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

258 Non-baryonic Matter

so that, using equation (12.3.9), we have

More accurate expressions from full numerical calculations are given in Chapter 15.

Before znν zeq the Jeans mass MJν practically coincides with MJ(a), the Jeans mass corresponding to adiabatic perturbations in a plasma of baryons and radiation. As we have seen above, MJ(a) grows after zeq and reaches a maximum value at zrec. In cases in which znX > zeq, the di erence between MJX,max and MJ(a)(zrec) is large.

Now we turn to cold thermal relics. One can show that

in the four redshift intervals z znX, znX z zdX, zdX z zeq and z zeq, respectively. The maximum value of the Jeans mass for typical cold-dark-matter particles is too small to be of interest in cosmology.

As we have already explained, in a collisionless fluid perturbations on scales less than the Jeans mass do not just oscillate but can be damped by two physical processes: in the ultrarelativistic regime, when the particle velocities are all of order v c, the amplitude of a perturbation decays because particles move with a large ‘directional’ dispersion from overdense to underdense regions, and vice versa; in the non-relativistic regime there is also a considerable spread in the particle velocities which tends to smear out the perturbation. This second damping mechanism is similar to the Landau damping that occurs in plasma physics, and is also known as phase mixing. In either case, to order of magnitude, after a time t perturbations are dissipated on a scale λ λfX, with

The scale λfX is called the free-streaming scale. We introduce here the freestreaming mass:

Let us again turn to the case of hot thermal relics with znX > zeq. In this case we find

260 Non-baryonic Matter

models of structure formation with collisionless relic particles. Historically, there have been two important scenarios involving: hot dark matter (HDM) in which the collisionless dark matter takes the form of a hot thermal relic; and cold dark matter (CDM) in which the dark matter is either a cold thermal relic, or perhaps a non-thermal relic such as an axion.

12.6.1 Hot Dark Matter

Recall that hot dark matter corresponds to thermal relics with znX zeq and therefore with a maximum value of MJX of order 1014M or greater. A typical HDM candidate particle is a neutrino species with mass of the order of 10 eV. When a perturbation enters the cosmic horizon in a universe dominated by such particles it will have δr δm δX. Fluctuations in the relic component δX with M > MJX(znX) can enjoy a period of uninterrupted growth (apart from a brief interval of stagnation due to the action of the Meszaros e ect ending at zeq). If the primordial spectrum of perturbations has an amplitude decreasing with scale, as we shall explain in the next chapter, one will first form structure in the collisionless component on the scale M MJX(znX). The first structures to form are called pancakes, as in the adiabatic baryon model. In the range of scales between MJX(znX) and MJ(a)(zrec) the fluctuations in the matter component undergo oscillations like acoustic waves until recombination. At zrec, in this range of scales, we therefore have

with AX(M) 1. The factor AX, of order unity for M MJ(a)(zrec), has a maximum value

for the scale M MJX(znX). After recombination the perturbations in the baryonic matter component again become unstable and begin to grow like the perturbations δX. The latter fluctuations, being more than an order of magnitude larger than δm, dominate the self-gravity of the system so that after recombination the baryonic material follows the behaviour of the dark matter: δm δX. This happens very quickly, as the following argument demonstrates. If there is more than one matter component, then equation (10.6.14) becomes

where the sum is taken over all the matter components; see also equations (11.9.3 a) and (11.9.3 b). This can be derived from a two-fluid model ignoring the factors of 43 and 2 corresponding to radiation pressure and the gravitational e ect of pressure, respectively, and letting τeγ = τγe → ∞. In this case the two fluids are baryons, b, and dark matter, X, and the initial conditions are such that δX δb at

Implications 261

trec. In an Einstein–de Sitter model Equation (12.6.3) for the baryonic component can be written

This equation is easily solved, since we know that δX t2/3, by the ansatz δb = Atp. One thus finds that

so that the baryonic fluctuations catch up the dark matter virtually instantaneously.

