Proper-time hypersurface of nonrelativistic matter flows: Galaxy bias in general relativity

Abstract

We compute the second-order density fluctuation in the proper-time hypersurface of nonrelativistic matter flows and relate it to galaxy number density fluctuation, providing physical grounds for galaxy bias in the context of general relativity. At the linear order, the density fluctuation in the proper-time hypersurface is equivalent to the density fluctuation in the comoving synchronous gauge, in which two separate gauge conditions coincide. However, at the second order, the density fluctuations in these gauge conditions differ, while both gauge conditions represent the same proper-time hypersurface. Compared to the density fluctuation in the temporal comoving and spatial $C$-gauge conditions, the density fluctuation in the commonly used gauge condition ($N=1$ and ${N}^{\ensuremath{\alpha}}=0$) violates the mass conservation at the second order. We provide their physical interpretations in each gauge condition by solving the geodesic equation and the nonlinear evolution equations of nonrelativistic matter. We apply this finding to the second-order galaxy biasing in general relativity, which complements the second-order relativistic description of galaxy clustering in Yoo and Zaldarriaga [Phys. Rev. D 90, 023513, (2014)].