Just Six Numbers

THE COSMOS AND THE MICROWORLD

Man is … related inextricably to all reality, known and unknowable … plankton, a shimmering phosphorescence on the sea and the spinning planets and an expanding universe, all bound together by the elastic string of time. It is advisable to look from the tide pool to the stars and then back to the tide pool again.

John Steinbeck, The Log from the Sea of Cortez

SIX NUMBERS

Mathematical laws underpin the fabric of our universe — not just atoms, but galaxies, stars and people. The properties of atoms — their sizes and masses, how many different kinds there are, and the forces linking them together — determine the chemistry of our everyday world. The very existence of atoms depends on forces and particles deep inside them. The objects that astronomers study — planets, stars and galaxies — are controlled by the force of gravity. And everything takes place in the arena of an expanding universe, whose properties were imprinted into it at the time of the initial Big Bang.

??? Science advances by discerning patterns and regularities in nature, so that more and more phenomena can be subsumed into general categories and laws. Theorists aim to encapsulate the essence of the physical laws in a unified set of equations, and a few numbers. There is still some way to go, but progress is remarkable.

??? This book describes six numbers that now seem especially significant. Two of them relate to the basic forces; two fix the size and overall `texture’ of our universe and determine whether it will continue for ever; and two more fix the properties of space itself:

? The cosmos is so vast because there is one crucially important huge number N in nature, equal to 1,000,000, 000,000,000,000,000,000,000,000,000,000. This number measures the strength of the electrical forces that hold atoms together, divided by the force of gravity between them. If N had a few less zeros, only a short-lived miniature universe could exist: no creatures could grow larger than insects, and there would be no time for biological evolution.

? Another number, [Epsilon], whose value is 0.007, defines how firmly atomic nuclei bind together and how all the atoms on Earth were made. Its value controls the power from the Sun and, more sensitively, how stars transmute hydrogen into all the atoms of the periodic table. Carbon and oxygen are common, whereas gold and uranium are rare, because of what happens in the stars. If [Epsilon] were 0.006 or 0.008, we could not exist.

? The cosmic number [Omega] (omega) measures the amount of material in our universe — galaxies, diffuse gas, and `dark matter’. [Omega] tells us the relative importance of gravity and expansion energy in the universe. If this ratio were too high relative to a particular `critical’ value, the universe would have collapsed long ago; had it been too low, no galaxies or stars would have formed. The initial expansion speed seems to have been finely tuned.

? Measuring the fourth number, [Lambda] (lambda), was the biggest scientific news of 1998. An unsuspected new force — a cosmic `antigravity’ — controls the expansion of our universe, even though it has no discernible effect on scales less than a billion light-years. It is destined to become ever more dominant over gravity and other forces as our universe becomes ever darker and emptier. Fortunately for us (and very surprisingly to theorists), [Lambda] is very small. Otherwise its effect would have stopped galaxies and stars from forming, and cosmic evolution would have been stifled before it could even begin.

? The seeds for all cosmic structures — stars, galaxies and clusters of galaxies — were all imprinted in the Big Bang. The fabric of our universe depends on one number, Q which represents the ratio of two fundamental energies and is about 1/100,000 in value. If Q were even smaller, the universe would be inert and structureless; if Q were much larger, it would be a violent place, in which no stars or solar systems could survive, dominated by vast black holes.

? The sixth crucial number has been known for centuries, although it’s now viewed in a new perspective. It is the number of spatial dimensions in our world, D, and equals three. Life couldn’t exist if D were two or four. Time is a fourth dimension, but distinctively different from the others in that it has a built-in arrow: we `move’ only towards the future. Near black holes, space is so warped that light moves in circles, and time can stand still. Furthermore, close to the time of the Big Bang, and also on microscopic scales, space may reveal its deepest underlying structure of all: the vibrations and harmonies of objects called `superstrings’, in a ten-dimensional arena.

Perhaps there are some connections between these numbers. At the moment, however, we cannot predict any one of them from the values of the others. Nor do we know whether some `theory of everything’ will eventually yield a formula that interrelates them, or that specifies them uniquely. I have highlighted these six because each plays a crucial and distinctive role in our universe, and together they determine how the universe evolves and what its internal potentialities are; moreover, three of them (those that pertain to the large-scale universe) are only now being measured with any precision.

