Lagrangian Perturbation Theory for Biased Tracers: Halo numbers are conserved?

The Lagrangian perturbation theory (LPT) 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. Biased tracers in contrast evolve through mergers and accretion, violating the assumption of number conservation. Furthermore, a sample of halos with a narrow mass bin at initial time evolves to become halos with very different masses at late time, making it difficult to directly connect predictions of the LPT to real observations, as observed samples have similar properties. Here we use N-body simulations to test these core predictions of the LPT for biased tracers and demonstrate that the LPT predictions for halos overestimates the power spectrum at $z=0$ by a factor of three for the mass bin sample Δ log Mₕ~(h⁻¹M_)= 0.5 at~z 3, while the predictions can be made consistent if we impose by hand the number conservation of halos in the simulations throughout the entire evolution. In reality, LPT modeling of biased tracers involves marginalization of unknown bias parameters, alleviating the problem. We discuss the implications for field-level models based on the LPT applications.