Proper-Time Hypersurface of Non-Relativistic Matter Flows: Galaxy Bias in General Relativity

Understand

We compute the second-order density fluctuation in the proper-time hypersurface of non-relativistic matter flows and relate it to the galaxy number density fluctuation in general relativity.

• At the linear order, it 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 proper-time hypersurface.
• Compared to the density fluctuation in the temporal comoving and the spatial C-gauge conditions, the density fluctuation in the commonly used gauge condition ($N=1$ and $N^\alpha=0$) violates the mass conservation at the second order.