The Big Bang

Introduction

In this unit, we present the three main lines of experimental evidence pointing to the big bang origin of the Universe: (i) the recession of the galaxies; (ii) the microwave remnant of the early fireball; and (iii) the comparison between the calculated primordial nuclear abundances and the present-day composition of matter in the Universe.

A data sheet of useful information is provided as a pdf for your use. You may wish to print out a copy to keep handy as you progress through the unit.

Click ‘View document’ to open the S357_1 data sheet.

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1 Introducing cosmology

General relativity has a very different conceptual basis from that of Newtonian mechanics. Its success in accounting for the precession of Mercury’s orbit, and the bending of light by massive objects like the Sun, gives us confidence that our picture of space and time should be Einstein’s rather than Newton’s. In this and the following units, we turn our attention to the study of the large-scale structure of spacetime. We see how spacetime as a whole is curved by the gross distribution of mass and energy in the Universe. This distribution, together with the question of how the Universe has developed over time, is the subject of cosmology.

Astronomy and cosmology are subjects that merge into one another with the single combined aim of understanding the structure and history of the Universe. The basis and motivation for the whole subject area comes ultimately from astronomers’ observations. Since the days of Galileo, optical telescopes of ever greater size have been made. In the last 60 years, it has been possible to study an increasing range of the electromagnetic spectrum as different types of telescope have become available. The first radio telescopes were made just before World War II. Infrared, ultraviolet, X-ray and gamma ray telescopes then followed, often operating from spacecraft above the Earth’s atmosphere. These methods have become so complicated that we have attempted to outline only certain results; a far longer course would have been necessary to do anything like justice to the delicacy and sophistication of the techniques involved.

What have we learned from these new techniques? Firstly, matter has been detected in a wide variety of forms – interstellar gas as well as the stars themselves, for instance. Secondly, it has become possible to perform detailed studies of the radiation from the very ancient Universe. These developments have given cosmologists a more comprehensive list of the forms of mass and energy that govern the spacetime of the Universe. Next, matter has been detected at much greater distances – the more distant view providing cosmologists with more telling tests of their models. Then there has been the increasingly detailed information on spectra, with its evidence on the compositions of the emitting bodies. This information is of particular concern to the astrophysicist, who tries to understand the evolution of stars and other types of matter. Such evolution is governed by stellar dynamics and the processes of nuclear physics. The latter will be touched on later, though with reference more to nuclear reactions taking place in the very early Universe than to those occurring in the stars today.

We hope to build a bridge between the two extremes – the raw data obtained by the astronomer, and the metric parameters derived by the cosmologist. Thus, we examine the reasons for believing that the Universe had its origins in the big bang. We shall find that there are three independent pieces of evidence all pointing to the same conclusion. Sections 2 to 4 introduce you to the first of these.

2 Radiation from the galaxies

Stars occur in great collections called galaxies. The distribution and motion of galaxies provide us with the first important experimental information on which we shall build our understanding of the type of universe we inhabit. So, what do we know about galaxies?

All the stars that can be distinguished by the naked eye – a few thousand in number – belong to one galaxy: our own Milky Way Galaxy. Sometimes it is just written Galaxy, with a capital G, to distinguish it from all the billions of other galaxies in the observable Universe.

Our Galaxy has the overall shape of a flattened spiral. As shown in Figure 1, we are halfway inside the Galaxy so the spiral form is not obvious to us as we look at the sky. In fact it has only been revealed by detailed radio mapping. This is because there are large sections of the

Galaxy that are obscured from optical observations by the intervening dust and gas which scatter and absorb visible light. (However, they are essentially transparent to radio waves.) If there were an unobscured view through telescopes, we would be able to see to the far side of the Galaxy, and in doing so we would record about 10¹¹ stars within it.

Figure 1: The Solar System is about halfway out from the centre of the Milky Way galaxy. We have a relatively clear view outwards from the disc to other galaxies

Fortunately for astronomers, the stars, though vast in number, display a degree of uniformity which makes their classification and study possible. Their masses are broadly similar – most are contained within a couple of orders of magnitude, with our Sun having a typical average mass. They do differ considerably in size and therefore in density (since their masses are similar), but these differences are thought principally to relate to different stages in the life of a star rather than to different types of star. These various stages correspond to different types of nuclei taking part in the nuclear reactions that provide the energy output. It is one of the features of astrophysics that one is able to take nuclear data obtained in the laboratory and use them to understand the various stages of evolution seen in different stars. The main process in young stars is a sequence of reactions leading to the fusion of hydrogen nuclei to make helium nuclei. Later, helium fuses to form carbon, then carbon undergoes reactions which build up heavier nuclei. This can continue up to iron. The different reactions take place at different temperatures and pressures, so this progression of reactions governs the sequence in which a star changes its size and appearance.

Stellar evolution is too slow a process for us to see any particular star undergo change in one human lifetime, apart from a few exceptionally young stars, and some very old stars undergoing gravitational collapse leading to supernova explosions. But by observing different stars at their respective different stages of development, it is possible to piece together the whole of a typical stellar life cycle.

The general idea of an evolutionary sequence – one that can be reconstructed on a computer – is of concern in this unit for the following reason: when astronomers look at a very distant galaxy, they are receiving radiation that left its source long ago. The galaxy will, therefore, seem younger than it actually is now. The travel time for the light may have been hundreds or thousands of millions of years. To interpret the observations, one needs to know how the power output of the galaxy evolves with time.

The light from a typical galaxy derives mostly from the stars it contains, with only a small amount from interstellar matter. So, to understand the power output of a galaxy one has to add together the light from about 10¹¹ stars at various stages of development. We cannot assume that the stars of the distant galaxy will be emitting the same amount of light as those of our own Galaxy today; the distant stars will be seen at an earlier stage in their development when perhaps their power output was different from what it actually is now. Indeed, there is an added complication. The evolution of a star depends critically on its mass. A very massive star will shine much more brightly than a less massive one, but over a much shorter time period. It could be that when we look at a young galaxy, we see many more massive stars living out their brief active lives than we observe today in our own Galaxy. So the mass distribution of active stars in our Galaxy might not be representative of what we see happening in the distant younger galaxy. It is difficult to know how to compensate for this. The mass distribution of the stars in a galaxy depends on the way the galaxy was formed, and unfortunately, the formation and growth of galaxies remains an unsolved, or at least poorly understood, problem.

In summary, we know little about the way the power output of a galaxy changes with time, and this represents a severe limitation on the cosmologist’s use and interpretation of astronomical data on power output. The frustration this causes will become apparent later.

The light from a galaxy can tell us more than just its overall power output. Additional information comes from the analysis of its spectrum. Let us assume that light comes mainly from the stars and that we can ignore interstellar matter. The majority of galaxies are too far away for it to be possible to resolve individual stars, and therefore the best one can do is to take the spectrum of the galaxy as a whole: 10¹¹ stars summed together. How can we relate this to a stellar spectrum?

The light from a star comes from the hot layers of gas near its surface. This light filters out through the dilute layers above. As it does so, characteristic patterns of absorption lines are imprinted on the spectrum.

