Gravitational lensing statistics with extragalactic surveys. III. Joint constraints on lambda_ {0} and Omega_ {0} from lensing statistics and the m-z relation for type IA supernovae

I present constraints on cosmological parameters in the lambda_ {0}-Omega_ {0} plane from a joint analysis of gravitational lensing statistics \citep{PHelbigMQWBK99a} and the magnitude-redshift relation for Type Ia supernovae \citep{SPerlmutteretal99a,ARiessetal98a}. I discuss reasons why this particular combination of tests is important and how the constraints can be improved in the future. The lensing statistics and supernova results are not inconsistent, thus it is meaningful to determine joint constraints on lambda_ {0} and Omega_ {0} by combining the results from both tests. The quantity measured by the lens statistics and the m-z relation for type Ia supernovae discussed here is approximately lambda_ {0}-Omega_ {0}. At 95% confidence, the upper limit on lambda_ {0}-Omega_ {0} from lensing statistics alone is 0.45 and from supernovae alone is in the range 0.65-0.81 (depending on the data set). For joint constraints, the upper limit on lambda_ {0}-Omega_ {0} is in the range 0.55-0.60 (again depending on the data set). For a flat universe with lambda_ {0} + Omega_ {0} = 1, this corresponds to upper limits on lambda_ {0}, taking the top of the range from different data sets, of 0.72, 0.90 and 0.80 for lensing statistics alone, supernovae alone and the joint analysis, respectively. This is perfectly consistent with the current `standard cosmological model' with lambda_ {0}~ 0.7 and Omega_ {0}~ 0.3 and is consistent with a flat universe but, neglecting other cosmological tests, does not require it.