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  • Open Access

Global structure of the time delay likelihood

Namu Kroupa1,2,3,* and Will Handley2,4

  • 1Astrophysics Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
  • 2Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, United Kingdom
  • 3Engineering Laboratory, University of Cambridge, Cambridge CB2 1PZ, United Kingdom
  • 4Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, United Kingdom

  • *Contact author: nk544@cam.ac.uk
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Phys. Rev. Research 8, 023175 – Published 18 May, 2026

DOI: https://doi.org/10.1103/kj4s-nnyx

Abstract

We identify a fundamental pathology in the likelihood for time delay inference that challenges standard inference methods. By analyzing the likelihood for time delay inference with Gaussian process light curve models, we show that it generically develops a boundary-driven “W” shape with a global maximum at the true delay and gradual rises toward the edges of the observation window. This arises because time delay estimation is intrinsically extrapolative. In practice, global samplers such as nested sampling are steered toward spurious edge modes unless strict convergence criteria are adopted. We demonstrate this with simulations and show that the effect strengthens with higher data density over a fixed time span. To ensure convergence, we provide concrete guidance, notably increasing the number of live points. Further, we show that methods implicitly favoring small delays, for example, optimizers and local Markov chain Monte Carlo, induce a bias toward larger H0. Our results clarify failure modes and offer practical remedies for robust fully Bayesian time delay inference.

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References (89)

