The expansion of the universe

9.4 The expansion of the universe

In the late 1920s, Edwin Hubble, using the 100 in. Hooker Telescope on Mount Wilson, measured the distances of galaxies in which he could observe a type of very bright variable stars, called Cepheid variables, which vary in brightness with very regular periods. (The method was described in Chapter 8.) He combined these measurements with those of their speed of approach or recession (provided by Slipher) of their host galaxies (measured from the blue or red shifts in their spectral lines) to produce a plot of speed against distance (Figure 9.4). All, except the closest galaxies, were receding from us and he found that the greater the dis- tance, the greater the apparent speed of recession. From this he derived Hubble’s Law in which the speed of recession and distance were directly proportional and

related by Hubble’s constant (H 0 ). The value that is derived from his original data was ∼500 km s Mpc . Such a linear relationship is a direct result of observing

a universe that is expanding uniformly, so Hubble had shown that we live within an expanding universe. The use of the word ‘constant’ is perhaps misleading. It would only be a real constant if the universe expanded linearly throughout the

whole of its existence. It has not – which is why the subscript is used. H 0 is the

Cosmology – the Origin and Evolution of the Universe

Figure 9.4 Hubble’s plot of recession velocity against distance.

Consider the very simple one-dimensional universe shown in Figure 9.5. Initially the three components are 10 miles apart. Let this universe expand uniformly by a factor of two in 1 h. As seen from the left-hand component, the

middle components will have appeared to have moved 10 miles in 1 h whilst the right-hand component will have appeared to move 20 miles – the apparent

recession velocity is proportional to the distance. If one makes the simple assumption that the universe has expanded at a uniform rate throughout its existence, then it is possible to backtrack in time until the universe would have had no size – its origin – and hence estimate the age, known as the Hubble age, of the universe. This is very simply given by 1/H 0 and, using 500 km s Mpc , one derives an age of about 2000 million years:

1/H 0 ⫽ 1 Mpc/500 km s ⫽ 3.26 million light years/500 km s

⫽ 3.26 ⫻ 10 6 ⫻ 365 ⫻ 24 ⫻ 3600 ⫻ 3 ⫻ 10 5 s/500

⫽ 3.26 ⫻ 10 6 ⫻ 3 ⫻ 10 5 years/500 ⫽ 1.96 ⫻ 10 9 years

⫽ ∼2 billion years

Introduction to Astronomy and Cosmology

In fact, in all the Friedmann models, the real age must be less than this as the universe would have been expanding faster in the past as you can see in Figure 9.2. In the case of the ‘fl at’ universe the actual age would be two-thirds that of the Hubble age or ∼1300 million years old.

9.4.1 A problem with age

This result obviously became a problem as the age of the Solar System was determined (∼4500 million years) and calculations relating to the evolution of stars made by Hoyle and others indicated that some stars must be much older than that, ∼10–12 thousand million years old. During the blackouts of World War II, Walter Baade used the 100 in. Hooker Telescope on Mount Wilson to study the stars in the Andromeda Galaxy and discovered that there were, in fact, two types of Cepheid variable. Those observed by Hubble were four times brighter than those that had been used for the distance calibration, and this led to the doubling of the measured galaxy distances. As a result, Hubble’s constant reduced to ∼250 km s Mpc . There still remained many problems in estimating distances, but gradually the observations have been refi ned and, as a result, the estimate of Hubble’s constant has reduced in value to about

70 km s Mpc . One of the best determinations is that made by a ‘key project’ of the Hubble Space Telescope that observed almost 800 Cepheid variable stars in 19 galaxies out to a distance of 108 million light years. Combined with some other measurements the following value was derived:

H 0 Mpc .

Observations of gravitational lenses give a totally independent method of deter- mining the Hubble constant and the best value to date is:

H 0 Mpc . (Error is controversial!)

Observations by the WMAP spacecraft (see Sections 9.10 and 9.11) have given another independent value:

H 0 Mpc .

Cosmology – the Origin and Evolution of the Universe

Combining WMAP data with other cosmological data gives:

H 0 Mpc .

These are all in very good agreement so it is unlikely that the true value of Hubble’s constant will differ greatly from 71 km s -1 Mpc .

Exercise: Use the method given above to show that the Hubble age based on the current value of Hubble’s constant is ∼14 000 million years

A Hubble age of ∼14 000 million years corresponds to the age of a ‘fl at’ universe of only ∼9300 million years. From observations of globular clusters (which contain some of the oldest stars in the universe) and of the white dwarf remnants of stars, we suspect that the universe is somewhat older than 12 000 million years. Hence, if we believe the current value of Hubble’s constant, there is still an age problem with the standard Friedmann Big Bang models.

The cosmological redshift

It is possible to characterize the redshift (or blueshift) of a galaxy by the relative difference between the observed and emitted wavelengths of an object. This is given the dimensionless quantity, z. From the defi nitions of z it is then possible to derive the alternate forms shown below:

In the above, the blueshift and redshift were regarded as being due to the Doppler effect, and this would be perfectly correct when considering the blue shifts shown by the galaxies in the Local Group. However, in the cases of galaxies beyond our Local Group there is a far better way of thinking about the cause of the redshifts that are observed by us. It is not right to think of the galaxies (beyond the move- ments of those in our Local Group) moving through space but, rather, that they are being carried apart by the expansion of space. A nice analogy is that of baking

a currant bun. The dough is packed with currants and then baked. When taken out of the oven the bun will (hopefully) be bigger and thus the currants will be further apart. They will not have moved through the dough, but will have been carried apart by the expansion of the dough.

Introduction to Astronomy and Cosmology

As Hubble showed, the universe is expanding so that it would have been smaller in the past. When a photon was emitted in a distant galaxy correspond- ing to a specifi c spectral line, the universe would have been smaller. In the time it has taken that photon to reach us whilst the photon has travelled through space, the universe has expanded and this expansion has stretched, by exactly the same ratio, the wavelength of the photon. This increases the wavelength so giving rise to a redshift that we call the ‘cosmological redshift’. A simple analogy is that of drawing a sine wave (representing the wavelength of a photon) onto a slightly blown up balloon. If the balloon is then blown up further, the length between the peaks of the sine wave (its wavelength) will increase.

This gives a very nice interpretation of the parameter z. If we fi nd that a galaxy is observed at a redshift of z, then, by adding 1 to that value we can fi nd the ratio of the wavelengths of the photons observed to those emitted. This simply means that we ‘see’ the galaxy at a time when the universe was smaller by just this ratio, (1 ⫹ z).

For example, if we observe a quasar whose redshift is 6.4 (the highest quasar redshift known at the time of writing), then 1 ⫹ z ⫽ 7.4, which means that we see

the quasar at a time when the universe was 7.4 times smaller than it is now. For small redshifts, the apparent velocity of recession is given by the simple formula:

v ⫽ zc

As an example, consider the object 3C273, the fi rst quasar that was discovered. Its measured redshift was 0.158 – a 16% shift in wavelength. This implies that it is moving away from us at a speed given by zc ⫽ 0.158 ⫻ 3 ⫻ 10 5 km s ⫽ ∼ 47 000 km s . The distance can then be found from Hubble’s Law and is

given by d ⫽ v/H 0 ⫽ 47 000/72 Mpc. This is ∼650 Mpc corresponding to ∼2100 million light years – about 1000 times further away than the Andromeda Galaxy.

However, at high redshifts, such as that of the quasar referred to above with a redshift of 6.4, the simple formula v ⫽ cz no longer applies and a more complex

analysis must be used.