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Register a new userAbundance of primordial black holes with local non-Gaussianity in peak theory
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We discuss the effect of local type non-Gaussianity on the abundance of primordial black holes (PBH) based on the peak theory. We provide the PBH formation criterion based on the so-called compaction function and use the peak theory statistics associated with the curvature perturbation with the local type non-Gaussianity. Providing a method to estimate the PBH abundance, we demonstrate the effects of non-Gaussianity. It is explicitly shown that the value of non-linear parameter $|f_{rm NL}| sim 1$ induces a similar effect to a few factors of difference in the amplitude of the power spectrum.
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Primordial black holes (PBHs) cannot be produced abundantly enough to be the dark matter in canonical single-field inflation under slow roll. This conclusion is robust to local non-Gaussian correlations between long- and short-wavelength curvature modes, which we show have no effect in slow roll on local primordial black hole abundances. For the prototypical model which evades this no go, ultra-slow roll (USR), these squeezed non-Gaussian correlations have at most an order unity effect on the variance of PBH-producing curvature fluctuations for models that would otherwise fail to form sufficient PBHs. Moreover, the transition out of USR, which is necessary for a successful model, suppresses even this small enhancement unless it causes a large increase in the inflaton kinetic energy in a fraction of an e-fold, which we call a large and fast transition. Along the way we apply the in-in formalism, the delta N formalism, and gauge transformations to compute non-Gaussianities and illuminate different aspects of the physical origin of these results. Local non-Gaussianity in the squeezed limit does not weaken the Gaussian conclusion that PBHs as dark matter in canonical single-field inflation require a complicated and fine-tuned potential shape with an epoch where slow roll is transiently violated.
In this paper, we update the peak theory for the estimation of the primordial black hole (PBH) abundance, particularly by implementing the critical behavior in the estimation of the PBH mass and employing the averaged compaction function for the PBH formation criterion to relax the profile dependence. We apply our peak theory to a specific non-Gaussian feature called the exponential tail, which is characteristic in ultra slow-roll models of inflation. With this type of non-Gaussianity, the probability of a large perturbation is not suppressed by the Gaussian factor but decays only exponentially, so the PBH abundance is expected to be much enhanced. Not only do we confirm this enhancement even compared to the case of the corresponding nonlinearity parameter $f_mathrm{NL}=5/2$, but also we find that the resultant PBH mass spectrum has a characteristic maximal mass which is not seen in the simple Press--Schechter approach.
We modify the procedure to estimate PBH abundance proposed in arXiv:1805.03946 so that it can be applied to a broad power spectrum such as the scale-invariant flat power spectrum. In the new procedure, we focus on peaks of the Laplacian of the curvature perturbation $triangle zeta$ and use the values of $triangle zeta$ and $triangle triangle zeta $ at each peak to specify the profile of $zeta$ as a function of the radial coordinate while the values of $zeta$ and $triangle zeta$ are used in arXiv:1805.03946. The new procedure decouples the larger-scale environmental effect from the estimate of PBH abundance. Because the redundant variance due to the environmental effect is eliminated, we obtain a narrower shape of the mass spectrum compared to the previous procedure in arXiv:1805.03946. Furthermore, the new procedure allows us to estimate PBH abundance for the scale-invariant flat power spectrum by introducing a window function. Although the final result depends on the choice of the window function, we show that the $k$-space tophat window minimizes the extra reduction of the mass spectrum due to the window function. That is, the $k$-space tophat window has the minimum required property in the theoretical PBH estimation. Our procedure makes it possible to calculate the PBH mass spectrum for an arbitrary power spectrum by using a plausible PBH formation criterion with the nonlinear relation taken into account.
We consider the application of peaks theory to the calculation of the number density of peaks relevant for primordial black hole (PBH) formation. For PBHs, the final mass is related to the amplitude and scale of the perturbation from which it forms, where the scale is defined as the scale at which the compaction function peaks. We therefore extend peaks theory to calculate not only the abundance of peaks of a given amplitude, but peaks of a given amplitude and scale. A simple fitting formula is given in the high-peak limit relevant for PBH formation. We also adapt the calculation to use a Gaussian smoothing function, ensuring convergence regardless of the choice of power spectrum.
Here we review the present status of modelling of and searching for primordial non-Gaussianity of cosmological perturbations. After introducing the models for non-Gaussianity generation during inflation, we discuss the search for non-Gaussian signatures in the Cosmic Microwave Background and in the Large-Scale Structure of the Universe.
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