Morphology of Inflationary Gravitational Wave Spectra imprinted by a Sequence of Post-Inflationary Epochs $via$ ${\rm GWInSpect}$

TL;DR

This paper explores how multiple post-inflationary epochs with different equations of state shape the spectrum of primordial gravitational waves, revealing diverse spectral features and providing a computational tool for future observational studies.

Contribution

It extends an analytical framework to systematically analyze the morphological diversity of inflationary GW spectra from multi-epoch histories and introduces the GWInSpect Python package for practical computations.

Findings

Spectral shapes can be convex, concave, or multi-peaked depending on epoch sequences.

A wide variety of spectral profiles naturally emerge from different post-inflationary histories.

The GWInSpect tool enables efficient computation of spectra for arbitrary EoS sequences.

Abstract

The expansion history of the Universe prior to Big Bang Nucleosynthesis (BBN) remains largely unconstrained. The high-energy post-inflationary era may involve multiple distinct epochs, each characterized by a different equation of state (EoS). A key prediction of inflation is the generation of tensor perturbations that later manifest as a stochastic background of primordial gravitational waves (GWs). The large-scale amplitude and small-scale spectral tilt ($n_{\rm GW}$) of these GWs encode the inflationary energy scale and the subsequent expansion history, respectively. A soft post-inflationary EoS ($w<1/3$) yields red-tilted GW spectra ($n_{\rm GW}<0$), while a stiff EoS ($w>1/3$) results in a blue-tilt ($n_{\rm GW}>0$). In our previous work [arXiv:2407.07956], we developed an analytical framework for computing the GW spectral energy density, $\Omega_{\rm GW}(f)$, for multiple post-inflationary transitions ($w_1 \to w_2 \to \cdots \to w_n \to 1/3$), focusing on the parameter space relevant for future GW observations. In this paper, we extend that framework to systematically investigate the $morphological~diversity$ of inflationary GW spectra generated by multi-epoch post-inflationary histories. Remaining model agnostic, we demonstrate that a wide variety of spectral shapes, ranging from convex and concave monotonic profiles to multi-peaked non-monotonic spectra, can naturally emerge depending on the sequence and duration of these epochs. We also introduce GWInSpect, a publicly available Python package that computes $\Omega_{\rm GW}(f)$ for arbitrary sequences of EoS transitions, providing a practical tool to study the pre-BBN expansion history of the Universe.