The Science of Discworld IV Judgement Da

EIGHTEEN



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BYE-BYE BIG BANG?





Viewed from Discworld, Roundworld is a puzzle. It sits neatly on a shelf in Rincewind’s office, but the wizards know that its outside has to be far smaller than its inside, because Rincewind (among many others) has visited the inside in person. Indeed, the whole of the inside is absolutely gigantic. The wizards have a theory about how this can be. Roundworld runs on mysterious rules, and its shape, size, and even its origin seem to be consequences of those rules. But the rules apply only on the inside. On the outside, magic takes over.

In chapter 16 we examined how Roundworld’s rules affect not just the answer to the question of its shape, but what we mean by the question itself. Now we turn to its origins.

When the Archchancellor told Marjorie about the origins of Roundworld, he naturally did so from the wizards’ perspective, in which the entire human universe is somehow located inside a small sphere, about the size of a football. And was brought into being by HEX saving Discworld from annihilation, and the Dean poking an exploratory finger into the resulting magical containment field.

What about the perspective of its inhabitants? Since ancient times, they have wondered how (or whether) their world began, but until recently their answers were resolutely human-centred: mainly stories about creator gods. In contrast, today’s scientific theories of the origins of the universe are (surprise!) universe-centred. They are based not on stories about gods, but on the rules that the universe seems to obey.

Since the rules are not written down in a book of magical spells, the wizards have to infer the rules from what Roundworld does. Roundworld scientists are in the same position, but their difficulty is compounded, being stuck inside the thing and confined to the present. Even so, they have outperformed the wizards by working out many of the rules that govern their world and their universe.

When it comes to the origin of the universe, what we really need is a time machine. Despite some hints at the frontiers of today’s physics, discussed in The Science of Discworld II and III, no practical time machine exists, or is likely to. But that doesn’t stop us wanting to know exactly how the universe began, or taking a serious stab at working it out from the evidence it left behind.

The origin of the universe is a weighty philosophical question, leading to profound scientific and mathematical ideas. Mathematics, after all, is humanity’s best-developed and most powerful system of logical inference, and if you can’t go back to the distant past to take a look, you have to remain in the present and infer.

We’ve already seen that questions of shape and origin tend to go hand in hand. That is especially true of the universe, because it is dynamic: what it looks like now depends on what happened to it in the past. So cosmology and cosmogony are intimately intertwined, just as they were in ancient mythology. The current theory of the origins of the universe is of course the Big Bang, which emerged unexpectedly from astronomical observations that were intended to sort out its size and shape. So before we examine the universe’s origins, we’ll take a quick look at these observations and what they led to.

In ancient times, the world and the universe were pretty much identified. The Sun, Moon, planets and stars were little more than added decoration in the sky; the world on which we lived dominated everything else. Now we realise that our planet is a very tiny lump of rock in a universe so huge that we find it hard to comprehend.

Humanity first caught a glimpse of the sheer vastness of the universe in 1838, when the astronomer Friedrich Bessel measured the distance to the star 61 Cygni. Until then, people who did not believe that the Earth goes round the Sun could offer a fairly plausible argument to show that the Earth had to be fixed. If it moved round the Sun, then the nearer stars would appear to move slightly against the background of more distant stars, an effect called parallax. But they didn’t. Bessel found out why: even the nearest stars are a tremendous distance away, making this apparent motion too small to detect. He used a sensitive new telescope to observe the star 61 Cygni. This had been nicknamed the ‘flying star’ by Giuseppe Piazzi in 1804 because its apparent motion across the sky, though very small, was unusually large compared to that of most other stars. This suggested that the star might be unusually close to the Earth. Bessel discovered that it is 11.4 light years away, roughly 1014 (one hundred trillion) kilometres. The modern figure is 11.403, so Bessel was spot-on.

The diminution of humanity had barely begun. As well as twinkling points of light, the night sky boasted a glowing river of light, the Milky Way. In fact it is a disc of stars, most of them too far away to be resolved as individuals, and we are inside it. We now call such a disc a galaxy. Hints that other galaxies might exist appeared when astronomers detected distant nebulas, fuzzy wisps of light. In 1755 the philosopher Immanuel Kant called such nebulas ‘island universes’; eventually they became known as galaxies, from the Latin for ‘milk’. Charles Messier compiled the first systematic catalogue of nebulas (a few were genuine wisps, not galaxies) in 1774. A prominent one, in the constellation of Andromeda, was the 31st in his list, and it was therefore designated M31. It showed no parallax, so it was presumably a long way away. The big question was: how far?

