PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 5. THE SCALAR BI-SPECTRUM DURING PREHEATING
whereγ(b,x) is the incomplete Gamma function (see, for example, Refs. [115, 116]), while
the quantity C is a dimensionless constant of integration. Then, the relation between the
incomplete Gamma and the Gamma functions allows us to express the function K(t) as
follows:
K(t) ≃ t e
4a 2 e
∫ t
[
d¯t 6
cos 2 (m¯t+∆) 5
(
tē
t
) 1/3 ( ) ( ) 2 5
+e 2i∆ (−2imt e ) −5/3 tē
Γ
3 t
( ) 4/3 ( ) ( ) 2
−e 2i(m¯t+∆) (−2imt e ) −1 tē 5
+e −2i∆ (2imt e ) −5/3 tē
Γ
t 3 t
( ) 4/3 ( ) ] 2
−e −2i(m¯t+∆) (2imt e ) −1 tē tē
+C
t t
≃ 3t4/3 e
10a 2 e
∫ t
d¯t ¯t −1/3
+··· . (5.23)
cos 2 (m¯t+∆)
In arriving at the final equality, we have used the asymptotic property of the incomplete
Gamma function [115, 116] and have retained only the dominant term in inverse power
ofmt. The final expression above can be integrated by parts to arrive at
∫ t
K(t) ≃ 3t4/3 e
t −1/3 tan(mt+∆)+ t4/3 e
d¯t ¯t −4/3 tan(m¯t+∆). (5.24)
10ma 2 e 10ma 2 e
The second term containing the integral in this expression is of the order of the other
terms that we have already neglected and, hence, it too can be ignored. As a result, the
growing mode of the curvature perturbation can be written as
[
]
f k ≃ A k 1− 3 k 2 t 4/3
e
10 a 2 e mt tan(mt+∆) 1/3
[
= A k 1− 1 ( ) ]
2 k H
5 aH m tan(mt+∆) , (5.25)
in perfect agreement with the result that has been obtained recently in the literature [64].
It is evident from the above expression that the evolution of the curvature perturbation
will contain sharp spikes during preheating, a feature that is clearly visible in Figure 5.3
wherein we have plotted the above analytical expression as well as the corresponding
numerical result (in this context, also see Figure 4 in the first reference in Refs. [63], where
the spikes are also clearly visible).
It is important that we make a couple of remarks concerning the appearance of the
spikes in the evolution of the curvature perturbation. Firstly, as the spikes are encountered
both analytically and numerically, evidently, they are not artifacts of the adopted
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