Figure 2 shows part of the absorption spectrum of the Sun, compared to an emission spectrum produced in the laboratory.

Figure 2: The absorption spectrum of light from the Sun, for wavelengths between 390 nm and 450 nm. It is complicated because of the many elements involved. The bright lines above and below the absorption spectrum belong to the emission spectrum of a laboratory reference source

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Question 1

A beam of white light is passed through a bulb containing calcium vapour. Explain, in broad terms, with the aid of diagrams, the nature of the light transmitted, (a) in the direction of the beam, and (b) at an angle to it.

Now read the answer

(a) At position A (Figure 3), the spectrum of the observed light consists of a flat continuum with absorption lines (corresponding to the transitions between the ground state and other states) as in Figure 4(a).

(b) At position B, no direct light is seen. The light that is observed comes from calcium atoms that have been excited, and is therefore an emission spectrum of lines on a dark background, with the wavelengths of the lines the same as those of the absorption lines seen at position A. This is shown in Figure 4(b).

Figure 3: White light is passed through a bulb containing calcium vapour and observed at A (in the direction of the beam) and at B (at an angle to the beam)

Figure 4: The spectra observed at the points A and B of Figure 3

The spectral lines in the light from a galaxy enable the astronomer to identify the elements emitting the light. Any of the ninety or so stable elements may be present, but the lighter elements, especially hydrogen, are usually the more abundant. Because some elements are common to most stars, the absorption lines of these particular elements will be visible in the light of the galaxy as a whole. Also, the absorption lines of magnesium, sodium, calcium and iron are often easy to distinguish, even though other elements, such as hydrogen and helium, are more abundant than these.

Usually the absorption lines are sharp and identifiable, despite several effects which can broaden them. One of these effects is a Doppler shift caused by the rotation of the source. For example, the spectral lines of calcium atoms moving towards us would have their apparent frequencies systematically increased (blueshifted) according to the equation

where f₀ is the observed frequency, f₁ is the emitted frequency and is the speed of appoach. The spectral lines of atoms moving away from us would be correspondingly redshifted. This Doppler shift causes the width of a given spectral line to broaden if we are viewing the rotating galaxy along the plane of its disc so that the light-absorbing atoms belonging to stars on one side are moving away from us, while those on the opposite side are moving towards us (refer back to Figure 1). In addition, the star itself might be rotating so that different parts of the star would have different components of velocity towards or away from us, thereby increasing the broadening. Random motions of stars can also cause line broadening in a similar way.

The shifts and broadenings of the lines are not usually bad enough to mask the spectrum. Therefore, although a galaxy is a very complicated light source, its light is not just a meaningless jumble of fuzzy lines. There are some features, such as the calcium lines, which stand out sharply.

Question 2

(a) It can be shown that when radiation rises a height H near the Earth’s surface, there is a shift in frequency towards the red end of the spectrum given by Δf/f = −gH/c², where g is the acceleration due to gravity. When radiation starting from the surface of a body of mass M and radius R escapes to a large distance, it suffers a fractional frequency shift Δf/f = −GM/Rc², where G is the gravitational constant. Outline the steps you would need to take to establish the connection between these two statements without necessarily giving the derivation. (Hint: Bear in mind that g = GM/R².)

(b) Imagine a star with a mass M = 2 × 10³⁰ kg, a radius R = 7 × 10⁸ m and a period of rotation T = 2 × 10⁶ s (similar to the Sun). Suppose also that the typical speed of turbulent motion in the atmosphere of the star is 6000 m s⁻¹. How is the frequency of the hydrogen line whose wavelength in the laboratory is 656 nm, affected by:

(i) gravitational redshift;

(ii) Doppler shift due to rotation;

(iii) Doppler shift due to turbulence?

(The values of G and c are given in the data sheet, and the formula for the Doppler shift of light is given in the equation above.)

Now read the answer

(a) The result Δf/f = −gH/c² applies only to small changes in height, H. For a large change, one must take account of the fact that g changes with height. The value of g is given by equating the weight of a body, mg, to the gravitational attraction. Thus at the surface of the Earth

where M and R are the mass and radius of the Earth respectively. At any distance r (greater than R), we may write more generally

At each distance r, the frequency shift caused by rising to r + dr is given by the formula .

where we write dr in place of H. Thus the total frequency shift suffered in escaping to a very large distance (‘infinity’) is determined by:

where f₁ is the emitted frequency and f₀ is the observed frequency. This gives the method asked for in the Question.

Performing the integration (which the Question did not ask for) gives the stated result. In fact,

Provided the magnitude of Δf = f₀ − f₁ ≪ f₁, we can approximate the left-hand side of this equation by

and conclude that, for any emitted frequency f.

(b) (i) Using the result of part (a), we can find the effect of the gravitational redshift on light leaving the star as follows:

All the radiation leaving the star has been ‘redshifted’ by this amount by the time it reaches the Earth, which we may certainly regard as at an ‘infinite’ distance from the star.

(ii) The speed at the edge of the star is

The Doppler shift is

when is a speed of approach. Here,

This could be used directly in the formula but since /c is so small, we can use the approximation

so the fractional change in frequency, Δf/f, is simply:

Light from the advancing limb is blueshifted by this amount, and from the receding limb is redshifted by this amount. The width imparted to lines is thus the fraction 14.7 ×10⁻⁶ of their central frequency.

(iii) The turbulent motion again produces both blueshifts and redshifts, of amount ±6000 m s⁻¹/c = ±20 × 10⁻⁶.

The frequency of the 656 nm line is given by

The frequency shifts are therefore as shown in Table 1.

Table 1

		Δf/f		Δf/MHz
gravitation		−2.12 × 10−6		−970
rotation		±7.33 × 10−6		±3350
turbulence		±20 × 10−6		±9150

In many galaxies, spectral lines have been observed whose relative positions correspond exactly with those of a known element (so that the spectrum is confidently identified), but whose absolute frequencies are all noticeably shifted. This can have nothing to do with the random motions of the stars within the galaxy, or the rotations of either the individual stars or of the galaxy as a whole; all these motions (being as often towards us as away from us) would merely broaden the line. A net shift of all the lines in a spectrum seems to suggest that the galaxy itself has a line-of-sight motion (i.e. a component of velocity towards or away from us). The shift is nearly always to longer wavelengths, i.e. towards the red end of the visible spectrum, so it is referred to as the ‘redshift’. If the redshift is interpreted as a Doppler shift (an interpretation we shall reconsider later) due to the motion of the galaxies, then it follows that most of the galaxies are receding from us. In other words, the Universe is expanding.

The term ‘redshift’ has a precise definition: it is equal to Δλ/λ, where Δλ is the shift in wavelength of a line whose emitted wavelength is λ. The value of Δλ/λ is the same for all the lines in the spectrum of an object receding at a given speed, and is normally denoted by z:

where λ₀ is the observed wavelength now, and λ₁ is the wavelength at emission. These redshifts in the spectra of galaxies are generally far larger than the stellar effects considered in Question 2, as can be judged from Figure 5.

