PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

5.2. THE CONTRIBUTIONS TO THE BI-SPECTRUM DURING PREHEATING
continue even after inflation, provided the background continues to be dominated by the
scalar field, these arguments will hold even during preheating. Therefore, it is evident
that the corresponding contribution due to the fourth term can be obtained to be
G 4 (k 1 ,k 2 ,k 3 ) ≃ − 1 [
]
2 [ǫ 2(η f )−ǫ 2 (η e )] |A k1 | 2 |A k2 | 2 + two permutations . (5.26)
Though the second slow roll parameter ǫ 2 grows extremely large during preheating [see
Eq. (5.7) as well as Figure 5.2], as in the case of super-Hubble modes during inflation,
the first of these terms [involving ǫ 2 (η f )] exactly cancels the contribution G 7 (k 1 ,k 2 ,k 3 )
[cf. Eq. (4.12)] that arises due to the field redefinition (with f k set to A k ). In other words,
though individual contributions turn out to be large, the sum of the contributions due to
the fourth and the seventh terms prove to be insignificant during preheating.
Before we go on to discuss the behavior of the other contributions, we should emphasize
here that the above result for the fourth and the seventh terms applies to all single
field models. It is important to appreciate the fact that we have made no assumptions
whatsoever about the inflationary potential in arriving at the above conclusion. However,
one should keep in mind that, regarding its behavior near the minima, we have made use
of the fact that the potential can be approximated by a parabola. Indeed, it is with this explicit
form that we have been able to identify a solution to the Mukhanov-Sasaki equation
that leads to a constant curvature perturbation.
5.2.2 The second term
Upon using the behavior (4.14) of the large scale modes, it is straightforward to show that,
during preheating, the contribution to the bi-spectrum due to the second term is given by
G 2 (k 1 ,k 2 ,k 3 ) = −2iM 2 Pl
(k 1 ·k 2 +two permutations)|A k1 | 2 |A k2 | 2 |A k3 | 2
× [I 2 (η f ,η e )−I ∗ 2 (η f,η e )], (5.27)
where the quantity I 2 (η f ,η e ) is described by the integral
I 2 (η f ,η e ) =
∫ ηf
η e
dη a 2 ǫ 2 1 . (5.28)
Clearly, as in the case of the super-Hubble contributions during inflation, G 2 (k 1 ,k 2 ,k 3 )
identically vanishes since I 2 (η f ,η e ) is real. Needless to add, this implies that the second
term does not contribute to the bi-spectrum during preheating. Again we should emphasize
the fact that, as in the case of the fourth and the seventh terms, this result holds good
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