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3.1. Linear perturbations

For simplicity, in the following discussion we will restrict attention to the case of an Einstein-de Sitter (k = 0) model composed of radiation, considered as a perfect fluid with an equation of state p = 1/3 rho c2, together with a pressureless baryon fluid. These conditions are easily relaxed but represent an adequate approximation to the standard hot big bang models during the radiation era and whilst perturbations on scales corresponding to galaxies and galaxy clusters are larger than the particle horizon.

The unperturbed metric may be written as (6)

Equation 3.1 (3.1)

and the unperturbed energy momentum tensor

Equation 3.2 (3.2)

where p0 is the unperturbed pressure and rho0 is the unperturbed energy density and the four velocity uµ has components

Equation 3.3 (3.3)

The scale factor a(tau) is related to p0 and rho0 by the usual Friedmann equations (Weinberg, 1972, Section 15),

Equation 3.4a (3.4a)



where a dot denotes a derivative with respect to tau. In addition, let

Equation 3.5 (3.5)

Now, consider the perturbed metric,

Equation 3.6 (3.6)

where the perturbations are assumed to be small | hµnu| << a2(tau). Since the hµnu are not physically measurable quantities one may make an infinitesimal coordinate transformation,

Equation 3.7 (3.7)

to find an equally valid solution

Equation 3.8 (3.8)

(Weinberg, 1972, Section 15.10). The ability to add terms to hµnu which correspond merely to infinitesimal (gauge) transformations has led to some problems in the interpretation of the results of the perturbation calculations. In particular since the quantity of most interest, the fractional change in energy density delta = delta rho / rho0, is not a gauge invariant quantity, various results for the growth rates of delta may be found in the literature (Lifshitz, 1946; Harrison, 1967; Silk, 1968; Sakai, 1969; Peebles, 1980a). This need not be a problem, however, since with sufficient care fictitious "gauge mode" solutions may be eliminated (Press and Vishniac, 1980a). An independent and interesting approach to the problem has recently been presented by Bardeen (1980). Following earlier work of Hawking (1966) and Olson (1976), Bardeen discusses a formalism in which the time development of gauge-invariant measures of small perturbations to the homogeneous background may be followed.

We shall discuss the case of a scalar (i.e. density) plane-wave perturbation (Lifshitz and Khalatnikov, 1963), in which

Equation 3.9 (3.9a)




Equation 3.10 (3.10)

and k is the comoving wave vector. The four-velocity may be written,

Equation 3.11 (3.11a)


Equation 3.11b (3.11b)

where (3.11b) results from the normalization condition uµuµ = - 1. The perturbed energy-momentum tensor may be written,

Equation 3.12 (3.12a)




We can now investigate the way in which the perturbation amplitudes in Eqs. (3.8)-(3.11) change under a coordinate transformation. Since the perturbation analysis is only to first order, only the first order effects of the transformation need be considered. The most general coordinate transformation may be written [Eq. (3.7)],

Equation 3.13 (3.13a)


Hence the perturbation amplitudes transform as (Weinberg, 1972, Section 10.9; Bardeen, 1980),

Equation 3.14 (3.14a)







The amplitudes in Eqs. (3.14) may be combined to produce gauge invariant amplitudes. For example, one of the quantities considered by Bardeen,

Equation 3.15 (3.15)

is clearly gauge invariant by Eqs. (3.14b, e, f). The quantity varepsilonm is equal to delta in a gauge in which v = B, i.e. if the matter worldlines are orthogonal to the hypersurface of constant time tau. Even in a gauge-invariant formalism there remains some ambiguity since one can define various gauge-invariant quantities like varepsilonm which may be interpreted as delta by some appropriate choice of hypersurface of simultaneity. In practice, for a sensible choice of hypersurface, the ambiguity disappears once the perturbation is well within the horizon, hence varepsilonm appeq delta for ktau >> 1. The motivation behind Bardeen's discussion of varepsilonm is that the equations governing its time evolution are particularly simple.

If the entropy per baryon is constant (adiabatic fluctuations), then [Eqs. (3.5)]

Equation 3.16 (3.16)

and wpi = cs2 delta. It is also interesting to include "entropy" perturbations, i.e. delta rhoR / rhoR neq 4/3 delta rhoB / rhoB. This may be done by considering the amplitude.

Equation 3.17 (3.17)

which is gauge invariant by Eqs. (3.14f, g). Bardeen derives the equation governing the time evolution of varepsilonm which, in the radiation era when w approx cs2 approx 1/3 takes the simple form,

Equation 3.18 (3.18)

Ignoring the source term eta(tau), it can be seen that the solutions to Eq. (3.18) will show oscillatory behavior on scales smaller than the horizon (ktau >> 1) varepsilonm propto exp(iktau / 31/2) and power law behaviour, varepsilonm = fg tau2 + fd tau-1 on scales greater than the horizon. These results agree with those obtained by Sakai (1969) using the comoving gauge (ui = h0i = 0) but differ from the growth rates obtained using the synchronous gauge (h00 = h0i = 0, Weinberg, 1972; Peebles, 1980a; Press and Vishniac, 1980a). These authors find solutions, delta = fg1 tau2 + fg2 tau + fd tau-2 for perturbations with ktau << 1. The decaying mode fd tau-2 has no physical significance since it may be eliminated by adjusting the initial hyper-surface of constant time (see Press and Vishniac). On scales smaller than the horizon, the synchronous gauge gives delta propto exp[iktau / 31/2] which agrees with the behaviour of varepsilonm for ktau >> 1.

Now consider the term eta(tau), this acts as a source term in Eq. (3.18) and one might speculate that a non-zero eta(tau) might generate fluctuations in varepsilonm. Consider for example, the case in which some of the matter is suddenly converted into a pressureless form, perhaps black holes. This will generate a pressure inhomogeneity (Meszaros, 1975: Carr, 1977),

Equation 3.19 (3.19)

but we can see from Eq. (3.18) that the induced varepsilonm will be of order ~ k2 tau2 eta(tau) and is due to pressure gradients transmitted at the sound speed. Thus if the perturbation comes within the horizon during the radiation era, varepsilonm approx pi(tau). This result is discussed in more detail by Press and Vishniac (1980a) who show that it is also true for perturbations which enter the horizon during the matter-dominated era, and by Bardeen (1980) who also considers the possible role of anisotropic stress perturbations. Their results lead to an important conclusion concerning the possibility of spontaneously forming density perturbations at early times which we discuss below.

6 Here, Greek indices run from 0 to 3 whilst Latin indices run from 1 to 3 and covariant derivatives will be denoted by a semi-colon. Units are chosen so that c = 1. Back.

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