Annals of Applied Statistics

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

Hyungsuk Tak, Kaisey Mandel, David A. van Dyk, Vinay L. Kashyap, Xiao-Li Meng, and Aneta Siemiginowska

Full-text: Open access

Enhanced PDF (1614 KB)

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 appear in the sky. Since the light in each gravitationally lensed image traverses a different path length from the quasar to the Earth, fluctuations 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 extrinsic 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 likelihood of the time delay as an approximation of its marginal posterior distribution. The Bayesian and profile likelihood approaches complement 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.

Article information

Source
Ann. Appl. Stat., Volume 11, Number 3 (2017), 1309-1348.

Dates
Received: February 2016
Revised: January 2017
First available in Project Euclid: 5 October 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoas/1507168831

Digital Object Identifier
doi:10.1214/17-AOAS1027

Mathematical Reviews number (MathSciNet)
MR3709561

Zentralblatt MATH identifier
1380.62275

Keywords
Gravitational lensing microlensing Ornstein–Uhlenbeck process Gibbs sampler profile likelihood ancillarity-sufficiency interweaving strategy adaptive MCMC Q0957$+$561 J1029$+$2623 LSST quasar

Citation

Tak, Hyungsuk; Mandel, Kaisey; van Dyk, David A.; Kashyap, Vinay L.; Meng, Xiao-Li; Siemiginowska, Aneta. Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves. Ann. Appl. Stat. 11 (2017), no. 3, 1309--1348. doi:10.1214/17-AOAS1027. https://projecteuclid.org/euclid.aoas/1507168831


Export citation

References

  • Berger, J. O., Bernardo, J. M. and Sun, D. (2015). Overall objective priors. Bayesian Anal. 10 189–221.
  • Berger, J. O., Liseo, B. and Wolpert, R. L. (1999). Integrated likelihood methods for eliminating nuisance parameters. Statist. Sci. 14 1–28.
  • Berk, D. E. V., Wilhite, B. C., Kron, R. G., Anderson, S. F., Brunner, R. J., Hall, P. B., Ivezić, Ž., Richards, G. T., Schneider, D. P., York, D. G., Brinkmann, J. V., Lamb, D. Q., Nichol, R. C. and Schlegel, D. J. (2004). The ensemble photometric variability of $\sim$25,000 quasars in the sloan digital sky survey. Astrophys. J. 601 692.
  • Blandford, R. and Narayan, R. (1992). Cosmological applications of gravitational lensing. Annu. Rev. Astron. Astrophys. 30 311–358.
  • Brooks, S. P., Morgan, B. J., Ridout, M. S. and Pack, S. (1997). Finite mixture models for proportions. Biometrics 53 1097–1115.
  • Brooks, S., Gelman, A., Jones, G. L. and Meng, X.-L., eds. (2011). Handbook of Markov Chain Monte Carlo. CRC Press, Boca Raton, FL.
  • Chang, K. and Refsdal, S. (1979). Flux variations of QSO 0957$+$561 A, B and image splitting by stars near the light path. Nature 282 561–564.
  • Courbin, F., Chantry, V., Revaz, Y., Sluse, D., Faure, C., Tewes, M., Eulaers, E., Koleva, M., Asfandiyarov, I., Dye, S., Magain, P., van Winckel, H., Coles, J., Saha, P., Ibrahimov, M. and Meylan, G. (2013). COSMOGRAIL: The COSmological MOnitoring of GRAvItational lenses IX. time delays, lens dynamics and baryonic fraction in He 0435-1223. Astron. Astrophys. 536 A53.
  • Davison, A. C. (2003). Statistical Models. Cambridge Series in Statistical and Probabilistic Mathematics 11. Cambridge Univ. Press, Cambridge.
  • Dobler, G., Fassnacht, C., Treu, T., Marshall, P. J., Liao, K., Hojjati, A., Linder, E. and Rumbaugh, N. (2015). Strong lens time delay challenge. I. Experimental design. Astrophys. J. 799 168.
  • Fassnacht, C., Pearson, T., Readhead, A., Browne, I., Koopmans, L., Myers, S. and Wilkinson, P. (1999). A determination of h$_{o}$ with the CLASS gravitational lens B1608$+$656. I. time delay measurements with the VLA. Astrophys. J. 527 498.
  • Fischer, P., Bernstein, G., Rhee, G. and Tyson, J. A. (1997). The mass distribution of the cluster Q0957+561 from gravitational lensing. Astron. J. 113 521.
  • Fohlmeister, J., Kochanek, C. S., Falco, E. E., Wambsganss, J., Oguri, M. and Dai, X. (2013). A two-year time delay for the lensed quasar SDSS J1029+ 2623. Astrophys. J. 764 186.
  • Gelman, A. and Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statist. Sci. 7 457–472.
  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A. and Rubin, D. B. (2014). Bayesian Data Analysis, 3rd ed. CRC Press, Boca Raton, FL.
  • Hainline, L. J., Morgan, C. W., Beach, J. N., Kochanek, C., Harris, H. C., Tilleman, T., Fadely, R., Falco, E. E. and Le, T. X. (2012). A new microlensing event in the doubly imaged quasar Q 0957$+$561. Astrophys. J. 744 104.
  • Harva, M. and Raychaudhury, S. (2006). Bayesian Estimation of Time Delays Between Unevenly Sampled Signals. IEEE, New York.
  • Hojjati, A., Kim, A. G. and Linder, E. V. (2013). Robust strong lensing time delay estimation. Phys. Rev. D 87 123512.
  • Inada, N., Oguri, M., Morokuma, T., Doi, M., Yasuda, N., Becker, R. H., Richards, G. T., Kochanek, C. S., Kayo, I., Konishi, K. et al. (2006). SDSS J1029$+$2623: A gravitationally lensed quasar with an image separation of 225. Astrophys. J. Lett. 653 L97.
  • Kelly, B. C., Bechtold, J. and Siemiginowska, A. (2009). Are the variations in quasar optical flux driven by thermal fluctuations? Astrophys. J. 698 895.
  • Kochanek, C., Morgan, N., Falco, E., McLeod, B., Winn, J., Dembicky, J. and Ketzeback, B. (2006). The time delays of gravitational lens He 0435–1223: An early-type galaxy with a rising rotation curve. Astrophys. J. 640 47.
  • Kozłowski, S. and Kochanek, C. S. (2009). Discovery of 5000 active galactic nuclei behind the magellanic clouds. Astrophys. J. 701 508.
  • Kozłowski, S., Kochanek, C. S., Udalski, A., Soszyński, I., Szymański, M., Kubiak, M., Pietrzyński, G., Szewczyk, O., Ulaczyk, K. and Poleski, R. (2010). Quantifying quasar variability as part of a general approach to classifying continuously varying sources. Astrophys. J. 708 927.
  • Kumar, S. R., Stalin, C. and Prabhu, T. (2014). H$_{0}$ from 11 well measured time-delay lenses. Astron. Astrophys. 580 A38.
  • Liao, K., Treu, T., Marshall, P., Fassnacht, C. D., Rumbaugh, N., Dobler, G., Aghamousa, A., Bonvin, V., Courbin, F., Hojjati, A., Jackson, N., Kashyap, V., Rathna Kumar, S., Linder, E., Mandel, K., Meng, X. L., Meylan, G., Moustakas, L. A., Prabhu, T. P., Romero-Wolf, A., Shafieloo, A., Siemiginowska, A., Stalin, C. S., Tak, H., Tewes, M. and van Dyk, D. (2015). Strong lens time delay challenge: II. Results of TDC1. Astrophys. J. 800 11.
