The Effect of $\alpha$-Vacua on the Scalar and Tensor Spectral Indices: Slow-Roll Approximation

TL;DR

This paper investigates how $oldsymbol{eta}$-vacua initial conditions and a physical momentum cutoff influence the scalar and tensor spectral indices during inflation, revealing $q$-dependent oscillatory corrections up to first order in slow-roll parameters.

Contribution

It introduces a $q$-dependent correction to spectral indices arising from $oldsymbol{eta}$-vacua with a momentum cutoff, extending previous inflationary perturbation analyses.

Findings

Spectral index corrections depend on wave number $q$ with oscillatory behavior.

Leading order corrections involve $q$-dependent amplitude and oscillations.

Corrections are calculated up to first order in slow-roll parameters.

Abstract

Since the duration of inflation is finite, imposing the initial condition in infinite past, i.e. the Bunch-Davies vacuum, is inherently ambiguous. In this paper, we resort to the mixed states as initial condition which are called the $\alpha$-vacua and then introduce a physical momentum cutoff $\Lambda$ [Danielsson:2002kx], in which the evolution of perturbations begins. We show that the initial time $t_i$, when the initial condition is imposed, depends on the wave number of fluctuation, as it is for the time of horizon crossing, $t_q$. Then we calculate the corrections to the scalar and tensor power spectra and their corresponding spectral indices. Throughout this work, the calculation is done up to the first order in slow-roll parameters. We indicate that the leading order corrections to the spectral indices have a $q$-dependent amplitude, $2\epsilon_{f}[2\epsilon_{f}(\frac{q}{q_{f}})^{4\epsilon+4\eta}-\eta_{f}(\frac{q}{q_{f}})^{3\epsilon+\xi}]$ times a $q$-dependent oscillatory part, $\cos(\frac{2\Lambda(q/q_{f})^{\epsilon}}{H_{f}})$, where $H$, $\epsilon$, $\eta$, and $\xi$ are the Hubble and slow-roll parameters respectively, and the subscript $f$ denotes that these quantities are evaluated at the time when the first scale, $q_{f}$, satisfies the initial condition, i.e. $q=a(t_i)\Lambda$.