The magnitude-redshift relation for Type Ia supernovae is beginning to provide strong constraints on the cosmic densities contributed by matter , \Omega _ { m } , and a cosmological constant , \Omega _ { \Lambda } , though the results are highly degenerate in the \Omega _ { m } – \Omega _ { \Lambda } plane .
Here we estimate the constraints that can be placed on a cosmological constant or quintessence-like component by extending supernovae samples to high redshift .
Such measurements , when combined with constraints from anisotropies in the cosmic microwave background , could provide an important consistency check of systematic errors in the supernovae data .
A large campaign of high-z supernovae observations with 10 metre class telescopes could constraint \Omega _ { m } to a ( 1 \sigma ) accuracy of 0.06 and \Omega _ { \Lambda } to 0.15 .
A sample of supernovae at redshift z \sim 3 , as might be achievable with a Next Generation Space Telescope , could constrain \Omega _ { m } to an accuracy of about 0.02 independently of the value of \Omega _ { \Lambda } .
The constraints on a more general equation of state , w _ { Q } = p / \rho , converge slowly as the redshift of the supernovae data is increased .
The most promising way of setting accurate constraints on w _ { Q } is by combining high-z supernovae and CMB measurements .
With feasible measurements it should be possible to constrain w _ { Q } to a precision of about 0.06 , if the Universe is assumed to be spatially flat .
We use the recent supernovae sample of Perlmutter et al . and observations of the CMB anisotropies to constraint the equation of state in quintessence-like models via a likelihood analysis .
The 2 \sigma upper limits are w _ { Q } < -0.6 if the Universe is assumed to be spatially flat , and w _ { Q } < -0.4 for universes of arbitrary spatial curvature .
The upper limit derived for a spatially flat Universe is close to the lower limit ( w _ { Q } \approx - 0.7 ) allowed for simple potentials , implying that additional fine tuning may be required to construct a viable quintessence model .