Gravity and Large-Scale Non-local Bias

TL;DR

The work demonstrates that any initially local galaxy bias becomes non-local under gravitational evolution, with the non-local structure naturally organized by Galileon invariants of the velocity potential. It develops a perturbative, multi-fluid framework that accommodates conserved and non-conserved tracers, and linear and higher-order bias evolution, revealing quadrupole and dipole contributions (the latter arising with velocity bias). The authors validate the theory with large-N-body simulations, showing clear non-local signatures in halo bias and a measurable impact on the halo bispectrum, thereby refining bias modeling crucial for cosmology and primordial non-Gaussianity constraints. Overall, incorporating non-local bias terms is essential for correct interpretation of large-scale structure observables, particularly the bispectrum, and reduces systematic biases in parameter estimation for luminous tracers.

Abstract

The relationship between galaxy and matter overdensities, bias, is most often assumed to be local. This is however unstable under time evolution, we provide proofs under several sets of assumptions. In the simplest model galaxies are created locally and linearly biased at a single time, and subsequently move with the matter (no velocity bias) conserving their comoving number density (no merging). We show that, after this formation time, the bias becomes unavoidably non-local and non-linear at large scales. We identify the non-local gravitationally induced fields in which the galaxy overdensity can be expanded, showing that they can be constructed out of the invariants of the deformation tensor (Galileons). In addition, we show that this result persists if we include an arbitrary evolution of the comoving number density of tracers. We then include velocity bias, and show that new contributions appear, a dipole field being the signature at second order. We test these predictions by studying the dependence of halo overdensities in cells of fixed matter density: measurements in simulations show that departures from the mean bias relation are strongly correlated with the non-local gravitationally induced fields identified by our formalism. The effects on non-local bias seen in the simulations are most important for the most biased halos, as expected from our predictions. The non-locality seen in the simulations is not fully captured by assuming local bias in Lagrangian space. Accounting for these effects when modeling galaxy bias is essential for correctly describing the dependence on triangle shape of the galaxy bispectrum, and hence constraining cosmological parameters and primordial non-Gaussianity. We show that using our formalism we remove an important systematic in the determination of bias parameters from the galaxy bispectrum, particularly for luminous galaxies. (abridged)

Figures (9)

• Figure 1: The evolution of linear density bias $b_1$ (top three lines) and velocity bias $b_{\rm v}$ (bottom lines) as a function of the scale factor $a$ with the initial values $b^*_1=2$ and $b_{\rm v}^* =1.1$, 1 and 0.9 respectively. A velocity bias larger than 1 slows down the decay of density bias slightly, while velocity bias less than 1 speeds it up.
• Figure 2: Emergence of non-local bias from local-bias initial conditions, as quantified by the evolution of the ratio of the galaxy multipoles $\chi^{(2)}_\ell$ (Eqs. \ref{['chi20']}-\ref{['chi22']}) to the corresponding matter multipoles (Eqs. \ref{['F20']}-\ref{['F22']}). At formation ($a=0.2$), bias is local with $b_1^* = 2$ and $b_2^* =0.5$, i.e. there is only a monopole at second-order. However, a quadrupole (three bottom lines) is generated at later times. If there is velocity bias, then a dipole is also generated (three middle lines). The three lines for each multipole correspond to different choices for the initial velocity bias: $b_{\rm v}^*=1.1$ (solid), 1 (dashed) and 0.9 (dotted).
• Figure 3: Local bias parameters at formation $b_1^*$ (top-left panel) and $b_2^*$ (top-center), comoving number density (normalized by present value, top-right), linear bias (bottom-left), second-order galaxy bias quadrupole $\chi^{(2)}_2$ and monopole $\chi^{(2)}_0$ (normalized by dark matter values, bottom center and right panels) as a function of scale factor $a$. Each panel shows three sets of values of $\alpha_1$ and $\alpha_2$, corresponding to $\{\alpha_1,\alpha_2\}=\{4, 1\}$ (solid), $\{1, 1\}$ (dashed), and $\{1, 4\}$ (dot-dashed). To describe galaxy formation, we have used the toy model in Eq. (\ref{['profileA']}) with $\sigma_0=0.2$ and characteristic galaxy formation time $a_0$ equal to 0.3 (blue), 0.5 (red) and 0.7 (green) respectively.
• Figure 4: Illustration of non-local large-scale bias in numerical simulations for high-mass halos at $z=1$ (see HMz1 in Table \ref{['HaloSample']}). The plot shows surfaces of constant $\delta_h=-0.3,0.1,0.5,0.9$ (from left to right, or red, blue, yellow, and green, respectively) as a function $\delta$, ${\mathcal{G}}_2$ and ${\mathcal{G}}_3$. If large-scale bias were a local function of $\delta$, surfaces of constant $\delta_h$ would be $\delta={\rm const.}$ planes (see next figure). Instead, there is significant tilt ($\nabla \delta_h$ is not parallel to the $\delta$-axis) showing a non-negligible dependence on ${\mathcal{G}}_2$. All fields ($\delta$, ${\mathcal{G}}_2$, ${\mathcal{G}}_3$ and $\delta_h$) have been smoothed with a top-hat window of radius $R_s=40 \, h^{-1} \, {\rm Mpc}$.
• Figure 5: Same as Fig. \ref{['4DHMz1']} but for low-mass halos at $z=0$ (see LMz0 in Table \ref{['HaloSample']}). For the least biased objects in our samples, bias becomes local.