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Time delay cosmography


Gravitational time delays, observed in strong lens systems where the variable background source is multiply imaged by a massive galaxy in the foreground, provide direct measurements of cosmological distance that are very complementary to other cosmographic probes. The success of the technique depends on the availability and size of a suitable sample of lensed quasars or supernovae, precise measurements of the time delays, accurate modeling of the gravitational potential of the main deflector, and our ability to characterize the distribution of mass along the line of sight to the source. We review the progress made during the last 15 years, during which the first competitive cosmological inferences with time delays were made, and look ahead to the potential of significantly larger lens samples in the near future.

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  1. 1.

    Additional distance dependencies appear in the multi-plane formalism, but always as dimensionless ratios with weaker cosmological dependence. The inverse proportionality to the Hubble constant is the same as in the single plane case.

  2. 2.

    The COSMOGRAIL curve-shifting analysis code is available from

  3. 3.

    In the case of the shapelet basis set, regularization can effectively be achieved through choosing the number of basis functions to use as well as the scale of the underlying Gaussian. Most analyses using shapelets have taken this approach to date, with Tagore and Jackson (2016) being a notable exception. A promising alternative scheme would be to assign a less physically motivated prior for the shapelet coefficients.

  4. 4.

    The original investigation by Refsdal (1964) involved the “assumption that the linear distance–redshift relation is valid”.

  5. 5.

    Importantly, the authors agreed to publish the unblinded results, no matter what.

  6. 6.

    For simplicity, we refer to equivalent uncertainty on an average time delay distance at the typical redshift of the deflector and source. In practice of course, there will be a distribution of redshifts and thus of individual distances. As the way in which the time delay distance depends on cosmological parameters varies slightly with redshift, the analysis of a real sample of lenses will have the added benefit of breaking some of the degeneracies between the cosmological parameters, and reducing the uncertainties more rapidly than if all the systems were at the same redshift.

  7. 7.

    The time delay distance referred to here is the same as ensemble average quantity that Coe and Moustakas (2009) call \(\mathcal {\tau }_\mathrm{C}\).


  1. Aghamousa A, Shafieloo A (2015) Fast and reliable time delay estimation of strong lens systems using the smoothing and cross-correlation methods. ApJ 804:39. doi:10.1088/0004-637X/804/1/39. arXiv:1410.8122

  2. Agnello A, Treu T, Ostrovski F, Schechter PL, Buckley-Geer EJ, Lin H, Auger MW, Courbin F, Fassnacht CD, Frieman J, Kuropatkin N, Marshall PJ, McMahon RG, Meylan G, More A, Suyu SH, Rusu CE, Finley D, Abbott T, Abdalla FB, Allam S, Annis J, Banerji M, Benoit-Lévy A, Bertin E, Brooks D, Burke DL, Rosell AC, Kind MC, Carretero J, Cunha CE, D’Andrea CB, da Costa LN, Desai S, Diehl HT, Dietrich JP, Doel P, Eifler TF, Estrada J, Neto AF, Flaugher B, Fosalba P, Gerdes DW, Gruen D, Gutierrez G, Honscheid K, James DJ, Kuehn K, Lahav O, Lima M, Maia MAG, March M, Marshall JL, Martini P, Melchior P, Miller CJ, Miquel R, Nichol RC, Ogando R, Plazas AA, Reil K, Romer AK, Roodman A, Sako M, Sanchez E, Santiago B, Scarpine V, Schubnell M, Sevilla-Noarbe I, Smith RC, Soares-Santos M, Sobreira F, Suchyta E, Swanson MEC, Tarle G, Thaler J, Tucker D, Walker AR, Wechsler RH, Zhang Y (2015) Discovery of two gravitationally lensed quasars in the dark energy survey. MNRAS 454:1260–1265. doi:10.1093/mnras/stv2171. arXiv:1508.01203

  3. Auger MW, Fassnacht CD, Abrahamse AL, Lubin LM, Squires GK (2007) The gravitational lens-galaxy group connection. II. Groups associated with B2319+051 and B1600+434. AJ 134:668–679. doi:10.1086/519238. arXiv:astro-ph/0603448

  4. Auger MW, Treu T, Bolton AS, Gavazzi R, Koopmans LVE, Marshall PJ, Moustakas LA, Burles S (2010) The sloan lens ACS survey. X. Stellar, dynamical, and total mass correlations of massive early-type galaxies. ApJ 724:511–525. doi:10.1088/0004-637X/724/1/511. arXiv:1007.2880

  5. Bar-Kana R (1996) Effect of large-scale structure on multiply imaged sources. ApJ 468:17. doi:10.1086/177666. arXiv:astro-ph/9511056

  6. Barnabè M, Nipoti C, Koopmans LVE, Vegetti S, Ciotti L (2009) Crash-testing the CAULDRON code for joint lensing and dynamics analysis of early-type galaxies. MNRAS 393:1114–1126. doi:10.1111/j.1365-2966.2008.14208.x. arXiv:0808.3916

  7. Bartelmann M (2010) Topical review gravitational lensing. Class Quant Grav 27(23):233001. doi:10.1088/0264-9381/27/23/233001. arXiv:1010.3829

  8. Bennett CL, Larson D, Weiland JL, Jarosik N, Hinshaw G, Odegard N, Smith KM, Hill RS, Gold B, Halpern M, Komatsu E, Nolta MR, Page L, Spergel DN, Wollack E, Dunkley J, Kogut A, Limon M, Meyer SS, Tucker GS, Wright EL (2013) Nine-year wilkinson microwave anisotropy probe (WMAP) observations: final maps and results. ApJs 208:20. doi:10.1088/0067-0049/208/2/20. arXiv:1212.5225

  9. Birrer S, Amara A, Refregier A (2015) Gravitational lens modeling with basis sets. ApJ 813:102. doi:10.1088/0004-637X/813/2/102. arXiv:1504.07629

  10. Birrer S, Amara A, Refregier A (2015) The mass-sheet degeneracy and time-delay cosmography: analysis of the strong lens RXJ1131-1231. JCAP. arXiv:1511.03662 (submitted)

  11. Blackburne JA, Pooley D, Rappaport S, Schechter PL (2010) Sizes and temperature profiles of quasar accretion disks from chromatic microlensing. arXiv:1007.1665v1 [astro-ph.CO]

  12. Blandford RD, Narayan R (1992) Cosmological applications of gravitational. ARA A 30:311–358. doi:10.1146/annurev.aa.30.090192.001523

  13. Bonvin V, Tewes M, Courbin F, Kuntzer T, Sluse D, Meylan G (2016) COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses. XV. Assessing the achievability and precision of time-delay measurements. A A 585:A88. doi:10.1051/0004-6361/201526704. arXiv:1506.07524

  14. Boroson TA, Moustakas LA, Romero-Wolf A, McCully C (2016) Using the LCOGT network to measure a high-precision time delay in the four-image gravitational lens HE0435-1223. In: American Astronomical Society Meeting Abstracts, vol 227, p 338.04

  15. Brewer BJ, Lewis GF (2006) The Einstein ring 0047–2808 revisited: a Bayesian inversion. ApJ 651:8–13. doi:10.1086/507475. arXiv:astro-ph/0606714

  16. Browne IWA et al (2003) The cosmic lens all-sky survey–II. Gravitational lens candidate selection and follow-up. MNRAS 341:13–32. doi:10.1046/j.1365-8711.2003.06257.x. arXiv:astro-ph/0211069

  17. Burud I, Courbin F, Magain P, Lidman C, Hutsemékers D, Kneib JP, Hjorth J, Brewer J, Pompei E, Germany L, Pritchard J, Jaunsen AO, Letawe G, Meylan G (2002) An optical time-delay for the lensed BAL quasar HE 2149–2745. A A 383:71–81. doi:10.1051/0004-6361:20011731. arXiv:astro-ph/0112225

  18. Chen GCF, Suyu SH, Wong KC, Fassnacht CD, Chiueh T, Halkola A, Hu I, Auger MW, Koopmans LVE, Lagattuta DJ, McKean JP, Vegetti S (2016) SHARP—III: first use of adaptive optics imaging to constrain cosmology with gravitational lens time delays. arXiv:1601.01321

