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

248 Gravitational Instability of Baryonic Matter

11.10 Summary

We have chosen to investigate the behaviour of density perturbations in a baryonradiation universe in some detail mainly for pedagogical reasons, that is to illustrate the important physics and display the required machinery. In fact, it is not thought possible that structure in the Universe grew in such a scenario. We shall explain why this is so and make some comments about the development of baryon-only models during the 1970s in Chapter 15.

We end by summarising the most important consequences for structure formation of the physics we have discussed in this chapter. First is the e ect of the evolution of the characteristic mass scales MJ(a), MJ(i) and MD(a). The behaviour of an adiabatic perturbation depends upon its characteristic mass scale. For perturbations on scales M > MJ(a)(zeq) 4 × 1015(Ωh2)−2M , i.e. 1015–1018M for acceptable values of the parameter Ωh2, we have a wavelength greater than the Jeans length either before decoupling or after, when the Jeans mass drops to MJ 105(Ωh2)−1/2M . Such scales therefore experience uninterrupted growth (we shall neglect the decaying modes in this study). The growth law is

in the period, if it exists, between equivalence and recombination. After decoupling, the radiation must be treated like a ‘gas’ of collisionless particles and the evolution of its perturbations must be handled in a more sophisticated manner than the classical gravitational instability treatment. We described this approach briefly in Section 11.10. As far as δm is concerned, the growth law is still given by Equation (11.10.2) for Ω = 1 and also for Ωz 1 if Ω < 1. More precise formulae are given in Section 11.4.

In the case of perturbations with mass in the interval

for acceptable values of Ωh2, we have the following evolutionary sequence. In the period before their entry into the cosmological horizon defined by zH(M), the perturbations evolve according to Equation (11.10.1); in the period between zH(M) and zrec they oscillate like acoustic waves with a sound speed vs(a) and with constant amplitude for z > zeq and amplitude decreasing as t−1/6 between equivalence and recombination; after decoupling they become unstable again and evolve like masses with M > MJ(a). Perturbations with masses M < MD(a)(zrec) evolve as before until the time tD(M) at which M = MD(a). After tD(M) these fluctuations become rapidly dissipated. The bottom line is that only the perturbations with M > MD(zrec) can survive from the plasma epoch until the period after recombination. It is interesting to note that this characteristic scale is similar to that of a rich cluster of galaxies.

Summary 249

As we have seen, isothermal perturbations with

are frozen-in until the epoch defined by zi = min(zeq, zrec). After this time, they are unstable and can grow according to the same law that applies to adiabatic perturbations at late times. We shall not worry about the evolution of perturbations on scales less than MJ(i)(zrec), because these have no real cosmological relevance. It is interesting to note that MJ(i)(zrec) is of the same order as the mass of a globular cluster.

Bibliographical Notes on Chapter 11

Historically important papers relevant to this chapter are Peebles and Yu (1970), Wilson and Silk (1981) and Wilson (1983). An alternative formulation of the kinetic approach is given by Efstathiou (1990).

Problems

1.What is the energy stored in a primordial acoustic wave? When these waves are dissipated by Silk damping, where does this energy go?

2.Derive the dispersion relation (11.8.5).

3.Derive the Equations (11.9.7) using the definitions given in Section 11.9.

12

Non-baryonic

Matter

12.1 Introduction

We shall now extend the analyses of the previous two chapters to study the evolution of perturbations in models of the Universe dominated by dark matter which is not in the form of baryons. As we saw in Section 4.4, dynamical considerations suggest that the value of Ω at the present epoch is around Ωdyn 0.2 and may well be higher. Given that modern observations of the light-element abundances require Ωbh2 0.02 to be compatible with cosmological nucleosynthesis calculations, at least part of this mass must be in the form of non-baryonic particles (or perhaps primordial black holes which formed before nucleosynthesis and therefore did not participate in it). As we have seen, most examples of the inflationary universe predict flat spatial sections which, in the absence of a cosmological constant, implies Ω very close to unity at the present time. If this is true, then the Universe must be dominated by non-baryonic material to such an extent that the baryons constitute only a fraction of a percent of the total amount of matter.

One of the problems in these models is that we do not know enough about high-energy particle physics to know for sure which kinds of particles can make up the dark matter, nor even what mass many of the predicted particles might be expected to have. Our approach must therefore be to keep an open mind about the particle physics, but to place constraints where appropriate using astrophysical considerations.

We begin by running briefly through the physics of particle production in the early Universe, and then go on to describe the e ect of di erent kinds of particles on the evolution of perturbations. Theories of galaxy formation based on the properties of di erent kinds of dark matter are then discussed in a qualitative way.

where τcoll = 1/ σAv neq is the mean time between collisions and τH = a/a˙ is the characteristic time for the expansion of the Universe; we have dropped the subscript X for clarity. Equation (12.2.4) has the approximate solution

where td is the moment of ‘freezing out’ of the creation and annihilation reactions, defined by

More exact solutions to Equation (12.2.4) behave in a qualitatively similar way to this approximation.

12.3 Hot Thermal Relics

As we have explained, hot thermal relics are those that decouple while they are still relativistic. Let us assume that the particle species X becomes non-relativistic at some time tnX, such that

(A 3.1 or 2.7 is a statistical-mechanical factor which takes these two values according to whether X is a fermionfermions or a boson). For simplicity we take A = 3 to get rough estimates. Hot relics are thus those for which tnX > tdX, where tdX is defined by Equation (12.2.6).

