Parameterization of Stochasticity in Galaxy Clustering and Reconstruction of Tomographic Matter Clustering

The stochasticity in galaxy clustering, the mismatch between galaxy and underlying matter distribution, suppresses the matter clustering amplitude reconstructed by the combination of galaxy auto-correlation and galaxy-galaxy lensing cross-correlation. In this work, we solve the stochasticity systematics by parameterizing the cross correlation coefficient $r(k)$ between galaxy and matter. We investigate the performance of 12 kinds of parameterization schemes, against the cosmoDC2 $\&$ TNG300-1 galaxy samples over a wide range of redshift and flux cut. The 2-parameter fits are found to describe the stochasticity up to $k_{\rm max}=0.9\,{\rm Mpc^{-1}}h$, while the best performing quadratic scheme $r^2_s(k) = 1+c_1 k+c_2 k^2$ reaches better than $1\%$ accuracy for both the direct ${r}^2_s(k)$ fit and reconstructing matter clustering. Then, we apply the accurate quadratic scheme to forecast the tomographic matter clustering reconstruction by the combination DESI-like LRG $\times$ CSST-like cosmic shear. Depending on 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 scale cut $k_{\rm max}=0.1\,{\rm Mpc^{-1}}h$. We demonstrate the necessity of including stochasticity in reconstruction, and forecast that the reconstruction alone enables a $S_8$ constraint at about $1.5\%$ precision, free from galaxy bias and stochasticity. We will validate our method for DESI spectroscopic survey, and the analysis is expected to be complementary to DESI cosmological constraint by BAO and RSD.