12.6.2 Cold Dark Matter

Particles of cold dark matter correspond to cold thermal relics (or non-thermal relics such as axions), with znX zeq. For such particles the maximum value of MJX is quite small compared with scales of cosmological interest. Perturbations in the collisionless component δX are frozen-in by the Meszaros e ect until zeq, but enjoy uninterrupted growth on scales M > MJX after zeq. In this case, assuming as before that the spectrum of initial fluctuations decreases with mass scale, as discussed in the next chapter, the first structure to form has a mass of order M MJ(i)(zrec) 105M ; the limit here is essentially provided by the pressure of the baryons after recombination. Although fluctuations are not dissipated in this model on small scales, the stagnation e ect does suppress their growth compared with large scales, so the spectrum of fluctuations is severely modified: see Chapter 15, where we discuss these e ects in detail. More detailed computations, based on kinetic theory, have shown that in both the CDM and HDM models, the residual fluctuations in the microwave radiation background are much smaller than those in the adiabatic baryon picture. This result can be understood from a qualitative point of view, by simply recognising that fluctuations on the scales MJX,max < M < MJ(a)(zrec) are roughly a factor AX smaller in this case than in the old adiabatic picture. As an example, in Figure 12.1 we show the results of a full numerical computation of the evolution of the perturbations δX, δm and δr corresponding to a mass scale M 1015M for a CDM model with a Hubble parameter h = 0.5. One can compare this result with the similar computations shown in the previous chapter for baryonic models. The CDM model in particular produces rather low fluctuations in the CMB radiation. Until relatively recently, this was considered an asset, but with the COBE discovery of the radiation it seems to be a weakness: COBE seems to have detected larger fluctuations than CDM would predict, as discussed in Chapter 17.

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12.6.3 Summary

By a relatively simple consideration of time and length scales, we have shown in this chapter how the presence of a significant component of non-baryonic material alters the growth rate of perturbations under gravitational instability. It has not been our aim in this chapter to develop complete models of structure formation based on this idea, but simply to explain the physical origin of the di erence with respect to models with baryons only. The two main points to remember are that

1.models with non-baryonic dark matter typically induce smaller fluctuations in the radiation background than those with only baryons;

2.structure can survive on scales less than the Silk mass in a cold-dark- matter universe (because fluctuations in the dark-matter component are not a ected by photon di usion);

3.structure is destroyed on small scales in a hot-dark-matter universe because of the free streaming of the non-baryonic component.

In Chapter 15 we will explain how these ingredients manifest themselves in more complete models of structure formation.

Bibliographic Notes on Chapter 12

The standard manifesto for structure formation within CDM models is Blumenthal et al. (1984), while the first detailed numerical computations were by Davis et al. (1985). This basic model has been developed much further; see, for example, Frenk et al. (1988). A detailed account of the evolution of CDM perturbations is given by Liddle and Lyth (1993). Neutrino-dominated universes are discussed by, for example, White et al. (1983). This general material is covered well by Padmanabhan (1993) and Peacock (1999). The possibility of directly detecting dark-matter candidates is discussed, for example, in Klapdor-Kleingrothaus and Zuber (1997).

Problems

1.Derive the approximate solutions (12.2.5 a) and (12.2.5 b).

2.Derive the approximate solution (12.4.5).

3.Compare the solutions obtained in Questions 1 and 2 with numerical solutions of Equations (12.2.4) and (12.2.6).

13

Cosmological

Perturbations

13.1 Introduction

In the previous chapters we have studied the linear evolution of a perturbation described as a plane wave with corresponding wave vector k. This representation is useful because a generic perturbation can be represented as a superposition of such plane waves (by the Fourier representation theorem) which, while they are evolving linearly, evolve independently of each other. In general we expect fluctuations to exist on a variety of mass or length scales and the final structure forming will depend on the growth of perturbations on di erent scales relative to each other. In this chapter we shall therefore look at perturbations in terms of their spectral composition and explain how the various spectral properties might arise.

A particularly important problem connected with the primordial spectrum of perturbations is to understand its origin. In the 1970s the form of the spectrum was generally assumed in an ad hoc fashion to have the properties which seemed to be required to explain the origin of structure in either the adiabatic or isothermal scenario. A particular spectrum, suggested independently by Peebles and Yu (1970), Harrison (1970) and Zel’dovich (1972), but now usually known as the

Harrison–Zel’dovich or scale-invariant spectrum, was taken to be the most ‘natural’ choice for initial fluctuations according to various physical arguments. Further motivation for this choice arrived in 1982 in the form of inflationary models, which, as we shall see in Section 13.6, usually predict a spectrum of the scaleinvariant form. The details of these fluctuations, which are generated by quantum oscillations of the scalar field driving the inflationary epoch, were first worked out by Guth and Pi (1982), Hawking (1982) and Starobinsky (1982). This result was very

264 Cosmological Perturbations

important, because it represented the first time that any particular choice of the spectrum of initial perturbations has been strongly motivated by physics.

As far as the evolution of the perturbation spectra is concerned, it is clear that the theory must depend on the nature of the particles which dominate the Universe, baryonic or non-baryonic, hot or cold, and on the nature of the fluctuations themselves, adiabatic or isothermal, curvature or isocurvature. We shall explain how these factors alter or ‘modulate’ the primordial spectrum later in this chapter. Because the fluctuations are, in some sense, ‘random’ in origin, we shall also need to introduce some statistical properties which can be used to describe density fluctuations, namely the power spectrum, variance, probability distribution and correlation functions.

