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

268 Cosmological Perturbations

where jl are spherical Bessel functions, Pl|m| are the associated Legendre polynomials, and r, ϑ and ϕ are spherical polar coordinates. The integral I in Equation (13.3.3 f) then becomes

(the integrals over ϑ and ϕ are zero unless m = l = 0); in this way the window function is just

its behaviour is such that W(x) 1 for x 1 and |W(x)| x−2 for x 1. Passing to a continuous distribution of plane waves, i.e. in the limit expressed

by Equation (13.2.5), the mass variance is

which, as it must be, is a function of R and therefore of M.

The significance of the window function is the following: the dominant contribution to σM2 is from perturbation components with wavelength λ k−1 > R, because those with higher frequencies tend to be averaged out within the window volume; we have tacitly assumed that the spectrum is falling with decreasing k, so waves with much larger λ contribute only a small amount. We will return to this point in Section 14.4, where we discuss e ects occurring at the edge of the window.

13.3.2 Properties of the filtered field

One can think of the result expressed by Equation (13.3.8) also as a special case of a more general situation. It is often interesting to think of the fluctuation field as being ‘filtered’ with a low-pass filter. The filtered field, δ(x; Rf), may be obtained by convolution of the ‘raw’ density field with some function F having a characteristic scale Rf:

δ(x; Rf) = δ(x )F(|x − x |; Rf) dx . (13.3.9)

The filter F has the following properties: F = const. Rf−3 if |x − x | Rf, F 0 if |x − x | Rf, F(y; Rf) dy = 1. For example, the ‘top-hat’ filter, with a sharp cut o , is defined by the relation

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13.3.3 Problems with filters

One of the reasons why one might prefer a Gaussian filter over the apparently simpler top hat is illustrated by applying Equation (13.3.17) to a power-law spectrum of the form (13.2.6). As we have said, in order for σ2 to converge, the spectrum P(k) must have an asymptotic behaviour as k → ∞ of the form kn∞ , with n∞ < −3. For this reason we can only take Equation (13.2.6) to be valid for wavenumbers smaller than a certain value k∞, after which the spectral index either changes slope to n∞ or there is a rapid cut-o in P(k). The convergence for small k, however, requires that n > −3. If one puts Equation (13.2.6) directly into (13.3.17) and assumes a top-hat filter, so that W(kR) = 1 for k 1/R ≡ kM, |W(kR)| (k/kM)−2 for kM k k∞, and P(k) = 0 for k > k∞, one obtains, for the interval −3 < n < 1,

we call the exponent α = 16 (3 + n) the mass index. For values n > 1 one finds, however, that

the mass index does not depend on the original spectral index. The result (13.3.21) is also obtained if n = 1, apart from a logarithmic term. The reason for this result is that we have taken for the definition of σM the variance of fluctuations inside a sphere with sharp edges. This corresponds to an extended window function in Fourier space. When n 1 the spectral components which enter the integral at the edges of the window function become significant contributors to the variance: σM2 defined by Equation (13.3.17) is no longer a useful measure of the mass fluctuations on a particular scale R, but is dominated by edge e ects which are sensitive to fluctuations on a much smaller scale than R. These e ects are a form of surface noise which depends on the number of ‘particles’ at the boundary; a

statistical fluctuation arises according to whether a particle happens to lie just inside, on or just outside the boundary. If the expected number of particles on a surface of area S is NS, then we clearly have

in accordance with Equation (13.3.21). This misleading result can be corrected if one makes a more realistic definition of the volume corresponding to the mass scale M. If one smears out the edges of the sphere such as, for example, via a Gaussian filter (13.3.11), one obtains

the new window function passes sharply from a value of order unity, for k < 1/R = kM, to a vanishingly small value for k > kM: the blurring out of the sphere has therefore made the window function sharper. With the new definition one finds, for any n,

(Γ is the Euler gamma function), which has a dependence on R which is now in accord with Equation (13.3.18). The behaviour of σM is therefore generally valid if one uses a Gaussian filter function.

13.4 Types of Primordial Spectra

Having established the description of a primordial stochastic density field in terms of its power spectrum and related quantities, we should now indicate some possibilities for the form of this spectrum. It is also important to develop some kind of intuitive understanding of what the spectrum means physically.

