PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM
the fourth term [54]. Upon making use of the behavior of the mode f k on super-Hubble
scales in the integral (4.11b), we have
G se
2 (k 1,k 2 ,k 3 ) = −2i (k 1 ·k 2 +two permutations) A ∗ k 1
A ∗ k 2
A ∗ k 3
I 2 (η e ,η s ), (4.24)
where I 2 (η e ,η s ) denotes the integral
I 2 (η e ,η s ) =
∫ ηe
η s
dη a 2 ǫ 2 1, (4.25)
so that the corresponding contribution to the bi-spectrum is given by
G se
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 e ,η s )−I ∗ 2 (η e,η s )]. (4.26)
Note that, due to quadratic dependence on the scale factor, actually, I 2 (η e ,η s ) is a rapidly
growing quantity at late times.
However, the complete super-Hubble contribution to
the bi-spectrum vanishes identically since the integral I 2 (η e ,η s ) is a purely real quantity.
Hence, in the case of the second term, it is sufficient to evaluate the contribution to the
bi-spectrum due to G is
2 (k 1,k 2 ,k 3 ).
The remaining terms
Let us now compute the contributions due to the remaining
terms, viz. the first, the third, the fifth and the sixth.
Notice that, the first term
G 1 (k 1 ,k 2 ,k 3 ) and the third termG 3 (k 1 ,k 2 ,k 3 ) involve the same integrals. Therefore, these
two contributions to the bi-spectrum can be clubbed together. Similarly, the fifth and the
sixth terms, viz. G 5 (k 1 ,k 2 ,k 3 ) and G 6 (k 1 ,k 2 ,k 3 ), also contain integrals of the same type,
and hence their contributions too can be combined. On making use of the super-Hubble
behavior (4.14) and (4.15) of the mode f k and its derivative, we obtain that
G se
1 (k 1,k 2 ,k 3 ) ≃ 2i ( A ∗ k 1
¯B∗ k2
¯B∗ k3
+ two permutations ) I 13 (η e ,η s ) (4.27)
and
G se
3 (k 1,k 2 ,k 3 ) ≃ −2i
[( )
k1 ·k 2
k 2 2
]
A ∗ k
¯B∗ 1 k2
¯B∗ k3
+ five permutations I 13 (η e ,η s ), (4.28)
where the quantity I 13 (η e ,η s ) represents the integral
I 13 (η e ,η s ) =
69
∫ ηe
η s
dη
a2. (4.29)