Inflationary Gravitational Waves as a probe of the unknown post-inflationary primordial Universe

TL;DR

This work assesses inflationary gravitational waves as probes of the unknown post-inflationary primordial Universe by modeling the reheating era as a sequence of sharp, piecewise-constant equation-of-state epochs $w_i$. It derives analytic tensor-mode solutions across epochs using Bessel functions and Israel junction conditions, yielding the present-day spectrum $\Omega_{\rm GW}(f)$ in terms of the $w_i$ and their durations, with the tilt in each epoch given by $n_{\rm GW} = 2\left(\frac{w_i-1/3}{w_i+1/3}\right)$. The paper forecasts detectability for upcoming GW observatories under BBN and CMB constraints, showing that multi-epoch stiff phases can produce blue-tilted spectra accessible to detectors such as LISA, BBO, DECIGO, CE, and ET, especially when the reheating temperature is favorable. It illustrates these ideas with a String-inspired scenario featuring a kination, RD, and early matter-dominated sequence, and discusses the broader implications for early-Universe cosmology and the limitations of single-EoS assumptions. Overall, the results highlight how a richer, multi-epoch post-inflationary history can expand the viable parameter space for observable inflationary GWs and motivate Bayesian inference to disentangle degeneracies among epochs.

Abstract

One of the key predictions of the standard inflationary paradigm is the quantum mechanical generation of the transverse and traceless tensor fluctuations due to the rapid accelerated expansion of space, which later constitute a stochastic background of primordial gravitational waves (GWs). The amplitude of the (nearly) scale-invariant inflationary tensor power spectrum at large scales provides us with crucial information about the energy scale of inflation in the case of the minimal inflaton coupling to gravity. Furthermore, the spectral energy density, $Ω_{_{\rm GW}}(f)$, of the GWs at sufficiently small scales (or, large frequencies $f$) serves as an important observational probe of post-inflationary primordial dynamics. In fact, the small-scale spectral tilt, $n_{_{\rm GW}} = \frac{{\rm d}\log{Ω_{_{\rm GW}}}}{{\rm d}\log{f}}$, of the spectral energy density of GWs is sensitive to the (unknown) post-inflationary equation of state (EoS), $w$, of the universe; with a softer EoS ($w < 1/3$) leading to a red tilt: $n_{_{\rm GW}} < 0$, while a stiffer EoS ($w > 1/3$) resulting in a blue tilt: $n_{_{\rm GW}} > 0$. The post-inflationary dynamics, however, is generically expected to be quite complex, potentially involving a number of distinct phases. Hence, in this work, we discuss the possibility of multiple sharp transitions, namely $w_1 \to w_2 \to w_3 \to ... \to w_n$, in the EoS of the post-inflationary universe and compute the corresponding spectral energy density of the inflationary GWs. We explicitly determine the region of the parameter space $\lbrace{ w_1, \, w_2, \, w_3, ..., w_n\rbrace}$ which leads to a potentially detectable signal in the upcoming GW detectors, without violating the current constraints.

Figures (14)

• Figure 1: A schematic depiction of the timeline of the universe with a post-inflationary reheating phase consisting of multiple equations of state. Note, the horizontal length is not a representation of the actual duration.
• Figure 2: The spectral energy density of first-order inflationary GWs is shown as a function of the present-day GW frequencies for a single constant post-inflationary EoS during reheating; along with the sensitivity curves of various GW detectors schmitz_2020_3689582, as well as the BBN constraint. The spectrum is shown for three different possibilities of reheating EoS above $f \gtrsim 10^{-8}\,{\rm Hz}$, corresponding to a blue-tilted stiff matter phase with $w_{_{\rm re}} = 0.8$ (solid black), radiative phase with $w_{_{\rm re}} = 1/3$ (dotted black), and softer matter-like phase with $w_{_{\rm re}} = 0$ (dashed black). The tensor-to-scalar ratio has been set to $r = 0.001$ and the reheating energy scale is fixed at $E_{\rm r*} = 1$ GeV. Note, the spectrum depicted in solid line ($w_{_{\rm re}}= 0.8$) violates the BBN constraint, in line with Ref. Figueroa:2019paj. The maximum frequency considered here (the rightmost point in the plot) is the UV cut-off frequency, $f_e = 6.37 \times 10^7$ Hz, which corresponds to $k_e = 4.05 \times 10^{22} \, {\rm Mpc}^{-1}$ for $r = 0.001$.
• Figure 3: The spectral energy density ($\Omega_{_{\rm GW}}$) of inflationary GWs associated with two (top row), three (middle row) and four (bottom row) epochs of piece-wise constant EoS parameters during reheating. Plots in the left column correspond to scenarios where $\Omega_{_{\rm GW}}$ has a minimal overlap with the LISA sensitivity curve, while those in the right column belong to scenarios where $\Omega_{_{\rm GW}}$ has a substantial overlap with LISA sensitivity region, without violating the existing BBN and aLIGO constraints. The energy scales $E_i$ (with $i \in \mathbb{Z}^+$) corresponding to the end of each epoch, are given in the top labels of each plot. The energy scales at the end of reheating and inflation are the same in all plots, namely, $E_{\rm r*} = 1$ GeV and $E_{\rm inf} = 5.76 \times 10^{15}$ GeV respectively, so as the tensor-to-scalar ratio, $r = 0.001$. The maximum frequency considered here (the rightmost point in the plot) is the UV cut-off frequency, $f_e = 6.37 \times 10^7$ Hz, which corresponds to $k_e = 4.05 \times 10^{22} \, {\rm Mpc}^{-1}$ for $r = 0.001$.
• Figure 4: Spectral energy density of inflationary GWs at the present epoch, for the case of a two-epoch reheating scenario, with $r=0.001$, $E_1 = 10^{10}\,{\rm GeV}$ and $E_{{\rm r}*} = 1\,{\rm GeV}$. The solid line corresponds to $w_1 = 1/3$ which easily satisfies the BBN constraint (\ref{['eq; BBN constraint integral']}). Therefore, $w_1 \leq 1/3$, the BBN constraint is guaranteed to be satisfied. However, we find that the BBN constraint is bound to be violated when $w_1 \geq 0.417$, (the dashed line is plotted for $w_1 =0.416$. The UV cut-off frequency is $f_e = 6.37 \times 10^7$ Hz which corresponds to $k_e = 4.05 \times 10^{22} \, {\rm Mpc}^{-1}$ for $r = 0.001$.
• Figure 5: The parameter space for two-epoch reheating phase with EoS parameters $w_1$ and $w_2$ that leads to a potentially detectable GW signal in the LISA, BBO, DECIGO, CE, ET, and aLIGO detectors. The grey-shaded region in each plot corresponds to the combination of $w_1$ and $w_2$ that violate the BBN constraint, while the maroon-shaded region represents those that are ruled out by the aLIGO null detection. Energy scale of the universe at the end of the first post-inflationary epoch is labelled above in each plot. Note that, the duration of each epoch (number of $e$-folds) has been provided opposite to the corresponding EoS axis.