The ubiquitous microwave background

In the 1950s, a popular theory of the universe was one referred to as the steady state model, a proposal first put forward by Thomas Gold and Hermann Bondi in 1948, and soon taken up in more detail by Fred Hoyle,[213] who were all at Cambridge University at the time. The theory required there to be a continual creation of material throughout space, at an extremely low rate. This material would have to be in the form of hydrogen molecules—each being a pair consisting of one proton and one electron, created out of the vacuum—at the extremely tiny rate of about one such atom per cubic metre per thousand million years. This would have to be at just the right rate to replenish the reduction of the density of material due to the expansion of the universe.

In many respects, this is a philosophically attractive and aesthetically pleasing model, as the universe requires no origin in time or space, and many of its properties can be deduced from the requirement that it should be self-propagating. It was fairly soon after this theory was being proposed that I entered Cambridge University, in 1952, as a young graduate student (researching in pure mathematics, but with a keen interest in physics and cosmology[214]), and I returned later, in 1956, as a research fellow. While at Cambridge, I got to know all three of the steady-state theory's originators, and I had certainly found this model to be appealing and the arguments fairly persuasive. However, towards the end of my time at Cambridge, detailed counts of distant galaxies carried out at the Mullard Radio

Observatory by (Sir) Martin Ryle (also in Cambridge) were beginning to provide clear observational evidence against the steady-state model.[215]

But the real death-blow was the accidental observation by the Americans Arno Penzias and Robert W. Wilson, in 1964, of microwave electromagnetic radiation, coming from all directions in space. Such radiation had in fact been predicted, in the later 1940s by George Gamow, and by Robert Dicke on the basis of what was then the more conventional 'Big-Bang theory', such presently observable radiation being sometimes described as 'the flash of the Big Bang', the radiation having been cooled from some 4000K to a few degrees above absolute zero[216] by an enormous red-shift effect due to the vast expansion of the universe since the emission of the radiation. After Penzias and Wilson had convinced themselves that the radiation they were observing (of around 2.725 K) was genuine, and must actually be coming from deep space, they consulted Dicke, who was quick to point out that their puzzling observations could be explained as what he and Gamow had previously predicted. This radiation has gone under various different names ('relic radiation', 3-degree background, etc.); nowadays it is commonly referred to simply as the 'CMB', which stands for 'cosmic microwave background'.[217] In 1978, Penzias and Wilson were awarded the Nobel Prize in Physics for its discovery.

The source of the photons which actually constitute the CMB that we now 'see' is not really the 'actual Big Bang', however, as these photons come to us directly from what is called the 'surface of last scattering' which occurred some 379000 years following the moment of the Big Bang (i.e. when the universe was about 1/36 000 of its present age). Earlier than this, the universe was opaque to electromagnetic radiation because it would have been inhabited by large numbers of separate charged parti-cles—mainly protons and electrons—milling around separately from each other, constituting what is referred to as a 'plasma'. Photons would have scattered many times in this material, being absorbed and created copiously, and the universe would have been very far from transparent. This 'foggy' situation would have continued until the time referred to as 'decoupling' (where 'last scattering' occurs) at which the universe became transparent because it had cooled down sufficiently for the separate electrons and protons to be able to pair up, largely in the form of hydrogen (with a few other atoms produced, mainly about 23% helium, whose nuclei— called 'a-particles'—would have been among the products of the first few minutes of the universe's existence). The photons were then able to decouple from these neutral atoms, to travel essentially undisturbed from then on, to become the radiation which is now perceived as the CMB.

Since its initial observation in the 1960s, many experiments have been performed to get better and better data concerning the nature and distribution of the CMB, there being so much detailed information now, that the subject of cosmology has been completely transformed—from one in which there was much speculation and very little data to bear on this speculation—to a precision science, in which, although there is still much speculation, there is now a very great deal of detailed data to modulate this speculation! One particularly noteworthy experiment was the COBE satellite (Cosmic Background Explorer), launched by NASA in November 1989. Its remarkable observations earned George Smoot and John Mather the 2006 Nobel Prize in Physics.