In 1924 Edwin Hubble showed that M31 lies far beyond the Milky Way, thanks to some brilliant work by Henrietta Leavitt, a human ‘computer’ hired to carry out the repetitive task of measuring and cataloguing how bright the stars were. At that time astronomers were looking for a ‘standard candle’ – a type of star whose intrinsic brightness could be inferred from other observations. This could then be compared to its apparent brightness, and the way brightness decreases with distance could be used to calculate how far away the star was. Leavitt was observing Cepheid variables – stars whose light output changes in a periodic cycle – and in 1908 she discovered that the light output of a Cepheid is related to the period of its cycle. Therefore its intrinsic brightness can be calculated from observations, so it can be used as a standard candle. In 1924 Hubble observed Cepheid variables in M31, and calculated that this galaxy is a million light years away. The current estimate is 2.5 million light years.

Most galaxies are much further away than that; so distant that there is no prospect of distinguishing Cepheids, indeed any individual stars. However, Hubble overcame this obstacle too. Vesto Slipher and Milton Humason discovered that the spectra of many galaxies were shifted towards the red end of the spectrum. The most plausible explanation was the Doppler effect, in which a wave changes frequency if its source moves. It is most familiar for sound waves: the pitch of a police car’s siren gets lower when it passes, changing from moving towards us to going away from us. The Doppler effect implies that the galaxies concerned must be receding at significant speeds. Hubble plotted the amount of redshift against estimates of distance for forty-six galaxies in which Cepheids had been spotted. The graph was approximately a straight line, suggesting that the velocity of recession (deduced from redshift) and distance were proportional. In 1929 he formulated this as a formula, which we now call Hubble’s law. The constant of proportionality, Hubble’s constant, is currently thought to be about 21 km/s per million light years. Hubble’s initial estimate was about seven times as big.

It is now realised that the Swedish astronomer Knut Lundmark had the same idea in 1924, five years before Hubble. He used the apparent sizes of galaxies to infer how far away they were, and his figure for the ‘Hubble’ constant was within 1% of today’s – far better than Hubble’s. However, his work was ignored because his methods had not been cross-checked using independent measurements.

These developments tied the size of the universe and its dynamic behaviour together, and led to a surprising inference. If all of the galaxies are moving away from us, then either the Earth is near the centre of some expanding region, or the entire universe is getting bigger.

Astronomers were already aware that the universe might be expanding. Einstein’s field equations, the basis of general relativity, predicted it. In 1924 Aleksandr Friedmann derived three types of solution of the field equations, depending on whether the curvature of space is positive, zero or negative. Mathematicians working in non-Euclidean geometry had already discovered such spaces: respectively, they are said to be elliptic, Euclidean or hyperbolic (like the Escherverse). Elliptic space is finite, a hypersphere – like the surface of a sphere but three-dimensional. The other two are of infinite extent. (The Escherverse is like Roundworld: from the outside it appears to be finite, but from inside, measured using its own metric, it is infinite in extent. That’s how it can contain infinitely many angels or devils, all the same size.) The field equations specified a range of shapes for the universe, but did not pin the shape down uniquely.

The field equations also allow the shape of the universe to change as time passes. In 1927 Georges Lemaître derived an expanding universe from Einstein’s field equations, and estimated the rate of expansion. His 1931 paper ‘A homogeneous universe of constant mass and growing radius accounting for the radial velocity of extragalactic nebulae’ went largely unread, because he published it in an obscure Belgian journal, but eventually it became a classic.

Lemaître’s solution conflicted with the prevailing cosmological wisdom, but the popular (and populist) astronomer Sir Arthur Eddington believed that Lemaître’s theory solved many of the major problems in cosmology. In 1930 he invited Lemaître to a meeting in London about physics and spirituality. By then, Lemaître had realised that if you ran the universe’s expansion backwards, everything converged to a single point some lengthy period into the past.fn1 He called this initial singularity the primeval atom, and published the idea in the leading scientific journal Nature. A huge controversy ensued. Lemaître may not have helped his cause by referring to the idea as ‘the Cosmic Egg exploding at the moment of the creation’.

