PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

4.2. THE SCALAR BI-SPECTRUM IN THE MALDACENA FORMALISM
parameter ǫ 1 remains small in the Starobinsky model as the field crosses the transition,
the second slow parameter ǫ 2 turns very large briefly [54, 55]. In the case of punctuated
inflation,ǫ 1 itself grows to a large value thereby actually interrupting inflation for about a
e-fold. It is this property that results in a sharper spike in the case of punctuated inflation
than the Starobinsky model. The overall step in these models is easier to understand, and
it simply arises due to the difference in the Hubble scales associated with the slow roll
epoch before and after the period of fast roll. Both these models also lead to oscillations
before the spectra turn nearly scale invariant on small scales. The spectra that arises in
punctuated inflation, in addition to leading to a better fit to the outliers at very small multipoles
(because of the drop in power on these scales), also provides an improvement in
the fit to the outlier atl ≃ 22 [77]. It is interesting to notice that the spectra of punctuated
inflation and the model with a step in the potential match briefly as they oscillate near
scales corresponding to l ≃ 22.
4.2 The scalar bi-spectrum in the Maldacena formalism
As we had mentioned in the introductory chapter, there now exists a standard formalism,
initially proposed by Maldacena, that allows one to evaluate the bi-spectrum in a given
inflationary model. In the Maldacena formalism [49], the bi-spectrum is evaluated using
the standard rules of perturbative quantum field theory, based on the interaction Hamiltonian
that depends cubically on the curvature perturbation. For the case of the canonical
scalar field of our interest, the action at the cubic order in the curvature perturbation is
found to be [49, 51, 52]
∫ ∫ [
S 3 [R] = M 2 dη d 3 x a 2 ǫ 2 Pl
1 RR′2 +a 2 ǫ 2 1 R(∂R)2 −2aǫ 1 R ′ (∂ i R)(∂ i χ)
+ a2
2 ǫ 1ǫ ′ 2 R2 R ′ + ǫ 1
2 (∂i R)(∂ i χ)(∂ 2 χ)+ ǫ 1
4 (∂2 R)(∂χ) 2 +aF
( ) ] δL2
, (4.4)
δR
where the Latin indices denote the spatial coordinates, while the function χ is defined
through the relation
∂ 2 χ ≡ aǫ 1 R ′ . (4.5)
The quantity δL 2 /δR denotes the variation of the Lagrangian density corresponding to
the following quadratic action:
S 2 [R] = 1 2
∫
dη
∫
d 3 x z 2 [R ′2 −(∂R) 2] , (4.6)
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