3. Maximum likelihood analysis
We assume that the SNe are independent events. The likelihood function for the combined sample is then just the product of the individual SN likelihood functions,
Redshift information could serve as an important confirmation that the effects observed are due to gravitational lensing, since the shapes and positions of the pdf 's evolve with redshift in a known manner. This redshift dependence does not, however, significantly increase the statistical power of the determination, compared to a sample where all of the SNe are at the mean redshift of the sample. To simplify the analysis we therefore assume that all of the SN are at a fixed redshift, z=1, and drop the redshift dependence of p. Taking the log and ensemble averaging we find
where is the number of observed SNe and is the assumed true value of .
An estimate of the unknown parameter is given by maximizing the log-likelihood function. The ensemble average of this gives the solution (i.e. the estimator is asymptotically unbiased). The error on the determination of the parameter is given by the curvature of the negative log-likelihood function around its maximum. The ensemble average of this minimum variance is
which is to be evaluated at . According to the Cramer-Rao theorem, gives the smallest attainable error on for an unbiased estimator. As expected, the error decreases as the inverse square root of the number of SNe. Eq. 5 determines the number of SNe required to achieve a given level of confidence in the measurement of , given a true value . The case of gives the sensitivity in the case of a Universe with a small fraction of matter in compact objects. In this case, for CDM model we find . This includes information from the full pdf, so any large differences between the models in the tail of the pdf would be statistically significant. Although this tail is susceptible to systematic effects generated by a lack of knowledge of the pdf, it will not be probed by small numbers of SNe. To exclude the tails we redid the integral in Eq. 5 including only the information within around the mean. The resulting error increases to . This means that we need on the order of 100 SNe to determine to 20%, and around 1000 to determine to 5%, all with one-sigma errors and assuming . The variance gradually increases with , and at the required number of SN increases by a factor of 2.
The variance in scales roughly linearly with the rms noise variance , so if the scatter in SN is larger the corresponding error in increases. Variance also scales roughly inversely with the separation between mean and empty beam . For an open Universe, is the value in the flat () Universe, so one needs roughly twice the number of SNe to reach the same sensitivity. For a flat Universe with and the statistical significance increases, and one needs one quarter the SNe for the same sensitivity.
One can also test for the systematic effects introduced by using an incorrect model for the smooth component. To do this we did an analysis of a CDM Universe, "mistakenly" assuming it was an OCDM one. This results in a bias on the parameter which can be as high as 20%, so that a precision test with this method is only possible once the parameters of the cosmological model are known precisely.
© European Southern Observatory (ESO) 1999
Online publication: November 2, 1999