Uses of a small field value which falls from a metastable maximum over   cosmological times
  We consider a small, metastable maximum vacuum expectation value $b_0$ of order of a few eV, for a pseudoscalar Goldstone-like field, which is related to the scalar inflaton field $\phi$ in an idealized model of a cosmological, spontaneously-broken chiral symmetry. The b field allows for relating semi-quantitatively three distinct quantities in a cosmological context.   (1) A very small, residual vacuum energy density or effective cosmological constant of ~ lambda b_0^4 ~ 2.7 x 10^{-47}GeV^4, for lambda ~ 3 x 10^{-14}, the same as an empirical inflaton self-coupling.   (2) A tiny neutrino mass, less then b_0.   (3) A possible small variation downward of the proton to electron mass ratio over cosmological time. The latter arises from the motion downward of the $b$ field over cosmological time, toward a nonzero limiting value as $t \to \infty$. Such behavior is consistent with an equation of motion.   We argue that hypothetical b quanta, potentially inducing new long-range forces, are absent, because of negative, effective squared mass in an equation of motion for $b$-field fluctuations. 
Saul Barshay and Georg Kreyerhoff