Physical Foundations of Cosmology

3.3 Rudiments of thermodynamics 85

whereO≡O

(

α^2 ,β^2

)

.Similarly we obtain

J∓(^3 )=

⎧

⎪⎪

⎨

⎪⎪

⎩

2

15

π^4 +

1

2

π^2

(

2 β^2 −α^2

)

+π

(

α^2 −β^2

) 3 / 2

−A+α^4 O,

7

60

π^4 +

1

4

π^2

(

2 β^2 −α^2

)

+A−

3

4

(ln 2)α^4 +α^4 O,

(3.45)

where

A=

1

8

(

2 β^4 − 6 α^2 β^2 − 3 α^4 ln

(

eC

4 πe^3 /^4

α

))

.

Low-temperature expansionIn the limit of small temperatures we haveα=
m/T1 andK 0 (nα)∝exp(−nα).Therefore, forα−β 1 ,all terms on the
right hand side in (3.38) are negligible compared to the first term:

J∓(−^1 ) 2 K 0 (α)coshβ. (3.46)

Integrating (3.39) and taking into account thatJ±(ν)must vanish asα→∞,we
obtain

J∓(^1 ) 2 αK 1 (α)coshβ=

√

2 παe−αcoshβ

[

1 +

3

8 α

+O

(

α−^2

)]

, (3.47)

and

J∓(^3 ) 6

(

α^2 K 0 (α)+ 2 αK 1 (α)

)

coshβ

√

18 πα^3 e−αcoshβ

(

1 +

15

8 α

)

. (3.48)

These formulae allow us to calculate the basic thermodynamic properties of nonrel-
ativistic particles whenα−β 1 .In such cases the exponential term dominates
the denominator of the integrand in (3.34) and the difference between Fermi and
Bose statistics becomes insignificant because the occupation numbers are much
less than unity. We will see in the next section that this case is the situation most
relevant for cosmological applications.

3.3.4 Ultra-relativistic particles

BosonsFor bosons the maximal value of the chemical potential cannot exceed
the mass,μb≤m.Assuming that bothαandβare much smaller than unity and
substituting (3.45) into (3.37), we find that at high temperatures the excess of
particles over antiparticles to leading order is

nb−nb ̄

gT^3

3

μb

T

. (3.49)