The Time Delay of Gravitational Lens 0957+561. I. Methodology and Analysis of Optical Photometric Data

When a data set's underlying covariance function (autocorrelation) can be estimated, one can construct from the data set a global X^2^ statistic that takes into account the expected statistical relationship between all pairs of points. Incorporating a projection operator, the statistic is independent of the underlying process mean and total variance, which are poorly determined (or ill-defined) for low-frequency divergent signals. Minimizing this X^2^ gives (1) an optimal reconstruction of the underlying signal, (2) standard errors on that reconstruction, and (3) an optimal determination of additional model parameters, such as the time delay and magnitude differences of two different sets of observations. Applying this methodology to the gravitational lens 0957+561, and using previously published optical data, we obtain the value 536^+14^_-12_ days (95% confidence interval) for the delay, consistent with the less precise radio value of Lehar et al., but inconsistent with previous optical determinations, including those using the same optical data. We find that the existence of a time delay is highly significant (1% level). Monte Carlo simulations demonstrate the effect of seasonal and monthly windowing on the optical data, and show why previous analyses are likely to have given erroneous results.