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

198 The Plasma Era

the expansion creates a variation of ν a(t)−1 and, because Nν must be conserved, T must also vary as a(t)−1. In fact, one can use a similar argument to show that a thermal Maxwell–Boltzmann distribution of particle velocities also remains constant during the expansion of the Universe but the e ective temperature varies as T a(t)−2.

The FIRAS instrument on the COBE satellite (Mather et al. 1994) obtained the results shown in Figure 9.1, together with results in di erent wavelength regions from other experiments. The fit to the black-body spectrum is extremely good, providing clear evidence that this radiation is indeed relic thermal radiation from a primordial fireball.

In fact, the quality of the fit of the observed CMB spectrum to a black-body curve does more than confirm the Big Bang picture. It places important constraints on processes which might be expected to occur within the Big Bang model itself and which would lead to slight distortions in the black-body shape. For example, even in the idealised equilibrium model of hydrogen recombination, the physical nature of this process is expected to produce distortions of the spectrum. Recombination occurs when Tr 4000 K. Although the number-density of photons is some 109 times greater than the number-density of baryons at this time, the density of photons with hν > 13.6 eV is less than the number-density of baryons. Since the optical depth for absorption of Lyman series photons is very high, recombination occurs mainly through two-photon decay, which is relatively slow. (This is one of the reasons why the ionisation fraction is somewhat higher than the Saha equation predicts.) Although each recombination therefore produces several photons, since the number-density of baryons is so much smaller than that of the photons, these recombination photons cannot change the spectral shape very much near its peak. They can, however, lead to strong distortions in the far Wien (hν kBT) and far Rayleigh–Jeans (hν kBT) parts of the spectrum. Unfortunately, the spectrum is quite weak in this region and galactic dust makes it di cult to make observations to test these ideas.

A more significant distortion mechanism is associated with the injection of some form of energy into the plasma at some time. As we have explained, the relaxation time for non-thermal energy injection to be thermalised is usually very short. Nevertheless, certain types of energy release cannot be thermalised and could therefore lead to observable distortions.

After energy injection, the first thing that happens is that the electrons adjust their temperature to whatever the non-equilibrium spectrum is. This happens on a timescale determined by the number-density of electrons, which is much smaller than the number-density of photons. Next, the radiation spectrum is adjusted by multiple scattering processes which conserve the total number of photons. As a result, the total number of photons does not match the e ective temperature of the spectrum; one finds instead a form

200 The Plasma Era

Physical processes operating at z < zrec can also distort the CMB spectrum, but here the distortion takes a slightly di erent form. If there exists a period of reionisation of the Universe, as indeed seems to be the case (see Section 19.3), Compton scattering of CMB photons by ionised material can distort the shape of the spectrum in a way that depends upon when the secondary heating occurred and how it a ected the intergalactic gas. In many circumstances only one parameter is needed to describe the distortion, because the electron temperature Te is greater than the radiation temperature Tr. The relevant parameter is the y-parameter

where the integral is taken over the time the photon takes to traverse the ionised medium. This is usually called the Sunyaev–Zel’dovich e ect (Sunyaev and Zel’dovich 1970).

When CMB photons scatter through material which has been heated in this way the shape of the spectrum is distorted in both Rayleigh–Jeans and Wien regions. If y < 0.25 the shape of the Rayleigh–Jeans part of the spectrum does not change, but the e ective temperature changes according to T = Tr exp(−2y). At high frequencies the intensity actually increases. This can be understood in terms of low-frequency CMB photons being boosted in energy by Compton scattering and transferred to high-frequency parts of the spectrum. Strong constraints on the allowed y-distortions are also placed by the COBE satellite: y 3 × 10−5. In Chapter 19 we explain how these observations can constrain theories of structure formation.

Bibliographic Notes on Chapter 9

A classic reference for the behaviour of the ionisation of the expanding Universe is Wyse and Jones (1985); Kaiser and Silk (1987) also contains an accessible discussion of optical depths and reionisation. Much of the other material is covered by standard texts; see in particular Peebles (1971, 1993).

