1.1 The expanding Universe

The first piece of evidence for the hot big bang was the discovery in 1927 by Edwin Hubble and (independently) by Georges Lemaître that the further away galaxies are, the faster they appear to be receding from us. The apparent speed of recession of a galaxy can be measured by observing the shift to longer wavelengths (i.e. the redshift, represented by the dimensionless quantity ) of absorption or emission lines in the object’s spectrum. The redshift is calculated as the shift in wavelength of a given spectral line divided by the laboratory (or rest) wavelength of that line, and the apparent recession speed is then equal to the redshift multiplied by the speed of light. The distance to a galaxy can be measured by any of a number of methods that rely on comparing an object’s observed brightness to its known luminosity.

This discovery by Hubble and Lemaître is understood as implying that space itself is expanding and the cosmological redshift is therefore the result of an expansion of the intervening space between us and distant galaxies: it is not the result of those galaxies moving through space, as in a conventional Doppler shift. The result can be expressed by the linear relationship known as the Hubble-Lemaître law:

where is the galaxy distance, typically measured in units of Mpc (where 1 Mpc = km), is the speed of light (which is about 300,000 km s^-1) and is a quantity known as the Hubble constant, which has a value of 67.7 km s^-1 Mpc^-1. A modern Hubble diagram showing how galaxies’ redshifts vary with distance is shown in Figure 2.

Figure 2 The relationship between distance and redshift from a compilation of supernova measurements made by different surveys

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The figure illustrates the relationship between distance and redshift for various types of supernova measurements, as a graph. The horizontal axis is labelled ‘redshift in z’. The following points are marked on the horizontal axis, from left to right in logarithmic spacing: 0.01, 0.1, 1.0. The vertical axis is labelled ‘distance per Mpc’. The following points are marked on the vertical axis, from bottom to top in logarithmic spacing: 10, 10 squared, 10 cubed, 10 raised to the fourth power. On the graph, a series of points are plotted, which are shown in the form of different coloured dots. The best-fit line begins at the origin and ends at (1.0, 10 raised to the fourth power). The points which are yellow in colour, labelled ‘low redshift’, are shown from the start, up to approximately one-third of the length of the best-fit line. The points which are blue in colour, labelled ‘SDSS’, are shown from approximately after the yellow points, up to approximately two-thirds of the best-fit line. The points which are pink in colour, labelled ‘SNLS’, are shown from approximately after the blue points, up to approximately the end of the best-fit line. Near the end of the best-fit line, a few points are labelled ‘HST’. Each point has a vertical error bar.

Figure 2 The relationship between distance and redshift from a compilation of supernova measurements made by different surveys

The Hubble constant therefore measures the rate of expansion over a fixed distance: observers at any location at the current time will measure that, over a distance of 1 Mpc, the Universe expands at a rate of 67.7 km s^-1. This implies that, for every megaparsec further away galaxies are situated, they appear to recede 67.7 km s^-1 faster.

Mathematically the Hubble constant can be expressed as

where is referred to as the scale factor of the Universe and is its rate of change with time. In fact it is now apparent that the expansion rate of the Universe is not constant in time, so we can refer to the time-varying Hubble parameter , which is related to time-varying value of the scale factor by

For many years it was believed that the expansion rate of the Universe was decelerating (slowing down) but in the 1990s, observations of distant type Ia supernovae showed that the expansion rate of the Universe is currently accelerating (speeding up). This is indicated by the fact that the trend-line shown in Figure 2 is not a straight line, but bends upwards at large distances and redshifts. Note that, by looking at objects further away in the Universe, we are also looking further back in time. This is because the light from these distant objects was emitted by them when the Universe was much younger than it is now, and it has taken the intervening time for the light to reach us. Therefore an alternative way of describing how far away galaxies are is by referring to their lookback time.

Type Ia supernovae occur when a white dwarf star accretes enough material from a companion star to exceed the Chandrasekhar limit of about 1.4 solar masses and no longer has enough internal pressure to support itself against collapse. The white dwarf explodes in a cataclysmic explosion called a supernova. Because the explosions are so violent, they release a huge amount of energy and can be observed at very large distances away; and because they all result from the explosion of a white dwarf at the same mass, they all emit the same amount of energy or have the same luminosity. Measuring their observed brightness and comparing this to their known luminosity allows the distance to the supernova to be calculated. The redshifts of the supernovae host galaxies can also be measured from their spectra. By observing type Ia supernovae at a range of redshifts, it was discovered that, although the initial expansion rate of the Universe was indeed decelerating, at a lookback time of around 6 billion years ago (when galaxies are now observed with a redshift around ) the expansion rate of the Universe instead began accelerating.