It is the usual practice to suppose that some mechanism, perhaps inflation, lays down the initial spectrum of perturbations at some very early time, say t = tp, which one is tempted to identify with the earliest possible physical timescale, the Planck time. The cosmological horizon at this time will be very small, so the fluctuations on scales relevant to structure formation will be outside the horizon. As time goes on, perturbations on larger and larger scales will enter the horizon as they grow by gravitational instability, become modified by the various damping and stagnation processes discussed in the previous chapters and, eventually, after recombination, give rise to galaxies and larger structures. The final structures which form will therefore depend upon the primordial spectrum to a large extent,

272 Cosmological Perturbations

but also upon the cosmological parameters and the form of any dark matter. It is common to assume a primordial spectrum of a power-law form:

In general, one would expect the amplitude Ap and the spectral index np to depend on k so that Equation (13.4.1) defines the e ective amplitude and index for a given k. In most models, however, np is e ectively constant over the entire range of scales relevant to the observable Universe. The mass variance corresponding to Equation (13.4.1) is

where MH(tp) is some reference mass scale which, for convenience, we take to be the horizon mass at time tp.

Clearly the discussion in Section 13.2 demonstrates that a perfectly homogeneous distribution of mass in which δ(x) = 0 has a power spectrum which is identically zero for all k and therefore has zero mass variance on any scale. To interpret other behaviours of σM2 it is perhaps helpful to think of the mass distribution as being composed of point particles with identical mass m. If these particles are distributed completely randomly throughout space, then the fluctuations in a volume V – which contains on average N particles and, therefore, on average, a mass M = mN – will be due simply to statistical fluctuations in the number of particles from volume to volume. For random (Poisson) distributions this means that δN2 1/2 N1/2, so that the RMS mass fluctuation is given by

corresponding, by Equation (13.4.2), to a value of the mass index α = 12 and therefore to a spectral index n = 0. Since P(k) is independent of k this is usually called a white-noise spectrum.

Alternatively, if the distribution of particles is not random throughout space but is instead random over spherical ‘bubbles’ with sharp edges, the RMS mass fluctuations becomes

as we have mentioned above; the mass fluctuation expressed by Equation (13.4.4) corresponds to a mass index α = 23 and to a spectral index n = 1. If the edges of the spheres are blurred, then the ‘surface e ect’ is radically modified and it is then possible to show that

corresponding to a mass index α = 56 and a spectral index n = 2. Equation (13.4.5) can be found if one assumes that one can create the perturbed distribution from a homogeneous distribution by some rearrangement of the matter which conserves mass. It would be reasonable to infer that this rearrangement can only take place over scales less than the horizon scale when the fluctuations were laid down, which gives a natural scale to the ‘bubbles’ we mentioned above. From Equation (13.2.3) one obtains

In calculating the mass variance σM2 , as we have explained, one counts only the waves with k < R−1, for which the term k · x is small: in the series (13.4.6) the higher and higher terms are smaller and smaller. Conservation of mass requires that the first term is zero, or that δk k and therefore σM M−5/6. If one also requires that linear momentum is conserved or, in other words, that the centre of mass of the system does not move, then the second term in (13.4.6) is also zero and we obtain δk k2, corresponding to a spectral index n = 4 and therefore to a mass index α = 76 :

It is tempting to imagine that fluctuations in the number of particles inside the horizon might lead to a ‘natural’ form for the initial spectrum. Such a spectrum has some severe problems, however. If one takes the time tp to be the Planck time, for example, the horizon contains on average only one ‘Planck particle’ and one cannot think of the spatial distribution within this scale as random in the sense required above. Moreover, the white-noise spectrum actually predicts a very chaotic cosmology in which a galactic-scale perturbation would arrive at the nonlinear growth phase (Chapter 14) much before teq. Let us consider a perturbation with a typical galaxy mass, 1011M , which contains Nb 1069 baryons corresponding to N Nbσ0r 1078 particles and therefore characterised by σM N−1/2 10−39. This perturbation would arrive at the nonlinear regime at a time tc given, approximately, by

in Equation (13.4.8) we have supposed that tc < teq, and this is confirmed a posteriori by the result Tc 1012 K. Such collapses would have a drastic e ect on the isotropy and spectrum of the microwave background radiation and on nucleosynthesis, so would consequently not furnish an acceptable theory of galaxy formation.

