Large and strong scale dependent bispectrum in single field inflation from a sharp feature in the mass

Abstract

We study an inflationary model driven by a single minimally coupled standard kinetic term scalar field with a step in its mass modeled by an Heaviside step function. We present an analytical approximation for the mode function of the curvature perturbation, obtain the power spectrum analytically and compare it with the numerical result. We show that, after the scale set by the step, the spectrum contains damped oscillations that are well described by our analytical approximation. We also compute the dominant contribution to the bispectrum in the equilateral and the squeezed limits and find new shapes. In the equilateral and squeezed limits the bispectrum oscillates and it has a linear growth envelope towards smaller scales. The bispectrum size can be large depending on the model parameters. * arrojaf@ewha.ac.kr

On the scalar consistency relation away from slow roll

“…One finds that the transition leads to large values for the second slow roll parameter ǫ 2 and, importantly, the quantityǫ 2 grows to be even larger, in fact, behaving as a Dirac delta function at the transition. As we shall discuss, it is this behavior that leads to the most important contribution to the scalar bi-spectrum in the model [36,42,43]. Clearly, it would be convenient to divide the evolution of the background quantities and the perturbation variables into two phases, before and after the transition at φ 0 .…”

“…The above bispectrum goes to a constant value at large scales, while it is found to oscillate with a constant amplitude in the small scale limit. In the equilateral limit, the contribution at the transition is known to lead to a term that grows linearly with k at large wavenumbers [42,43]. This essentially arises due to the infinitely sharp transition in the Starobinsky model.…”

“…It should be clear that the first term in the above expression involving V φφ will lead to a Dirac delta function due to the discontinuity in the first derivative of the potential in the case of the Starobinsky model. Hence, the dominant contribution toǫ 2 at the transition can be written as [42,43] 14) where a 0 denotes the value of the scale factor when η = η 0 . Post transition, the dominant contribution toǫ 2 is found to be [36] …”

“…The first slow roll parameter before and after the transition is found to be [36,[41][42][43] 11b) where ∆A = A − − A + , H 0 is the Hubble parameter determined by the relation…”

“…However, due to the transition, the modes after the transition are related by the Bogoliubov transformations to the modes before the transition. Therefore, the scalar mode and its time derivative before the transition can be written as [36,[41][42][43][44]:…”