[Paper Review] Primordial black holes from null energy condition violation during inflation

Yong Cai, Mian Zhu+1 more|arXiv (Cornell University)|May 18, 2023

Cosmology and Gravitation TheoriesPhysics and Astronomy112 references10 citations

TL;DR

This paper proposes a single-field inflation scenario with an intermediate NEC-violation stage that enhances the scalar power spectrum, producing primordial black holes (PBHs) and associated scalar-induced gravitational waves (SIGWs). It analyzes PBH abundance and SIGW signals within EFT beyond Horndeski framework and discusses observational prospects.

ABSTRACT

Primordial black holes (PBHs) and the violation of the null energy condition (NEC) have significant implications for our understanding of the very early universe. We present a novel approach to generate PBHs via the NEC violation in a single-field inflationary scenario. In our scenario, the universe transitions from a first slow-roll inflation stage with a Hubble parameter $H = H_{ ext{inf}1}$ to a second slow-roll inflation stage with $H = H_{ ext{inf}2}\gg H_{ ext{inf}1}$, passing through an intermediate stage of NEC violation. The NEC violation naturally enhances the primordial scalar power spectrum at a certain wavelength, leading to the production of PBHs with masses and abundances of observational interest. We also investigate the phenomenological signatures of scalar-induced gravitational waves (SIGWs) resulting from the enhanced density perturbations. Our work highlights the potential of utilizing a combination of PBHs, SIGWs, and primordial gravitational waves as a valuable probe for studying NEC violation during inflation, opening up new avenues for exploring the early universe.

Motivation & Objective

• Motivate the role of NEC violation in the early universe and its potential to generate PBHs during inflation.
• Propose a single-field inflation scenario with an NEC-violating intermediate phase that boosts the Hubble parameter and power spectrum.
• Show how a narrow enhancement in the curvature perturbation leads to PBH formation and compute resulting PBH abundances.
• Examine scalar-induced gravitational waves (SIGWs) and assess consistency with current constraints and future detectors.
• Discuss EFT framework with beyond-Horndeski terms to stabilize NEC-violating dynamics and outline observational probes.

Proposed method

• Use an EFT action with P(φ,X) plus an operator L_{δg^{00}R^{(3)}} to avoid ghost and gradient instabilities.
• Model background: a first slow-roll inflation with H_inf1 transitions to a second slow-roll with H_inf2>>H_inf1 via an NEC-violating intermediate stage.
• Derive the quadratic action for scalar perturbations and set the sound speed c_s^2 = 1 through parameter choices.
• Compute the scalar power spectrum P_ζ(k) by solving the perturbation equation with NEC-violation induced H(t).
• Apply Press-Schechter-like formalism to relate P_ζ to PBH abundance f_PBH(M) with a fixed δ_c = 0.5 and a top-hat window function.
• Calculate the transfer function T(k,τ) for radiation domination and the variance σ_R^2 to evaluate PBH formation.
• Compute scalar-induced gravitational wave spectrum P_h(τ,k) and Ω_GW today to compare with PTA constraints and future detector sensitivities.

Figure 1: In our scenario, the universe begins with a period of slow-roll inflation, and then transitions into a second stage of slow-roll inflation with a higher energy scale, after passing through a phase of violating the null energy condition.

Experimental results

Research questions

• RQ1 Can NEC violation during a transient stage of inflation efficiently enhance curvature perturbations to seed PBHs while remaining consistent with CMB and PTA constraints?
• RQ2 What PBH mass ranges and abundances are natural outcomes of an intermediate NEC-violation stage in single-field EFT inflation?
• RQ3 Do scalar-induced gravitational waves associated with the enhanced perturbations respect current bounds and lie within reach of future GW observatories?
• RQ4 How does the beyond-Horndeski EFT framework stabilize NEC-violating dynamics without affecting background evolution?
• RQ5 What observational signatures (PBHs, SIGWs, primordial GWs) jointly probe NEC violation during inflation?

Key findings

• The NEC-violation stage can significantly boost PBH abundance across multiple mass scales.
• The model yields a sharp peak in the curvature power spectrum, enabling PBH production in observationally interesting ranges.
• PBH abundances f_PBH(M) can accommodate OGLE ultrashort-timescale microlensing events for certain parameters.
• SIGW spectra corresponding to the enhanced perturbations are consistent with current EPTA constraints and may be detectable by PTA and space-borne detectors.
• The framework allows independent parameterization of scalar and tensor perturbations at quadratic order, reducing some instabilities, but covariant realizations may modify GW spectra.

Figure 2: A numerical solution of the Hubble parameter $H$ with respect to time $t$ in our model, which results in the blue curve of $P_{\zeta}$ in Fig. 3 . We have set the Planck scale $M_{\text{P}}=1$ .