Primordial black holes in Randall-Sundrum: Cosmological signatures

We reconsider primordial black hole physics in Randall-Sundrum Type-II universes, focusing on constraints from cosmological and astrophysical observables. We pay particular attention to scenarios that allow the entirety of dark matter to be in the form of higher-dimensional primordial black holes. This is possible for a range of AdS radii and black hole masses. Observable constraints are generally modified due to the changes in the higher-dimensional gravitational sector, and come from low-energy $e^{\pm}$ emission, microlensing, and possibly from contributions to unresolved radiation backgrounds. We discuss constraints from the cosmic microwave background due to injection of Hawking quanta into the intergalactic medium. Finally, we comment on recent discussions on the compatibility of higher-dimensional black holes and the KM3-230213A event.

💡 Research Summary

This paper revisits the physics of primordial black holes (PBHs) in the Randall‑Sundrum Type‑II (RS‑II) braneworld, with the explicit goal of delineating the observationally allowed window in which PBHs could constitute the entirety of dark matter. The authors begin by reviewing the RS‑II setup: a (3+1)‑dimensional brane embedded in a five‑dimensional anti‑de Sitter (AdS) bulk characterized by a curvature radius ℓ. The bulk cosmological constant Λ₅ = –6/ℓ² and the brane tension λ relate the fundamental five‑dimensional Planck scale M₅ to the effective four‑dimensional Planck mass M₄. Table‑top gravity tests bound ℓ ≲ 10⁻⁶ m.

The Friedmann equation on the brane acquires an extra quadratic term ρ²/(2λ) and a dark‑radiation term ρ_E. At early times, when the energy density ρ ≫ λ, the universe expands in a “quadratic regime” with H ∝ ρ, while at later times it reverts to the standard H ∝ √ρ. The transition occurs at a cosmic time t_c ≈ ℓ². PBHs that form before t_c are necessarily “small” (horizon radius r₀ ≪ ℓ); those forming after t_c are “large” (r₀ ≫ ℓ).

For small PBHs the authors adopt the five‑dimensional Schwarzschild–Tangherlini metric. The horizon radius scales as r₀ ∝ √(M ℓ) and the Hawking temperature is T_BH = 1/(2π r₀), which is higher than the four‑dimensional case for the same mass. Consequently, small PBHs evaporate more rapidly and emit a spectrum enriched in bulk modes and extra degrees of freedom. Large PBHs are treated with the usual four‑dimensional Schwarzschild (or Kerr) metric; their evaporation proceeds as in standard cosmology. The paper emphasizes that a smooth analytic solution interpolating between the two regimes is still lacking, so the authors use the Tangherlini solution for near‑horizon processes (evaporation) and the Garriga‑Tanaka far‑field metric for lensing calculations.

The PBH population is modeled with a monochromatic mass function and the assumption that the mass remains essentially constant after formation, apart from an O(1) accretion boost for t > t_c and negligible mass loss until the final evaporation stage. This simplification allows the authors to focus on the observable signatures of Hawking radiation.

Using modern Hawking‑radiation tools (e.g., BlackHawk), the paper computes the full emission spectra for small PBHs, including photons, electrons/positrons, neutrinos, and gravitons, taking into account the extra dimensional phase space. The authors then confront these predictions with several observational probes:

1. Low‑energy e⁺e⁻ flux – The emitted electrons and positrons at MeV energies could be detected by spacecraft at the heliopause (Voyager, New Horizons). Comparing the predicted flux with existing measurements yields constraints on (M, ℓ) that exclude a swath of low‑mass PBHs.

2. Cosmic microwave background (CMB) – Energy injection from Hawking quanta modifies the ionization history and produces spectral distortions (y‑type). By feeding the calculated injection histories into recombination codes and comparing with Planck/ACT data, the authors derive bounds that are particularly strong for PBHs evaporating around recombination (M ≈ 10¹⁴–10¹⁵ g).

3. Unresolved electromagnetic backgrounds – The cumulative photon emission contributes to the diffuse X‑ray and gamma‑ray backgrounds measured by Fermi‑LAT and INTEGRAL. The analysis shows that PBHs can account for at most a few percent of these backgrounds, tightening the allowed parameter space.

4. Microlensing – For large PBHs (M ≳ 10⁻⁹ M_⊙) the paper uses OGLE, MACHO, and Gaia microlensing event rates. The modified far‑field potential (Garriga‑Tanaka) introduces a 1/r³ correction, but the effect on lensing observables is subdominant. The resulting limits largely overlap with existing constraints from standard PBH studies.

Putting all constraints together, the authors identify a viable region where PBHs could be all of dark matter: curvature radii ℓ in the range 10⁻⁸–10⁻⁶ m and masses M ≈ 10¹⁶–10²⁰ g. This window is distinct from the four‑dimensional case because the higher temperature of small PBHs relaxes some of the CMB and gamma‑ray bounds, while the extra dimensional dynamics tighten the low‑energy e⁺e⁻ limits.

Finally, the paper discusses the recent high‑energy neutrino event KM3‑230213A. The event’s energy and arrival direction are examined for compatibility with a final‑burst Hawking emission from a small RS‑II PBH. Although the coincidence is intriguing, the current statistics are insufficient to claim a detection; the authors suggest that future neutrino observatories (IceCube‑Gen2, KM3NeT) could provide decisive tests.

In summary, this work provides a comprehensive, up‑to‑date assessment of primordial black holes in the RS‑II braneworld, integrating theoretical modeling with a broad suite of cosmological and astrophysical observations, and delineates the parameter space where such objects could serve as the sole dark‑matter component.