Are black holes overproduced during preheating?

We provide a simple but robust argument that primordial black hole production generically does not exceed astrophysical bounds during the resonant preheating phase after inflation. This conclusion is supported by fully nonlinear lattice simulations of various models in two and three dimensions which include rescattering but neglect metric perturbations. We examine the degree to which preheating amplifies density perturbations at the Hubble scale and show that, at the end of the parametric resonance, power spectra are universal, with no memory of the power spectrum at the end of inflation. In addition, we show how the probability distribution of density perturbations changes from exponential on very small scales to Gaussian when smoothed over the Hubble scale—the crucial length for studies of primordial black hole formation—hence justifying the standard assumption of Gaussianity.

Figure 1

Evolution of the power spectrum of scalar fields and the total density perturbation for $g^2/\lambda =50$. The horizontal axis is comoving wave number normalized by the (effective) inflaton mass at the end of inflation $m_{\varphi }$. Time flows vertically. Lines labeled as a, b, c, d, e, and f are for times $\tilde{\eta }=0$, 50, 60, 70, 80, and 100, respectively. Thus, during our range of simulation, the inflaton field oscillates about $100/7.4\sim 14$ times. An arrow pointing upward represents a rapid increase of the amplitude of fluctuations of the inflaton field due to the initial parametric resonance, and the arrow pointing rightward shows rescattering: High energy particles are generated by mode-mode coupling and scattering of low energy particles.

Figure 2

The power spectrum of density perturbation after preheating for two-dimensional space. We see that the slope of the power spectrum for small $k$ is two, as expected.

Figure 3

Power spectrum of the total density perturbation after preheating for three cases in three dimensions. Lines labeled as a, b, and c correspond to $g^2/\lambda =50$, $g^2/\lambda =50$ with scale-invariant power spectrum, and $g^2/\lambda =2$, respectively. We see that the slope of the power spectrum for small $k$ is universal with a value of three. The dashed line has $k^3$ power and crosses the threshold at the horizon scale. All three lines lie significantly under the dashed line, showing that PBHs are not overproduced in these cases. The peak of the spectrum and subsequent decay are not resolved by our three-dimensional simulations but are resolved in two dimensions.

Figure 4

Evolution of the power spectrum of $\delta$ for the massive inflaton model. Lines labeled as a, b, c, d, e, and f correspond to $mt=0$, 50, 60, 70, 100, and 120, respectively. The dashed line represents the threshold for PBH overproduction which lies well above any of the curves.

Figure 5

Time evolution of the ratio of the interaction energy $\frac{g^2}{2}{\varphi }^2{\chi }^2$ to the total energy density $\rho$. Time is normalized in $1/m$. We see that $ϵ$ after preheating is about $3\times {10}^{-2}$, i.e., the total system behaves approximated as dust.

Figure 6

The probability distribution of density fluctuations smoothed over the scale $m_{\varphi }$ (the Compton wavelength of the inflaton) in the conformal model. The dotted line is the spectrum at initial time and the solid line is the spectrum at the end of preheating. The dashed line is an appropriately scaled Gaussian distribution. At the end of preheating, the distribution of large fluctuations is significantly more amplified than the Gaussian prediction.

Figure 7

The probability distribution of density fluctuation smoothed over the scale $50\times m_{\varphi }$ in the conformal model. The dotted line is the distribution at the initial time and the solid one is at the end of preheating. Contrary to Fig. 6, the distribution is still Gaussian even at the end of preheating. Hence, on the horizon scale relevant to PBH production the Gaussian assumption is a good approximation.

Figure 8

Time evolution of $\sqrt{{\mathcal{P}}_{\delta }}$ at $k=k_{\max }=q^{1/4}m$. From this, we see that backreaction terminates the parametric amplification at around $mt=70$.