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

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, $Ω_{\rm GW}(f)$, 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 $Ω_{\rm GW}(f)$ for arbitrary sequences of EoS transitions, providing a practical tool to study the pre-BBN expansion history of the Universe.

Figures (7)

• Figure 1: A schematic depiction of the timeline of the universe featuring multiple post-inflationary epochs, each described by a nearly constant equation of state parameter. Note that the horizontal length is not a representation of the actual duration.
• Figure 2: Schematic illustration of representative morphologies of the present-day spectral energy density of inflationary gravitational waves, $\Omega^{(0)}_{\rm GW}(f)$, as a function of frequency $f$ (for various sequences of 7 pre-hot Big Bang epochs after inflation). The solid (dashed) blue curve represents a convex (concave), monotonically increasing spectrum arising from a decreasing (increasing) sequence of post-inflationary EoS parameters $w_1 > w_2 > \cdots > w_n$ (for concave the sequence $w_1 < w_2 < \cdots < w_n$ ), corresponding to an overall blue-tilted spectrum. Similarly the solid (dashed) brown curve denotes a convex (concave), monotonically decreasing spectrum produced by a decreasing (increasing) sequence of EoS parameters $w_1 > w_2 > \cdots > w_{n}$ (for concave shape $w_1 < w_2 < \cdots < w_n$), resulting in a red-tilted spectrum. Non-monotonic spectra are illustrated by the green curve for a single peak and the purple curve for two peaks (and a dip). The solid (dashed) orange curve corresponds to monotonically increasing (decreasing) spectrum which is neither convex nor concave. The solid black curve illustrates an approximately scale-invariant spectrum corresponding to a radiation-like EoS $w=1/3$. The grey-shaded regions indicate the approximate sensitivities of space-based detectors (LISA) and ground-based interferometers (aLIGO), while the red-shaded region marks the BBN constraint. The figure serves to highlight the broad variety of inflationary GW spectral shapes possible for different post-inflationary expansion histories.
• Figure 3: Spectral energy density of inflationary GWs, $\Omega^{(0)}_{\rm GW}(f)$, for the case of monotonic blue-tilted, convex spectra generated by strictly decreasing sequences of post-inflationary EoS parameters $\left(w_1>w_2>\cdots>w_n>w_{\rm rad}=1/3\right)$. The top row contains 5 different post-inflationary epochs, while the bottom row contains 10 epochs. The left column corresponds to an energy scale of $E_{\rm r*}=1 \, \mathrm{GeV}$ at the beginning of hot Big Bang, while the right column corresponds to $E_{\rm r*}=100 \, \mathrm{GeV}$. Transitions between successive epochs are marked by grey vertical lines. The BBN constraint is illustrated by the red-shaded region, and the sensitivity curves for LISA, Planck, and aLIGO are displayed by grey-shaded regions. [None of the curves violate the existing constraints from BBN and aLIGO.]
• Figure 4: Spectral energy density of inflationary GWs, $\Omega^{(0)}_{\rm GW}(f)$, for the case of monotonic blue-tilted, concave spectra generated by strictly increasing sequences of post-inflationary EoS parameters $\left(\bm{w_1 < w_2 < \cdots < w_n} \right)$. The top row contains 5 different post-inflationary epochs, while the bottom row contains 10 epochs. The left column corresponds to an energy scale of $E_{\rm r*}=1 \, \mathrm{GeV}$ at the beginning of hot Big Bang, while the right column corresponds to $E_{\rm r*}=100 \, \mathrm{GeV}$. Transitions between successive epochs are marked by grey vertical lines. The BBN constraint is illustrated by the red-shaded region, and the sensitivity curves for LISA, Planck, and aLIGO are displayed by grey-shaded regions. [None of the curves violate the existing constraints from BBN and aLIGO, except for the red-dashed curve in the bottom-left panel.]
• Figure 5: Spectral energy density of inflationary GWs, $\Omega^{(0)}_{\rm GW}(f)$, for the case of non-monotonic spectra with peak(s), arising when the post-inflationary universe begins with a softer EoS $(w_1<1/3)$, but later evolves to become stiffer $(w > 1/3)$ at some epoch. The top row contains 5 different post-inflationary epochs, while the bottom row contains 10 epochs. The left column corresponds to an energy scale of $E_{\rm r*}=1 \, \mathrm{GeV}$ at the beginning of hot Big Bang, while the right column corresponds to $E_{\rm r*}=100 \, \mathrm{GeV}$. Transitions between successive epochs are marked by grey vertical lines. The BBN constraint is illustrated by the red-shaded region, and the sensitivity curves for LISA, Planck, and aLIGO are displayed by grey-shaded regions. [None of the curves violate the existing constraints from BBN and aLIGO.]