  19. Coe D, Moustakas LA (2009) Cosmological constraints from gravitational lens time delays. ApJ 706:45–59. doi:10.1088/0004-637X/706/1/45. arXiv:0906.4108

  20. Coles J (2008) A new estimate of the hubble time with improved modeling of gravitational lenses. ApJ 679:17–24. doi:10.1086/587635. arXiv:0802.3219

  21. Collett TE, Cunnington S (2016) Selection biases in time-delay cosmography. Mon Not R Astron Soc (submitted). arXiv:1605.08341

  22. Collett TE, Marshall PJ, Auger MW, Hilbert S, Suyu SH, Greene Z, Treu T, Fassnacht CD, Koopmans LVE, Bradač M, Blandford RD (2013) Reconstructing the lensing mass in the universe from photometric catalogue data. Mon Not R Astron Soc 432:679. doi:10.1093/mnras/stt504

  23. Conley A, Goldhaber G, Wang L, Aldering G, Amanullah R, Commins ED, Fadeyev V, Folatelli G, Garavini G, Gibbons R, Goobar A, Groom DE, Hook I, Howell DA, Kim AG, Knop RA, Kowalski M, Kuznetsova N, Lidman C, Nobili S, Nugent PE, Pain R, Perlmutter S, Smith E, Spadafora AL, Stanishev V, Strovink M, Thomas RC, Wood-Vasey WM, Supernova Cosmology Project (2006) Measurement of \(\varOmega _{m}\), \(\varOmega \) from a blind analysis of type Ia supernovae with CMAGIC: using color information to verify the acceleration of the universe. ApJ 644:1–20. doi:10.1086/503533. arXiv:astro-ph/0602411

  24. Coupon J, Arnouts S, van Waerbeke L, Moutard T, Ilbert O, van Uitert E, Erben T, Garilli B, Guzzo L, Heymans C, Hildebrandt H, Hoekstra H, Kilbinger M, Kitching T, Mellier Y, Miller L, Scodeggio M, Bonnett C, Branchini E, Davidzon I, De Lucia G, Fritz A, Fu L, Hudelot P, Hudson MJ, Kuijken K, Leauthaud A, Le Fèvre O, McCracken HJ, Moscardini L, Rowe BTP, Schrabback T, Semboloni E, Velander M (2015) The galaxy-halo connection from a joint lensing, clustering and abundance analysis in the CFHTLenS/VIPERS field. MNRAS 449:1352–1379. doi:10.1093/mnras/stv276. arXiv:1502.02867

  25. Courbin F, Magain P, Keeton CR, Kochanek CS, Vanderriest C, Jaunsen AO, Hjorth J (1997) The geometry of the quadruply imaged quasar PG 1115+080: implications for H_0_. A A 324:L1–L4 astro-ph/9705093

  26. Courbin F, Saha P, Schechter PL (2002) Quasar lensing. In: Courbin F, Minniti D (eds) Gravitational lensing: an astrophysical tool, Lecture notes in physics, vol 608. Springer Verlag, Berlin, p 1. arXiv:astro-ph/0208043

  27. Courbin F, Eigenbrod A, Vuissoz C, Meylan G, Magain P (2005) COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses. In: Mellier Y, Meylan G (eds) Gravitational lensing impact on cosmology, IAU Symposium, vol 225, pp 297–303. doi:10.1017/S1743921305002097

  28. 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, Meylan G (2011) COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses. IX. Time delays, lens dynamics and baryonic fraction in HE 0435–1223. A A 536:A53. doi:10.1051/0004-6361/201015709, arXiv:1009.1473

  29. Courteau S, Cappellari M, de Jong RS, Dutton AA, Emsellem E, Hoekstra H, Koopmans LVE, Mamon GA, Maraston C, Treu T, Widrow LM (2014) Galaxy masses. Rev Mod Phys 86:47–119. doi:10.1103/RevModPhys.86.47, arXiv:1309.3276

  30. Dahle H, Gladders MD, Sharon K, Bayliss MB, Rigby JR (2015) Time delay measurements for the cluster-lensed sextuple quasar SDSS J2222+2745. ApJ 813:67. doi:10.1088/0004-637X/813/1/67, arXiv:1505.06187

  31. Dalal N, Kochanek CS (2002) Direct detection of cold dark matter substructure. ApJ 572:25–33. doi:10.1086/340303. arXiv:astro-ph/0111456

  32. Diego JM, Broadhurst T, Chen C, Lim J, Zitrin A, Chan B, Coe D, Ford HC, Lam D, Zheng W (2016) A free-form prediction for the reappearance of supernova Refsdal in the Hubble frontier fields cluster MACSJ1149.5+2223. MNRAS 456:356–365. doi:10.1093/mnras/stv2638, arXiv:1504.05953

  33. Dobler G, Fassnacht CD, Treu T, Marshall P, Liao K, Hojjati A, Linder E, Rumbaugh N (2015) Strong lens time delay challenge. I. Experimental design. ApJ 799:168. doi:10.1088/0004-637X/799/2/168

  34. Efstathiou G (2014) H\(_{0}\) revisited. MNRAS 440:1138–1152. doi:10.1093/mnras/stu278. arXiv:1311.3461

  35. Eigenbrod A, Courbin F, Vuissoz C, Meylan G, Saha P, Dye S (2005) COSMOGRAIL: The COSmological MOnitoring of GRAvItational Lenses. I. How to sample the light curves of gravitationally lensed quasars to measure accurate time delays. A A 436:25–35. doi:10.1051/0004-6361:20042422. arXiv:astro-ph/0503019

  36. Eigenbrod A, Courbin F, Meylan G, Agol E, Anguita T, Schmidt RW, Wambsganss J (2008a) Microlensing variability in the gravitationally lensed quasar QSO 2237+0305 \(\equiv \) the Einstein cross. II. Energy profile of the accretion disk. A A 490:933–943. doi:10.1051/0004-6361:200810729. arXiv:0810.0011

  37. Eigenbrod A, Courbin F, Sluse D, Meylan G, Agol E (2008b) Microlensing variability in the gravitationally lensed quasar QSO 2237+0305 \(\equiv \) the Einstein Cross. I. Spectrophotometric monitoring with the VLT. A A 480:647–661. doi:10.1051/0004-6361:20078703. arXiv:0709.2828

  38. Ellis RS (2010) Gravitational lensing: a unique probe of dark matter and dark energy. Philos Trans R Soc Lond Ser A 368:967–987. doi:10.1098/rsta.2009.0209

  39. Eulaers E, Tewes M, Magain P, Courbin F, Asfandiyarov I, Ehgamberdiev S, Rathna Kumar S, Stalin CS, Prabhu TP, Meylan G, Van Winckel H (2013) COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses XII Time delays of the doubly lensed quasars SDSS J1206+4332 and HS 2209+1914. A A 553:A121. doi:10.1051/0004-6361/201321140. arXiv:1304.4474

  40. Falco EE (2005) A most useful manifestation of relativity: gravitational lenses. N J Phys 7:200. doi:10.1088/1367-2630/7/1/200

  41. Falco EE, Gorenstein MV, Shapiro II (1985) On model-dependent bounds on H(0) from gravitational images application of Q0957 + 561A, B. ApJL 289:L1–L4. doi:10.1086/184422

  42. Fassnacht CD, Pearson TJ, Readhead ACS, Browne IWA, Koopmans LVE, Myers ST, Wilkinson PN (1999) A determination of \(H_0\) with the CLASS gravitational lens B1608+656. I. Time delay measurements with the VLA. ApJ 527:498–512. doi:10.1086/308118. arXiv:astro-ph/9907257

  43. Fassnacht CD, Xanthopoulos E, Koopmans LVE, Rusin D (2002) A determination of \(H_0\) with the CLASS gravitational lens B1608+656. III. A significant improvement in the precision of the time delay measurements. ApJ 581:823–835. doi:10.1086/344368. arXiv:astro-ph/0208420

  44. Fassnacht CD, Gal RR, Lubin LM, McKean JP, Squires GK, Readhead ACS (2006) Mass along the line of sight to the gravitational lens B1608+656: galaxy groups and implications for \(H_0\). ApJ 642:30–38. doi:10.1086/500927. arXiv:astro-ph/0510728

  45. Fassnacht CD, Koopmans LVE, Wong KC (2011) Galaxy number counts and implications for strong lensing. MNRAS 410:2167–2179. doi:10.1111/j.1365-2966.2010.17591.x. arXiv:0909.4301

  46. Fiacconi D, Madau P, Potter D, Stadel J (2016) Cold dark matter substructures in early-type galaxy halos. arXiv:1602.03526

  47. Fohlmeister J, Kochanek CS, Falco EE, Wambsganss J, Morgan N, Morgan CW, Ofek EO, Maoz D, Keeton CR, Barentine JC, Dalton G, Dembicky J, Ketzeback W, McMillan R, Peters CS (2007) A time delay for the cluster-lensed quasar SDSS J1004+4112. ApJ 662:62–71. doi:10.1086/518018

  48. Freedman WL, Madore BF, Scowcroft V, Burns C, Monson A, Persson SE, Seibert M, Rigby J (2012) Carnegie Hubble program: a mid-infrared calibration of the Hubble constant. ApJ 758:24. doi:10.1088/0004-637X/758/1/24. arXiv:1208.3281

  49. Gavazzi R (2005) Projection effects in cluster mass estimates: the case of MS2137-23. A A 443:793–804. doi:10.1051/0004-6361:20053166

  50. Greene ZS, Suyu SH, Treu T, Hilbert S, Auger MW, Collett TE, Marshall PJ, Fassnacht CD, Blandford RD, Bradač M, Koopmans LVE (2013) Improving the precision of time-delay cosmography with observations of galaxies along the line of sight. ApJ 768:39. doi:10.1088/0004-637X/768/1/39. arXiv:1303.3588

  51. Grillo C, Lombardi M, Bertin G (2008) Cosmological parameters from strong gravitational lensing and stellar dynamics in elliptical galaxies. A A 477:397–406. doi:10.1051/0004-6361:20077534. arXiv:0711.0882

  52. Grillo C, Karman W, Suyu SH, Rosati P, Balestra I, Mercurio A, Lombardi M, Treu T, Caminha GB, Halkola A, Rodney SA, Gavazzi R, Caputi KI (2016) The story of supernova Refsdal told by muse. ApJ 822:78. doi:10.3847/0004-637X/822/2/78, arXiv:1511.04093

  53. Hainline LJ, Morgan CW, MacLeod CL, Landaal ZD, Kochanek CS, Harris HC, Tilleman T, Goicoechea LJ, Shalyapin VN, Falco EE (2013) Time delay and accretion disk size measurements in the lensed quasar SBS 0909+532 from multiwavelength microlensing analysis. ApJ 774:69. doi:10.1088/0004-637X/774/1/69. arXiv:1307.3254

  54. Heymans C, Van Waerbeke L, Bacon D, Berge J, Bernstein G, Bertin E, Bridle S, Brown ML, Clowe D, Dahle H, Erben T, Gray M, Hetterscheidt M, Hoekstra H, Hudelot P, Jarvis M, Kuijken K, Margoniner V, Massey R, Mellier Y, Nakajima R, Refregier A, Rhodes J, Schrabback T, Wittman D (2006) The shear testing programme–I. Weak lensing analysis of simulated ground-based observations. MNRAS 368:1323–1339. doi:10.1111/j.1365-2966.2006.10198.x. arXiv:astro-ph/0506112

  55. Hezaveh Y, Dalal N, Holder G, Kuhlen M, Marrone D, Murray N, Vieira J (2013a) Dark matter substructure detection using spatially resolved spectroscopy of lensed dusty galaxies. ApJ 767:9. doi:10.1088/0004-637X/767/1/9. arXiv:1210.4562

  56. Hezaveh YD, Marrone DP, Fassnacht CD, Spilker JS, Vieira JD, Aguirre JE, Aird KA, Aravena M, Ashby MLN, Bayliss M, Benson BA, Bleem LE, Bothwell M, Brodwin M, Carlstrom JE, Chang CL, Chapman SC, Crawford TM, Crites AT, De Breuck C, de Haan T, Dobbs MA, Fomalont EB, George EM, Gladders MD, Gonzalez AH, Greve TR, Halverson NW, High FW, Holder GP, Holzapfel WL, Hoover S, Hrubes JD, Husband K, Hunter TR, Keisler R, Lee AT, Leitch EM, Lueker M, Luong-Van D, Malkan M, McIntyre V, McMahon JJ, Mehl J, Menten KM, Meyer SS, Mocanu LM, Murphy EJ, Natoli T, Padin S, Plagge T, Reichardt CL, Rest A, Ruel J, Ruhl JE, Sharon K, Schaffer KK, Shaw L, Shirokoff E, Stalder B, Staniszewski Z, Stark AA, Story K, Vanderlinde K, WeißA Welikala N, Williamson R (2013b) ALMA observations of SPT-discovered, strongly lensed, dusty, star-forming galaxies. ApJ 767:132. doi:10.1088/0004-637X/767/2/132. arXiv:1303.2722

  57. Hicken M, Wood-Vasey WM, Blondin S, Challis P, Jha S, Kelly PL, Rest A, Kirshner RP (2009) Improved dark energy constraints from \({\sim }\)100 new CfA supernova type Ia light curves. ApJ 700:1097–1140. doi:10.1088/0004-637X/700/2/1097. arXiv:0901.4804

  58. Hilbert S, Hartlap J, White SDM, Schneider P (2009) Ray-tracing through the millennium simulation: born corrections and lens-lens coupling in cosmic shear and galaxy-galaxy lensing. A A 499:31–43. doi:10.1051/0004-6361/200811054. arXiv:0809.5035

  59. Hjorth J, Burud I, Jaunsen AO, Schechter PL, Kneib JP, Andersen MI, Korhonen H, Clasen JW, Kaas AA, Østensen R, Pelt J, Pijpers FP (2002) The time delay of the quadruple quasar RX J0911.4+0551. APJL 572:L11–L14. doi:10.1086/341603. arXiv:astro-ph/0205124

  60. Hojjati A, Linder EV (2014) Next generation strong lensing time delay estimation with Gaussian processes. Phys. Rev. D 90(12):123501. doi:10.1103/PhysRevD.90.123501. arXiv:1408.5143

  61. Holz DE (2001) Seeing double: strong gravitational lensing of high-redshift supernovae. ApJL 556:L71–L74. doi:10.1086/322947. arXiv:astro-ph/0104440

  62. Hu W (2005) Dark energy probes in light of the CMB. In: Wolff SC, Lauer TR (eds) Observing dark energy, Astronomical Society of the Pacific Conference Series, vol 339, p 215, arXiv:astro-ph/0407158

  63. Jackson N (2013) Quasar lensing. arXiv:1304.4172

  64. Jackson N (2015) The Hubble constant. Living Rev Relativ, p 18. doi:10.1007/lrr-2015-2

  65. Jakobsson P, Hjorth J, Burud I, Letawe G, Lidman C, Courbin F (2005) An optical time delay for the double gravitational lens system FBQ 0951+2635. A A 431:103–109. doi:10.1051/0004-6361:20041432. arXiv:astro-ph/0409444

  66. Jauzac M, Richard J, Limousin M, Knowles K, Mahler G, Smith GP, Kneib JP, Jullo E, Natarajan P, Ebeling H, Atek H, Clément B, Eckert D, Egami E, Massey R, Rexroth M (2016) Hubble frontier fields: predictions for the return of SN Refsdal with the MUSE and GMOS spectrographs. MNRAS 457:2029–2042. doi:10.1093/mnras/stw069, arXiv:1509.08914

  67. Jee I, Komatsu E, Suyu SH (2015) Measuring angular diameter distances of strong gravitational lenses. JCAP 11:033. doi:10.1088/1475-7516/2015/11/033. arXiv:1410.7770

