Steepest growth of the power spectrum and primordial black holes (1811.11158v3)

Abstract: We derive analytic bounds on the shape of the primordial power spectrum in the context of single-field inflation. In particular, the steepest possible growth has a spectral index of $n_s - 1 = 4$ once transients have died down. Its primary implication is that any constraint on the power spectrum at a particular scale can be extrapolated to an upper bound over an extended range of scales. This is important for models which generate relics due to an enhanced amplitude of the primordial scalar perturbations, such as primordial black holes. In order to generate them, the power spectrum needs to grow many orders of magnitude larger than its observed value on CMB scales - typically achieved through a phase of ultra slow-roll inflation - and is thus subject to additional constraints at small scales. We plot all relevant constraints including CMB spectral distortions and gravitational waves sourced by scalar perturbations at second order. We show how this limits the allowed mass of PBHs, especially for the large masses of interest following recent detections by LIGO and prospects for constraining them further with future observations. We show that any transition from approximately constant $\epsilon$ slow-roll inflation to a phase where the power spectrum rapidly rises necessarily implies an intervening dip in power. We also show how to reconstruct a potential that can reproduce an arbitrary time-varying $\epsilon$, offering a complementary perspective on how ultra slow-roll can be achieved.

• The paper establishes that the steepest allowed growth in the primordial power spectrum is n_s - 1 = 4, setting universal constraints on single-field inflation models.
• It demonstrates that observational limits from CMB distortions and gravitational wave backgrounds restrict the scale for primordial black hole formation and affect their mass estimates.
• The analysis motivates the exploration of multi-field models and future high-precision observations to further probe inflationary dynamics and primordial black hole origins.

Analytical Constraints on the Primordial Power Spectrum and Its Implications for Primordial Black Holes

This paper, authored by Byrnes et al., presents a comprehensive analytical investigation into the behavior of the primordial power spectrum within the framework of single-field inflation. The paper primarily focuses on deriving constraints on the spectral growth rate and examining its implications, particularly in the context of primordial black hole (PBH) formation as potential dark matter candidates. This work is especially pertinent given the renewed interest in PBHs following gravitational wave detections by LIGO.

The authors initiate their analysis by establishing that the steepest permissible growth for the primordial power spectrum is characterized by a spectral index of $n_s - 1 = 4$, a result derived using analytical techniques that evaluate transitions through various phases of inflation. This approach underscores a crucial finding: constraints applied to the power spectrum at specific scales can be extrapolated across a broader range of scales, providing a universal upper limit on power growth.

By incorporating observational constraints, such as those from CMB spectral distortions and gravitational wave backgrounds produced by scalar perturbations, the authors effectively narrow down the permissible range for PBH formation. They delineate that a transition to ultra-slow-roll inflation, which is necessary for substantial amplification of perturbations leading to PBH creation, must inherently involve an intervening dip in power. Furthermore, their analysis indicates that while the PBH mass function appears largely insensitive to the precise slope of the power spectrum, limits on the allowed mass of PBHs can still be significantly affected by these constraints.

In exploring the broader implications of their results, Byrnes et al. employ a set of matching calculations to model potential transitions between phases of different $\eta$ values during inflation. Through a series of analytic matches, they demonstrate that the power spectrum cannot exceed a growth index of $k^4$, even when instantaneous transitions are considered. These insights serve to further constrain inflationary models that require a rapid growth in the power spectrum to account for PBHs.

The research also ventures into potential avenues for future exploration. The limitations on the growth of the primordial power spectrum underscore the need to consider multi-field models of inflation or additional physics that might circumvent the derived bounds. Furthermore, the prospective role of advanced observational instruments, such as the Square Kilometre Array (SKA) and space-based probes like LISA, are acknowledged for their potential to probe these constraints further, particularly in the context of stochastic gravitational wave backgrounds and spectral distortions.

Ultimately, this paper provides a robust framework for understanding the constraints on power spectrum growth within single-field inflation and its implications for PBH formation. By delineating the steepest possible growth rate and linking it to observational constraints, the authors offer key insights into the feasibility of various inflation models. Future exploration could potentially involve extending these analyses to multi-field contexts or leveraging forthcoming astronomical data to refine our understanding of primordial power spectrum dynamics and PBH genesis.