Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves

Abstract

The gravitational field of a galaxy can act as a lens and deflect the light emitted by a more distant object such as a quasar. Strong gravitational lensing causes multiple images of the same quasar to ap- pear in the sky. Since the light in each gravitationally lensed image traverses a different path length from the quasar to the Earth, fluc- tuations in the source brightness are observed in the several images at different times. The time delay between these fluctuations can be used to constrain cosmological parameters and can be inferred from the time series of brightness data or light curves of each image. To estimate the time delay, we construct a model based on a state- space representation for irregularly observed time series generated by a latent continuous-time Ornstein-Uhlenbeck process. We account for microlensing, an additional source of independent long-term ex- trinsic variability, via a polynomial regression. Our Bayesian strategy adopts a Metropolis-Hastings within Gibbs sampler. We improve the sampler by using an ancillarity-sufficiency interweaving strategy and adaptive Markov chain Monte Carlo. We introduce a profile likeli- hood of the time delay as an approximation of its marginal posterior distribution. The Bayesian and profile likelihood approaches comple- ment each other, producing almost identical results; the Bayesian method is more principled but the profile likelihood is simpler to implement. We demonstrate our estimation strategy using simulated data of doubly- and quadruply-lensed quasars, and observed data from quasars Q0957+561 and J1029+2623

Rapid optical variations correlated with X-rays in the 2015 second outburst of V404 Cygni (GS 2023+338)

“…8 The 1-s binned X-ray light curves were derived with the tool ii_light described in Sec. 2.2. ally lensed stochastic light curves (Tak et al 2016a). 9 This method assumes that the irregularly sampled light curves are generated by a latent continuous-time damped random walk (DRW) process (Kelly et al 2009) and that one of the latent light curves is a shifted version of the other by the time lag in the horizontal axis and by the magnitude offset 9 The 5-s binned X-ray light curves were derived with the tool ii_light described in Sec.…”

“…10 Our data, however, do not completely meet the second model assumption (i.e., one of the latent light curves is a parallel-shifted version of the other). This is because our optical light curves have smaller amplitudes than the Xray ones and the origin of the optical light curves may be different from that of the X-ray ones unlike gravitationally lensed light curves as originally applied in Tak et al (2016a). Thus we scaled the X-ray light curve to the optical one using the results of the power law regression (see Sec.…”

“…We use a Bayesian model (Tak et al 2016a) to estimate the time lag between optical and X-ray light curves. The notation x = (x 1 , .…”

“…We choose the shape and scale parameters of the inverse Gamma distributions of τ and σ 2 in a way to constrain ∆, considering shorter timescale of the observed data than gravitationally lensed light curves. Further details of the motivation for the choice of prior distributions are given in Sections 2.5 of Tak et al (2016a).…”

Disentangling Time-series Spectra with Gaussian Processes: Applications to Radial Velocity Analysis

“…However, for the purposes of inferring intrinsic spectra and stellar radial velocities, simple Gaussian processes offer an attractive framework to model stellar spectra in a purely data-driven manner. Gaussian processes have been used successfully for other time-series applications in astronomy such as, for example, modeling lensed quasar time delays (Hojjati et al 2013;Tak et al 2016), inferring stellar rotation periods (Angus et al 2015), and modeling correlated noise in photometric observations of planet transits and eclipses (Evans et al 2015;Montet et al 2016).…”