Parametrization of stochasticity in galaxy clustering and reconstruction of tomographic matter clustering

The stochasticity in galaxy clustering, the mismatch between the galaxy and underlying matter distributions, suppresses the matter-clustering amplitude reconstructed by the combination of galaxy autocorrelation and galaxy-galaxy lensing cross-correlation. In this work, we solve the stochasticity systematics by parametrizing the cross-correlation coefficient $r(k)$ between the galaxy and matter distributions. We investigate the performance of 12 kinds of parametrization schemes against the cosmoDC2 and TNG300-1 galaxy samples over a wide range of redshifts and flux cuts. The two-parameter fits are found to describe the stochasticity up to ${k}_{\mathrm{max}}=0.9\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}\text{ }h$, while the best-performing quadratic scheme, ${r}_{s}^{2}(k)=1+{c}_{1}k+{c}_{2}{k}^{2}$, reaches better than 1% accuracy for both the direct ${r}_{s}^{2}(k)$ fit and reconstructing matter clustering. Then, we apply the accurate quadratic scheme to forecast the tomographic matter-clustering reconstruction by the combination of DESI-like $\mathrm{LRG}\ifmmode\times\else\texttimes\fi{}\mathrm{CSST}$-like cosmic shear. Depending on the assumption of stochasticity, we find that the neglect of a serious stochasticity would result in significant systematic bias in both the reconstruction and the inferred cosmological parameters, even if we adopt a scale cut of ${k}_{\mathrm{max}}=0.1\text{ }\text{ }{\mathrm{Mpc}}^{\ensuremath{-}1}\text{ }h$. We demonstrate the necessity of including stochasticity in reconstruction and forecast that the reconstruction alone enables an ${S}_{8}$ constraint at about 1.5% precision, free from galaxy bias and stochasticity. We will validate our method for the DESI spectroscopic survey, and the analysis is expected to be complementary to the DESI cosmological constraint by baryonic acoustic oscillations and redshift space distortion.