PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 4. BI-SPECTRA ASSOCIATED WITH LOCAL AND NON-LOCAL FEATURES
From these results, one can easily show that the super-Hubble contributions due to the
first and the third terms to the bi-spectrum can be written as
[ (
G se
1 (k 1 ,k 2 ,k 3 )+G se
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 ∗ ¯Bk2 k 2
A ∗ ¯Bk3 k
] 3
+ two permutations
I 13 (η e ,η s ). (4.30)
The corresponding contributions due to the fifth and the sixth terms can be arrived at in
a similar fashion. We find that
{[
G se
5 (k 1,k 2 ,k 3 )+G se
6 (k 1,k 2 ,k 3 ) = iM2 Pl k1 ·k 2
+ k ]
1 ·k 3
+ k2 1 (k 2 ·k 3 )
2 k2
2 k3
2 k2 2k2 3
×|A k1 | ( )
2 A k2
¯B∗ k2
A k3
¯B∗ k3
−A ∗ k
¯Bk2 2
A ∗ k
¯Bk3
}
3
+two permutations I 56 (η e ,η s ), (4.31)
where the quantity I 56 (η e ,η s ) is described by the integral
I 56 (η e ,η s ) =
∫ ηe
η s
dη
a 2 ǫ 1. (4.32)
Hence, the non-zero, super-Hubble contribution to the bi-spectrum is determined by the
sum of the contribution due to the first, the third, the fifth and the sixth terms arrived
at above. In order to illustrate that this contribution is insignificant, we shall now turn
to estimating the amplitude of the corresponding contribution to the non-Gaussianity
parameterf NL
.
4.3.2 An estimate of the super-Hubble contribution to the
non-Gaussianity parameter
Let us restrict ourselves to the equilateral limit, i.e. when k 1 = k 2 = k 3 = k, for simplicity.
In such a case, the super-Hubble contributions to the bi-spectrum, say, G es
eq(k), due to the
first, the third, the fifth and the sixth terms, as given by the expressions (4.30) and (4.31),
add up to be
G es
eq(k) = iM 2 |A Pl
k| 2( A 2 ¯B
) [
k k ∗2 −A ∗ k 2 ¯B2 k 12I 13 (η e ,η s )− 9 ]
4 I 56(η e ,η s ) . (4.33)
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