13.2 The Perturbation Spectrum

To describe the distribution of matter in the Universe at a given time and its subsequent evolution one might try to divide it into volumes which initially evolve independently of each other. Fairly soon, however, this independence would no longer hold as the gravitational forces between one cell and its neighbours become strong. It is therefore not a good idea to think of a generic perturbation as a sum of spatial components. It is a much better idea to think of the perturbation as a superposition of plane waves which have the advantage that they evolve independently while the fluctuations are still linear. This e ectively means that one represents the distribution as independent components not in real space, but in Fourier transform space, or reciprocal space, in terms of the wavevectors of each component k.

Let us consider a volume Vu, for example a cube of side L ls, where ls is the maximum scale at which there is significant structure due to the perturbations; Vu can be thought of as a ‘fair sample’ of the Universe if this is the case. It is possible therefore to construct, formally, a ‘realisation’ of the Universe by dividing it into cells of volume Vu with periodic boundary conditions at the faces of each cube. This device will be convenient for many applications but should not be taken too literally. Indeed, one can take the limit Vu → ∞ in most cases, as we shall see later.

Let us denote by ρ the mean density in a volume Vu and ρ(x) to be the density at a point specified by the position vector x with respect to some arbitrary origin. As usual we define the fluctuation δ(x) = [ρ(x) − ρ ]/ ρ . In light of the above comments we take this to be expressible as a Fourier series:

where the assumption of periodic boundary conditions δ(L, y, z) = δ(0, y, z), etc., requires that the wavevector k has components

with nx, ny and nz integers. The Fourier coe cients δk are complex quantities given, as it is straightforward to see, by

because of conservation of mass in Vu we have δk=0 = 0; because of the reality of

δ(x) we have δk = δ−k.

If, instead of the volume Vu, we had chosen a di erent volume Vu, the perturbation within the new volume would again be represented by a series of the form (13.2.1), but with di erent coe cients δk. If one imagines a large number N of such volumes, i.e. a large number of ‘realisations’ of the Universe, one will find that δk varies from one to the other in both amplitude and phase. If the phases are random, not only across the ensemble of realisations, but also from node to node within each realisation, then the density field has Gaussian statistics which we shall discuss in detail in Section 13.7. For the moment, however, it suffices to note the following property. Although the mean value of the perturbation δ(x) ≡ δ across the statistical ensemble is identically zero by definition, its mean square value, i.e. its variance σ2, is not. It is straightforward to show that

where the average is taken over an ensemble of realisations. The quantity δk is defined by the relation (13.2.4) and its meaning will become clearer later, in Section 13.8. One can see from Equation (13.2.4) that |δk|2 is the contribution to the variance due to waves of wavenumber k. If we now take the limit Vu → ∞ and assume that the density field is statistically homogeneous and isotropic, so that there is no dependence on the direction of k but only on k = |k|, we find

where we have, for simplicity, put δ2k = P(k) in the limit Vu → ∞. The quantity P(k) is called the power spectral density function of the field δ or, more loosely, the power spectrum. The variance does not depend on spatial position but on time, because the perturbation amplitudes δk evolve. The quantity σ2 therefore tells us about the amplitude of perturbations, but does not carry information about their spatial structure.

As we shall see, it is usual to assume that the perturbation power spectrum P(k), at least within a certain interval in k, is given by a power law

the exponent n is usually called the spectral index. The exponent need not be constant over the entire range of wave numbers: the convergence of the variance in (13.2.5) requires that n > −3 for k → 0 and n < −3 for k → ∞.

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Equation (13.2.5) can also be written in the form

represents the contribution to the variance per unit logarithmic interval in k. We shall find this quantity useful to compare with observations of galaxy clustering on large scales in Section 16.6. If ∆(k) has only one pronounced maximum at kmax, then the variance is given approximately by

Some other useful properties of the spectrum P(k) are its spectral moments

where the index l (which is an integer) is the order; the zeroth-order moment is just the variance σ2. Typically, such as for power-law spectra, these moments do not converge and it is necessary to filter the spectrum to get meaningful results; we discuss this in Section 13.3 and thereafter. Higher-order moments of the (filtered) spectrum contain information about the shape of P(k) just as moments of a probability distribution contain information about its shape. As we shall see in Section 14.8, many interesting properties of the fluctuation field δ(x) can be expressed in terms of the spectral moments or combinations of them such as

where γ and R are usually called the spectral parameters.

13.3 The Mass Variance

13.3.1 Mass scales and filtering

The problem with the variance σ2 is that it contains no information about the relative contribution to the fluctuations from di erent k modes. It may also be formally infinite, if the integral in Equation (13.2.5) does not converge. It is convenient therefore to construct a statistical description of the fluctuation field as a function of some ‘resolution’ scale R. Let M be the mean mass found inside a spherical volume V of radius R: