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A A Highly Subjective Summary of the Gravitational Lens Time Delays to Date and the Derived Values of the Hubble Constant

All discussion is based on parametric models (though see the text for non-parametric models). For B0218+357 the mass-model is reasonably well known, thanks to an Einstein ring and VLBI structure [176Jump To The Next Citation Point] but the galaxy centre astrometry is not. York et al. [178] find 70 ± 5 (2σ) based on an ACS determination of the galaxy centre, and Wucknitz et al. [176] find 78 ± 6 (2σ) using the ring and VLBI structure. I double the errors and take 74 ± 5. For HE0435–1223 the image positions of [174] with the lens redshift of [99] and the longest time delay of [79], together with an isothermal sphere model with external shear, gives H0 = 90, similar to the value inferred by Kochanek et al. A 20% error for the lack of knowledge of the mass profile and another 20% for the presence of nearby galaxies are added in quadrature to the time delay error to give H0 = 90 ± 26. For SBS0909+532 the system is very underconstrained, especially as the flux ratios are not well known, possibly due to extinction [166Jump To The Next Citation Point]. Lehár et al. [88Jump To The Next Citation Point] find a predicted time delay of 49 ± 38 days for H0 = 70, by considering the allowed space where χ2 < 1, the time delay of 45 days [166] translates into H0 = 76 ± 59; one might argue that the errors should be slightly smaller but this lens is not in any case going to contribute significantly to the overall total. In RXJ0911+0551 the situation is complicated by the presence of a nearby cluster. Hjorth et al. [55] obtain 71 ± 9 (2σ) based on a velocity dispersion for the cluster from [74] (but see also discussion in [77]). As with B0218+357 I double the errors and take 71 ± 9. In FBQ0951+2635 an isothermal model gives 60+9+2 −7−2 [68]. The same source claims that in this lens the variation of H0 with mass slope is relatively weak, but I nevertheless add an extra 20% to the error and take 60 ± 15. For Q0957+561, a system with a huge cluster contribution, probably the most compelling analysis (of the vast literature on this object) is that in [70] where the lensed host galaxy is used to disentangle the cluster shear from the internal shear produced by the lensed galaxy shape. Assuming a cluster convergence of 0.2, using the range of Keeton et al.’s sample models and increasing the errors by 20% gives 90 ± 25. For SDSS J1004+4112 the cluster contribution makes modelling so problematic that it is difficult to sort out sensible H0 estimates from simple models. However, once the cluster mass profile is thoroughly understood using the large number of constraints and the remaining time delays are known, this may give the best H0 estimate of all [42]. HE1104–185 is also a bizarre system in which the closer image is brighter; Lehár et al. [88] predict much larger time delays than are actually found, and typical isothermal models plus the time delay in [101] give H0 ∼ 95. For consistency I adopt this value, together with a 20% error added in quadrature to the time delay error to reflect the fact that we do not know the galaxy mass profile, giving 95 ± 25. In PG1115+080 traditional fits of isothermal models give H0 in the forties [135Jump To The Next Citation Point], but a more recent stellar dynamical measurement [159] suggests a steeper-than-isothermal profile and with the time delay in [135] gives H0 = 59+12+3 −7− 3, so I adopt 59 ± 12. In RXJ1131–1231 almost any smooth isothermal model fails (and H0 < 10), so Morgan et al. [98] suggest that a large piece of substructure or satellite galaxy falls close to one of the images. While plausible, this means that the system is almost impossible to model for H0. There is a measured time delay for JVAS B1422+231 [110] but in my view it needs confirmation before being used for H0 determination. SBS 1520+530 is modelled in [18], who also provide the time delay, by an isothermal model together with consideration of a nearby cluster. I add 20% to their error, due to the isothermal assumption, and get 51 ± 13. In B1600+434 an isothermal model gives 60+1−612 (random errors [81], but adjusted for a (Ωm, Ω Λ = 0.3, 0.7) model). I add 20% for systematics involved in lack of knowledge of the mass profile and get 60 ± 20. In B1608+656 Koopmans et al. [83] assemble an impressive array of data, including three time delays, stellar velocity measurements and Einstein ring fitting to get +7 75−6, to which I add another 10% due to a feeling of unease about the lens being two interacting galaxies and consequent effects on the mass profile, and take 75 ± 10. For SDSS J1650+4251 I adopt the value of H0 = 52 from the time delay and isothermal models of [167], plus an extra 20% error for the mass slope, and use H0 = 52 ± 11. In PKS1830–211 models in [173] using their improved galaxy centre position give 44 ± 9, to which I add the now traditional 20% for ignorance of the mass slope and 10% (cf. B1608+656) for the possibility of associated secondaries [25], to give 44 ± 13. Finally, for HE2149-2745, a simple SIE model gives H0 = 49, in agreement with the average of 48+7 − 4 quoted in [76] on analysis of this plus four other lenses; Burud et al. [17] derive a higher value due to restrictions on assumed ellipticity of the lens galaxy. Adding the usual 20% for ignorance of the mass slope to the time delay error we obtain 49 ± 11. At last, the overall average is 66 ± 3 (or 61 ± 4 without B0218+357). We should also consider the effect of mass along the line of sight, which is likely to add a further 5 – 10% to the error budget of each lens system and which will probably not reduce as √n-; much of it is systematic if the error is normally in the sense of ignoring nearby groups of matter which add to the local convergence.

It should be emphasised that this is an attempt to work out one observer’s view of the situation based on parametric models, and that the reader will have noticed a lot of subjective judgements and arbitrary manipulation of errors going on, but you did ask.


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