The Primordial Hot Plasma 117adiabatically is certainly very good during the radiation-dominated era when the fluid
was composed of photons and elementary particles in thermal equilibrium. However,
the Universe, black holes and for instance voids are generalized thermodynamic sys-
tems for which the equations of thermodynamics must be written in a complete form
involving a factor due to the expansion of the Universe and a redefinition of entropy.
This is also true during the matter-dominated era before matter clouds start to
contract into galaxies under the influence of gravity. Even on a very large scale we
may consider the galaxies forming a homogeneous ‘fluid’, an idealization as good as
the cosmological principle that forms the basis of all our discussions. In fact, we have
already relied on this assumption in the derivation of the Einstein equation and in
the discussion of equations of state. However, the pressure in the ‘fluid’ of galaxies of
density푁is negligibly small, because it is caused by their random motion, just as the
pressure in a gas is due to the random motion of the molecules. Since the average
peculiar velocities⟨푣⟩of the galaxies are of the order of 10−^3 푐, the ratio of pressure
푝=푚⟨푣⟩^2 푁to matter density휌gives an equation of state (Problem 5) of the order of
푤≈푚⟨푣⟩^2 푁
휌푐^2
=
⟨푣⟩^2
푐^2
≈ 10 −^6.
We have already relied on this value in the case of a matter-dominated universe when
deriving Equation (5.30).
Energy Density. Let us compare the energy densities of radiation and matter. The
energy density of electromagnetic radiation corresponding to one photon in a vol-
ume푉is
휌r푐^2 ≡휀r=ℎ휈
푉= hc
푉휆. (6.14)
In an expanding universe with cosmic scale factor푎, all distances scale as푎and so
does the wavelength휆. The volume푉then scales as푎^3 ; thus휀rscales as푎−^4 .Hereand
in the following the subscript ‘r’ stands for radiation and relativistic particles, while
‘m’ stands for nonrelativistic (cold) matter.
Statistical mechanics tells us that the pressure in a nonviscous fluid is related to
the energy density by the equation of state [Equation (5.32)]
푝=1
3
휀, (6.15)
where the factor^13 comes from averaging over the three spatial directions. Thus pres-
sure also scales as푎−^4 , so that it will become even more negligible in the future than
it is now. The energy density of matter,
휌m푐^2 =푚푐^2
푉
, (6.16)
also decreases with time, but only with the power푎−^3. Thus the ratio of radiation
energy to matter scales as푎−^1 :
휀r
휌m