Figure 5: The absorption spectra of five different galaxies. Only the most strongly absorbed lines, common to the bulk of the stars, are visible, but this is enough to give a recognisable pattern

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In summary then, when dealing with light from stars, there are three ways in which the frequency can be shifted:

(i) The Doppler shift due to motion – whether that arises through the rotation of the star, its bodily motion along the line of sight, or turbulence in its atmosphere. This type of shift, which can be red or blue, is accounted for by special relativity.

(ii) The gravitational redshift, arising from general relativity.

(iii) A new type of redshift due to the recession of the galaxies, which is also a consequence of general relativity.

(Note that it is customary to reserve the suffix, 0, for the values of quantities as they are at the present time, t₀. Values at other times must carry some other suffix. In this way, we end up with the rather counterintuitive situation where λ₁ applies to an earlier time, t₁, than λ₀.)

Question 3

Hydrogen has emission lines at 434, 486 and 656 nm. A galaxy is observed to have these three lines redshifted in wavelength by 2% (i.e. z = Δλ/λ = 0.02). What wavelengths will be observed? What is the frequency shift Δf/f ?

Now read the answer

If λ₁ is the emitted wavelength and λ₀ is the observed wavelength, then

Therefore

Thus the observed wavelengths are:

The relative frequency shift is given by

But

Therefore

and

Note that the shift in frequency is the same for all lines of the spectrum, but differs from the shift in wavelength, which is z.

As we shall see later in this unit, the redshift in galactic light has provided one of the main clues to the nature of the large-scale structure of spacetime. But to discover how the pattern of the redshifts observed in different galaxies reveals this structure, we need further information about the distances of galaxies.

3 Distances of galaxies

3.1 First steps towards a distance scale

As you will see from Table 2, when it comes to astronomy and cosmology, one is called on to deal with a wide range of distances. (Note that a light-year (ly) is the distance light travels in one year, i.e. 9.46 × 10¹⁵ m. The distances are also quoted in a very commonly used astronomical unit of distance: the megaparsec, Mpc, where a parsec (pc) is 3.26 ly or 3.09 × 10¹⁶m.)

To measure the distance of a far-off galaxy clearly requires a series of steps. The first of these, the Earth–Sun distance, is based on our knowledge of the orbital period of the Earth about the Sun, and that of some other planet. One can then readily obtain an estimate of their relative distances from the Sun by using Kepler’s third law. To convert this to an absolute measurement, one needs a determination of the actual distance between the Earth and the planet at some time. In practice, this fix is gained by measuring the distances to Mercury, Venus and Mars, using radar. These then allow one to compute the Earth–Sun distance. It is currently estimated, with obvious high precision, to be 149 597 870.66 km.

Knowing this, the diameter of the Earth’s orbit can now be used as a baseline for measuring the distance to nearby stars, using the surveyor’s triangulation method. One makes angular measurements on a star from opposite ends of an Earth-orbital diameter, i.e. at intervals of 6 months. At the time of writing (2004), data from the Hipparcos satellite surpass in precision and scope all previous measurements of nearby stellar distances.

But no matter how good this satellite-gathered data, there soon comes a stage where the angles become too small to measure accurately using the triangulation method. Some other method must be employed to extend the distance scale to the more distant stars.

If a source has a known total light output, i.e. known luminosity, then it can be used as a standard source. The distance of the source can be found from the received light power. The further away the star is, the fainter it appears. The relationship follows an inverse square law (allowing for various corrections, such as that due to the absorption of light by interstellar dust). The light power received per unit area of detector is called the flux density. It follows, summing over the area of a sphere centred on the star and of radius, r, equal to the distance to the star, that in the absence of absorption:

Unfortunately stars differ widely in their luminosities, so if we simply look up at the sky and pick out a faint star, it may be either an intrinsically dim star, or a particularly distant star – or partly dim and distant.

This ambiguity can be partly overcome by recognising that, while it is impossible to directly measure the luminosity of a particular source, one can estimate the temperature of the star. This is done from measurements on the overall shape of its emitted spectrum – stars emitting predominantly in the red part of the spectrum being cooler than those that emit more in the blue region. Thus the spectrum, and hence temperature, can be determined with a fair degree of confidence. Now, it turns out that when observations are made on a compact cluster of stars (the cluster being small enough for all its stars to be considered equidistant from us), a plot of flux density versus temperature generally gives the same distribution no matter which cluster is chosen – apart from an overall constant factor depending on the distance to the cluster. Thus, the most likely luminosity of a star is related to its temperature. But in order to use the temperatures of stars to obtain luminosities, and hence distances, from Equation 2, we need to progress beyond a plot of flux density versus temperature to one of luminosity versus temperature.

In order to achieve this, studies have been made of a group of about 100 stars called the ‘Hyades star cluster’. This group is close enough to us for its distance to be determined by the Hipparcos satellite. Knowing this distance and the flux density of the stars, the luminosities could be established, and the plot of luminosity versus temperature calibrated. By studying the flux densities and colours of stars more distant than the Hyades star cluster, we can use this calibrated plot to infer their distances from us.

This method applies to stars confined to our own Galaxy. As for stars in other galaxies, most of these cannot be distinguished separately. So, the next step is to try to find a particularly bright type of star – one that can be recognised not only in our Galaxy but also in neighbouring galaxies. If this were possible, it would give us a method of extending the distance scale out to other galaxies. Fortunately there does exist a type of star that can be recognised at least in neighbouring galaxies – a star known as a

Cepheid variable. The important characteristic of a Cepheid is that its light output varies in a regular fashion, with a period which is directly related to its mean luminosity (see Figure 6a). The variation is due to cyclical changes in diameter. The relationship between mean luminosity and period can be calibrated by studying Cepheids that are sufficiently close for their distances to have been measured by methods mentioned earlier. So whenever such stars can be distinguished from the other types of variable star in the more distant galaxies, their luminosities can be deduced (see Figure 6b). Comparing luminosity with the measured flux density then enables the distance of the star to be calculated. Cepheids have been used extensively to measure the distances to nearby galaxies -those belonging to a cluster of galaxies known as the Local Group (to be described later). Following the advent of the Hubble Space Telescope with its superior resolution of individual stars, it became possible in 1996 to extend Cepheid-based measurements as far as a galaxy known as M100, in the Virgo cluster, yielding a distance of 5.6 × 10⁷ ly.

