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.
