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

168 The Lepton Era

temperature was around T 1015 GeV. These baryons have a number density n given by the Boltzmann distribution:

where the su xes ‘n’ and ‘p’ denote neutrons and protons, respectively. In Equation (8.1.1) we have neglected the chemical potential of the protons and neutrons µp(n); we shall return to this matter in Section 8.2. From Equation (8.1.1) one finds that the ratio between the numbers of protons and neutrons is

is the di erence in rest-mass energy between ‘n’ and ‘p’, corresponding to a temperature Tpn ≡ Q/kB 1.5 × 1010 K. For T Tpn, the number of protons is virtually identical to the number of neutrons.

8.2 Chemical Potentials

Throughout this chapter we shall need to keep track of the e ective number of particle species which are relativistic at temperature T. This is done through the quantity g (T), the number of degrees of freedom as a function of temperature. We need to consider thermodynamic aspects of the particle interactions in order to make progress. In particular we need to consider the chemical potentials µ relevant to the di erent particle species. Recall that the chemical potential, roughly speaking, defines the way in which the internal energy of a system changes as the number of particles is changed.

In the case of an ideal gas the chemical potential µi for the ith particle type (which we assume to have statistical weight gi) a ects the equilibrium number density ni according to

where the ‘+’ sign applies to fermions, and the ‘−’ sign to bosons. The existence of a non-zero chemical potential signifies the existence of degeneracy. It is a basic tenet of the theory of statistical mechanics that one conserves the chemical potentials of ingoing and outcoming particles during a reaction when the reaction is in equilibrium; also, the chemical potential of photons is zero.

In what follows we shall assume that the appropriate chemical potentials describing the thermodynamics of the particle interactions are zero. It is necessary to make some remarks to justify this assumption. As we shall see, the

170 The Lepton Era

which is symmetrical about the origin and in which the dimensionless quantities xi = µi/kBT are called the degeneracy parameters, Equation (8.2.6) becomes

The conservation of B, valid from the epoch we are considering until the present time, yields

Introducing the radiation entropy per baryon σrad we discussed in Chapter 5, this becomes

because the high value of σrad means that T0ra0 Ta. This relation is therefore equivalent to

As far as Le and Lµ are concerned, we shall assume that the density of the appropriate lepton numbers are very small, as is the baryon number density. We shall justify this approximation for the leptons only partially, and in an a posteriori manner, when we look at nucleosynthesis. The assumption is nevertheless quite strongly motivated in the framework of GUT theories in which one might expect the lepton and baryon asymmetries to be similar. In analogy with Equation (8.2.11) we therefore have

where the factor 12 comes from the relation gµ = ge = 2gν = 2. From Equa-

tion (8.2.3) and from the relation µi = −µi¯ we have

which, with Equations (8.2.9)–(8.2.12), furnishes a set of six equations for the six

unknowns xp, xn, xe+ , xµ+ , xν¯e , xνµ . If this system has a solution xi (i = p, n, e+, µ+, ν¯e, νµ), then it also admits the symmetric solution −xi . To have physical

significance, however, the solution must be unique; this means that xi = 0. The six chemical potentials we have mentioned and, therefore, the others related to them by symmetry, are all zero.

Before ending this discussion it is appropriate to underline again the fact that the hypothesis that we can neglect the lepton number density with respect to nγ is only partially justified by the observations of cosmic abundances which the standard nucleosynthesis model predicts and which we discuss later in this

chapter. The greatest justification for this hypothesis is actually the enormous simplification one achieves by using it, as well as a theoretical predisposition towards vanishing Le and Lµ (as with B) on grounds of symmetry, particularly in the framework of GUTs. One can, however, obtain a firm upper limit on the chemical potential of the cosmic neutrino background from the condition that the global value of Ω0 cannot be greater than a few. Assuming that there are only three neutrino flavours, and that neutrinos are massless, one can derive the following constraint:

This limit corresponds to a present value of the degeneracy parameter which is much greater than we suggested above: if the µνi,0 are all equal, and if T0νi 2 K (as we will find later), this limit corresponds to a degeneracy parameter of the order of 40.

8.3 The Lepton Era

The lepton era lasts from the time the pions either annihilate or decay into photons, i.e. from Tπ 130 MeV 1012 K, to the time in which the e+ − e− pairs annihilate at a temperature Te 5 × 109 K 0.5 MeV. At the beginning of the lepton era the Universe comprises photons, a small number of baryons and the leptons e−, e+, µ+, µ− (and probably τ+ and τ−), with their respective neutrinos. If the τ particles are much more massive than muons, then they will already have annihilated by this epoch, but the corresponding neutrinos will remain. Neglecting the (non-relativistic) baryon component, the number of degrees of freedom at the start of the lepton era is g (T < Tπ ) = 4 × 2 × 78 + Nν × 2 × 78 + 2 14.25 (if the number of neutrino types is Nν = 3), corresponding to a cosmological time tπ 10−5 s. We will study the Universe during the lepton era under the hypothesis which we have just discussed in the previous section, namely that all the relevant chemical potentials are zero.

