Introduction to Cosmology

The Primordial Hot Plasma 119

Note that푇mis not the temperature of matter in thermal equilibrium, but rather a
bookkeeping device needed for dimensional reasons. The equation of state differs
from that of radiation and relativistic matter, Equation (6.15), by a factor of 2:

푝=^2

3

휀kin.

Including the mass term of the푛particles, the energy density of nonrelativistic matter
becomes

휌m≡휀m=nm푐^2 +

3

2

nk푇m. (6.24)

Substituting Equations (6.23) and (6.24) into Equation (6.20) we obtain

d(푎^3 nm푐^2 )+^3

2

d(푎^3 nk푇m)=−nk푇md푎^3. (6.25)

Let us assume that the total number of particles always remains the same: in a
scattering reaction there are then always two particles coming in, and two going out,
whatever their types. This is not strictly true because there also exist other types of
reactions producing more than two particles in the final state. However, let us assume
that the total number of particles in the volume푉under consideration is푁=Vn,and
that푁is constant during the adiabatic expansion,

d푁=d(Vn)=

4

3

휋d(푎^3 푛)= 0. (6.26)

The first term in Equation (6.25) then vanishes and we are left with

3

2

푎^3 d푇m=−푇md(푎^3 ),

or

3

2

d푇m

푇m

=−

d(푎^3 )

푎^3

.

The solution to this differential equation is of the form

푇m∝푎−^2. (6.27)

Thus we see that the temperature of nonrelativistic matter has a different dependence
on the scale of expansion than does the temperature of radiation. This has profound
implications for one of the most serious problems in thermodynamics in the nine-
teenth century.
The number density of relativistic particles other than photons is given by distri-
butions very similar to the Planck distribution. Let us replace the photon energyℎ휈in
Equation (6.10) by퐸, which is given by the relativistic expression in Equation (6.18).
Noting that the kinematic variable is now the three-momentum푝=|p|(since for rel-
ativistic particles we can ignore the mass), we can replace Planck’s distribution by the
number density of particle species푖with momentum between푝and푝+d푝,

푛푖(푝)d푝=^8 휋

ℎ^3

푛spin,푖

2

푝^2 d푝

e퐸푖(푝)∕푘푇푖± 1

. (6.28)