  1. S. Refsdal, On the possibility of determining Hubble's parameter and the masses of galaxies from the gravitational lens effect, Mon. Not. R. Astron. Soc. 128, 307 (1964).
  2. T. Treu and P. J. Marshall, Time delay cosmography, Astron. Astrophys. Rev. 24, 11 (2016).
  3. S. Birrer, M. Millon, D. Sluse, A. Shajib, F. Courbin, L. Koopmans, S. Suyu, and T. Treu, Time-delay cosmography: Measuring the Hubble constant and other cosmological parameters with strong gravitational lensing, Space Sci. Rev. 220, 48 (2024).
  4. T. Treu, S. H. Suyu, and P. J. Marshall, Strong lensing time-delay cosmography in the 2020s, Astron. Astrophys. Rev. 30, 8 (2022).
  5. S. Birrer, G. P. Smith, A. J. Shajib, D. Ryczanowski, and N. Arendse, Challenges and opportunities for time-delay cosmography with multi-messenger gravitational lensing, Phil. Trans. A 383, 20240130 (2025).
  6. B. M. Peterson, Reverberation mapping of active galactic nuclei, Publ. Astron. Soc. Pac. 105, 247 (1993).
  7. E. M. Cackett, M. C. Bentz, and E. Kara, Reverberation mapping of active galactic nuclei: From X-ray corona to dusty torus, iScience 24, 102557 (2021).
  8. H. Netzer and B. M. Peterson, Reverberation mapping and the physics of active galactic nuclei, in Astronomical Time Series. Astrophysics and Space Science Library, edited by D. Maoz, A. Sternberg, and E. M. Leibowitz (Springer, Dordrecht, 1997), Vol. 218, pp. 85–108.
  9. P. A. Abell, J. Allison, S. F. Anderson, J. R. Andrew, J. R. P. Angel, L. Armus, D. Arnett, S. J. Asztalos, T. S. Axelrod, S. Bailey, et al., LSST science book, version 2.0, Tech. Rep. (LSST Corporation, Tucson, AZ, 2009).
  10. Ž. Ivezić, S. M. Kahn, J. A. Tyson, B. Abel, E. Acosta, R. Allsman, D. Alonso, Y. AlSayyad, S. F. Anderson, J. Andrew, et al., LSST: From science drivers to reference design and anticipated data products, Astrophys. J. 873, 111 (2019).
  11. J. Kollmeier, S. Anderson, G. Blanc, M. Blanton, K. Covey, J. Crane, N. Drory, P. Frinchaboy, C. Froning, J. Johnson, et al., SDSS-V: Pioneering panoptic spectroscopy, Bull. Am. Astron. Soc. 51, 274 (2019).
  12. A. Almeida, S. F. Anderson, M. Argudo-Fernández, C. Badenes, K. Barger, J. K. Barrera-Ballesteros, C. F. Bender, E. Benitez, F. Besser, J. C. Bird, et al., The eighteenth data release of the Sloan Digital Sky Surveys: Targeting and first spectra from SDSS-V, Astrophys. J. Suppl. Ser. 267, 44 (2023).
  13. C. Grier, J. R. Trump, Y. Shen, K. Horne, K. Kinemuchi, I. D. McGreer, D. Starkey, W. Brandt, P. Hall, C. Kochanek, et al., The Sloan Digital Sky Survey reverberation mapping project: Hα and Hβ reverberation measurements from first-year spectroscopy and photometry, Astrophys. J. 851, 21 (2017).
  14. C. Grier, Y. Shen, K. Horne, W. Brandt, J. Trump, P. Hall, K. Kinemuchi, D. Starkey, D. Schneider, L. C. Ho, et al., The Sloan Digital Sky Survey reverberation mapping project: Initial C iv lag results from four years of data, Astrophys. J. 887, 38 (2019).
  15. I. Jennifer, H. Li, Y. Shen, W. Brandt, C. Grier, P. Hall, L. Ho, Y. Homayouni, K. Horne, D. Schneider, et al., The Sloan Digital Sky Survey reverberation mapping project: Comparison of lag measurement methods with simulated observations, Astrophys. J. 884, 119 (2019).
  16. Y. Homayouni, J. R. Trump, C. Grier, K. Horne, Y. Shen, W. Brandt, K. S. Dawson, G. F. Alvarez, P. J. Green, P. Hall, et al., The Sloan Digital Sky Survey reverberation mapping project: Mg ii lag results from four years of monitoring, Astrophys. J. 901, 55 (2020).
  17. H. McDougall, B. J. Pope, and T. M. Davis, LITMUS: Bayesian lag recovery in reverberation mapping with fast differentiable models, Publ. Astron. Soc. Aust. 43, e018 (2026).
  18. A. Hojjati, A. G. Kim, and E. V. Linder, Robust strong lensing time delay estimation, Phys. Rev. D 87, 123512 (2013).
  19. M. Tewes, F. Courbin, and G. Meylan, COSMOGRAIL: The cosmological monitoring of gravitational lenses-XI. Techniques for time delay measurement in presence of microlensing, Astron. Astrophys. 553, A120 (2013).
