Separating the Universe into the Real and Fake
  The separate universe technique provides a means of establishing consistency
relations between short wavelength observables and the long wavelength matter
density fluctuations within which they evolve by absorbing the latter into the
cosmological background. We extend it to cases where non-gravitational forces
introduce a Jeans scale in other species like dynamical dark energy or massive
neutrinos. The technique matches the synchronous gauge matter density
fluctuations to the local expansion using the acceleration equation and
accounts for the temporal nonlocality and scale dependence of the long
wavelength response of small scale matter observables, e.g. the nonlinear power
spectrum, halo abundance and the implied halo bias, and $N$-point correlation
functions. Above the Jeans scale, the local Friedmann equation relates the
expansion to real energy densities and a curvature that is constant in comoving
coordinates. Below the Jeans scale, the curvature evolves and acts like a fake
density component. In all cases, the matter evolution on small scales is
correctly modeled as we illustrate using scalar field dark energy with
adiabatic or isocurvature initial conditions across the Jeans scale set by its
finite sound speed.
