Why Primordial Non-Gaussianity is Very Small? (Quantum Physics / Maths)

Inflation in the early universe has been a part of standard cosmology not only to solve the horizon, flatness, and monopole problems, but also to account for the origin of large-scale structures. Inflation is well described by a homogeneous scalar field dubbed as inflaton φ in quasi de-Sitter (dS) space. Properties of the inflaton, such as the forms of its kinetic and potential terms as well as its coupling to gravity are yet to be clarified both from theoretical and observational viewpoints. Quantum fluctuations of scalar field and gravitational field generated during inflation serve as probes of its physics that can be tested by observations of cosmic microwave background (CMB) and large-scale structures..

The simplest class of inflation models is the potential driven canonical slow-roll inflation where the kinetic term of the inflaton has the canonical form and its potential energy drives inflation. In this model, fluctuations can be expressed approximately as a massless free scalar field in dS space minimally-coupled to gravity. As a result, its linear perturbation calculation predicts a nearly scale-invariant spectrum with highly Gaussian distribution, in good agreement with observations. In these canonical models, deviations from the scale-invariant Gaussian distribution are controlled by the slow-roll parameters. Observationally, exact scale-invariant power spectrum has already been ruled out with more than two-σ confidence level with the red-tilted spectral index n_s < 1, but the primordial non-Gaussianity characterized by the bispectrum has not been detected so far, and only constraints on the non-linearity parameter “f_NL” of various types have been obtained so far.

Primordial non-Gaussianity is a potentially powerful discriminant of the physical mechanisms that generated the cosmological fluctuations observed today.”

In this situation, a number of extended inflation models has been proposed so far which can realize sizable non-Gaussianity while reproducing the observed red-tilted spectrum. Theoretically, the simplest local type non-Gaussianity may be produced by curvaton or modulated reheating scenarios, both of which require another fluctuating field in addition to the inflaton. Since there is no observational evidence requiring multiple fluctuating fields during inflation, we do not consider such models but stick to the single-field inflation models, whose non-Gaussian signature is mostly the equilateral one. To realize sizable non-Gaussianity in single field models, some models modify the kinetic terms as in k- or G-inflation, ghost condensate, Dirac-Born Infeld, and other models extend the gravitational sector, or both. In these non-canonical models, enhanced non-Gaussianity may be realized due to the smallness of sound speed during inflation and still consistent with the observation. Hence primordial non-Gaussianity serves as a good probe of new physics these extended models are based.

In the language of quantum field theory (QFT), power spectrum of perturbation corresponds to the vacuum expectation value (VEV) of the fluctuation two-point functions. The lowest order of non-Gaussianity is the VEV of the fluctuation three-point functions. Technically, the non-Gaussianity exists because of higher order interaction terms and it is evaluated by in-in perturbation theory with two-point function as the input to the calculation. However, such interaction terms also generate higher-order corrections to the two-point function which are called “loop corrections” in QFT terminology. Such correction must be analyzed carefully to ensure its smallness compared with the tree-level amplitude, or the result of calculations based on the linear perturbation theory, on which most cosmologists rely to set the initial condition of the post-inflationary universe, would lose its significance.

Now, Kristiano and Yokoyama calculated one-loop corrections to the power spectrum in generic single-field inflation, arising from the three-body interactions of perturbations which also generate primordial non-Gaussianity, using soft effective field theory.

The effective field theory (EFT) for soft limit of scalar field in dS space was proposed by Cohen and Green. Kristiano and Yokoyama applied their framework to inflation, because the fluctuations in inflation can be approximated as a scalar field in dS space. In particular, they used the EFT power counting method to classify interaction terms based on their significance on super-horizon scale. Then, they calculated the loop correction to the power spectrum by using the standard in-in perturbation theory.

They found that spectral index plays a role of a regulator of the divergence. Because of it, the loop correction is enhanced by an inverse factor of 1 − n_s which is a small positive number according to the latest observation. As a result in order for the loop correction to be small enough to warrant the validity of the standard perturbation theory, the amplitude of equilateral non-Gaussianity must be much smaller than the current observational bound.

“Due to the enhancement inversely proportional to the observed red-tilt of the spectral index of curvature perturbation, the correction turns out to be much larger than previously anticipated. As a result, the primordial non-Gaussianity must be much smaller than the current observational bound in order to warrant the validity of cosmological perturbation theory.”

— told Kristiano, first author of the study

Reference: J. Kristiano, Jun’ichi Yokoyama, “Why Primordial Non-Gaussianity is Very Small?”, ArXiv, pp. 1-7, 2021. https://arxiv.org/abs/2104.01953

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