Observable induced gravitational waves from an early matter phase

TL;DR

This work investigates gravitational waves induced by second-order scalar perturbations in a Universe that undergoes an early matter-dominated epoch after inflation, with reheating temperatures $T_r<10^9$ GeV. By employing hilltop and running-mass inflation models that boost small-scale power, the authors compute the resulting GW spectrum ${\cal P}_h(k)$ during the early matter and subsequent radiation-dominated eras, including analytical estimates and full numerical results, and connect these signals to detector sensitivities. They relate PBH constraints to the permissible small-scale power and determine how $T_r$ and the nonlinear cutoff $k_{NL}$ shape the observable GW signal across experiments such as DECIGO, BBO, and LIGO/KAGRA. A key finding is that lower $T_r$ can yield induced GW energy densities within current or near-future detector reach for specific model parameters, offering a powerful probe of early-Universe dynamics; the results are considered upper bounds due to the simplifying sudden-transition approximation. The study thus links small-scale inflationary processes, PBH production, and gravitational-wave observables in a testable framework.

Abstract

Assuming that inflation is succeeded by a phase of matter domination, which corresponds to a low temperature of reheating $T_r<10^9\rm{GeV}$, we evaluate the spectra of gravitational waves induced in the post-inflationary universe. We work with models of hilltop-inflation with an enhanced primordial scalar spectrum on small scales, which can potentially lead to the formation of primordial black holes. We find that a lower reheat temperature leads to the production of gravitational waves with energy densities within the ranges of both space and earth based gravitational wave detectors.

Figures (12)

• Figure 1: The PBH bound on the primordial spectrum for various cut-off masses, as defined in the figure legend.
• Figure 2: A schematic of the evolution of tensor modes on different scales with respect to the logarithm of the scale factor. The solid red line depicts the evolution of the source term between epochs. $a_r$ is the scale factor at reheating and $a_{eq}$ is the scale factor at radiation-matter equality. Grey lines represent super-horizon modes. The black dash-dot line represents the evolution of a mode which enters during EMD. The blue dash-dot line represents the amplitude of the mode which enters during RD and the green line, which is barely visible at the right hand side of the plot, represents the mode which enters during the current epoch (assuming no acceleration).
• Figure 3: The solid lines represent the source term, the red is the sudden transition approximation Eq. (\ref{['eq:sudden']}), the green is the smooth turnover with $n=1$, and the black is $n=4$. The dashed lines represent the tensor modes. As we can see the tensor modes do approach the freely oscillating limit, but there is some loss of amplitude with respect to the sudden transition approximation
• Figure 4: We plot the spectrum of induced gravitational waves for a flat primordial spectrum with $T_r=10^9\rm{GeV}$. The spectrum for scales that re-enter the horizon deep in the radiation era have a flat spectrum, due to the fact that the source term is a decaying function and thus the modes oscillate freely, and is represented by the solid blue line. The solid black line represents the modes that re-enter the horizon during the early matter phase ($k>k_R$). The The red dashed line is the complete spectrum, assuming an early matter phase followed by a phase of radiation domination. The spectrum for $k>k_{NL}$ behaves as $\mathcal{P}_h\propto 1/k^4$, as $\mathcal{P}_h\propto 1/k$ for $k_r<k<k_{NL}$, as $\mathcal{P}_h\propto k^3$ for $k\lesssim k_r$ and as $\mathcal{P}_h\sim \rm{constant}$ for $k\ll k_r$. One can think of this as follows: modes that re-enter the horizon during the radiation phase but with $k\sim k_r$, i.e. near the EMD phase, will interact with modes that re-entered during EMD and hence their behaviour/characteristics are modified from the instant reheating scenario. We have utilised a simplified analysis, in that $\mathcal{P}_h(k)=\mathcal{P}_{h_{matter}}(k)+\mathcal{P}_{h_{rad}}(k)$.
• Figure 5: We plot the spectrum of induced gravitational waves for a flat primordial spectrum with various reheat temperatures and the cutoff scale $k_{max}=k_{NL}$. The black dashed lines correspond to the PBH bound assuming (from left pseudo-vertical line to right pseudo-vertical line) $T_r=1\rm{GeV},10\rm{GeV},10^2\rm{GeV},10^3\rm{GeV},10^4\rm{GeV},10^5\rm{GeV},10^6\rm{GeV},10^7\rm{GeV}, 10^8\rm{GeV}$ and $10^9\rm{GeV}$. The cluster of solid lines in the top right corner are the sensitivity ranges of ground based detectors, LIG0 S5 and S6 ligo, and KAGRA kagra, while the thick horizontal salmon pink line is the forecast sensitivity of Advanced LIGO adv_ligoAbbott:2009ws. Also shown is the sensitivity limit of the Square Kilometre Array (SKA) JenetHobbsYardleyPPTA. The green, red, blue and purple solid lines correspond to taking $T_r=1\rm{MeV}$, $1\rm{GeV}$, $10^4\rm{GeV}$, and $10^8\rm{GeV}$. The blue dashed line is the spectrum for $T_r=10^4\rm{GeV}$ without terminating at $k_{NL}$. It is interesting to note that even a flat primordial spectrum can lead to a spectrum of induced gravitational waves detectable by cross-correlated DECIGO