PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

5.1. BEHAVIOR OF BACKGROUND AND PERTURBATIONS DURING PREHEATING
potentialz ′′ /z is no longer given by the slow roll expression (5.8). It is clear that the effective
potential will contain oscillatory functions and, hence, it can even possibly vanish.
So, it is not a priori obvious that one can use the same approach as in the inflationary
epoch and simply ignore the k 2 term in the differential equation (1.16) for arriving at the
behavior of the super-Hubble modes. Moreover, it is known that, during the preheating
phase, one has to deal with the resonant behavior exhibited by the equation of motion
under certain conditions [62, 63]. As a consequence, at this stage, it becomes necessary
that we remain cautious and analyze equation (1.16) more carefully.
In order to study the perturbations during the preheating phase, it proves to be more
convenient to work in terms of cosmic time and use a new rescaled variable V k that is
related to the Mukhanov-Sasaki variable as follows: V k ≡ a 1/2 v k . Then, one finds that
Eq. (1.16) takes the form [62, 63]
[
k
¨V 2
k +
a + d2 V
2 dφ + 3 ˙φ 2 ˙φ 4
−
2 M 2 2H 2 M 4 Pl Pl
( )
+ 3 ˙φ2
4M 2 2 −V + 2 ˙φ
HM 2 Pl
Pl
]
dV
V k = 0. (5.10)
dφ
Recall that, in the quadratic potential of our interest, soon after inflation, the evolution
of the scalar field φ(t) is given by Eq. (5.4). Using this solution and its derivative (5.5),
it is then easy to show that, while the third, fourth and the fifth terms within the square
brackets in the above differential equation decay asa −3 , the last term decays more slowly
as it scales asa −3/2 . Upon retaining only the first, second and the last terms and neglecting
the others, one arrives at an equation of the form
[
]
d 2 V k
dσ + 1+ k2
2 m 2 a − 4 ( ae
) 3/2
cos (2σ+2∆) V 2 k = 0, (5.11)
mt e a
where the new independent variableσ is a dimensionless quantity which we have defined
to beσ ≡ mt+π/4. We can rewrite the above equation as
with A k andq being given by
d 2 V k
dσ 2 +[A k −2q cos (2σ+2∆)] V k = 0, (5.12)
A k = 1+ k2
m 2 a2, (5.13)
2
( ae
) 3/2,
q =
(5.14)
mt e a
where, as we mentioned, t e and a e denote the cosmic time and the scale factor when
inflation ends. The above equation is similar in form to the Mathieu equation (see, for
93