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

188 The Lepton Era

they can begin to fuse. The net result is less 4He and more D than in the standard model, for the same value of η. The observed limits on cosmological abundances do not therefore imply such a strong upper limit on Ωb. It has even been suggested that such a mechanism may allow a critical density of baryons, Ωb = 1, to be compatible with observed elemental abundances. This idea is certainly interesting, but to find out whether it is correct one needs to perform a detailed numerical solution of the neutron transport and nucleosynthesis reactions, allowing for a strong spatial variation. In recent years, attempts have been made to perform such calculations but they have not been able to show convincingly that the standard model needs to be modified and the limits (8.6.25) weakened.

In conclusion we would like to suggest that, even if the standard model of nucleosynthesis is in accord with observations (which is quite remarkable, given the simplicity of the model), the constraints particularly on Ωb emerging from these calculations are so fundamental to so many things that one should always keep an open mind about alternative, non-standard models which, as far as we are aware, are not completely excluded by observations.

Bibliographic Notes on Chapter 8

Bernstein (1988) is a detailed monograph on relativistic statistical mechanics, which is also well covered by Kolb and Turner (1990). The physics of the quark– hadron transition is discussed by Applegate and Hogan (1985) and Bonometto and Pantano (1993).

For more extensive discussions of both theoretical and observational aspects of cosmological nucleosynthesis, see the technical review articles of Schramm and Wagoner (1979), Merchant Boesgaard and Steigman (1985), Bernstein et al. (1988), Walker et al. (1991) and Smith et al. (1993) and the book by Börner (1988). An important paper in the historical development of this field is Hoyle and Tayler (1964).

Problems

1.Cross-sections for weak interactions at an energy E increase with E as E2. Show that the rate of weak interactions in the early Universe depends on the temperature T as σwk T5. Using an appropriate model, estimate the temperature at which weak interactions freeze out in the Big Bang.

2.Let t1 be the epoch when electron–positron annihilation is completed and t2 be the epoch when helium fusion begins. You may assume that these two events take place at temperatures of 5 × 109 and 109 K, respectively. Assuming a simplified

model in which Λ = k = 0 and which is radiation dominated before teq = 3 × 105 years and matter dominated from teq until the present time (which you can take to be 1010 years), use the present temperature of the cosmic microwave background, 2.7 K, to infer values of t1 and t2.

3.If the abundance of neutrons, Xn, declines by beta decay in the interval between t1 and t2 (given in Question 2) according to

derive an estimate of Xn at the time helium fusion begins.

9

The Plasma Era

9.1 The Radiative Era

The radiative era begins at the moment of the annihilation of electron–positron pairs (e+–e−). This occurs, as we have explained, at a temperature Te 5 × 109 K, corresponding to a time te 10 s. After this event, the contents of the Universe are photons and neutrinos (which have already decoupled from the background and which in this chapter we shall assume to be massless) and matter (which we take to be essentially protons, electrons and helium nuclei after nucleosynthesis; the possible existence of non-baryonic dark matter is not relevant to the following considerations and we shall therefore use Ω0 to mean Ω0b throughout this chapter).

The density of photons and neutrinos (the relativistic particles) is

The end of the radiative era occurs when the density of matter coincides with that of the relativistic particles, corresponding to a redshift

192 The Plasma Era

At high temperatures both the hydrogen and helium are fully ionised, and exist in the form of ions (H+, He++). Gradually, as the temperature cools, the number of He+ ions and neutral H and He atoms grows according to the equilibrium reactions

H+ + e− H + γ, He++ + e− He+ + γ, He+ + e− He + γ, (9.1.5)

in which the density of the individual components is governed by the Saha equation which we saw in a di erent context in Section 8.6. We shall study in Section 9.3 in particular the equilibrium with regard to hydrogen recombination. It has been calculated that at T 104 K the helium content is 50% in the form He++ and 50% He+, while the hydrogen is 100% H+; at T 7 × 103 K one has 50% He+ and 50% He but still 100% H+; at T 4 × 103, corresponding to z 1500, one has 100% He, 50% H+ and 50% H. One usually takes the epoch of recombination to be that corresponding to a temperature of around Trec 4000 K when 50% of the matter is in the form of neutral atoms to a good approximation. Usually, in fact, one ignores the existence of helium during the period in which T > Trec; this period is usually called the plasma epoch.

