Primordial Black Hole Seeds for Little Red Dots in $f(R)$ Gravity

The discovery by the James Webb Space Telescope (JWST) of an abundant population of compact little red dots (LRDs) and supermassive black holes at high redshifts poses a severe timing challenge for standard Eddington-limited growth from stellar remnants. In this work, we present a unified hybrid assembly framework in which intermediate-mass seed black holes are formed via hierarchical primordial black hole (PBH) mergers within dense clusters in Hu-Sawicki $f(R)$ gravity, followed by galactic gas accretion. We demonstrate that the screened $f(R)$ fifth force nonlinearly accelerates early seed formation through a quadratic enhancement of the gravitational-wave radiation-capture kernel, while environmental chameleon screening dynamically self-regulates the growth. Because subsequent gas accretion scales multiplicatively with seed mass, the $f(R)$ seed enhancement is preserved throughout the accretion history, drastically reducing the time-averaged Eddington ratios required to assemble $10^{6}\text{-}10^{9}\,M_{\odot}$ LRDs by cosmic dawn. Confronting our active mass functions with JWST data reveals that a small clustered PBH fraction ($f_{\rm PBH} \lesssim 10^{-3}$) under a realistic active galactic nucleus duty cycle ($\delta \approx 10^{-2}$) naturally matches observed LRD space densities at $z \sim 5.5\text{-}6.5$ while remaining fully compliant with LIGO-Virgo-KAGRA limits, with modified gravity naturally driving the high-mass tail. Finally, we show that parameter degeneracies between modified gravity and cluster density can be cleanly broken by combining three multi-messenger diagnostics: remnant spin state signatures, a stochastic GW background cutoff, and a scale-dependent inversion in the PBH spatial correlation function.