Different effects of the Lorentz and Gaussian bump functions on the formation of primordial black holes and secondary gravitational waves

Abstract

Scalar perturbations in the inflation can be amplified when the base inflation potential $V_b(φ)$ incorporates a local bump $f(φ)$ such as $V(φ)=V_b(φ)(1+f(φ))$. This modification will lead to a peak in the curvature power spectrum, increasing a significant abundance of primordial black holes (PBHs). However, since there is no underlying physical reason for the choice of $f(φ)$, it is essential to investigate the effects of various bump functions on PBH generation. In this paper, we choose the well-known Starobinsky potential as the base inflation potential to compare the effects produced by different bumps, specifically focusing on the Lorentz and Gaussian bumps which are widely used. To clearly illustrate the differences between these two bumps, we keep parameters in bump functions the same. We find an interesting and novel result that the Lorentz cases manifest a stronger ability to enhance the power spectrum and produce more abundance of PBHs than Gaussian cases. Moreover, we also investigate the different effects of bump functions on the scalar-induced gravitational waves (SIGWs). The results indicate that the Lorentz bump generates SIGWs with a higher energy density, which can be potentially detected in the future. Our study gives valuable insights into the choice and constraints on the bump functions, and the different effects may distinguish the two bump cases for practical purposes in future experiments.

Figures (10)

• Figure 1: Gaussian function and Lorentz function. We represent the Gaussian function and Lorentz function with red dotted lines and blue solid lines, respectively, where the parameter selection is $b=4\times10^{-4},c=0.00999227,\phi_0=5.1$.
• Figure 2: The evolution of the potential functions with base potential (black dot), Gaussian bump (red dotted lines), and Lorentz bump (blue solid lines), and the parameter selection is $b=4\times10^{-4},c=0.00999227,\phi_0=5.1$.
• Figure 3: (a) shows the evolutions of e-folds number $N$. (b) shows the evolutions of the slow-roll parameter $\epsilon_\text{H}$, where the parameters of bump are $b=4\times10^{-4},c=0.00999227,\phi_0=5.1$.
• Figure 4: The scalar power spectrums under the same parameters $b=4\times10^{-4}, c=0.00999227, \phi_0=5.1$. The $P_L$ and $P_G$ represent the Lorentz and Gaussian cases, respectively. The shaded regions represent the observation constraint Planck:2018jriInomata:2018epaInomata:2016uipFixsen:1996njChluba:2012weJeong:2014gna.
• Figure 5: The scalar power spectrums for two other parameters (sets II and III). The shaded regions are consistent with Fig. \ref{['Pr-GL1']}.