Conformal Fermi coordinates and the local universe formalism
In an inhomogeneous Universe, the physical effect of long-wavelength perturbation on short distances should be such that short-wavelength perturbations effectively evolve in a modified homogeneous universe. We explicitly construct the so-called conformal Fermi normal coordinates (CFNC) through an expansion around the observer's geodesic, which describe the local spacetime as a quasi-FRW metric and are valid at all times. The CFNC formalism demonstrates that the zeroth-order picture is that local expansion rate and spatial curvature are renormalized by long-wavelength perturbations, and the general condition for the spatial curvature to be a constant is derived. Beyond this "separate universe" picture, CFNC allows for systematic extraction of additional local effects from long-wavelength perturbations that cannot be attributed to a re-definition of the background FRW cosmology. The formalism can be useful in the studies of tracer bias, intrinsic alignment and gravitational-wave "fossil" effect.