  68. Jee I, Komatsu E, Suyu SH, Huterer D (2016) Time-delay cosmography: increased leverage with angular diameter distances. JCAP 4:031. doi:10.1088/1475-7516/2016/04/031, arXiv:1509.03310

  69. Kawamata R, Oguri M, Ishigaki M, Shimasaku K, Ouchi M (2016) Precise strong lensing mass modeling of four hubble frontier field clusters and a sample of magnified high-redshift galaxies. ApJ 819:114. doi:10.3847/0004-637X/819/2/114. arXiv:1510.06400

  70. Keeton CR (2011) GRAVLENS: computational methods for gravitational lensing. Astrophysics Source Code Library. arXiv:1102.003

  71. Keeton CR, Moustakas LA (2009) A new channel for detecting dark matter substructure in galaxies: gravitational lens time delays. ApJ 699:1720–1731. doi:10.1088/0004-637X/699/2/1720. arXiv:0805.0309

  72. Keeton CR, Zabludoff AI (2004) The importance of lens galaxy environments. ApJ 612:660–678. doi:10.1086/422745. arXiv:astro-ph/0406060

  73. Keeton CR, Falco EE, Impey CD, Kochanek CS, Lehár J, McLeod BA, Rix HW, Muñoz JA, Peng CY (2000) The host galaxy of the lensed quasar q0957+561. Astrophys J 542:74. doi:10.1086/309517

  74. Kelly PL, Rodney SA, Treu T, Foley RJ, Brammer G, Schmidt KB, Zitrin A, Sonnenfeld A, Strolger LG, Graur O, Filippenko AV, Jha SW, Riess AG, Bradac M, Weiner BJ, Scolnic D, Malkan MA, von der Linden A, Trenti M, Hjorth J, Gavazzi R, Fontana A, Merten JC, McCully C, Jones T, Postman M, Dressler A, Patel B, Cenko SB, Graham ML, Tucker BE (2015) Multiple images of a highly magnified supernova formed by an early-type cluster galaxy lens. Science 347:1123–1126. doi:10.1126/science.aaa3350. arXiv:1411.6009

  75. Kelly PL, Rodney SA, Treu T, Strolger LG, Foley RJ, Jha SW, Selsing J, Brammer G, Bradač M, Cenko SB, Graur O, Filippenko AV, Hjorth J, McCully C, Molino A, Nonino M, Riess AG, Schmidt KB, Tucker B, von der Linden A, Weiner BJ, Zitrin A (2016) Deja Vu all over again: the reappearance of supernova Refsdal. ApJL 819:L8. doi:10.3847/2041-8205/819/1/L8, arXiv:1512.04654

  76. Kim AG, Padmanabhan N, Aldering G, Allen SW, Baltay C, Cahn RN, D’Andrea CB, Dalal N, Dawson KS, Denney KD, Eisenstein DJ, Finley DA, Freedman WL, Ho S, Holz DE, Kasen D, Kent SM, Kessler R, Kuhlmann S, Linder EV, Martini P, Nugent PE, Perlmutter S, Peterson BM, Riess AG, Rubin D, Sako M, Suntzeff NV, Suzuki N, Thomas RC, Wood-Vasey WM, Woosley SE (2015) Distance probes of dark energy. Astropart Phys 63:2–22. doi:10.1016/j.astropartphys.2014.05.007. arXiv:1309.5382

  77. Klein JR, Roodman A (2005) Blind analysis in nuclear and particle physics. Annu Rev Nucl Part Sci 55:141–163

  78. Kneib JP, Bonnet H, Golse G, Sand D, Jullo E, Marshall P (2011) LENSTOOL: a gravitational lensing software for modeling mass distribution of galaxies and clusters (strong and weak regime). Astrophysics Source Code Library. arXiv:1102.004

  79. Kochanek CS (2002) What do gravitational lens time delays measure? ApJ 578:25–32. doi:10.1086/342476. arXiv:astro-ph/0205319

  80. Kochanek CS, Schechter PL (2004) The Hubble constant from gravitational lens time delays. Measuring and modeling the universe, p 117, arXiv:astro-ph/0306040

  81. Kochanek CS, Keeton CR, McLeod BA (2001) The importance of Einstein rings. APJ 547:50–59. doi:10.1086/318350. arXiv:astro-ph/0006116

  82. Kolatt TS, Bartelmann M (1998) Gravitational lensing of type IA supernovae by galaxy clusters. MNRAS 296:763–772. doi:10.1046/j.1365-8711.1998.01466.x. arXiv:astro-ph/9708120

  83. Koopmans LVE (2005) Gravitational imaging of cold dark matter substructures. MNRAS 363:1136–1144. doi:10.1111/j.1365-2966.2005.09523.x

  84. Koopmans LVE, Fassnacht CD (1999) A Determination of H\(_0\) with the CLASS gravitational lens B1608+656. II. Mass models and the Hubble constant from lensing. ApJ 527:513–524. doi:10.1086/308120. arXiv:astro-ph/9907258

  85. Koopmans LVE, Treu T, Fassnacht CD, Blandford RD, Surpi G (2003) The Hubble constant from the gravitational lens B1608+656. ApJ 599:70–85. doi:10.1086/379226. arXiv:astro-ph/0306216

  86. Koopmans LVE, Bolton A, Treu T, Czoske O, Auger MW, Barnabè M, Vegetti S, Gavazzi R, Moustakas LA, Burles S (2009) The structure and dynamics of massive early-type galaxies: on homology, isothermality, and isotropy inside one effective radius. ApJL 703:L51–L54. doi:10.1088/0004-637X/703/1/L51. arXiv:0906.1349

  87. Kundic T, Turner EL, Colley WN, Gott JRI, Rhoads JE, Wang Y, Bergeron LE, Gloria KA, Long DC, Malhotra S, Wambsganss J (1997) A robust determination of the time delay in 0957+561A, B and a measurement of the global value of Hubble’s constant. APJ 482:75. doi:10.1086/304147. arXiv:astro-ph/9610162

  88. Lewis GF, Ibata RA (2002) An investigation of gravitational lens determinations of H\(_0\) in quintessence cosmologies. MNRAS 337:26–33. doi:10.1046/j.1365-8711.2002.05797.x. arXiv:astro-ph/0206425

  89. Liao K, Treu T, Marshall P, Fassnacht CD, Rumbaugh N, Dobler G, Aghamousa A, Bonvin V, Courbin F, Hojjati A, Jackson N, Kashyap V, Rathna Kumar S, Linder E, Mandel K, Meng XL, Meylan G, Moustakas LA, Prabhu TP, Romero-Wolf A, Shafieloo A, Siemiginowska A, Stalin CS, Tak H, Tewes M, van Dyk D (2015) Strong lens time delay challenge. II. Results of TDC1. ApJ 800:11. doi:10.1088/0004-637X/800/1/11, arXiv:1409.1254

  90. Linder EV (2011) Lensing time delays and cosmological complementarity. Phys. Rev. D 84(12):123529. doi:10.1103/PhysRevD.84.123529. arXiv:1109.2592

  91. Linder EV (2015) Tailoring strong lensing cosmographic observations. Phys. Rev. D 91(8):083511. doi:10.1103/PhysRevD.91.083511, arXiv:1502.01353

  92. MacLeod CL, Morgan CW, Mosquera A, Kochanek CS, Tewes M, Courbin F, Meylan G, Chen B, Dai X, Chartas G (2015) A consistent picture emerges: a compact X-ray continuum emission region in the gravitationally lensed quasar SDSS J0924+0219. ApJ 806:258. doi:10.1088/0004-637X/806/2/258, arXiv:1501.07533

  93. Magain P, Courbin F, Sohy S (1998) Deconvolution with correct sampling. ApJ 494:472–477. doi:10.1086/305187. arXiv:astro-ph/9704059

  94. Marshall PJ et al (2007) Superresolving distant galaxies with gravitational telescopes: keck laser guide star adaptive optics and hubble space telescope imaging of the lens system SDSS J0737+3216. ApJ 671:1196–1211. doi:10.1086/523091. arXiv:0710.0637

