On primordial black holes from an inflection point

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.

• The paper identifies inaccuracies in previous analyses by correcting oversimplified slow-roll assumptions near the inflection point.
• It refines the inflationary model to achieve a pronounced peak in the power spectrum through precise parameter tuning and a sustained ultra-slow-roll phase.
• The findings advocate exploring alternative potentials that naturally exhibit tuned inflection points for viable primordial black hole dark matter candidates.

Analyzing the Dynamics of Primordial Black Hole Formation Near an Inflection Point

The study examines the theoretical framework in which primordial black holes (PBHs) could potentially form during the inflationary period of the early universe, positioning these objects as viable candidates for dark matter. Specifically, it investigates the scenario in which an inflection point in the inflationary scalar potential could induce a resonant enhancement in the power spectrum of curvature perturbations, leading to PBH formation.

Critical Evaluation of Previous Analyses

The paper begins with a critical assessment of existing literature, asserting that prior analyses contain inaccuracies. Specifically, it points out two key misconceptions: the overestimation of the inflaton's trajectory consistency with the slow-roll attractor and the inadequate consideration of the rapid transition to an ultra-slow-roll phase near or at the inflection point. These oversights result in erroneous predictions of the power spectrum's behavior.

Reevaluation of the Model and Prescription for Peak Enhancement

Upon correcting these analytical missteps, the authors revisit the model, introducing a refined approach to facilitate a significant peak in the matter power spectrum. This involves fine-tuning inflationary parameters to prevent the inflaton from overshooting and exiting the inflationary phase prematurely. They underscore the necessity of maintaining a prolonged ultra-slow-roll phase, which promotes exponential growth in the power spectrum, albeit requiring careful adjustment of model parameters.

Implications of Findings and Computational Results

The paper presents detailed calculations using the prototype potential models, illustrating that after correcting previous errors, a notable amplification of the power spectrum becomes achievable. However, despite achieving a substantially higher peak than previously reported, the resultant density of PBHs remains insufficient for a significant contribution to dark matter as calculated by their model.

Implications and Speculative Future Directions

The authors suggest that while their chosen potential is suboptimal for PBH-driven dark matter, other potentials could possess the necessary characteristics for significant PBH formation. They propose that future research should aim to identify such potentials that naturally exhibit an appropriately tuned inflection-point feature. This quest requires balancing between physical plausibility and the level of fine-tuning required, suggesting that mechanisms generating these potentials should be well-founded or impervious to quantum corrections.

Conclusion

Ultimately, this analysis emphasizes the importance of accurate inflationary dynamics modeling in the context of PBH formation, highlighting the challenges in achieving substantial PBH numbers under the specific conditions of an inflection point in a scalar potential. The insights offered by Germani and Prokopec pave the way for more refined models in the quest to elucidate the enigmatic nature of dark matter through the study of primordial black holes, bridging ongoing theoretical efforts with cosmological structures.