The spectrum (13.4.5), often called the particles-in-boxes spectrum, also has problems. It only makes sense to treat the perturbations from a statistical point of view when the horizon contains a reasonably large number of particles, say Ni 100. This happens at a time ti corresponding to a temperature Ti 2×1018 GeV. A

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fluctuation on a scale M of the order of the horizon mass at Ti has σM (ti) Ni−1/2 if the particles are distributed randomly, but, as we have explained above, the ‘surface e ect’ might produce an RMS mass fluctuation of the form

for N > Ni. The constant Bp is obtained in a first approximation by putting σM (N = Ni) = Ni−1/2; one thus finds Bp 5. However, even in this case, the variance on a scale M 1011M yields a completely unsatisfactory result. Taking, as in the previous case, N 1078 and allowing the perturbation to grow uninterruptedly (σM t, for t < teq, and σM t2/3, for t > teq), i.e. without taking account of periods of damping or oscillation, one finds

the fluctuation would not yet have arrived at the nonlinear regime and could not therefore have formed structure. Equation (13.4.10) is valid for Ω = 1 and things get worse if Ω < 1. On the scales of galaxies the amplitude of the whitenoise spectrum, np = 0, is too high, while that of the particles-in-boxes spectrum, np = 2, is much too low.

The problems arising from spectra obtained by reshu ing matter within a horizon volume have led most cosmologists to abandon such an origin and appeal to some process which occurs apparently outside the horizon to lay down some appropriate spectrum. As already mentioned, in the early 1970s, Peebles and Yu (1970), Harrison (1970) and Zel’dovich (1970), working independently, suggested a spectrum with np = 1, corresponding to

(the value of Kp proposed by Zel’dovich was of the order of 10−4, so as to produce fluctuations in the cosmic microwave background at a lower level than the observational limits of that time, while still allowing galaxy formation by the present epoch). This spectrum, called the Harrison–Zel’dovich spectrum, is of the same form as Equation (13.4.4), but is not interpreted as a surface e ect. One of its properties is that fluctuations in the gravitational potential, δϕ, or, in relativistic terms, in the metric, are independent of length scale r. In fact

if Equation (13.4.11) holds. The Equation (13.4.11) therefore characterises a spectrum which has a metric containing ‘wrinkles’ with an amplitude independent of scale. As we shall see in Section 14.5, fluctuations of this form enter the cosmological horizon with a constant value of the variance, equal to Kp2. For these reasons this spectrum is often called the scale-invariant spectrum. We shall see in

Section 14.6 that a spectrum of density fluctuations close to this form is in fact a common feature of inflationary models.

As a final remark in this section, we should mention that the spectrum of the density perturbation δ can also be used to construct the spectrum of the perturbations to the gravitational potential, δϕ, and to the velocity field v in linear theory. The results are particularly simple. Since 2δϕ δ, one has k2ϕk δk, where ϕk is the Fourier transform of δϕ, so that Pϕ(k) P(k)k−4. For a density fluctuation spectrum with spectral index n one therefore has nϕ = n − 4 so that, for n = 1, one has nϕ = −3. This spectrum is generally, i.e. whether it refers to a potential, velocity or density field, called the flicker-noise spectrum, and the associated variance has a logarithmic divergence at small k. The velocity field is the gradient of a velocity potential which is just proportional to the gravitational potential so that vk kϕk and Pv(k) P(k)k−2. We discuss velocity and potential perturbations in more detail in Chapter 18, where the exact expressions for the appropriate power spectra are also given.

13.5 Spectra at Horizon Crossing

In Section 11.5 we defined the time at which a perturbation of mass M enters the horizon; we found that, for M MH(zeq) 5 × 1015(Ωh2)−2M , this moment corresponds to a redshift

this relation is valid for a flat universe or an open universe for z Ω−1; in this section we shall assume the simplest case of Ω = 1.