  95. McCully C, Keeton CR, Wong KC, Zabludoff AI (2014) A new hybrid framework to efficiently model lines of sight to gravitational lenses. MNRAS 443:3631–3642. doi:10.1093/mnras/stu1316. arXiv:1401.0197

  96. McCully C, Keeton CR, Wong KC, Zabludoff AI (2016) Quantifying environmental and line-of-sight effects in models of strong gravitational lens systems. arXiv:1601.05417

  97. Meng XL, Treu T, Agnello A, Auger MW, Liao K, Marshall PJ (2015) Precision cosmology with time delay lenses: high resolution imaging requirements. JCAP 9:059. doi:10.1088/1475-7516/2015/09/059, arXiv:1506.07640

  98. Metcalf RB (2005) Testing cdm with gravitational lensing constraints on small-scale structure. Astrophys J 622(72):2005. doi:10.1086/427864, (c) 2005:The American Astronomical Society

  99. Metcalf RB, Madau P (2001) Compound gravitational lensing as a probe of dark matter substructure within galaxy halos. ApJ 563:9–20. doi:10.1086/323695. arXiv:astro-ph/0108224

  100. Momcheva I, Williams K, Keeton C, Zabludoff A (2006) A spectroscopic study of the environments of gravitational lens galaxies. ApJ 641:169–189. doi:10.1086/500382. arXiv:astro-ph/0511594

  101. Momcheva IG, Williams KA, Cool RJ, Keeton CR, Zabludoff AI (2015) A spectroscopic survey of the fields of 28 strong gravitational lenses. ApJs 219:29. doi:10.1088/0067-0049/219/2/29, arXiv:1503.02074

  102. More A, Oguri M, Kayo I, Zinn J, Strauss MA, Santiago BX, Mosquera AM, Inada N, Kochanek CS, Rusu CE, Brownstein JR, da Costa LN, Kneib JP, Maia MAG, Quimby RM, Schneider DP, Streblyanska A, York DG (2016) The SDSS-III BOSS quasar lens survey: discovery of 13 gravitationally lensed quasars. MNRAS 456:1595–1606. doi:10.1093/mnras/stv2813, arXiv:1509.07917

  103. Moustakas LA, Bolton AJ, Booth JT, Bullock JS, Cheng E, Coe D, Fassnacht CD, Gorjian V, Heneghan C, Keeton CR, Kochanek CS, Lawrence CR, Marshall PJ, Metcalf RB, Natarajan P, Nikzad S, Peterson BM, Wambsganss J (2008) The observatory for multi-epoch gravitational lens astrophysics (OMEGA). In: Space telescopes and instrumentation 2008: optical, infrared, and millimeter, PROC SPIE, vol 7010, p 70101B. doi:10.1117/12.789987, arXiv:0806.1884

  104. Newton ER, Marshall PJ, Treu T, Auger MW, Gavazzi R, Bolton AS, Koopmans LVE, Moustakas LA (2011) The sloan lens ACS survey. XI. Beyond Hubble resolution: size, luminosity, and stellar mass of compact lensed galaxies at intermediate redshift. ApJ 734:104. doi:10.1088/0004-637X/734/2/104, arXiv:1104.2608

  105. Nierenberg AM, Treu T, Wright SA, Fassnacht CD, Auger MW (2014) Detection of substructure with adaptive optics integral field spectroscopy of the gravitational lens B1422+231. MNRAS 442:2434–2445. doi:10.1093/mnras/stu862. arXiv:1402.1496

  106. Nightingale JW, Dye S (2015) Adaptive semi-linear inversion of strong gravitational lens imaging. MNRAS 452:2940–2959. doi:10.1093/mnras/stv1455. arXiv:1412.7436

  107. Nordin J, Rubin D, Richard J, Rykoff E, Aldering G, Amanullah R, Atek H, Barbary K, Deustua S, Fakhouri HK, Fruchter AS, Goobar A, Hook I, Hsiao EY, Huang X, Kneib JP, Lidman C, Meyers J, Perlmutter S, Saunders C, Spadafora AL, Suzuki N (2014) Supernova cosmology project lensed type Ia supernovae as probes of cluster mass models. MNRAS 440:2742–2754. doi:10.1093/mnras/stu376. arXiv:1312.2576

  108. Oguri M (2007) Gravitational lens time delays: a statistical assessment of lens model dependences and implications for the global Hubble constant. ApJ 660:1–15. doi:10.1086/513093. arXiv:astro-ph/0609694

  109. Oguri M (2015) Predicted properties of multiple images of the strongly lensed supernova SN Refsdal. MNRAS 449:L86–L89. doi:10.1093/mnrasl/slv025. arXiv:1411.6443

  110. Oguri M, Marshall PJ (2010) Gravitationally lensed quasars and supernovae in future wide-field optical imaging surveys. MNRAS 405:2579–2593. doi:10.1111/j.1365-2966.2010.16639.x. arXiv:1001.2037

  111. Oguri M, Inada N, Pindor B, Strauss MA, Richards GT, Hennawi JF, Turner EL, Lupton RH, Schneider DP, Fukugita M, Brinkmann J (2006) The sloan digital sky survey quasar lens search. i. candidate selection algorithm. Astron J 132:999. doi:10.1086/506019

  112. Paraficz D, Hjorth J (2009) Gravitational lenses as cosmic rulers: \(\varOmega _{m}\), \(\varOmega \) from time delays and velocity dispersions. A A 507:L49–L52. doi:10.1051/0004-6361/200913307. arXiv:0910.5823

  113. Paraficz D, Hjorth J (2010) The Hubble constant inferred from 18 time-delay lenses. ApJ 712:1378–1384. doi:10.1088/0004-637X/712/2/1378. arXiv:1002.2570

  114. Patel B, McCully C, Jha SW, Rodney SA, Jones DO, Graur O, Merten J, Zitrin A, Riess AG, Matheson T, Sako M, Holoien TWS, Postman M, Coe D, Bartelmann M, Balestra I, Benítez N, Bouwens R, Bradley L, Broadhurst T, Cenko SB, Donahue M, Filippenko AV, Ford H, Garnavich P, Grillo C, Infante L, Jouvel S, Kelson D, Koekemoer A, Lahav O, Lemze D, Maoz D, Medezinski E, Melchior P, Meneghetti M, Molino A, Moustakas J, Moustakas LA, Nonino M, Rosati P, Seitz S, Strolger LG, Umetsu K, Zheng W (2014) Three gravitationally lensed supernovae behind CLASH galaxy clusters. ApJ 786:9. doi:10.1088/0004-637X/786/1/9. arXiv:1312.0943

  115. Pelt J, Kayser R, Refsdal S, Schramm T (1996) The light curve and the time delay of QSO 0957+561. A A 305:97 arXiv:astro-ph/9501036

  116. Percival WJ, Reid BA, Eisenstein DJ, Bahcall NA, Budavari T, Frieman JA, Fukugita M, Gunn JE, Ivezić Ž, Knapp GR, Kron RG, Loveday J, Lupton RH, McKay TA, Meiksin A, Nichol RC, Pope AC, Schlegel DJ, Schneider DP, Spergel DN, Stoughton C, Strauss MA, Szalay AS, Tegmark M, Vogeley MS, Weinberg DH, York DG, Zehavi I (2010) Baryon acoustic oscillations in the sloan digital sky survey data release 7 galaxy sample. MNRAS 401:2148–2168. doi:10.1111/j.1365-2966.2009.15812.x. arXiv:0907.1660

  117. Perlick V (1990a) On Fermat’s principle in general relativity. I. The general case. Class Quant Grav 7:1319–1331. doi:10.1088/0264-9381/7/8/011

  118. Perlick V (1990b) On Fermat’s principle in general relativity. II. The conformally stationary case. Class Quant Grav 7:1849–1867. doi:10.1088/0264-9381/7/10/016

