MODERN COSMOLOGY

(P)reheating after inflation 173

At the stage of inflation all energy is concentrated in a classical slowly
moving inflaton fieldφ. Soon after the end of inflation this field begins to
oscillate near the minimum of its effective potential. Eventually it produces many
elementary particles, they interact with each other and come to a state of thermal
equilibrium with some temperatureTr.
Elementary theory of this process was developed many years ago [15]. It was
based on the assumption that the oscillating inflaton field can be considered as a
collection of non-interacting scalar particles, each of which decays separately in
accordance with perturbation theory of particle decay. However, it was recently
understood that in many inflationary models the first stages of reheating occur
in a regime of a broad parametric resonance. To distinguish this stage from
the subsequent stages of slow reheating and thermalization, it was calledpre-
heating[16]. The energy transfer from the inflaton field to other bose fields and
particles during pre-heating is extremely efficient.
To explain the main idea of the new scenario we will consider first the
simplest model of chaotic inflation with the effective potentialV(φ)=^12 m^2 φ^2 ,

and with the interaction Lagrangian−^12 g^2 φ^2 χ^2. We will takem= 10 −^6 MP,as
required by microwave background anisotropy [7] and, in the beginning, we will
assume for simplicity thatχparticles do not have a bare mass, i.e.mχ(φ)=g|φ|.
In this model inflation occurs at|φ|> 0. 3 MP[7]. Suppose for definiteness
that initiallyφis large and negative, and inflation ends atφ∼− 0. 3 MP.After
that the fieldφrolls toφ=0, and then it oscillates aboutφ=0 with a gradually
decreasing amplitude.
For the quadratic potentialV(φ) =^12 mφ^2 the amplitude after the first
oscillation becomes only 0. 04 MP, i.e. it drops by a factor of ten during the first
oscillation. Later on, the solution for the scalar fieldφasymptotically approaches
the regime

φ(t)=(t)sinmt

(t)=

MP

√

3 πmt

∼

MP

2 π

√

3 πN

. (4.27)

Here(t)is the amplitude of oscillations,Nis the number of oscillations since
the end of inflation. For simple estimates which we will make later one may use

(t)≈

MP

3 mt

≈

MP

20 N

. (4.28)

The scale factor averaged over several oscillations grows asa(t)≈a 0 (t/t 0 )^2 /^3.
Oscillations ofφin this theory are sinusoidal, with the decreasing amplitude

(t)=

MP

3

(

a 0

a(t)

) 3 / 2

.

The energy density of the fieldφdecreases in the same way as the density of
non-relativistic particles of massm:

ρφ=^12 φ ̇^2 +^12 m^2 φ^2 ∼a−^3.