We propose to calculate the variance σM2 corresponding to a scale M at the time defined by zH(M) if the primordial fluctuation spectrum is of the power-law form (13.4.2). The perturbation grows without interruption from the moment of its origin, which we called tp, to the time in which it enters the cosmological horizon, with a law σM t (1+z)−2 before equivalence and σM t2/3 (1+z)−1 after equivalence. If zH(M) > zeq, we therefore have

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again identical to (13.5.3). The index αH is the mass index of fluctuations at their entry into the cosmological horizon. This has a corresponding spectral index nH, in accord with (13.4.1), which one finds from

one therefore has

The Equation (13.5.5) indicates that the Harrison–Zel’dovich scale-invariant spectrum with np = 1 arrives at the cosmological horizon with a mass variance which is independent of M and equal to Kp2. Steeper spectra (np > 1, αp > 23 ) have a variance which decreases with increasing M at horizon entry; shallower spectra (np < 1, αp < 23 ) have variance increasing with M. For this latter type, there is the problem that, on su ciently large scales, one has a universe with extremely large fluctuations which would include separate closed mini-universes. There is clearly then a strong motivation for having a spectrum which, whatever its origin, produces a mass index αp 23 on the very largest scales. As a final comment, notice that the spectral index of fluctuations at horizon entry (13.5.6) is precisely the same as the spectral index for fluctuations in the gravitational potential field, defined in Section 13.4.

13.6 Fluctuations from Inflation

We have already mentioned that one of the virtues of the inflationary cosmology is that it predicts a spectrum of perturbations which might be adequate for the purposes of structure formation. The source for these fluctuations is the quantum field Φ which drives inflation in the manner described in Section 7.10. A full treatment of the origin of these fluctuations is outside the scope of this book since it requires advanced techniques from quantum field theory. Here we shall merely give an outline; Brandenberger (1985) gives a nice review. In this section we use units where = c = kB = 1.

Suppose that the expectation value of the scalar field Φ(x, t) is homogeneous in space, i.e. Φ(x, t) = Φ(t). It then follows an equation of motion of the form

cf. Equation (7.10.5), where V is the e ective potential and the prime denotes a derivative with respect to Φ. As we mentioned in Section 7.10, most inflationary models satisfy the ‘slow-rolling’ conditions which we shall assume here because these simplify the calculations. Let us introduce these conditions again in a more quantitative way. In the slow-rolling approach the motion of the field is damped so

first slow-rolling condition. The second slow-rolling condition in fact corresponds to two requirements: firstly that the parameter H, defined by

˙2

which e ectively means that V Φ , the condition for inflation to occur; secondly that

which, together with (13.6.3), implies that the scale factor is evolving approximately exponentially: a exp(Ht). The third condition is that η, defined by

which can be thought of as a consistency requirement on the other two conditions, since it can be obtained from them by di erentiation.

We now have to understand what happens when we perturb the equation (13.6.1). Assuming, as always, that the spatial fluctuations in the Φ field, δΦ = φ, can be decomposed into Fourier modes φk by analogy with (13.2.1), we obtain

It turns out, for reasons we shall not go into, that the V term in Equation (13.6.7) is negligible when a given fluctuation scale is pushed out beyond the horizon. The resulting equation then looks just like a damped harmonic oscillator for any particular k mode. Applying some quantum theory, it is possible to calculate the expected fluctuations in each ‘mode’ of this system in much the same way as one calculates the ground-state oscillations in any system of quantum oscillators. One finds the solution

One can think of this e ect as similar to the Hawking radiation from the event horizon of a black hole: there is an event horizon in de Sitter space and one therefore sees a thermal background at a temperature TH = H/2π which corresponds to fluctuations in the Φ field in the same manner as the thermal fluctuations at the Planck epoch we discussed in Chapter 6.

From (13.6.8) we can define a quantity ∆φ(k) by (13.2.8) so that ∆φ = const. H. These fluctuations are therefore of the same amplitude (in an appropriately defined sense), i.e. independent of scale as long as H is constant.

These considerations establish the form of the spectrum appropriate to the fluctuations in Φ but we have not yet arrived at the spectrum of the density perturbations themselves. The resolution of this step requires some technicalities