On separate universes

Abstract

Abstract. The separate universe conjecture states that in General Relativity a density perturbation behaves locally (i.e. on scales much smaller than the wavelength of the mode) as a separate universe with different background density and curvature. We prove this conjecture for a spherical compensated tophat density perturbation of arbitrary amplitude and radius in ΛCDM. We then use Conformal Fermi Coordinates to generalize this result to scalar perturbations of arbitrary configuration and scale in a general cosmology with a mixture of fluids, but to linear order in perturbations. In this case, the separate universe conjecture holds for the isotropic part of the perturbations. The anisotropic part on the other hand is exactly captured by a tidal field in the Newtonian form. We show that the separate universe picture is restricted to scales larger than the sound horizons of all fluid components. We then derive an expression for the locally measured matter bispectrum induced by a long-wavelength mode of arbitrary wavelength, a new result which in standard perturbation theory is equivalent to a relativistic second-order calculation. We show that nonlinear gravitational dynamics does not generate observable contributions that scale like local-type non-Gaussianity f loc NL , and hence does not contribute to a scale-dependent galaxy bias ∆b ∝ k −2 on large scales; rather, the locally measurable long-short mode coupling assumes a form essentially identical to subhorizon perturbation theory results, once the long-mode density perturbation is replaced by the synchronous-comoving gauge density perturbation. Apparent f loc NL -type contributions arise through projection effects on photon propagation, which depend on the specific large-scale structure tracer and observable considered, and are in principle distinguishable from the local mode coupling induced by gravity. We conclude that any observation of f loc NL beyond these projection effects signals a departure from standard single-clock inflation.

How Gaussian can our Universe be?

“…Higher-order corrections in ∆x F to the distance between points would now encode the intrinsic curvature of spatial slices. This generalization of the FNC is called Conformal Fermi Coordinates (CFC) [7][8][9]: they are the coordinates that a local observer uses to describe physics in an expanding universe. They are naturally suited to the case where there is a separation of scales, such as the one described in Fig.…”

“…We can now split ζ in a long-and short-wavelength part, ζ(x) = ζ s (x) + ζ (x): because we are interested in the bispectrum only, it will be sufficient to consider the linear response of the short-scale modes to the coordinate transformation (that is, we can work at linear order in ζ ). Now, given that the background is FLRW, we can straightforwardly write down the (normalized) time-like geodesic congruence U µ as [8,9] 12) where the first order perturbations V i are the peculiar velocities of the observers U µ . Neglecting vorticity (which is not sourced in single-field models), the corresponding spatial vectors of the tetrad are [8,9] (e i )…”

“…We can then use the additional freedom in the definition of spatial coordinates to bring the spatial part h F ij in conformal Newtonian form, following [9]. …”

“…3 and Sec. 4 can be used as initial condition for the study of the dynamics of small-scale perturbations in the CFC frame when they re-enter the horizon, which has been carried out in [8,9]. In this section we sketch how this can be done, leaving the details for future work.…”

“…In fact, if we wanted to do the same calculation, but working in FNC, we could not have Figure 2. For a local observer U µ , the effect of a long mode k is that of making the short modes evolve in a separate universe of size ∼ 1/k , described by a curved FLRW metric with time-dependent tidal corrections [8,9]. Long modes become classical on super-Hubble scales, and we can describe their effect on small-scale perturbations by going to CFC.…”