Primordial black holes and scalar induced gravitational waves from sound speed resonance in non-minimal derivative coupling inflation model

We investigate an inflationary model with a non-minimal derivative coupling, where the coupling function contains both constant and periodic components. On large scales, the model is in excellent agreement with the latest Planck-ACT-LiteBIRD-BICEP/Keck 2018 (P-ACT-LB-BK18) observations. On small scales, the periodic component induces a sound-speed resonance mechanism that significantly amplifies curvature perturbations, resulting in the production of primordial black holes (PBHs). By incorporating nonlinear effects in the PBH abundance calculation, we find that the resulting PBHs can account for the majority of dark matter in the Universe. Furthermore, the PBH formation process generates scalar-induced gravitational waves (SIGWs) with a characteristic multi-peak spectral shape, which may be detectable by future space-based detectors such as LISA, Taiji, and TianQin. The model also predicts a high-frequency stochastic gravitational-wave background (SGWB) from PBH binary mergers. A combined detection of SIGWs and high-frequency gravitational waves (GWs) in future experiments would provide a direct and testable probe of this inflationary scenario.

💡 Research Summary

The authors propose an inflationary scenario in which the inflaton field is coupled derivatively to the Einstein tensor through a function θ(φ) that contains both a constant term and a small sinusoidal modulation, θ(φ)=m+w sin(nφ) Θ(φ_s−φ) Θ(φ−φ_e). The constant part reproduces the standard non‑minimal derivative coupling (NMDC) model on large scales, while the periodic part is confined to a short interval of the inflaton trajectory (φ_e<φ<φ_s) and induces a tiny oscillation in the scalar sound speed c_s.

On CMB scales (≈50 e‑folds before the end of inflation) the model yields a scalar spectral index n_s≈0.9758 and a tensor‑to‑scalar ratio r≈0.03, both well within the latest Planck‑ACT‑LiteBIRD‑BICEP/Keck 2018 constraints. The small periodic term does not affect the background dynamics appreciably, preserving the agreement with large‑scale observations.

When the inflaton passes through the oscillatory window, the time‑dependent sound speed can be written as c_s²≈1+δc_s with |δc_s|≪1 and δc_s∝w cos(nφ). This periodic modulation turns the mode equation for the Mukhanov‑Sasaki variable u_k into a Mathieu equation. In the narrow‑resonance regime (q≪1) the first instability band (α=1) is excited for wave numbers k≈k_n(1±|C|), where k_n≡n |φ̇| and C∝w H φ̇/(M_pl⁶ H² m+…). The Floquet exponent μ_k≈√(q²−(k/k_n−1)²) leads to an exponential amplification factor
E(k)=exp