Using Supernovae to Determine the Equation of State of the Dark Energy: Is Shallow Better than Deep?

Measurements of the flux and redshifts of Type Ia supernovae have provided persuasive evidence that the expansion of the universe is accelerating. If true, then in the context of standard FRW cosmology this suggests that the energy density of hte universe is dominated by "dark energy" -- a component with negative pressure of magnitude comparable to its energy density. To further investigate this phenomenon, more extensive surveys of supernovae are being planned. Given the likely timescales for completion, by the time data from these surveys are available some important cosmological parameters will be known to high precision from CMB measurements. Here we consider the impact of that foreknowledge on the design of supernova surveys. In particular we show that, despite greater opportunities to multiplex, purely from the point of view of statistical errors, a deep survey may not obviously be better than a shallow one.