  119. Perlmutter S, Aldering G, Goldhaber G, Knop RA, Nugent P, Castro PG, Deustua S, Fabbro S, Goobar A, Groom DE, Hook IM, Kim AG, Kim MY, Lee JC, Nunes NJ, Pain R, Pennypacker CR, Quimby R, Lidman C, Ellis RS, Irwin M, McMahon RG, Ruiz-Lapuente P, Walton N, Schaefer B, Boyle BJ, Filippenko AV, Matheson T, Fruchter AS, Panagia N, Newberg HJM, Couch WJ (1999) The supernova cosmology project measurements of omega and lambda from 42 high-redshift supernovae. ApJ 517:565–586. doi:10.1086/307221. arXiv:astro-ph/9812133

  120. Petters AO, Levine H, Wambsganss J (2001) Singularity theory and gravitational lensing, Progress in mathematical physics v.21. Birkhäuser, Boston

  121. Planck Collaboration, Ade PAR, Aghanim N, Arnaud M, Ashdown M, Aumont J, Baccigalupi C, Banday AJ, Barreiro RB, Bartlett JG, et al (2015) Planck 2015 results. XIII. Cosmological parameters. arXiv:1502.01589

  122. Poindexter S, Morgan N, Kochanek CS, Falco EE (2007) Mid-IR observations and a revised time delay for the gravitational lens system quasar HE 1104–1805. ApJ 660:146–151. doi:10.1086/512773. arXiv:astro-ph/0612045

  123. Poindexter S, Morgan N, Kochanek CS (2008) The spatial structure of an accretion disk. ApJ 673:34–38. doi:10.1086/524190. arXiv:0707.0003

  124. Press WH, Rybicki GB, Hewitt JN (1992) The time delay of gravitational lens 0957+561. II. Analysis of radio data and combined optical-radio analysis. APJ 385:416. doi:10.1086/170952

  125. Quimby RM, Oguri M, More A, More S, Moriya TJ, Werner MC, Tanaka M, Folatelli G, Bersten MC, Maeda K, Nomoto K (2014) Detection of the gravitational lens magnifying a type Ia supernova. Science 344:396–399. doi:10.1126/science.1250903. arXiv:1404.6014

  126. Rathna Kumar S, Tewes M, Stalin CS, Courbin F, Asfandiyarov I, Meylan G, Eulaers E, Prabhu TP, Magain P, Van Winckel H, Ehgamberdiev S (2013) COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses. XIV. Time delay of the doubly lensed quasar SDSS J1001+5027. A A 557:A44. doi:10.1051/0004-6361/201322116. arXiv:1306.5105

  127. Rathna Kumar S, Stalin CS, Prabhu TP (2015) H\(_{0}\) from ten well-measured time delay lenses. A A 580:A38. doi:10.1051/0004-6361/201423977. arXiv:1404.2920

  128. Read JI, Saha P, Macciò AV (2007) Radial density profiles of time-delay lensing galaxies. ApJ 667:645–654. doi:10.1086/520714. arXiv:0704.3267

  129. Refsdal S (1964) On the possibility of determining Hubble’s parameter and the masses of galaxies from the gravitational lens effect. MNRAS 128:307

  130. Riess AG, Filippenko AV, Challis P, Clocchiatti A, Diercks A, Garnavich PM, Gilliland RL, Hogan CJ, Jha S, Kirshner RP, Leibundgut B, Phillips MM, Reiss D, Schmidt BP, Schommer RA, Smith RC, Spyromilio J, Stubbs C, Suntzeff NB, Tonry J (1998) Observational evidence from supernovae for an accelerating universe and a cosmological constant. Astron J 116:1009. doi:10.1086/300499

  131. Riess AG, Macri LM, Hoffmann SL, Scolnic D, Casertano S, Filippenko AV, Tucker BE, Reid MJ, Jones DO, Silverman JM, Chornock R, Challis P, Yuan W, Foley RJ (2016) A 2.4% determination of the local value of the Hubble constant. arXiv:1604.01424

  132. Rigault M, Aldering G, Kowalski M, Copin Y, Antilogus P, Aragon C, Bailey S, Baltay C, Baugh D, Bongard S, Boone K, Buton C, Chen J, Chotard N, Fakhouri HK, Feindt U, Fagrelius P, Fleury M, Fouchez D, Gangler E, Hayden B, Kim AG, Leget PF, Lombardo S, Nordin J, Pain R, Pecontal E, Pereira R, Perlmutter S, Rabinowitz D, Runge K, Rubin D, Saunders C, Smadja G, Sofiatti C, Suzuki N, Tao C, Weaver BA (2015) Confirmation of a star formation bias in type Ia supernova distances and its effect on the measurement of the Hubble constant. ApJ 802:20. doi:10.1088/0004-637X/802/1/20. arXiv:1412.6501

  133. Rodney SA, Patel B, Scolnic D, Foley RJ, Molino A, Brammer G, Jauzac M, Bradač M, Broadhurst T, Coe D, Diego JM, Graur O, Hjorth J, Hoag A, Jha SW, Johnson TL, Kelly P, Lam D, McCully C, Medezinski E, Meneghetti M, Merten J, Richard J, Riess A, Sharon K, Strolger LG, Treu T, Wang X, Williams LLR, Zitrin A (2015) Illuminating a dark lens : a type Ia supernova magnified by the frontier fields galaxy cluster abell 2744. ApJ 811:70. doi:10.1088/0004-637X/811/1/70, arXiv:1505.06211

  134. Rodney SA, Strolger LG, Kelly PL, Bradač M, Brammer G, Filippenko AV, Foley RJ, Graur O, Hjorth J, Jha SW, McCully C, Molino A, Riess AG, Schmidt KB, Selsing J, Sharon K, Treu T, Weiner BJ, Zitrin A (2016) SN Refsdal: photometry and time delay measurements of the first Einstein cross supernova. APJ 820:50. doi:10.3847/0004-637X/820/1/50, arXiv:1512.05734

  135. Rusu CE, Oguri M, Minowa Y, Iye M, Inada N, Oya S, Kayo I, Hayano Y, Hattori M, Saito Y, Ito M, Pyo TS, Terada H, Takami H, Watanabe M (2016) Subaru Telescope adaptive optics observations of gravitationally lensed quasars in the sloan digital sky survey. MNRAS 458:2–55. doi:10.1093/mnras/stw092, arXiv:1506.05147

  136. Saha P, Coles J, Macciò AV, Williams LLR (2006) The Hubble time inferred from 10 time delay lenses. ApJL 650:L17–L20. doi:10.1086/507583. arXiv:astro-ph/0607240

  137. Schechter PL, Bailyn CD, Barr R, Barvainis R, Becker CM, Bernstein GM, Blakeslee JP, Bus SJ, Dressler A, Falco EE, Fesen RA, Fischer P, Gebhardt K, Harmer D, Hewitt JN, Hjorth J, Hurt T, Jaunsen AO, Mateo M, Mehlert D, Richstone DO, Sparke LS, Thorstensen JR, Tonry JL, Wegner G, Willmarth DW, Worthey G (1997) The quadruple gravitational lens PG 1115+080: time delays and models. APJL 475:L85–L88. doi:10.1086/310478. arXiv:astro-ph/9611051

  138. Schechter PL, Pooley D, Blackburne JA, Wambsganss J (2014) A calibration of the stellar mass fundamental plane at z \({\sim }\)0.5 using the micro-lensing-induced flux ratio anomalies of macro-lensed quasars. ApJ 793:96. doi:10.1088/0004-637X/793/2/96, arXiv:1405.0038

  139. Schmidt KB, Rix HW, Shields JC, Knecht M, Hogg DW, Maoz D, Bovy J (2012) The color variability of quasars. ApJ 744:147. doi:10.1088/0004-637X/744/2/147. arXiv:1109.6653

  140. Schneider P (1985) A new formulation of gravitational lens theory, time-delay, and Fermat’s principle. A A 143:413–420

  141. Schneider P (2014) Generalized multi-plane gravitational lensing: time delays, recursive lens equation, and the mass-sheet transformation. arXiv:1409.0015

  142. Schneider P, Sluse D (2013) Mass-sheet degeneracy, power-law models and external convergence: Impact on the determination of the Hubble constant from gravitational lensing. A A 559:A37. doi:10.1051/0004-6361/201321882. arXiv:1306.0901