  • Linder, E. V. (2011). Lensing time delays and cosmological complementarity. Phys. Rev. D 84 123529.
  • Liu, J. S. (2008). Monte Carlo Strategies in Scientific Computing. Springer, New York.
  • LSST Science Collaboration (2009). LSST Science Book, Version 2.0. Available at arXiv:0912.0201.
    arXiv: 0912.0201
  • MacLeod, C., Ivezić, Ž., Kochanek, C., Kozłowski, S., Kelly, B., Bullock, E., Kimball, A., Sesar, B., Westman, D., Brooks, K., Gibson, R., Becker, A. C. and de Vries, W. H. (2010). Modeling the time variability of SDSS stripe 82 quasars as a damped random walk. Astrophys. J. 721 1014.
  • Morgan, C. W., Hainline, L. J., Chen, B., Tewes, M., Kochanek, C. S., Dai, X., Kozlowski, S., Blackburne, J. A., Mosquera, A. M., Chartas, G., Courbin, F. and Meylan, G. (2012). Further evidence that quasar X-ray emitting regions are compact: X-ray and optical microlensing in the lensed quasar q J0158-4325. Astrophys. J. 756 52.
  • Mosquera, A. M. and Kochanek, C. S. (2011). The microlensing properties of a sample of 87 lensed quasars. Astrophys. J. 738 96.
  • Munoz, J., Falco, E., Kochanek, C., Lehár, J., McLeod, B., Impey, C., Rix, H.-W. and Peng, C. (1998). The CASTLES project. Astrophys. Space Sci. 263 51–54.
  • Oguri, M. and Marshall, P. J. (2010). Gravitationally lensed quasars and supernovae in future wide-field optical imaging surveys. Mon. Not. R. Astron. Soc. 405 2579–2593.
  • Oscoz, A., Mediavilla, E., Goicoechea, L. J., Serra-Ricart, M. and Buitrago, J. (1997). Time delay of QSO 0957$+$561 and cosmological implications. Astrophys. J. Letters 479 L89.
  • Oscoz, A., Alcalde, D., Serra-Ricart, M., Mediavilla, E., Abajas, C., Barrena, R., Licandro, J., Motta, V. and Munoz, J. (2001). Time delay in QSO 0957$+$561 from 1984-1999 optical data. Astrophys. J. 552 81.
  • Pelt, J., Hoff, W., Kayser, R., Refsdal, S. and Schramm, T. (1994). Time delay controversy on QSO 0957+ 561 not yet decided. Astron. Astrophys. 286 775–785.
  • Pelt, J., Kayser, R., Refsdal, S. and Schramm, T. (1996). The light curve and the time delay of QSO 0957$+$561. Astron. Astrophys. 305 97–106.
  • R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
  • Refsdal, S. (1964). The gravitational lens effect. Mon. Not. R. Astron. Soc. 128 295–306.
  • Roberts, G. O. and Rosenthal, J. S. (2007). Coupling and ergodicity of adaptive Markov chain Monte Carlo algorithms. J. Appl. Probab. 44 458–475.
  • Schneider, P., Ehlers, J. and Falco, E. (1992). Gravitational Lenses. Springer, Berlin.
  • Schneider, P., Wambsganss, J. and Kochanek, C. S. (2006). Gravitational Lensing: Strong, Weak and Micro. Springer, Berlin.
  • Serra-Ricart, M., Oscoz, A., Sanchís, T., Mediavilla, E., Goicoechea, L. J., Licandro, J., Alcalde, D. and Gil-Merino, R. (1999). BVRI photometry of QSO 0957$+$561A, B: Observations, new reduction method, and time delay. Astrophys. J. 526 40.
  • Shalyapin, V., Goicoechea, L. and Gil-Merino, R. (2014). A 5.5-year robotic optical monitoring of Q0957$+$561: Substructure in a non-local cD galaxy. Astron. Astrophys. 540 A132.
  • Suyu, S., Auger, M., Hilbert, S., Marshall, P., Tewes, M., Treu, T., Fassnacht, C., Koopmans, L., Sluse, D., Blandford, R., Courbin, F. and Meylan, G. (2013). Two accurate time-delay distances from strong lensing: Implications for cosmology. Astrophys. J. 766 70.
  • Tak, H., Kelly, J. and Morris, C. N. (2017). Rgbp: An R package for Gaussian, Poisson, and Binomial Random Effects Models with Frequency Coverage Evaluations. J. Stat. Softw. 78 1–33.
  • Tak, H., Mandel, K., van Dyk, D. A., Kashyap, V. L., Meng, X.-L and Siemiginowska, A. (2017). Supplement to “Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves.” DOI:10.1214/17-AOAS1027SUPP.
  • Tewes, M., Courbin, F. and Meylan, G. (2013). COSMOGRAIL: The COSmological MOnitoring of GRAvItational lenses XI. techniques for time delay measurement in presence of microlensing. Astrophys. J. 605 58.
  • Tierney, L. (1994). Markov chains for exploring posterior distributions. Ann. Statist. 22 1701–1762.
  • Treu, T. and Marshall, P. J. (2016). Time delay cosmography. Astron. Astrophys. Rev. 24 11.
  • Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of the Brownian motion. Phys. Rev. 36 823–841.
  • van Dyk, D. A. and Meng, X.-L. (2001). The art of data augmentation. J. Comput. Graph. Statist. 10 1–111.
  • Walsh, D., Carswell, R. and Weymann, R. (1979). 0957$+$561 A, B- twin quasistellar objects or gravitational lens. Nature 279 381–384.
  • Yu, Y. and Meng, X.-L. (2011). To center or not to center: That is not the question—an ancillarity-sufficiency interweaving strategy (ASIS) for boosting MCMC efficiency. J. Comput. Graph. Statist. 20 531–570.
  • Zu, Y., Kochanek, C., Kozłowski, S. and Udalski, A. (2013). Is quasar optical variability a damped random walk? Astrophys. J. 765 106.

