Nonlinear perturbation theory with halo bias and redshift-space distortions via the Lagrangian picture

TL;DR

This work extends nonlinear gravitational perturbation theory by incorporating biasing and redshift-space distortions within a Lagrangian framework. By treating halo bias as a local Lagrangian function $F(\delta_R)$ and using a cumulant-expanded, partially resummed LPT approach, it provides analytic one-loop predictions for biased power spectra and correlation functions in both real and redshift space. The results show a weak, smooth scale dependence of halo bias with BAO features largely preserved, and clarify the non-equivalence between local Lagrangian and local Eulerian bias schemes. The framework offers insight into bias evolution and redshift-space effects without relying on heavy simulations, while suggesting paths to extend to nonlocal bias or higher nonlinearities.

Abstract

The nonlinear perturbation theory of gravitational instability is extended to include effects of both biasing and redshift-space distortions, which are inevitable in predicting observable quantities in galaxy surveys. The precise determination of scales of baryon acoustic oscillations is crucial to investigate the nature of dark energy by galaxy surveys. We find that a local Lagrangian bias and redshift-space distortions are naturally incorporated in our formalism of perturbation theory with a resummation technique via the Lagrangian picture. Our formalism is applicable to any biasing scheme which is local in Lagrangian space, including the halo bias as a special case. Weakly nonlinear effects on halo clustering in redshift space are analytically given.We assume only a fundamental idea of the halo model: haloes form according to the extended Press-Schechter theory, and the spatial distributions are locally biased in Lagrangian space. There is no need for assuming the spherical collapse model to follow the dynamical evolution, which is additionally assumed in standard halo prescriptions. One-loop corrections to the power spectrum and correlation function of haloes in redshift space are explicitly derived and presented. Instead of relying on expensive numerical simulations, our approach provides an analytic way of investigating the weakly nonlinear effects, simultaneously including the nonlinear biasing and nonlinear redshift-space distortions. Nonlinearity introduces a weak scale dependence in the halo bias. The scale dependence is a smooth function in Fourier space, and the bias does not critically change the feature of baryon acoustic oscillations in the power spectrum. The same feature in the correlation function is less affected by nonlinear effects of biasing.

Figures (10)

• Figure 1: Local Lagrangian bias parameters $\langle F'\rangle$, $\langle F"\rangle$ as functions of halo mass. Different curves correspond to different redshifts ($z=0, 0.5, 1, 3$ from bottom to top in each panel).
• Figure 2: Dependencies on halo mass and redshift of nonlinear power spectrum in real space. In the top panels, each power spectrum is divided by a smoothed, no-wiggle linear power spectrum $P_{\rm nw}(k)$EH99, and by a squared linear bias factor $b^2$. Values of redshifts and halo masses are shown in each panel. Solid lines: nonlinear power spectra of haloes with different masses with increasing order from thinner to thicker lines; dotted lines: linear theory; dashed lines: nonlinear power spectra of dark matter. In the bottom panels, halo power spectra are divided by corresponding mass power spectra and by squared linear bias factor, presenting the scale dependence of halo bias. Vertical short-dashed lines correspond to the scale $k_{\rm NL}/2$ to indicate the validity range $k < k_{\rm NL}/2$, where our result is expected to be accurate within a few percent.
• Figure 3: Same as Fig. \ref{['fig:realps']}, but in redshift space. Spherically averaged power spectra are plotted. Linear redshift-space enhancement factor $R = 1 + 2\beta/3 + \beta^2/5$ is also scaled out.
• Figure 4: Dependencies on halo mass and redshift of nonlinear correlation function in real space. Correlation functions with a fixed redshift and with different halo masses are presented in each column. Mass of the halo varies in increasing order from thinner to thicker solid lines. Dotted lines correspond to the prediction of linear theory and dashed lines correspond to nonlinear correlation functions of dark matter. In the top rows, the bare values of correlation function are plotted. In the middle rows, the correlation functions are normalized by linear bias factors and linear growth factors. In the bottom rows, residual values in the normalized correlation function of haloes (plotted in middle rows), relative to that of dark matter, are plotted.
• Figure 5: Same as Fig. \ref{['fig:realxi']}, but in redshift space. Spherically averaged correlation functions are plotted. Linear redshift-space enhancement factor $R$ is also scaled out.