Observable Spectra of Induced Gravitational Waves from Inflation

TL;DR

This work targets the problem of constraining the primordial power spectrum on very small scales by leveraging induced gravitational waves (GWs) generated at second order by scalar perturbations during the radiation era, with PBH bounds providing additional constraints on $ ext{P}_ ext{ζ}$. It analyzes two inflationary scenarios—Hilltop-type and running mass—that can enhance small-scale power without violating large-scale constraints, computing the induced GW spectrum $P_h(k)$ and comparing with sensitivity curves for LISA, BBO/DECIGO, and related detectors. The main findings are that Hilltop-type models with integral couplings (notably $p=2$, $q=3$) can produce observable $ ext{Ω}_{GW}$ within BBO/DECIGO bands for a reasonable number of e-folds, while running mass models can approach detector sensitivities but typically require an unusually small $N$ and can yield PBHs in the dark matter mass range under favorable thermal histories. These results provide testable predictions for upcoming GW observatories and establish a bridge between small-scale inflationary features, induced GWs, and PBH dark matter scenarios, highlighting how future measurements can discriminate between Hilltop-type and running mass mechanisms.

Abstract

Measuring the primordial power spectrum on small scales is a powerful tool in inflation model building, yet constraints from Cosmic Microwave Background measurements alone are insufficient to place bounds stringent enough to be appreciably effective. For the very small scale spectrum, those which subtend angles of less than 0.3 degrees on the sky, an upper bound can be extracted from the astrophysical constraints on the possible production of primordial black holes in the early universe. A recently discovered observational by-product of an enhanced power spectrum on small scales, induced gravitational waves, have been shown to be within the range of proposed space based gravitational wave detectors; such as NASA's LISA and BBO detectors, and the Japanese DECIGO detector. In this paper we explore the impact such a detection would have on models of inflation known to lead to an enhanced power spectrum on small scales, namely the Hilltop-type and running mass models. We find that the Hilltop-type model can produce observable induced gravitational waves within the range of BBO and DECIGO for integral and fractional powers of the potential within a reasonable number of e-folds. We also find that the running mass model can produce a spectrum within the range of these detectors, but require that inflation terminates after an unreasonably small number of e-folds. Finally, we argue that if the thermal history of the Universe were to accomodate such a small number of e-folds the Running Mass Model can produce Primordial Black Holes within a mass range compatible with Dark Matter, i.e. within a mass range 10^{20}g< M_{BH}<10^{27}g.

Figures (12)

• Figure 1: Upper limits on $\mathcal{P}_\zeta$ from PBH constraints. For simplicity, $\gamma$ is set to be unity. The different coloured solid lines correspond to bounds from different phenomena and experiments as summarised in table \ref{['tab:pbh']}. For comparison, the dotted lines are included to indicate CMB constraints on accreting PBHs; the thin-dashed line illustrates a potential constraint from future $21\,\mathrm{cm}$ line experiments; the thick-dashed line is the limit necessary to avoid excessive generation of entropy and Planck mass relics ($k \lessgtr 2.1 \times 10^{23}\,\mathrm{Mpc}^{-1}$). For the latter the reheating temperature is assumed to be higher than $10^{16}\,\mathrm{GeV}$. See Carr:2009jm for details.
• Figure 2: An illustration of the Hilltop-type and running models. In our scenario scales of cosmological interest during the hilltop regime (indicated with red), and the end of inflation occurs once the inflaton has reached a flatter region of the potential (indicated with blue).
• Figure 3: The scalar spectra of the hilltop model terminating at $N=65$ ($\log_{10}(k/(0.002[\rm{Mpc}^{-1}]))\sim28.2$) while ensuring that $\mathcal{P}(N=60)$ is less than the PBH bound at that scale. The cross-hatched region is the PBH constraint.
• Figure 4: Scalar spectra from hilltop inflation with self-coupling powers labelled in the legend in Fig. (\ref{['spectra_hill3']}). The plot corresponds to maximising the spectrum at $N=60$ ($\log_{10}(k/(0.002[\rm{Mpc}^{-1}]))\sim26)$. The cross-hatched region is the PBH constraint.
• Figure 5: Scalar spectra from hilltop inflation with self-coupling powers labelled in the legend in Fig. (\ref{['spectra_hill3']}). The plot corresponds to maximising the first order spectrum at $N=55$ ($\log_{10}(k/(0.002[\rm{Mpc}^{-1}]))\sim23.9$), and the cross hatched region is the PBH bound