9.2 The Plasma Epoch

The plasma we consider is composed of protons, electrons and photons at a temperature T > Trec. In this situation the plasma is an example of a ‘good plasma’, in the sense that the energy contributed by Coulomb interactions between the particles is much less than their thermal energy. This criterion is expressed by the inequality

in which ne is the number-density of ions from which one can obtain the mean separation

In these equations, and throughout this section, e is expressed in electrostatic units. In the cosmological case we find that

An equivalent way to express (9.2.1) is to assert that the number of ions ND inside a sphere of radius λD is large (‘screening’ e ects are negligible). One can show that

The Coulomb interaction between an electron and a proton is felt only while the electron traverses the Debye sphere of radius λD around an ion. The typical time taken to cross the Debye sphere is

where ωe is the plasma frequency. The time τe can be compared with the characteristic time for an electron to lose its momentum by electron–photon scattering

the result is that τe τeγ for z 2 × 107(Ω0h2)1/5, which is true for virtually the entire period in which we are interested here. The fact that τe τeγ means that collective plasma e ects are insignificant in this case, i.e. there is a very small probability of an electron–photon collision during the time of an electron–proton collision. On the other hand, for z 2 ×107(Ω0h2)1/5 electrons and photons are e ectively ‘glued’ together (τeγ τe in this period). One must therefore assign the electron an ‘e ective mass’ me = me + (ρr + pr/c2)/ne 43 ρr/ne me when describing an electron–proton collision. Returning to the case where z 2 × 107(Ω0h2)1/5, the electrons and protons are strongly coupled and e ectively stuck together; the characteristic time for electron–photon scattering is

which we refer to in Section 12.8. One should mention here that the factor 34 in Equations (9.2.7) and (9.2.8) comes from the fact that, as well as the inertia ρrc2 of the radiation, one must also include the pressure pr = ρrc2/3. Another timescale of interest is the timescale for photon–electron scattering; this is of order

The relaxation time for thermal equilibrium between the protons and electrons to be reached is

which is much smaller than the characteristic time for the expansion of the Universe during this period. One can therefore assume that protons and electrons have the same temperature. In the cosmological plasma, Compton scattering is the dominant form of interaction. In the absence of sources of heat, this scattering maintains the plasma in thermal equilibrium with the radiation. This is the basic reason why we expect to see a thermal black-body radiation spectrum. As we shall discuss in Section 9.5, energy injected into the plasma at a redshift z > zt 107–108 will be completely thermalised on a very short timescale. One

194 The Plasma Era

cannot therefore obtain information about energy sources at z > zt from the observed spectrum of the radiation. On the other hand, energy injected after zt may not be thermalised, and one might expect to see some signal of this injection in the spectrum of relic radiation.

9.3 Hydrogen Recombination

During the final stages of the plasma epoch, the particles p, e−, H and γ (ignoring the helium for simplicity) are coupled together via the reactions (9.1.5). Supposing that these reactions hold the particles in thermal equilibrium, we can study the process of hydrogen recombination, which marks the end of the plasma era and the beginning of the era of neutral matter. Let us concentrate on the ionisation fraction

Neutral hydrogen has a binding energy BH 13.6 eV (corresponding to a temperature TH 1.6 × 105 K). At a temperatures of the order of T 104 K all the particles involved are non-relativistic, and one can therefore apply simple Boltzmann statistics to the plasma. We therefore obtain the number-density of the ith particle species in the form

(cf. Section 8.6). The relevant chemical potentials are related by

the photons are in equilibrium and therefore have zero chemical potential. The statistical weights of the particles we are considering are gp = ge− = 12 gH = 2. The masses of the proton, the electron and the neutral hydrogen atoms are related by

From the preceding equations, noting that global charge neutrality requires ne = np, we obtain the relation

which is called the Saha formula corresponding to the hydrogen recombination reaction. In Table 9.1 we give some examples of the behaviour of the hydrogen ionisation fraction x as a function of redshift z and temperature T = T0r(1 + z) for various values of the density parameter in the form Ω0h2. As one can see from Table 9.1, the process of hydrogen recombination does not begin at TH because

Table 9.1 Ionisation fractions as function of z (or T) and Ω0h2.

of the relatively large numerical factor appearing in front of the exponential in Equation (9.3.5). The redshift at which the ionisation fraction falls to 0.5 does not vary much with the parameter Ω0h2 and is always contained in the interval 1400–1600. It is a good approximation therefore to assume a redshift zrec 1500 as characteristic of the recombination epoch.

The Saha formula is valid as long as thermal equilibrium holds. In an approximate way, one can say that this condition is true as long as the characteristic timescale for recombination τrec x/x˙ is much smaller than the timescale for the expansion of the Universe, τH. This latter condition is true for z > 2000(Ω0h2)−1, only when the ionisation fraction is still of order unity. It is possible therefore that physical processes acting out of thermal equilibrium could have significantly modified the cosmological ionisation history. For this reason, many authors have investigated non-equilibrium thermodynamical processes during the plasma epoch. These studies are much more complex than the quasi-equilibrium treatment we have described here, and to make any progress requires certain approximations. There is nevertheless a consensus that the value of x during recombination (z 1000) is probably a factor of order 100 greater than that predicted by the Saha Equation (9.3.5). In fact, in the interval 900 < z < 1500, the following approximate expression for x(z), due to Sunyaev and Zel’dovich, holds:

All calculations predict that the ionisation fraction tends to a value in the range 10−4–10−5 for z → 0. As we shall see in Chapter 19, the ionisation fraction of intergalactic matter at t = t0 is actually much higher than this, probably due to the injection of energy by early structure formation after zrec.

