An asymptotic decrease of (m_p/m_e) with cosmological time, from a decreasing, small effective vacuum expectation value moving from a potential maximum in the early universe

The empirical, possible small variation downward by about 10^-5, of the ratio of the proton mass to the electron mass, over a characteristic time interval estimated here to be about a billion years, is related to the decrease with time of a small, effective vacuum expectation value for a Goldstone-like pseudoscalar field which is present in the early universe, and is related to the scalar inflaton field. The same vacuum expectation value controls the magnitude of a very small, residual vacuum energy density today, which has also slowly decreased. The present time variation is estimated to be near to a definite limit, that is asymptotically approached as t->oo.