??? These six numbers constitute a `recipe’ for a universe. Moreover, the outcome is sensitive to their values: if any one of them were to be `untuned’, there would be no stars and no life. Is this tuning just a brute fact, a coincidence? Or is it the providence of a benign Creator? I take the view that it is neither. An infinity of other universes may well exist where the numbers are different. Most would be stillborn or sterile. We could only have emerged (and therefore we naturally now find ourselves) in a universe with the `right’ combination. This realization offers a radically new perspective on our universe, on our place in it, and on the nature of physical laws.

??? It is astonishing that an expanding universe, whose starting point is so `simple’ that it can be specified by just a few numbers, can evolve (if these numbers are suitable `tuned’) into our intricately structured cosmos. Let us first set the scene by viewing these structures on all scales, from atoms to galaxies.

THE COSMOS THROUGH A ZOOM LENS

Start with a commonplace `snapshot’ — a man and woman — taken from a distance of a few metres. Then imagine the same scene from successively more remote viewpoints, each ten times further away than the previous one. The second frame shows the patch of grass on which they are reclining; the third shows that they are in a public park; the fourth reveals some tall buildings; the next shows the whole city; and the next-but-one a segment of the Earth’s horizon, viewed from so high up that it is noticeably curved. Two frames further on, we encounter a powerful image that has been familiar since the 1960s: the entire Earth — continents, oceans, and clouds — with its biosphere seeming no more than a delicate glaze and contrasting with the arid features of its Moon.

??? Three more leaps show the inner Solar System, with the Earth orbiting the Sun further out than Mercury and Venus; the next shows the entire Solar System. Four frames on (a view from a few light-years away), our Sun looks like a star among its neighbours. After three more frames, we see the billions of similar stars in the flat disc of our Milky Way, stretching for tens of thousands of light-years. Three more leaps reveal the Milky Way as a spiral galaxy, along with Andromeda. From still further, these galaxies seem just two among hundreds of others — outlying members of the Virgo Cluster of galaxies. A further leap shows that the Virgo Cluster is itself just one rather modest cluster. Even if our imaginary telephoto lens had the power of the Hubble Space Telescope, our entire galaxy would, in the final frame, be a barely detectable smudge of light several billion light-years distant.

??? The series ends there. Our horizon extends no further, but it has taken twenty-five leaps, each by a factor of ten, to reach the limits of our observable universe starting with the `human’ scale of a few metres.

??? The other set of frames zooms inward rather than outward. From less than one metre, we see an arm; from a few centimetres — as close as we can look with the unaided eye — a small patch of skin. The next frames take us into the fine textures of human tissue, and then into an individual cell (there are a hundred times more cells in our body than there are stars in our galaxy). And then, at the limits of a powerful microscope, we probe the realm of individual molecules: long, tangled strings of proteins, and the double helix of DNA.

??? The next `zoom’ reveals individual atoms. Here the fuzziness of quantum effects comes in: there is a limit to the sharpness of the pictures we can get. No real microscope can probe within the atom, where a swarm of electrons surrounds the positively charged nucleus, but substructures one hundred times smaller than atomic nuclei can be probed by studying what happens when other particles, accelerated to speeds approaching that of light, are crashed into them. This is the finest detail that we can directly measure; we suspect, however, that the underlying structures in nature may be `superstrings’ or `quantum foam’ on scales so tiny that they would require seventeen more zooms to reveal them.

??? Our telescopes reach out to a distance that is bigger than a superstring (the smallest substructure postulated to exist within atoms) by a sixty-figure number: there would be sixty frames (of which present measurements cover forty-three) in our `zoom lens’ depiction of the natural world. Of these, our ordinary experience spans nine at most — from the smallest things our eyes can see, about a millimetre in size, to the distance logged on an intercontinental flight. This highlights something important and remarkable, which is so obvious that we take it for granted: our universe covers a vast range of scales, and an immense variety of structures, stretching far larger, and far smaller, than the dimensions of everyday sensations.

