Quantum diffusion during inflation and primordial black holes (1707.00537v3)

Abstract: We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-$\delta N$ formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than $\sim 1$ $e$-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

• The paper introduces a stochastic-δN framework that rigorously calculates the PDF of curvature perturbations with significant quantum diffusion effects.
• It employs dual computational methods—using the characteristic function and direct heat equation solutions—to quantify non-Gaussian deviations in the perturbation distribution.
• The analysis constrains quantum diffusion-dominated inflation to under one e-fold, ensuring primordial black hole production remains within observational limits.

Overview of Quantum Diffusion During Inflation and Primordial Black Holes

In the paper titled "Quantum diffusion during inflation and primordial black holes", the authors present an analytical framework involving the stochastic-$\delta N$ formalism to rigorously calculate the probability density function (PDF) of curvature perturbations generated during inflation, notably accounting for significant quantum backreaction effects. This approach constitutes a noteworthy advancement in calculating primordial black hole (PBH) formation accurately by integrating stochastic quantum diffusion effects.

Main Contributions

The paper addresses the critical dynamics of inflation with quantum effects impacting the production of PBHs. It employs two complementary computational approaches to solve the stochastic equations: one focusing on the characteristic function of the PDF and the other on the direct solutions of a heat equation for the PDF.

The authors analyze the classical limit, wherein quantum diffusion plays a subdominant role in field dynamics. They demonstrate that the standard Gaussian PDF is recoverable at leading order while establishing methods to evaluate higher-order corrections contributing non-Gaussian characteristics to the distribution of curvature perturbations. In contrast, they identify that large quantum diffusion yields a significantly non-Gaussian PDF described by an elliptic theta function, heavily influenced by the ratio of the squared width to the height of the potential region.

Key results include the constraint that no more than approximately one e-fold can be spent in quantum diffusion-dominated regions of inflationary potentials without exceeding observational limits on PBH abundance. The paper also shows a pronounced non-Gaussian behavior where the PDF decays exponentially rather than following a Gaussian distribution, which has substantial implications for estimating PBH mass fractions.

Implications and Future Directions

This work holds substantive implications for refining the understanding of PBH formation during inflation. Since quantum diffusion dramatically alters the behavior of curvature perturbations, existing models relying solely on classical assumptions may require reevaluation. The authors highlight that certain models may need adjustment due to non-observations of expected PBHs under previous classical-only assessments.

The paper opens avenues for numerous future investigations. It suggests exploring quantum diffusion effects beyond slow-roll conditions, incorporating multi-field models where quantum diffusion could present more extreme perturbations. Additionally, assessing the role of quantum diffusion in producing other cosmological structures, such as ultra-compact mini-halos, is an enticing prospect.

In conclusion, this research underscores the necessity of incorporating quantum effects into inflationary models to more precisely predict cosmological phenomena and constraints. The introduced methods provide a basis for potentially redefining certain inflationary potential constraints through updated assessments on the quantum diffusion impacts on structure formation.