Two Ways of Biasing Galaxy Formation

TL;DR

The paper demonstrates that local Eulerian and local Lagrangian galaxy-bias prescriptions yield fundamentally different predictions for the galaxy bispectrum and skewness, both in real and redshift space, when evolved from Gaussian initial conditions in the mildly nonlinear regime. By deriving explicit expressions for the real-space bispectra $B_g^E$ and $B_g^L$ (and their skewness $S_g$) and introducing $Q$-amplitudes to capture shape dependence, the authors show that no choice of bias parameters can make the two schemes identical; a distinctive inertia-driven term in the Lagrangian case provides a measurable signature. In redshift space, the differences persist through modified kernels and a simple relation between the Eulerian and Lagrangian redshift-space bispectra, making bispectrum shape a powerful discriminator in survey data. The work argues that upcoming galaxy catalogs (e.g., SDSS, 2dF) should be able to distinguish between these biasing schemes, guiding future modeling of galaxy formation and large-scale structure analyses.

Abstract

We calculate the galaxy bispectrum in both real and redshift space adopting the most common prescriptions for local Eulerian biasing and Lagrangian evolving-bias model. We show that the two biasing schemes make measurably different predictions for these clustering statistics. The Eulerian prescription implies that the galaxy distribution depends only on the present-day local mass distribution, while its Lagrangian counterpart relates the current galaxy distribution to the mass distribution at an earlier epoch when galaxies first formed. Detailed measurement of the galaxy bispectrum (of its reduced amplitude) can help establish whether galaxy positions are determined by the current mass distribution or an earlier mass distribution.

Figures (1)

• Figure 1: The halo bispectrum amplitude $Q$ for configurations with sides $k_1 = 0.05\,h_{65}/{\rm Mpc}$ and $k_2 = 0.1\,h_{65}/{\rm Mpc}$ separated by an angle $\theta$ for a linear $\Lambda-$CDM power spectrum ($\Omega_m=0.3$, $\Omega_\Lambda=0.7$, $n=1$, $\sigma_8=0.9$). The prediction of the local Eulerian bias model with the bias parameters estimated from the IRAS QDOT 2 Jy redshift catalogue ($b_1^E=0.76$, $b_2^E=-0.33$; Scoccimarro et al. 2000) is represented by a dashed line. The continuos and dotted lines represents two local Lagrangian bias models with $b_0^L=0$ and $b_1^L=b_1^E-1=-0.24$. The value of $b_2^L$ has been fixed to match the Eulerian bias prediction for $Q$ at its minimum and maximum value (respectively, $b_2^L=-0.39$ and $b_2^L=-0.58$): for these specific configuration and choice of parameters, discrepancies between the predictions of the two biasing schemes are about $10 \%$ for the $Q$-tails and about 50% for the $Q$-trough.