LARGE NUMBERS AND DIVERSE SCALES

We are each made up of between [10.sup.28] and [10.sup.29] atoms. This `human scale’ is, in a numerical sense, poised midway between the masses of atoms and stars. It would take roughly as many human bodies to make up the mass of the Sun as there are atoms in each of us. But our Sun is just an ordinary star in the galaxy that contains a hundred billion stars altogether. There are at least as many galaxies in our observable universe as there are stars in a galaxy. More than [10.sup.78] atoms lie within range of our telescope.

??? Living organisms are configured into layer upon layer of complex structure. Atoms are assembled into complex molecules; these react, via complex pathways in every cell, and indirectly lead to the entire interconnected structure that makes up a tree, an insect or a human. We straddle the cosmos and the microworld — intermediate in size between the Sun, at a billion metres in diameter, and a molecule at a billionth of a metre. It is actually no coincidence that nature attains its maximum complexity on this intermediate scale: anything larger, if it were on a habitable planet, would be vulnerable to breakage or crushing by gravity.

??? We are used to the idea that we are moulded by the microworld: we are vulnerable to viruses a millionth of a metre in length, and the minute DNA double-helix molecule encodes our total genetic heritage. And it’s just as obvious that we depend on the Sun and its power. But what about the still vaster scales? Even the nearest stars are millions of times further away than the Sun, and the known cosmos extends a billion times further still. Can we understand why there is so much beyond our Solar System? In this book I shall describe several ways in which we are linked to the stars, arguing that we cannot understand our origins without the cosmic context.

??? The intimate connections between the `inner space’ of the subatomic world and the `outer space’ of the cosmos are illustrated by the picture in Figure 1.1 — an ouraborus , described by Encyclopaedia Britannica as the ’emblematic serpent of ancient Egypt and Greece, represented with its tail in its mouth continually devouring itself and being reborn from itself … [It] expresses the unity of all things, material and spiritual, which never disappear but perpetually change form in an eternal cycle of destruction and re-creation’.

??? On the left in the illustration are the atoms and subatomic particles; this is the `quantum world’. On the right are planets, stars and galaxies. This book will highlight some remarkable interconnections between the microscales on the left and the macroworld on the right. Our everyday world is determined by atoms and how they combine together into molecules, minerals and living cells. The way stars shine depends on the nuclei within those atoms. Galaxies may be held together by the gravity of a huge swarm of subnuclear particles. Symbolized `gastronomically’ at the top, is the ultimate synthesis that still eludes us — between the cosmos and the quantum.

??? Lengths spanning sixty powers of ten are depicted in the ouraborus . Such an enormous range is actually a prerequisite for an `interesting’ universe. A universe that didn’t involve large numbers could never evolve a complex hierarchy of structures: it would be dull, and certainly not habitable. And there must be long timespans as well. Processes in an atom may take a millionth of a billionth of a second to be completed; within the central nucleus of each atom, events are even faster. The complex processes that transform an embryo into blood, bone and flesh involve a succession of cell divisions, coupled with differentiation, each involving thousands of intricately orchestrated regroupings and replications of molecules; this activity never ceases as long as we eat and breathe. And our life is just one generation in humankind’s evolution, an episode that is itself just one stage in the emergence of the totality of life.

??? The tremendous timespans involved in evolution offer a new perspective on the question `Why is our universe so big?’ The emergence of human life here on Earth has taken 4.5 billion years. Even before our Sun and its planets could form, earlier stars must have transmuted pristine hydrogen into carbon, oxygen and the other atoms of the periodic table. This has taken about ten billion years. The size of the observable universe is, roughly, the distance travelled by light since the Big Bang, and so the present visible universe must be around ten billion light-years across.

??? This is a startling conclusion. The very hugeness of our universe, which seems at first to signify how unimportant we are in the cosmic scheme, is actually entailed by our existence! This is not to say that there couldn’t have been a smaller universe, only that we could not have existed in it. The expanse of cosmic space is not an extravagant superfluity; it’s a consequence of the prolonged chain of events, extending back before our Solar System formed, that preceded our arrival on the scene.

