Gravitational wave background as a probe of the primordial black hole abundance (0812.4339v2)

Abstract: Formation of significant number of primordial black holes (PBHs) is realized if and only if primordial density fluctuations have a large amplitude, which means that tensor perturbations generated from these scalar perturbations as a second order effect are also large and comparable to the observational data. We show that pulsar timing observation could find/rule out PBHs with \sim 10^2 M_solar which are considered as a candidate of intermediate-mass black holes and that PBHs with mass range 10^{20-26} g, which serves as a candidate of dark matter, may be probed by future space-based laser interferometers and atomic interferometers.

• The paper demonstrates that the GW background can serve as a probe for PBH abundance through detailed analysis of second-order tensor perturbations.
• It quantifies how pulsar timing arrays and space-based detectors can constrain PBH mass ranges, particularly for candidates of dark matter and intermediate-mass black holes.
• It provides numerical estimates linking primordial density fluctuations to induced gravitational waves, setting observational thresholds for validating PBH models.

Primordial Black Holes and Gravitational Wave Background

The paper by Saito and Yokoyama addresses the potential role of the gravitational wave (GW) background as a probe for the abundance of primordial black holes (PBHs). This paper explores the intricate interrelations between primordial density fluctuations, PBH formation, and induced gravitational waves as explored through observational means. Here, I provide a focused examination of the primary methodologies, results, and implications of their work.

PBHs are hypothesized to form in the early universe from density fluctuations that enter the cosmological horizon during the radiation-dominated phase. These black holes are categorized by their mass spectrum, with those below $10^{15}$ g having evaporated due to Hawking radiation by the present epoch. PBHs with a mass range above this could play crucial astrophysical roles, such as candidates for intermediate-mass black holes (IMBHs) and potential contributors to the dark matter content of the universe.

The formation of PBHs requires significant primordial density fluctuations. Such conditions also lead to large second-order tensor perturbations, potentially comparable to first-order gravitational perturbations. The authors calculate that pulsar timing observations could help identify or exclude PBHs with masses around 100 solar masses, believed to be candidates for IMBHs. They also highlight that PBHs in the mass range of $10^{20}$ to $10^{26}$ grams, possible dark matter candidates, could be probed by future space-based laser interferometers and atomic interferometers.

Key theoretical insights from this work include the exploration of conditions under which the power spectrum of primordial fluctuations shows a peak, which leads to the formation of PBH through subsequent collapse. The authors posit a scenario where the peak amplitude of density fluctuations allows the second-order effects to dominate over the first-order tensor perturbations. Consequently, they suggest the potential for future observational techniques—such as the pulsar timing arrays and space-based GW detectors—to serve as a means to infer the abundance and mass range of PBHs.

Importantly, Saito and Yokoyama provide quantitative estimates of the stochastic GW background produced as a second-order effect. They describe how specific observational thresholds relate to PBH mass constraints, employing the equation of state during the radiation-dominated era to account for tensor perturbations induced by scalar modes. Their computations indicate that the induced stochastic background has a greater amplitude than those generated by the corresponding PBH collapse, thus offering a tangible method to probe the PBH size spectrum.

The potential ability to constrain PBH abundance through GW detection represents a major implication of their research. The spectrum of induced GWs reveals insights into the hyper-local properties of primordial density fluctuations. The authors postulate that long-duration pulsar timing observations could detect these GWs, especially those linked to IMBH-mass PBHs. Furthermore, technical advancements in space-based interferometry could enhance the detection sensitivity across a broader mass range, with LISA, DECIGO, and BBO providing coverage for the entire suggested dark matter PBH spectrum.

In conclusion, the paper's examination of the interplay between tensor perturbations and primordial fluctuations provides a basis for potential empirical validation of PBHs as a component of non-baryonic dark matter. Future enhancements in observational astrophysical tools might enable the successful differentiation and identification of these PBHs, which could profoundly influence our understanding of the early universe's formation and the current dark matter puzzle.