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Signatures of the Very Early Universe: Inflation, Spatial Curvature and Large Scale Anomalies

A short inflationary phase may not erase all traces of the primordial universe. Associated observables include both spatial curvature and "anomalies" in the microwave background or large scale structure. The present curvature $Ω_{K,0}$ reflects the initial curvature, $Ω_{K,\mathrm{start}}$, and the angular size of anomalies depends on $k_\mathrm{start}$, the comoving horizon size at the onset of inflation. We estimate posteriors for $Ω_{K,\mathrm{start}}$ and $k_\mathrm{start}$ using current data and simulations, and show that if either quantity is measured to have a non-zero value, both are likely to be observable. Mappings from $Ω_{K,\mathrm{start}}$ and $k_\mathrm{start}$ to present-day observables depend strongly on the primordial equation of state; $Ω_{K,0}$ spans ten orders of magnitude for a given $Ω_{K,\mathrm{start}}$ while a simple and general relationship connects $Ω_{K,0}$ and $k_\mathrm{start}$. We show that current bounds on $Ω_{K,0}$ imply that if $k_\mathrm{start}$ is measurable, the curvature was already small when inflation began. Finally, since the energy density changes slowly during inflation, primordial gravitational wave constraints require that a short inflationary phase is preceded by a nontrivial pre-inflationary phase with critical implications for the expected value of $Ω_{K,\mathrm{start}}$.