Figure 6: (a) An 18-day section of the light curve of the typical Cepheid variable, Delta Cephei, which has a 5.37-day oscillation. (b) The observed relationship between the period and luminosity (in watts) of Cepheid variables (1 W = 1 J s⁻¹). Notice that both scales are logarithmic, so the straight line implies that period ∝ (luminosity)^m, where 1/m is the slope of the line.

based on H. Arp (1960) in Astronomical J., 65, 426

Another type of star that can be recognised in other galaxies is a Type Ia supernova. What happens is this: when a medium-sized star, such as the Sun, approaches the end of its active life, it shrinks down to a small star called a ‘white dwarf’. If the white dwarf happens to belong to a binary system of two stars, it can, from time to time, capture material from the atmosphere of its companion, so increasing its own mass. But this is a process that cannot continue indefinitely. The maximum mass for a white dwarf is 1.4 solar masses; anything above that limit and its inner forces cannot resist the inward pull of gravity, and the star has to collapse down to the next stable form (called a ‘neutron star’). Thus, the white dwarf in the binary system can capture material only up to this limiting mass. Once it exceeds this limit, the collapse occurs with the excess energy emitted as an explosion – the Type Ia supernova. Because white dwarfs are always of the same limiting mass when this happens, the explosions are similar, yielding essentially the same spectrum and variation of light intensity over time. (It is these characteristics that allow the Type Ia supernovae to be distinguished from other types.) Why they are important in the present context is that, to within ±30%, they have the same luminosity. Taking into account that the variation in luminosity with time does show a difference in characteristic decay times, and these are correlated to somewhat different values of peak luminosity, allowance can be made for this, and the uncertainty spread in peak luminosity reduced to ±20%. This in turn leads to relative distances measurable to ±10%.

Type Ia supernovae provide us with standard lamps (or standard explosions!), that, at their peak, are 100 000 times brighter than a Cepheid, and are visible hundreds of times further away. Having established their luminosity by measuring the few that occur in galaxies for which the distance is already known from measurements on Cepheids, they can be used to extend the distance scale to very great distances.

This however still does not go far enough. Since normal stars cannot be resolved in the farther galaxies, the additional methods of estimating distance will have to be based on the properties of galaxies as a whole. We therefore interrupt the distance story to describe some properties of whole galaxies, showing first why they are important to cosmology, and then how they have led to new ways of measuring distance.

Question 4

Suppose an astronomer using a telescope of 2 m diameter has a detector whose limit of sensitivity is 3.2 ×10⁻¹⁷ W. Use Figure 6 to deduce, for this instrument, the period of the faintest Cepheid variable that can be observed at a distance of 2 × 10⁶ ly [light years].

(Remember: The value of a light-year in SI units is given on the data sheet)

Now read the answer

At 2 × 10⁶ ly, the fraction of light intercepted is:

The energy received from the faintest star is 3.2 × 10⁻¹⁷ W.

Therefore the output of the star must be

From Figure 6(b), the corresponding period of the Cepheid variable is about 3 days.

3 Distances of galaxies

3.2 Some general properties of galaxies

Firstly, we note that galaxies tend to occur in clusters rather than singly. The mutual gravitational attraction of galaxies naturally tends to hold them on paths that remain close to each other. Typically a cluster contains tens or hundreds of galaxies. There are, however, large clusters with thousands of galaxies, and there are some solitary galaxies. Our own Galaxy is a member of a smallish cluster of about 36 galaxies called the Local Group (see Figure 7). A typical cluster of moderate size is shown in Figure 8.

Figure 7: Our cluster of galaxies, called the Local Group. M31, the Andromeda galaxy, can be seen with binoculars. The Large Magellanic Cloud is one of our nearest neighbours, and is visible to the naked eye from the Southern Hemisphere

from Le Grand Atlas de l’Astronomie, Encyclopaedia Universalis editeur

Figure 8: A cluster of galaxies whose position in the sky is behind the constellation of stars in our Galaxy which we call Hercules. This cluster is not visible to the naked eye – not because it is too small but because it is too faint

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How are the galaxy clusters distributed? Are they close enough to affect each other? If they are, then the motion of a given cluster would depend mostly on the distortion of spacetime caused by its nearest neighbours. Because the distances between clusters vary, this would correspond to a large spread in relative speeds. At the other extreme, if the clusters of galaxies were very far apart, then the attraction of neighbouring clusters would be negligible, and the motion would be dominated by the overall spacetime curvature due to the matter of the whole Universe – a much simpler situation.

In fact, galaxy clusters are loosely associated in superclusters. However, they are far enough apart for one to regard clusters as essentially independent (see Figure 9). The fact that clusters are, in effect, independent of one another is of central importance to cosmology. It means that a cluster of galaxies can be taken as the basic ‘particle’ of cosmological dynamics, and the motion of individual galaxies within a cluster can be ignored on the grand scale of cosmology. Thus, we conclude:

Figure 9: A very schematic view of three clusters, showing typical diameters of galaxies (single dots) and clusters (groups of dots), and typical distances between clusters

In other words, a galaxy cluster plays a similar role on the grand scale to that of a planet (e.g. Mercury) mapping out the local region of spacetime in the Solar System, or Newton’s falling apple doing the same thing closer to the Earth’s surface.

The next important property of galaxies is that there is a statistical predictability about their masses and luminosities. Figure 10 shows the proportion of galaxies having a given luminosity. It varies over quite a wide range, the distribution falling off steeply on the high-luminosity side. (Note that the luminosity is expressed in terms of (absolute) ‘magnitudes’, this being a parameter such that the smaller or more negative the value of the magnitude, the more luminous the object. This rather odd choice arises for historical reasons.) There are large numbers of the least luminous galaxies (shown by the rising curve on the left-hand side), but in practice these tend to be invisible in the more distant clusters which are of greater interest to cosmologists. No doubt it would have been easier for cosmology if galaxies had been more similar. However, because galaxies occur in clusters, one can at least use statistical methods. For instance, the distribution of luminosities is more or less the same for all clusters, resulting in the average luminosity of galaxies in a cluster being fairly standard even though the individual galaxies vary greatly. This statement assumes, of course, that we are dealing with galaxy clusters of the same age. As we have already pointed out, this might not be the case when observing galaxies at a great distance, and hence as they were some time in the distant past.

Figure 10: Number of galaxies per magnitude class per cubic megaparsec, as a function of absolute magnitude. Different symbols represent the results of different observers

based on Erik Holmberg in E.M. and G.R. Burbidge (1975) Galaxies and the Universe, University of Chicago Press

3 Distances of galaxies

3.3 Extending the distance scale

Having reviewed some of the properties of galaxies, we are now in a position to return to the question of how we are to develop further our methods of measuring distance.

The various steps taken in determining larger distances from known smaller ones are often called ‘rungs in the distance ladder’. The process of constructing a rung has been:

1. Find a measurable quantity associated with a class of objects.
2. Observe how the measurable quantity depends on distance for objects close enough to have had their distances measured by the method of a previous rung.
3. Assume the same relationship holds for more distant objects of the same class, and hence calculate their distances.
4. Return to step 1, with a new measurable quantity.

The classes of objects (and distance indicators) for the first four rungs of the distance ladder were:

These distance indicators all depended on recognising a particular type of star. But, as was mentioned in Section 3.1, individual stars can be resolved only in galaxies that are not too distant. For most galaxies, a method is needed which depends on recognising, or deducing, the luminosity of the galaxy as a whole. Although, as already noted, individual galaxies vary considerably in their luminosity, they occur in clusters. A simple rule which seems to work in practice is to assume that the third-brightest galaxy in all clusters has the same luminosity (a standard 10³⁷ W lamp).