  143. Schneider P, Sluse D (2014) Source-position transformation: an approximate invariance in strong gravitational lensing. A A 564:A103. doi:10.1051/0004-6361/201322106. arXiv:1306.4675

  144. Schneider P, Ehlers J, Falco EE (1992) Gravitational lenses. Springer-Verlag, Berlin, Heidelberg, New York

  145. Schneider P, Kochanek CS, Wambsganss J (2006) Gravitational lensing: strong, weak and micro. doi:10.1007/978-3-540-30310-7

  146. Sharon K, Johnson TL (2015) Revised lens model for the multiply imaged lensed supernova Refsdal in MACS J1149+2223. ApJL 800:L26. doi:10.1088/2041-8205/800/2/L26. arXiv:1411.6933

  147. Skidmore W, TMT International Science Development Teams, Science Advisory Committee (2015) Thirty meter telescope detailed science case: 2015. Res Astron Astrophys 15:1945. doi:10.1088/1674-4527/15/12/001, arXiv:1505.01195

  148. Sonnenfeld A, Bertin G, Lombardi M (2011) Direct measurement of the magnification produced by galaxy clusters as gravitational lenses. A A 532:A37. doi:10.1051/0004-6361/201016309. arXiv:1106.1442

  149. Sonnenfeld A, Treu T, Gavazzi R, Marshall PJ, Auger MW, Suyu SH, Koopmans LVE, Bolton AS (2012) Evidence for dark matter contraction and a salpeter initial mass function in a massive early-type galaxy. ApJ 752:163. doi:10.1088/0004-637X/752/2/163. arXiv:1111.4215

  150. Sonnenfeld A, Treu T, Marshall PJ, Suyu SH, Gavazzi R, Auger MW, Nipoti C (2015) The SL2S galaxy-scale lens sample. V Dark matter Halos and Stellar IMF of massive early-type galaxies out to redshift 0.8. ApJ 800:94. doi:10.1088/0004-637X/800/2/94, arXiv:1410.1881

  151. Spergel DN, Flauger R, Hložek R (2015) Planck data reconsidered. Phys. Rev. D 91(2):023518. doi:10.1103/PhysRevD.91.023518. arXiv:1312.3313

  152. Sun YH, Wang JX, Chen XY, Zheng ZY (2014) The discovery of timescale-dependent color variability of quasars. ApJ 792:54. doi:10.1088/0004-637X/792/1/54. arXiv:1407.4230

  153. Suyu SH (2012) Cosmography from two-image lens systems: overcoming the lens profile slope degeneracy. MNRAS 426:868–879. doi:10.1111/j.1365-2966.2012.21661.x. arXiv:1202.0287

  154. Suyu SH, Halkola A (2010) The halos of satellite galaxies: the companion of the massive elliptical lens SL2S J08544–0121. A A 524:A94. doi:10.1051/0004-6361/201015481. arXiv:1007.4815

  155. Suyu SH, Marshall PJ, Hobson MP, Blandford RD (2006) A Bayesian analysis of regularized source inversions in gravitational lensing. MNRAS 371:983–998. doi:10.1111/j.1365-2966.2006.10733.x. arXiv:astro-ph/0601493

  156. Suyu SH, Marshall PJ, Blandford RD, Fassnacht CD, Koopmans LVE, McKean JP, Treu T (2009) Dissecting the gravitational lens B1608+656. I. Lens potential reconstruction. ApJ 691:277–298. doi:10.1088/0004-637X/691/1/277. arXiv:0804.2827

  157. Suyu SH, Marshall PJ, Auger MW, Hilbert S, Blandford RD, Koopmans LVE, Fassnacht CD, Treu T (2010) Dissecting the gravitational lens B1608+656. II. Precision measurements of the Hubble constant, spatial curvature, and the dark energy equation of state. ApJ 711:201–221. doi:10.1088/0004-637X/711/1/201. arXiv:0910.2773

  158. Suyu SH, Treu T, Blandford RD, Freedman WL, Hilbert S, Blake C, Braatz J, Courbin F, Dunkley J, Greenhill L, Humphreys E, Jha S, Kirshner R, Lo KY, Macri L, Madore BF, Marshall PJ, Meylan G, Mould J, Reid B, Reid M, Riess A, Schlegel D, Scowcroft V, Verde L (2012) The Hubble constant and new discoveries in cosmology. arXiv:1202.4459

  159. Suyu SH, Auger MW, Hilbert S, Marshall PJ, Tewes M, Treu T, Fassnacht CD, Koopmans LVE, Sluse D, Blandford RD, Courbin F, Meylan G (2013) Two accurate time-delay distances from strong lensing: implications for cosmology. ApJ 766:70. doi:10.1088/0004-637X/766/2/70

  160. Suyu SH, Treu T, Hilbert S, Sonnenfeld A, Auger MW, Blandford RD, Collett T, Courbin F, Fassnacht CD, Koopmans LVE, Marshall PJ, Meylan G, Spiniello C, Tewes M (2014) Cosmology from gravitational lens time delays and Planck data. ApJL 788:L35. doi:10.1088/2041-8205/788/2/L35. arXiv:1306.4732

  161. Tagore AS, Jackson N (2016) On the use of shapelets in modelling resolved, gravitationally lensed images. MNRAS 457:3066–3075. doi:10.1093/mnras/stw057, arXiv:1505.00198

  162. Tak H, Mandel K, van Dyk DA, Kashyap VL, Meng XL, Siemiginowska A (2016) Bayesian estimates of astronomical time delays between gravitationally lensed stochastic light curves. arXiv:1602.01462

  163. Tewes M, Courbin F (2013) COSMOGRAIL: the COSmological MOnitoring of GRAvItational lenses XI. Techniques for time delay measurement in presence of microlensing. A A 553:A120. doi:10.1051/0004-6361/201220123. arXiv:1208.5598

  164. Tewes M, Courbin F, Meylan G, Kochanek CS, Eulaers E, Cantale N, Mosquera AM, Magain P, Van Winckel H, Sluse D, Cataldi G, Vörös D, Dye S (2013) COSMOGRAIL: the COSmological MOnitoring of GRAvItational lenses. XIII. Time delays and 9-yr optical monitoring of the lensed quasar RX J1131–1231. A A 556:A22. doi:10.1051/0004-6361/201220352. arXiv:1208.6009

  165. The Dark Energy Survey Collaboration, Abbott T, Abdalla FB, Allam S, Amara A, Annis J, Armstrong R, Bacon D, Banerji M, Bauer AH, Baxter E, Becker MR, Benoit-Lévy A, Bernstein RA, Bernstein GM, Bertin E, Blazek J, Bonnett C, Bridle SL, Brooks D, Bruderer C, Buckley-Geer E, Burke DL, Busha MT, Capozzi D, Carnero Rosell A, Carrasco Kind M, Carretero J, Castander FJ, Chang C, Clampitt J, Crocce M, Cunha CE, D’Andrea CB, da Costa LN, Das R, DePoy DL, Desai S, Diehl HT, Dietrich JP, Dodelson S, Doel P, Drlica-Wagner A, Efstathiou G, Eifler TF, Erickson B, Estrada J, Evrard AE, Fausti Neto A, Fernandez E, Finley DA, Flaugher B, Fosalba P, Friedrich O, Frieman J, Gangkofner C, Garcia-Bellido J, Gaztanaga E, Gerdes DW, Gruen D, Gruendl RA, Gutierrez G, Hartley W, Hirsch M, Honscheid K, Huff EM, Jain B, James DJ, Jarvis M, Kacprzak T, Kent S, Kirk D, Krause E, Kravtsov A, Kuehn K, Kuropatkin N, Kwan J, Lahav O, Leistedt B, Li TS, Lima M, Lin H, MacCrann N, March M, Marshall JL, Martini P, McMahon RG, Melchior P, Miller CJ, Miquel R, Mohr JJ, Neilsen E, Nichol RC, Nicola A, Nord B, Ogando R, Palmese A, Peiris HV, Plazas AA, Refregier A, Roe N, Romer AK, Roodman A, Rowe B, Rykoff ES, Sabiu C, Sadeh I, Sako M, Samuroff S, Sánchez C, Sanchez E, Seo H, Sevilla-Noarbe I, Sheldon E, Smith RC, Soares-Santos M, Sobreira F, Suchyta E, Swanson MEC, Tarle G, Thaler J, Thomas D, Troxel MA, Vikram V, Walker AR, Wechsler RH, Weller J, Zhang Y, Zuntz J (2015) Cosmology from cosmic shear with DES science verification data. arXiv:1507.05552