Let us denote by gX the statistical weight of the particle X and by gX the e ective number of degrees of freedom of the Universe at tdX. Following the same kind of reasoning as in Chapter 8, based on the conservation of entropy per unit comoving volume, we have

where T0X is the present value of the e ective temperature defined by the mean particle momentum via

T0r is the present temperature of the photon background and T0ν = (114 )1/3T0r takes account of the Nν neutrino families; g0 3.9 for Nν = 3. We thus obtain from (12.3.2)

254 Non-baryonic Matter

(photons, neutrinos and electrons all contribute to gν ). In this case we obtain the well-known relation

where B = 34 or 1 according to whether the particle X is a fermion or a boson. The density parameter corresponding to these particles is then just

Equations (12.3.7) and (12.3.8) are to be compared with Equations (8.5.5) and (8.5.10). For example, consider hypothetical particles with mass mX 1 KeV, which decouple at T 102–103 MeV when gX 102; these have ΩX 1.

Let us now apply Equation (12.3.8) to an example: the case of a single massive neutrino species with mν 1 MeV, which decouples at a temperature of a few MeV when gX = 10.75 (taking account of photons, electrons and three types of massless neutrinos). The condition that the cosmic density of such relics should not be much greater than the critical density requires that mν < 90 eV: this bound was obtained by Cowsik and McClelland (1972). If, instead, all the neutrino types have mass around 10 eV, then their density will be given by the equation already presented in Section 8.5:

Equations (12.3.1) and (12.3.4) can be used to calculate the redshift corresponding to tnX:

The moment of equivalence, teq, between the relativistic components (photons, massless neutrinos) and the non-relativistic particles (X after tnX and baryons) is given by

if one assumes that ΩX Ωb, and neglects the contribution of baryons to Ω. In Equation (12.3.11) we have K0 1 +0.227Nν taking account of the massless neutrinos. It is clear that we cannot have znX < zeq; in the case where the collisionless component dominates at tnX one assumes znX = zeq.

Because ΩX is proportional to mX by Equation (12.3.8), one can write

which complement Equations (12.3.10) and (12.3.11). In particular, if the X particles are massive neutrinos, we can obtain

mν is the average neutrino mass.

12.4 Cold Thermal Relics

Calculating the density of cold thermal relics is much more complicated than for hot relics. At the moment of their decoupling the number density of particles in this case is given by a Boltzmann distribution:

The present density of cold relics is therefore

The problem is to find TdX, that is to say the temperature at which Equation (12.2.6) is true. The characteristic time for the expansion of the Universe at tdX is

which is the same as appeared in Equation (7.1.6), while the characteristic time for collisional annihilations is given by

256 Non-baryonic Matter

q = 0 or 1 for most kinds of reaction. Introducing the variable x = mXc2/kBT, the condition τcoll(x) = τH(x) is true when x = xdX = mXc2/kBTdX 1. The value of xdX must be found by an approximate solution of Equation (12.2.6), which reads

As an application of Equation (12.4.4), one can consider the case of a heavy neutrino of mass mν 1 MeV. If the neutrino is a Dirac particle (i.e. if the particle and its antiparticle are not equivalent), then the cross-section in the nonrelativistic limit varies as v−1 corresponding to q = 0 in (12.4.4), for which σ0 = const. 0.8gwk2 (mν2 c/ 4) (gwk is the weak interaction coupling constant). Putting gν = 2 and gν 60 one finds that xdν 15, corresponding to a temperature Tdν 70 (mν /GeV) MeV. Placing this value of xdν in Equation (12.4.7), the condition that Ων h2 < 1 implies that mν > 1 GeV: this limit was found by Lee and Weinberg (1977), amongst others. If, on the other hand, the neutrino is a Majorana particle (i.e. if the particle and its antiparticle are equivalent), the annihilation rateσAv has terms in x−q with q = 0 and 1, thus complicating matters considerably. Nevertheless, the limit on mν we found above does not change. In fact we find mν > 5 GeV. If the neutrino has mass mν 100 GeV, the energy scale of the electroweak phase transition, the cross-section is of the form σA T−2 and all the previous calculations must be modified.

The relations (12.3.10) and (12.3.11) which supply znX and zeq remain substantially unchanged, except that in the expression for znX one should replace gX by gnX, the value of g at tnX.

12.5 The Jeans Mass

In this section we shall study the evolution of the Jeans mass MJX and the freestreaming mass MfX for a fluid of collisionless particles. As we have explained in Section 10.3 and Chapter 11, we need first to determine the behaviour of the mean particle velocity vX in the various relevant cosmological epochs. These epochs are the two intervals t < tnX and t > tnX for hot relics; the three intervals t < tnX, tnX t tdX and t > tdX for cold relics. In the first case (hot relics) we have, roughly,

One defines the Jeans mass for the collisionless component to be the quantity

the Jeans length λJX is given by Equation (10.3.11) where one replaces v by vX from above:

The total density ρ includes contributions from a relativistic component ρr (photons and massless neutrinos), the collisionless component ρX and the baryonic component ρb which, in the first approximation, can be neglected. One can put

ρρr for z > zeq and ρ ρX for z < zeq.

Now let us consider the case of hot thermal relics. Assuming that znX > zeq we

easily obtain

for z zeq. The mass MJX(znX) represents the maximum value of MJX. Its value depends on the type of collisionless particle. The highest value of this mass is obtained for particles having znX zeq, such as neutrinos with a mass aroundmν 10 eV. In this case we have

which corresponds to a length scale