## 1 Introduction

### 1.1 A brief history

The last century saw an expansion in our view of the world from a static, Galaxy-sized Universe, whose constituents were stars and “nebulae” of unknown but possibly stellar origin, to the view that the observable Universe is in a state of expansion from an initial singularity over ten billion years ago, and contains approximately 100 billion galaxies. This paradigm shift was summarised in a famous debate between Shapley and Curtis in 1920; summaries of the views of each protagonist can be found in [43] and [195].

The historical background to this change in world view has been extensively discussed and whole books have been devoted to the subject of distance measurement in astronomy [176*]. At the heart of the change was the conclusive proof that what we now know as external galaxies lay at huge distances, much greater than those between objects in our own Galaxy. The earliest such distance determinations included those of the galaxies NGC 6822 [93], M33 [94] and M31 [96], by Edwin Hubble.

As well as determining distances, Hubble also considered redshifts of spectral lines in galaxy spectra which had previously been measured by Slipher in a series of papers [197, 198]. If a spectral line of emitted wavelength is observed at a wavelength , the redshift is defined as

For nearby objects and assuming constant gravitational tidal field, the redshift may be thought of as corresponding to a recession velocity which for nearby objects behaves in a way predicted by a simple Doppler formula,^{1}. Hubble showed that a relation existed between distance and redshift (see Figure 1*); more distant galaxies recede faster, an observation which can naturally be explained if the Universe as a whole is expanding. The relation between the recession velocity and distance is linear in nearby objects, as it must be if the same dependence is to be observed from any other galaxy as it is from our own Galaxy (see Figure 2*). The proportionality constant is the Hubble constant , where the subscript indicates a value as measured now. Unless the Universe’s expansion does not accelerate or decelerate, the slope of the velocity–distance relation is different for observers at different epochs of the Universe. As well as the velocity corresponding to the universal expansion, a galaxy also has a “peculiar velocity”, typically of a few hundred kms

^{–1}, due to groups or clusters of galaxies in its vicinity. Peculiar velocities are a nuisance if determining the Hubble constant from relatively nearby objects for which they are comparable to the recession velocity. Once the distance is 50 Mpc, the recession velocity is large enough for the error in due to the peculiar velocity to be less than about 10%.

Recession velocities are very easy to measure; all we need is an object with an emission line and a spectrograph. Distances are very difficult. This is because in order to measure a distance, we need a standard candle (an object whose luminosity is known) or a standard ruler (an object whose length is known), and we then use apparent brightness or angular size to work out the distance. Good standard candles and standard rulers are in short supply because most such objects require that we understand their astrophysics well enough to work out what their luminosity or size actually is. Neither stars nor galaxies by themselves remotely approach the uniformity needed; even when selected by other, easily measurable properties such as colour, they range over orders of magnitude in luminosity and size for reasons that are astrophysically interesting but frustrating for distance measurement. The ideal object, in fact, is one which involves as little astrophysics as possible.

Hubble originally used a class of stars known as Cepheid variables for his distance determinations. These
are giant blue stars, the best known of which is UMa, or Polaris. In most normal stars, a self-regulating
mechanism exists in which any tendency for the star to expand or contract is quickly damped out. In a small
range of temperature on the Hertzsprung–Russell (H-R) diagram, around 7000 – 8000 K, particularly at high
luminosity,^{2}
this does not happen and pulsations occur. These pulsations, the defining property of Cepheids, have a
characteristic form, a steep rise followed by a gradual fall. They also have a period which is directly
proportional to luminosity, because brighter stars are larger, and therefore take longer to pulsate. The
period-luminosity relationship was discovered by Leavitt [123] by studying a sample of Cepheid
variables in the Large Magellanic Cloud (LMC). Because these stars were known to be all at
the same distance, their correlation of apparent magnitude with period therefore implied the
P-L relationship.

The Hubble constant was originally measured as [95] and its subsequent
history was a more-or-less uniform revision downwards. In the early days this was caused by
bias^{3}
in the original samples [12], confusion between bright stars and H ii regions
in the original samples [97, 185] and differences between type I and II
Cepheids^{4}
[7]. In the second half of the last century, the subject was dominated by a lengthy dispute between investigators
favouring values around and those preferring higher values of .
Most astronomers would now bet large amounts of money on the true value lying between these extremes,
and this review is an attempt to explain why and also to try and evaluate the evidence for the best-guess
current value. It is not an attempt to review the global history of determinations, as this has been
done many times, often by the original protagonists or their close collaborators. For an overall review of
this process see, for example, [223] and [210]. Compilations of data and analysis of them are
given by Huchra (http://cfa-www.harvard.edu/~huchra/hubble), and Gott ([77], updated
by [35]).^{5}
Further reviews of the subject, with various different emphases and approaches, are given by [212, 68*].

In summary, the ideal object for measuring the Hubble constant:

- Has a property which allows it to be treated as either as a standard candle or as a standard ruler
- Can be used independently of other calibrations (i.e., in a one-step process)
- Lies at a large enough distance (a few tens of Mpc or greater) that peculiar velocities are small compared to the recession velocity at that distance
- Involves as little astrophysics as possible, so that the distance determination does not depend on internal properties of the object
- Provides the Hubble constant independently of other cosmological parameters.

