PHYS08200605006 D.K. Hazra - Homi Bhabha National Institute

CHAPTER 1. INTRODUCTION
1.3 The generation and imprints of non-Gaussianities
So far, we have discussed the generation of correlations during inflation and the comparison
of the theoretical predictions with the observational evidence at the level of the
two point function, viz. the primordial and the CMB angular power spectra. Over the last
decade, there has been frenetic activity towards gaining a theoretical understanding of the
amplitude and the form of the three and the four point functions that are produced during
inflation. In particular, a large fraction of these investigations have been dedicated to
analyzing inflationary and post-inflationary dynamics that lead to deviations from Gaussianity
(see Refs. [49, 50, 51, 52, 53, 54, 55]; for earlier efforts in this direction, see Refs. [56]).
At the same time, there has also been a constant effort to arrive at increasingly tighter constraints
on the actual extent of primordial non-Gaussianities observed in the CMB and the
LSS data [57, 58]. It is now commonly recognized that non-Gaussianities, if detected, can
play a significant role in helping us discriminate between the various inflationary as well
as the post-inflationary scenarios.
In a linear theory described by a quadratic action, when one works with vacuum initial
states, the correlations prove to be Gaussian in nature. It is the non-linearities in the
action governing the variable of interest that leads to deviations from Gaussianities in the
correlation functions. The simplest signature of non-Gaussianity is a non-zero three point
function, which, in a given theory, will be determined by the cubic terms in the action.
Then, evidently, the first step towards evaluating the three point correlation functions
generated in an inflationary model would be to arrive at the action describing the scalar
and tensor perturbations at the next higher, i.e. the cubic, order [49, 51]. Having obtained
such an action, one can make use of the standard procedures of quantum field theory
to arrive at the corresponding scalar and tensor bi-spectra, viz. the three point functions
in Fourier space, as well as cross-correlations between the scalars and the tensors, corresponding
to the cubic order action. Based on these arguments, there now exists a standard
formalism, originally due to Maldacena [49], to evaluate the bi-spectrum generated
during inflation (for more details, see Chapter 4).
The scalar bi-spectrum, say, B S
(k 1 ,k 2 ,k 3 ), associated with the modes ˆR k of the curvature
perturbation is defined as
〈 ˆR k1
ˆRk2 ˆRk3 〉 = (2π) 3 B S
(k 1 ,k 2 ,k 3 ) δ (3) (k 1 +k 2 +k 3 ). (1.26)
Ideally, one would have liked to compute the CMB angular bi-spectrum corresponding
to the primordial bi-spectrum generated during inflation and compare with the observa-
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