Non-Gaussian Fluctuations and Primordial Black Holes from Inflation

TL;DR

The paper tackles the question of whether non-Gaussian fluctuations significantly affect primordial black hole formation during inflation, challenging the standard Gaussian assumption. It adopts a stochastic inflation framework with a Langevin equation $\dot{\phi} = -\frac{V'}{3H} + \frac{H^{3/2}}{2\pi} g(t)$ to compute the full probability distribution of inflaton fluctuations, applied to toy potentials that generate PBHs. The results show that non-Gaussian tails can drastically alter PBH abundances depending on the potential shape: plateau and wiggle potentials tend to suppress PBH production relative to Gaussian predictions, while cliff-like regions can restore Gaussianity; overall, PBH constraints become highly model-dependent. The work demonstrates a practical method to obtain model-specific fluctuation distributions, implying that PBH-based limits on inflation must be revisited with the correct non-Gaussian statistics.

Abstract

We explore the role of non-Gaussian fluctuations in primordial black hole (PBH) formation and show that the standard Gaussian assumption, used in all PBH formation papers to date, is not justified. Since large spikes in power are usually associated with flat regions of the inflaton potential, quantum fluctuations become more important in the field dynamics, leading to mode-mode coupling and non-Gaussian statistics. Moreover, PBH production requires several sigma (rare) fluctuations in order to prevent premature matter dominance of the universe, so we are necessarily concerned with distribution tails, where any intrinsic skewness will be especially important. We quantify this argument by using the stochastic slow-roll equation and a relatively simple analytic method to obtain the final distribution of fluctuations. We work out several examples with toy models that produce PBH's, and show that the naive Gaussian assumption can result in errors of many orders of magnitude. For models with spikes in power, our calculations give sharp cut-offs in the probability of large positive fluctuations, meaning that Gaussian distributions would vastly over-produce PBH's. The standard results that link inflation-produced power spectra and PBH number densities must then be reconsidered, since they rely quite heavily on the Gaussian assumption. We point out that since the probability distributions depend on the nature of the potential, it is impossible to obtain results for general models. However, calculating the distribution of fluctuations for any specific model seems to be relatively straightforward, at least in the single inflaton case.

Figures (12)

• Figure 1: The unnormalized probability distribution (42) for the driftless $\phi^{4}$ example, where $x \equiv \phi/m_{pl}$. We use the the value $x_{o} = 1$ arbitrarily and pick $C^{2}t = 1$, which is unnaturally large, in order to demonstrate the intrinsic non-Gaussianity resulting from non-linear diffusion.
• Figure 2: The unnormalized probability distribution (69) for the non-linear drift example. We have used $\sigma^{2} = 0.02$. Note that the non-linear drift has caused negative skewing, in contrast to Fig. 1, due to the tendency for negative fluctuations to fall down the hill more quickly than positive ones.
• Figure 3: The plateau potential (70): $\tilde{V}(x > 0) = 1 + \arctan(x)$, $\tilde{V}(x < 0) = 1 + 4\times10^{33}x^{21}$, where $\tilde{V}$ is defined to be dimensionless and of order unity, $\tilde{V} \equiv V/(\lambda m_{pl}^{4})$. This potential produces the fluctuation spectrum shown in Fig. 4 and the distribution of fluctuations shown in Fig. 5.
• Figure 4: The spectrum of density fluctuations at horizon crossing, $\delta_{H}$, associated with the plateau model (70) as a function of the logarithm of the length scale $L$ in units of pc. The plateau region seen in Fig. 3 produces the rapid rise in power, corresponding to $\sim 10^{32}$g PBH production.
• Figure 5: The solid line is the calculated final distribution of fluctuations (82) associated with PBH production from the plateau potential (70), Fig. 3. The dashed line is a stochastic numerical simulation of the the same model which consists of $4\times10^{4}$ points. The calculated distribution is normalized to to the number of points and bin size of the numerical result.