PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM
grals
G 1 (k 1 ,k 2 ,k 3 ) = 2i
∫ ηe
η i
dηa 2 ǫ 2 1
(
f
∗
k1
f k ′∗
2
f k ′∗
3
+two permutations ) ,
(4.11a)
G 2 (k 1 ,k 2 ,k 3 ) = −2i (k 1 ·k 2 +two permutations) dηa 2 ǫ 2 1fk ∗ 1
fk ∗ 2
fk ∗ 3
, (4.11b)
η i
∫ [
ηe
(k1 )
]
G 3 (k 1 ,k 2 ,k 3 ) = −2i dηa 2 ǫ 2 ·k 2
1 fk ∗ 1
f k ′∗
2
f k ′∗
3
+five permutations , (4.11c)
η i
G 4 (k 1 ,k 2 ,k 3 ) = i
G 5 (k 1 ,k 2 ,k 3 ) = i 2
G 6 (k 1 ,k 2 ,k 3 ) = i 2
∫ ηe
k 2 2
∫ ηe
(
f
∗
k1
f ∗ k 2
f ′∗
k 3
+two permutations ) ,
dηa 2 ǫ 1 ǫ ′ 2
η i
∫ [
ηe
(k1 )
dηa 2 ǫ 3 ·k 2
1
η i
k2
2
∫ {
ηe
[k ]
2
1 (k 2 ·k 3 )
η i
dηa 2 ǫ 3 1
k 2 2 k2 3
f ∗ k 1
f ′∗
k 2
f ′∗
k 3
+five permutations
f ∗ k 1
f ′∗
k 2
f ′∗
k 3
+two permutations
]
(4.11d)
, (4.11e)
}
, (4.11f)
whereǫ 2 is the second slow roll parameter that is defined with respect to the first through
the expression (1.8). The lower limit of the above integrals, viz. η i , denotes a sufficiently
early time when the initial conditions [say, the Bunch-Davies conditions (1.23)] are imposed
on the modes f k . The additional, seventh term G 7 (k 1 ,k 2 ,k 3 ) arises due to a field
redefinition (in this context, see, Refs. [49, 51, 53]), and its contribution to G(k 1 ,k 2 ,k 3 ) is
given by
G 7 (k 1 ,k 2 ,k 3 ) = ǫ 2(η e )
2
(
|fk2 (η e )| 2 |f k3 (η e )| 2 +two permutations ) . (4.12)
4.3 The numerical computation of the scalar bi-spectrum
In this section, after illustrating that the super-Hubble contributions to the complete bispectrum
during inflation proves to be negligible, we shall outline the methods that we
adopt to numerically evolve the equations governing the background and the perturbations,
and eventually evaluate the inflationary scalar power and bi-spectra. Also, we shall
illustrate the extent of accuracy of the numerical methods by comparing them with the
expected form of the bi-spectrum in the equilateral limit in power law inflation and the
analytical results that are available in the case of the Starobinsky model [54, 55].
4.3.1 The contributions to the bi-spectrum on super-Hubble scales
It is clear from the above expressions that the evaluation of the bi-spectrum involves
integrals over the mode f k and its derivative f ′ k as well as the slow roll parameters ǫ 1,
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