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

278 Cosmological Perturbations

concerning gauge choices which we shall skip in this case. What we are interested in at the end is the amplitude of the fluctuations when they enter the cosmological horizon after inflation has finished. If we define ∆2H(k) to be the value of ∆2(k) for the fluctuations in the density at scale k when they reenter the horizon after inflation, one can find

where the ‘ ’ denotes the value of V or H at the time when the perturbation left the horizon during inflation. One therefore sees the fluctuation on reentry which was determined by the conditions just as it left, which is physically reasonable. One does not know the values of these parameters a priori, however, so they cannot be used to predict the spectral amplitude. In an exactly exponential inflationary epoch V and H are constant so that ∆2H(k) is constant. Since ∆2 k3P(k), and PH(k) P(k)k−4 from (13.5.6), we therefore have P(k) k, which is the Harrison–Zel’dovich spectrum we mentioned before in Section 13.4.

In fact, the generic inflationary prediction is not for a pure de Sitter expansion, so that the quantity ∆2H is not exactly independent of scale. It is straightforward to show that the actual spectral index is related to the slow-roll parameters H (13.6.2) and η (13.6.5) when the perturbation scale k leaves the horizon via

which gives n = 1 in the slow-rolling limit, as expected.

The quantum oscillations in Φ also lead to the generation of a stochastic background of gravitational waves with a spectrum and amplitude which depends on a di erent combination of slow-roll parameters from the scalar density fluctuation spectrum (in fact, the gravitational wave spectrum depends only on H). The relative amplitudes of the gravitational waves and scalar perturbations also depend on the shape of the potential. Since gravitational waves are of no direct relevance to structure formation, we shall not discuss them in more detail here. Gravitational waves can, in principle, also generate temperature fluctuations in the cosmic microwave background, so we shall discuss them briefly in Section 17.4 and they may ultimately be detectable, a possibility we discuss in Chapter 21.

We should also mention that the quantum fluctuations in φk have random phases and therefore should be Gaussian (see Section 14.7) in virtually all realistic inflationary models (except perhaps those with multiple scalar fields or where the field evolution is nonlinear). This is because one usually assumes the field Φ to be in its ground state: zero point fluctuations are then those of a ground-state harmonic oscillator in quantum mechanics, i.e. Gaussian. Along with the computational advantages we shall mention later, this is a strong motivation for assuming that δ(x) is a Gaussian random field.

13.7Gaussian Density Perturbations

In Section 13.2 we defined the power spectrum P(k) of density perturbations, which measures the amplitude of the fluctuations as a function of wavenumber k or, equivalently, mass scale M. For some purposes, however, it is necessary to know not only the spectrum, that is the mean square fluctuation of a given wavenumber, but also the (probability) distribution of the fluctuations in either real space or Fourier space. Returning to the discussion we made in Section 13.2, consider a (large) number N of realisations of our periodic volume and label these realisations by Vu1, Vu2, Vu3, . . . , VuN . It is meaningful to consider the probability distribution P(δk) of the relevant coe cients

from realisation to realisation across this ensemble. Let us assume that the distribution is statistically homogeneous and isotropic (as it must be if the Cosmological Principle holds), and that the real and imaginary parts have a Gaussian distribution and are mutually independent, so that

where w stands for either the real part or the imaginary part of δk and α2k = δ2k/2; δ2k is the spectrum (see Section 13.2). This is the same as the assumption that the phases ϑk in Equation (13.7.1) are mutually independent and randomly distributed over the interval between ϑ = 0 and ϑ = 2π. In this case the moduli of the Fourier amplitudes have a Rayleigh distribution:

Because of the assumption of statistical homogeneity and isotropy of the Universe, the quantity δk depends only on the modulus of the wavevector k, denoted k, and not on its direction. It is fairly simple to show that, if the Fourier quantities |δk| have the Rayleigh distribution, then the probability distribution P(δ) of

In fact, Gaussian statistics in real space do not require the distribution (13.7.3) for the Fourier component amplitudes. One can see that δ(x) is simply a sum over a large number of Fourier modes. If the phases of each of these modes are random, then the central limit theorem will guarantee that the resulting superposition will be close to a Gaussian distribution if the number of modes is large. While (13.7.3) provides the formal definition of a Gaussian random field, the main requirement in practice is simply that the phases are random. As we explained in Section 14.6,

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Gaussian fields are strongly motivated by inflation. This class of field is the generic prediction of inflationary models where the density fluctuations are generated by quantum fluctuations in a scalar field during the inflationary phase.

