A separate universe view of the asymmetric sky

We provide a unified description of the hemispherical asymmetry in the cosmic microwave background generated by the mechanism proposed by Erickcek, Kamionkowski, and Carroll, using a δ formalism that consistently accounts for the asymmetry-generating mode throughout. We derive a general form for the power spectrum which explicitly exhibits the broken translational invariance. This can be directly compared to cosmic microwave background observables, including the observed quadrupole and fNL values, automatically incorporating the Grishchuk-Zel'dovich effect. Our calculation unifies and extends previous calculations in the literature, in particular giving the full dependence of observables on the phase of our location in the super-horizon mode that generates the asymmetry. We demonstrate how the apparently different results obtained by previous authors arise as different limiting cases. We confirm the existence of non-linear contributions to the microwave background quadrupole from the super-horizon mode identified by Erickcek et al. and further explored by Kanno et al., and show that those contributions are always significant in parameter regimes capable of explaining the observed asymmetry. We indicate example parameter values capable of explaining the observed power asymmetry without violating other observational bounds.