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

228 Introduction to Jeans Theory

which, in a matter-dominated universe with p ρc2 a−3, becomes

corresponding to (10.6.8), while for a radiation-dominated universe we have

The Equation (10.12.4) can, in a certain sense, be interpreted as a kind of conservation law for angular momentum L, in which one replaces the matter density by (ρ + p/c2). Equation (10.12.4) can then be written in the form

which is known as Loytsianski’s theorem, an extension of Equation (10.6.10). The final perturbation type, the scalar mode, actually represents the longitudi-

nal compressional density wave we have been discussing in most of this chapter. One finds in the relativistic approach the same results as we have introduced in a Newtonian approximation.

In modern cosmological theories involving inflation the relativistic treatment is extremely important; while we can handle the growth of fluctuations inside the horizon Rc adequately using the Newtonian treatment we have described, fluctuations outside the horizon must be handled using general relativity. In particular, in inflationary theories one must consider the super-horizon evolution of scalar fluctuations, i.e. when λ > Rc, in a model where the equation of state is of the form p = wρc2, with w < −13 . We mention this problem again in Section 13.6.

Bibliographic Notes on Chapter 10

The pioneering works by Silk (1967, 1968), as well as Doroshkevich et al. (1967), Peebles and Yu (1970), Weinberg (1971), Chibisov (1972) and Field (1971) are all still worth reading. Weinberg (1972) summarises much of this historical work; see also Zel’dovich (1965). For detailed perturbation theory and alternative formulations of the material we have covered in this chapter, see Efstathiou and Silk (1983), Kodama and Sasaki (1984), Efstathiou (1990) and Peacock (1999).

Problems

1.Calculate the Jeans length for air at room temperature.

2.How is the expression for the Jeans length modified in the presence of a magnetic field?

3.Derive Equations (10.6.6 a) and (10.6.6 b).

4.Show that the solutions to (10.7.3) for finite λ > λJ have the form given by equation (10.7.6). Thus obtain the correct form in the limit λ → ∞, i.e. δ+ t2/3, δ− t−1.

5.Derive Equation (10.11.3) and obtain the growing mode solution (10.11.4).

11

Gravitational

Instability of

Baryonic Matter

11.1 Introduction

In this chapter we shall apply the principle of the Jeans instability to models of the Universe in which the dominant matter component is baryonic. As we shall see, the adoption of a realistic physical fluid brings in many more complications than we found in our previous analyses of gravitational instability in purely dust or radiation universes. The interaction of matter with radiation during the plasma epoch is one such complication which we have not addressed so far. Although the baryon-dominated models are in this sense more realistic than the simple ones we have used in our illustration of the basic physics, we should make it clear at the outset that these models are not successful at explaining the origin of the structure observed in our Universe. In the next chapter we shall explain why this is so and why models including non-baryonic weakly interacting dark matter may be more successful than the baryon-dominated ones. Nevertheless, we feel it is important to study the baryonic situation in some detail. Our primary reason for this is pedagogical. Although it is believed that there is non-baryonic matter, there certainly are baryons in our Universe. Whatever the dominant form of the matter, we must in any case understand the behaviour of baryons in the presence of radiation during the cosmological expansion. The simplest way to understand this behaviour is to study a model which includes only these two ingredients. Once we have understood the physics here, we can go on to study

230 Gravitational Instability of Baryonic Matter

the e ect of other components. The baryon-dominated models also provide an interesting insight into the history of the study of large-scale structure, and their analysis is an interesting part of the development of the subject in the late 1960s and in the 1970s. We begin with some comments on the form of perturbations in baryonic models.

11.2 Adiabatic and Isothermal Perturbations

Before recombination, the Universe was composed of a plasma of ionised matter and radiation, interacting via Compton scattering with characteristic times given by τeγ and τγe, described in Section 9.2. For simplicity we neglect the presence of helium nuclei in this plasma, and take it to be composed entirely of protons and electrons. We shall also neglect the role of neutrinos in most of this discussion.

As we have seen in Chapter 10, there exist a number of possible perturbation modes in a self-gravitating fluid. There are vortical perturbations (transverse waves) which do not interest us here. There are also perturbations of adiabatic or entropic type, the first time dependent, the second independent of time in the static case studied in Chapter 10. The distinction between these two latter types of perturbation remains when one moves to the cosmological case of an expanding background model.

The entropy per unit mass of a fluid composed of matter and radiation in a volume V has a very high value because of the enormous value of the entropy per baryon σr. In other words, the entropy is carried almost entirely by the radiation:

A perturbation which leaves S invariant – an adiabatic perturbation – is made up of perturbations in both the matter density ρm and the radiation density ρr (or, equivalently, T, the radiation temperature) such that

As we have seen in Section 7.4, the value of σrad may be explained by microscopic physics involving a GUT or electroweak phase transition. If such a microphysical explanation is correct, one might expect small inhomogeneities to have the same value of σr and therefore be of adiabatic type.

