Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

which in the static case and with δ exp(iωt) corresponds to the dispersion relation (10.2.7).

As we shall see, for wavelengths λ such that the second term in the parentheses in (10.6.14) is much less than the first, i.e. for λ λJ, where

we have two oscillating solutions, while for wavelengths λ λJ we have two solutions which involve the phenomenon of gravitational instability.

10.7 Solutions for the Flat Dust Case

The solutions of Equation (10.6.13) depend on the background model relative to which the perturbations are defined. The simplest model we can look at is the flat, matter-dominated Einstein–de Sitter universe which we shall use first to derive some key results. In this model,

and the velocity of sound, assuming that the matter comprises monatomic particles of mass m, is given by

This equation, for k → 0, is solved with a trial solution of the form δ tn, with n constant; one gets the exact result that there are two modes, one growing,

One can try to solve Equation (10.7.3) in the case k ≠ 0 using the same trial solution. We obtain

This power-law solution is, in fact, only correct with constant n for k → 0, but the approximate solution (10.7.6) yields important physical insights. When the expression inside the square root in Equation (10.7.6) is positive, that is for

the solutions of (10.7.3) represent the gravitational instability of the system according to which the density fluctuations grow with time. When λ < λJ, there are oscillating solutions.

As we mentioned above, the solutions for λ ≠ 0 are approximate because they are derived under the assumption that the index n of the trial power-law solution is constant in time. In general, however, it will depend on time through the behaviour of the ratio λJ/λ. We shall discuss this fact in more detail later, in § 10.10. The exponent n does not depend on time if the equation of state is of the form p ρ4/3 (i.e. in the plasma epoch with z < zeq). In this case the Equation (10.7.4) is exact, and the term in t−1/6 which comes from (10.7.7) can be obtained using the theory of adiabatic invariants in the manner discussed in Section 10.6.

It is also worth noting the fact that the Jeans length λJ is identical to that introduced in Equation (10.2.6) of the previous chapter. In this respect, no new physics is involved when one moves to the expanding (or contracting) case.

10.8 The Growth Factor

The Equation (10.6.13) admits analytic solutions for λ λJ also in models where Ω0 ≠ 1. Using the parametric variables ϑ and ψ introduced in Section 2.4 and substituting in (10.6.14) yields the equations

for Ω0 < 1. The relationship between proper time t and the parametric variables ψ and ϑ is given in Section 2.4. In both cases one can verify that, for small values of ϑ or ψ, that is for t t0, one obtains Equation (10.5.3), so that all these cases are identical at early times when the curvature terms in the Friedmann equations are negligible. It is interesting to note that in open universes the growing solution δ+ remains practically constant for cosh ψ 5, which corresponds to a redshift z z 25 Ω if Ω 1; we shall also come across this result later in this section.

Now that we have obtained a number of solutions for di erent cosmological models, it is helpful to introduce a general notation to describe the growth of fluctuations. The name growth factor is given to the relative size of the solution δ+ as a function of t: thus, the growth factor in the interval (ti, t0) is Ai0 = δ+(t0)/δ+(ti). For reasons which will become clearer later on, the most interesting value of the growth factor will be that relative to ti = trec. From Equations (10.8.3 a), (10.7.3) and (10.8.4 a) concerning δ+ and (2.4.6), (2.2.6 a) and (2.4.2) we obtain:

for Ω0 < 1, where cosh ψ0 = (2Ω0−1 − 1). The growth factor Ar0 is an increasing function of the density parameter Ω: it varies from a value of 10 for Ω0 10−2, to a value of order 300 for Ω0 10−1, to 1500 for Ω0 = 1, and 3000 for Ω0 4.

To give a more succinct summary of the e ect of cosmology on the growth of perturbations, it is helpful to introduce the quantity f, defined by

This gives the growth factor relative to the Einstein–de Sitter case with the advantage that it does not require a translation between scale factor and time. It is an extremely helpful approximation to take

for models with Λ = 0 (Peebles 1980). If there is a cosmological constant, it actually does not make much di erence to f. A better fit in such cases is

