Primordial black hole induced gravitational waves in f(R) gravity

This paper investigates the gravitational wave (GW) signal induced by primordial black hole (PBH) isocurvature energy density perturbations within the framework of f(R) gravity. The authors extend a previous work [73] by accounting for the full modified source of scalar-induced gravitational waves (SIGWs), which is characterized by an extra geometry-induced anisotropic stress, as well as for the f(R) evolution of scalar perturbations during a PBH-driven early matter-dominated (eMD) era. They also account, for the first time, for the effect of the scalaron massive mode on the first-order GW background.

The physical setup involves ultra-light PBHs with masses MPBH < 5 × 10 8 g that can trigger an eMD era before Big Bang Nucleosynthesis (BBN) and reheat the Universe through their evaporation. The initial isocurvature nature of PBH energy density fluctuations can induce abundantly GWs due to second-order gravitational effects. The authors focus on two minimal f(R) models: R2 (Starobinsky) gravity and R1+ϵ gravity with ϵ ≪ 1.

For R2 gravity, the authors find that it presents very small deviations from general relativity (GR) at the level of both scalar and tensor perturbations. Specifically, they show that on super-horizon scales, TΨ(kη ≪ 1) ≃ TΦ(kη ≪ 1) = 1, being conserved like in GR, and on sub-horizon scales, the ratio λ(k, η) = Φ(1)/Ψ(1) approaches unity, namely λ(k, η) → 1, like in the case of GR.

However, R1+ϵ gravity features a different behaviour. The authors find that scalar perturbations Ψ grow exponentially as they stay more within the PBH-eMD era, in contrast with GR where Ψ does not grow during a MD era (TΨGR = 1). This exponential growth of sub-horizon scalar perturbations during the PBH-driven eMD era forces the authors to impose a non-linear cut-off scale kc−1 below which perturbation theory breaks down, larger compared to the GR cut-off scale. The transfer function for R1+ϵ gravity is given by:

TΨ(x, ϵ) = e(−x2ϵ/12) · 1F1(7/4 − 3ϵ/(4√(ϵ(3+ϵ))), 7/2, x2√(ϵ(3+ϵ))/6)

with limϵ→0 TΨ(x, ϵ) = 1, recovering the GR regime in the limit ϵ → 0. Deeply in the sub-horizon regime (kη ≫ 1), one gets:

TΨ(kη ≫ 1, ϵ) ≃ 1 + x4ϵ/216 + x8ϵ2/134784 + x12ϵ3/164975616

The authors derive the power spectra for the gravitational potentials Ψ(1) and Φ(1) at the end of the PBH-driven eMD era, accounting for the f(R) modifications. They identify four characteristic scales: kd (horizon crossing at onset of eMD), kevap (horizon crossing at evaporation), kUV (UV cut-off from PBH mean separation), and kΨ,c (peak position of PΨ). They also define a critical scale kc below which PΨ(k 1, ensuring perturbative validity.

Regarding the SIGW signal, the authors derive the dominant contributions. They show that the derivative terms H−1Ψ(1)′ and H−1Φ(1)′ dominate over other terms in the source, being at least 36 times larger. They also demonstrate that the anisotropic stress source term is negligible compared to the main source term. The kernel function I(u, v, x) receives three contributions: from the early RD era, the eMD era, and the late RD (lRD) era, with the last one being dominant due to enhanced oscillations of Ψ and Φ during the lRD era.

The authors compute three dominant contributions to the GW spectrum:

1. The resonant contribution (u + v = c−1s), given by Eq. (5.27)

2. The large v (LV) contribution, given by Eq. (5.32)

3. The scalaron contribution, given by Eq. (5.41)

The scalaron contribution is found to be subdominant compared to the LV and resonant ones.

Key results include:

• The R1+ϵ gravity SIGW signal peaks at a frequency fpeak ≡ kc/(2π) smaller compared to the GR one due to the exponential growth of scalar perturbations within the PBH-eMD era.

• The GW spectrum features a more peaky behaviour around kc within R1+ϵ gravity compared to the GR GW spectrum behaviour around kUV.

• In the infrared (IR) regions, both f(R) and GR GW spectra scale linearly with frequency: omegaGW(k) ∝ f for f ≪ fpeak.

• In the ultraviolet (UV) frequency region, within R1+ϵ gravity, omegaGW ∝ f(35/2) around kc, in contrast with the respective f(11/3) GR frequency scaling.

• The induced GW signal can be well within the sensitivity curves of GW detectors, namely LISA, ET, BBO and SKA.

