Primordial black hole abundance from random Gaussian curvature perturbations and a local density threshold

Yoo, Chul-Moon; Harada, Tomohiro; Garriga, Jaume; Kohri, Kazunori

doi:10.1093/ptep/pty120

Primordial black hole abundance from random Gaussian curvature perturbations and a local density threshold

Chul-Moon Yoo
(
- Nagoya U.
)
,
Tomohiro Harada
(
- Rikkyo U.
)
,
Jaume Garriga
(
- ICC, Barcelona U.
)
,
Kazunori Kohri
(
- KEK, Tsukuba and
- Oxford U., Theor. Phys. and
- Sokendai, Tsukuba
)

May 10, 2018

31 pages

Published in:

PTEP 2018 (2018) 12, 123E01,
PTEP 2024 (2024) 4, 049202 (erratum)

Published: Dec 1, 2018
and
Published: Mar 30, 2024

e-Print:

1805.03946 [astro-ph.CO]

DOI:

10.1093/ptep/pty120 (publication),
10.1093/ptep/ptae037 (erratum)

Report number:

RUP-18-15,
KEK-Cosmo-225,
KEK-TH-2052

View in:

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Abstract: (Oxford University Press)

The production rate of primordial black holes (PBHs) is often calculated by considering a nearly Gaussian distribution of cosmological perturbations, and assuming that black holes will form in regions where the amplitude of such perturbations exceeds a certain threshold. A threshold |

\zeta_{\rm th}

| for the curvature perturbation is somewhat inappropriate for this purpose, because it depends significantly on environmental effects, not essential to the local dynamics. By contrast, a threshold |

\delta_{\rm th}

| for the density perturbation at horizon crossing seems to provide a more robust criterion. On the other hand, the density perturbation is known to be bounded above by a maximum limit |

\delta_{\rm max}

| at the horizon entry and, given that |

\delta_{\rm th}

| is comparable to |

\delta_{\rm max}

|⁠, the density perturbation will be far from Gaussian near or above the threshold. In this paper, we provide a new plausible estimate for the primordial black hole abundance based on peak theory. In our approach, we assume that the curvature perturbation is given as a random Gaussian field with the power spectrum characterized by a single scale, while an optimized criterion for PBH formation is imposed, based on the locally averaged density perturbation around the nearly spherically symmetric high peaks. Both variables are related by the full non-linear expression derived in the long-wavelength approximation of general relativity. We do not introduce a window function, which is usually introduced to obtain the scale dependence of the spectrum. The scale of the inhomogeneity is introduced as a random variable in the peak theory, and the scale-dependent PBH fraction is automatically induced. We find that the mass spectrum is shifted to larger mass scales by one order of magnitude or so, compared to a conventional calculation. The abundance of PBHs becomes significantly larger than the conventional one, by many orders of magnitude, mainly due to the optimized criterion for PBH formation and the removal of the suppression associated with a window function.

Note:

31 pages, 11 figures, significant modification from the first version, comments on the effect of critical behavior and related refs are added, errata are reflected

E01 Relativity
E31 Black holes
E80 Inflation and cosmology in general
density: perturbation
curvature: perturbation
black hole: primordial
threshold: density
density: local
horizon: crossing
mass: scale

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