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 flocNL, and hence does not contribute to a scale-dependent galaxy bias Δ 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 flocNL-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 flocNL beyond these projection effects signals a departure from standard single-clock inflation.

Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and "projection" terms.

We present high signal-to-noise galaxy-galaxy lensing measurements of the Baryon Oscillation Spectroscopic Survey constant mass (CMASS) sample using 250 deg2 of weak-lensing data from Canada-France-Hawaii Telescope Lensing Survey and Canada-France-Hawaii Telescope Stripe 82 Survey. We compare this signal with predictions from mock catalogues trained to match observables including the stellar mass function and the projected and twodimensional clustering of CMASS. We show that the clustering of CMASS, together with standard models of the galaxy-halo connection, robustly predicts a lensing signal that is 20-40 per cent larger than observed. Detailed tests show that our results are robust to a variety of systematic effects. Lowering the value of S8 = σ8 √ Ωm/0.3 compared to Planck Collaboration XIII reconciles the lensing with clustering. However, given the scale of our measurement (r < 10 h-1 Mpc), other effects may also be at play and need to be taken into consideration. We explore the impact of baryon physics, assembly bias, massive neutrinos and modifications to general relativity on ΔΣ and show that several of these effects may be non-negligible given the precision of our measurement. Disentangling cosmological effects from the details of the galaxy-halo connection, the effect of baryons, and massive neutrinos, is the next challenge facing joint lensing and clustering analyses. This is especially true in the context of large galaxy samples from Baryon Acoustic Oscillation surveys with precise measurements but complex selection functions.