10.9 Solution for Radiation-Dominated Universes

The procedure followed in Section 10.6 for a matter-dominated universe can also be followed, with appropriate modifications, for a universe which is radiation dominated. As we have already noted, in radiation universes the gravitational ‘source’ in the Einstein equations must include pressure terms, so a Newtonian treatment will not su ce. For pure radiation we have that ρ + 3p/c2 = 2ρ. As well as the equations of energy and momentum conservation, we must also take account of the e ect of radiation pressure. One can demonstrate that the relativistic analogues of Equations (10.5.1) can be written in the form

we have not bothered to write down the appropriate law of conservation of entropy, since we shall only be interested from now on in longitudinal adiabatic perturbations. Following the same method as we did in Section 10.6, we arrive at equations which are analogous to Equation (10.6.13):

in which the velocity of sound is now vs = c/√3. Let us concentrate upon finding the solution for a flat universe, which will be a good approximation to our Universe

222 Introduction to Jeans Theory

before matter–radiation equivalence. For this model we have

which, upon substitution in (10.9.2), gives

For k → 0 Equation (10.9.4) is solved by δ tn, with n constant; one again gets a growing mode, but in the form

while the decaying mode is again of the form

Looking for solutions of the power-law form also for k ≠ 0, one finds similar (non-exact) results to those in Section 10.7, but with λJ given by

Going further still, one can extend these analyses to models with a general equation of state of the form p = wρc2, with w constant and vs ≠ 0. In general one now has vs = w1/2c for w > 0, but for a matter-dominated universe (w 0) the value of vs must be defined in an appropriate manner. For example in the case w = 0, which corresponds either to dust or a collisionless fluid, vs2 is of order the mean square velocity of the particles. In any case, the general result for λJ can be written

and the increasing and decreasing modes for scale λ λJ are of the exact form

10.10 The Method of Autosolution

There is another method which can be used to study the evolution of perturbations in the regime with λ λJ: the method of autosolution, pioneered in a paper by Zel’dovich and Barenblatt (1958). This method is based on the property that a spherical perturbation with diameter λ λJ evolves in exactly the same manner as a universe model. This is essentially a consequence of Birkho ’s theorem in general relativity, which is the relativistic analogue of Newton’s famous Spherical Theorem. In the simplest case of a sphere which is homogeneous and isotropic, the evolution is just that of a Friedmann model with parameters di ering slightly from the surrounding (unperturbed) universe. In particular, the density ρp inside the perturbation will be di erent from the density of the universe ρ; the di erence between ρp and ρ evolves with time because the interior and exterior universe evolve according to di erent equations.

The Friedmann equations regarding the evolution of a universe comprised of a fluid with equation of state p = wρc2 can be written in the form

It is clear that Equation (10.10.1) just represents conservation of energy. To obtain the evolution of ρp we consider perturbations of the total energy or, alternatively, of the time of origin of the expansion of the model described by Equation (10.10.1).

Concerning the energy, we have

this quantity is proportional to the perturbation to the energy. We can easily obtain, from (10.10.1) and (10.10.3), that

which can be approximated by

Using the fact that 0ap f(a) da− 0af(a) da (ap−a)f(a), from equation (10.10.5) we find

224 Introduction to Jeans Theory

which gives

The sign of H has the opposite sense to that of δ: an underdense region has an excess of energy compared with the background universe, and vice versa. In the special case of a flat universe the total energy, which is related to B, is exactly zero, and Equation (10.10.7) becomes

which coincides with the result given in Equation (10.9.6). In the case of an open universe, for t t (see Section 2.3), we have instead that A 0 and, from (10.10.7), we obtain

in accordance with the result found for w = 0 in Section 11.4. This result can also be obtained by observing that, for t t , the characteristic time for the Jeans instability to grow, τJ (Gρ)−1/2, is much greater than the characteristic time of the expansion of the universe, τH = a/a˙. In fact, one can easily show, using the formulae derived in Section 2.3, that

Equation (10.10.7) represents the solution that increases with respect to time, δ+. To obtain the decreasing solution δ−, one must perturb the time at which the expansion begins. We have, respectively, that

where the parameter τ represents the time lag (either positive or negative) between the perturbed and unperturbed solutions. From the preceding equations one obtains

The sign of δ is this time the same as the sign of τ. In the special case of the flat Einstein–de Sitter model we have, in accordance with our previous calculations,

with a behaviour as a function of time which is in this case independent of w. In general, however, Equation (10.10.15) represents a decreasing perturbation with a behaviour that depends upon w.