An alternative method is to separate galaxies into different types, with the assumption, or at least the hope, that the types have characteristic luminosities. There are certainly generic differences between spiral galaxies (Figure 11) and elliptical galaxies (Figure 12). But it is also well established that there is a useful correlation between the luminosities of spiral galaxies and their rotation speeds, which can be determined from (radio) observations of Doppler broadening.

Figure 11: Spiral galaxy M81 (NGC 3031), taken with the 200-inch telescope at the Palomar Observatory in California. This galaxy, like our own, has tightly wound arms and a prominent bulge (known as a nuclear bulge)

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Figure 12: M59 (NGC 4261), an elliptical galaxy about 19 Mly (i.e. 1.9 × 10⁷ ly) away. Like all elliptical galaxies, M59 has no spiral arms

NASA/IPAC Extragalactic Database (NED), Jet Propulsion Laboratory, California Institute of Technology

Another method exploits the fact that some types of radio galaxy (so-called because they are strong emitters at radio frequencies) are fairly uniform in size, and radio interferometers can resolve very small angular separations – a thousand times smaller than those resolved by optical telescopes. The apparent size leads to an estimate of the distance of the galaxy. Because some radio galaxies are also visible at optical wavelengths, the optical and radio distance scales can be intercalibrated.

Yet another method should also be mentioned. It involves the behaviour of a single star in a galaxy, though one too distant to be resolved. We have already mentioned Type Ia supernovae. These are events that occur when a white dwarf (which itself originally resulted from the last stages in the active life of a medium-sized star) captures material from a companion star, and undergoes collapse to a neutron star. If instead of a medium-sized star one begins with a very massive star, then at the end of its active life it catastrophically collapses directly to a neutron star or black hole. This leads to a gigantic explosion even more energetic than a Type Ia supernova; it is called a Type II supernova. These explosions are bright enough to be visible at large distances – in some cases briefly shining more brightly than all the other stars of the galaxy put together. Unfortunately they are rather rare, occurring only once every hundred years or so in a typical galaxy. A Type II supernova explosion causes a spherical shell of hot gas to expand out of the star at high speed – thousands of kilometres per second. The spectral lines in the observed light from this shell (mostly from hydrogen and therefore easily identified) are blueshifted by its velocity towards us. Knowing this velocity from the amount of blueshift, the increase in size of the shell, month by month, can be calculated. Thus the shell is a source of known size, even though this cannot be resolved from an observed angular width. The temperature can be found from the overall shape of the continuous spectrum between the spectral lines. Knowing both the size and temperature of the shell, its total light output (that is, its luminosity) can be found. From this and the observed flux this and the observed flux density, the distance can be calculated using the inverse square law.

Remember, luminosity = 4r² × flux density, where r is the distance.

Figure 13: NGC 5457, a spiral galaxy in our Local Group, having looser arms and a far less noticeable nuclear bulge than the galaxy shown in Figure 11

Hale Observatories

Question 5

Comment on the truth or otherwise of the following statements:

(i) Averaging statistically over the luminosities of its constituent galaxies, each cluster of galaxies can be assumed to have the same overall luminosity.

(ii) The distance-measuring method involving Type II supernovae relies on the fact that the shell of material thrown out by the explosion greatly exceeds the parent star in size, to the extent that it can be optically resolved.

Now read the answer

4 The variation of redshift with distance

4.1 Hubble’s discoveries

In this section, we bring together two important features of galaxies – their redshifts and their distances.

This crucial development owes its origins to Edwin Hubble. His pioneering work in 1923 first led to the confirmation that certain of the fuzzy patches in the sky, loosely called ‘nebulae’, were in fact galaxies like our own.

Figure 14: (a) Nebula NGC 6514, a cloud of gas and dust in our own Galaxy; (b) Hubble showed that objects like M31 (NGC 224, now called the Andromeda galaxy) were galaxies like our own, and they ceased to be called nebulae. The two large bright patches near M31 are satellite galaxies, NGC 205 and NGC 221

(a) courtesy of Uppsala University; (b) Hale Observatories

It was immediately realised that the Universe was enormously bigger than had previously been thought. Also the number of galaxies was large. In fact, it is now known that the number of galaxies accessible to our telescopes is comparable with the number of stars in our Galaxy – about 100 billion (i.e. 10¹¹).

A second significant discovery made by Hubble concerned the spectra of galaxies, nearly all of which are redshifted. This redshift was a systematic shift of all the lines to the red end of the spectrum (it was discovered by another US astronomer, Vesto Slipher). Using his measurements of distance, Hubble showed that the redshift increased with distance. As far as he could tell, the redshift of a galaxy was proportional to its distance.

Figure 15: The redshift–distance relationship for galaxies, as plotted by Hubble in 1929. The solid line represents the relationship inferred from individual galaxies (solid circles), the dashed line the relationship when the galaxies are combined into groups (open circles). Hubble’s distance scale has been omitted since it is now known that it was in error.

based on E. P. Hubble (1929) in Proc. Nat. Acad. Sci., 15, 168

Hubble’s original measurements, shown in Figure 15, exhibited a large scatter about a straight line. This was partly due to the inevitable observational uncertainties, especially in distance measurements. But even if one were able to remove the observational uncertainties, there would still have been a considerable scatter about a straight line. This comes from the fact that most of the galaxies Hubble looked at were members of clusters, and each was moving about within its cluster. Because the galaxies were rather close to us, the speeds associated with this random motion were comparable to the recessional speed Hubble was trying to measure. Sometimes the motion of the galaxy relative to the cluster centre was directed towards us (giving a blueshift component), sometimes away from us (adding a further redshift component). If the structure of the cluster were well enough known, these effects could, in principle, be estimated and a correction applied. Allowance must also be made for small gravitational redshifts (see Question 2). Finally, the motion of the Earth with respect to our Local Group must be subtracted. (This will be discussed in Section 6.2.4.) Suppose that all these corrections could be perfectly made in all cases. Then a redshift (z) against distance (r) plot would, as far as we know, appear as in Figure 16.

Figure 16: An idealised Hubble diagram with all sources of scatter removed

The straight line in Figure 16 would represent the underlying cosmological redshift. Thus, for this cosmological redshift, we have

Interpreting the observed redshift as a Doppler shift implies that each galaxy is receding at a speed proportional to its distance from us. To see this, consider the Doppler shift formula and the relationship between redshift, wavelength change and frequency shift given in the answer to Question 3 . It follows that the redshift, z, is given by

Hence, Equations 3 and 4 together imply that

This provides the basis for one of the common ways of writing Hubble’s law,

The factor of proportionality, H, is sometimes called the Hubble constant. But the term parameter is perhaps preferable since the word ‘constant’ might lead one to think that it should remain constant in time – instead of being a constant of proportionality between two variables as those variables are at a particular point in time. H must, we now realise, vary slowly with time.

If the speed of recession is proportional to distance, this implies all distances between galaxies are increasing at the same rate. Not only do all clusters of galaxies appear to be receding from us here on Earth, they would appear to be receding in exactly the same manner from whatever vantage point an observer adopted.

Hubble’s work has been continued, refined, and extended to much more distant and fainter galaxies.