Much later Fred Hoyle, by then a leading advocate of the steady-state theory – that the universe is in equilibrium, aside from local fluctuations, and it has always been that way – dismissed Lemaître’s theory as the ‘Big Bang’. The name stuck. So did the theory, to Hoyle’s discomfort. Hoyle had developed the steady-state theory in 1948, aided by Thomas Gold, Hermann Bondi and others. It required matter to be continuously created, gently, particle by particle, in the voids between the stars, to prevent the density of matter decreasing as the universe got bigger. The necessary rate of production was low, about one hydrogen atom per cubic metre every billion years.

Unfortunately for Hoyle, indirect evidence against the steady-state theory, and in favour of the Big Bang, kept piling up. The smoking gun was the discovery of cosmic background radiation in 1965 – a sizzle of random radio noise that we now think originated when the universe first became transparent to radio waves, shortly after the Big Bang. Moreover, its temperature agreed with theory. Hawking has called this observation ‘the final nail in the coffin of the steady-state theory’.

Einstein, in private, was unimpressed by Lemaître’s expanding-universe solution. He accepted the mathematics, but not the physical reality. But when Hubble’s results were published two years later, Einstein immediately changed his mind and gave Lemaître strong public support. In 1935 Howard Robertson and Arthur Walker proved that every homogeneous, isotropic universe – one that is the same at every point and in every direction – corresponds to a particular family of solutions of Einstein’s field equations. The resulting universes could be static, expanding or contracting; their topology could be simple or complex. The family is called the Friedmann-Lemaître-Robertson-Walker metric, or the ‘standard model of cosmology’ if that’s too big a mouthful. It now dominates mainstream cosmological thinking.

Narrativium now took over, and led many cosmologists into the realms of scientific mythology. The correct statement that ‘there exist solutions of Einstein’s field equations corresponding to the classical non-Euclidean geometries’ transmogrified into the false statement ‘these are the only possible solutions of constant curvature’. The mistake may have arisen because mathematicians weren’t paying enough attention to astronomy and astronomers weren’t paying enough attention to mathematics. Robertson and Walker’s uniqueness theorem proves that the metric is unique, and it is easy to imagine that the space must also be unique. After all, doesn’t the metric define the space?

No, it doesn’t.

The metric is local; the space is global. Both infinite Euclidean space and the flat torus have the same metric, because the geometry of small regions is identical. The computer screen remains flat; what changes are the rules about going off the edge. But globally, the flat torus has special geodesics – ones that form closed loops – whereas Euclidean space does not. So the metric does not define the space. But cosmologists thought it did. In 1999, writing in Scientific American, Jean-Pierre Luminet, Glenn Starkman, and Jeffrey Weeks, wrote: ‘The decades from 1930 to 1990 were the dark ages of the subject. Most astronomy textbooks, quoting one another for support, stated that the universe must be either a hypersphere, an infinite Euclidean space, or an infinite hyperbolic space. Other topologies were largely forgotten.’

In fact, more than one topology is possible in each of the three cases. Friedmann had said as much in his 1924 paper, for negative curvature, but this remark somehow became forgotten. Finite spaces of zero curvature had already been discovered, the most obvious being the flat torus. Elliptic space was finite anyway. But even that space was not the only possibility with constant positive curvature, a fact known to Poincaré in 1904. Unfortunately, once the misconception had set in, it was very hard to root it out again, and it obscured the question of the shape of the universe for decades.

However, at that time cosmologists were after bigger game: the origin of the universe. According to the Big Bang solution of the field equations, both space and time sprang into existence from nothing, and then evolved into today’s universe. Physicists were ready for this radical theory because quantum mechanics had already softened them up for the idea that particles can arise spontaneously from nothing. If a particle can do it, why not a universe? If space can do it, why not time?

Looking back at Einstein again. He could even have predicted an expanding spherical universe, but he got it into his head that the static one was the right one. To obtain a static solution, he modified his field equations to include an extra term depending on a ‘cosmological constant’. By choosing this constant suitably, the universe could be rendered static. Precisely why the cosmological constant would have that value was less clear, but the new term in the equations obeyed all of the deep symmetry principles that drove Einstein’s philosophy of how the universe ought to behave. It would actually take a lot of special pleading to eliminate that term. When telescopic observations of the spectra of galaxies revealed an expanding universe, Einstein decided that including the cosmological constant had been his ‘biggest blunder’. If he had left it out, he could have predicted the expansion.

