Lecture Notes for Astronomy 321, W 2004 1 Stellar Energy ...

Table 1: Properties of cosmic fluids
R(t) ρ eqn. of state
matter R ∝ t 2/3 ρ ∝ R −3 P = 2 3 ρc2 (v/c) 2
radiation R ∝ t 1/2 ρ ∝ R −4 P = 1 3 ρc2
vacuum R ∝ e αt ρ =const P = −ρc 2
The three types of cosmic fluids which we have input to our general
relativistic description of matter and energy are listed in Table 1, along with
a few of their important properties. The time-dependence t 2/3 for matter was
determined in the previous section.
Figure 15 shows the cooling of the constituents due to cosmic expansion
as a function of time, for the radiation and matter dominated eras. The
transition between the radiation and matter eras occurs at the decoupling
time, t dec ≈ 3 × 10 5 yr ≈ 10 13 s. Before this time, photons, electrons, and
protons were in thermal equilibrium, with
e − + p ↔ H .
At t dec , cooling has gone sufficiently below the ionization energy for hydrogen
(13.6 eV) that the reaction above goes primarily from left to right, thus
producing predominantly neutral matter which is “suddenly” transparent
to photons. These photons are what we currently observe as the cosmic
microwave background (CMBR) with a blackbody temperature of 2.7 K.
This cooling corresponds to a redshift in the wavelength of 1100. So the
decoupling time is:
t dec = 3 × 10 5 yr; (1 + z dec ) = 1100 (31)
This last relation implies that the size of the universe was about 10 3 smaller
at decoupling than today.
Finally, a useful relation is that between R and T :
R ∝ 1/T . (32)
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