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If the inflationary epoch lasted only 10 e-foldings longer than required to solve the horizon problem, observable "remnants" of the preinflationary Universe may exist. They include dipole and quadrupole anisotropies of the cosmic microwave background radiation (CMBR). These "remnants" arise due to preinflationary fluctuations in scalar fields such as the inflaton, the axion, and the ilion, or due to preinflationary density perturbations. The dipole anisotropy can lead to the illusion of a "tilted universe": Viewed from the rest frame of the CMBR galaxies throughout the entire observable Universe would have a uniform streaming velocity. A dipole CMBR anisotropy could provide a very unconventional explanation for the large peculiar velocities measured for galaxies in our 50h^-1 Mpc neighborhood. Among things very unlikely to be a "remnant" of inflation is a value of today that is significantly different from unity.
Michael S. Turner (Sun,) studied this question.