  Many cosmological models invoke rolling scalar fields to account for the
observed acceleration of the expansion of the universe. These theories
generally include a potential V(phi) which is a function of the scalar field
phi. Although V(phi) can be represented by a very diverse set of functions,
recent work has shown the under some conditions, such as the slow roll
conditions, the equation of state parameter w is either independent of the form
of V(phi) or is part of family of solutions with only a few parameters. In
realistic models of this type the scalar field couples to other sectors of the
model leading to possibly observable changes in the fundamental constants such
as the fine structure constant alpha and the proton to electron mass ratio mu.
This paper explores the limits this puts on the validity of various cosmologies
that invoke rolling scalar fields. We find that the limit on the variation of
mu puts significant constraints on the product of a cosmological parameter w+1
times a new physics parameter zeta_mu^2, the coupling constant between mu and
the rolling scalar field. Even when the cosmologies are restricted to very slow
roll conditions either the value of zeta_mu must be at the lower end of or less
than its expected values or the value of w+1 must be restricted to values
vanishingly close to 0. This implies that either the rolling scalar field is
very weakly coupled with the electromagnetic field, small zeta_mu, very weakly
coupled with gravity, w+1 ~ 0 or both. These results stress that adherence to
the measured invariance in mu is a very significant test of the validity of any
proposed cosmology and any new physics it requires. The limits on the variation
of mu also produces a significant tension with the reported changes in the
value of alpha.
