Long gradient mode and large-scale structure observables

We extend the study of long-mode perturbations to other large-scale observables such as cosmic rulers, galaxy-number counts, and halo bias. The long mode is a pure gradient mode that is still outside an observer’s horizon. We insist that gradient-mode effects on observables vanish. It is also crucial that the expressions for observables are relativistic. This allows us to show that the effects of a gradient mode on the large-scale observables vanish identically in a relativistic framework. To study the potential modulation effect of the gradient mode on halo bias, we derive a consistency condition to the first order in gradient expansion. We find that the matter variance at a fixed physical scale is not modulated by the long gradient mode perturbations when the consistency condition holds. This shows that the contribution of long gradient modes to bias vanishes in this framework.

Figure 1

The apparent position of a galaxy is shown relative to its real position for an observer with four velocity $u\mu$.