Problems

1.Use the Saha formula (9.3.5) to compute the ionisation fraction of a pure hydrogen plasma at T = 3000 K if Ω0bh2 = 0.01.

2.Derive Equation (9.4.5), i.e. show that

and derive an expression for the constant A in terms of physical constants and cosmological parameters.

4.Low-energy photons from the cosmic microwave background pass through a cloud of hot plasma (at a temperature of order 108 K) before arriving at the observer. Show that the observer sees a fractional reduction in the temperature T of the microwave background in the direction of the cloud given by

∆T −2 σTPe dt. T mec

PART 3

Theory of

Structure Formation

10

Introduction to

Jeans Theory

10.1 Gravitational Instability

In an attempt to understand the formation of stars and planets, Jeans (1902) demonstrated the existence of an important instability in evolving clouds of gas. This instability, now known as the gravitational Jeans instability, gravitational instability, or simply Jeans instability, is now the cornerstone of the standard model for the origin of galaxies and large-scale structure.

Jeans demonstrated that, starting from a homogeneous and isotropic ‘mean’ fluid, small fluctuations in the density, δρ, and velocity, δv, could evolve with time. His calculations were done in the context of a static background fluid; the expansion of the Universe was not known at the time he was working and, in any case, is not relevant for the formation of stars and planets. In particular, he showed that density fluctuations can grow in time if the stabilising e ect of pressure is much smaller than the tendency of the self-gravity of a density fluctuation to induce collapse. It is not surprising that such an e ect should exist: gravity is an attractive force so, as long as pressure forces are negligible, an overdense region is expected to accrete material from its surroundings, thus becoming even more dense. The denser it becomes the more it will accrete, resulting in an instability which can ultimately cause the collapse of a fluctuation to a gravitationally bound object. The simple criterion needed to decide whether a fluctuation will grow with time is that the typical lengthscale of a fluctuation should be greater than the Jeans length, λJ, for the fluid. Before we calculate the Jeans length in mathematical detail, we first give a simple order-of-magnitude argument to demonstrate its physical significance.

The equations of motion of such a fluid, in the Newtonian approximation, are

These are the continuity equation, the Euler equation and the Poisson equation, respectively. Throughout this chapter and the next we shall neglect any dissipative terms arising from viscosity or thermal conductivity. For this reason we must add another equation to the ones above, describing the conservation of entropy per unit mass s:

The system of Equations (10.2.1) admits the static solution with ρ = ρ0, v = 0, s = s0, p = p0 and ϕ = 0. Unfortunately, however, according to the system of Equations (10.2.1), if ρ0 ≠ 0, then the gravitational potential ϕ must vary spatially; in other words, a homogeneous distribution of ρ cannot be stationary, and must be globally either expanding or contracting. There is therefore nothing necessarily relativistic about the expansion of the Universe: the incompatibility of a static universe with the Cosmological Principle is also apparent in Newtonian gravity. This same e ect is also the reason why the Einstein static universe is unstable. As we shall see, however, when we consider the case of an expanding universe, the results of Jeans remain qualitatively unchanged. We shall therefore proceed with Jeans’ treatment, even though it does have this problem. It turns out to be an incorrect theory, which nevertheless can be ‘reinterpreted’ to give correct results. Its great advantage is that Newtonian gravity is more familiar to most students than general relativity.

Now let us look for a solution to (10.2.1) that represents a small perturbation of the (erroneous) static solution: ρ = ρ0 + δρ, v = δv, p = p0 + δp, s = s0 + δs, ϕ = ϕ0 + δϕ. Introducing these small quantities into the Equations (10.2.1) and neglecting terms of higher order in small quantities, we find

We now have to study all the solutions to this perturbed system of equations. Indeed, as we shall see, there are five solutions: two of adiabatic type, one of