We use Monte Carlo techniques to simulate the ability of future large high-redshift galaxy surveys to measure the temporal evolution of the dark energy equation-of-state w(z), using the baryonic acoustic oscillations in the clustering power spectrum as a standard ruler. Our analysis only utilizes the oscillatory component of the power spectrum and not its overall shape, which is potentially susceptible to broadband tilts induced by a host of model-dependent systematic effects. Our results are therefore robust and conservative. We show that baryon oscillation constraints can be thought of, to high accuracy, as a direct probe of the distance - redshift and expansion rate - redshift relations where distances are measured in units of the sound horizon. Distance precisions of 1% are obtainable for a fiducial redshift survey covering 10,000 deg^2 and redshift range 0.5 < z < 3.5. If the dark energy is further characterized by w(z) = w_0 + w_1 z (with a cut-off in the evolving term at z = 2), we can then measure the parameters w_0 and w_1 with a precision exceeding current knowledge by a factor of ten: 1 sigma accuracies Delta w_0 ~ 0.03 and Delta w_1 ~ 0.06 are obtainable (assuming a flat universe and that the other cosmological parameters Omega_m} and h could be measured independently to a precision of +/- 0.01 by combinations of future CMB and other experiments). We quantify how this performance degrades with redshift/areal coverage and knowledge of Omega_m and h, and discuss realistic observational prospects for such large-scale spectroscopic redshift surveys, with a variety of diverse techniques. We also quantify how large photometric redshift imaging surveys could be utilized to produce measurements of (w_0,w_1) with the baryonic oscillation method which may be competitive in the short term.
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