[Paper Review] Primordial Black Hole Formation in Non-Minimal Curvaton Scenario

Shi Pi, Misao Sasaki|arXiv (Cornell University)|Dec 23, 2021

Cosmology and Gravitation TheoriesPhysics and Astronomy153 references18 citations

TL;DR

This paper proposes a non-minimal curvaton model with a non-trivial field metric that enhances curvature perturbations on small scales, leading to significant primordial black hole (PBH) formation. Using a full nonlinear probability distribution function (PDF), it shows that the resulting non-Gaussian curvature perturbation can be well approximated by quadratic local non-Gaussianity, enabling detectable induced gravitational waves if PBHs constitute all dark matter.

ABSTRACT

In the curvaton scenario, the curvature perturbation is generated after inflation at the curvaton decay, which may have a prominent non-Gaussian effect. For a model with a non-trivial kinetic term, an enhanced curvature perturbation on a small scale can be realized, which can lead to copious production of primordial black holes (PBHs) and induce secondary gravitational waves (GWs). Using the probability distribution function (PDF) which takes full nonlinear effects into account, we calculate the PBH formation. We find that under the assumption that thus formed PBHs would not overclose the universe, the non-Gaussianity of the curvature perturbation can be well approximated by the local quadratic form, which can be used to calculate the induced GWs. In this model the limit of large non-Gaussianity can be reached when the curvaton energy fraction $r$ is small at the moment of curvaton decay. We also show that in the $r o1$ limit the PDF is similar to that of ultraslow-roll inflation.

Motivation & Objective

• To construct a non-minimal curvaton model that enhances curvature perturbations on small scales through a non-trivial field metric.
• To study primordial black hole (PBH) formation in the presence of strong non-Gaussianity using a fully nonlinear probability distribution function (PDF).
• To determine whether the non-Gaussian curvature perturbation in this model can be approximated by the local quadratic form for practical cosmological applications.
• To calculate the energy spectrum of induced gravitational waves (GWs) from the peaked curvature perturbation and assess detectability by space-based interferometers.
• To explore the regime where the quadratic approximation breaks down and compare with ultra-slow-roll inflation behavior.

Proposed method

• Introduces a non-minimal curvaton model with a field metric that suppresses the curvaton kinetic term at a specific scale $k_* = 10^{13}~\text{Mpc}^{-1}$, enhancing curvature perturbations on small scales.
• Uses the full nonlinear probability distribution function (PDF) of the curvature perturbation $\zeta$ to compute PBH formation, accounting for all nonlinear effects.
• Derives the non-Gaussian structure of $\zeta$ and shows it can be well approximated by the local quadratic form $\zeta = \zeta_g + F_{\text{NL}}(\zeta_g^2 - \langle\zeta_g^2\rangle)$ under the condition $\langle\zeta^2\rangle \lesssim 0.1$.
• Calculates the induced gravitational wave (GW) energy spectrum using the quadratic non-Gaussian approximation, with $F_{\text{NL}} = 3/(4r)$, where $r$ is the curvaton energy fraction at decay.
• Evaluates the detectability of induced GWs by LISA, Taiji, and TianQin, comparing the predicted spectrum to their sensitivity curves.
• Analyzes the $r \to 1$ limit, showing the PDF approaches that of ultra-slow-roll inflation, with a logarithmic tail $\zeta \sim (2/3)\ln|1+\delta|$ when higher-order terms dominate.

Experimental results

Research questions

• RQ1 Can a non-minimal curvaton model with a non-trivial field metric generate a peaked, small-scale curvature perturbation sufficient for copious PBH formation?
• RQ2 To what extent can the fully nonlinear curvature perturbation in this model be approximated by the local quadratic non-Gaussian form?
• RQ3 What is the resulting energy spectrum of induced gravitational waves from the enhanced curvature perturbation, and is it detectable by future space-based GW observatories?
• RQ4 How does the non-Gaussianity parameter $F_{\text{NL}}$ behave in the limit of small curvaton energy fraction $r$, and can it reach large values?
• RQ5 What happens to the PBH formation PDF in the $r \to 1$ limit, and how does it compare to ultra-slow-roll inflation?

Key findings

• The non-Gaussian curvature perturbation in the non-minimal curvaton model can be well approximated by the local quadratic form $\zeta = \zeta_g + F_{\text{NL}}(\zeta_g^2 - \langle\zeta_g^2\rangle)$ when $\langle\zeta^2\rangle \lesssim 0.1$, justifying the use of this approximation for cosmologically relevant parameters.
• The non-Gaussianity parameter $F_{\text{NL}} = 3/(4r)$ can become very large as $r \to 0$, enabling strong non-Gaussian effects that enhance PBH production.
• For PBHs to constitute all dark matter, the required curvature perturbation power spectrum must satisfy $\sigma_\zeta^2 \gtrsim 2 \times 10^{-3}$, leading to a minimum induced GW energy density $\Omega_{\text{GW}} \gtrsim 3 \times 10^{-11}$.
• The induced GW spectrum peaks at $\sim 0.03$ Hz with $\Omega_{\text{GW}} \sim 2.95 \times 10^{-11}$ in the small-$r$ limit, which exceeds the sensitivity curves of LISA, Taiji, and TianQin at $10^{-2}$ Hz.
• In the $r \to 1$ limit, the PDF of $\zeta$ deviates from the quadratic form and takes a logarithmic form similar to ultra-slow-roll inflation, $\zeta \sim (2/3)\ln|1+\delta|$, indicating breakdown of the quadratic approximation.
• The model predicts that induced GWs from PBH formation in this scenario must be detectable by planned space-based interferometers like LISA if PBHs are the dominant dark matter component.