The Thermal History of the Universe

1. Some facts about gravitation

We can use Newton's theory, because the relevant formulae turn out to be the same in Einstein's theory (general relativity) provided that space and time are properly defined.

a.
The mutual gravitational force between a sphere of mass M and an external sphere of mass m (say, the earth and an orange) is radial, attractive, and has magnitude GMm/r2, where r is the distance between the centers of the two spheres and G = 6.67 × 10-8 cm3/g sec2. The radial mass distribution within each sphere is arbitrary: for instance, M could be the mass of a spherical shell.

b.
On the other hand, a spherical mass distribution (such as a spherical mass shell) exerts no gravitational force on a body located inside it.

c.
The radial equation of motion m dr2/dt2 = - GMm/r2 has the first integral 

½mv2 - GMm/r = E,       (1)
where E is the total energy and v = dr/dt.

d.
If the spherical mass M is contained within a radius r such that the escape velocity at r exceeds the speed of light c, nothing can escape the gravitational pull of M, which is then called a black hole. Putting v = c in Eq. (2), we find that the critical radius R for black hole formation (Schwarzschild radius) is

R = 2GM/c2.       (3)

2. Some facts about cosmology

By visible universe, we mean the region of space within a sphere, centered on earth, of radius 10 billion light years or so (l ly is about 1015 km). Within the visible universe now, luminous matter is mostly organized in galaxies. Our galaxy, which is about average, contains some 10 billion stars and has a diameter of 100 thousand ly. The average distance between galaxies is about 5 million ly. Thus, there are only about 10 billion galaxies in the visible universe, and only two thousand or so on a radial line to the edge of the visible universe. Prior to 1980, it was generally assumed that the visible universe encompasses a large fraction of the entire universe; today's cosmologies envision grander scenarios.

a.
Observation shows that the visible universe is expanding. On the average, the relation between distance r and relative speed v for any two galaxies is v = Hr, where H is Hubble's constant. The inverse of H has the following meaning: if the same velocities (not the same H) prevailed in the past, then all galaxies were together at time t = l/H before the present. The observed value of (1/H) is about 18 billion years.

b.
The equation of motion for an exploding homogeneous universe is obtained as follows. Draw an arbitrary sphere of radius r with an arbitrarily chosen center, and consider a "marker" galaxy on the surface of this sphere. It is attracted to the center of the sphere by the mass M that is inside the sphere. (The outside mass gives zero force, see lb.) The equation of motion, as in 1c, is 

d2r/dt2 = - GM/r2.       (4)

c.
At present, the dominant contribution to M comes from nucleons and electrons, which are indestructible. Thus, M is constant. The first integral of eq. (4) is then given by eq. (1). The future of the universe (continued expansion or recontraction) depends on the sign of E, as discussed in lc. Using v = Hr and M = 4prr3/3, where r is the mass density, we see that E is positive if H2 > 8prG/3. The "visible" mass density (r = 2·10 -31 g/cm3) is such that 3H 2/8Gpr = 10, approximately, and therefore E > 0; but the mass density of "invisible" matter (dust, spent stars) could be enough to reverse the sign of E. We take E = 0 for simplicity.

d.
We are then led to the "escape velocity" condition, eq. (2), which can be written as dr/dt = (2GM/r)1/2 and integrated to give 

r3/2 = 3 GMt       (5)
This is valid for t > 3·106 years (see 4a below). Thus, it is valid at the present time, t = t0, when r = r0. Eliminating M, we have 
r/r0 = (t/t0)2/3       (6)

e.
Hubble's law, v = Hr, can be rewritten as H = v/r = dln r /dt. Using eq. (6), H = (2/3)t. The present value of the "Hubble time", 1/H0 = 18 billion years, corresponds then to a "true age" t0 = (2/3)H0 = 12 billion years.

