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

178The Lepton Era

5.The neutrinos have a negligible degeneracy parameter.

6.The Universe is not composed in such a way that some regions contain matter and others antimatter.

7.There is no appreciable magnetic field at the epoch of nucleosynthesis.

8.The density of any exotic particles (photinos, gravitinos, etc.) at Te is negligible compared with the density of the photons.

As we shall see, these hypotheses agree pretty well with such facts as we know. The hypothesis (3) is made because at the moment of nucleosynthesis, T 109 K (t 300 s), the mass of baryons contained within the horizon is very small, i.e. 103M , while the light-element abundances one measures seem to be the same over scales of order tens of Mpc; the hypotheses (4) and (8) are necessary because an increase in the density of the Universe at the epoch of nucleosynthesis would lead, as we shall see, to an excessive production of helium; the hypothesis (6) is made because the gamma rays which would be produced at the edges where such regions touch would result in extensive photodissociation of the 2H, and therefore a decrease in the production of 4He.

Later on, we shall discuss briefly some of the consequences on the nucleosynthesis process of relaxing or changing some of these assumptions.

8.6.3 The neutron–proton ratio

In Section 8.1 we stated that the ratio between the number densities of neutrons and protons is given by the relation

as long as the protons and neutrons are in thermal equilibrium. This equilibrium is maintained by the weak interactions

which occur on a characteristic timescale τcoll of order that given by Equation (8.4.2); this timescale is much smaller than τH for T Tdν 1010 K, i.e. until the time when the neutrinos decouple. At tdν the ratio

More accurate calculations (taking into account the only partial e ciency of the above reactions) lead one to the conclusion that the ratio Xn remains equal to the

equilibrium value until Tn 1.3 × 109 K (tn 20 s), after which the neutrons can only transform into protons via the β-decay, n → p + e− + ν¯e, which has a mean lifetime τn of order 900 s. After tn the ratio Xn thus varies according to the law of radioactive decay:

for t − tn t < τn; the value of Xn remains frozen at the value Xn(0) 0.17 for the entire period we are interested in. As we shall see, nucleosynthesis e ectively begins at t 102 s.

When the temperature is of order Tn, the relevant components of the Universe are photons, protons and neutrons in the ratio n/p exp(−1.5) 0.2, corresponding to the value Xn(0), and small amounts of heavier particles (besides the neutrinos which have already decoupled). The electrons and positrons annihilate at Te 5 × 109 K; the annihilation process is not very important for nucleosynthesis, it merely acts as a marker of the end of the lepton era and the beginning of the radiative era.

8.6.4Nucleosynthesis of Helium

To build nuclei with atomic weight A 3 one needs to have a certain amount of deuterium. The amount created is governed by the equation

one can easily verify that this reaction has a characteristic timescale τcoll τH in the period under consideration. The particles n, p, d and γ therefore have a number density given by the statistical equilibrium relations under the Boltzmann approximation:

with i = n, p, d and gn = gp = 2gd/3 = 2. For the chemical potentials we take the relationship already mentioned in Section 8.2, giving

It is perhaps a good time to stress that the chemical potentials of these particles are negligible when nn nn¯ and np np¯, but this is certainly not the case at the present epoch, because the thermal conditions are now very di erent.

It is useful to introduce, alongside Xn, another quantity Xp = p/ntot 1 − Xn and Xd = d/ntot. From Equations (8.6.7) and (8.6.8), one can derive the equilibrium relations between n, p and d:

180 The Lepton Era

which can be expressed as

where T9 = (T/109 K), Ω0b is the present density parameter in baryonic material. In (8.6.12), Bd is the binding energy of deuterium:

The function Xd depends only weakly on Ωh2.

For T9 10 the value of Xd is negligible: all the nucleons are still free because the high energy of the ambient photons favours the photodissociation reaction. The fact that nucleosynthesis cannot proceed until Xd grows su ciently large is usually called the deuterium bottleneck and is an important influence on the

eventual helium abundance. The value of Xd is no longer negligible when T9 1. At T9 0.9 for Ω = 1 (t 300 s) or at T9 0.8 for Ω 0.02 (t 200 s) Xd XnXp. For T < T9 the value of Xd becomes significant. At lower temperatures all

the neutrons might be expected to be captured to form deuterium. This deuterium does not appear, however, because reactions of the form

which have a large cross-section and are therefore very rapid, mop up any free neutrons into 4He. Thus, the abundance of helium that forms is

in reasonable accord with that given by observations. In Equation (8.6.13), the factor 2 takes account of the fact that, after helium synthesis, there are practically only free protons and helium nuclei, so that

The value of Y obtained is roughly independent of Ω. This is essentially due to two reasons:

1.the value of Xn before nucleosynthesis does not depend on Ω because it is determined by weak interactions between nucleons and leptons and not by strong interactions between nucleons; and

2.the start of nucleosynthesis is determined by the temperature rather than the density of the nucleons.

density of ordinary matter (baryons) (g cm−3)

Figure 8.1 Light-element abundance determined by numerical calculations as functions of the matter density, as explained in the text. The arrows mark the possible deuterium abundance. From Schramm and Turner (1996). Picture courtesy of Mike Turner.