  20. Y. Zu, C. Kochanek, and B. M. Peterson, An alternative approach to measuring reverberation lags in active galactic nuclei, Astrophys. J. 735, 80 (2011).
  21. Y.-R. Li, J.-M. Wang, L. C. Ho, P. Du, and J.-M. Bai, A Bayesian approach to estimate the size and structure of the broad-line region in active galactic nuclei using reverberation mapping data, Astrophys. J. 779, 110 (2013).
  22. A. Pancoast, B. J. Brewer, and T. Treu, Modelling reverberation mapping data—I. Improved geometric and dynamical models and comparison with cross-correlation results, Mon. Not. R. Astron. Soc. 445, 3055 (2014).
  23. A. Pancoast, B. J. Brewer, T. Treu, D. Park, A. J. Barth, M. C. Bentz, and J.-H. Woo, Modelling reverberation mapping data—II. Dynamical modelling of the Lick AGN Monitoring Project 2008 data set, Mon. Not. R. Astron. Soc. 445, 3073 (2014).
  24. D. Starkey, K. Horne, M. Fausnaugh, B. Peterson, M. C. Bentz, C. Kochanek, K. Denney, R. Edelson, M. Goad, G. De Rosa, et al., Space Telescope and Optical Reverberation Mapping Project. VI. Reverberating disk models for NGC 5548, Astrophys. J. 835, 65 (2017).
  25. C. Vanderriest, J. Schneider, G. Herpe, M. Chevreton, M. Moles, and G. Wlerick, The value of the time delay δt (A, B) for the “double” quasar 0957+561 from optical photometric monitoring, Astron. Astrophys. 215, 1 (1989).
  26. J. Lehár, J. N. Hewitt, D. H. Roberts, and B. F. Burke, The radio time delay in the double quasar 0957+561, Astrophys. J. 384, 453 (1992).
  27. W. H. Press, G. B. Rybicki, and J. N. Hewitt, The time delay of gravitational lens 0957+561. I. Methodology and analysis of optical photometric data. II. Analysis of radio data and combined optical-radio analysis, Astrophys. J. 385, 404 (1992).
  28. J. Pelt, R. Kayser, S. Refsdal, and T. Schramm, The light curve and the time delay of QSO 0957+561, arXiv:astro-ph/9501036.
  29. T. Kundic, E. L. Turner, W. N. Colley, J. R. Gott, J. E. Rhoads, Y. Wang, L. E. Bergeron, K. A. Gloria, D. C. Long, S. Malhotra, et al., A robust determination of the time delay in 0957+561A, B and a measurement of the global value of Hubble's constant, Astrophys. J. 482, 75 (1997).
  30. S. Suyu, P. Marshall, R. Blandford, C. Fassnacht, L. Koopmans, J. McKean, and T. Treu, Dissecting the gravitational lens B1608+656. I. Lens potential reconstruction, Astrophys. J. 691, 277 (2009).
  31. S. Suyu, P. Marshall, M. Auger, S. Hilbert, R. Blandford, L. Koopmans, C. Fassnacht, and T. Treu, Dissecting the gravitational lens B1608+656. II. Precision measurements of the Hubble constant, spatial curvature, and the dark energy equation of state, Astrophys. J. 711, 201 (2010).
  32. L. Koopmans, T. Treu, C. Fassnacht, R. Blandford, and G. Surpi, The Hubble constant from the gravitational lens B1608+656, Astrophys. J. 599, 70 (2003).
  33. G. Dobler, C. D. Fassnacht, T. Treu, P. Marshall, K. Liao, A. Hojjati, E. Linder, and N. Rumbaugh, Strong lens time delay challenge. I. Experimental design, Astrophys. J. 799, 168 (2015).
  34. K. Liao, T. Treu, P. Marshall, C. D. Fassnacht, N. Rumbaugh, G. Dobler, A. Aghamousa, V. Bonvin, F. Courbin, A. Hojjati, et al., Strong lens time delay challenge. II. Results of TDC1, Astrophys. J. 800, 11 (2015).
  35. S. R. Kumar, M. Tewes, C. Stalin, F. Courbin, I. Asfandiyarov, G. Meylan, E. Eulaers, T. Prabhu, P. Magain, H. Van Winckel, et al., COSMOGRAIL: The cosmological monitoring of gravitational lenses-XIV. Time delay of the doubly lensed quasar SDSS J1001+5027, Astron. Astrophys. 557, A44 (2013).
  36. F. Courbin, V. Bonvin, E. Buckley-Geer, C. Fassnacht, J. Frieman, H. Lin, P. Marshall, S. Suyu, T. Treu, T. Anguita, et al., COSMOGRAIL XVI: Time delays for the quadruply imaged quasar DES J0408−5354 with high-cadence photometric monitoring, Astron. Astrophys. 609, A71 (2018).
  37. J. Skilling, Nested sampling for general Bayesian computation, Bayesian Anal. 1, 833 (2006).