  166. Treu T (2010) Strong lensing by galaxies. ARA A 48:87–125. doi:10.1146/annurev-astro-081309-130924. arXiv:1003.5567

  167. Treu T, Ellis RS (2015) Gravitational lensing: Einsteins unfinished symphony. Contemp Phys 56(1):17–34. doi:10.1080/00107514.2015.1006001

  168. Treu T, Koopmans LVE (2002a) The internal structure and formation of early-type galaxies: the gravitational lens system MG 2016+112 at z = 1.004. ApJ 575:87–94. doi:10.1086/341216. arXiv:astro-ph/0202342

  169. Treu T, Koopmans LVE (2002) The internal structure of the lens PG1115+080: breaking degeneracies in the value of the Hubble constant. MNRAS 337:L6–L10. doi:10.1046/j.1365-8711.2002.06107.x. arXiv:astro-ph/0210002

  170. Treu T, Koopmans LVE (2004) Massive dark matter halos and evolution of early-type galaxies to z \({\sim }\) 1. ApJ 611:739–760. doi:10.1086/422245. arXiv:astro-ph/0401373

  171. Treu T, Marshall PJ, Clowe D (2012) Resource letter GL-1: gravitational lensing. Am J Phys 80:753–763. doi:10.1119/1.4726204. arXiv:1206.0791

  172. Treu T, Marshall PJ, Cyr-Racine FY, Fassnacht CD, Keeton CR, Linder EV, Moustakas LA, Bradac M, Buckley-Geer E, Collett T, Courbin F, Dobler G, Finley DA, Hjorth J, Kochanek CS, Komatsu E, Koopmans LVE, Meylan G, Natarajan P, Oguri M, Suyu SH, Tewes M, Wong KC, Zabludoff AI, Zaritsky D, Anguita T, Brunner RJ, Cabanac R, Falco EE, Fritz A, Seidel G, Howell DA, Giocoli C, Jackson N, Lopez S, Metcalf RB, Motta V, Verdugo T (2013) Dark energy with gravitational lens time delays. arXiv:1306.1272

  173. Treu T, Brammer G, Diego JM, Grillo C, Kelly PL, Oguri M, Rodney SA, Rosati P, Sharon K, Zitrin A, Balestra I, Bradač M, Broadhurst T, Caminha GB, Halkola A, Hoag A, Ishigaki M, Johnson TL, Karman W, Kawamata R, Mercurio A, Schmidt KB, Strolger LG, Suyu SH, Filippenko AV, Foley RJ, Jha SW, Patel B (2016) Refsdal meets popper: comparing predictions of the re-appearance of the multiply imaged supernova behind MACSJ1149.5+2223. ApJ 817:60. doi:10.3847/0004-637X/817/1/60. arXiv:1510.05750

  174. Vanderriest C, Felenbok P, Schneider J, Wlerick G, Bijaoui A, Lelievre G (1982) The photometry of 0957 plus 561—detection of short-period variations. A A 110:L11–L14

  175. Vanderriest C, Schneider J, Herpe G, Chevreton M, Moles M, Wlerick G (1989) The value of the time delay Delta t(A, B) for the ‘double’ quasar 0957+561 from optical photometric monitoring. A A 215:1–13

  176. Vegetti S, Koopmans LVE (2009) Bayesian strong gravitational-lens modelling on adaptive grids: objective detection of mass substructure in galaxies. MNRAS 392:945–963. doi:10.1111/j.1365-2966.2008.14005.x. arXiv:0805.0201

  177. Vegetti S, Koopmans LVE, Auger MW, Treu T, Bolton AS (2014) Inference of the cold dark matter substructure mass function at z = 0.2 using strong gravitational lenses. MNRAS 442:2017–2035. doi:10.1093/mnras/stu943. arXiv:1405.3666

  178. Vuissoz C, Courbin F, Sluse D, Meylan G, Ibrahimov M, Asfandiyarov I, Stoops E, Eigenbrod A, Le Guillou L, van Winckel H, Magain P (2007) COSMOGRAIL: the COSmological MOnitoring of GRAvItational lenses. V. The time delay in SDSS J1650+4251. A A 464:845–851. doi:10.1051/0004-6361:20065823. arXiv:astro-ph/0606317

  179. Vuissoz C, Courbin F, Sluse D, Meylan G, Chantry V, Eulaers E, Morgan C, Eyler ME, Kochanek CS, Coles J, Saha P, Magain P, Falco EE (2008) COSMOGRAIL: the COSmological MOnitoring of GRAvItational lenses. VII. Time delays and the Hubble constant from WFI J2033–4723. A A 488:481–490. doi:10.1051/0004-6361:200809866. arXiv:0803.4015

  180. Walsh D, Carswell RF, Weymann RJ (1979) 0957 + 561 A, B–twin quasistellar objects or gravitational lens. Nature 279:381–384. doi:10.1038/279381a0

  181. Warren SJ, Dye S (2003) Semilinear gravitational lens inversion. ApJ 590:673–682. doi:10.1086/375132. arXiv:astro-ph/0302587

  182. Weinberg DH, Mortonson MJ, Eisenstein DJ, Hirata C, Riess AG, Rozo E (2013) Observational probes of cosmic acceleration. Phys. Rep. 530:87–255. doi:10.1016/j.physrep.2013.05.001. arXiv:1201.2434

  183. Wong KC, Keeton CR, Williams KA, Momcheva IG, Zabludoff AI (2011) The effect of environment on shear in strong gravitational lenses. ApJ 726:84. doi:10.1088/0004-637X/726/2/84. arXiv:1011.2504

  184. Wucknitz O (2002) Degeneracies and scaling relations in general power-law models for gravitational lenses. MNRAS 332:951–961. doi:10.1046/j.1365-8711.2002.05426.x. arXiv:astro-ph/0202376

  185. Wucknitz O, Biggs AD, Browne IWA (2004) Models for the lens and source of B0218+357: a LENSCLEAN approach to determine \(H_0\). MNRAS 349:14–30. doi:10.1111/j.1365-2966.2004.07514.x. arXiv:astro-ph/0312263

  186. Xu D, Sluse D, Schneider P, Springel V, Vogelsberger M, Nelson D, Hernquist L (2016) Lens galaxies in the Illustris simulation: power-law models and the bias of the Hubble constant from time delays. MNRAS 456:739–755. doi:10.1093/mnras/stv2708. arXiv:1507.07937

  187. Xu DD, Mao S, Wang J, Springel V, Gao L, White SDM, Frenk CS, Jenkins A, Li G, Navarro JF (2009) Effects of dark matter substructures on gravitational lensing: results from the Aquarius simulations. MNRAS, p 1108. doi:10.1111/j.1365-2966.2009.15230.x. arXiv:0903.4559

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We are grateful to S. Suyu and E. Komatsu for insightful discussions about the cosmological distance information content of time delay lenses, and to S. Suyu for making the B1608 \(+\) 656 MCMC chains available for us to make Fig. 7. We thank A. Agnello, M. Bartelmann, S. Birrer, V. Bonvin, D. Coe, T. Collett, F. Courbin, I. Jee, C. Kochanek, E. Linder, D. Sluse, S.Suyu, and M. Tewes for very valuable feedback on a draft of this review. T.T. thanks the Packard Foundation for generous support through a Packard Research Fellowship, the NSF for funding through NSF Grant AST-1450141, “Collaborative Research: Accurate cosmology with strong gravitational lens time delays”. P.J.M. acknowledges support from the U.S. Department of Energy under Contract Number DE-AC02-76SF00515.

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Treu, T., Marshall, P.J. Time delay cosmography. Astron Astrophys Rev 24, 11 (2016).

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  • Cosmology
  • Gravitational lensing
  • Gravity
  • Dark energy