PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 5. THE SCALAR BI-SPECTRUM DURING PREHEATING
terms during preheating can be written as
[ (
G 1 (k 1 ,k 2 ,k 3 )+G 3 (k 1 ,k 2 ,k 3 ) = 2iM 2 1− k 1 ·k 2
− k )
1 ·k 3
|A
Pl
k2
2 k3
2 k1 | 2
× ( )
A k2
¯B∗ k2
A k3
¯B∗ k3
−A ∗ k
¯Bk2 2
A ∗ k
¯Bk3
] 3
+two permutations
I 13 (η f ,η e ), (5.31)
where the quantity I 13 (η f ,η e ) represents the integral
I 13 (η f ,η e ) =
∫ ηf
This integral can be trivially carried out during preheating to yield
η e
dη
a2. (5.32)
I 13 (η f ,η e ) = t e
a 3 e
[
1−e
−3(N f −N e)/2 ] . (5.33)
Since the second term in this expression for I 13 (η f ,η e ) dies quickly with growing N f , the
corresponding contribution to the bi-spectrum proves to be negligible.
The contributions due to the fifth and the sixth terms during preheating can be arrived
at in a similar fashion. We obtain that
{[
G 5 (k 1 ,k 2 ,k 3 )+G 6 (k 1 ,k 2 ,k 3 ) = iM2 Pl k1 ·k 2
+ k 1 ·k 3
+ k2 1 (k ]
2 ·k 3 )
|A
2 k2
2 k3
2 k2k 2 3
2 k1 | 2
( )
Ak2 ¯B∗ k2
A k3
¯B∗ k3
− A ∗ ¯Bk2 k 2
A ∗ ¯Bk3 k 3
}
+ two permutations I 56 (η f ,η e ), (5.34)
with I 56 (η f ,η e ) denoting the integral
I 56 (η f ,η e ) =
∫ ηf
η e
dη
a 2 ǫ 1. (5.35)
This integral too can be evaluated rather easily to arrive at the following expression:
I 56 (η f ,η e ) = 3t e
a 3 e
(
cos 2 (mt e +∆)−e −3(N f−N e)/2 cos 2[ mt e e 3(N f−N e)/2 +∆ ]
+mt e cos(2∆) { Si(2mt e )−Si [ 2mt e e 3(N f−N e)/2 ]}
+mt e sin(2∆) { Ci(2mt e )−Ci [ 2mt e e 3(N f−N e)/2 ]}) , (5.36)
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