For a Gaussian field δ, not only can the distribution function of values of δ at individual spatial positions be written in the form (13.7.4), but also the N-variate joint distribution of a set of δi ≡ δ(xi) can be written as a multivariate Gaussian distribution:

where M is the inverse of the correlation matrix C = δiδj , V is a column vector made from the δi, and VT is its transpose. An example for N = 2 will be given in equation (14.8.2). This expression (13.7.5) is considerably simplified by the fact that δi = 0 by construction. The expectation value δiδj can be expressed in terms of the covariance function, ξ(rij),

where the averages are taken over all spatial positions with |xi − xj| = rij, and the second equality follows from the assumption of statistical homogeneity and isotropy. We shall see in the next section that ξ(r) is intimately related to the power spectrum, P(k). This means that the power spectrum or, equivalently, the covariance function of the density field is a particularly important statistic because it provides a complete statistical characterisation of the density field as long as it is Gaussian.

The ability to construct not only the N-dimensional joint distribution of values of δ, but also joint distributions of spatial derivatives of δ of arbitrary order, ∂nδ/∂xin, all of the form (13.7.5), but which involve spectral moments (13.2.10), is what makes Gaussian random fields so useful from an analytical point of view. The properties of Gaussian random fields are also interesting in the framework of biased galaxy-formation theories, which we discuss in Section 15.7. In this context one is particularly interested in regions of particularly high density which one might associate with galaxies. For example, one can show that the number of peaks of the density field per unit volume with height δ(x)/σ0 in the range ν to ν + dν, with ν 1, is

while the total number of peaks per unit volume with height exceeding νσ is

the quantities R and γ are defined by Equation (13.2.11). The mean distance between peaks of any height is of order 4R . The ratio R0 = σ0/σ1 R /γ represents the order of magnitude of the coherence length of the field, i.e. the value of r at which the covariance function ξ(r) becomes zero.

In a homogeneous and isotropic universe the function ξ(r) does not depend on either the origin or the direction of r, but only on its modulus; the result (13.8.9) implies therefore that

in general the covariance function must change sign – from positive at the origin, at which (13.8.1) guarantees ξ(0) = σ2 0, to negative at some r – to make the overall integral (13.8.10) converge in the correct way. A perfectly homogeneous distribution would have P(k) ≡ 0 and ξ(r) would be identically zero for all r.

The meaning of the function ξ(r) can be illustrated by the following example. Imagine that the material in the Universe is distributed in regions of the same size r0 with density fluctuations δ > 0 and δ < 0. In this case the product δ(x)δ(x+r) will be, on average, positive for distances r < r0 and negative for r > r0. This means that the function ξ(r) reaches zero at a value r r0, which represents the mean size of regions and therefore the coherence length of the fluctuation field. Inside the regions themselves, where ξ(r) > 0, there is correlation, while, outside the regions, where ξ(r) < 0, there is anticorrelation.

The function ξ(r) is the two-point covariance function. In an analogous manner it is possible to define spatial covariance functions for N > 2 points. For example, the three-point covariance function is

where the mean is taken over all the points x and over all directions of r and s such that |r − s| = t: in other words, over all points defining a triangle with sides r, s and t.

The generalisation of (13.8.12) to N > 3 is obvious. It is convenient to define quantities related to the N-point covariance functions called the cumulants, κN , which are constructed from the moments of order up to and including N. The cumulants are defined as the part of the expectation value δ1 . . . δN (δ1 ≡ δ(x1), etc.), of which (13.8.12) is the special case for N = 3, which cannot be expressed in terms of expectation values of lower order. Cumulants are also sometimes called the connected part of the corresponding covariance function. To determine them in terms of δ1δ2 . . . δN for any order, one simply expresses the required expectation value as a sum over all distinct possible partitions of the set {1, . . . , N}, ignoring the ordering of the components of the set; the cumulant is just the part of this sum which corresponds to the unpartitioned set. This definition makes use

of the cluster expansion. For example, the possible partitions of the set {1, 2, 3} are ({1}, {2, 3}), ({2}, {1, 3}) ({3}, {1, 2}), ({1}, {2}, {3}) and the unpartitioned set ({1, 2, 3}). This means that the expectation value can be written

δ1δ2δ3 = δ1 c δ2δ3 c + δ2 c δ1δ3 c

+ δ3 c δ1δ2 c + δ1 c δ2 c δ3 c + δ1δ2δ3 c. (13.8.13)

The cumulants are κ3 ≡ δ1δ2δ3 c, κ2 = δ1δ2 c, etc. Since δ = 0 by construction, κ1 = δ1 c = δ1 = 0. Moreover, κ2 = δ1δ2 c = δ1δ2 . The secondand third-order cumulants are simply the same as the covariance functions. The fourthand higher-order quantities are di erent, however. The particularly useful aspect of the cumulants which motivates their use is that all κN for N > 2 are zero for a Gaussian random field; for such a field the odd N expectation values are all zero, and the even ones can be expressed as combinations of δiδj in such a way that the connected part is zero.

