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 Omega(K,0) reflects the initial curvature, Omega(K,start), and the angular size of anomalies depends on k(start), the comoving horizon size at the onset of inflation. We estimate posteriors for Omega(K,start) and k(start) using current data and simulations, and show that if either quantity is measured to have a nonzero value, both are likely to be observable. Mappings from Omega(K,start) and k(start) to present-day observables depend strongly on the primordial equation of state; Omega(K,0) spans 10 orders of magnitude for a given Omega(K,start), while a simple and general relationship connects Omega(K,0) and kstart. We show that current bounds on Omega(K,0) imply that if k(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 be preceded by a nontrivial preinflationary phase with critical implications for the expected value of Omega(K,start).