10.11 The Meszaros E ect

As we shall see later on, in a universe composed of non-relativistic matter and relativistic particles (radiation, massless neutrinos, etc.), there can exist a mode of perturbation in which the non-relativistic component is perturbed with respect to a homogeneous distribution while the relativistic component remains unperturbed. If the matter component is entirely baryonic, this type of perturbation is often called isothermal, and a picture of structure formation based on this type of fluctuation was popular in the 1970s. In the 1980s, alternative scenarios were developed in which an important role is played by various forms of nonbaryonic matter (massive neutrinos, axions, photinos, etc.): perturbations which involve this component and not the others (baryons, photons, massless neutrinos) are usually termed isocurvature fluctuations, because these fluctuations do not modify the local spatial curvature. It is consequently important to study the evolution of perturbations of a non-relativistic component with density ρnr in a universe dominated by a fluid of relativistic particles of density ρr. The Universe is dominated by such a fluid at redshifts given by the inequality (5.3.4).

The problem of the evolution of perturbations through zeq has been studied by various authors, the first being Meszaros (1974): one finds that the growingmode perturbation δnr remains ‘frozen’ until zeq even when λ λJ. This e ect of freezing-in of perturbations or ‘stagnation’ or the Meszaros e ect is very important for models in which galaxies and clusters of galaxies are formed by the growth

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of primordial fluctuations in a universe dominated by cold dark matter. We should point out that this e ect does not require perturbations of isocurvature form: it is a generic feature of models with a period of domination by relativistic particles. To form structure one requires at the very least that the perturbations to the nonrelativistic particle distribution, δnr, should be of order unity. The time available for fluctuations to grow from a small amplitude up to this is changed if there is an extended period of stagnation. The problem is exacerbated if Ω 1 because of the freezing out of perturbations when the universe becomes dominated by curvature. We shall describe the detailed consequences of this e ect later; for the moment let us just describe the basic physics.

Let us begin with a qualitative argument. The characteristic time for a gravitational instability process to boost the perturbations in the non-relativistic component δnr is given by the Jeans timescale, τJ (Gρnr)−1/2, while the characteristic time for the expansion of the universe is given by τH (Gρr)−1/2 before zeq; the two timescales are similar after zeq. Consequently, as long as the Universe is dominated by the relativistic component, the fluctuations in the other component remain frozen; the perturbation can only grow after zeq.

We can now study this e ect in an analytical manner, restricting ourselves for simplicity to the case of a flat universe and λ λJ. Introducing the variable

one finds that the equation describing the perturbation in the non-relativistic component δ = δρnr/ρnr becomes

which has, as usual, two solutions, one increasing and one decreasing. We shall forget about the decaying mode from now on: interested readers can calculate the relevant behaviour for the decaying mode themselves. We have

Before zeq (y < 1) the growing mode is practically frozen: the total growth in the interval (0, teq) is only

after zeq the solution rapidly matches the law in a matter-dominated Einstein– de Sitter universe:

10.12 Relativistic Solutions

As we have already explained, the solution of the linear evolution of perturbations, i.e. perturbations with |δ| 1, in Friedmann models within the framework of general relativity was studied for the first time by Lifshitz (1946). In the relativistic approach one proceeds in a quite di erent manner from the Newtonian treatment we have concentrated upon so far. The fundamental object one should treat perturbatively is usually taken to be the metric gij, to which one adds small perturbations hij. One problem that arises immediately is to distinguish between real physical perturbations, and those that arise purely from the choice of reference coordinate system. These latter perturbation modes are called ‘gauge modes’ and one can avoid them by choosing a particular gauge and then finding the gauge modes by hand, or by choosing gauge-invariant combinations of physical variables. In any case, the perturbed metric becomes

For the energy–momentum tensor one adopts a tensor Tij, which is perturbed relative to an ideal fluid, so that ρ, p and Ui are perturbed relative to their values in the background Friedmann model. One then writes down the Einstein equations in terms of the (perturbed) metric gij and the (perturbed) energy–momentum tensor Tij. The procedure is complicated from an analytical point of view, so we just summarise the results here. We find there are three perturbation types which can be classified as tensor, vector and scalar modes.

There are in fact two solutions of tensor type, both corresponding to the propagation of gravitational waves. Gravitational waves are described by an equation of state of radiative type and their amplitude hij varies with time according to

for a matter-dominated Einstein–de Sitter universe, and according to

for the analogous radiation-dominated universe. The solutions (10.12.2) correspond to wavelengths λ ct; for λ ct we have instead two oscillating solutions:

hij t5/8J±3/2(3ckt), hij t3/4J±1/2(2ckt), (10.12.3)

where J are Bessel functions.

While the tensor modes have no Newtonian analogue, the vector modes are similar to phenomena which appear in the Newtonian analysis. They correspond to rotational modes in the velocity field, which have velocity v perpendicular to