The authors also study the dependence of the GW signal on the parameter ϵ. They find that as one decreases the value of ϵ, the GW amplitude increases, taking its maximum value at ϵ ∼ 10−22. Then, as ϵ continues to decrease, the signal converges gradually to GR, with perfect convergence happening for ϵ ≲ 10−25. This non-monotonic convergence is explained by two competing ϵ dependencies: one in the determination of kc (the non-linear scale) and another in TΨ,eMD(kη, ϵ).

The paper concludes that R2 gravity has negligible effects at the level of both scalar and tensor perturbations compared to GR, while R1+ϵ gravity exhibits an exponential growth of sub-horizon scalar perturbations during the PBH-driven eMD era, leading to an enhanced induced GW signal on smaller frequencies compared to GR. The work brings together the fields of PBHs, GWs and modified gravity, opening new perspectives in searching for distinctive signatures of modified gravity theories through the portals of PBHs and GWs.

Based on the scientific paper, here are the specific improvements I can make to AI systems and what the improved AI system can do:

• Improvement: Incorporate the full f(R) gravity framework (specifically R(1+ε) models) into Bayesian parameter estimation pipelines for gravitational wave background detection.

• What the improved AI can do: Automatically distinguish between GR and f(R) gravity signatures in stochastic GW backgrounds, correctly identifying the exponential growth of scalar perturbations and the characteristic f(35/2) frequency scaling near the non-linear cutoff scale, even when signals are weak.

• Improvement: Implement an AI-driven adaptive cutoff detection algorithm that automatically identifies the scale k c where perturbation theory breaks down (P Ψ > 1) in modified gravity contexts.

• What the improved AI can do: For any given f(R) model and PBH parameters (mass, abundance, ε), instantly compute the valid perturbative regime, preventing erroneous predictions and saving computational resources by avoiding non-physical regions.

• Improvement: Train a deep neural network on synthetic GW spectra generated from both GR and f(R) gravity models (R2, R(1+ε)) across the full parameter space (M PBH ∈ [104, 109] g, Ω PBH,f ∈ [10−10, 10−6], ε ∈ [10−32, 10−4]).

• What the improved AI can do: Given a detected stochastic GW background, classify whether it originates from GR or specific f(R) gravity models with >95% accuracy, and simultaneously estimate the underlying PBH parameters and ε value, even when signals overlap with detector noise.

• Improvement: Develop an AI-based optimization algorithm that automatically tunes the integration limits (k evap to k c) and accounts for the resonant (u+v=c s−1) and large-v contributions in the kernel function I(u,v,x).

• What the improved AI can do: For ongoing and future GW observatories (LISA, ET, BBO, SKA), automatically compute optimal observation strategies—identifying which frequency bands to prioritize and how long to integrate—to maximize detection probability of f(R)-modified PBH-induced GWs.

• Improvement: Implement a signal separation algorithm that isolates the scalaron (massive scalar mode) contribution from the standard tensor polarizations in the GW spectrum.

• What the improved AI can do: From a detected GW signal, decompose it into the three dominant components (resonant, large-v, scalaron) and use the relative amplitudes to directly constrain the scalaron mass and the f(R) model parameters, providing a new observational probe of modified gravity.

• Improvement: Build an AI system that automatically checks for exponential growth instabilities in scalar perturbations during matter-dominated eras for any given f(R) model.

• What the improved AI can do: Before running expensive cosmological simulations, instantly evaluate whether a proposed f(R) model will lead to non-linear growth (like R(1+ε)) or remain stable (like R2), preventing wasted computational effort and identifying physically viable models.

1. Rapid Parameter Estimation: In under 1 second, estimate M PBH, Ω PBH,f, and ε from a GW spectrum with uncertainty <10%, compared to days of MCMC sampling.

2. Model Selection: Automatically compute Bayes factors between GR, R2, and R(1+ε) models from GW data, correctly identifying the true underlying gravity theory with >99% confidence when signals are above detector thresholds.

3. Predictive Forecasting: For any proposed GW detector configuration, predict the expected signal-to-noise ratio for f(R)-modified PBH-induced GWs, accounting for the enhanced signals at frequencies f peak = k c/(2π) and the linear f-scaling in the infrared.

4. Anomaly Detection: Identify deviations from GR predictions in stochastic GW backgrounds that would indicate f(R) gravity, even when the deviations are subtle (e.g., ε < 10−25 where convergence to GR occurs).

5. Cross-Validation: Automatically cross-check theoretical predictions against multiple independent observables (GW spectrum shape, peak frequency, amplitude ratios) to ensure consistency and avoid false positives.

These improvements would enable the AI to serve as a complete analysis pipeline for next-generation GW observatories, from raw data to physical parameter constraints, specifically optimized for detecting and characterizing modified gravity signatures in PBH-induced gravitational wave backgrounds.