Stringent $\sigma_8$ constraints from small-scale galaxy clustering using a hybrid MCMC+emulator framework

We present a novel simulation-based hybrid emulator approach that maximally derives cosmological and Halo Occupation Distribution (HOD) information from non-linear galaxy clustering, with sufficient precision for DESI Year 1 (Y1) analysis. Our hybrid approach first samples the HOD space on a fixed cosmological simulation grid to constrain the high-likelihood region of cosmology+HOD parameter space, and then constructs the emulator within this constrained region. This approach significantly reduces the parameter volume emulated over, thus achieving much smaller emulator errors with fixed number of training points. We demonstrate that this combined with state-of-the-art simulations result in tight emulator errors comparable to expected DESI Y1 LRG sample variance. We leverage the new AbacusSummit simulations and apply our hybrid approach to CMASS non-linear galaxy clustering data. We infer constraints on $\sigma_8 = 0.762\pm0.024$ and $f\sigma_8 (z_{eff} = 0.52) = 0.444\pm0.016$, the tightest among contemporary galaxy clustering studies. We also demonstrate that our $f\sigma_8$ constraint is robust against secondary biases and other HOD model choices, a critical first step towards showcasing the robust cosmology information accessible in non-linear scales. We speculate that the additional statistical power of DESI Y1 should tighten the growth rate constraints by at least another 50-60%, significantly elucidating any potential tension with Planck. We also address the "lensing is low" tension, where we find that the combined effect of a lower $f\sigma_8$ and environment-based bias lowers the predicted lensing signal by 15%, accounting for approximately 50% of the discrepancy between the lensing measurement and clustering-based predictions.