Primordial Black Hole Formation from Power Spectrum with Finite-width

Abstract

Primordial black holes (PBHs) can form from gravitational collapse of large overdensities in the early Universe, giving rise to rich phenomena in astrophysics and cosmology. We develop a novel, general, and systematic method based on theory of density contrast peaks to calculate the abundance of PBHs for a broad power spectrum of curvature perturbations with Gaussian statistics. We introduce a window function to account for the relevant perturbation scales associated with PBHs of different masses, along with a filter function that removes unphysical contributions from super-horizon-scale overdensities. While some uncertainties remain due to the limited understanding of the nonlinear collapse process, our approach substantially reduces the discrepancy previously observed between peaks theory and the Press-Schechter formalism.

Figures (30)

• Figure 1: Algorithm flowchart of our method for calculating PBH abundance.
• Figure 2: PBH mass function for a monochromatic power spectrum obtained from the fiducial expression Eq. \ref{['eq:fid-fPBH-mono']} (red curve), and from the complete expression Eq. \ref{['eq:fPBHPT']} with different $\Xi$'s (dashed, dotted and solid black curves), considering a Gaussian window function of Eq. \ref{['eq:winG']}. Adjusting $\Xi$ such that central mass from the complete formula in Eq. \ref{['eq:fPBHPT']} is the same as the one from the fiducial expression Eq. \ref{['eq:fid-fPBH-mono']}, we have $\Xi_{\mathrm{G}}=1/2.82$. Note that we normalize all the curves by $f_{\mathrm{PBH}}^{\mathrm{tot}}=1$, which have different variances: $\mathcal{A}_{\mathcal{R}}=1.47\times 10^{-2}$ for the red curve and $\mathcal{A}_{\mathcal{R}}=1.55\times 10^{-2}$ for the black curve.
• Figure 3: Illustration of $(\Sigma_2 R_s^2)^2$ as a function of $R_s$ for different choices of log-normal power spectrum widths $\Delta$, from Eq. \ref{['eq:sigmasqu-logN']}.
• Figure 4: Normalized peak profile $\hat{\mathcal{R}}/\mu$ of Eq. \ref{['eq:profilefull']} for different choices of log-normal spectrum widths $\Delta$, considering $K=\sqrt{\gamma_3}$ and $R_s^{-1}=k_*$. Window-function-dominated approximation (WFD) of Eq. \ref{['eq:app-w-dom']} is displayed for reference (gray dashed line).
• Figure 5: [Left] Normalized peak profile $\hat{\mathcal{R}}/\mu(q_1+q_2)$ of Eq. \ref{['eq:profilefull']} for different choices of $R_s$, considering $K=\sqrt{\gamma_3}$. [Right] Normalized peak profile $\hat{\mathcal{R}}/\mu(q_1+q_2)$ of Eq. \ref{['eq:profilefull']} for different choices of $K$, considering $R_s^{-1}=k_*$.