Primordial black holes and smooth coarse-graining in excursion set theory

Abstract

The excursion-set formalism enables us to infer the mass distribution of collapsed objects, such as primordial black holes (PBHs), by the language of stochastic processes. Within the framework, this article investigates how a smooth coarse-graining procedure affects the resulting PBH mass function. As a demonstrative example, we employ a Gaussian window function, for which the stochastic noise becomes fully correlated across scales. It is found that these correlated noises result in a mass function of PBHs, whose maximum and its neighbourhood are predominantly determined by the probability that the density contrast exceeds a given threshold at each mass scale. Our results clarify the role of noise correlations induced by smooth coarse-graining and highlight their importance in predicting the abundance of PBHs.

Figures (6)

• Figure 1: (Left) The three functions in the equal-time limit on a base-$10$ logarithmic scale, appearing in the variance of the noise (\ref{['eq:ncovH_ms']}) for the Heaviside window function and for $\widetilde{k}_{\star} = 8$. (Right) The function $\widetilde{\tau}^{2} I_{1} ( \widetilde{\tau} ) / 4$, proportional to the variance of the coarse-grained density contrast for several $\widetilde{k}_{\star}$. Contrary to the standard excursion-set method, the variance is no longer a monotonic function.
• Figure 2: (Left) The diagonal elements of the functions appearing in the covariance of the noise, the limit $\widetilde{\tau}_{1} = \widetilde{\tau}_{2} = \widetilde{\tau}$ in Eq. (\ref{['eq:ncovG_ms']}). (Right) The combination $I_{1} - 2 I_{2} + I_{3}$ including the non-diagonal components. The parameters are fixed to be $k_{\star} = 8$ and $\mathcal{P}_{0} = 1$ in both panels.
• Figure 3: The analytical and numerically reconstructed variance of the correlated noise, for $\widetilde{\mathrm{W}} (z) = \Theta (1 - z)$ and $\widetilde{\mathrm{W}} (z) = \exp \, (- z^{2} / 2)$ in the left and right panels, respectively. The numerical curve was generated using the Cholesky factorisation with $N = 10^{5}$ noise realisations averaged. The parameters are fixed to be $k_{\star} = 8$ and $\mathcal{P}_{0} = 1$ in both panels.
• Figure 4: Twenty samples of the stochastic trajectories for $\widetilde{\mathrm{W}} (z) = \Theta (1 - z)$ and $\widetilde{\mathrm{W}} (z) = \exp \, (- z^{2} / 2)$, in the left and right panels respectively. The parameters are fixed to be $k_{\star} = 8$ and $\mathcal{P}_{0} = 1$ in both panels.
• Figure 5: The number count of the realisations out of $N = 10^{5}$ generated stochastic realisations for the two kinds of the window functions, $\widetilde{\mathrm{W}} (z) = \Theta (1 - z)$ and $\widetilde{\mathrm{W}} (z) = \exp \, ( - z^{2} / 2 )$ in the left and right panels respectively. The parameters are fixed to be $\widetilde{k}_{\star} = 8$ and $\mathcal{P}_{0} = 1$.