> 1 these equations become:

Equation 84 (84)

These have a simple solution which we found earlier (see Eq. (69)):

Equation 85 (85)

In this limit, the matter perturbations grow linearly with expansion: deltam propto y propto a. In fact this is the most dominant growth mode in the linear perturbation theory.

4.4. An alternative description of matter-radiation system

Before proceeding further, we will describe an alternative procedure for discussing the perturbations in dark matter and radiation, which has some advantages. In the formalism we used above, we used perturbations in the energy density of radiation (deltaR) and matter (deltam) as the dependent variables. Instead, we now use perturbations in the total energy density, delta and the perturbations in the entropy per particle, sigma as the new dependent variables. In terms of deltaR, deltam, these variables are defined as:

Equation 86 (86)
Equation 87 (87)

Given the equations for deltaR, deltam, one can obtain the corresponding equations for the new variables (delta, sigma) by straight forward algebra. It is convenient to express them as two coupled equations for Phi and sigma. After some direct but a bit tedious algebra, we get:

Equation 88 (88)
Equation 89 (89)

where we have defined

Equation 90 (90)

These equations show that the entropy perturbations and gravitational potential (which is directly related to total energy density perturbations) act as sources for each other. The coupling between the two arises through the right hand sides of Eq. (88) and Eq. (89). We also see that if we set sigma = 0 as an initial condition, this is preserved to O(k4) and - for long wave length modes - the Phi evolves independent of sigma. The solutions to the coupled equations obtained by numerical integration is shown in Fig. (2) right panel. The entropy perturbation sigma approx 0 till the mode enters Hubble radius and grows afterwards tracking either deltaR or deltam whichever is the dominant energy density perturbation. To illustrate the behaviour of Phi, let us consider the adiabatic perturbations at large scales with sigma approx0, k -> 0; then the gravitational potential satisfies the equation:

Equation 91 (91)

which has the two independent solutions:

Equation 92 (92)

both of which diverge as y -> 0. We need to combine these two solutions to find the general solution, keeping in mind that the general solution should be nonsingular and become a constant (say, unity) as y ->0. This fixes the linear combination uniquely:

Equation 93 (93)

Multiplying by Phii we get the solution that was found earlier (see Eq. (78)). Given the form of Phi and sigma appeq 0 we can determine all other quantities. In particular, we get:

Equation 94 (94)

The corresponding velocity field, which we quote for future reference, is given by:

Equation 95 (95)

We conclude this section by mentioning another useful result related to Eq. (88). When sigma approx 0, the equation for Phi can be re-expressed as

Equation 96 (96)

where we have defined:

Equation 97 (97)

(The i factor arises because of converting a gradient to the k space; of course, when everything is done correctly, all physical quantities will be real.) Other equivalent alternative forms for zeta, which are useful are:

Equation 98 (98)

For modes which are bigger than the Hubble radius, Eq. (96) shows that zeta is conserved. When zeta = constant, we can integrate Eq. (98) easily to obtain:

Equation 99 (99)

This is the easiest way to obtain the solution in Eq. (78).

The conservation law for zeta also allows us to understand in a simple manner our previous result that Phi only deceases by a factor (9/10) when the mode remains bigger than Hubble radius as we evolve the equations from y << 1 to y >> 1. Let us compare the values of zeta early in the radiation dominated phase and late in the matter dominated phase. From the first equation in Eq. (98), [using Phi' approx 0 we find that, in the radiation dominated phase, zeta approx (1/2) Phii + Phii = (3/2) Phii; late in the matter dominated phase, zeta approx (2/3)Phif + Phif = (5/3) Phif. Hence the conservation of zeta gives Phif = (3/5)(3/2)Phii = (9/10) Phii which was the result obtained earlier. The expression in Eq. (99) also works at late times in the lambda dominated or curvature dominated universe.

One key feature which should be noted in the study of linear perturbation theory is the different amount of growths for Phi, deltaR and deltam. The Phi either changes very little or decays; the deltaR grows in amplitude only by a factor of few. The physical reason, of course, is that the amplitude is frozen at super-Hubble scales and the pressure prevents the growth at sub-Hubble scales. In contrast, deltam, which is pressureless, grows logarithmically in the radiation dominated era and linearly during the matter dominated era. Since the later phase lasts for a factor of 104 in expansion, we get a fair amount of growth in deltam.

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