[Paper Review] Primordial black holes and uncertainties in the choice of the window function

Kenta Ando, Keisuke Inomata+1 more|arXiv (Cornell University)|Feb 18, 2018

Cosmology and Gravitation TheoriesPhysics and Astronomy84 references140 citations

TL;DR

This paper investigates how uncertainties in the choice of window function—commonly used to relate inflationary power spectra to primordial black hole (PBH) formation in real space—affect predictions for PBHs with masses around 30M⊙, relevant to LIGO detections. Using three window functions (real-space top-hat, Gaussian, and k-space top-hat), it shows that these uncertainties significantly alter predictions for stochastic gravitational waves, potentially invalidating pulsar timing array constraints for the real-space top-hat window function.

ABSTRACT

Primordial black holes (PBHs) can be produced by the perturbations that exit the horizon during inflationary phase. While inflation models predict the power spectrum of the perturbations in Fourier space, the PBH abundance depends on the probability distribution function (PDF) of density perturbations in real space. In order to estimate the PBH abundance in a given inflation model, we must relate the power spectrum in Fourier space to the PDF in real space by coarse-graining the perturbations with a window function. However, there are uncertainties on what window function should be used, which could change the relation between the PBH abundance and the power spectrum. This is particularly important in considering PBHs with mass $30 M_\odot$ that account for the LIGO events because the required power spectrum is severely constrained by the observations. In this paper, we investigate how large influence the uncertainties on the choice of a window function have over the power spectrum required for LIGO PBHs. As a result, it is found that the uncertainties significantly affect the prediction for the stochastic gravitational waves (GWs) induced by the second order effect of the perturbations. In particular, the pulsar timing array constraints on the produced GWs could disappear for the real-space top-hat window function.

Motivation & Objective

• To assess the impact of window function uncertainty on PBH abundance predictions for 30M⊙ black holes.
• To evaluate how different window functions alter constraints from stochastic gravitational waves and µ-distortion.
• To compare the effects of real-space top-hat, Gaussian, and k-space top-hat window functions on observable signatures.
• To determine whether pulsar timing array constraints on induced gravitational waves can be evaded due to window function choice.

Proposed method

• Uses three window functions: real-space top-hat, Gaussian, and k-space top-hat, to coarse-grain density perturbations.
• Applies the standard PBH formation formalism with threshold δc = 0.4 and mass-redshift relation M ∝ (k / 4.2×10⁶ Mpc⁻¹)⁻².
• Calculates the PBH production rate β(M) using the Gaussian PDF and σ²(M) derived from the power spectrum and window function.
• Computes the stochastic gravitational wave spectrum induced at second order from the same perturbations.
• Compares the resulting GW amplitude and constraints from pulsar timing arrays (PTA) across window functions.
• Uses the transfer function T(k,η) to evolve sub-horizon modes during radiation domination.

Experimental results

Research questions

• RQ1 How do different window functions affect the required power spectrum for LIGO-mass PBHs?
• RQ2 To what extent do window function choices alter predictions for stochastic gravitational waves from second-order effects?
• RQ3 Can pulsar timing array constraints on induced GWs be invalidated by choosing a specific window function?
• RQ4 How sensitive are PBH abundance predictions to the choice of coarse-graining window function?
• RQ5 Does the real-space top-hat window function lead to a different PBH mass function compared to Gaussian or k-space top-hat?

Key findings

• The choice of window function significantly alters the predicted amplitude of stochastic gravitational waves induced by second-order effects.
• For the real-space top-hat window function, pulsar timing array constraints on induced gravitational waves can be completely evaded.
• The Gaussian window function leads to a sharp peak in the PBH mass function around 30M⊙, while other window functions allow broader or shifted distributions.
• The required power spectrum must be rapidly damped on both large and small scales to satisfy µ-distortion constraints, and this requirement is window-function dependent.
• The PBH production rate β(M) and the resulting dark matter fraction f(M) depend sensitively on the window function, especially for high-mass PBHs.
• The uncertainty in window function choice introduces a significant theoretical ambiguity in predicting observable signatures of PBHs.