Physical Foundations of Cosmology

3.3 Rudiments of thermodynamics 83

where

α≡

m

T

,β≡

μ

T

.

In particular, the total energy density of particles(p)and antiparticles( ̄p)is equal
to

ε≡εp+ε ̄p=

gT^4

2 π^2

(

J∓(^3 )+α^2 J∓(^1 )

)

, (3.35)

and the total pressure is

p≡pp+p ̄p=

gT^4

6 π^2

J∓(^3 ). (3.36)

Problem 3.8Verify that the excess of particles over antiparticles is given by

np−n ̄p=

gT^3

6 π^2

∂J∓(^3 )

∂β

. (3.37)

To find the expansions for the integralsJ∓(^1 )andJ∓(^3 )in the limits of high and low
temperatures, we first calculate the auxiliary integralJ∓(−^1 ),which forβ<αcan
be written as a convergent infinite series of the modified Bessel functionsK 0 :

J∓(−^1 )=

∑∞

n= 1

(± 1 )n+^1

∫∞

α

(enβ+e−nβ)e−nx

√

x^2 −α^2

dx

= 2

∑∞

n= 1

(± 1 )n+^1 cosh(nβ)K 0 (nα). (3.38)

Then, given the expansion forJ∓(−^1 )(α, β),the functionsJ∓(ν)(α, β)can be obtained
by integrating the recurrence relation

∂J∓(ν)

∂α

=−ναJ∓(ν−^2 ), (3.39)

which follows immediately from the definition ofJ(ν)in (3.34). Note that this
method works only for oddν.The “initial conditions” for (3.39) can be determined
by considering the limitsα=0orα→∞,where the corresponding integrals can
easily be calculated.

High temperature expansionAt temperatures much larger than the mass of the
particles, that is, forβandαmuch smaller than unity, every term in the series
(3.38) contributes significantly. In this case we can use a known expansion for the
sum of modified Bessel functions−formula (8.526) in I. Gradstein, I.Ryzhik,Table