Question 6

Taking typical intercluster distances to be approximately 2 × 10⁸ ly, and the value of H to be about 2 × 10⁻¹⁸ s⁻¹, estimate the minimum z value that can reliably be ascribed to the expansion of the Universe. Compare this with the redshifts measured by Hubble in Figure 15.

Now read the answer

To get a redshift that can confidently be regarded as due to expansion, we must look to another cluster.

Intercluster distances are approximately

The Hubble constant is about 2 × 10⁻¹⁸ s⁻¹

Hence the cosmological redshift of a neighbouring cluster is

Compare this with Figure 15 – Hubble was not able to look far enough!

Since it is hard to measure the distance to a far-off galaxy, it is not surprising that there has been a good deal of controversy about the reliability of distance estimates. The redshift measurements are much easier to make and are more direct, so there has been much less uncertainty over them. Nevertheless, it is useful to have a check, and this has been provided by radio astronomers measuring the redshift of lines in the radio spectrum, such as that due to the emission of hydrogen at a wavelength of 21 cm. This confirms that the redshift is the same over a large range of frequencies.

Based on a range of recent results, the value of H is currently reckoned to be 2.3 × 10⁻¹⁸ s⁻¹, with an observational uncertainty of about 10%.

Hubble, who interpreted redshifts in terms of recessional velocities, would quote this as 23 km s⁻¹ for every million light-years of distance of a galaxy. Estimates of H are often quoted in terms of km s⁻¹ Mpc⁻¹. In these units, H is about 70 km s⁻¹ Mpc⁻¹.

It has become common usage to write H = h × 100 km s⁻¹ Mpc⁻¹, in which case h currently has a best estimate of 0.7.

We end this section with a more up-to-date version of the Hubble diagram (Figure 17) for some galaxies and clusters of galaxies.

Figure 17: A plot of redshift against distance for a selection of galaxies and clusters of galaxies

adapted from A. Sandage (1970) Physics Today, vol. 34, February 1970, American Institute of Physics

Question 7

Use the sum of the shifts calculated for the star specified in Question 2 to give an estimate of the uncertainty in the cosmological redshift z for a star in a galaxy at 4 × 10⁶ ly from us. (This may be an overestimate because the centre of a line can be estimated quite well even if the line is fuzzy. However, in a real galactic spectrum there will be a further component of broadening due to galactic rotation.)

Now read the answer

From Question 2(b), parts (ii) and (iii), the minimum width of the lines is

with a systematic fractional shift of about −2 × 10⁻⁶ (Question 2(b), (i)). Thus the contributions to Δf/f due to turbulence, rotation and gravitational redshift vary between (+20 + 7 − 2) × 10⁻⁶ and (−20 – 7 – 2) × 10⁻⁶. This gives an error of about ±3 × 10⁻⁵. For a galaxy at 4 × 10⁶ ly (≈4 × 10⁶ × 10¹⁶ m), the expected cosmological redshift is

Thus the line-broadening effects are approximately a fraction

of the expected cosmological shift.

Notice that a distance of 4 × 10⁶ ly corresponds to nearby galaxies in our Local Group and the relative uncertainties in redshift will decrease with distance. They are commonly less than 1%. The uncertainties in distance, however, will increase with distance, because less accurate estimation procedures have to be used. Also note that a distance of 4 × 10⁶ ly may be unreasonably small for cosmological consideration.

Question 8

By taking appropriate readings off the graph of Figure 17, estimate the value of the Hubble constant, H, indicated by this set of data.

Now read the answer

One can use any pair of points on the straight line graph that has been drawn, but to minimise reading errors we take the extreme points. The difference in z is 1.0000 − 0.0025 = 0.9975. The difference in distance, r, is 4000 − 10 = 3990 Mpc.

Thus

But

So

4 The variation of redshift with distance

4.2 Evidence for a big bang

Having interpreted the redshift as indicating a recessional speed proportional to distance, one may extrapolate into the future to predict how the positions of the galaxies will evolve with time. One can also run the sequence backwards, so to speak, to discuss what their positions were in the past. Clearly, at former times the galaxies were closer to each other.

But not only that. Because of the proportional relationship between speed and distance (Equation 6), at a certain time in the past, all the matter of the Universe must have been together at a point of extraordinarily high density. It was from this condition that it subsequently expanded giving the matter of the Universe its present-day distribution. This is our first indication that the history of the Universe featured an explosive event.

This has become known as the big bang. It is believed to have marked the beginning of the Universe. (Actually the phrase ‘big bang’ is used in two ways: (i) to denote the instant at which cosmic expansion begins; and (ii) to refer to that instant plus the sequence of events immediately following. It is usually clear from the context which of these meanings is intended.)

It is possible to deduce much about the nature of the big bang and how long ago it took place – in other words, how old the Universe is. But we need to be sure that there really was a big bang. What we seek is evidence that is independent of the observation of moving galaxies. The remainder of this unit is devoted to describing just such confirmatory observations. Not only do they add to our confidence that the Universe did indeed have a definite beginning, they also inform us that the beginning was exceedingly violent – the big bang was hot. This indication assumes great importance when we seek to get some understanding of the varying types of process that must have been taking place during the initial stages of expansion – during the first years, minutes, even fractions of a second after the instant of the big bang.

5 The microwave background radiation

5.1 A second major discovery

In the introduction to this unit, we said that there were three pillars of evidence for the big bang. We now turn to the second. It rests on a discovery that ranks in importance with that of Hubble’s law. It came about when observations in a new region of the electromagnetic spectrum – the microwave region – became possible. This was due to the invention of new detectors, working at frequencies as high as 30 000 MHz. In 1965, two Bell Telephone scientists, A. Penzias and R. Wilson, were investigating the radio noise found at wavelengths between a few millimetres and a few centimetres. These wavelengths were, at the time, a relatively untapped field for communications. (They are now very useful for satellites because even small antennae give narrow beams at these wavelengths. Penzias and Wilson were working on the Telstar/Echo satellite project at that time.)

They found that, once all known sources of noise had been accounted for, they were left with a residual signal which was coming equally from all directions. It was soon realised that because of this isotropy, it could not originate on the Earth, or in the Solar System. Nor could it be coming even from the Galaxy – the Galaxy being a thin disc, with us not at its centre.

Question 9

Consider our Galaxy to be a uniform disc which is generating radio waves uniformly throughout its volume (Figure 19). Assume for the purposes of this question that all other radio sources are negligible. Also assume that the Earth is in the central plane of the disc but off centre (nearer the edge than the centre) and that it is at rest in the Galaxy. Sketch graphs showing the way the signal picked up by a radio telescope on Earth will vary when the telescope rotates:

(a) in the plane of the Galaxy;

(b) in a plane perpendicular to the Galaxy.

Figure 19: The Earth inside the Galaxy, shown schematically as a disc

Now read the answer

(a) If φ is the angle between the direction of the telescope and the line from the centre of the Galaxy to the telescope, with φ = 0 corresponding to looking directly away from the centre of the Galaxy, then, if the telescope is swept round in the plane of the Galaxy, the variation of intensity W of radio waves with φ will be that shown in Figure 18(a).

