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Constraints on Pre-inflation Fluctuations in a Nearly Flat Open ΛCDM Cosmology

We analyze constraints on parameters characterizing the pre-inflating universe in an open inflation model with a present slightly open $Λ$CDM universe. We employ an analytic model to show that for a broad class of inflation-generating effective potentials, the simple requirement that some fraction of the observed dipole moment represents a pre-inflation isocurvature fluctuation allows one to set upper and lower limits on the magnitude and wavelength scale of pre-inflation fluctuations in the inflaton field, and the curvature of the pre-inflation universe, as a function of the fraction of the total initial energy density in the inflaton field as inflation begins. We estimate that if the pre-inflation contribution to the current CMB dipole is near the upper limit set by the {\it Planck} Collaboration then the current constraints on $Λ$CDM cosmological parameters allow for the possibility of a significantly open $Ω_{i} \le 0.4$ pre-inflating universe for a broad range of the fraction of the total energy in the inflaton field at the onset of inflation. This limit to $Ω_{i}$ is even smaller if a larger dark-flow tilt is allowed.