There are two very striking and important features of the CMB which were made particularly evident by COBE, and I want to concentrate on both of these. The first is the extraordinary closeness by which the observed frequency spectrum matches that explained by Max Planck in 1900 to account for the nature of what is called 'black-body radiation' (and which marked the starting point of quantum mechanics). The second is the extremely uniform nature of the CMB over the whole sky. Each of these two facts will be telling us something very fundamental about the nature of the Big Bang, and its curious relation to the Second Law. Much of modern cosmology has moved on from this now, and is concerned more with the slight and subtle deviations from uniformity in the CMB that are also seen. I shall be coming to some of these later (see §3.6), but for the moment I shall need to address these two more blatant facts in turn, as we shall find that they both have a very great significance for us.

Figure 2.6 depicts the frequency spectrum of the CMB, essentially as initially measured by COBE, but where now greater precision is obtained from later observations. The vertical axis measures the intensity of the radiation, as a function of the different frequencies, these being marked off along the horizontal axis with increasing frequency off to the right.

The continuous line is Planck's 'black-body curve', which is given by a specific formula,[218] and it is what quantum mechanics tells us is the radiation spectrum of thermal equilibrium, for any particular temperature T. The little vertical bars are error bars, telling us roughly the range within which the observed intensities lie. It should be noted, however, that these error bars are exaggerated by a factor of 500, so the actual observation points lie much more closely on the Planck curve than would appear—in fact, so closely that, to the eye, even the observations on the very far right, where the error is greatest, concur with the Planck curve to within the thickness of the ink line! Indeed, the CMB provides us with the most precise agreement between an observed intensity spectrum and the calculated Planck black-body curve that is known in observational science.

Fig. 2.6 Frequency spectrum of the CMB, as initially observed by COBE, but supplemented by later more precise observations. Note that the 'error bars' are exaggerated by a factor of 500. This shows precise agreement with the Planck spectrum.

What does this tell us? It appears to tell us that what we are looking at comes from a state that must effectively be thermal equilibrium. But what does 'thermal equilibrium' actually mean? I refer the reader back to Fig. 1.15, where we find the words 'thermal equilibrium' labelling the coarse-graining region of phase space which is (by far) the largest of all. In other words, this is the region representing maximum entropy. But we must recall the thrust of the arguments given in §1.6. These arguments told us that the whole basis of the Second Law must be explained by the fact that the initial state of the universe—which we evidently must take as being the Big Bang—must be a (macroscopic) state of extraordinarily tiny entropy. What we appear to have found is essentially the complete opposite, namely a (macroscopic) state of maximum entropy!

One point must be addressed here, namely the fact that the universe is expanding, so what we are looking at can hardly be an actual 'equilibrium' state. However, what is evidently happening here is an adia-batic expansion, where 'adiabatic' here refers, effectively, to a 'reversible' change in which the entropy remains constant. The fact that this kind of 'thermal state' is actually preserved in the early universe's expansion was pointed out by R.C. Tolman in 1934.[219] We shall be seeing some more of Tolman's contributions to cosmology in §3.3. In terms of phase space, the picture is more like Fig. 2.7 than Fig. 1.15, where the expansion is described as a succession of maximal coarse-graining regions of essentially equal volume. In this sense, the expansion can still be viewed as a kind of thermal equilibrium.

Fig. 2.7 Adiabatic expansion of the universe depicted as a succession of maximal coarse-graining regions of equal volume.

Fig. 2.7 Adiabatic expansion of the universe depicted as a succession of maximal coarse-graining regions of equal volume.

So we still seem to be seeing maximum entropy. Something appears to have gone seriously wrong with the arguments. It is not even just that the observations of the universe have come up with a surprise. Not at all: in a certain sense, the observations are closely in accord with what was expected. Given that there was actually a Big Bang, and that this initial state is to be described as being in accord with the standard picture presented by general-relativistic cosmology, then a very hot and uniform initial thermal state is what would be expected. So where does the resolution of this conundrum lie? Perhaps rather surprisingly, the issue has to do with the assumption that the universe is indeed in accordance with the standard picture of relativistic cosmology! We shall need to examine this assumption very carefully indeed, to see what has eluded us.

First, we must remind ourselves what Einstein's general theory of relativity is all about. It is, after all, an extraordinarily accurate theory of gravity, where the gravitational field is described in terms of a curvature of space-time. I shall have a lot to say about this theory in due course, but for the moment let us think in terms of the older—and still extraordinarily accurate—Newtonian gravitational theory, and try to understand, in rough general terms, how it fits in with the Second Law— of thermodynamics, that is; I do not mean Newton's second law!

Often, considerations of the Second Law might be discussed in terms of a gas constrained to lie within a sealed box. In accordance with such discussions, let us imagine that there is a small compartment in one corner of the box, and the gas is initially constrained to be within that compartment. When the door to the compartment is opened and the gas is allowed to move freely within the box, we expect that it will rapidly spread itself out evenly within the box, and the entropy would indeed be increasing throughout this process, in accordance with the Second Law. The entropy is thus much higher for the macroscopic state in which the gas is distributed uniformly than it was when the gas was all together in the compartment. See Fig. 2.8(a). But let us now consider a similar-looking situation, but with an imaginary box of galactic size, and where the individual molecules of gas are replaced by individual stars moving within this box. The difference between this situation and that of the gas is not just a matter of scale, and I shall take size to be irrelevant for the present purposes. What is relevant is the fact that the stars attract each other, through the relentless force of gravity. We might imagine that the distribution of stars is initially spread fairly uniformly throughout our galactic-sized box. But, now, as time progresses, we find a tendency for the stars to collect together in clumps (and generally to move more rapidly as they do so). Now the uniform distribution is not the one of highest entropy, increasing entropy being accompanied by an increase in the clumpiness of the distribution. See Fig. 2.8(b).

Gas in a box

time entropy

Gravitating bodies

(b) black hole

Fig. 2.8 (a) Gas is initially constrained within a small compartment in the corner of a box before being released and distributing itself uniformly throughout the box. (b) In a galactic-sized box, stars are initially uniformly distributed but collect together in clumps over time: a uniform distribution in this case is not the one with highest entropy.

We may ask what now is the analogue of thermal equilibrium, where the entropy has increased to its maximum? It turns out that this question cannot be properly addressed within the confines of Newtonian theory. If we consider a system that consists of massive point particles attracting each other according to Newton's inverse square law, then we can envisage states in which some of the particles get progressively closer and closer to each other, moving more and more rapidly, so that there is no limit to the degree of clumpiness and rapidity of motion, and the proposed state of 'thermal equilibrium' simply does not exist. The situation turns out to be much more satisfactory in Einstein's theory, because the 'clumpiness' can saturate, when the matter conglomerates into a black hole.

We shall come to black holes in more detail in §2.4, where we learn that the formation of a black hole represents an enormous increase in the entropy. Indeed, at the present epoch of the universe's evolution, the greatest entropy contribution, by far, lies in large black holes, like the one at the centre of our own Milky Way galaxy, with a mass of around 4000000 times the mass of our Sun. The total entropy in such objects completely swamps that in the CMB, which had previously been thought to represent the dominant contribution to the entropy present in the universe. Thus, the entropy has greatly increased via gravitational condensation from what it was at the creation of the CMB.

