PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM
The various contributions
To begin with, note that, each of the integrals G C
(k 1 ,k 2 ,k 3 ), where C = (1,6), can be
divided into two parts as follows:
G C
(k 1 ,k 2 ,k 3 ) = G is
C (k 1,k 2 ,k 3 )+G se
C (k 1,k 2 ,k 3 ). (4.16)
The integrals in the first term G is
C (k 1,k 2 ,k 3 ) run from the earliest time (i.e. η i ) when the
smallest of the three wavenumbers k 1 , k 2 and k 3 is sufficiently inside the Hubble radius
[typically corresponding to k/(aH) ≃ 100] to the time (say, η s ) when the largest of the
three wavenumbers is well outside the Hubble radius [say, when k/(aH) ≃ 10 −5 ]. Then,
evidently, the second term G se
C (k 1,k 2 ,k 3 ) will involve integrals which run from the latter
time η s to the end of inflation at η e . In what follows, we shall discuss the various
contributions to the bi-spectrum due to the terms G se
C (k 1,k 2 ,k 3 ). We shall show that the
corresponding contribution either remains small or, when it proves to be large, it is exactly
canceled by another contribution to the bi-spectrum.
The contributions due to the fourth and the seventh terms
Let us first focus on the
fourth term G 4 (k 1 ,k 2 ,k 3 ) since it has often been found to lead to the largest contribution
to the bi-spectrum when deviations from slow roll occur [53, 54, 55]. As the slow roll
parameters turn large towards the end of inflation, we can expect this term to contribute
significantly at late times. However, as we shall quickly illustrate, such a late time contribution
is exactly canceled by the contribution from G 7 (k 1 ,k 2 ,k 3 ) which arises due to the
field redefinition.
Upon using the form (4.14) of the mode f k and its derivative (4.15) on super-Hubble
scales in the expression (4.11d), one obtains that
G se
4 (k 1,k 2 ,k 3 ) ≃ i ( A ∗ k 1
A ∗ k 2
¯B∗ k3
+ two permutations ) ∫ η e
This expression can be trivially integrated to yield
η s
dη ǫ ′ 2 . (4.17)
G se
4 (k 1,k 2 ,k 3 ) ≃ i ( A ∗ k 1
A ∗ k 2
¯B∗ k3
+ two permutations ) [ǫ 2 (η e )−ǫ 2 (η s )], (4.18)
so that the corresponding contribution to the bi-spectrum can be expressed as
[
G se
4 (k 1,k 2 ,k 3 ) ≃ iM 2 [ǫ
Pl 2 (η e )−ǫ 2 (η s )] |A k1 | 2 |A k2 | 2( )
A k3
¯B∗ k3
−A ∗ ¯Bk3 k 3
]
+ two permutations . (4.19)
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