[Paper Review] On primordial black holes from an inflection point

Cristiano Germani, Tomislav Prokopec|arXiv (Cornell University)|Jun 13, 2017

Cosmology and Gravitation Theories23 references4 citations

TL;DR

This paper challenges the claim that inflation near an inflection point in the scalar potential can produce a large resonance in the curvature power spectrum, leading to primordial black hole (PBH) formation. It shows that standard slow-roll approximations fail due to inflaton overshooting and transition into ultra-slow-roll, but demonstrates that a significant power spectrum peak—sufficient for PBHs—can still emerge via fine-tuned parameters that minimize overshooting and extend the ultra-slow-roll phase.

ABSTRACT

Recently, it has been claimed that inflationary models with an inflection point in the scalar potential can produce a large resonance in the power spectrum of curvature perturbation. In this paper however we show that the previous analyses are incorrect. The reason is twofold: firstly, the inflaton is over-shot from a stage of standard inflation and so deviates from the slow-roll attractor before reaching the inflection. Secondly, on the (or close to) the inflection point, the ultra-slow-roll trajectory supersede the slow-roll one and thus, the slow-roll approximations used in the literature cannot be used. We then reconsider the model and provide a recipe for how to produce nevertheless a large peak in the matter power spectrum via fine-tuning of parameters.

Motivation & Objective

• To re-examine the claim that inflation near an inflection point in the scalar potential can produce a large resonance in the curvature power spectrum, as proposed in recent literature.
• To identify the failure of standard slow-roll approximations near an inflection point due to inflaton overshooting and transition into ultra-slow-roll dynamics.
• To determine under what conditions a significant amplification of the power spectrum can still occur, enabling PBH formation as a dark matter candidate.
• To provide a systematic recipe for achieving a large peak in the matter power spectrum through parameter tuning, despite the breakdown of slow-roll methods.

Proposed method

• Analyzes the dynamics of the inflaton field near an inflection point in the scalar potential, where the first derivative vanishes.
• Identifies that the inflaton typically overshoots the slow-roll attractor before reaching the inflection point, invalidating standard slow-roll approximations.
• Demonstrates that the system instead enters an ultra-slow-roll regime where $\dot{\phi} \propto a^{-3}$, leading to $\epsilon_2 \propto a^{-6}$ and exponential growth of the power spectrum.
• Uses the ultra-slow-roll formalism to compute the power spectrum amplification as $\mathcal{P} \propto e^{6N_{\rm usr}}$, where $N_{\rm usr}$ is the number of e-foldings in the ultra-slow-roll phase.
• Performs numerical analysis to show that a large peak in the power spectrum can be achieved only with fine-tuned initial conditions and potential parameters.
• Considers the role of minimizing overshooting and flattening the potential near the inflection point to extend the ultra-slow-roll phase and enhance the spectrum.

Experimental results

Research questions

• RQ1 Why do previous analyses claiming a large resonance in the power spectrum near an inflection point fail?
• RQ2 What happens to the inflaton dynamics when it approaches an inflection point in the scalar potential, and how does this affect the validity of slow-roll approximations?
• RQ3 Can a significant amplification of the curvature power spectrum still occur despite the breakdown of slow-roll dynamics near the inflection point?
• RQ4 What parameter tuning is required to achieve a large enough power spectrum peak to produce PBHs as a dark matter candidate?
• RQ5 How does the duration and strength of the ultra-slow-roll phase influence the final amplitude of the power spectrum?

Key findings

• The standard slow-roll approximation breaks down near an inflection point because the inflaton overshoots the slow-roll attractor before reaching the inflection point.
• Near the inflection point, the system transitions into an ultra-slow-roll regime where $\epsilon_2 \simeq -6$, leading to exponential growth of the curvature power spectrum as $\mathcal{P} \propto e^{6N_{\rm usr}}$.
• A large peak in the matter power spectrum can be achieved only through fine-tuning of potential parameters to minimize overshooting and maximize the duration of the ultra-slow-roll phase.
• The amplification is most effective when the potential is flattened near the inflection point and the slow-roll region is placed as close as possible to it.
• Numerical results (figure 4) confirm that a significant power spectrum enhancement is possible only under specific, finely tuned conditions.
• The study concludes that while PBHs can in principle form with sufficient tuning, the required fine-tuning may be substantial, and the question of naturalness remains open.