[Paper Review] Scale dependent non gaussianity from generalized features of the inflaton potential

Antonio Enea Romano, Alexander Gallego Cadavid|arXiv (Cornell University)|Apr 11, 2014

Cosmology and Gravitation Theories3 citations

TL;DR

This paper investigates how discontinuities in the n-th derivative of the inflaton potential generate scale-dependent non-Gaussianity in primordial curvature perturbations. Using both numerical simulations and analytical approximations, it shows that the power spectrum exhibits oscillations around scale $k_0$, while the bispectrum displays scale-dependent oscillatory behavior in squeezed and equilateral limits, with amplitude inversely proportional to scale and quadratic suppression on small scales.

ABSTRACT

We study the effects of a general class of features of the inflaton potential on the spectrum and bispectrum of primordial curvature perturbations. These features correspond to a discontinuity in the $n-th$ order derivative of the potential. We provide fully numerical calculations and analytical approximations for the spectrum and the bispectrum, which are in good agreement with each other. The spectrum of primordial perturbations has oscillations around the scale $k_0$ which leaves the horizon at the time $ au_0$ when the feature occurs, with the amplitude and phase of the oscillations determined by the size and the order of the discontinuity. Both in the squeezed and equilateral large scale limit the bispectrum has an oscillatory behavior whose phase depends on the parameters determining the discontinuity, and whose amplitude is inversely proportional to the scale. The small scale bispectrum in the squeezed and equilateral limits have a very similar form and are quadratically suppressed. Given the generality of this class of features it could be used to model phenomenologically different types of non gaussian features encountered in observational data such as the cosmic microwave background radiation or large scale structure.

Motivation & Objective

• To understand the impact of general features in the inflaton potential—specifically discontinuities in the n-th derivative—on primordial curvature perturbations.
• To model phenomenological non-Gaussian features observed in cosmic microwave background and large-scale structure data.
• To provide a unified framework for analyzing spectrum and bispectrum responses to such features using both numerical and analytical techniques.
• To characterize the scale dependence and shape of non-Gaussianity in squeezed and equilateral limits.
• To determine the dependence of oscillation amplitude and phase on the size and order of the discontinuity in the potential.

Proposed method

• Model the inflaton potential with a discontinuity in the n-th order derivative, representing a general class of features.
• Perform fully numerical calculations of the power spectrum and bispectrum using standard inflationary perturbation theory.
• Develop analytical approximations for the spectrum and bispectrum based on the sudden approximation and effective field theory techniques.
• Compare numerical and analytical results to validate the accuracy of the analytical approach across different scales.
• Analyze the squeezed and equilateral limits of the bispectrum to extract shape and scale dependence of non-Gaussianity.
• Derive the scaling behavior of the small-scale bispectrum, showing quadratic suppression in both limits.

Experimental results

Research questions

• RQ1 How do discontinuities in the n-th derivative of the inflaton potential affect the power spectrum of primordial curvature perturbations?
• RQ2 What is the scale dependence and shape of the bispectrum in the squeezed and equilateral limits for such features?
• RQ3 How do the amplitude and phase of oscillations in the spectrum depend on the size and order of the discontinuity?
• RQ4 To what extent do the analytical approximations match the numerical results for the spectrum and bispectrum?
• RQ5 What is the small-scale behavior of the bispectrum, and how does it compare between the squeezed and equilateral configurations?

Key findings

• The power spectrum exhibits oscillations centered at scale $k_0$, with amplitude and phase determined by the size and order of the discontinuity in the potential.
• The bispectrum shows oscillatory behavior in both the squeezed and equilateral limits, with phase dependent on the parameters of the discontinuity.
• The amplitude of the bispectrum is inversely proportional to the scale, indicating a suppression at smaller scales.
• The small-scale bispectrum in both the squeezed and equilateral limits has a very similar form and is quadratically suppressed.
• Numerical and analytical calculations of the spectrum and bispectrum are in good agreement, validating the analytical approach.
• The general class of features studied can serve as a phenomenological model for diverse non-Gaussian features observed in CMB and large-scale structure data.