A perturbation of entropic type or an isothermal perturbation is such that a nonzero perturbation in the matter component δm ≠ 0 is not accompanied by any fluctuation in the radiation component. In other words there is no inhomogeneity in the radiation temperature, hence the word isothermal. This type of fluctuation

is closely related, but not identical, to the isocurvature fluctuations discussed in the previous chapter and also in the next one. The physical reason why δT 0 rests on the fact that such fluctuations are more or less independent of time; the high thermal conductivity of the cosmological medium allows the temperature to be levelled out by heat conduction. A perturbation with δρm ≠ 0 is held frozen and therefore time independent by the strong frictional ‘drag’ forces between the matter and radiation fluid. An exact treatment of this problem confirms, at least to a first approximation, this division into two main types of perturbation.

After recombination, and the consequent decoupling of matter and radiation, the perturbations δρm in the total matter density evolve in the same way regardless of whether they were originally of adiabatic or isothermal type. Because there is essentially no interaction between the matter and radiation, and the radiation component is dynamically negligible compared with the matter component, the Universe behaves as a single-fluid dust model.

Before recombination a generic perturbation can be decomposed into a superposition of adiabatic and isothermal modes which evolve independently; the two modes can be thought of as similar to the normal modes of a dynamical system. To understand what is going on it is therefore useful, as a first approximation, to study the behaviour of each mode separately.

11.3 Evolution of the Sound Speed and Jeans Mass

As we have already explained, the distinction between adiabatic and isothermal perturbations only has meaning before recombination. In this period we shall denote the relevant sound speeds for the adiabatic and isothermal modes by vs(a) and vs(i), respectively.

The adiabatic sound speed, vs(a), is that of a plasma with density ρ = ρm + ρr and pressure p = pr + pm pr 13 ρrc2. We assume the neutrinos are massless. Recalling Equation (11.2.3), we therefore have

This equation gives vs(a) c/√3 for t teq, while vs(a) 0.76c/√3 for t = teq and during the interval trec > t teq, which exists only if Ωbh2 4 × 10−2, we have

between these two regimes will be much smoother than this.

232 Gravitational Instability of Baryonic Matter

The isothermal sound speed vs(i) is that appropriate for a gas of monatomic particles of mass mp (the proton mass) and temperature Tm Tr = T0r(1 +z), i.e.

where we have assumed that Trec = T(zrec) 4000 K. The velocity of sound associated with matter perturbations after zrec is given by v(i) and one finds that Tm Tr in this period only for z 300; see Section 9.4. After this, until the moment of reheating, Tm (1+z)2, so that Equation (11.3.4) should be modified. However, as far as the origin of galaxies and clusters is concerned, the value of vs(i) for z zrec is not important so we shall not discuss it further here.

We have already introduced the Jeans length, λJ. An alternative way of specifying the physical scale appropriate for gravitational instability is to deal with a mass scale. For this reason, we shall define the Jeans mass to be the mass contained in a sphere of radius 12 λJ

in this expression we have assumed that, for any value of the equation-of-state parameter w, the relation

is a good approximation. More accurate expressions can be found in Section 10.9, but we shall not use them in this order-of-magnitude analysis. It is useful to note the obvious relation between mass and length scales M ρλ3 so that, for example, 1 Mpc corresponds to 1011(Ω0h2)−1M .

Before recombination we must distinguish between adiabatic and isothermal perturbations. We begin with the Jeans mass associated with adiabatic perturbations, MJ(a), for which one must insert the quantity vs(a) in place of vs in the Equation (11.4.2). One should also use ρ = ρm + ρr because the total density is included in the terms describing the self-gravity of the perturbation. For simplicity we can adopt the approximate relations that ρ ρr for z > zeq and ρ ρm for z < zeq. Together with the other approximations we have introduced above for vs(a) we find that, for z zeq,

while in the interval zeq > z > zrec, if it exists, we have

This is an approximate relation. In reality, if zeq zrec, because ρr is small at zrec, the value of the Jeans mass at recombination, MJ(a)(zrec), will be about a factor three higher than MJ(a)(zeq).

Now turning to the isothermal perturbations, we must use the expression given in Equation (11.3.4) for vs(i) in place of vs. We then find that, in the interval zeq > z > zrec,

It is interesting to note that both MJ(a) and MJ(i) remain roughly constant during the interval (if it exists) between equivalence and recombination. After recombination, since we are only interested in the matter perturbations, the Jeans mass MJ can be taken to coincide with MJ(i) while Tm Tr, and then thereafter the behaviour is roughly proportional to (1 + z)3/2.

11.4 Evolution of the Horizon Mass

An important concept which we have not yet come across in the study of gravitational instability is that of the cosmological horizon. Essentially this defines the scale over which di erent parts of a perturbation can be in causal contact with each other at a particular epoch. We shall not worry too much here about the technical issue of whether we should use the particle horizon, RH, or the radius of the speed of light sphere, Rc, to characterise the horizon. In the case we are considering here, these di er only by a factor of order unity anyway, so we shall use the radius of the particle horizon, RH, to define the horizon mass by analogy with the definition of the Jeans mass:

which represents the total mass inside the particle horizon which of course includes the e ective mass contributed by the radiation. It is often more interesting to consider only the baryonic part of this mass, since that is the part that will dominate any structures that form after zrec. Thus we have

Before equivalence, the Universe is well described by an Einstein–de Sitter model of pure radiation for which, using results from Chapters 2 and 5 and with the assumption that ρ ρr,

evolution of perturbations. We shall begin by considering adiabatic fluctuations in some detail.

