The nonlinear redshift-space power spectrum of galaxies

TL;DR

This paper develops a third-order perturbative framework for the galaxy power spectrum in redshift space, incorporating nonlinear local Eulerian bias. By mapping real-space fluctuations to redshift space and expanding the biased density to third order, the authors derive one-loop corrections to the redshift-space power spectrum and bispectrum, revealing a constant, shot-noise-like term from the quadratic bias and a nearly scale-independent effective bias on large scales that deviates from the linear bias. They also show that redshift-space distortions introduce nontrivial angular dependencies beyond the Kaiser model, affecting the quadrupole-to-monopole ratio and the estimation of $\beta$. Theoretical predictions agree with biased N-body simulations when an appropriate smoothing scale is applied, supporting the perturbative approach for mildly nonlinear regimes and highlighting caveats for cosmological parameter inference from redshift surveys.

Abstract

We study the power spectrum of galaxies in redshift space, with third order perturbation theory to include corrections that are absent in linear theory. We assume a local bias for the galaxies: i.e. the galaxy density is sampled from some local function of the underlying mass distribution. We find that the effect of the nonlinear bias in real space is to introduce two new features: first, there is a contribution to the power which is constant with wavenumber, whose nature we reveal as essentially a shot-noise term. In principle this contribution can mask the primordial power spectrum, and could limit the accuracy with which the latter might be measured on very large scales. Secondly, the effect of second- and third-order bias is to modify the effective bias (defined as the square root of the ratio of galaxy power spectrum to matter power spectrum). The effective bias is almost scale-independent over a wide range of scales. These general conclusions also hold in redshift space. In addition, we have investigated the distortion of the power spectrum by peculiar velocities, which may be used to constrain the density of the Universe. We look at the quadrupole-to-monopole ratio, and find that higher-order terms can mimic linear theory bias, but the bias implied is neither the linear bias, nor the effective bias referred to above. We test the theory with biased N-body simulations, and find excellent agreement in both real and redshift space, providing the local biasing is applied on a scale whose fractional r.m.s. density fluctuations are $< 0.5$.

Figures (7)

• Figure 1: Real-space power spectrum. The (upper) solid line is the final power spectrum; the dashed lines are the linear power spectrum, unsmoothed and smoothed, and the dotted lines are (from the bottom at $k=10^{-3}$) the contributions to the power from $P_{22}$, $P_{13}$ and their sum. The dot-dashed line is the second-order mass power spectrum, which merges with the dashed linear power spectrum to form the lower solid line. For model details, see text.
• Figure 2: Effective bias in real space, from perturbation theory (solid). This is approximately constant over a fairly wide range of $k$, yet the bias is far from linear. The dotted line shows the approximate analytic formula (\ref{['beff']}). The filter erases power at $k\mathrel{\hbox{$>$} \hbox{$\sim$}} 0.2$ and the spike at $k=1$ is due to the matter power spectrum crossing zero, by which time third-order perturbation theory has broken down.
• Figure 3: The effective bias factor $\sqrt{ P_{biased}/P_{mass}}$ for a Hydra N-body simulation (solid), along with the approximate formula (\ref{['beff']}), for $b_1=b_2=b_3=1$ and $R_f=2 h^{-1}$ Mpc, which gives an r.m.s. fractional overdensity of 0.41. For other details, see text. This figure also differs from Fig. 2 in that the biased field has been deconvolved with a Gaussian, which affects the results at high $k$.
• Figure 4: Redshift-space power spectrum for the model of Fig. 1. The curves are for values of $\mu$ from 0 (bottom) to 1 in steps of 0.1. The dashed and dotted lines are the unsmoothed linear and nonlinear power spectrum in real space.
• Figure 5: Hydra simulations in redshift space (points), along with the theoretical curve from perturbation theory ($b_1=b_2=b_3=1.0$, and $\sigma_0=0.41$). The dotted line is the smoothed linear theory matter power spectrum in real space.