Well … that’s the standard story, but it requires an unstated assumption. In order to derive a formula for how the shape and size of the universe changes over time, the mathematical physicists of the early twentieth century looked only for spherically symmetric solutions of Einstein’s field equations. This assumption reduces the spatial variables from three to one: the distance from the centre. As a bonus, it simplifies the Einstein field equations, which can now be solved by an explicit formula. Although there is a hand-waving justification of spherical symmetry, ‘the universe should be the same everywhere’, it doesn’t have a solid basis. Einstein had insisted that the laws should be the same everywhere, but that doesn’t imply the same behaviour everywhere. Both planets and the vacuum obey the same laws, for example.

With the advent of computers, it turned out that the Einstein field equations have zillions of solutions – infinitely many, depending on the choice of initial conditions – almost all of which are not spherically symmetric. Space might expand in some regions, contract in others, or swirl round and round. It could change its behaviour as time passed. So although an expanding universe is one possible solution of the Einstein field equations, it no more constitutes a prediction of an expanding universe than the possibility of rain tomorrow, as a solution of the weather equations, constitutes a firm forecast of rain.

A few years ago, all was sweetness and light. The Big Bang fitted all the important observations. In particular, it predicted that the cosmic microwave background radiation should have a temperature of about 3 degrees absolute. Score one to the Big Bang.

As research continued, however, difficulties emerged. Today’s universe has a lot of large-scale structure – vast skeins of galaxies surrounding even vaster voids, like the foam in a glass of beer, with galaxies forming on the surfaces of beer bubbles, and voids corresponding to the air inside them. Backtracking from its present state and using current theories, the universe should be about 13.5 billion years old. On the one hand, that’s too short a time to explain the current clumpiness of matter. On the other hand, it’s not long enough to explain the current flatness of space.

A second difficulty emerges from the observed ‘rotation curves’ of galaxies. Galaxies do not rotate like a rigid object: stars at different distances from the centre move with different speeds. Stars in the galaxy’s central bulge move quite slowly; those further out are faster. However, the stars outside the central bulge all move with much the same speed. This is a puzzle for theorists, because both Newtonian and Einsteinian gravity require the stars to move more slowly in the outer reaches of the galaxy. Virtually all galaxies behave in this unexpected manner, which conflicts with numerous observations.

The third problem is the 1998 discovery that the expansion of the universe is accelerating, which is consistent with a positive non-zero cosmological constant. This was based on the High-z Supernova Search Team’s observations of the redshift in Type Ia supernovae, and won the Nobel Prize for Physics in 2011.

The prevailing cosmological wisdom deals with these problems by bolting on three additional assumptions. The first is inflation, in which the entire universe expanded to a huge size in an extraordinarily short time. The figures are shocking: between 10-36 and 10-32 seconds after the Big Bang the volume of the universe multiplied by a factor of at least 1078. The cause of this rapid growth – an explosion far more impressive than the wimpy Big Bang that started it all – is, we are told, an inflaton field. (Not ‘inflation field’: an inflaton is – well, a quantum field that causes inflation.) This theory fits many observations very well. The main snag is the absence of any direct evidence for the existence of an inflaton field.

To solve the problem of galactic rotation curves, cosmologists propose the existence of dark matter. This is a form of matter that can’t be observed by the radiation it emits, because it doesn’t, not in any quantity that can be observed from here. It’s entirely reasonable that a lot of the matter in the universe might not be observable, but what we can infer indirectly leads to the conclusion that whatever dark matter may be, it’s not made from the fundamental particles that we know about on Earth. It’s a very alien form of matter, which mainly interacts with everything else through the force of gravity. No such particles have ever been observed, but there are several competing suggestions for what they ought to be, the front runner being WIMPs (weakly interacting massive particles). Despite a lot of theorising, the precise nature of dark matter is up for grabs.

The acceleration of the expansion of the universe is attributed to ‘dark energy’, which is little more than a name for ‘stuff that makes the expansion accelerate’ – though, to be fair, supplemented by detailed analyses of what kind of effect this stuff must have, and suggestions for what it might be. One possibility is Einstein’s cosmological constant.