The most important physical phenomenon we have to deal with is the interaction between matter and radiation during the plasma epoch and the consequent dissipation due to viscosity and thermal conduction. We shall study the basic physics in this section and the more detailed ramifications in Section 12.7. As we shall see, dissipative processes act significantly on sound waves with a wavelength λ, or an e ective mass scale M = 16 πρmλ3, less than a certain characteristic scale λD, called the dissipation scale whose corresponding mass scale, MD, is called the dissipation mass. During the period in which we are interested (the period before recombination), it turns out that MD MJ for both adiabatic and isothermal perturbations; however, the dissipation mass for isothermal perturbations has no practical significance for cosmology.

The e ect of these dissipative processes upon an adiabatic perturbation is to decrease its amplitude. From a kinetic point of view this is because of the phenomenon of di usion, which slowly moves particles into the region outside the perturbation. One can assume for all practical purposes that, after a time t, a perturbation of wavelength λ < λD(t), where λD(t) is the mean di usion length for particles in a time t, is totally dissipated. Given that the particles travel in an arbitrary direction, the e ect is a complete randomisation of the original fluctuation so that it becomes smeared out and dissipated. The distance λD is obviously

¯

connected with the mean free path l of the particles.

¯

On scales λ < l the fluctuation is dissipated in a time of order the wave period and over a distance of order the wavelength λ. In this case it does not make sense

¯

to talk about di usion, and the role of λD is taken by l. We therefore have free streaming of particles, which is important in the models we discuss in the next chapter, which have perturbations in a fluid of collisionless particles. On scales

¯

λl, it is more illuminating to employ a macroscopic model, where dissipation is attributed to the presence of viscosity η and thermal conductivity Dt. Evidently, however, there is a strict connection between the coe cients of viscosity and

thermal conductivity on the one hand, and the coe cient of di usion D and its

¯

related length scale λD on the other. On scales λ l the model for dissipation we must use cannot be a fluid model, but must be based on kinetic theory.

Let us elaborate these concepts in more mathematical detail. The phenomenon

where Jm is the matter flux caused by the density gradient ρm and D is called the coe cient of di usion. Together with the continuity equation, Equation (11.5.1) furnishes Fick’s second law

There is a formal analogy of this relation with the equation of heat conduction

236 Gravitational Instability of Baryonic Matter

(Dt = λt/ρct is the coe cient of thermal di usion; ct is the specific heat; λt is the thermal conductivity), which is obtained easily from the Fourier postulate about conduction, similar to Equation (11.5.1), and from the calorimetric equation.

It is obvious from Equations (11.5.2) and (11.5.3) that the coe cients D and Dt have the same dimensions as each other, and also the same as those of the kinematic viscosity ν = η/ρ which appears in the Navier–Stokes equation

according to this formal analogy, one is invited to interpret ν as a sort of coe - cient of velocity di usion. We have that

Adopting a kinetic treatment to confirm these relations, one finds that

where v¯ is the mean particle velocity and τ is the mean time between two consecutive particle collisions.

¯ ¯

From a dimensional point of view, the mean distance d l a ected after a time t τ by the three ‘di usion’ processes described above are, respectively,

This relationship is easy to demonstrate by assuming that all these di usion processes can be attributed to the di usion of particles by a simple random walk.

Following on from (11.5.8), the dissipation scale (or the di usion scale) of an acoustic wave at time t is therefore

We define the dissipation time of a perturbation of wavelength λ by the quantity

i.e. the time when λD[τD(λ)] = λ. In particular, the times for dissipation through thermal conduction and viscosity are, respectively,

In a situation where both these phenomena are present, the characteristic time for dissipation τdis(λ) is given by the relation

characteristic of processes acting in parallel.

The full (non-relativistic) theory of dissipation of acoustic waves through viscosity and thermal conduction yields the following result

where E is the mechanical energy transported by the sound wave, ζ is the second viscosity, and γ is the adiabatic index. The Equation (11.5.13) verifies the applicability of Equation (11.5.12).

11.6 Dissipation of Adiabatic Perturbations

We now apply the physics described in the previous section to adiabatic perturbations in the plasma epoch of the expanding Universe described in Chapter 9. In the period prior to recombination, when τep τeγ(τγe), one can treat the plasma–photon system as an imperfect radiative fluid, where the e ect of dissipation manifests itself as an imperfect thermal equilibrium between matter and radiation. In this situation, the kinematic viscosity and the coe cient of thermal di usion are given by

Equation (11.6.1) cannot be used in Equation (11.5.13), which was obtained in a non-relativistic treatment. There are special processes which modify Equation (11.5.13) in the relativistic limit: for example the thermal conduction is not proportional to T, but to T −[T/(p+ρc2)] p. In particular, Equation (11.5.11) becomes