(b) If θ is the angle between the direction of the telescope and a line drawn perpendicular to the plane of the Galaxy at the telescope, with θ = 0 corresponding to looking directly ‘upwards’ through the disc, then the variation of intensity W of radio waves with θ will be as shown in Figure 18 (b).

Figure 18: Variation of intensity of radio waves with angle (a) in the plane of the Galaxy, and (b) in a plane perpendicular to the Galaxy

Having separated out these other sources of background noise, it was concluded that the isotropic component had to be of cosmic origin. It is called the cosmic microwave background radiation.

Figure 20(a) shows how complicated the overall microwave spectrum is, due largely to a set of lines generated in the Earth’s atmosphere. Figure 20(b) focuses on the region at the left-hand side of Figure 20(a). This region is of interest because it is at lower frequencies than most of the atmospheric lines. The atmospheric interference varies with the thickness of the atmosphere, and hence with the angle of observation. This noise can therefore be separated out from the cosmic signal in which we are interested. Figure 20(b) shows the microwave spectrum after correcting for the effects of the atmosphere.

Figure 20: (a) The overall ground-based microwave spectrum. (b) Ground-based measurements by Penzias and Wilson, and by other observers, of the microwave intensity at various frequencies; an enlargement of the spectrum shown at the left-hand side of (a), corrected to remove the effects of the atmosphere. The shaded area represents the limits given by measurements made with a detector that covered a wide band of frequencies. (The solid line shows a thermal spectrum corresponding to a temperature of 2.8 K. The term ‘thermal spectrum’ is explained a little later in the text.)

(a) based on D.P. Woody et al. (1975) in Phys. Rev. Lett., 34, 1036–9; (b) based on R.A. Muller (1978) in Scientific American, 238, 64–74

In 1989, the COBE satellite was launched. It was able to make measurements from above the Earth’s atmosphere and was therefore not subject to some of the problems encountered by Penzias and Wilson, and by other ground-based observers. However, a major contaminant – radiation originating from within the Galaxy – remained. The COBE results, after correction for this effect, are shown in Figure 21.

The shape revealed by this closer look is identical to that found for the spectrum inside a hot cavity – for example the spectrum in an oven or furnace (Figure 22). It is called a ‘black body’ spectrum (because it is that which is given out by an idealised heated black surface), or simply a ‘thermal’ spectrum.

Figure 21: Data on the microwave background radiation taken by the COBE satellite. The curve through the data points is that of a thermal spectrum of 2.73 K. Note that Figure 20(b) had a logarithmic frequency scale, while this one is linear

courtesy of NASA/CSFC and COBE Science Working Group

Figure 22: Thermal spectra for various temperatures, T, based on laboratory measurements of relative intensity, W, of radiation with frequency f

One of the basic features of thermal radiation is that, regardless of the temperature of the surface or enclosure generating it, the shape of the spectrum is always the same. The peak of the spectrum moves up to higher frequencies as the temperature rises (it might glow red hot or white hot). But the basic shape remains the same. What this means is that if the spectra for two different temperatures are drawn on two graphs, we can always choose scales – linear but different – so that the graphs can be superimposed (Figures 23 and 24). This single, characteristic shape arises from, and is an indicator of, the equilibrium conditions inside the oven. The rescaling in Figure 23 works only for thermal spectra, and so is a true indicator of equilibrium.

As an example of the opposite extreme, take the line spectra from a hydrogen lamp and a sodium lamp. No amount of rescaling would fit these two together; they are quite distinct.

Figure 23: What is meant by the shapes of thermal spectra being independent of temperature: (a) is the measured spectrum at 998 K (from Figure 22);

(b) is obtained from (a) by expanding the vertical scale by a factor of (1646/998)³;

(c) is obtained from (b) by expanding the horizontal scale by a factor of

(1646/998);

(d) is the measured spectrum at 1646 K (from Figure 22) plotted using the same scales as those in (a), and is the same curve as (c).

Figure 24: As suggested by Figure 23, a plot of W/T³ as a function of f/T is a single curve, for all temperatures; that is, if W/T³ is plotted against f/T for a number of frequencies and temperatures, the plotted points for all frequencies and temperatures lie on one and the same curve

Since the cosmic microwave spectrum has a thermal shape; the conclusion is that it was generated in equilibrium conditions – in some sort of cavity. But the radiation fills the Universe: the ‘cavity’ is the Universe itself, a cavity with no walls.

The word ‘equilibrium’ is used here with some reservations. The Universe has not reached overall equilibrium, and strictly speaking never will. This is because of the expansion of the Universe. We saw in Section 4.1 how the wavelength of light from distant galaxies is redshifted due to the expansion of the Universe. It turns out that the same thing happens to the microwave background radiation; its wavelength is also increased. The peak frequency of the spectrum is reduced, and this means the radiation is progressively cooling. Hence the radiation has not strictly speaking reached equilibrium. However, the processes which transferred energy between radiation and matter in the early Universe were so rapid that it is meaningful to think of a quasi-equilibrium state having been attained at an early time, with the temperature gradually falling subsequently because of the expansion of the Universe.

A second feature of the thermal spectrum is that if the detector is situated within the ‘oven’ generating it (as distinct from looking at a distant black surface or opening to an oven), the intensity of the radiation at any particular frequency uniquely identifies the thermal curve to which it belongs, i.e. what the temperature of the ‘oven’ is. So, if Figure 22 referred to thermal spectra picked up by a detector situated inside an oven, the ordinate of the graph could be expressed in terms of absolute, rather than relative intensities. Under these circumstances, a measurement of intensity at a single frequency, say 60 × 10¹² Hz, would be sufficient to identify which curve that data point belonged to, and hence what the temperature of the oven was.

Inasmuch as the Universe can be regarded as an ‘oven’, and we are in it, Penzias and Wilson were able to estimate from the intensity of the radiation at the single frequency they were detecting that the temperature was about 3 K. For this reason, the cosmic microwave background radiation is often called the 3 K radiation. The spectrum observed by COBE allowed a more precise estimate of the temperature, namely 2.73 K. This is based on the results presented in Figure 21, where it should be noted that the data points do not deviate from the thermal curve by more than 0.03%, this being consistent with measurement precision.

In the 1940s, some theoreticians already had an inkling that this radiation should exist, from the predictions of their cosmological models. The Abbe Georges Lemaitre, a Belgian cosmologist, was the first to see clearly (around 1927) that the expansion of the Universe pointed back to a ‘big bang’. But he could not get much further because not enough was known about nuclear physics at that time. It was George Gamow and his colleagues, in 1948, who first saw that very high temperatures must be involved at early times in an expanding Universe. They sketched out some nuclear reactions that must therefore have taken place. By 1953, their reconstruction had been refined and gave definite predictions for nuclear abundances – and radiation intensity. It was at this stage they realised that radiation had been a vital component in the early Universe and that this same radiation, albeit substantially redshifted and cooled, should still be around today (there being no other place for it to go!). In the 1950s, radio equipment already existed which was sensitive enough to detect this radiation. Indeed, radio astronomy was developing fast, based at first on the technology of military radar in World War II. So by 1953 the stage was set for this radiation to be discovered. But as it so happened, the two groups, theoretical and experimental, did not stumble into each other for another 12 years. Despite conferences and journals, scientific communication with New Jersey was not as effective as communication over 10¹⁰ light-years! When theory and observation finally came together, the question of priority took some sorting out, as indicated in Gamow’s letter, reproduced as Figure 25.

