Measuring time delays - I. Using a flux time series that is a linear combination of time-shifted light curves

Several phenomena in astrophysics generate light curves with time delays. Among these are reverberation mapping and lensed quasars. In these systems, the measurement of the time delay is complicated by the fact that the delayed components are unresolved and that the light curves are generated from a stochastic process. We derive the likelihood function of the observations given a model of either a combination of time-delayed light curves or a single light curve. This likelihood function is different from the auto-correlation-function-based investigated by previous studies. We demonstrate that given a single-band light curve that is a linear combination of two (or more) time-shifted copies of an original light curve, generated from a probability distribution with some roughly known power spectrum, it is possible to test if the total-flux light curve is a composition of time-delayed copies or, alternatively, is consistent with being a single copy of the original light curve. Furthermore, it is possible to measure the time delays between these components even when the flux ratio is about 1/10. In the era of synoptic sky surveys, this method is useful for identifying lensed quasars and simultaneously measuring their time delays, and also for estimating the reverberation time scales of active galactic nuclei. In a companion paper, we derive another method that uses the centre-of-light astrometric position (e.g. of a lensed quasar) along with the combined flux. We also present the proper likelihood function for fitting a power-law model to a power spectrum. We test the new method on simulations and provide python and matlab implementations.