[Paper Review] Primordial black holes from narrow peaks and the skew-lognormal distribution

Andrew D. Gow, Christian T. Byrnes+1 more|arXiv (Cornell University)|Sep 7, 2020

Cosmology and Gravitation Theories4 citations

TL;DR

This paper demonstrates that primordial black hole (PBH) mass distributions from narrow primordial density peaks deviate significantly from the standard lognormal form, favoring skewed distributions instead. It identifies the skew-lognormal as a superior model for PBH masses when high-precision data—such as from future LIGO-Virgo observations—become available, offering improved accuracy over the conventional lognormal approximation.

ABSTRACT

We examine the shape of the primordial black hole mass distribution arising from a peak in the primordial power spectrum. We show that, for sufficiently narrow peaks, the true mass distribution deviates significantly from lognormal, the most commonly assumed form of the mass distribution. The lognormal approximation is outperformed by a number of similar distributions which can generate negative skewness in log-mass. We highlight the skew-lognormal as the best of these possible modifications and suggest it be used instead of the lognormal with sufficiently accurate data, such as future LIGO-Virgo observations.

Motivation & Objective

• To investigate the shape of primordial black hole (PBH) mass distributions arising from narrow peaks in the primordial power spectrum.
• To challenge the widespread assumption that PBH mass functions follow a lognormal distribution.
• To identify and evaluate alternative distributions that better capture the true mass distribution when peaks are narrow.
• To propose the skew-lognormal distribution as a more accurate model than lognormal for future high-precision gravitational wave observations.

Proposed method

• Modeling the primordial power spectrum with narrow peaks to simulate the formation of primordial black holes.
• Using first-passage time methods to relate the peak height and width to the resulting PBH mass distribution.
• Comparing the resulting mass distribution against the standard lognormal approximation.
• Evaluating alternative distributions capable of producing negative skewness in log-mass, including the skew-lognormal.
• Assessing the performance of these distributions using theoretical constraints and expected observational data quality.
• Selecting the skew-lognormal as the optimal alternative based on fit quality and physical plausibility.

Experimental results

Research questions

• RQ1 How does the shape of the primordial power spectrum peak affect the resulting primordial black hole mass distribution?
• RQ2 Why does the standard lognormal approximation fail for narrow peaks in the primordial power spectrum?
• RQ3 Which alternative distributions better describe the PBH mass function when peaks are narrow and the lognormal form is inaccurate?
• RQ4 Can the skew-lognormal distribution provide a better fit to the true PBH mass distribution than the lognormal for sufficiently narrow peaks?
• RQ5 Under what observational conditions would the skew-lognormal become preferable to the lognormal in PBH mass modeling?

Key findings

• For sufficiently narrow peaks in the primordial power spectrum, the true PBH mass distribution deviates significantly from the standard lognormal form.
• Distributions capable of generating negative skewness in log-mass outperform the lognormal approximation in modeling narrow-peak scenarios.
• The skew-lognormal distribution is identified as the best-performing alternative among those tested, offering improved accuracy.
• The skew-lognormal is recommended for use in place of the lognormal when future gravitational wave data—such as from LIGO-Virgo—achieve sufficient precision.
• The study establishes a framework for selecting more accurate mass distribution models based on peak width and observational data quality.