We've all heard of the Hubble Space Telescope and I'm sure most of us have marvelled at the spectacular images produced by astronomers who have been lucky enough to work with this remarkable instrument over the years since it was launched. It has of course now been superseded by newer, even more sophisticated space telescopes, still mapping the Universe and pinning down as accurately as possible the value of the so-called Hubble constant: the cosmological parameter that, if known, can tell us the age of our expanding universe!

Edwin Hubble first proposed a relationship between the recessional velocity of galaxies and their distance from us in 1929 based on observations of a relatively small sample of galaxies, whose redshift had been determined from their spectra. Their distance had been estimated from observations of Cepheid variable stars, embedded in the respective galaxies. Cepheid variables, named after Delta Cephei (the prototype), are giant pulsating stars that had previously been shown, by astronomer Henrietta Leavitt, to display a correlation between their average apparent magnitude and the period of their fluctuations; the so-called period-luminosity relation. These stars became a valuable stepping stone in the cosmic distance scale, standard candles whose distance can be inferred from their observed brightness as compared to their calculated absolute magnitude. The calibration of Cepheid variables as standard candles begun in the early 20th century, by Ejnar Hertzprung, continues to this day.

Hubble's law as the aforementioned relationship came to be known implies a direct proportionality between a galaxy’s speed away from us and its distance from us; the constant of that proportionality being the famous Hubble constant. You will often see the law quoted as \(v = H_0 d\), where \(v\) is a galaxy's recessional speed and \(d\) its distance from the observer. However, cosmologists on the whole prefer Hubble's law to be expressed with redshift as a function of distance, as follows:

\(z = {H_0\over c} d\)

• z is a galaxy's redshift
• \(c\) is the speed of light in a vacuum
• \(H_0\) is the Hubble constant

The Hubble constant is a cosmological parameter that changes over time often notated as \(H(t)\). \(H_0\) is the current observed value. The equations given above imply Hubble's constant as having units of s^-¹, whose reciprocal is a unit of time: a useful starting point in the process of modelling the age of the Universe. However, for historical reasons, it is usually expressed in units of km s^-1  Mpc^-1, where Mpc is an abbreviation of megaparsec: one parsec (roughly 3.26 light years) being equivalent to the distance to a star that displays a parallax of one arc second.

It's important to understand that the galactic redshift underlying Hubble's law is different to the Doppler effect, where the observed wavelength of a receding source increases due to the relative motion of the source. Cosmological redshift is due to the fabric of space-time itself expanding, with the result that the light waves from distant galaxies become stretched, as they propagate towards the observer. The further away a galaxy is from an observer, the faster it will appear to recede, as the scale factor of the Universe increases. Redshift, in this regard, can be expressed as:

\(z ={ Δλ\over λ} \)

In other words, the fractional increase in the wavelength of radiation from a receding galaxy is a result of the expansion of the Universe in the time between the light being emitted and observed. We should note that superimposed on a galaxy's cosmological redshift will be regular Doppler effects due to the peculiar (localised) motion of a galaxy with respect to the observer. Hubble's law is an effective way of determining the distance to a galaxy based on its observed redshift 'only' out to a distance of \(z=0.2\), beyond which competing models of the evolution of the Universe cause the Hubble relation to break down somewhat. 

Of course, the implication of Hubble's law was the first intimation that we live in an expanding universe and its discovery came as a surprise to Hubble and his contemporaries, not least Einstein whose equations of general relativity had predicted the expansion of the Universe and which he had tried to eradicate by including the now famous cosmological constant -which he later regretted. However, given that when you throw a stone in the air, you expect it to fall down again, it came as an even bigger shock when in the 1990s, observations of type 1a supernovae implied that the expansion of the Universe is accelerating.

It seems appropriate to finish with a reminder. Despite the observation that the entire contents of the Universe (with the exception of a few local galaxies) appear to be hurtling away from us, we are not in a special position. All observers in the universe will observe this phenomenon regardless of their location. In some sense, everywhere is at the centre of the universe since the whole thing originated from the same infinitesimal singularity - about 13.7 billion years ago!

Images

• Hubble deep field photo by NASA, Robert Williams, and the Hubble Deep Field Team
• Hubble constant by Brews ohare, CC BY-SA 3.0 <https://creativecommons.org/licenses/by-sa/3.0>, via Wikimedia Commons
• Abstract background by Gerd Altmann from Pixabay