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

5

Thermal History

of the Hot Big

Bang Model

5.1 The Standard Hot Big Bang

The hot Big Bang is the name usually given to the standard cosmological model: a homogeneous, isotropic universe whose evolution is governed by the Friedmann equations obtained from general relativity (with or without a cosmological constant), whose main constituents can be described by matter and radiation fluids, and whose kinematic properties (i.e. the Hubble constant) match those we observe in the real Universe. It is further assumed that the radiation component of the energy density is of cosmological origin: this is why the term ‘hot’ is given to the model. Of course, our real Universe is not exactly homogeneous and isotropic, so this model is to some extent an abstraction. However, as we shall see later, this standard model does provide us with a framework within which we can study the emergence of structures like the observed galaxies and clusters of galaxies from small fluctuations in the density of the early Universe. In this chapter, we give a brief overview of the evolution the basic physical properties of this model; more detailed treatment will be deferred to Chapters 8 and 9.

As we have already seen in Chapter 4, the present-day matter density is

In the following, as in Chapter 4, we shall drop one of the subscripts and use Ω0 to quantify the density of non-relativistic matter. Observations tell us that

110 Thermal History of the Hot Big Bang Model

Ω0 is somewhere in the range 0.01 < Ω0 < 2. The luminous material in galaxies and clusters is primarily hydrogen and a small part of helium. Cosmological nucleosynthesis provides an explanation for the relative abundances of these, and other, light elements: see Chapter 8. As we have seen, however, the Universe is probably dominated by unseen dark matter, whose nature is yet to be clarified.

The energy-density contributed by the radiation background at 2.73 K is

where σr is the radiation density constant. We discussed this before, in Chapter 4. The standard model also predicts the existence of a cosmological background of neutrinos, which we discuss more fully in Chapter 8, with an energy density

Nν is the number of neutrino species, which is now known from particle physics experiments at LEP/CERN to be very close to Nν = 3. Equation (5.1.3) applies if the neutrinos are massless, which we shall assume to be the case in this chapter; the idea that they might have a mass of order mν 10 eV would have important implications for cosmology, as we shall discuss in Chapters 8 and 13.

If the neutrinos are massless, then their contribution to the density parameter is

Ω0ν Ω0r 10−5h−2.

From the point of view of the Friedmann models, the real Universe is well approximated as a dust or matter-dominated model, with total energy density

where T0m is the present temperature of the intergalactic gas (assumed to be hydrogen) and mp is the proton mass. This temperature is di erent from the temperature of the radiative component, T0r, because matter and radiation are completely decoupled from each other at the present epoch. In fact the neutrino component is also decoupled from the other two (matter and photons). Matter and radiation are decoupled because the characteristic timescale for collisions between photons and neutral hydrogen atoms, τ0c = mp/(ρ0mσHc), where σH is the scattering cross-section of a hydrogen atom, is much larger than the characteristic time for the expansion of the Universe: τH ≡ (a/a)˙ 0 = H0−1.

An important quantity is the ratio, η0, between the present mean numberdensity of nucleons (or baryons), n0b, and the corresponding quantity for photons, n0γ. The present density in baryons is

while the corresponding number for the photons is obtained by integrating over a Planck spectrum at a temperature of T0r = 2.73 K:

the quantity ζ(3) 1.202, where ζ is the Riemann zeta function which crops up in the integral over the black-body spectrum. We therefore have

we prefer to give the value η−0 1 rather than η0 because, as we shall see, η−0 1 practically coincides with the entropy per baryon, σ0r, which will figure prominently later on. The fact that η−0 1 is so large is of particular importance in the analysis of the standard model; we shall return to it later.

5.2 Recombination and Decoupling

During the period in which matter and radiation are decoupled, the matter temperature, Tm, and the radiation temperature, Tr, evolve independently of each other. If the gas component expands adiabatically, and is assumed to consist only of hydrogen, standard thermodynamics gives us

Given that ρma3 is constant, because of mass conservation, Equation (5.2.1) leads to

If σc, the collision cross-section between photons and atoms, is constant, then the collision time τc simply scales as the inverse of the number-density of atoms and therefore decreases with redshift much more rapidly than the characteristic timescale for the expansion τH: for example, in a flat universe,