At the start of the lepton era, all the constituent particles mentioned above are still in thermal equilibrium because the relevant collision time τcoll is much smaller than τH, the Hubble time. For example, at T 1011 K (t 10−4 s) the collision time between photons and electrons is τcoll (σTnec)−1 10−21 s. The same can be said for the neutrinos for T > 1010 K, which is the temperature at which they decouple from the rest of the Universe as we shall show.

Other important facts during the lepton era are the annihilation of muons at Tµ < 1012 K, which happens early on, the annihilation of the electron–positron pairs, which happens at the end, and cosmological nucleosynthesis, which begins at around T 109 K, at the beginning of the radiative era. Because the conditions for nucleosynthesis are prepared during the lepton era, we shall cover nucleosynthesis in this chapter, rather than in the next.

During the evolution of the Universe we assume that entropy is conserved for components still in thermal equilibrium. This hypothesis is justified by the slow

where nl is the number density of a generic lepton, given by

neutrino decoupling is then at Tdν 3 × 1010 K. It is noteworthy that in any case the decoupling of the neutrinos happens after the annihilation of the µ+–µ− pairs and before the annihilation of the e+–e− pairs: this is important for calculating the properties of the cosmic neutrino background, as we show in the next section.

8.5 The Cosmic Neutrino Background

At the time of their decoupling, the temperature of the neutrinos coincides with the temperature T of the other constituents of the Universe which are still in thermal equilibrium: e+, e− and γ. The neutrino ‘gas’ then expands adiabatically because no other component is in thermal contact with it: for such a gas one can assume an equation of state appropriate for radiative matter and one therefore finds the relation

Until the moment of e+–e− annihilation, the ‘gas’ composed of e−, e+ and γ also follows a law identical to Equation (8.5.1). The temperature T su ers an increase at the moment of pair annihilation, as was explained in Section 8.3. Applying Equation (8.3.3) one finds that at Te 5 × 109 K the temperature T (which now is just Tr) becomes

because for T > Te one has g(−) = 112 , while for T < Te we have g(+) = 2 (just photons). After pair annihilation the photon gas expands adiabatically and, for high values of σ0r, we get

One thus finds that the temperature of the radiation background remains a factor of (11/4)1/3 higher than the temperature of the neutrino background. One therefore finds

174 The Lepton Era

corresponding to a number density

As we have explained, the number of neutrino species is probably Nν = 3; considerations based on cosmological nucleosynthesis have for some time ruled out the possibility that Nν > 4–5. In the case Nν = 3, where we have νe, νµ and ντ along with their respective antineutrinos, we get n0γ n0ν and ρ0γ ρ0ν . We stress again that all these results are obtained under the assumption that the neutrinos are not degenerate and that they are massless.

Let us now discuss what happens to the cosmic neutrino background if the neutrinos have a mean mass of order 10 eV, parametrised by mν = Ni=ν1 mνi /Nν . After decoupling, the number of neutrinos in a comoving volume does not change so that Equation (8.5.5) is still valid; this is due to the fact that for T Tdν the neutrinos are still ultrarelativistic, so that the above considerations are still valid. We therefore obtain

corresponding to a density parameter

the Universe would be dominated by neutrinos.

In the case of massive neutrinos, the quantity T0ν is not so much a physical temperature, but more a kind of ‘counter’ for the number of particles; we shall come back to this shortly. The distribution function for neutrinos (number of particles per unit volume in a unit range of momentum) fν before the time tdν (which we suppose, for simplicity, is the same for all types) is the relativistic one because Tdν mν c2/3kB = Tnν 1.3 × 105(mν /10 eV) K (the epoch in which T Tnν indicates the passage from the era when the neutrinos are relativistic to the era when they are no longer relativistic; in the above approximation this happens a little before equivalence). We therefore obtain

where pν is the neutrino momentum. After decoupling, because the neutrinos undergo a free expansion, one has pν a−1 and the neutrino distribution is still described by Equation (8.5.11) if one uses the counter

Notice that the ‘temperature’ varies as a−1 for the neutrinos, just as it does for radiation. As we mentioned above, this is not really a true physical temperature because the neutrinos are no longer relativistic at low redshifts, though their ‘temperature’ still varies in the same way as radiation. On the other hand, initially cold (non-relativistic) particles would have T a−2 in this regime due to the adiabatic expansion.

The energy density of neutrinos for T < Tdν is given by

for Tν Tnν .