9.4 The Matter Era

The matter era begins at zeq. As we have already explained, assuming a value of zrec 1500, one concludes that zeq > zrec for Ω0h2 0.04. During the matter era the relations (9.1.1) and (9.1.2) are still valid for the radiation and matter densities, respectively, and the radiation temperature is given by Tr = T0r(1 + z). As

196 The Plasma Era

far as the matter temperature is concerned, this remains approximately equal to the radiation temperature until z 300, thanks to the residual ionisation which allows an exchange of energy between matter and radiation via Compton di usion. The characteristic timescale di ers by a factor 1/x from that given by Equation (9.2.9) due to the partial ionisation. The timescale τeγ can be compared with the characteristic time for the expansion of the Universe which, for zeq z Ω0−1, is given by

τH = 32 t0c(Ω0h2)−1/2(1 + z)−3/2 3.15 × 1017(Ω0h2)−1/2(1 + z)−3/2 s (9.4.1)

(cf. Equation (5.6.11)). One finds that τH < τeγ for z < 102(Ω0h2)5. After this redshift the thermal interaction between matter and radiation becomes insignificant, so that the matter component cools adiabatically with a law Tm (1 + z)2. The epoch zd 300 is the order of magnitude of the epoch of decoupling.

After decoupling, any primordial fluctuations in the matter component that survive the radiative era can grow and eventually give rise to cosmic structures: stars, galaxies and clusters of galaxies. The part of the gas that does not end up in such structures may be reheated and partly reionised by star and galaxy formation. This partial reionisation is called reheating, but should not be confused with the process of reheating which happens at the end of inflation.

An important consideration in the post-recombination epoch is the issue of the optical depth τ of the Universe due to Compton scattering. This is a dimensionless quantity such that exp(−τ) (often called the visibility) describes the attenuation of the photon flux as it traverses a certain length. The probability dP that a photon has su ered a scattering event from an electron while travelling a distance c dt is given by

I(t0, z) is the intensity of the background radiation reaching the observer at time t0 with a redshift z if it is incident on a region at a redshift z with intensity I[t(z)]; τ(z) is called the optical depth of such a region. The probability that a photon, which arrives at the observer at the present epoch, su ered its last scattering event between z and z − dz is

The quantity g(z) is called the di erential visibility or e ective width of the surface of last scattering; with a behaviour of the ionisation fraction given by (9.3.6) for z > 900 and a residual value x(z) 10−4–10−5 for z < 900, one finds that g(z)

is well approximated by a Gaussian with peak at zls 1100 and width ∆z 400, which corresponds to a (comoving) length scale of around 40h−1 Mpc or to an angular scale subtended on the last scattering surface of 10Ω01/2 arcmin. (Incidentally, at zrec the horizon is of order 200h−1 Mpc, which corresponds to an angular scale of around 2◦.) The value of zls is not very sensitive to variations in Ω0h2. The integral of g(z) over the range 0 z ∞ is clearly unity. At redshift zls we also have τ(z) 1. One usually takes the ‘surface’ of last scattering to be defined by the distance from the observer from which photons arrive with a redshift zls, due to the expansion of the Universe.

If there is a reionisation of the intergalactic gas, in the manner we have described above, at zreh < zrec, we can put x = 1 in the interval 0 z zreh and obtain, from Equations (2.4.16) and (9.4.2),

in this case τ(z) is unity at zls 20(Ω0h2)1/3, which is reasonably exact for acceptable values of Ω0h2. In conclusion, we can see that, if zreh > 20(Ω0h2)1/3, then the redshift of last scattering is given by zls 20(Ω0h2)−1/3; if, however, zreh < 2, the redshift of last scattering is of order 103 and we have a ‘standard’ ionisation history. In either case the study of the isotropy of the radiation background can give information on the state of the Universe only as far as regions at distances corresponding to zls.

9.5 Evolution of the CMB Spectrum

Assuming that radiation is held in thermal equilibrium at some temperature Ti, the intensity of the radiation (defined as power received per unit frequency per unit area per steradian) is given by a black-body spectrum:

One can easily show that in the course of an adiabatic expansion of the Universe, after all processes creating or absorbing photons have become insignificant, the form of the spectrum I(t, ν) remains the same with the replacement of Ti by

This can be understood because the number of photons per unit frequency in volume V a(t)3 is given by