List of Footnotes

1		See [105 ] and references therein, e.g., [27 ], for discussion of the details of the interpretation of redshift in an expanding Universe. The first level of sophistication involves maintaining the GR principle that space is locally Minkowskian, so that in a small region of space all effects must reduce to SR (for instance, in Peacock’s [155 ] example, the expansion of the Universe does not imply that a long-lived human will grow to four metres tall in the next $1010$ years). Redshift can be thought of as a series of transformations in photon wavelengths between an infinite succession of closely-separated observers, resulting in an overall wavelength shift between two observers with a finite separation and therefore an associated “velocity”. The second level of sophistication is to ask what this velocity actually represents. [105 ] calculates the ratio of photon wavelength shifts between pairs of fundamental observers to the shifts in their proper separation in the presence of arbitrary gravitational fields, and shows that this ratio only corresponds to the purely dynamical result if the gravitational tide is constant.
2		This is known in the literature as the “instability strip” and is almost, but not quite, parallel to the luminosity axis on the H-R diagram. In normal stars, any compression of the star, and the associated rise in temperature, results in a decrease in opacity; the resulting escape of photons produces expansion and cooling. For stars in the instability strip, a layer of partially ionized He close to the surface causes opacity to rise instead of falling with an increase in temperature, producing a degree of positive feedback and consequently oscillations. The instability strip has a finite width, which causes a small degree of dispersion in period–luminosity correlations among Cepheids.
3		There are numerous subtle and less-subtle biases in distance measurement; see [213 ] for a blow-by-blow account. The simplest bias, the “classical” Malmquist bias, arises because, in any population of objects with a distribution in intrinsic luminosity, only the brighter members of the population will be seen at large distances. The result is that the inferred average luminosity is greater than the true luminosity, biasing distance measurements towards the systematically short. The Behr bias [12 *] from 1951 is a distance-dependent version of the Malmquist bias, namely that at higher distances, increasingly bright galaxies will be missing from samples. This leads to an overestimate of the average brightness of the standard candle which becomes worse at higher distance.
4		Cepheids come in two flavours: type I and type II, corresponding to population I and II stars. Population II stars are an earlier metal-poor generation of stars, which formed after the hypothetical, truly primordial Population III stars, but before later-generation Population I stars like the Sun which contain significant extra amounts of elements other than hydrogen and helium due to enrichment of the ISM by supernovae in the meantime. The name “Cepheid” derives from the fact that the star $δ$ Cephei was the first to be identified (by Goodricke in 1784). Population II Cepheids are sometimes known as W Virginis stars, after their prototype, W Vir, and a W Vir star is typically a factor of 3 fainter than a classical Cepheid of the same period.
5		The conclusion of the latter, that based on median statistics of the Huchra compilation, $− 1 −1 H0 = 67± 2 km s Mpc$ , is slightly scary in retrospect given the Planck value of $−1 −1 67.2± 1.3 km s Mpc$ for a flat Universe [2 *].
6		Historically, the Hubble constant has often been quoted as $H0 = 100h km s− 1 Mpc −1$ , as a way of maintaining agnosticism in an era where observations allowed a wide range in $h$ . This is largely disappearing, but papers using $h$ can be hard to interpret [42 ]
7		For example, one topic that may merit more than a footnote in the future is the study of cosmology using gravitational waves. In particular, a coalescing binary system consisting of two neutron stars produces gravitational waves, and under those circumstances the measurement of the amplitude and frequency of the waves determines the distance to the object independently of the stellar masses [193 ]. This was studied in more detail by [36 ] and extended to more massive black-hole systems [90 , 45 ]. More massive coalescing signals produce lower-frequency gravitational-wave signals, which can be detected with the proposed LISA space-based interferometer (http://lisa.nasa.gov/documentation.html). The major difficulty is obtaining the redshift measurement to go with the distance estimate, since the galaxy in which the coalescence event has taken place must be identified. Given this, however, the precision of the $H0$ measurement is limited only by weak gravitational lensing along the line of sight, and even this is reducible by observations of multiple systems or detailed investigations of matter along the line of sight. $H0$ determinations to $∼$ 2% should be possible, but depend on the launch of LISA or a similar mission. This is an event that is probably decades away, although a pathfinder mission to test some of the technology is due for launch in 2015.
8		Strictly speaking, provided we ignore effects to do with curvature of the Universe.
9		An isothermal model is one in which the projected surface mass density decreases as $1∕r$ . An isothermal galaxy will have a flat rotation curve, as is observed in many galaxies.
10		Essentially all radio time delays have come from the VLA, although monitoring programmes with MERLIN have also been attempted.
11		The redshifts of the lens and source also need to be known, as does the position of the centre of the lens galaxy; this measurement is not always a trivial proposition [241 ].
12		As discussed extensively in [113 , 117 ], this is not a global degeneracy, but arises because the lensed images tell you about the mass distribution in the annulus centred on the galaxy and with inner and outer radii defined by the inner and outer images. Kochanek [113 ] derives detailed expressions for the time delay in terms of the central underlying and controlling parameter, the surface density in this annulus [76 ].
13		The programme, known as H0LiCOW, is now continuing in order to measure time delays, improve models and derive further $H 0$ values for more lenses.
14		Because of the expansion of the Universe, there is a time dilation of a factor $(1 + z)−1$ which must be applied to timescales measured at cosmological distances before these are used for such comparisons.
15		The effective radius is the radius from within which half the galaxy’s light is emitted.
16		Nearly all Cepheids measured in galaxies containing SN Ia have periods $> 20$ days, so the usual sense of the effect is that Galactic Cepheids of a given period are brighter than LMC Cepheids.
17		Here, as elsewhere in astronomy, the term “metals” is used to refer to any element heavier than helium. Metallicity is usually quoted as 12+log(O/H), where O and H are the abundances of oxygen and hydrogen.
18		The details are discussed in more detail in an earlier version of this review [102 ].
19		This characteristic size is about $4× 105$ light years at recombination, corresponding to an angular scale of about $1∘$ on the sky. The fact that the CMB is homogeneous on scales much larger than this is an illustration of the “horizon problem” discussed in Section 1.2, and which inflation may solve.
20		See http://background.uchicago.edu/~whu/intermediate/intermediate.html for a much longer exposition and tutorial on all these areas.
21		See Table 2 of [2 *] for a full list.

	Abstract
1	Introduction
	1.1	A brief history
	1.2	A little cosmology
2	One-Step Distance Methods
	2.1	Megamaser cosmology
	2.2	Gravitational lenses
	2.3	The Sunyaev–Zel’dovich effect
	2.4	Gamma-ray propagation
3	Local Distance Ladder
	3.1	Preliminary remarks
	3.2	Basic principle
	3.3	Problems and comments
4	The CMB and Cosmological Estimates of the Distance Scale
	4.1	The physics of the anisotropy spectrum and its implications
	4.2	Degeneracies and implications for H₀
5	Conclusion
	Acknowledgements
	References
	Footnotes
	Updates
	Figures
	Tables