  38. G. Ashton, N. Bernstein, J. Buchner, X. Chen, G. Csányi, A. Fowlie, F. Feroz, M. Griffiths, W. Handley, M. Habeck, et al., Nested sampling for physical scientists, Nat. Rev. Methods Primers 2, 39 (2022).
  39. J. Buchner, Nested sampling methods, Statist. Surv. 17, 169 (2023).
  40. N. Kroupa, D. Yallup, W. Handley, and M. Hobson, Kernel-, mean-, and noise-marginalized Gaussian processes for exoplanet transits and H0 inference, Mon. Not. R. Astron. Soc. 528, 1232 (2024).
  41. S. Chib, Marginal likelihood from the Gibbs output, J. Am. Stat. Assoc. 90, 1313 (1995).
  42. A. Gelman and X.-L. Meng, Simulating normalizing constants: From importance sampling to bridge sampling to path sampling, Stat. Sci. 13, 163 (1998).
  43. N. Lartillot and H. Philippe, Computing Bayes factors using thermodynamic integration, Syst. Biol. 55, 195 (2006).
  44. R. J. Urick, Principles of Underwater Sound-2 (McGraw-Hill, New York, 1975).
  45. G. L. Turin, Introduction to spread-spectrum antimultipath techniques and their application to urban digital radio, Proc. IEEE 68, 328 (2005).
  46. J. Chen, J. Benesty, and Y. Huang, Time delay estimation in room acoustic environments: An overview, EURASIP J. Adv. Signal Process. 2006, 026503 (2006).
  47. K. Gedalyahu and Y. C. Eldar, Time-delay estimation from low-rate samples: A union of subspaces approach, IEEE Trans. Signal Process. 58, 3017 (2010).
  48. A. Quazi, An overview on the time delay estimate in active and passive systems for target localization, IEEE Trans. Acoust. Speech Signal Process. 29, 527 (1981).
  49. S. Heerah, R. Molinari, S. Guerrier, and A. Marshall-Colon, Granger-causal testing for irregularly sampled time series with application to nitrogen signalling in Arabidopsis, Bioinformatics 37, 2450 (2021).
  50. D. P. Schaff and F. Waldhauser, Waveform cross-correlation-based differential travel-time measurements at the Northern California Seismic Network, Bull. Seismol. Soc. Am. 95, 2446 (2005).
  51. M. Hoffmann, M. Rosenbaum, and N. Yoshida, Estimation of the lead-lag parameter from non-synchronous data, Bernoulli 19, 426 (2013).
  52. S. Bjorklund and L. Ljung, A review of time-delay estimation techniques, in Proceedings of the 42nd IEEE International Conference on Decision and Control (IEEE cat. no. 03CH37475) (IEEE, Piscataway, NJ, 2003), Vol. 3, pp. 2502–2507.
  53. L. B. Pártay, G. Csányi, and N. Bernstein, Nested sampling for materials, Eur. Phys. J. B 94, 159 (2021).
  54. P. Y. Chew and A. Reinhardt, Phase diagrams—why they matter and how to predict them, J. Chem. Phys. 158, 030902 (2023).
  55. C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning (The MIT Press, Cambridge, MA, 2005).
  56. C. L. MacLeod, Ž. Ivezić, C. Kochanek, S. Kozłowski, B. Kelly, E. Bullock, A. Kimball, B. Sesar, D. Westman, K. Brooks, et al., Modeling the time variability of SDSS Stripe 82 quasars as a damped random walk, Astrophys. J. 721, 1014 (2010).
  57. Y. Zu, C. Kochanek, S. Kozłowski, and A. Udalski, Is quasar optical variability a damped random walk? Astrophys. J. 765, 106 (2013).
  58. N. Kroupa, D. Yallup, and W. Handley, Time delays and stationarity in quasar light curves, Phys. Rev. D 113, 063549 (2026).
  59. L. T. Hergt, W. J. Handley, M. P. Hobson, and A. N. Lasenby, Bayesian evidence for the tensor-to-scalar ratio r and neutrino masses mν: Effects of uniform versus logarithmic priors, Phys. Rev. D 103, 123511 (2021).
  60. H. Jeffreys, An invariant form for the prior probability in estimation problems, Proc. R. Soc. London A 186, 453 (1946).
  61. W. Handley, M. Hobson, and A. Lasenby, POLYCHORD: Nested sampling for cosmology, Mon. Not. R. Astron. Soc.: Lett. 450, L61 (2015).
  62. W. J. Handley, M. P. Hobson, and A. N. Lasenby, PolyChord: Next-generation nested sampling, Mon. Not. R. Astron. Soc. 453, 4385 (2015).
  63. W. Handley, anesthetic: Nested sampling visualisation, J. Open Source Software 4, 1414 (2019).