Until recently, these three dei ex machina resolved most significant discrepancies between the naive Big Bang theory and increasingly sophisticated observations. The introduction of these three items of novel physics, all produced out of a hat and without much independent observational support (other than what they were invented to explain), could be justified pragmatically: they worked, and nothing else seemed to. But there is now a growing realisation that the first of those statements no longer holds, but unfortunately the second still does. A growing minority of cosmologists suspect that three dei ex machina is at least two too many for comfort.

It is now realised that if an inflaton field exists, it doesn’t conveniently switch on once and then cease to operate, which is assumed in the usual explanation of the structure of our universe. Instead, the inflaton field can swing into action anywhere, and at any time, repeatedly. This leads to a scenario called eternal inflation, with our region of the universe being just one inflated bubble in a bubble-bath of cosmic foam. A new period of inflation might start in your living room this afternoon, instantaneously blowing up your television set and the cat by a factor of 1078.

Another problem is that almost all inflationary universes fail to match ours, and if you restrict initial conditions to get the ones that do, then a non-inflationary universe that performs just as well is vastly more probable. According to Roger Penrose, suitable initial conditions not requiring inflation outnumber those for an inflationary universe by a factor of one googolplex – ten to the power ten to the power one hundred. So an explanation not involving inflation, although it requires an extraordinarily unlikely initial state, is massively more plausible than an explanation that does involve inflation.

A few mavericks have been devising alternatives to the standard model all along, but now mainstream cosmologists are also having to rethink the theory. There is no shortage of ideas. In some, there is no Big Bang; instead, there is a kind of revival of the steady-state universe, in which a suitably clumpy distribution of matter can survive for hundreds of billions of years, perhaps indefinitely. The redshift is not caused by expansion, but by gravity. Dark matter is not needed to explain rotation curves: instead, relativistic inertial dragging, in which rotating matter carries space along with it, might do the job.

Perhaps more radical is the proposal that either our theory of gravity, or our theory of motion, need to be modified slightly. In 2012 the particle physicist and Nobel prize-winner Martinus Veltman, when asked ‘Will super-symmetry explain dark matter?’, replied: ‘Of course it won’t. People have been looking for this stuff since the 1980s and are just talking ballyhoo. Isn’t it more likely that we don’t understand gravity all that well? Astrophysicists believe in Einstein’s theory of gravity with a fervour that is unbelievable. Do you know how much of Einstein’s theory has been tested at the distances of galaxies where we “see” dark matter? None of it.’fn2

The best known proposal here is MOND, Modified Newtonian Dynamics, suggested in 1983 by Mordehai Milgrom. The basic idea is that Newton’s second law of motion may not be valid for very small accelerations, so that acceleration is not proportional to the force of gravity when that force is very weak. There is a tendency to assume that MOND is the only alternative to general relativity; the correct statement is that it is the most extensively explored one. Robert Caldwell,fn3 in a special issue of a Royal Society journal devoted to cosmological tests of general relativity, wrote: ‘To date, it appears entirely reasonable that the observations may be explained by new laws of gravitation.’ In the same issue Ruth Durrerfn4 pointed out that the evidence for dark energy is weak: ‘Our single indication for the existence of dark energy comes from distance measurements and their relation to redshift.’ The rest of the evidence, she says, merely establishes that distances estimated by redshift measurements are larger than those expected from the standard cosmological model. Something unexpected is going on, but it might not be dark energy.

Our confidence that we know how our universe began is being shaken. Some modified version of the Big Bang may well be correct – but then again, maybe not. When new evidence comes along, scientists change their minds.

Though perhaps not quite yet.

fn1 Actually, Lemaître’s doesn’t, not in its original formulation. Instead of a point singularity at a finite time in the past, it has a hyperspherical singularity infinitely far in the past.

fn2 Martinus Veltman, coming to terms with the Higgs, Nature 490 (2012) S10-S11.

fn3 Robert R. Caldwell, A gravitational puzzle, Philosophical Transactions of the Royal Society of London A (2011) 369, 4998-5002.

fn4 Ruth Durrer, What do we really know about dark energy? Philosophical Transactions of the Royal Society of London A (2011) 369, 5102-5114.