244 Inflation I: homogeneous limit
φχχφψψFig. 5.5.5.5.1 Elementary theory
We consider an inflaton fieldφof massmcoupled to a scalar fieldχand a spinor field
ψ. Their simplest interactions are described by three-legged diagrams (Figure 5.5),
which correspond to the following terms in the Lagrangian:
Lint=−gφχ^2 −hφψψ. ̄ (5.57)We have seen that these kinds of couplings naturally arise in gauge theories with
spontaneously broken symmetry, and they are enough for our illustrative purposes.
To avoid a tachyonic instability we assume that|gφ|is smaller than the squared
“bare” massm^2 χ. The decay rates of the inflaton field intoχχandψψ ̄ pairs are
determined by the coupling constantsg andhrespectively. They can easily be
calculated and the corresponding results are cited in books on particle physics:
χ≡(φ→χχ)=
g^2
8 πm,ψ≡(φ→ψψ)=
h^2 m
8 π. (5.58)
Let us apply these results in order to calculate the decay rate of the inflaton. As
we have noted, anoscillatinghomogeneous scalar field can be interpreted as a
condensate of heavy particles of massm“at rest,” that is, their 3-momentakare
equal to zero. Keeping only the leading term in (5.45), we have
φ(t) (t)cos(mt), (5.59)where (t)is the slowly decaying amplitude of oscillations. The number density
ofφparticles can be estimated as
nφ=
εφ
m=
1
2 m(
φ ̇^2 +m^2 φ^2)
1
2
m 2. (5.60)This number is very large. For example, form∼ 1013 GeV,wehavenφ∼ 1092
cm−^3 immediately after the end of inflation, when ∼1 in Planckian units.
One can show that quantum corrections do not significantly modify the inter-
actions (5.57) only ifg<mandh<m^1 /^2. Therefore, formmPl, the highest
decay rate intoχparticles,χ∼m, is much larger than the highest possible rate
for the decay into fermions,ψ∼m^2 .Ifg∼m, then the lifetime of aφparticle
is about−χ^1 ∼m−^1 and the inflaton decays after a few oscillations. Even if the