Physical Foundations of Cosmology

246 Inflation I: homogeneous limit

particles per unit volume in the three-dimensional space. Taking into account that
nk=n−k≡nkandnφ1, we infer that the number densitiesnφandnχsatisfy
(5.61), where

effχ(1+ 2 nk). (5.63)

Given a number densitynχ, let us calculatenk.Aφparticle “at rest” decays into
twoχparticles, both having energym/2. Because of the interaction term (5.57),
the effective squared mass of theχparticle depends on the value of the inflaton
field and is equal tom^2 χ+ 2 gφ(t). Therefore the corresponding 3-momentum of
the producedχparticle is given by

k=

((

m

2

) 2

−m^2 χ− 2 gφ(t)

) 1 / 2

, (5.64)

where we assume thatm^2 χ+ 2 gφm^2. The oscillating term,

gφg cos(mt),

leads to a “scattering” of the momenta in phase space. Ifg m^2 /8, then all
particles are created within a thin shell of width

km

(

4 g

m^2

)

m (5.65)

located near the radiusk 0 m/2 (Figure 5.6(a)). Therefore

nk=m/ 2 

nχ

(

4 πk 02 k

)

/( 2 π)^3



2 π^2 nχ

mg

=

π^2

g

nχ

nφ

. (5.66)

∆k∼−(^4 mgΦ )

k*/√π

(a) (b)

k 0 ∼−m 2

Fig. 5.6.