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

238 Gravitational Instability of Baryonic Matter

Before equivalence, when ρr > ρm, we have

Thus, before equivalence, the dissipation can be attributed mainly to the e ect of radiative viscosity and, after equivalence, it is mainly due to thermal conduction. In any case the quantity τdis does not change by much between these two epochs: Equations (11.6.5) and (11.6.7) show that, in the final analysis, the dissipation of acoustic waves in the plasma epoch is due to the di usion of photons.

As we have explained, we must consider dissipation after a time t on scales characterised by a mass M < MD(t) or by a length λ < λD(t). It is straightforward to verify, within the framework of the approximations introduced above, that the condition λ < λD(t) is identical to the condition that τdis(λ) < t. It therefore emerges that

For adiabatic perturbations of mass M < MJ(a)(zeq), the time t is the interval of time ∆t in which such perturbations evolve like acoustic waves: given that MHb MJ(a) before equivalence, this interval is approximated by ∆t(M) t − t(zH(M)) t, where now t stands for cosmological proper time; t(zH(M)) is negligible with respect to t for the range of masses we are interested in.

Before equivalence the dissipation scale for adiabatic perturbations is, from Equations (11.6.8) and (11.6.5),

where t is given by Equation (5.6.7) and τγe is given by Equation (9.2.9). The corresponding dissipation mass scale is then given by

which yields

If Ωh2 4 × 10−2, then zrec zeq and the mass scale for dissipation at recombination becomes MD(a)(zrec) 1017M .

If the Universe is su ciently dense so that zeq > zrec, we can obtain in a similar manner, using Equations (11.6.8), (11.6.7) and (2.2.6b) for the period zeq > z >

At recombination we have MD(a)(zrec) 1012(Ωh2)−5/4M .

As we shall see, the value of MD(a)(zrec) is of great significance for structure formation. Its magnitude depends on the density parameter through the quantity Ω0h2. Approximate numerical values for 4 × 10−2 Ω0h2 2 are 1017M MD(a)(zrec) 4 × 1011M . The first to calculate the value of MD(a)(zrec) was Silk (1967) – for this reason the quantity MD(a) is also known as the Silk mass. It is interesting to note that

where Mγ is the mass contained within a sphere of diameter lγ = cτγe. The reason for the relation (11.6.13) is implicit in Equation (11.5.9).

In the case where there is a significant amount of non-baryonic matter so that Ω ≠ Ωb, which is the case we shall discuss in the next chapter, Silk damping of course still occurs, but the damping mass scale changes. It is a straightforward exercise to show that, in this case, the corresponding value at zrec can be obtained

from the above case if zrec > zeq by changing Ω to Ωb and, if zrec < zeq, by changing Ω to (ΩbΩ9)1/10.

The importance of the Silk mass can be explained as follows. Without taking account of dissipative processes, the amplitude of an acoustic wave on a mass scale M < MJ(a) would remain constant in time during radiation domination and would decay according to a t−1/6 law in the period between equivalence and recombination. The dissipative processes we have considered cause a decrease of the amplitude of such waves, with a rate of attenuation that depends upon M. In fact the energy of the wave E A2 is damped exponentially. The time for a wave to damp away completely is therefore much less than the timescale for the next scale to enter the horizon. The upshot of this is that fluctuations on all scales less than the Silk mass are completely obliterated by photon di usion almost immediately. No structure will therefore be formed on a mass scale less than this.

240 Gravitational Instability of Baryonic Matter

11.7 Radiation Drag

We now turn our attention to physical processes which are important for isothermal rather than adiabatic fluctuations. We have already mentioned that isothermal perturbations on a scale M > MJ(i) are frozen-in because of a kind of viscous friction force acting on particles trying to move through a smooth radiation background. This force is essentially due to radiation drag. We can show schematically that this freezing-in e ect is relevant if the viscous forces on the perturbation Fv per unit mass dominate the self-gravitational force Fg per unit mass. This condition is that

where we have used the fact that ρm (Gt2)−1, and we are now interested in the period defined by zeq z Ω−1. The inequality (11.7.1) holds for t > τeγ, which is true before recombination. Now let us treat this phenomenon in a more precise way. If a perturbation in the ionised component (plasma) moves with a velocity v c relative to an unperturbed radiation background, any electron encounters a force opposing its motion that has magnitude

This applies also to electron–proton pairs because for z > zrec the protons are always strictly coupled to the motion of the electrons. In fact, because of the Doppler e ect, an electron moving through the radiation background experiences a radiation temperature which varies with the angle ϑ between its velocity and the line of sight:

which corresponds to an energy flux in the solid angle dΩ of

and a momentum flux in the direction of the velocity of

The momentum acquired by an electron per unit time, which is caused by the anisotropic radiation field experienced by it, is therefore

since the Thomson cross-section of a proton is a factor (mp/me)2 smaller than that of an electron, the force su ered by the protons is negligible. Equation (11.7.6)

where kJm and kJr are the wavenumbers appropriate to the wavelengths given by equations (10.6.15) and (10.9.7). The dispersion relation is of fourth order in ω. For a given k there exist four solutions ωi(k), with i = 1, 2, 3, 4, and there are also four perturbation modes. Next one puts an expression of the form (11.8.4) in the equations for Vm and Vr, (11.8.1 b) and (11.8.2 b), with the same restriction on τω. Then substituting in these four equations the solutions ωi(k) one obtains the four perturbation modes:

The analytical study of the acoustic modes described by the Equations (11.8.7) is very complicated, except in special cases where one can simplify the dispersion relation to transform it into a cubic equation or, most usefully, a quadratic equation. In general the ith root of (11.8.5) is complex:

One has wavelike propagation when Re ωi(k) ≠ 0; in this case one can easily see that ωj(k) = −ωi (k) is also a solution: these two solutions represent waves propagating in opposite sense to each other, with phase velocity vs(k) = |Re ωi(k)|/k and amplitude which decreases with time when Im ωi(k) < 0; the characteristic time for the wave to decay is given by τi = |Im ωi(k)|−1.

