Little red dots as a cosmological probe: constraining H₀ with quasi-periodic pulsations

The James Webb Space Telescope (JWST) has uncovered a population of ``little red dots'' (LRDs) at $z \gtrsim 4$, potentially representing early supermassive black holes embedded in dense gaseous envelopes. The recent discovery of the lensed LRD RXJ2211-RX1 reveals significant variability on rest-frame timescales of decades, which may be interpreted as quasi-periodic variation that has a potential physical parallel to stellar pulsations. In this work, we derive an idealized, self-consistent period-luminosity-temperature ($P$-$L$-$T_{\rm eff}$) relation based on the hydrostatic envelope model. If this theoretical relation holds and can be empirically validated/calibrated, it would offer a novel framework for constraining the Hubble constant ($H_0$). The current sparse sampling of \tgta\ yields a preliminary $H_0 = 120.7_{-46.5}^{+47.0} \text{ km s}^{-1}\text{ Mpc}^{-1}$ as a proof-of-concept, with the error budget dominated by the uncertainty of the pulsation period. Our forecasting analysis shows that continuous monitoring over a 10-year baseline can reduce the $H_0$ uncertainty to 3-20\%, depending on the intrinsic pulsation period, while the systematic uncertainty floor remains to be fully characterized. This method offers a potential independent probe to measure luminosity distances in the early universe.