Next Contents Previous


Having discussed the evolution of the background universe, we now turn to the study of structure formation. Before discussing the details, let us briefly summarise the broad picture and give references to some of the topics that we will not discuss. The key idea is that if there existed small fluctuations in the energy density in the early universe, then gravitational instability can amplify them in a well-understood manner leading to structures like galaxies etc. today. The most popular model for generating these fluctuations is based on the idea that if the very early universe went through an inflationary phase [9], then the quantum fluctuations of the field driving the inflation can lead to energy density fluctuations [10 , 11]. It is possible to construct models of inflation such that these fluctuations are described by a Gaussian random field and are characterized by a power spectrum of the form P(k) = A kn with n appeq 1 (see Sec. 7). The models cannot predict the value of the amplitude A in an unambiguous manner but it can be determined from CMBR observations. The CMBR observations are consistent with the inflationary model for the generation of perturbations and gives A appeq (28.3 h-1 Mpc)4 and n = 0.97 ± 0.023 (The first results were from COBE [12] and WMAP has reconfirmed them with far greater accuracy). When the perturbation is small, one can use well defined linear perturbation theory to study its growth. But when delta approx (delta rho / rho) is comparable to unity the perturbation theory breaks down. Since there is more power at small scales, smaller scales go non-linear first and structure forms hierarchically. The non linear evolution of the dark matter halos (which is an example of statistical mechanics of self gravitating systems; see e.g. [13]) can be understood by simulations as well as theoretical models based on approximate ansatz [14] and nonlinear scaling relations [15]. The baryons in the halo will cool and undergo collapse in a fairly complex manner because of gas dynamical processes. It seems unlikely that the baryonic collapse and galaxy formation can be understood by analytic approximations; one needs to do high resolution computer simulations to make any progress [16]. All these results are broadly consistent with observations.

As long as these fluctuations are small, one can study their evolution by linear perturbation theory, which is what we will start with [17]. The basic idea of linear perturbation theory is well defined and simple. We perturb the background FRW metric by gikFRW -> gikFRW + hik and also perturb the source energy momentum tensor by TikFRW -> TikFRW + delta Tik. Linearising the Einstein's equations, one can relate the perturbed quantities by a relation of the form L(gikFRW) hik = delta Tik where L is second order linear differential operator depending on the back ground metric gikFRW . Since the background is maximally symmetric, one can separate out time and space; for e.g, if k = 0, simple Fourier modes can be used for this purpose and we can write down the equation for any given mode, labelled by a wave vector k as:

Equation 42 (42)

To every mode we can associate a wavelength normalized to today's value: lambda(t) = (2 pi / k)(1 + z)-1 and a corresponding mass scale which is invariant under expansion:

Equation 43 (43)

The behaviour of the mode depends on the relative value of lambda(t) as compared to the Hubble radius dH(t) ident (dot{a} / a)-1. Since the Hubble radius: dH(t) propto t while the wavelength of the mode: lambda(t) propto a(t) propto (t1/2, t2/3) in the radiation dominated and matter dominated phases it follows that lambda(t) > dH(t) at sufficiently early times. When lambda(t) = dH(t), we say that the mode is entering the Hubble radius. Since the Hubble radius at z = zeq is

Equation 44 (44)

it follows that modes with lambda0 > lambdaeq enter Hubble radius in MD phase while the more relevant modes with lambda < lambdaeq enter in the RD phase. Thus, for a given mode we can identify three distinct phases: First, very early on, when lambda > dH, z > zeq the dynamics is described by general relativity. In this stage, the universe is radiation dominated, gravity is the only relevant force and the perturbations are linear. Next, when lambda < dH and z > zeq one can describe the dynamics by Newtonian considerations. The perturbations are still linear and the universe is radiation dominated. Finally, when lambda < dH, z < zeq we have a matter dominated universe in which we can use the Newtonian formalism; but at this stage - when most astrophysical structures form - we need to grapple with nonlinear astrophysical processes.

Let us now consider the metric perturbation in greater detail. When the metric is perturbed to the form: gab -> gab + hab the perturbation can be split as hab = (h00, h0 alpha ident walpha, halphabeta). We also know that any 3-vector w(x) can be split as w = wperp + w|| in which w|| = nabla Phi|| is curl-free (and carries one degree of freedom) while wperp is divergence-free (and has 2 degrees of freedom). This result is obvious in k-space since we can write any vector w(k) as a sum of two terms, one along k and one transverse to k:

Equation 45 (45)

Fourier transforming back, we can split w into a curl-free and divergence-free parts. Similar decomposition works for halphabeta by essentially repeating the above analysis on each index. We can write:

Equation 46 (46)

The ualphaperp is divergence free and halphaperp betaperp is traceless and divergence free. Thus the most general perturbation hab (ten degrees of freedom) can be built out of

Equation 47 (47)

We now use the freedom available in the choice of four coordinate transformations to set four conditions: Phi|| = Phi1 = 0 and ualphaperp = 0 thereby leaving six degrees of freedom in (h00 ident 2Phi, psi, wperp, halphaperp betaperp) as nonzero. Then the perturbed line element takes the form:

Equation 48 (48)

To make further simplification we need to use two facts from Einstein's equations. It turns out that the Einstein's equations for wperp and halphaperp betaperp decouple from those for (Phi, psi). Further, in the absence of anisotropic stress, one of the equations give psi = Phi. If we use these two facts, we can simplify the structure of perturbed metric drastically. As far as the growth of matter perturbations are concerned, we can ignore walphaperp and halphaperp betaperp and work with a simple metric:

Equation 49 (49)

with just one perturbed scalar degree of freedom in Phi. This is what we will study.

