How accurate are the time delay estimates in gravitational lensing?

Abstract

We present a novel approach to estimate the time delay between light curves of multiple images in a gravitationally lensed system, based on Kernel methods in the context of machine learning. We perform various experiments with artificially generated irregularlysampled data sets to study the effect of the various levels of noise and the presence of gaps of various size in the monitoring data. We compare the performance of our method with various other popular methods of estimating the time delay and conclude, from experiments with artificial data, that our method is least vulnerable to missing data and irregular sampling, within reasonable bounds of Gaussian noise. Thereafter, we use our method to determine the time delays between the two images of quasar Q0957+561 from radio monitoring data at 4 cm and 6 cm, and conclude that if only the observations at epochs common to both wavelengths are used, the time delay gives consistent estimates, which can be combined to yield 408 ± 12 days. The full 6 cm dataset, which covers a longer monitoring period, yields a value which is 10% larger, but this can be attributed to differences in sampling and missing data.

A Kernel-Based Approach to Estimating Phase Shifts Between Irregularly Sampled Time Series: An Application to Gravitational Lenses

“…Elsewhere, we have empirically shown, using artificial irregularly sampled time series with noise and gaps (typical of radio observations), that a kernel-based approach to measure the time delay between two such time series outperforms typical statistical methods used by astronomers [7]. In this contribution, we extend, improve and test this approach to analyse actual optical observations (and artificial data representing such data), which shows high variability compared with radio observations.…”

Computational Intelligence in Astronomy – A Win-Win Situation

“…In [5,4] we introduced a model based technique for estimating time delays in fluxes from gravitationally lensed objects (such as quasars). The main idea of the method was to impose an internal model on the quasar variability in time and then expect that the multiple images will follow that model, up to observational noise and time delays.…”

“…Despite numerous claims, a universally agreed value for the time delay in the Q0957+561 system has not emerged [18,5]. Indeed, a major problem with time delay estimation in astrophysics literature has been that these estimates are routinely produced for individual quasars, for which we have no idea what the 'true' time delay is (e.g.…”

“…The data was highly sampled with periodic gaps [7], simulating a monitoring campaign of eight months; yielding 61 samples per time series. We imposed seven 'true' delays and 100 realisations for each value of true delay [5]. -The data was generated from a Bayesian model [13], simulating three levels of noise with 225 data sets per level of noise (there were 3 levels of noise).…”

Bayesian estimation of time delays between unevenly sampled signals

“…A recent study [3], involving one the authors of this paper, shows that the delay estimation problem with irregular sampling, mainly in the context of the determination of time lags in gravitationally lensed multiple images of distant celestial objects, is far from completely solved.…”