Figure 25: A letter from George Gamow to Arno Penzias

courtesy of Dr A.A. Penzias

The reference to ‘almighty Dicke’ at the end of this letter concerns R. H. Dicke, the leader of a group at Princeton University (also in New Jersey) which was pursuing both theoretical and observational research into the background radiation in the 1960s. The paper by Penzias and Wilson announcing their discovery, and a paper by Dicke’s group providing a possible cosmological interpretation, were published back to back in volume 142 of the Astrophysical Journal in 1965.

The spectrum of thermal radiation at a given temperature T can be expressed in terms of a function W(f, T) which gives the intensity of radiation of frequency f. The energy density of that part of the radiation with frequencies lying between f and f + Δf is given by W(f, T) Δf. The formula for W(f, T) was derived by Max Planck in 1900:

where A and B are constants. At low frequencies (f ≪ T/B), one can use a simpler approximate formula:

Question 10

(a) Verify the result represented by Figure 24, that W/T³ is a function of f/T only, which applies for all frequencies and temperatures.

(b) Use the result of (a) to show that maximum intensity occurs at a frequency, f_max, which is proportional to the temperature.

(c) Imagine that you receive a radio message from a very distant galaxy, informing you about measurements of the cosmic background radiation at various frequencies (defined in terms of fractions of the frequency of a standard spectral line). These results do not agree with your own measurements of the radiation. In particular, you find that the quoted value of the maximum intensity is eight times your value and occurs at a frequency that is twice your frequency for maximum intensity. You find that the quoted intensities at low frequencies are twice your values at the same frequencies. How can you explain these discrepancies? (You may assume that the discrepancy is not due to calibration or other errors.)

Now read the answer

(a) Let y = W/T³ and x = f/T. Then

which shows that W/T³ is a function of f/T only.

(b) From the above expression for y, we see there must be a universal value of x which gives a maximum value of y. But a maximum value of y implies a maximum value of W at a given temperature. If we denote this value of x by the constant x_max, it follows that

and hence that

Thus the frequency, f_max which gives maximum intensity is proportional to the temperature T. (You might have heard of this relation referred to as Wien’s displacement law.)

(c) The explanation is that the cosmic background radiation has cooled during the time it took for the radio message to reach you. The discrepancies between their results and yours can all be explained by this difference in temperature of the radiation.

In part (b), it was shown that the frequency, f_max, corresponding to maximum intensity, is proportional to T. Since their value of f_max is twice yours, the temperature of the cosmic background radiation then must have been twice what it is here and now, i.e. it must have been just under 6 K.

In part (a), it was shown that:

If the maximum intensity is W_max, it follows that

which is independent of T since x_max is a universal constant from part (b). Hence

which is why their value of W_max is 2³ = 8 times yours.

Finally, Equation 8 shows that the intensity at a given low frequency is proportional to the temperature and hence will be twice what you measure now at the same low frequency.

Question 11

(a) On the basis of your solutions to parts (a) and (b) of Question 10, which would be hotter, a red star or a yellow star?

(b) The curves of Figure 22 can be used to extrapolate results to lower temperatures. By taking a measurement off the figure, and using the result of Question 10(b), estimate the frequency of the intensity maximum for radiation emitted by a body at room temperature. What is the term used to describe electromagnetic radiation in this frequency range?

Now read the answer

5 The microwave background radiation

5.2 The origin of the 3 K radiation

In speaking of the radiation as having a cosmic origin, what do we have in mind? Essentially this:

In the violent conditions of the early evolution of the Universe, a stage was reached where the matter consisted of a plasma of electrons, protons, neutrons, and some light nuclei such as helium. There were no atoms as such for the simple reason that atoms would have been too fragile to withstand the violence of the collisions that were taking place at the temperature that then existed. As electromagnetic radiation passed through the plasma, it interacted with the matter, exchanging energy in packets or ‘quanta’ of magnitude

where f is the frequency of the radiation and h is the Planck constant.

The radiation was mainly affected by its collisions with the electrons. This is because such collisions cause much bigger energy changes to the photons than collisions with the far more massive nucleons (just as a table tennis ball may lose all its energy in a collision with another table tennis ball, but will bounce off a relatively massive billiard ball with little change in energy). Thus there is a ready exchange of energy between the photons and the electrons, in the process of which, the radiation acquires the thermal spectrum characterised by the temperature of the electrons. The radiation and the electrons tend to come into thermal equilibrium with each other, and the electrons are said to have thermalised the radiation.

As the expansion of the Universe proceeded, the temperature of the radiation progressively fell, and so did that of the matter. This fall led to important changes in behaviour. From the earliest times, the Universe had been opaque to radiation, in the sense that it could not travel far before it interacted with the electrons. But as the temperature declined and photon energies decreased, a stage was reached where electrons could be bound to nuclei to form neutral atoms – atoms that were no longer likely to be disrupted in collisions with the reduced-energy photons. Later, the energy of the radiation reduced still further to the point where it could not even excite the atomic electrons to higher energy states. At this stage, the radiation could no longer be strongly absorbed by matter. This being so, the Universe became transparent. This stage we call the decoupling of radiation from matter. (You will find that some books refer to this stage in the development of the Universe as the ‘recombination’ era rather than the decoupling epoch.) It occurs when the radiation has cooled down to the point where the most probable photon energy corresponds to a temperature of 3000 K. This occurred some 4 × 10⁵ years after the instant of the big bang. Thus the radiation we now observe as 3 K radiation is today’s cooled-down remnant of that 3000 K big bang radiation.

How confident can we be that this was indeed the origin of the 3 K radiation? There are essentially four properties that lead to this conclusion:

1. As we have already mentioned, the isotropy of the radiation points to some global, cosmic origin.
2. The spectrum of the radiation is such that it could only have been produced by a sufficiently rapid interaction of the radiation with matter for the thermal energy distribution of the particles of matter to be imprinted on the radiation. Only in the early dense stages of the Universe were particles and radiation interacting fast enough for this to have been achieved within the time available.
3. The present temperature of the radiation of only 3 K is lower than that of most visible matter currently in the Universe. How could it be so low? The only reasonable explanation is that it has been strongly redshifted -indicating that it has been travelling towards us over an exceedingly long period of time, i.e. it was emitted soon after the big bang.
4. The density of photons corresponding to the 3 K radiation is enormous. In fact there are believed to be about 10⁹ times as many 3 K photons in any large region of the Universe as there are neutrons and protons. Clearly this radiation is no mere by-product of an obscure process; it is a ubiquitous feature of the Universe. This prompts us to ask at what stage of the Universe is radiation likely to have played a dominant role? The answer has to be: the violent early Universe.