[Paper Review] Smooth vs instant inflationary transitions: steepest growth re-examined and primordial black holes

Philippa S. Cole, Andrew D. Gow+2 more|arXiv (Cornell University)|Apr 15, 2022

Cosmology and Gravitation TheoriesPhysics and Astronomy83 references18 citations

TL;DR

This paper re-evaluates the steepest growth of primordial curvature perturbations in single-field inflation, showing that realistic, smoothed transitions—over at least one e-fold—suppress unphysical oscillations and reduce peak amplitude compared to idealized instant transitions. The key finding is that the primordial black hole (PBH) mass function is largely insensitive to the steepness of the power spectrum peak beyond $k^2$, making $k^4$ growth the effective limit for PBH formation, while plateau widths over two e-folds significantly boost the high-mass PBH tail.

ABSTRACT

Primordial black holes (PBHs) can be produced by a range of mechanisms in the early universe. A particular formation channel that connects PBHs with inflationary phenomenology invokes enhanced primordial curvature perturbations at small scales. In this paper, we examine how rapidly the background can transition between different values of the parameters of the Hubble hierarchy in the context of single-clock inflation, which must ultimately derive from a consistent derivative expansion for the background inflaton field. We discuss artefacts associated with instant or very rapid transitions, and consider the impact on the steepest power spectrum growth and the formation of PBHs. In particular, we highlight the robustness of the $k^4$ steepest growth previously found for single-field inflation with conservatively smoothed transitions and limits on how much the amplitude of the power spectrum can grow, and demonstrate that the PBH mass distribution is sensitive to the artefacts, which go away when the transitions are smoothed. We also show that the mass distribution is relatively insensitive to the steepness of the growth and subsequent decay of the power spectrum, depending primarily on the peak amplitude and the presence of any plateaus that last more than an e-fold. The shape of the power spectrum can of course be constrained by other tracers, and so understanding the physical limitations on its shape remains a pertinent question.

Motivation & Objective

• To assess the physical validity of instant transitions in inflationary models that produce enhanced primordial curvature perturbations.
• To determine how realistic, smoothed transitions affect the shape of the primordial power spectrum and its implications for PBH formation.
• To evaluate the robustness of the $k^4$ steepest growth limit in single-field inflation under physical consistency constraints.
• To investigate how the shape of the power spectrum peak—especially its growth rate and plateau duration—affects the resulting PBH mass function.
• To demonstrate that unsmoothed, idealized models overestimate power spectrum amplitude and introduce spurious features like post-peak oscillations.

Proposed method

• Uses a derivative expansion for the inflaton field to model smooth transitions between inflationary phases, avoiding unphysical instant jumps.
• Applies matching calculations with conservative smoothing over approximately one e-fold to simulate realistic parameter transitions.
• Compares PBH mass functions derived from unsmoothed (instant) transitions versus smoothed, physically plausible transitions.
• Analyzes power-law growth and decay of the primordial power spectrum, including cases with symmetric and asymmetric rise/fall rates.
• Evaluates the impact of plateau durations on the PBH mass function, particularly for widths exceeding one e-fold.
• Considers observational constraints from gravitational wave backgrounds and CMB spectral distortions to assess sensitivity to power spectrum shape.

Experimental results

Research questions

• RQ1 What is the maximum physically viable growth rate of the primordial power spectrum in single-field inflation under realistic, smoothed transitions?
• RQ2 How do unphysical instant transitions distort the power spectrum and PBH mass function compared to smooth, e-fold-scale transitions?
• RQ3 To what extent does the steepness of the power spectrum peak influence the PBH mass function, especially beyond the $k^4$ limit?
• RQ4 How do plateau durations in the power spectrum affect the PBH mass distribution, particularly in the high-mass regime?
• RQ5 Can observational probes like pulsar timing arrays or LISA detect features in the primordial power spectrum shape, and how does smoothing affect their sensitivity?

Key findings

• The steepest physically viable growth of the primordial power spectrum remains $k^4$, even in models claiming faster growth, as steeper rates are erased upon realistic smoothing.
• Unphysical oscillations in the power spectrum after the peak—common in instant-transition models—disappear when transitions are smoothed over one e-fold.
• The PBH mass function is insensitive to the steepness of the power spectrum peak as long as it exceeds $k^2$ before the peak and $k^{-1}$ after, rendering $k^4$ the effective threshold.
• A plateau in the power spectrum lasting more than one e-fold significantly boosts the high-mass tail of the PBH mass function, with a two-e-fold plateau having a substantial impact.
• Smoothed transitions reduce the peak amplitude of the power spectrum compared to instant transitions, leading to a more accurate PBH abundance estimate.
• Observational constraints on the primordial power spectrum shape—via gravitational wave backgrounds or CMB distortions—can be sensitive to its form, underscoring the need for realistic spectral modeling.