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 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 vanishes
identically in a relativistic frame work. 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 frame work.