3. The adiabatic expansion

It is important to realize that the cosmic expansion, although by no means slow, is ideally adiabatic in the following sense. When a volume is expanded in the laboratory at a finite rate (by lifting a piston, for instance), irreversibility is introduced by the fact that the added volume is localized near the moving walls (near the piston, for instance) and is filled by an irreversible flow of matter. In the universe, however, space is expanding everywhere and matter-energy is carried along by the expansion.

a.
If the universe is filled by a homogeneous one-component fluid, no entropy is generated by the expansion, which is then reversible. New entropy can be generated by processes analogous to the nucleation of liquid droplets in an expanding gas, or to the precipitation of compounds from a solution. Currently, entropy is generated mostly by star formation and by the "cooking" of the heavy elements in stars out of the "primordial" hydrogen and helium. Starlight carries away the "latent heat" of star formation and evolution. This heat, however, is not sufficient to compensate for the adiabatic cooling due to the expansion. It seems obvious then, that the universe was hotter and more homogeneous in the past.

b.
A simple calculation can be done by neglecting stars altogether. We have then a homogeneous mixture of H and He (27% He by weight) in thermal equilibrium (why in thermal equilibrium? Because this is the most likely distribution). The adiabatic expansion of this monatomic gas obeys TV 2/3 = constant, or Tr2 = constant. Combining this with eq. (6), we obtain T/T0 = (t0/t)4/3, where T0 is the present temperature of the gas that has not been swept up to form stars.

4. The blackbody radiation.

Clearly, T was much higher in the past. When T was above 3000 K, hydrogen was ionized and radiation was in thermal equilibrium with matter. Amazingly, we can compute rather accurately the time when radiation decoupled from matter (see b below): it is half a million years. Since then, radiation and matter have undergone separately the process of adiabatic expansion, and are now at different temperatures.

a.
What is the present radiation temperature? We know that the entropy density of radiation is proportional to T 3 and that the entropy of the radiation contained in an expanding sphere of radius r has not changed since decoupling occurred. Thus the radiation temperature T is such that T 3r3 is constant, or T/T0 = r0/r. We can use eq. (6) to compute the expansion ratio r0/rd, where rd was the radius at the time of radiation decoupling (td = 5·105 y). We can do so because the matter density has been dominant from the time of radiation decoupling to the present (t0 = 12 Gy). We obtain r0/rd = (t0/td)2/3 = 1000, and therefore T0 = 10-3·3000 K = 3 K, in agreement with observation.

b.
It turns out, coincidentally, that the decoupling time td is close to the time when the radiation energy equaled the rest mass-energy of matter. Therefore for t < td we can treat radiation as the dominant source of gravitation in the universe. We can still use eq. (2), but now M is given by c2 times the radiation energy density, times the volume: 

M = (4csT 4) (4pr3/3),
where s is the Stefan-Boltzmann constant. Inserting in eq. (2) we have 
dr/dt = (32psGc/3)1/2 T 2r       (7)
From Tr = constant, we find dr = -(r/T) dT and we can rewrite eq. (7) as 
- dT/dt = (32psGc/3)1/2 T 3.
An easy integration gives an explicit relation for the cosmic temperature during the first half million years: 
1/(2T 2) = (32psGc/3)1/2 t.       (8)

Putting in the numbers (only fundamental constants!) we find that the decoupling temperature 3000 K (ionization of H) corresponds to td = 0.67 My, which is close enough to 1/2 My (we left out the neutrinos, etc.).

5. The Big Bang.

According to eq. (8), the temperature of the universe was infinite at t = 0. Actually, at large T other phenomena intervene. For t < 3 minutes, the temperature is so high (T > 109 K), that He4 is no longer stable and matter consists mostly of protons, electrons and neutrons. From a knowledge of nuclear cross-sections, we can actually predict how much He4 and other light elements were formed at this stage. Neutrinos were also in thermal equilibrium, and in fact they are still around, adiabatically cooled like the photons.
At about t = 1 minute, T   was high enough to create electron-positron pairs, and before t = 10-4 sec there were plenty of proton-antiproton pairs, and all kinds of other particles. The equation of chemical equilibrium can be applied to this expanding mixture. At this stage, there were about as many protons and antiprotons as photons. Essentially all the photons have survived (redshifted to 3 K), but the particle-antiparticle pairs have disappeared. For mysterious reasons, there was a small excess of nucleons over antinucleons (about 1 in 109). These residual nucleons dominate the mass-energy of the universe today, but by far most of the entropy (by a factor 109) is in the photons of the blackbody background radiation and in the neutrinos. The entropy produced by stars, planets, and humans is insignificant by comparison.

 

Fig. 10.17 Eras in the evolution of the universe. (After Harrison, Annual Review of Astronomy and Astrophysics. 1973.)