[Paper Review] Constraints on power spectrum of density fluctuations from PBH evaporations

É. V. Bugaev, Peter Klimai|arXiv (Cornell University)|Dec 21, 2006

Cosmology and Gravitation Theories1 references5 citations

TL;DR

This paper constrains the power spectrum of primordial density fluctuations using extragalactic neutrino and photon backgrounds from evaporating primordial black holes (PBHs). By modeling PBH formation via critical or standard collapse and calculating energy spectra under a bump-shaped primordial fluctuation spectrum, it finds that neutrino and photon background data impose comparable, and in some cases stronger, constraints than traditional methods—especially at high redshifts for small-scale fluctuations.

ABSTRACT

We calculate neutrino and photon energy spectra in extragalactic space from evaporation of primordial black holes, assuming that the power spectrum of primordial density fluctuations has a strong bump in the region of small scales. The constraints on the parameters of this bump based on neutrino and photon cosmic background data are obtained.

Motivation & Objective

• To constrain the amplitude and shape of a localized bump in the primordial power spectrum of density fluctuations using PBH evaporation signatures.
• To assess the viability of models with enhanced small-scale power (e.g., two-step inflation or non-single-field inflation) that could produce PBHs.
• To compare constraints derived from extragalactic neutrino and photon diffuse backgrounds, evaluating their relative sensitivity.
• To evaluate the impact of different PBH formation models—Carr-Hawking vs. critical collapse—on the resulting constraints.

Proposed method

• Modeling the primordial power spectrum with a Gaussian bump: $\lg\delta_{H}(k) = A + (\lg\delta_{H}^{0} - A)\exp\left[-\frac{(\lg k - \lg k_0)^2}{2\Sigma^2}\right]$.
• Using the Press-Schechter formalism to compute the PBH mass function, incorporating smoothing scale dependence and horizon-crossing density contrasts.
• Calculating time-integrated energy spectra of photons and neutrinos from PBH evaporation using a photosphere model with $T_\nu = 100$ GeV and $T_f = 120$ MeV.
• Applying redshift-dependent absorption factors $e^{-\tau}$ for photons ($z_{\text{max}} \sim 700$) and neutrinos ($z_{\text{max}} \sim 10^6$) in a flat $\Lambda$CDM cosmology.
• Integrating over the PBH mass function to compute the total diffuse flux, comparing with observed extragalactic photon ($\sim 10^{-2}$ GeV$^{-1}$cm$^{-2}$s$^{-1}$sr$^{-1}$ at 10 MeV) and neutrino ($\Phi < 1.2$ cm$^{-2}$s$^{-1}$ at >19.3 GeV) backgrounds.
• Deriving constraints on $\delta_H(k_0)$ as a function of horizon mass $M_h^0$, fixing $\Sigma = 3$ and $T_{\text{RH}} = 10^{10}$ GeV.

Experimental results

Research questions

• RQ1 How strong are constraints on a localized bump in the primordial power spectrum derived from PBH evaporation, using observed extragalactic diffuse backgrounds?
• RQ2 What is the relative sensitivity of neutrino versus photon background measurements in constraining small-scale primordial fluctuations?
• RQ3 How do different PBH formation models—standard collapse versus critical collapse—affect the resulting constraints on the power spectrum?
• RQ4 At what scales and redshifts do neutrino backgrounds become more constraining than photon backgrounds for PBH evaporation?
• RQ5 Can the critical collapse model with $\delta_c = 0.45$, $\gamma_c = 0.36$, $k_c = 4$ yield detectable or constrained signatures in the diffuse background?

Key findings

• Constraints from neutrino backgrounds are comparable to those from photon backgrounds, with the former being stronger at high redshifts and for small-scale fluctuations.
• For the critical collapse model ($\delta_c = 0.45$, $\gamma_c = 0.36$, $k_c = 4$), the PBH mass function has a long tail of low-mass black holes, leading to broader redshift distributions of energy spectra.
• The constraints on $\delta_H(k_0)$ are stronger in the standard Carr-Hawking collapse model than in the critical collapse model due to differences in mass function shape.
• At horizon masses $M_h^0 \sim 10^{11}$ g (corresponding to $T_{\text{RH}} = 10^{10}$ GeV), the bump amplitude $\delta_H^0$ is constrained to be below $\sim 0.06$ for $\Sigma = 3$, depending on the collapse model.
• The inclusion of photosphere effects leads to steep $E^{-4}$ energy spectra, significantly enhancing low-energy fluxes compared to standard Hawking models.
• For $k_0 \sim 2.75 \times 10^{16}$ Mpc$^{-1}$, the constraints exclude large values of $\delta_H^0$ at small scales, disfavoring strong blue-tilted spectra unless fine-tuned.