This relates to the second feature of the CMB referred to above, namely its closely uniform temperature over the whole sky. How closely uniform is it? There is a slight temperature variation understood as a Doppler shift, coming from the fact that the Earth is not exactly at rest with respect to the mass distribution of the universe as a whole. The Earth's motion is composed of various contributions, such as its motion about the Sun, the Sun's motion around the Milky-Way galaxy, and the galaxy's motion due to local gravitational influences of other relatively nearby mass distributions. All combine together to provide what is referred to as the Earth's 'proper motion'. This leads to a very slight increase of the apparent temperature of the CMB in the direction in the sky that we are moving towards,[220] and a very slight decrease in the direction in the sky that we are moving away from, and an easily calculated pattern of slight temperature alterations over the whole sky. Correcting for this, we find a CMB sky that has an extraordinarily uniform temperature over the sky, with deviations of the order of only a few parts in 105.

This tells us that, at least over the surface of last scattering, the universe was extraordinarily uniform, like the right-hand picture of Fig. 2.8(a) and also like the left-hand picture Fig. 2.8(b). It is reasonable to assume, therefore, that so long as we can ignore the influences of gravity, the material content of the universe (at last scattering) was indeed at as high an entropy as it could achieve on its own. Gravitational influences would, after all, be small because of the uniformity, but it was this very uniformity in the matter distribution that provided the potential for enormous subsequent entropy increases when gravitational influences later come into play. Our picture of the entropy of the Big Bang is therefore completely changed once we consider the introduction of gravitational degrees of freedom. It is the assumption that our universe, overall, is very closely in accord with the spatial homogeneity and isotropy—sometimes referred to as the 'cosmological principle'[221], basic to FLRW cosmology and, in particular, central to the Friedmann models discussed in §2.1—that implies the huge suppression of gravitational degrees of freedom in the initial state. This early spatial uniformity represents the universe's extraordinarily low initial entropy.

A natural question to ask is: what on earth does that cosmological uniformity have to do with our familiar Second Law, which seems to permeate so much of the detailed physical behaviour in the world we know? There are multitudes of commonplace instances of the Second Law that would seem to bear no relation to the fact that gravitational degrees were suppressed in the early universe. Yet the connection is indeed there, and it is actually not so hard to trace back these commonplace instances of the Second Law to the uniformity of the early universe.

Let us consider, as an example, the egg of §1.1, perched on the edge of a table, about to fall off and smash on the floor below (see Fig. 1.1). The entropy-raising process of the egg rolling off the table and smashing is enormously favoured, probabilistically, provided that we are prepared to assume that the egg started in the very low-entropy state of being perched, unbroken, at the edge of the table. The puzzle of the Second Law is not the raising of the entropy following that event; the puzzle lies in the event itself, i.e. the question of how the egg happened to find itself in this extremely low-entropy state in the first place. The Second Law tells us that it must have arrived in this very improbable state through a sequence of other states that had been even more improbable prior to this, and getting more so the farther back in time we examine the system.

There are basically two things to explain. One is the question of how the egg got up on the table, and the other is how the low-entropy structure of that egg itself came about. Indeed, the material of an egg (taken to be a hen's egg) has been superbly organized into a perfect package of appropriate nourishment for an intended chick. But let us start with what may seem to be the easier part of the problem, namely that of how the egg found itself up on the table. The likely answer would be that some person put it there, perhaps a little carelessly, but human intervention was the probable cause. There is clearly a lot of highly organized structure in a functioning human being, which suggests a low entropy, and the placing of the egg on the table would have taken only a very little from the rather large reservoir of low entropy in the relevant system, consisting of a reasonably well-fed person and a surrounding oxygen-laden atmosphere. The situation with the egg itself is somewhat similar, in that the egg's highly organized structure, superbly geared to supporting the burgeoning life of a presumed embryo within it, is very much part of the grand scheme of things that keeps life going, on this planet. The entire fabric of life on Earth requires the maintaining of a profound and subtle organization, which undoubtedly involves entropy being kept at a low level. In detail, there is an immensely intricate and interconnected structure, which has evolved in keeping with the fundamental biological principle of natural selection and with many detailed matters of chemistry.