One has gravitational instability when Re ωi(k) = 0. This instability can be of either increasing or decreasing type according to whether Im ωi(k) is greater than or less than zero, and the characteristic time for the evolution of the instability is given by τi = |Im ωi(k)|−1.

In general, before decoupling there are two modes of approximately adiabatic nature, in the sense that δr/δm 43 . These modes are unstable for M > MJ(a), so that one increases and the other decreases; for M < MJ(a) they evolve like damped acoustic waves with the sound speed vs vs(a). A third mode, again of approximately adiabatic type, also exists but is non-propagating and always damped. The fourth and final mode is approximately isothermal (in the sense that |δr| |δm|), so that for M > MJ(i) it is an unstable growing mode, but with a characteristic growth time τ > τH, so it is e ectively frozen-in. During decoupling, the last two of these modes gradually transform themselves into two isothermal modes which oscillate like waves for M < MJ(i) with a sound speed vs vs(i), and are unstable (one growing and the other decaying) for M > MJ(i). The first two modes become purely radiative, i.e. δm 0, which are unstable for wavelengths greater than the appropriate Jeans√length for radiation λ(Jr) and which oscillate like waves propagating at a speed c/ 3 practically without damping for λ < λ(Jr). These last two modes are actually spurious, since in reality the radiation after decoupling behaves like a collisionless fluid which cannot be described by an equation of the

perturbations to both the matter and radiation but also of fluctuations in the metric. The Robertson–Walker metric describing the unperturbed background must be replaced by a metric whose components gik di er by infinitesimal quantities from the original gik: the deviations δgik are connected with the perturbations to the matter and radiation by the Einstein equations. There is also the problem referred to in Section 10.12 concerning the choice of gauge. This is a subtle problem which we shall not describe in detail at the moment, although we will return to it briefly in Chapter 17 where we discuss the cosmic microwave background. The simplest approach is to adopt a synchronous gauge characterised by the metric

where |hαβ| 1. The treatment is considerably simplified if the unperturbed metric is flat so that γαβ = δαβ, where δαβ is the Kronecker symbol: δαβ = 1 for α = β, δαβ = 0 for α ≠ β. This is also the case in an approximate sense if the Universe is not flat, but one is looking at scales much less than the curvature radius or at very early times.

The time evolution of the trace h of the tensor hαβ is related to the evolution of matter and radiation perturbations

The equations that describe the evolution of the time-dependent parts δm and Vm of the perturbations in the density and velocity of the matter are

the perturbation in the velocity of the radiation Vr will be defined a little later. As far as the radiation perturbations are concerned, one can demonstrate that

their evolution is described by a single equation involving the brightness function δ(r)(x, t), whose Fourier transform can be written

the quantity δ(kr) at any point involves contributions from photons with momenta directions specified by the spherical polar angles ϑ and ϕ. The di erential equation which describes the evolution of δ(kr), which was first derived from the Liouville equation by Peebles and Yu (1970), is

246 Gravitational Instability of Baryonic Matter

where ϑ is the angle between the photon momentum and the wave vector k, which we assume to define the polar axis of a local coordinate system. Given the rotational symmetry, one can expand δ(kr) in angular moments σl defined with respect to the Legendre polynomials

δ(kr) = (2l + 1)Pl(cos ϑ)σl(k, t). (11.9.6)

l

The perturbation δr coincides with the moment σ0, while the velocity perturbation Vr which appears in (11.9.3 b) is given by 14 σ1.

It is comparatively straightforward to show that the evolution of the brightness function is governed by a hierarchy of equations for the moments σl:

One can verify that the two-fluid approximation practically coincides with the system of Equations (11.9.2)–(11.9.3 b) and (11.9.7) if one puts σ3 = 0 and neglects σ˙2 in Equation (11.9.7 c), thus truncating the hierarchy. This approximation is good in the epoch during which τγe τH, which is in practice any time prior to recombination, and on large scales, such that λ cτγe. In the general situation, both during and after recombination, the system can be solved only by truncating the hierarchy at some suitably high value of l; the number of l-modes one has to take grows steadily as decoupling and recombination proceed. A couple of examples of a full numerical solution of the evolution of perturbations in the matter δm and radiation δr components in an adiabatic scenario are shown in Figures 11.1 and 11.2. The mass scale in both these calculations is of order 1015M . Notice how the matter and radiation perturbations oscillate together in both calculations until recombination, whereafter the radiation perturbation stays roughly constant and the matter perturbation becomes unstable and grows until the present epoch. Figure 11.1 shows a model with Ω = 1 so that the growth after recombination is a pure power law, while Figure 11.2 has Ω = 0.1, so that the e ect of the growth factor (Section 11.4) in flattening out the behaviour of the perturbations is clear. In the opposite limit to that of the validity of the two-fluid approach, one has τγe τH, which is much later than recombination or for small scales such that λ cτγe. In such a case we have