8.6.5Other elements

As far as the abundances of other light elements are concerned one needs to perform a detailed numerical integration of all the rate equations describing the reaction network involved in building up heavier nuclei than 4He. We have no space to discuss the details of these calculations here, but the main results are illustrated in Figure 8.1.

The figure shows the computed abundance of 4He (usually denoted by YP), depending on the number of neutrino types. Note that some helium is certainly made in stars so that a correction must be made to the observed abundance Y

the primordial helium abundance more accurately than this rough figure, it is necessary to correct for the processing of hydrogen into helium in stars. This is generally done by taking account of the fact that stars with higher metallicity have a slightly higher 4He abundance, and extrapolating to zero metallicity; metals are assumed to be a byproduct of the fusion of hydrogen into helium. One therefore generally requires an index of metallicity in the form of either O/H or N/H determinations. Good data on these abundances have been obtained for around 50 extragalactic HII regions (Pagel et al. 1992; Skillman et al. 1993; Izotov et al. 1994). Olive and Steigman (1995) and Olive and Scully (1995), for example, have found on the basis of these data that there is evidence for a linear correlation of Y with O/H and N/H; the intercept of this relation yields

The first error is purely statistical and the second is an estimate of the systematic uncertainty in the abundance determinations.

8.6.7 Observations: Deuterium

The abundance of deuterium has been the subject of intense investigation in recent months. Prior to this period, deuterium abundance information was based on interstellar medium (ISM) observations and Solar System data. From the ISM, one gets

with an uncertainty of about 10% (Linsky et al. 1993, 1995). This value may or may not be close to universal, as it is possible that the abundances in the ISM are inhomogeneous. Solar System investigations based on properties of meteoritic rock involve a more circuitous route through 3He (which one assumes was e ciently burned into D during the pre-main-sequence phase of the Sun). This argument leads to a value of

with an uncertainty of nearly 100% (Scully et al. 1996).

More recently, the rough consensus between these two estimates was shaken by claims of detections of deuterium absorption in the spectra of high-redshift quasars. The occurrence of gas at high redshift and in systems of low metallicity suggests that one might well expect to see a light-element abundance close to the primordial value. The first such observations yielded much higher values than (8.6.18) and (8.6.19) by about a factor of 10 (Carswell et al. 1994; Songaila et al. 1994), i.e.

other measurements seemed to confirm these high values (Rugers and Hogan 1996a,b; Carswell et al. 1996; Wampler et al. 1996).

8.6.9Lithium 7

In old hot stars (Population II), the lithium abundance is found to be nearly uniform (Molaro et al. 1995; Spite et al. 1996). Indeed there appears to be little variation from star to star in a sample of 100 halo stars, over and above that expected from the statistical errors in the abundance determinations. The problem with the interpretation of such data, however, is in the fact that astrophysical processes can both create and destroy lithium. Up to about half the primordial 7Li abundance may have been destroyed in stellar processes, while it is estimated that up to 30% of the observed abundance might have been produced by cosmic ray collisions. The resulting best guess for the primordial abundance is

but the uncertainty, dominated by unknown parameters of the model used to process the primordial abundance, is at least 50% and is itself highly uncertain (Walker et al. 1993; Olive and Schramm 1992; Steigman et al. 1993).

8.6.10 Observations versus theory

We have tried to be realistic about the uncertainties in both the observations and the extrapolation of those observations back to the primordial abundances. Going into the detailed models of galactic chemical evolution that are required to handle D, 3He and 7Li opens up a rather large can of model-dependent worms, so we shall simply sketch out the general consensus about what these results mean for η and

Ωb.

The estimates of the primordial values of the relative abundances of deuterium (D), 3He, 4He and 7Li all appear to be in accord with nucleosynthesis predictions, but only if the density parameter in baryonic material is

(e.g. Walker et al. 1991; Smith et al. 1993). This roughly corresponds to 3 η10 4. A baryon density higher than this would produce too much 7Li, while a lower value would produce too much deuterium and 3He. Copi et al. (1995a,b) suggest a somewhat wider range of allowed systematic errors, leading to 2 η10 6.5, which translates into

The dependence of 4He is so weak that it can really only be used as a consistency check on the scheme.

