Gravitational waves from a universe filled with primordial black holes (2010.11573v3)

Abstract: Ultra-light primordial black holes, with masses $m_\mathrm{PBH}<10^9\mathrm{g}$, evaporate before big-bang nucleosynthesis and can therefore not be directly constrained. They can however be so abundant that they dominate the universe content for a transient period (before reheating the universe via Hawking evaporation). If this happens, they support large cosmological fluctuations at small scales, which in turn induce the production of gravitational waves through second-order effects. Contrary to the primordial black holes, those gravitational waves survive after evaporation, and can therefore be used to constrain such scenarios. In this work, we show that for induced gravitational waves not to lead to a backreaction problem, the relative abundance of black holes at formation, denoted $ \Omega_\mathrm{PBH,f} $, should be such that $ \Omega_\mathrm{PBH,f} <10^{-4}(m_\mathrm{PBH}/10^9\mathrm{g})^{-1/4}$. In particular, scenarios where primordial black holes dominate right upon their formation time are all excluded (given that $m_\mathrm{PBH}>10\, \mathrm{g}$ for inflation to proceed at $\rho^{1/4}<10^{16}\mathrm{GeV}$). This sets the first constraints on ultra-light primordial black holes.

• The paper establishes the first constraints on ultra-light primordial black holes by analyzing the gravitational wave background they induce.
• It employs second-order cosmological perturbation theory with a monochromatic PBH distribution to calculate the stochastic gravitational wave spectrum.
• The results imply that PBHs above 10 g could not dominate the early universe, guiding future gravitational wave detection and cosmological modeling.

Overview of Gravitational Waves from a Universe Filled with Primordial Black Holes

This paper investigates the potential for ultra-light primordial black holes (PBHs) to have significant cosmological implications through their contribution to the production of gravitational waves (GWs). The paper is primarily concerned with PBHs that evaporate before the onset of big-bang nucleosynthesis (BBN), thereby circumventing direct observational constraints. Despite their ephemeral existence, these black holes could have been abundant enough to dominate the universe's energy content temporarily, which in turn could induce the formation of GWs.

Key Results

The authors provide the first constraints on the formation and abundance of ultra-light PBHs by evaluating the gravitational wave background they induce. The analysis shows that for induced GWs to avoid causing a backreaction problem, the relative abundance of PBHs at formation must be less than $10^{-4} \times (m/10^9\, \mathrm{g})^{-1/4}$. This constraint inherently excludes scenarios where PBHs dominate the universe immediately upon their formation if their masses are greater than $10\, \mathrm{g}$. Such a mass threshold is necessary for inflation with $\rho^{1/4} < 10^{16}\, \mathrm{GeV}$.

Methodology

The authors initially model the PBH distribution assuming a homogeneous, isotropic gas of monochromatically mass-distributed PBHs. This configuration enables them to calculate the initial density contrast fluctuations associated with the PBHs, which are of Poissonian type. These scalar fluctuations are further shown to induce tensor perturbations, resulting in a quantifiable stochastic background of GWs by employing second-order cosmological perturbation theory.

During a PBH-dominated epoch, the paper outlines how PBHs contribute increasingly to the universe's energy density until their evaporation through Hawking radiation. The gravitational potential's power spectrum, calculated post-PBH formation, serves as a baseline for estimating the corresponding tensor power spectrum of the GWs.

Implications and Future Work

The detected constraints imply that ultra-light PBHs cannot have dominated the universe from their formation onward if they exist today, nudging researchers toward exploring other evolutionary scenarios. This limitation provides a base for more precise cosmological models that incorporate PBHs as viable constituents.

Furthermore, the identified GW signatures hold the potential to guide observers in forthcoming experiments. Arrays such as the Einstein Telescope, LISA, and the Square Kilometre Array might detect these induced GWs, offering indirect evidence of such PBHs, thereby enriching our comprehension of the early universe scenarios.

Future research could focus on resolving the dynamics of induced gravitational waves during the transitional epoch between a PBH-dominated phase and radiation dominance more comprehensively. This could entail detailed numerical simulations or advanced analytical techniques to improve understanding.

Conclusion

By establishing a critical connection between PBH population dynamics and GW production, the research bridges theoretical predictions and empirical investigation frameworks. The derived constraints inject fresh perspectives into the plausible mass spectrum for PBHs and open avenues for leveraging GWs as a tool to probe unseen corners of the cosmological epoch post-inflation and pre-BBN. As observational technologies advance, such predictions will be invaluable for testing foundational aspects of cosmic evolution and the role of PBHs therein.