Having decided on the gauge, let us consider the evolution equations for the perturbations. While one can directly work with the Einstein's equations, it turns out to be convenient to use the equations of motion for matter variables, since we are eventually interested in the matter perturbations. In what follows, we will use the over-dot to denote (d / deta) so that the standard Hubble parameter is H = (1 / a)(da / dt) = dot{a} / a2. With this notation, the continuity equation becomes:

Equation 50 (50)

Since the momentum flux in the relativistic case is (rho + p) valpha, all the terms in the above equation are intuitively obvious, except probably the dot{Phi} term. To see the physical origin of this term, note that the perturbation in Eq. (49) changes the factor in front of the spatial metric from a2 to a2(1 - 2Phi) so that lna -> lna - Phi; hence the effective Hubble parameter from (dot{a} / a) to (dot{a} / a) - dot{Phi} which explains the extra dot{Phi} term. This is, of course, the exact equation for matter variables in the perturbed metric given by Eq. (49); but we only need terms which are of linear order. Writing the curl-free velocity part as valpha = nablaalpha v, the linearised equations, for dark matter (with p=0) and radiation (with p = (1/3) rho) perturbations are given by:

Equation 51 (51)

where nm and nR are the number densities of dark matter particles and radiation. The same equations in Fourier space [using the same symbols for, say, delta(t,x) or delta(t,k)] are simpler to handle:

Equation 52 (52)

Note that these equations imply

Equation 53 (53)

For long wavelength perturbations (in the limit of k -> 0), this will lead to the conservation of perturbation delta(s / nm) in the entropy per particle.

Let us next consider the Euler equation which has the general form:

Equation 54 (54)

Once again each of the terms is simple to interpret. The (rho + p) arises because the pressure also contributes to inertia in a relativistic theory and the factor 4 in the last term on the right hand side arises because the term valpha partialeta(rho + p) on the left hand side needs to be compensated. Taking the linearised limit of this equation, for dark matter and radiation, we get:

Equation 55 (55)

Thus we now have four equations in Eqs. (52), (55) for the five variables (deltam, deltaR, vm, vR, Phi). All we need to do is to pick one more from Einstein's equations to complete the set. The Einstein's equations for our perturbed metric are:

Equation 56 Equation 56 Equation 56 (56)
Equation 57 Equation 57 Equation 57 (57)
Equation 58 Equation 58 Equation 58 (58)

where A denotes different components like dark matter, radiation etc. Using Eq. (57) in Eq. (56) we can get a modified Poisson equation which is purely algebraic:

Equation 59 (59)

which once again emphasizes the fact that in the relativistic theory, both pressure and density act as source of gravity.

To get a feel for the solutions let us consider a flat universe dominated by a single component of matter with the equation of state p = w rho . (A purely radiation dominated universe, for example, will have w = 1/3.) In this case the Friedmann background equation gives rho propto a-3 (1 + w) and

Equation 60 (60)

The equation for the potential Phi can be reduced to the form:

Equation 61 (61)

The second term is the damping due to the expansion while last term is the pressure support that will lead to oscillations. Clearly, the factor k eta determines which of these two terms dominates. When the pressure term dominates (k eta >> 1), we expect oscillatory behaviour while when the background expansion dominates (k eta << 1), we expect the growth to be suppressed. This is precisely what happens. The exact solution is given in terms of the Bessel functions

Equation 62 (62)

From the theory of Bessel functions, we know that:

Equation 63 (63)

This shows that if we want a finite value for Phi as eta -> 0, we can set C2 = 0. This gives the gravitational potential to be

Equation 64 (64)

The corresponding density perturbation will be:

Equation 65 (65)

To understand the nature of the solution, note that dH = (dot{a} / a)-1 propto eta and kdH appeq dH / lambda propto keta. So the argument of the Bessel function is just the ratio (dH / lambda). From the theory of Bessel functions, we know that for small values of the argument Jnu(x) propto xnu is a power law while for large values of the argument it oscillates with a decaying amplitude:

Equation 66 (66)

Hence, for modes which are still outside the Hubble radius (k << eta-1), we have a constant amplitude for the potential and density contrast:

Equation 67 (67)

That is, the perturbation is frozen (except for a decaying mode) at a constant value. On the other hand, for modes which are inside the Hubble radius (k >> eta-1), the perturbation is rapidly oscillatory (if w neq 0). That is the pressure is effective at small scales and leads to acoustic oscillations in the medium.

A special case of the above is the flat, matter-dominated universe with w=0. In this case, we need to take the w -> 0 limit and the general solution is indeed a constant Phi = Phii(k) (plus a decaying mode Phidecay propto eta-5 which diverges as eta -> 0). The corresponding density perturbations is:

Equation 68 (68)

which shows that density perturbation is "frozen" at large scales but grows at small scales:

Equation 69 (69)

We will use these results later on.

Next Contents Previous