Recent experimental measurements, such as those from SuperKamiokande (Fukuda et al. 1999) suggest that at least one of the neutrino flavours must have a non-zero mass. The physics behind these measurements stems from the realisation that the energy (or mass) eigenstates of the neutrinos might not coincide with the states of pure lepton number; a similar phenomenon called Cabibbo mixing occurs with quarks. To illustrate, let us consider only the electron neutrino νe and the muon version νµ. These are the lepton states with Le = 1 and Lµ = 1, respectively. In general one might imagine that these are combinations of the

where θ is a mixing angle. That means that a state of pure electron neutrino is a superposition of the ν1 and ν2 states:

If the eigenvalues of the two energy eigenstates are E1 and E2, respectively, then the state will evolve according to

176 The Lepton Era

It then follows that the probability of finding a pure electron neutrino state a time t after it is set up is

hence the term neutrino oscillation: the particle precesses between electron– neutrino and mu–neutrino states. If both states have the same momentum, then the energy di erence is just

where E = (E1 + E2)/2. This then leads to a neat alternative form to (8.5.18),

for a beam of electron neutrinos travelling a distance R. The quantity L is the oscillation length

which gives the typical scale of the oscillations. Note that oscillations do not occur if the two neutrinos have equal mass. The mixing length (8.5.21) is typically very large, so the best experiments involve solar neutrinos (produced by nuclear reactions in the Sun’s core) or atmospheric neutrinos (produced by cosmic ray collisions in the atmosphere). Recent results agree on a positive detection, but there is some uncertainty in the neutrino masses that can be involved and also whether all three neutrino species (including the tau) can be massive. It seems unlikely, however, that the neutrinos have masses around 10 eV, which is the mass they would have to have in order to contribute significantly to the critical density.

8.6 Cosmological Nucleosynthesis

8.6.1 General considerations

We begin our treatment of cosmological nucleosynthesis in the framework of the Big Bang model with some definitions and orders of magnitude. We define the abundance by mass of a certain type of nucleus to be the ratio of the mass contained in such nuclei to the total mass of baryonic matter contained in a suitably large volume. The abundance of 4He, usually indicated with the symbol Y, has a value Y 0.25, obtained from various observations (stellar spectra, cosmic rays, globular clusters, solar prominences, etc.) or about 6% of all nuclei. The abundance of 3He corresponds to about 10−3Y, while that of deuterium D (2H or, later on, d), is of order 2 × 10−2Y.

In the standard cosmological model the nucleosynthesis of the light elements (that is, elements with nuclei no more massive than 7Li) begins at the start of the

radiative era. Nucleosynthesis of the elements of course occurs in stellar interiors, during the course of stellar evolution. Stellar processes, however, generally involve destruction of 2H more quickly than it is produced, because of the very large crosssection for photodissociation reactions of the form

Nuclei heavier than 7Li are essentially only made in stars. In fact there are no stable nuclei with atomic weight 5 or 8 so it is di cult to construct elements heavier than helium by means of p + α and α + α collisions (α represents a 4He nucleus). In stars, however, α + α collisions do produce small quantities of unstable 8Be, from which one can make 12C by 8Be +α collisions; a chain of synthesis reactions can therefore develop leading to heavier elements. In the cosmological context, at the temperature of 109 K characteristic of the onset of nucleosynthesis, the density of the Universe is too low to permit the synthesis of significant amounts of 12C from 8Be + α collisions. It turns out therefore that the elements heavier than 4He are made mostly in stellar interiors. On the other hand, the percentage of helium observed is too high to be explained by the usual predictions of stellar evolution. For example, if our Galaxy maintained a constant luminosity for the order of 1010 years, the total energy radiated would correspond to the fusion of 1% of the original nucleons, in contrast to the 6% which is observed.

It is interesting to note that the di culty in explaining the nucleosynthesis of helium by stellar processes alone was recognised by Gamow (1946) and by Alpher et al. (1948), who themselves proposed a model of cosmological nucleosynthesis. Di culties with this model, in particular an excessive production of helium, persuaded Alpher and Herman (1948) to consider the idea that there might have been a significant radiation background at the epoch of nucleosynthesis; they estimated that this background should have a present temperature of around 5 K, not far from the value it is now known to have (T0r 2.73 K), although some 15 years were to pass before this background was discovered. For this reason one can safely say that the satisfactory calculations of primordial element abundances which emerge from the theory represent, along with the existence of the cosmic microwave background, one of the central pillars upon which the Big Bang model is based.

8.6.2The standard nucleosynthesis model

The hypotheses usually made to explain the cosmological origin of the light elements are as follows.

1.The Universe has passed through a hot phase with T 1012 K, during which its components were in thermal equilibrium.

2.General Relativity and known laws of particle physics apply at this time.

3.The Universe is homogeneous and isotropic at the time of nucleosynthesis.

4.The number of neutrino types is not high (in fact we shall assume Nν 3).