This strong constraint on Ωb is the main argument for the existence of nonbaryonic dark matter, which we discuss in more detail in the second half of this book.

186 The Lepton Era

8.7 Non-standard Nucleosynthesis

We have seen that standard nucleosynthesis seems to account reasonably well for the observed light-element abundances and also places strong constraints on the allowed range of the density parameter. To what extent do these results rule out alternative models for nucleosynthesis, and what constraints can we place on models which violate the conditions (1)–(8) of the previous section? We shall make some comments on this question by describing some attempts that have been made to vary the conditions pertaining to the standard model.

First, one could change the expansion rate τH at the start of nucleosynthesis. A decrease of τH (i.e. a faster expansion rate) can be obtained if the Universe contains other types of particles in equilibrium at the epoch under consideration. These could include new types of neutrino, or supersymmetric particles like photinos and gravitinos: in general, τH t (g T4)−1/2. A small reduction of τH reduces the time available for the neutrons to decay into protons, so that the value of Xn tends to move towards its primordial value of Xn 0.5; the reduction of τH does not, however, influence the time of onset of nucleosynthesis to any great extent so that this still occurs at T 109 K. The net result is an increase in the amount of helium produced. As we have mentioned above, these results have for a long time led cosmologists to rule out the possibility than Nν might be larger than 4 or 5. Now we know that Nν = 3 from particle experiments; nucleosynthesis still rules out the existence of any other relativistic particle species at the appropriate epoch. A large reduction in τH, however, tends to reduce the abundance of helium: the reactions (8.6.12) have too little time to produce significant helium because the density of the Universe falls rapidly. A decrease in the expansion rate allows a larger number of neutrons to decay into protons so that the ratio Xn(T ) becomes smaller. Since basically all the neutrons end up in helium, the production of this element is decreased.

Another modification one can consider concerns the hypothesis that the neutrinos are not degenerate. If the chemical potential of νe is such that

(the upper limit was derived in Section 8.2), the obvious relation

(because at T 1010 K we have µe+ µe− 0 through the requirement of electrical neutrality) leads one to the conclusion that

for T Tdν 1010 K. For xνe 0 (degeneracy of the νe), the value of Xn(Tdν ) is much less than 0.5, so that one makes hardly any helium. If xνe 0 (degeneracy of

ν¯e), the high number of neutrons, because Xn(Tdν ) < 1, and the consequent low number of protons prevents the formation of deuterium and therefore helium. Deuterium would be formed much later when the expansion of the Universe had diluted the ν¯e and some neutrons could have decayed into protons. But at this point the density would be too low to permit significant nucleosynthesis, unless Ω 1. In the case when xνe −1, one can have Xn 0.5 at the moment of nucleosynthesis, so that all the neutrons end up in helium. This would mean that essentially all the baryonic matter in the Universe would be in the form of helium. In the case where the neutrinos or antineutrinos are degenerate there is another complication in the theory of nucleosynthesis: the total density of neutrinos and antineutrinos would be greater than one would think if there were such a degeneracy. For example, if |xνe | 1, we have

This fact gives rise to a decrease in the characteristic time for the expansion τH, with the corresponding consequences for nucleosynthesis. One can therefore conclude that the problems connected with a significant neutrino degeneracy are large, and one might be tempted to reject them on the grounds that models invoking such a degeneracy are also much more complicated than the standard model.

Even graver di culties face the idea of nucleosynthesis in a cold universe, i.e. a model in which the background radiation is not all of cosmological origin and in models where the universal expansion is not isotropic.

We should also mention that it has been suggested, and still is suggested by (the few remaining) advocates of the steady-state theory, that a radically alternative but possibly attractive model of nucleosynthesis might be one in which the light elements were formed in an initial highly luminous phase of galaxy formation or, perhaps, in primordial ‘stars’ of very high mass, the so-called Population III objects. The constraints on these models from observations of the infrared background are, however, severe.

Probably the best argument for non-standard nucleosynthesis is the suggestion that the standard model itself may be flawed. If the quark–hadron phase transition is a first-order transition, then, as the Universe cools, one would produce bubbles of the hadron phase inside the quark plasma. The transition proceeds only after the nucleation of these bubbles, and results in a very inhomogeneous distribution of hadrons with an almost uniform radiation background. In this situation, both protons and neutrons are strongly coupled to the radiation because of the e ciency of ‘charged-current’ interactions. These reactions, however, freeze out at T 1 MeV so that the neutrons can then diffuse while the protons remain locked to the radiation field. The result of all this is that the n/p ratio, which is one of the fundamental determinants of the 4He abundance, could vary substantially from place to place. In regions of relatively high proton density, every neutron will end up in a 4He nucleus. In neutron-rich regions, however, the neutrons have to undergo β-decay before