PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.3. THE NUMERICAL COMPUTATION OF THE SCALAR BI-SPECTRUM
In the equilateral limit,
f NL
(k 1 ,k 2 ,k 3 ) simplifies to
the expression (1.29) for the non-Gaussianity parameter
f eq
NL (k) = −10 9
1
(2π) 4 k 6 G eq (k)
P 2 S (k) , (4.34)
where P S
(k) is the scalar power spectrum defined in Eq. (1.21). It is straightforward to
show that the f NL
corresponding to the super-Hubble contribution to the bi-spectrum
G es
eq (k) above is given by
f eq(se) (k) ≃ − 5iM2 Pl
NL
18
( ) [ A
2
k ¯B∗ 2 k −A ∗ k 2 ¯B2 k
12I
|A k | 2 13 (η e ,η s )− 9 ]
4 I 56(η e ,η s ) , (4.35)
where we have made use of the fact that f k ≃ A k at late times in order to arrive at the
power spectrum.
To estimate the above super-Hubble contribution to the non-Gaussianity parameter
f eq(se) , let us choose to work with power law inflation because it permits exact calculations,
and it can also mimic slow roll inflation. During power law expansion, the
NL
scale
factor can be written as
a(η) = a 1
( η
η 1
) γ+1
, (4.36)
where a 1 and η 1 are constants, while γ is a free index. It is useful to note that, in such a
case, the first slow roll parameter is a constant and is given by ǫ 1 = (γ + 2)/(γ + 1). The
current observational constraints on the scalar spectral index suggest that γ −2, which
implies that the corresponding scale factor is close to that of de Sitter.
In power law inflation, the exact solution to Eq. (1.16) can be expressed in terms of the
Bessel function J ν (x) as follows (see, for instance, Refs. [105]):
v k (η) = √ −kη [C k J ν (−kη)+D k J −ν (−kη)], (4.37)
where ν = (γ +1/2), and the quantities C k and D k are constants that are determined by
the initial conditions. Upon demanding that the above solution satisfies the Bunch-Davies
initial condition (1.23), one obtains that
C k = −D k e −iπ(γ+1/2) ,
(4.38a)
√ π e iπγ/2
D k =
k 2 cos(πγ) . (4.38b)
Sincef k = v k /z, withz = √ 2ǫ 1 M Pl
a, and asǫ 1 is a constant in power law inflation, we can
arrive at the constants A k and B k [cf. Eqs. (4.14) and Eqs. (4.15)] from the super-Hubble
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