Supplemental materials

  • R codes and data. This zip file [Tak et al. (2017)] contains all the computer code (Rcode.R) and data (Data.zip) used in this article. An R package, timedelay, that implements the Bayesian and profile likelihood methods is publicly available at CRAN (https://cran.r-project.org/package=timedelay).
    Digital Object Identifier: doi:10.1214/17-AOAS1027SUPP
    Supplemental ZIP File (28 KB)

The Institute of Mathematical Statistics

Annals of Applied Statistics

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

  • Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis
    Kirch, Claudia, Edwards, Matthew C., Meier, Alexander, and Meyer, Renate, Bayesian Analysis, 2019
  • Gravitational Lensing Accuracy Testing 2010 (GREAT10) Challenge Handbook
    Kitching, Thomas, Balan, Sreekumar, Bernstein, Gary, Bethge, Matthias, Bridle, Sarah, Courbin, Frederic, Gentile, Marc, Heavens, Alan, Hirsch, Michael, Hosseini, Reshad, Kiessling, Alina, Amara, Adam, Kirk, Donnacha, Kuijken, Konrad, Mandelbaum, Rachel, Moghaddam, Baback, Nurbaeva, Guldariya, Paulin-Henriksson, Stephane, Rassat, Anais, Rhodes, Jason, Schölkopf, Bernhard, Shawe-Taylor, John, Gill, Mandeep, Shmakova, Marina, Taylor, Andy, Velander, Malin, van Waerbeke, Ludovic, Witherick, Dugan, Wittman, David, Harmeling, Stefan, Heymans, Catherine, Massey, Richard, Rowe, Barnaby, Schrabback, Tim, and Voigt, Lisa, Annals of Applied Statistics, 2011
  • Handbook for the GREAT08 Challenge: An image analysis competition for cosmological lensing
    Bridle, Sarah, Gill, Mandeep, Heavens, Alan, Heymans, Catherine, High, F. William, Hoekstra, Henk, Jarvis, Mike, Kirk, Donnacha, Kitching, Thomas, Kneib, Jean-Paul, Kuijken, Konrad, Shawe-Taylor, John, Lagatutta, David, Mandelbaum, Rachel, Massey, Richard, Mellier, Yannick, Moghaddam, Baback, Moudden, Yassir, Nakajima, Reiko, Paulin-Henriksson, Stephane, Pires, Sandrine, Rassat, Anais, Amara, Adam, Refregier, Alexandre, Rhodes, Jason, Schrabback, Tim, Semboloni, Elisabetta, Shmakova, Marina, van Waerbeke, Ludovic, Witherick, Dugan, Voigt, Lisa, Wittman, David, Applegate, Douglas, Balan, Sreekumar T., Berge, Joel, Bernstein, Gary, Dahle, Hakon, and Erben, Thomas, Annals of Applied Statistics, 2009

+ See more