5 Conclusion

Progress over the last few years in determining the Hubble constant to increasing accuracy has been encouraging and rapid. For the first time, in the form of megamaser studies, there is a one-step method available which does not have serious systematics. Simultaneously, gravitational lens time delays, also a one-step method but with a historical problem with systematics due to the mass model, has also made progress due to a combination of better simulations of the environment of the lens galaxies and better use of information which helps to ease the mass degeneracy. The classical Cepheid method has also yielded greatly improved control of systematics, mainly by moving to calibrations based on NGC 4258 and Galactic Cepheids which are much less sensitive to metallicity effects.

Identification of the current “headline” best distance determinations, by methods involving astrophysical objects, is a somewhat subjective business. However, most such lists are likely to include the following, including the likely update paths:

• Megamasers:  [24]. Further progress will be made by identification and monitoring of additional targets, since the systematics are likely to be well controlled using this method.
• Gravitational lenses:  [206] (best determination with systematics controlled, two lenses),  [194] (18 lenses, but errors from range of free-form modelling). Progress is likely by careful control of systematics to do with the lens mass model and the surroundings in further objects; a programme (H0LiCOW [204]) is beginning with precisely this objective.
• Cepheid studies:  [174] with corrected NGC 4258 distance from [100]; (parallax of Galactic Cepheids) and (mid-IR observations) [69]. The Carnegie Cepheid programme is continuing IR observations which should significantly reduce systematics of the method.

In parallel with these developments, the Planck satellite has given us much improved constraints on in combination with other cosmological parameters. The headline determinations are all from Planck in combination with other information, and are:

• For a flat-by-fiat Universe,  [2*] from Planck.
• For a Universe free to curve,  [2*] using Planck together with BAO data.
• Local BAO measurements:  [15*] using only the well-determined from the CMB, but independent of other cosmology [15, 19, 6, 148].

There is thus a mild tension between some (but not all) of the astrophysical measurements and the cosmological inferences. There are several ways of looking at this. The first is that a 2.5- discrepancy is nothing to be afraid of, and indeed is a relief after some of the clumped distributions of published measurements in the past. The second is that one or more methods are now systematics-limited; in other words, the subject is limited by accuracy rather than precision, and that careful attention to underestimated systematics will cause the values to converge in the next few years. Third, it is possible that new physics is involved beyond the variation of the dark energy index . This new physics could, for example, involve the number of relativistic degrees of freedom being greater than the standard value of 3.05, corresponding to three active neutrino contributions [2]; or a scenario in which we are living in a local bubble with a different  [130]. Most instincts would dictate taking these possibilities in this order, unless all of the high-quality astrophysical values differed from the cosmological ones.

The argument can be turned around, by observing that independent determinations of can be fed in as constraints to unlock a series of accurate measurements of other cosmological parameters such as . This point has been made a number of times, in particular by Hu [91], Linder [126] and Suyu et al. [207]; the dark energy figure of merit, which measures the dependence of dark energy and its redshift evolution, can be be improved by large factors using such independent measurements. Such measurements are usually extremely cheap in observing time (and financially) compared to other dark energy programmes. They will, however, require 1% determinations of , given the current state of play in cosmology. This is not impossible, and should be reachable with care quite soon.