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Inflationary cosmology
7.5 Reheating after inflation
Eventually, slow roll ends when the inflaton field ~ reaches a steeper region of the
potential. The inflaton then descends more rapidly towards the absolute minimum
of the potential, overshoots it and starts to oscillate about the absolute minimum.
Assuming that the inflaton possesses couplings to matter fields, the oscillation is
damped by quantum mechanical particle creation as vacuum energy is converted
into energy of particles [6-9]. Denote the decay rate of the inflaton by r;. We
shall assume that r; ;S HOIJI;, where H05I.: is the value of the Hubble constant when
the slow-roll period ends and the oscillating period begins.
A simple way of introducing the damping by particle emission into the
dynamics of the inflaton is to modify (7.33) to
j, + 3H~ + r;~ + V'(~) = o.
(7.59)
Multiplying by ~/2 and recalling (7.40) for the vacuum energy density Pv, we
find that
PV + (3H + r;)~2 = o.
(7.60)
For simple harmonic oscillations, the average of the kinetic energy over an
oscillation is equal to the average of the potential energy over an oscillation, so
that
I ·2
I
l(~ ) = (V(~» = l(PV).
(7.61)
Averaging (7.60) over oscillations, we write
PV + (3H + r ;)pv = 0
(7.62)
where Pv is now understood to refer to the time-averaged quantity. We shall
discuss later in this section the circumstances in which (7.62) is valid.
While t ;S r;I, neglecting the time between the big bang and the start of
oscillations, the development of Pv is given, to a good approximation, by
PV +3Hpv =0
(7.63)
which is identical to the energy conservation equation in a matter-dominated FRW
universe. Thus, we may regard the vacuum energy density as equivalent to a gas
of non-relativistic ~ particles. During this period,
Pv ex R- 3
R ex t 2 / 3
and
Pv ex ,-2.
(7.64)
When t "., r; I, rapid decay of the vacuum energy to emitted particles occurs
and the universe reheats to a temperature TR given by
",2 (
_I
7) 4
30 NB + gNF TR = pv(t = r; )
(7.65)
Inflationary cosmology
7.5 Reheating after inflation
Eventually, slow roll ends when the inflaton field ~ reaches a steeper region of the
potential. The inflaton then descends more rapidly towards the absolute minimum
of the potential, overshoots it and starts to oscillate about the absolute minimum.
Assuming that the inflaton possesses couplings to matter fields, the oscillation is
damped by quantum mechanical particle creation as vacuum energy is converted
into energy of particles [6-9]. Denote the decay rate of the inflaton by r;. We
shall assume that r; ;S HOIJI;, where H05I.: is the value of the Hubble constant when
the slow-roll period ends and the oscillating period begins.
A simple way of introducing the damping by particle emission into the
dynamics of the inflaton is to modify (7.33) to
j, + 3H~ + r;~ + V'(~) = o.
(7.59)
Multiplying by ~/2 and recalling (7.40) for the vacuum energy density Pv, we
find that
PV + (3H + r;)~2 = o.
(7.60)
For simple harmonic oscillations, the average of the kinetic energy over an
oscillation is equal to the average of the potential energy over an oscillation, so
that
I ·2
I
l(~ ) = (V(~» = l(PV).
(7.61)
Averaging (7.60) over oscillations, we write
PV + (3H + r ;)pv = 0
(7.62)
where Pv is now understood to refer to the time-averaged quantity. We shall
discuss later in this section the circumstances in which (7.62) is valid.
While t ;S r;I, neglecting the time between the big bang and the start of
oscillations, the development of Pv is given, to a good approximation, by
PV +3Hpv =0
(7.63)
which is identical to the energy conservation equation in a matter-dominated FRW
universe. Thus, we may regard the vacuum energy density as equivalent to a gas
of non-relativistic ~ particles. During this period,
Pv ex R- 3
R ex t 2 / 3
and
Pv ex ,-2.
(7.64)
When t "., r; I, rapid decay of the vacuum energy to emitted particles occurs
and the universe reheats to a temperature TR given by
",2 (
_I
7) 4
30 NB + gNF TR = pv(t = r; )
(7.65)