Physical Foundations of Cosmology

82 The hot universe

Note that massless particles (m=0) always have an ultra-relativistic equation of
state,

p=

ε

3

, (3.30)

independent of their spin and chemical potential.

Problem 3.7Substituting (3.20) into (3.16) and (3.22) into (3.21), verify that the
entropy densityis

s=

ε+p−μn

T

. (3.31)

(Hint Prove and then use the relation
p
T

=±

∑

gln( 1 ±n), (3.32)

where the plus and minus signs apply to bosons and fermions respectively. It follows
that forn1wehavepnT.)

Verify the following useful relations

n=

∂p

∂μ

, s=

∂p

∂T

(3.33)

The above integrals over energy cannot be calculated exactly when both the mass
and chemical potential are different from zero. Therefore, we consider the limits
of high and low temperature and expand the integral in terms of small parameters.
At temperatures much larger than the mass the calculation of the leading terms can
be performed by simply neglecting the mass. However, it is not so easy to derive
the subleading corrections. The problem is that these corrections are nonanalytic
in both the mass and the chemical potential. Because the corresponding results are
not readily available in the literature, we provide below a derivation of the high-
temperature expansion. The reader who is not interested in these mathematical
details can skip the next subsection and go directly to the final formulae.

3.3.3 Calculating integrals

Changing the integration variable in (3.27), (3.28) and (3.29) fromtox=/T
and taking into account the fact that the chemical potentials of particles and an-
tiparticles are equal in magnitudes and have opposite signs, the calculation of the
basic thermodynamical quantities reduces to computing the integrals

J∓(ν)(α, β)≡

∫∞

α

(x^2 −α^2 )ν/^2

ex−β∓ 1

dx+

∫∞

α

(x^2 −α^2 )ν/^2

ex+β∓ 1

dx, (3.34)