118 Thermal History of the Universe
The present radiation energy density is predominantly in the form of microwaves
and infrared light. The change from radiation domination to matter domination is
gradual: at푡=1000yr the radiation fraction was about 90%, at푡=2Myr only about
10% (see Figure 7.3).
Relativistic versus Nonrelativistic Particles. It is important to distinguish
between relativistic and nonrelativistic particles because their energy spectra in ther-
mal equilibrium are different. A coarse rule is that a particle is nonrelativistic when its
kinetic energy is small in comparison with its mass, and relativistic when퐸≳ 10 푚푐^2.
The masses of some cosmologically important particles are given in Table A.4. For
comparison, the equivalent temperatures are also given. This gives a rough idea of
the temperature of the heat bath when the respective particle is nonrelativistic.
The adiabaticity condition푆̇=0 can be applied to both relativistic and nonrela-
tivistic particles. Let us first consider the relativistic particles which dominate the
radiation era. Recall from Equation (3.9) that the energy of a particle depends on two
terms, mass and kinetic energy,
퐸=√
푚^2 푐^4 +푃^2 푐^2 , (6.18)
where푃is momentum. For massless particles such as the photons, the mass term is
of course absent; for relativistic particles it can be neglected.
When푝is a constant we can write the law of energy conservation
d퐸=−푝d푉. (6.19)
Replacing퐸by the energy density휀rtimes the volume푎^3 ≡푉, the above law becomes
d(푎^3 휀r)=−푝d(푎^3 ). (6.20)Substituting휀rfor the pressure푝from the equation of state Equation (6.15) we obtain
푎^3 d휀r+휀rd푎^3 =−^1
3휀rd푎^3 ,or
d휀r
휀r=−
4
3
d푎^3
푎^3. (6.21)
The solution to this equation is
휀r∝푎−^4 , (6.22)in agreement with our previous finding. We have in fact already used this result in
Equation (5.33).
For nonrelativistic particles the situation is different. Their kinetic energy휀kinis
small, so that the mass term in Equation (6.18) can no longer be neglected. The motion
of푛particles per unit volume is then characterized by a temperature푇m,causing
a pressure
푝=nk푇m. (6.23)