It is possible to define ξ(r) also in terms of a discrete distribution of masses rather than a continuous density field. Formally one can write the density field ρ(x) = i miδD(x − xi), where the sum is taken over all the mass points labelled by i and found at position xi; δD is the Dirac function. If all the mi = m, the mean density is ρ = nV m. The probability of finding a mass point in a randomly chosen volume δV at x is therefore δP = m−1ρ(x)δV; the joint probability of finding a point in δV1 and a point in δV2 separated by a distance r is

δ2P2 = ρ(x)ρ(x + r) δV1δV2 m2

= n2 ρ(x)ρ(x + r) δV δV

which defines ξ(r) to be the two-point correlation function of the mass points. The same result holds if we take the probability of finding a point in a small volume δV, where the density is ρ, to be proportional to ρ. This forms the so-called Poisson clustering model which we shall use later, in Section 16.6.

One can also extend the (discrete) correlations to orders N > 2 by a straightforward generalisation of equation (13.8.14):

where r stands for all the rij separating the N points. However, the function ξ(N)(r), which is called the total N-point correlation function, contains contributions from correlations of orders less than N. For example, the number of triplets is larger than a random distribution partly because there are more pairs than in a random distribution:

284 Cosmological Perturbations

The part of ξ(3) which does not depend on ξij, usually written ζ123, is called the irreducible or connected three-point function. The four-point correlation function ξ(4) will contain terms in ζijk, ξijξkl and ξij, which must be subtracted to give the connected four-point function η1234. The connected correlation functions are analogous to the cumulants defined above for continuous variables, and are constructed from the same cluster expansion. The only di erence is that, for discrete distributions, one interprets single partitions (e.g. δ1 c) as having the value unity rather than zero. For the two-point function there are only two partitions, ({1}, {2}) and ({1, 2}). The first term would correspond to δ1 δ2 = 0 in the continuous variable case because δ = 0, but the two expectation values are each assigned a value of unity in the discrete variable case, so that δ2P2 1 +ξ(r) and ξ(2)(r) = ξ(r), as expected. For the three-point function, the right-hand side of Equation (13.8.12) has, first, three terms corresponding to the three terms in ξij in Equation (13.8.16), then a product of three single-partitions each with the value unity, and finally a triplet which corresponds to the connected part ζ123. This reconciles the forms of (13.8.16) and (13.8.12) and shows that ξ(3) = ξ23 + ξ13 + ξ12 + ζ123. This procedure can be generalised straightforwardly to higher N.

13.9 Non-Gaussian Fluctuations?

As we have explained, the power spectrum of density fluctuations scales in the linear regime in such a way that each mode evolves independently according to the growth law. This means, for example, that σM t2/3 in an Einstein–de Sitter model. Since each mode evolves independently, the random-phase hypothesis of Section 13.7 continues to hold as the perturbations evolve linearly and the distribution of δ should therefore remain Gaussian.

Notice, however, that δ is constrained to have a value δ −1, otherwise the energy density ρ would be negative. The Gaussian distribution (13.7.3) always assigns a non-zero probability to regions with δ < −1. The error in doing this is negligible when σM is small because the probability of δ < −1 is then very small, but, as fluctuations enter the nonlinear regime with σM 1, the error must increase to a point where the Gaussian distribution is a very poor approximation to the true distribution function. What happens is that, as the fluctuations evolve into this regime, mode-coupling e ects cause the initial distribution to skew, generating a long tail at high δ while they are also bounded at δ = −1. Notice, however, that if the mass distribution is smoothed on a scale M, one should recover the regime where σM 1, where the field will still be Gaussian. Large scales therefore continue to evolve linearly, even when small scales have undergone nonlinear collapse in the manner described in the next chapter.

The generation of non-Gaussian features as a result of the nonlinear evolution of initially Gaussian perturbations is well known and can be probed using numerical simulations or analytical approximations. We shall not say much about this question here, except to remark that, on scales where such e ects are important,

the power spectrum, or, equivalently, the covariance function, does not furnish a complete statistical description of the properties of the density field δ.

Despite the strong motivation for the Gaussian scenario from inflationary models we should at least mention the possibility that either the primordial fluctuations are not Gaussian or that some later mechanism, apart from gravity, induces non-Gaussian behaviour during their evolution.

