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1.2 Expansion of the Universe

1.2.1 Hubble's Law

The Universe is so vast in extent and our location on a small planet is so remote, that investigating the physical nature of the almost infinite array of stars, is severely limited. Traditionally, men have observed the night sky, and more recently, with the aid of telescopes, have been able to make star maps, and classify stars and galaxies on the basis of their luminosity. The distance to nearby stars could be measured by triangulation from the base of the Earth’s orbit around the Sun.

However, most stars remained beyond the limits of this technique, and it was widely believed that all stars were part of our galaxy. Improvements in telescope power and photon detection after 1900, made it possible to collect enough light from distant stars to facilitate spectral analysis. It was then discovered that the light from distant stars had the usual hydrogen spectra but with increased wavelengths. The manifestation of this was that familiar lines in hydrogen spectra were shifted towards the red end of the spectrum. This phenomenon, known as ’Red-shift’, was interpreted as Doppler Effect which suggested that distant stars were moving away from Earth. More importantly, this shift was proportional to distance. This development provided Astronomers with a means of measuring distance to remote stars and galaxies.

Spectral measurements were difficult to make, however, as light detection technology and telescopes improved, by 1929, it was possible for the Astronomer, Edwin Hubble, to publish comprehensive evidence that Red-shift of distant stars was proportional to their distance from Earth. In addition, he showed that this effect was the same for stars in every direction. The inescapable conclusion was that Red-shift was caused by the high velocity of stars receding from Earth. Furthermore, for recession velocities to be proportional to distance in every direction, either the Earth was the center of an expanding ball, or, more likely, that space itself was expanding so that the distance between all stars was increasing.

This phenomenon, known as Hubble’s Law, is represented by the mathematical expression, \(v_r = Hr\), where \(r\) is the distance to a star, \(v_r\), is the recession velocity, and \(H\) is the Hubble constant.

The value of Hubble’s constant is of the order of \({10^{-18}}\) per second. For example, a star at a distance of \({10^{20}}\) meters, will have a recession velocity \({10^{20} \times 10^{-18} } = 100\) meters per second which can be measured from the Red-shift of its spectrum. Stars even further away will have correspondingly higher velocities up to a limit imposed by the speed of light, (\({3 \times 10^{8}}\)) meters per second. This limit corresponds to a distance of about \({10^{26}}\) meters.

When written in terms of distance, \(r\), and \(v_r\), the rate of change of \(r\), Hubble’s law becomes the differential equation,

$$recession\ velocity\ v_r = \frac{\partial r}{\partial t} = Hr \nonumber$$

which has the solution,

\(r = r_o e^{Ht}\)

where \(r_o\), is the distance at a time designated to be \( t=0\).

The exponential relationship between time and distance is a very important property of expansion. Unfortunately, this fact is not recognized in modern cosmology, where Hubble’s constant, \(H\), is believed to vary throughout time so that expansion varies, somewhat unrealistically, such that it allows the volume of the universe to be negative. In contrast, if the universe is expanding exponentially its size can never have been negative. Even if time is projected back to minus infinity, \((t=−inf)\), the size of the universe, \(e^{-inf}\), tends to zero, but never passes through zero into negative volume.

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