Many different methods are discussed in this review. We begin with one-step methods, and in particular with the use of megamasers in external galaxies – arguably the only method which satisfies all the above criteria. Two other one-step methods, gravitational lensing and Sunyaev–Zel’dovich measurements, which have significant contaminating astrophysical effects are also discussed. The review then discusses two other programmes: first, the Cepheid-based distance ladders, where the astrophysics is probably now well understood after decades of effort, but which are not one-step processes; and second, information from the CMB, an era where astrophysics is in the linear regime and therefore simpler, but where is not determined independently of other cosmological parameters in a single experiment, without further assumptions.

### 1.2 A little cosmology

The expanding Universe is a consequence, although not the only possible consequence, of general relativity coupled with the assumption that space is homogeneous (that is, it has the same average density of matter at all points at a given time) and isotropic (the same in all directions). In 1922, Friedman [72] showed that given that assumption, we can use the Einstein field equations of general relativity to write down the dynamics of the Universe using the following two equations, now known as the Friedman equations:

Here is the scale factor of the Universe. It is fundamentally related to redshift, because the quantity is the ratio of the scale of the Universe now to the scale of the Universe at the time of emission of the light (). is the cosmological constant, which appears in the field equation of general relativity as an extra term. It corresponds to a universal repulsion and was originally introduced by Einstein to coerce the Universe into being static. On Hubble’s discovery of the expansion of the Universe, he removed it, only for it to reappear seventy years later as a result of new data [157*, 169*] (see also [34*, 235*] for a review). is a curvature term, and is , , or , according to whether the global geometry of the Universe is negatively curved, spatially flat, or positively curved. is the density of the contents of the Universe, is the pressure and dots represent time derivatives. For any particular component of the Universe, we need to specify an equation for the relation of pressure to density to solve these equations; for most components of interest such an equation is of the form . Component densities vary with scale factor as the Universe expands, and hence vary with time.At any given time, we can define a Hubble parameter

which is obviously related to the Hubble constant, because it is the ratio of an increase in scale factor to the scale factor itself. In fact, the Hubble constant is just the value of at the current time.^{6}

If , we can derive the kinematics of the Universe quite simply from the first Friedman equation. For a spatially flat Universe , and we therefore have

where is known as the critical density. For Universes whose densities are less than this critical density, and space is negatively curved. For such Universes it is easy to see from the first Friedman equation that we require , and therefore the Universe must carry on expanding for ever. For positively curved Universes (), the right hand side is negative, and we reach a point at which . At this point the expansion will stop and thereafter go into reverse, leading eventually to a Big Crunch as becomes larger and more negative.For the global history of the Universe in models with a cosmological constant, however, we need to consider the term as providing an effective acceleration. If the cosmological constant is positive, the Universe is almost bound to expand forever, unless the matter density is very much greater than the energy density in cosmological constant and can collapse the Universe before the acceleration takes over. (A negative cosmological constant will always cause recollapse, but is not part of any currently likely world model). Carroll [34] provides further discussion of this point.

We can also introduce some dimensionless symbols for energy densities in the cosmological constant at the current time, , and in “curvature energy”, . By rearranging the first Friedman equation we obtain

The density in a particular component of the Universe , as a fraction of critical density, can be written as

where the exponent represents the dilution of the component as the Universe expands. It is related to the parameter defined earlier by the equation . For ordinary matter , and for radiation , because in addition to geometrical dilution as the universe expands, the energy of radiation decreases as the wavelength increases. The cosmological constant energy density remains the same no matter how the size of the Universe increases, hence for a cosmological constant we have and . is not the only possibility for producing acceleration, however. Any general class of “quintessence” models for which will do; the case is probably the most extreme and eventually results in the accelerating expansion becoming so dominant that all gravitational interactions become impossible due to the shrinking boundary of the observable Universe, finally resulting in all matter being torn apart in a “Big Rip” [32]. In current models will become increasingly dominant in the dynamics of the Universe as it expands. Note that by definition, because implies a flat Universe in which the total energy density in matter together with the cosmological constant is equal to the critical density. Universes for which is almost zero tend to evolve away from this point, so the observed near-flatness is a puzzle known as the “flatness problem”; the hypothesis of a period of rapid expansion known as inflation in the early history of the Universe predicts this near-flatness naturally. As well as a solution to the flatness problem, inflation is an attractive idea because it provides a natural explanation for the large-scale uniformity of the Universe in regions which would otherwise not be in causal contact with each other.We finally obtain an equation for the variation of the Hubble parameter with time in terms of the Hubble constant (see, e.g., [155]),

where represents the energy density in radiation and the energy density in matter.To obtain cosmological distances, we need to perform integrals of the form

where the right-hand side can be expressed as a “Hubble distance” , multiplied by an integral over dimensionless quantities such as the terms. We can define a number of distances in cosmology, including the “comoving” distance defined above. The most important for present purposes are the angular diameter distance , which relates the apparent angular size of an object to its proper size, and the luminosity distance , which relates the observed flux of an object to its intrinsic luminosity. For currently popular models, the angular diameter distance increases to a maximum as increases to a value of order 1, and decreases thereafter. Formulae for, and fuller explanations of, both distances are given by [87].