Small scale induced gravitational waves from primordial black holes, a stringent lower mass bound, and the imprints of an early matter to radiation transition

Nilanjandev Bhaumik and Rajeev Kumar Jain

Phys. Rev. D 104, 023531 – Published 28 July 2021

Abstract

In all inflationary scenarios of primordial black hole (PBH) formation, amplified scalar perturbations inevitably accompany an induced stochastic gravitational wave background (ISGWB) at smaller scales. In this paper, we study the ISGWB originating from the inflationary model, introduced in our previous paper [N. Bhaumik and R. K. Jain, Primordial black holes dark matter from inflection point models of inflation and the effects of reheating, J. Cosmol. Astropart. Phys. 01 (2020) 037] wherein PBHs can be produced with a nearly monochromatic mass fraction in the asteroid mass window accounting for the total dark matter in the universe. We numerically calculate the ISGWB in our scenario for frequencies ranging from nano-Hz to kHz that covers the observational scales corresponding to future space-based gravitational wave (GW) observatories such as IPTA, LISA, DECIGO, and Einstein Telescope. Interestingly, we find that ultralight PBHs ( $M_{PBH} \sim 10^{- 20} M_{⊙}$ ), which shall completely evaporate by today with an exceedingly small contribution to dark matter, would still generate an ISGWB that may be detected by a future design of the ground-based Advanced LIGO detector. Using a model-independent approach, we obtain a stringent lower mass limit for ultralight PBHs which would be valid for a large class of ultra slow roll inflationary models. Further, we extend our formalism to study the imprints of a reheating epoch on both the ISGWB and the derived lower mass bound. We find that any noninstantaneous reheating leads to an even stronger lower bound on the PBH mass and an epoch of a prolonged matter-dominated reheating shifts the ISGWB spectrum to smaller frequencies. In particular, we show that an epoch of an early matter-dominated phase leads to a secondary amplification of the ISGWB at a much smaller scale corresponding to the smallest comoving scale leaving the horizon during inflation or the end of the inflation scale. Finally, we discuss the prospects of the ISGWB detection by the proposed and upcoming GW observatories.

Received 15 April 2021
Accepted 9 July 2021

DOI:https://doi.org/10.1103/PhysRevD.104.023531

Physics Subject Headings (PhySH)

Dark matter Gravitational waves Inflation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Nilanjandev Bhaumik ^* and Rajeev Kumar Jain ^†

Department of Physics, Indian Institute of Science, Bangalore 560012, India

^*nilanjandev@iisc.ac.in
^†rkjain@iisc.ac.in

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Vol. 104, Iss. 2 — 15 July 2021

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Images

Figure 1
On the left, we plot the power spectra of primordial curvature perturbations $P_{R}$ (solid curves) and primordial tensor perturbations $P_{h}$ (dashed curves) for different parameters of the scenario proposed in our earlier work [69]. All these spectra show a similar enhancement at smaller scales, required for the abundant PBH production. Also shown are the relevant constraints from cosmic microwave background (CMB) spectral distortions [82, 84]. On the right, we plot the spectral energy density of the ISGWB corresponding to the spectra on the left. All the GW spectra also show similar behavior. In particular, a bump in $P_{R}$ on small scales leads to a peak in $Ω_{GW} h^{2}$ on larger frequencies which fall in the sensitivity regimes of various future space-based GW observatories such as IPTA, LISA, DECIGO, and ET. The color coding of different plots is consistent across the two figures.
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Figure 2
On the left, we plot the initial PBH mass fraction $β_{ini}$ (at formation) for two different cases, produced in our model [69] in the very low mass range corresponding to $M \sim 10^{- 20} - 10^{- 21} M_{⊙}$ , as well as the observational constraints arising from BBN and the extragalactic $γ$ -ray background. Note that while comparing the constraints on $β_{ini}$ , we are assuming the mass fractions of our model to be monochromatic. Such small-mass PBHs would have been completely evaporated by today due to Hawking radiation. However, they will still induce an observable ISGWB in the higher frequency range which falls in the design sensitivity contours of the Advanced LIGO detector and therefore can, in principle, be detected in future runs, as shown on the right.
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Figure 3
On the left, we have plotted the growing behavior of $ε$ before the transition, as obtained in Eq. (3.15). It is evident that for all values of $b_{6}$ in the range $\sqrt{3} < b_{6} \leq 5 / \sqrt{3}$ , $η$ saturates to a constant value greater than $- 1$ . On the right is the largest value of $ε_{s}$ and the corresponding number of $e$ -folds whenever $η$ reaches its asymptotic value $η_{s}$ .
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Figure 4
On the left, the ISGWB energy density $Ω_{GW} h^{2}$ has been plotted for different reheating histories, as shown in the inset. On the right is the ISGWB energy density for the same reheating history, but for two different scalar power spectra peaking at different wavenumbers $k$ , leading to the same amplitude at different frequencies for the first peak but different amplitudes at the same frequency for the second peak.
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Figure 5
Comparison of contributions from the pure RD approximation (light red) for the $k ≪ k_{r}$ limit and from the $k ≫ k_{r}$ limit result (light blue) along with the full numerical result (solid black).
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