Physical Foundations of Cosmology

3.3 Rudiments of thermodynamics 81

∆σ

∆Ω

∆t

n

R

v

Fig. 3.3.

where the minus sign applies to bosons and the plus to fermions. Theenergy density
is equal to

ε=

∑



ng=

g

2 π^2

∫∞

m

√

(^2 −m^2 )

exp((−μ)/T)∓ 1

^2 d. (3.28)

Let us now calculate the pressure. To do so we consider a small area element
σn,wherenis the unit normal vector. All particles with velocity|v|,striking this
area element in the time interval betweentandt+t,were located, att= 0 ,in a
spherical shell of radiusR=|v|twith width|v|t(Figure 3.3). The total number
of particles with energy(|v|)within a solid angle
of this shell is equal to

N=ngR^2 |v|t

,

wheregis the number of states per unit spatial volume. Not all particles in the
shell reach the target, only those with velocities directed to the area element. Taking
into account the isotropy of the velocity distribution, we find the total number of
particles striking the area elementσnwith velocityvis

Nσ=

(v·n)σ

|v| 4 πR^2

N=

(v·n)σ

4 π

ngt

.

If these particles are reflected elastically, each transfers momentum 2(p·n)tothe
target.Therefore, the contribution of particles with velocity|v|to the pressure is

p=

∫

2(p·n)Nσ

σt

=

|p|^2

2 π

ng

∫

cos^2 θsinθdθdφ=

|p|^2

3 

ng,

where we have used the relation|v|=|p|/and integrated over the hemisphere.
The totalpressureis then

p=

∑



|p|^2

3 

ng=

ε

3

−

m^2 g

6 π^2

∫∞

m

√

(^2 −m^2 )

exp((−μ)/T)∓ 1

d. (3.29)