The Evolution of Bias - Generalized

TL;DR

This work shows that galaxy bias generically evolves toward unity even when gravity is modified or dark energy clusters, provided galaxies and matter share the same gravitational force. The authors derive linear, stochastic, and nonlinear bias evolutions under mass and number conservation without assuming a specific Poisson equation, revealing that cosmic acceleration slows bias relaxation and can induce scale dependence if gravity is modified. A key contribution is the prediction that the large-scale bias can acquire scale dependence in modified gravity scenarios, offering observational tests such as relative bias across galaxy populations and direct Poisson-equation tests via lensing. These results provide a practical framework to probe deviations from general relativity with large-scale structure data, while highlighting the role of initial bias and the need for a more complete halo-bias theory in non-standard cosmologies.

Abstract

Fry (1996) showed that galaxy bias has the tendency to evolve towards unity, i.e. in the long run, the galaxy distribution tends to trace that of matter. Generalizing slightly Fry's reasoning, we show that his conclusion remains valid in theories of modified gravity (or equivalently, complex clustered dark energy). This is not surprising: as long as both galaxies and matter are subject to the same force, dynamics would drive them towards tracing each other. This holds, for instance, in theories where both galaxies and matter move on geodesics. This relaxation of bias towards unity is tempered by cosmic acceleration, however: the bias tends towards unity but does not quite make it, unless the formation bias were close to unity. Our argument is extended in a straightforward manner to the case of a stochastic or nonlinear bias. An important corollary is that dynamical evolution could imprint a scale dependence on the large scale galaxy bias. This is especially pronounced if non-standard gravity introduces new scales to the problem: the bias at different scales relaxes at different rates, the larger scales generally more slowly and retaining a longer memory of the initial bias. A consistency test of the current (general relativity + uniform dark energy) paradigm is therefore to look for departure from a scale independent bias on large scales. A simple way is to measure the relative bias of different populations of galaxies which are at different stages of bias relaxation. Lastly, we comment on the possibility of directly testing the Poisson equation on cosmological scales, as opposed to indirectly through the growth factor.

Figures (2)

• Figure 1: The evolution of the linear bias and relative bias as a function of wavenumber $k$. In the bottom panel is $b^A$, the linear bias of galaxies that formed at redshift $5$ with an initial scale independent bias of $b_0 = 2$. In the middle panel is $b^B$, the linear bias of galaxies that formed at redshift $2$ with an initial scale independent bias of $b_0 = 0.7$. The top panel shows the ratio of the two. The number labeling each curve is the corresponding redshift. This makes use of the Yukawa modification to the Poisson equation: Eq. (\ref{['yukawa']}).
• Figure 2: Analog of Fig. \ref{['bevolveB.yukawa']} except that a DGP motivated modification of growth rate is used: Eq. (\ref{['DDGP']}). Note that this does not account for the scale dependence arising from the transition from the scalar-tensor regime to the general relativity regime, the so called $r_*$ effect.