What, you might well ask, do such matters of biology and chemistry have to do with the uniformity of the early universe? Biological complication does not allow the system as a whole to violate the general laws of physics, such as the law of conservation of energy; moreover, it cannot provide escape from the constraints imposed by the Second Law. The structure of life on this planet would run rapidly down were it not for a powerful low-entropy source, upon which almost all life on Earth depends, namely the Sun.[222] One tends to think of the Sun as supplying the Earth with an external source of energy, but this is not altogether correct, as the energy that the Earth receives from the Sun by day is essentially equal to that which the Earth returns to the darkness of space![223] If this were not so, then the Earth would simply heat up until it reaches such an equilibrium. What life depends upon is the fact that the Sun is much hotter than the darkness of space, and consequently the photons from the Sun have a considerably higher frequency (namely that of yellow light) than the infra-red photons that Earth returns to space. Planck's formula E=hv

(see §2.3) then tells us that, on average, the energy carried in by individual photons from the Sun is considerably greater than the energy carried out by individual photons returning to space. Thus, there are many more photons carrying the same energy away from Earth than there are that carry it in from the Sun. See Fig. 2.9. More photons imply more degrees of freedom and therefore a larger phase-space volume. Accordingly, Boltzmann's S=k\ogV, (see §1.3) tells us that energy coming in from the Sun carries a considerably lower entropy than that returning to space.

Fig. 2.9 Photons arriving at the Earth's surface from the Sun have higher energy (shorter wavelength) than those returned to space by the Earth. Given an overall energy balance (the Earth does not get hotter over time), there must be more photons leaving than arriving; that is, the energy arriving has lower entropy than that departing.

Now, on Earth, the green plants have, by the process of photosynthesis, found a way of converting the relatively high-frequency photons coming from the Sun to photons of a lower frequency, using this gain in low entropy to build up their substance by extracting carbon from CO2 in the air and returning it as O2. When animals eat plants (or eat other animals that eat plants), they use this source of low entropy, and the O2, to keep down their own entropy.[224] This applies to humans, of course, and also to chickens, and it supplies the source of low entropy needed for the construction of our unbroken egg and for it to be placed on the table!

So what the Sun does for us is not simply to supply us with energy, but to provide this energy in a low-entropy form, so that we (via the

Fig. 2.9 Photons arriving at the Earth's surface from the Sun have higher energy (shorter wavelength) than those returned to space by the Earth. Given an overall energy balance (the Earth does not get hotter over time), there must be more photons leaving than arriving; that is, the energy arriving has lower entropy than that departing.

green plants) can keep our entropy down, this coming about because the Sun is a hot spot in an otherwise dark sky. Had the entire sky been of the same temperature as that of the Sun, then its energy would have been of no use whatever to life on Earth. This applies, also, to the Sun's ability to raise water from the oceans high up into the clouds, which again depends crucially on this temperature difference.

Why is the Sun a hot spot in the dark sky? Well, there are all sorts of complicated processes going on in the Sun's interior, and the thermonuclear reactions that result in hydrogen being converted to helium play an important part in this. However the key issue is that the Sun is there at all, and this has come about from the gravitational influence which holds the Sun together. Without thermonuclear reactions, the Sun would still shine, but shrink and get much hotter, and have a far shorter life. On Earth, we clearly gain from these thermonuclear reactions, but they would not even have the chance to take place were it not for the gravitational clumping that produced the Sun in the first place. Accordingly, it is the potential for stars to form (albeit via somewhat complicated processes in appropriate regions in space), through the relentless entropy-raising process of gravitational clumping, from initial material that started off in a very uniform gravitationally low-entropy state.

This all comes about, ultimately, from the fact that we have been presented with a Big Bang of a very special nature, the extreme (relative) lowness of its entropy being manifested in the fact that its gravitational degrees of freedom were indeed not initially activated. This is a curiously lop-sided situation, and to understand it better we shall try to dig a little more deeply, in the next three sections, into Einstein's beautiful curved-space-time description of gravity. Then, in §2.6 and §3.1, I shall return to the issue of the nature of this extraordinary specialness that is actually exhibited in our Big Bang.

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