Complementary Probes of Warped Extra Dimension: Colliders, Gravitational Waves and Primordial Black Holes from Phase Transitions

We study the formation of primordial black holes (PBHs) and stochastic gravitational waves background (SGWB) produced by the supercooled radion phase transition (PT) in warped extra-dimension models solving the gauge hierarchy problem. We first determine how the SGWB and the produced PBH mass and abundance depend on the warped model's infrared energy scale $\rho$, and the number of holographic colors $N$. With this finding, we recast on the plane $\{\rho, N\}$ the current SGWB and PBH constraints, as well as the expected parameter reaches of GW detectors, as LISA and ET, and the gravitational lensing ones, such as NGRST. On the same plane, we also map the collider bounds on massive graviton production, and cosmological bounds on the radion phenomenology. We find that, for $N \sim 10-50$, the considered PT predicts a PBH population mass in the range $M_{\rm PBH}\sim(10^{-1} - 10^{-25}) M_{\odot}$ for $\rho \sim (10^{-4} - 10^{8})\textrm{ TeV}$. In the range $\rho \simeq (0.05 - 0.5)$ GeV, it can explain the recent SGWB hint at nHz frequencies and generate PBH binaries with mass $M_{\rm PBH}\sim(0.1 - 1 ) M_\odot$ detectable at LISA and ET. The experimentally allowed mass region where PBHs can account for the whole dark matter abundance, and are produced with a tuning $\lesssim 10^{-4}$, corresponds to $10$ TeV $\lesssim \rho\lesssim$ $10^4$ TeV. These PBHs can compensate the lack of natural candidates for dark matter in warped extra dimensional models. Such a region represents a great science case where forthcoming and future colliders like HE-LHC and FCC-hh, gravitational-wave observatories and other PBHs probes play a key complementary role.