  64. D. Yallup, N. Kroupa, and W. Handley, Nested slice sampling, in Frontiers in Probabilistic Inference: Learning meets Sampling (ICLR 2025 Workshop, Singapore, 2025).
  65. D. Yallup, N. Kroupa, and W. Handley, Nested slice sampling: Vectorized nested sampling for GPU-accelerated inference, arXiv:2601.23252.
  66. R. Frostig, M. J. Johnson, and C. Leary, Compiling machine learning programs via high-level tracing, Syst. Mach. Learn. 4, 9 (2018).
  67. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, Cambridge, UK, 2007).
  68. K. C. Wong, S. H. Suyu, M. W. Auger, V. Bonvin, F. Courbin, C. D. Fassnacht, A. Halkola, C. E. Rusu, D. Sluse, A. Sonnenfeld, et al., H0LiCOW–IV. Lens mass model of HE 0435−1223 and blind measurement of its time-delay distance for cosmology, Mon. Not. R. Astron. Soc. 465, 4895 (2017).
  69. R. Barkana, FASTELL: Fast calculation of a family of elliptical mass gravitational lens models, Astrophysics Source Code Library, record ascl9910.003, 1999.
  70. N. D. Morgan, C. S. Kochanek, O. Pevunova, and P. L. Schechter, The lens redshift and galaxy environment for HE 0435-1223, Astron. J. 129, 2531 (2005).
  71. D. Sluse, D. Hutsemékers, F. Courbin, G. Meylan, and J. Wambsganss, Microlensing of the broad line region in 17 lensed quasars, Astron. Astrophys. 544, A62 (2012).
  72. S. Birrer and A. Amara, lenstronomy: Multi-purpose gravitational lens modelling software package, Phys. Dark Universe 22, 189 (2018).
  73. S. Birrer, A. J. Shajib, D. Gilman, A. Galan, J. Aalbers, M. Millon, R. Morgan, G. Pagano, J. W. Park, L. Teodori, et al., lenstronomy II: A gravitational lensing software ecosystem, J. Open Source Software 6, 3283 (2021).
  74. R. Narayan and M. Bartelmann, Lectures on gravitational lensing, arXiv:astro-ph/9606001.
  75. K. Hukushima and K. Nemoto, Exchange Monte Carlo method and application to spin glass simulations, J. Phys. Soc. Jpn. 65, 1604 (1996).
  76. D. J. Earl and M. W. Deem, Parallel tempering: Theory, applications, and new perspectives, Phys. Chem. Chem. Phys. 7, 3910 (2005).
  77. N. Chopin, A sequential particle filter method for static models, Biometrika 89, 539 (2002).
  78. P. Del Moral, A. Doucet, and A. Jasra, Sequential Monte Carlo samplers, J. R. Stat. Soc. B 68, 411 (2006).
  79. B. J. Brewer, L. B. Pártay, and G. Csányi, Diffusive nested sampling, Stat. Comput. 21, 649 (2011).
  80. A. Cabezas, A. Corenflos, J. Lao, R. Louf, A. Carnec, K. Chaudhari, R. Cohn-Gordon, J. Coullon, W. Deng, S. Duffield, et al., BlackJAX: Composable Bayesian inference in JAX, arXiv:2402.10797.
  81. S. Martiniani, J. D. Stevenson, D. J. Wales, and D. Frenkel, Superposition enhanced nested sampling, Phys. Rev. X 4, 031034 (2014).
  82. R. J. Baldock, L. B. Pártay, A. P. Bartók, M. C. Payne, and G. Csányi, Determining pressure-temperature phase diagrams of materials, Phys. Rev. B 93, 174108 (2016).
  83. N. Unglert, L. B. Pártay, and G. K. H. Madsen, Replica exchange nested sampling, J. Chem. Theory Comput. 21, 7304 (2025).
  84. N. Kroupa, G. Csányi, and W. Handley, Resonances in reflective Hamiltonian Monte Carlo, Phys. Rev. E 111, 045308 (2025).
  85. R. Salomone, L. F. South, C. Drovandi, D. P. Kroese, and A. M. Johansen, Unbiased and consistent nested sampling via sequential Monte Carlo, J. R. Stat. Soc. Ser. B Stat. Methodol. 87, 1221 (2025).
  86. http://www.dirac.ac.uk/.
  87. A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications (Now Publishers, Hanover, MA, 2004).
  88. T. Tao, Topics in Random Matrix Theory (American Mathematical Society, Providence, RI, 2012), Vol. 132.
  89. M. Potters and J.-P. Bouchaud, A First Course in Random Matrix Theory: For Physicists, Engineers and Data Scientists (Cambridge University Press, Cambridge, 2020).

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Phys. Rev. Research 8, 023175– Published 18 May, 2026
  • Received 13 September 2025
  • Accepted 11 April 2026
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