Yukawa coupling, and inflationary correlation functions for a spectator scalar via stochastic spectral expansion

TL;DR

The paper analyzes a stochastic spectator scalar field coupled to a massless fermion through a Yukawa interaction during inflation in de Sitter space, employing a quintessence-like potential $V(\phi)=\alpha|\phi|^p$ to ensure a bounded total potential. Using the stochastic spectral expansion, it computes the two-point and three-point correlation functions, obtaining a blue-tilted power spectrum $\mathcal{P}^{(2)}_\phi$ with $n_s-1=2E_1/H$ and a bispectrum characterized by a squeezed-limit shape function that depends on the Yukawa coupling $g$. The results show that increasing $g$ pushes the spectrum toward blue, flattens the squeezed-limit peak of the shape function, and enhances the local non-Gaussianity parameter $f^{\rm loc}_{NL}$, potentially exceeding Planck bounds for large $g$. The work provides a robust numerical framework for stochastic inflation with fermionic Yukawa interactions and suggests extensions to quasi-de Sitter dynamics, massive fermions, and curvature perturbations via stochastic $\delta N$ or RG-improved potentials.

Abstract

We consider a stochastic spectator scalar field coupled to fermion via the Yukawa interaction, in the inflationary de Sitter background. We consider the fermion to be massless, and take the one loop effective potential found earlier by using the exact fermion propagator in de Sitter spacetime. We take the potential for the spectator scalar to be quintessence-like, $V(φ)=α|φ|^p$ ($α\ensuremath{>} 0,\ p\ensuremath{>} 4$), so that the total effective potential is generically bounded from below for all values of the parameters and couplings, and a late time equilibrium state is allowed. Using next the stochastic spectral expansion method, we numerically investigate the two point correlation function, as well as the density fluctuations corresponding to the spectator field, with respect to the three parameters of the total effective potential, $α,\ p$ and the Yukawa coupling, $g$. In particular, we find that the power spectrum and the spectral index corresponds to blue tilt with increasing $g$. The three point correlation function and non-Gaussianity corresponding to the density fluctuation has also been investigated. The increasing Yukawa coupling is shown to flatten the peak of the shape function in the squeezed limit. Also in this limit, the increase in the same is shown to increase the local non-Gaussianity parameter.

Figures (14)

• Figure 1: Plot of the effective potential \ref{['cy11']}, \ref{['scpot']}, with parameters $\bar{\alpha}=0.1$, $p=4.001$ and different values of the Yukawa coupling.
• Figure 2: The variation of the quantity $\bar{W}(\bar{\alpha},y)$, \ref{['wphi2']}, which appears in the effective Schrödinger equation \ref{['eq:schrodinger equation']}. The double well behaviour comes from the symmetry of our effective potential, \ref{['eq:eff-pot']}.
• Figure 3: Plots of the lowest three of the wavefunctions $\psi_n(\bar{\alpha},p,g,y)$,\ref{['eq:schrodinger equation']}, \ref{['eq:eff-pot']}, \ref{['wphi2']}. Parameters $\bar{\alpha}$ and $p$ are held fixed and the Yukawa coupling $g$ is varied. The wavefunction is complex, and we have plotted its real part. The imginary part shows the same qualitative behaviour.
• Figure 4: Variation of the spectral index and the power spectrum, respectively \ref{['spec']}, \ref{['P1']}, with respect to the dimensionless parameter $\bar{\alpha}$, \ref{['wphi2']}. We have taken different values of the Yukawa coupling but the parameter $p$ is held fixed.
• Figure 5: Variation of the spectral index and the power spectrum, respectively \ref{['spec']}, \ref{['P1']}, with respect to the dimensionless parameter $\bar{\alpha}$, \ref{['wphi2']}. We have taken different values of the parameter $p$, whereas the Yukawa is held fixed.