112 Thermal History of the Hot Big Bang Model

where we have assumed matter domination to calculate τH; if the Universe were radiation dominated, this reasoning would still hold good. In fact, the crosssection for scattering of electrons by atoms does not behave as simply as this with z. The main mechanism by which photons interact with matter is Thomson scattering by electrons, but photons of su cient energy can also be absorbed by the atom, resulting in photo-ionisation. The ions thus produced may then recombine, with the usual cascades producing the Lyman and Balmer series. Photons of exactly the right wavelength can also cause upward transitions, leading to absorption lines. However, in the cosmological situation we are interested in, it su ces to take Thomson scattering by electrons as the dominant mechanism. As we shall see, as the photon energies increase to the energies relevant for the other processes mentioned here, the plasma becomes fully ionised and Thomson scattering is then indeed the dominant interaction between the matter and radiation. In any event, there clearly exists a time, say td, before which scattering occurs on a timescale much less than the expansion timescale, resulting in a tight coupling between matter and radiation. After td, a process of decoupling occurs and, for t td, matter and radiation e ectively evolve separately. As we shall see in Chapter 9, this process is not instantaneous and actually continues over a relatively large range of t (or z). Before decoupling, at t = td, matter and radiation are held in equilibrium with each other at the same temperature, and T varies with z in a manner intermediate between (5.2.2) and (5.2.4), which we can represent by Equation (5.3.3) below. At very high T (high z), the equilibrium state for the matter component has a very high state of ionisation. As T decreases, the fraction of atoms which are ionised (the degree of ionisation) falls. There exists therefore a time, say trec, before which the matter is fully ionised, and after which the ionisation is very small. This transition is usually called recombination, although it would be more accurate to call it simply combination. Recombination is also a relatively gradual process so it does not occur at a single definite t = trec. Notice, however, that in general td trec. We discuss recombination and decoupling in the context of realistic cosmological models in Section 5.4 and in Chapter 9.

5.3 Matter–Radiation Equivalence

Another important timescale in the thermal history of the Universe is that of matter–radiation equivalence, say t = teq, which we take to occur at zeq = z(teq). Remember that the matter density evolves according to

while the density of radiation follows

in the period after decoupling, and

114 Thermal History of the Hot Big Bang Model

which, unfortunately, cannot be integrated analytically, because σrad(T) depends on the unknown function T(a). It is easy to see that σrad(T) does not depend on a after decoupling if we interpret T as the temperature of the radiation. The value of σrad must therefore coincide with its present value σrad(t = t0), which can be calculated in terms of the present density of the Universe, ρ0m, and the present radiation temperature, T0r:

which is a very large number given the known bounds on the parameters Ω0b and h.

The Equation (5.4.4) is valid also at t = td. In a short interval of time at td, we can make use of the fact that σrad(t) σrad(td) = σrad(t0) 1, thus obtaining

which, upon integration, leads to Equation (5.2.4). This shows that we indeed expect H 0; it is virtually guaranteed by the very high actual value of σ0r.

At higher temperatures, the matter component also becomes relativistic and therefore assumes the equation of state p = 13 ρc2. In this regime the behaviour of T is very closely represented by Equation (5.2.4). The reason for this is as follows.

where p is a particle with mass m (for example an electron). In this situation the creation–annihilation reaction

has an equilibrium which lies to the right. A significant number of electron– positron (e+–e−) pairs are therefore created. At higher temperatures still, even more particle species might be created, of higher and higher masses.

The era contained between the two temperatures Te ( 5×109 K) and Tπ , where e and π are the electron and pion, respectively, is called the lepton era because, as besides the radiative fluid of photons and neutrinos, the background of leptons e+, e−, µ+, µ− and τ+ and τ− dominates the energy density. The brief interval with 200–300 MeV > kBT > Tπ 130 MeV is called the hadron era, because as well as photons, neutrinos and leptons, we now also have hadrons (π0, π+, π−, p, p,¯ n, n,¯ etc.); they do not, however, dominate the energy density. For kBT > 200– 300 MeV, the hadrons are separated into their component quarks. We shall discuss these phases in some detail in Chapter 8. There are so many relativistic particle species at such high energies, however, that for the moment it su ces to say that it is a good approximation to take the relativistic equation of state p = 13 ρc2 and ρc2 = AσT4 appropriate for pure radiation, which gives the Equation (5.2.4) exactly, but in which the constant A describes the fact that there are many di erent relativistic particles in addition to the photons.

116 Thermal History of the Hot Big Bang Model

of gamma rays than is observed.) In the epoch following TGUT 1015 GeV, which we will discuss in Chapter 7, we have

nb n¯ nγ T3, (5.5.8)

b

from which the baryon–antibaryon asymmetry is expected to be

The baryon–antibaryon asymmetry is very small, of the order of σrad−1, so that for every, say, 109 antibaryons there will be 109 + 1 baryons. The reason for this asymmetry, and why it is so small, is therefore the same as the reason why the value of σrad is large. Developments in the theory of elementary particles have led to some suggestions as to how cosmological baryosynthesis might occur; we shall discuss them in some detail in Chapter 7.

5.6 Timescales in the Standard Model

In the standard model, after the lepton era, the Friedmann Equation (1.12.6) becomes

where, as usual, the su x ‘0’ refers to the present epoch. The last bracket neglects contributions from relativistic particles which are small at the present time. Jumping the gun slightly (see Chapter 8 for details), we have replaced the purely radiation contribution Ωr by K0Ωr to take account of the contribution of light neutrinos to the relativistic part of the fluid; that is to say, the sum over i in Equation (5.3.4) now includes both photons and neutrinos. We shall see later, in Chapter 8, that

with Nν the number of types of light neutrino; K0 1.68 if Nν = 3. The second part of K0 derives from the neutrinos, and di ers from the photon contribution because they are fermions. The matter component is simply written Ω0 in Equation (5.6.1).

In light of Section 5.3, we can now calculate the equivalence redshift, zeq, at which ρm = K0ρr = ρeq. The result is