Galaxy Bias and non-Linear Structure Formation in General Relativity

TL;DR

This work establishes a General Relativistic framework for galaxy bias by embedding local small-scale dynamics in Fermi coordinates, where long-wavelength perturbations enter as local curvature of a curved FRW patch. It shows that GR-induced bias scales as $\left(\frac{k}{aH}\right)^2$ for horizon-scale modes and thus vanishes for super-horizon modes, while local-type primordial non-Gaussianity yields a non-vanishing, constant contribution to the bias, enabling horizon-scale probes of non-Gaussianity. The authors further provide a practical interpretation: long-wavelength information can be captured with small-box N-body simulations by incorporating curvature $\Omega_K(\zeta)$ and an appropriate coordinate rescaling, avoiding the need for prohibitively large simulations. They also highlight projection effects that can mimic local NG signatures and discuss strategies to disentangle GR corrections from primordial non-Gaussian signals, enhancing the robustness of LSS analyses on near-horizon scales.

Abstract

Length scales probed by large scale structure surveys are becoming closer to the horizon scale. Further, it has been recently understood that non-Gaussianity in the initial conditions could show up in a scale dependence of the bias of galaxies at the largest distances. It is therefore important to include General Relativistic effects. Here we provide a General Relativistic generalization of the bias, valid both for Gaussian and non-Gaussian initial conditions. The collapse of objects happens on very small scales, while long-wavelength modes are always in the quasi linear regime. Around every collapsing region, it is therefore possible to find a reference frame that is valid for all times and where the space time is almost flat: the Fermi frame. Here the Newtonian approximation is applicable and the equations of motion are the ones of the N-body codes. The effects of long-wavelength modes are encoded in the mapping from the cosmological frame to the local frame. For the linear bias, the effect of the long-wavelength modes on the dynamics is encoded in the local curvature of the Universe, which allows us to define a General Relativistic generalization of the bias in the standard Newtonian setting. We show that the bias due to this effect goes to zero as the squared ratio of the physical wavenumber with the Hubble scale for modes longer than the horizon, as modes longer than the horizon have no dynamical effects. However, the bias due to non-Gaussianities does not need to vanish for modes longer than the Hubble scale, and for non-Gaussianities of the local kind it goes to a constant. As a further application, we show that it is not necessary to perform large N-body simulations to extract information on long-wavelength modes: N-body simulations can be done on small scales and long-wavelength modes are encoded simply by adding curvature to the simulation and rescaling the coordinates.

Figures (4)

• Figure 1: Fermi Coordinates.
• Figure 2: Observed galaxy power spectrum for $z=1$, $b_{\Omega_K}=1.5$ ($b=2$) and $\partial \log n_p/\partial \log (1+z)=3$. We choose the following cosmological parameters: $\Omega_m=0.28\, ,\;\sigma_8=0.84\, , \; H_0=0.70$. Left panel: We show the spectra parallel to the line of sight (red) and transverse to the line of sight (blue). The solid line is for Gaussian initial conditions, whereas dot-dashed is $f_{\rm NL}^{loc.}=+0.5$ and dashed is $f_{\rm NL}^{loc.}=-0.5$. The lower black line is just the power spectrum of density in comoving gauge, the upper is multiplied by the redshift space distortion factor $(1+f/b)^2$ to give the power parallel to the line of sight. We see that the effects of non-Gaussianity and GR-effects on the power spectrum differ, because the latter depend also on the line of sight parameter $\mu$ through the peculiar velocity effects. Right panel: Same as left, but orange lines show non -Gaussian power spectrum without the GR-effects (just redshift space distortions).
• Figure 3: Geometrical Construction of the Fermi Coordinates.
• Figure 4: Time dependence of the local expansion history as a function of the global expansion factor. In all panels the solid black line represents the flat background model, whereas the red dashed and blue dash-dotted lines represent an over- or underdense region. Top left: Ratio of the local and global Hubble rate. Top right: Local matter density parameter. Bottom left: Local cosmological constant density parameter. Bottom right: Local curvature density parameter.