PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 4. BI-SPECTRA ASSOCIATED WITH LOCAL AND NON-LOCAL FEATURES
which, for instance, leads to the equation of motion (1.15) for the curvature perturbation
R, and can be written as
δL 2
δR = ˙Λ+HΛ−ǫ 1 (∂ 2 R). (4.7)
The term F(δL 2 /δR) that has been introduced in the above cubic order action refers to
the following expression [51, 54]:
F
( ) δL2
δR
=
{ [
1
a 2 Hǫ 2 R 2 +4aRR ′ +(∂ i R)(∂ i χ)− 1 ] δL2
2aH
H (∂R)2 δR
+ [ Λ(∂ i R)+(∂ 2 R)(∂ i χ) ] [ ( )]
δ ij ∂ j ∂ −2 δL2
+ 1 H δim δ jn (∂ i R)(∂ j R) ∂ m ∂ n
[
∂ −2 ( δL2
δR
δR
)] } . (4.8)
where, again, the Latin indices represent the spatial coordinates.
For convenience, we shall introduce a new quantity G(k 1 ,k 2 ,k 3 ) that is related to the
bi-spectrum B S
(k 1 ,k 2 ,k 3 ) by a constant factor as follows:
G(k 1 ,k 2 ,k 3 ) = (2π) 9/2 B S
(k 1 ,k 2 ,k 3 ). (4.9)
It can be shown that the quantity G(k 1 ,k 2 ,k 3 ), which results from the interaction Hamiltonian
corresponding to the cubic action (4.4), evaluated towards the end of inflation, say,
at the conformal time η e , can be expressed as [49, 51, 52, 53, 54]
7∑
G(k 1 ,k 2 ,k 3 ) ≡ G C
(k 1 ,k 2 ,k 3 )
C=1
≡ M 2 Pl
{
6∑
[f k1 (η e )f k2 (η e )f k3 (η e )] G C
(k 1 ,k 2 ,k 3 )
C=1
+ [ fk ∗ 1
(η e )fk ∗ 2
(η e )fk ∗ 3
(η e ) ] G ∗ (k C
1,k 2 ,k 3 ) +G 7 (k 1 ,k 2 ,k 3 ), (4.10)
}
wheref k are the Fourier modes associated with the curvature perturbation [cf. Eq. (1.19)]
that satisfy the differential equation (1.15). The quantities G C
(k 1 ,k 2 ,k 3 ) with C = (1,6)
correspond to the six terms in the interaction Hamiltonian, and are described by the inte-
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