Attempts to construct inflationary models with non-Gaussian fluctuations due to oscillations in Φ have largely been unsuccessful. It is necessary to have some kind of feature in the potential V(Φ) or to have more than one scalar field. There are, however, some other possibilities. First, as we mentioned briefly in Section 7.6, it is possible that some form of topological defect might survive a phase transition in the early Universe. These defects comprise regions of trapped energy density which could act as seeds for structure formation. However, in such pictures the seeds are very di erent from quantum fluctuations induced during inflation and would be decidedly non-Gaussian at very early times. One of the early favourites for a theory based on this idea was the cosmic-string scenario in which one-dimensional string-like defects act as seeds. The behaviour of a network of cosmic strings is di cult to handle even with numerical methods and this scenario did not live up to its early promise. The original idea was that the evolving network would form loops of string which shrink and produce gravitational waves; as they do so they accrete matter. More accurate simulations, however, showed that this does not happen and that small loops cannot be responsible for structure formation. A revised version of this theory has been suggested more recently, in which long pieces of string, moving relativistically, produce ‘wakes’ which can give rise to sheet-like inhomogeneities. Another possibility is that three-dimensional defects called textures, rather than one-dimensional strings, might be the required seed. Perhaps primordial black holes could also act as a form of zero-dimensional seed. These pictures do not seem as compelling as the ‘inflationary paradigm’ we have mentioned above, but they are not ruled out by present observations.

The second possibility is that some astrophysical mechanism might induce nonGaussian behaviour. A possible example is that some kind of cosmic explosion, perhaps associated with early formation of very massive objects, could form a blast wave which would push material around into a bubbly or cellular pattern at early times (e.g. Ostriker and Cowie 1981). This would be non-Gaussian and would subsequently evolve under its own gravity to form a distribution very dissimilar to that which would form in an inflationary model. Unfortunately, this model seems to be ruled out by the lack of any distortions in the spectrum of the microwave background radiation; see Chapter 19.

Although there is no strongly compelling physical motivation for non-Gaussian fluctuations, one should be sure to test the Gaussian assumption as rigorously as possible. One can do this in many ways, using the microwave background and galaxy-clustering statistics. Until non-Gaussian models are shown to be excluded by the observations, there is always the possibility that some physics we do not yet understand created initial fluctuations of a very di erent form to those predicted by inflation.

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Bibliographic Notes on Chapter 13

An interesting discussion of the properties of primordial power spectra is given by Gott (1980). Adler (1981) and Vanmarcke (1983) are useful texts on the general mathematical properties of Gaussian random fields; application of Gaussian random fields in a cosmological context are discussed by Bardeen et al. (1986), a famous paper known to the community as BBKS. Non-Gaussian perturbations are discussed by Brandenberger (1990) and Coulson et al. (1994).

Problems

1. Show that if a perturbation field has a power spectrum of the form k exp(−λ k), then

√ 0

the covariance function crosses zero at r = λ0 3. Give a physical interpretation of this result.

2.Calculate the spectral parameters (13.2.11) for the power spectrum defined in Question 1.

3.A lognormal field Y(r) is defined by Y(r) = exp[X(r)], where X is a Gaussian random field. Calculate the two-point covariance function of Y in terms of the covariance function of X.

4.For the lognormal field Y defined in Question 3 calculate the three-point function

(a) in terms of the two-point function of X, and (b) in terms of the two-point function of Y.

5.Repeat Questions 3 and 4 for the χ2 field defined by Z = X2, where X is a Gaussian random field.

14

Nonlinear

Evolution

After recombination, fluctuations in the matter component δ on a scale M > MJ(i)(zrec) 105M grow according to the theory developed in Chapters 10–12 while |δ| 1. This is obviously a start, but it cannot be used to follow the evolution of structure into the strongly nonlinear regime where overdensities can exist with δ 1. A cluster of galaxies, for example, corresponds to a value of δ of order several hundred or more. To account for structure formation we therefore need to develop techniques for studying the nonlinear evolution of perturbations. This is a much harder problem than the linear case, and exact solutions are di cult to achieve. We shall mention some analytical and numerical approaches in this chapter.

14.1 The Spherical ‘Top-Hat’ Collapse

The simplest approach to nonlinear evolution is to follow an inhomogeneity which has some particularly simple form. This is not directly relevant to interesting cosmological models, because the real fluctuations are expected to be highly irregular and random. Considering cases of special geometry can nevertheless lead to important insights. In this spirit let us consider a spherical perturbation with constant density inside it which, at an initial time ti trec, has an amplitude δi > 0 and |δi| 1. This sphere is taken to be expanding with the background universe in such a way that the initial peculiar velocity at the edge, Vi, is zero. As we have mentioned before, the symmetry of this situation means that we can treat the perturbation as a separate universe and, for simplicity, we assume that the background universe at ti is described by an Einstein–de Sitter model; in this case we