The values of the fundamental constants such as $mu = m_P/m_e$, the proton to electron mass ratio and $alpha$, the fine structure constant, are sensitive to the product $sqrt{zeta_x^2(w+1)}$ where $zeta_x$ is a coupling constant between a rolling scalar field responsible for the acceleration of the expansion of the universe and the electromagnetic field with x standing for either $mu$ or $alpha$. The dark energy equation of state $w$ can assume values different than $-1$ in cosmologies where the acceleration of the expansion is due to a scalar field. In this case the value of both $mu$ and $alpha$ changes with time. The values of the fundamental constants, therefore, monitor the equation of state and are a valuable tool for determining $w$ as a function of redshift. In fact the rolling of the fundamental constants is one of the few definitive discriminators between acceleration due to a cosmological constant and acceleration due to a quintessence rolling scalar field. $w$ is often given in parameterized form for comparison with observations. In this manuscript the predicted evolution of $mu$, is calculated for a range of parameterized equation of state models and compared to the observational constraints on $Delta mu / mu$. We find that the current limits on $Delta mu / mu$ place significant constraints on linear equation of state models and on thawing models where $w$ deviates from $-1$ at late times. They also constrain non-dynamical models that have a constant $w$ not equal to $-1$. These constraints are an important compliment to geometric tests of $w$ in that geometric tests are sensitive to the evolution of the universe before the epoch of observation while fundamental constants are sensitive to the evolution of the universe after the observational epoch. Abstract truncated.
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