Lagrangian Perturbation Theory for Biased Tracers: Significance of the Number Conservation
The Lagrangian perturbation theory provides a simple yet powerful way of computing the nonlinear matter power spectrum, and it has been applied to biased tracers such as halos and galaxies. The number conservation of matter particles allows a simple relation between the fluctuations at the initial and the late times, which is essential in deriving the exact expression for the nonlinear matter power spectrum. Here we investigate the significance of the number conservation in the Lagrangian perturbation theory for biased tracers. We use $N$-body simulations to test the significance of number conservation by tracing dark matter halo samples in time. For the mass bin sample $Δ\log M_h~(h^{-1}M_{\odot})= 0.5$ at $z\simeq3$, the theoretical predictions for the halos overestimates the power spectrum at $z=0$ by a factor of three, while the simulation results match the theoretical predictions if the number conservation of halos is imposed in the simulations throughout the evolution. Starting with a halo sample at $z=0$ as another test, we trace back in time the particles that belong to the halos at~$z=0$ and use their center-of-mass positions as halo positions at $z>0$. The halo power spectra at $z>0$ from the simulations agree with the theoretical predictions of the Lagrangian perturbation theory. This numerical experiment proves that the number conservation is crucial in the Lagrangian perturbation theory predictions. We discuss the implications for various applications of the Lagrangian perturbation theory for biased tracers.