??? This may seem a regression to an ancient `anthropocentric’ perspective — something that was shattered by Copernicus’s revelation that the Earth moves around the Sun rather than vice versa. But we shouldn’t take Copernican modesty (sometimes called the `principle of mediocrity’) too far. Creatures like us require special conditions to have evolved, so our perspective is bound to be in some sense atypical. The vastness of our universe shouldn’t surprise us, even though we may still seek a deeper explanation for its distinctive features.

CAN WE HOPE TO UNDERSTAND OUR UNIVERSE?

The physicist Max Born claimed that theories are never abandoned until their proponents are all dead — that science advances `funeral by funeral’. But that’s too cynical. Several long-running cosmological debates have now been settled; some earlier issues are no longer controversial. Many of us have often changed our minds — I certainly have. Indeed, this book presents a story that I would once myself have thought surprising. The cosmic perspective I’ll describe is widely shared, even though many would not go the whole way with my interpretation.

??? Cosmological ideas are no longer any more fragile and evanescent than our theories about the history of our own Earth. Geologists infer that the continents are drifting over the globe, about as fast as your fingernails grow, and that Europe and North America were joined together 200 million years ago. We believe them, even though such vast spans of time are hard to grasp. We also believe, at least in outline, the story of how our biosphere evolved and how we humans emerged. But some key features of our cosmic environment are now underpinned by equally firm data. The empirical support for a Big Bang ten to fifteen billion years ago is as compelling as the evidence that geologists offer on our Earth’s history. This is an astonishing turnaround: our ancestors could weave theories almost unencumbered by facts, and until quite recently cosmology seemed little more than speculative mathematics.

??? A few years ago, I already had ninety per cent confidence that there was indeed a Big Bang — that everything in our observable universe started as a compressed fireball, far hotter than the centre of the Sun. The case now is far stronger: dramatic advances in observations and experiments have brought the broad cosmic picture into sharp focus during the 1990s, and I would now raise my degree of certainty to ninety-nine per cent.

??? `The most incomprehensible thing about the universe is that it is comprehensible’ is one of Einstein’s best-known aphorisms, expressing his amazement that the laws of physics, which our minds are somehow attuned to understand, apply not just here on Earth but also in the remotest galaxy. Newton taught us that the same force that makes apples fall holds the Moon and planets in their courses. We now know that this same force binds the galaxies, pulls some stars into black holes, and may eventually cause the Andromeda galaxy to collapse on top of us. Atoms in the most distant galaxies are identical to those we can study in our laboratories. All parts of the universe seem to be evolving in a similar way, as though they shared a common origin. Without this uniformity, cosmology would have got nowhere.

??? Recent advances bring into focus new mysteries about the origin of our universe, the laws governing it, and even its eventual fate. These pertain to the first tiny fraction of a second after the Big Bang, when conditions were so extreme that the relevant physics isn’t understood — where we wonder about the nature of time, the number of dimensions, and the origin of matter. In this initial instant, everything was squeezed to such immense densities that (as symbolized in the ouraborus ) the problems of the cosmos and the microworld overlap.

??? Space can’t be indefinitely divided. The details are still mysterious, but most physicists suspect that there is some kind of granularity on a scale of [10.sup.-33] centimetres. This is twenty powers often smaller than an atomic nucleus: as big a decrease — as many frames in our `zoom lens’ depiction — as the increase in scale from an atomic nucleus to a major city. We then encounter a barrier: even if there were still tinier structures, they would transcend our concepts of space and time.

??? What about the largest scales? Are there domains whose light has not yet had time to reach us in the ten billion years or so since the Big Bang? We plainly have no direct evidence. However, there are no theoretical bounds on the extent of our universe (in space, and in future time), and on what may come into view in the remote future — indeed, it may stretch not just millions of times further than our currently observable domain, but millions of powers of ten further. And even that isn’t all. Our universe, extending immensely far beyond our present horizon, may itself be just one member of a possibly infinite ensemble. This `multiverse’ concept, though speculative, is a natural extension of current cosmological theories, which gain credence because they account for things that we do observe. The physical laws and geometry could be different in other universes, and this offers a new perspective on the seemingly special values that the six numbers take in ours.

Copyright ? 2000 Martin Rees. All rights reserved.