Principal Component Analysis of Galaxy Clustering in Hyperspace of Galaxy Properties

Ongoing and upcoming galaxy surveys are providing precision measurements of galaxy clustering. However a major obstacle in its cosmological application is the stochasticity in the galaxy bias. We explore whether the principal component analysis (PCA) of galaxy correlation matrix in hyperspace of galaxy properties (e.g. magnitude and color) can reveal further information on mitigating this issue. Based on the hydrodynamic simulation TNG300-1, we analyze the cross power spectrum matrix of galaxies in the magnitude and color space of multiple photometric bands. (1) We find that the first principal component $E_i^{(1)}$ is an excellent proxy of the galaxy deterministic bias $b_{D}$, in that $E_i^{(1)}=\sqrt{P_{mm}/\lambda(1)}b_{D,i}$. Here $i$ denotes the $i$-th galaxy sub-sample. $\lambda^{(1)}$ is the largest eigenvalue and $P_{mm}$ is the matter power spectrum. We verify that this relation holds for all the galaxy samples investigated, down to $k\sim 2h/$Mpc. Since $E_i^{(1)}$ is a direct observable, we can utilize it to design a linear weighting scheme to suppress the stochasticity in the galaxy-matter relation. For an LSST-like magnitude limit galaxy sample, the stochasticity $\mathcal{S}\equiv 1-r^2$ can be suppressed by a factor of $\ga 2$ at $k=1h/$Mpc. This reduces the stochasticity-induced systematic error in the matter power spectrum reconstruction combining galaxy clustering and galaxy-galaxy lensing from $\sim 12\%$ to $\sim 5\%$ at $k=1h/$Mpc. (2) We also find that $\mathcal{S}$ increases monotonically with $f_\lambda$ and $f_{\lambda^2}$. $f_{\lambda,\lambda^2}$ quantify the fractional contribution of other eigenmodes to the galaxy clustering and are direct observables. Therefore the two provide extra information on mitigating galaxy stochasticity.