How the fact that lots of information is needed to describe the world tells us chance played a key role in creating everyday reality
‘Information is a revolutionary new kind of concept and the recognition of this fact is one of the milestones of this age.’
Gregory Chaitin (The Unknowable)
‘We profess ourselves to be the slaves of chance.’
William Shakespeare (The Winter’s Tale)
The world is complex. Rain clouds scud across the sky. A tree sways gently in the breeze. A woman in a red coat walks her cream poodle down the street and stops at a pedestrian crossing. Describing such a scene precisely requires a vast amount of information. It is necessary, for instance, to specify the location, shape and composition of every cloud, the location and shape of every branch and leaf on the tree, and so on. The reason it takes so much information is that a lot of things have to be specified in order to ensure that the scene is uniquely distinguishable from myriad other possibilities. This is because there are an awful lot of ways the scene could be different, an awful lot of alternative ways its ‘stuff ’ could be arranged. A cloud could be in another place, a lamp post could substitute for the tree, a man walking his pet ferret could replace the woman and the dog. In fact, to be sure the scene cannot be mistaken for any other possible scene, it is necessary to specify the location and properties of every single atom in the scene. Every subatomic particle even.
The observation that a vast amount of information is needed to describe the Universe may seem trite and of little consequence. But actually it is telling us something profound about our Universe. According to physicist Stephen Hsu, of the University of Oregon in Eugene, it is telling us that the world around us is the way we find it and not some other way because of pure chance. It is telling us that the complexity of the world is the outcome of a long series of rolls of the dice extending all the way back to the beginning of time. Einstein famously declared that ‘God does not play dice with the universe.’ But, says Hsu, ‘Not only does God play dice with the universe but, if he did not, there would be no universe – at least, not one of the richness and complexity for life to have arisen.’1
How is it possible to come to such a startling conclusion merely from the fact that the Universe is complex and so requires a lot of information to describe it? Well, strictly speaking, it isn’t. Such a conclusion is possible only by comparing how much information there is in today’s Universe with how much there was when the Universe was born.2
According to the standard picture of cosmology, the Universe, with all its billions upon billions of galaxies and stars, inflated from a tiny piece of ‘vacuum’ far smaller than an atom. Estimating the information content of this pre-inflation patch is technical, but Hsu uses the following simplified argument. Before inflation, when the Universe was about 10–44 seconds old, the four fundamental forces of nature are believed to have been ‘unified’ into a single ‘super-force’. The Universe at this epoch was at the ‘Planck temperature’ – about 1032 degrees – and was no larger than the ‘Planck length’ – about 10–35 metres. Lacking as they do a quantum theory of the force of gravity, physicists can know nothing about this time. However, by the time the Universe had expanded to ten times the Planck length and its temperature had fallen to a tenth of the Planck temperature, the effects of quantum gravity were already minimal. It is therefore possible to say something about this time. The Universe can be imagined as made up of 1,000 cubes, each with sides of the Planck length. The reason for thinking this is that the Planck length is the minimum possible length, akin to the dot size on a newspaper photograph. As already pointed out, the information content of something is related to the number of distinguishable ways that the subcomponents of the thing can be arranged. So the key question is: how many distinguishable states were there in this 1,000-cube universe?
Each of the small cubes could be either filled with energy or be empty, just as a location on a photograph can be filled with black ink or be empty. A universe made of 1,000 filled or unfilled cubes is not easy to visualise, so think of a one-cube universe. The cube could be either filled or empty, making two distinguishable states. In a two-cube universe, the cubes could be empty–empty, empty–full, full–empty or full–full, making four possible arrangements, which is equivalent to 22. In a three-cube universe, there are 23 arrangements. In a four-cube universe, 24. See the pattern? It is therefore possible to say that in the 1,000-cube universe that existed prior to inflation, the number of distinguishable ways the vacuum could have been arranged was 21,000.
21,000 – 2 multiplied by itself 1,000 times – is approximately a billion. It may seem like a lot of possible arrangements for the pre-inflationary patch of vacuum, but actually it is ridiculously small. Think of a computer disk. When people say a disk can store N bits of data, they mean the disk can store 2N different strings of 0s and 1s, or binary digits (bits).3 Listing all the possible ways that the stuff of the pre-inflationary patch could have been arranged therefore requires a mere 1,000 bits of storage. In other words, the precise state of the early Universe would fill only a kilobit of disk space. A byte – 8 bits – is usually used to store a character such as an ‘A’ or a ‘5’, so specifying a whole universe requires less than 200 bytes. Imagine being given a piece of paper with 200 characters, or about 30 words, scrawled across it. Incredibly, this is sufficient to specify the state of an entire universe. If this is not mind-blowing enough, think of it another way. In Song of Myself, Walt Whitman wrote: ‘And I say to any man or woman, Let your soul stand cool and composed before a million universes.’ Well, today it is easy to ‘stand cool and composed before a million universes’. Just buy a 1 gigabit (Gb) key-ring flash memory. Believe it or not, you could store the information for a million universes on it.
The idea that the 21,000 possible arrangements of the pre-inflation vacuum can be stored in 1,000 bits highlights the definition of information. If the number of possible arrangements of something is 2N, then the information content is defined as N.4
So much for the piddling information content of the pre-inflation Universe – but how much information is there in today’s Universe? In calculating the figure, the key thing is to recognise where most of that information resides. Even if we possessed a super-telescope that could look out at the Universe and record the precise state of every atom in every star in every galaxy, there would still be an overwhelmingly large number of ways the Universe could be different from our specification. The reason is that space is permeated by a vast number of photons from the Big Bang – the leftover heat from the primordial fireball. The photons of the Big Bang outnumber microscopic particles of matter such as electrons by a factor of about 10 billion and even photons of starlight by a factor of 1,000. Their precise state is therefore the biggest unknown in the Universe.
The Big Bang photons are indistinguishable, so swapping them does not result in a new arrangement. However, a photon has two distinct states open to it because it can be ‘polarised’ in two different ways. Think of a photon flying through space as corkscrewing either clockwise or anticlockwise about its direction of motion. If we imagine each photon as traversing its own cube of space, then the situation is pretty similar to the pre-inflation vacuum, with each cube containing either a clockwise photon or an anticlockwise one rather than being either filled with energy or empty. The same result, therefore, holds. The number of distinguishable states of N Big Bang photons is 2N,5 so the information required to describe the states of all the Big Bang photons in the Universe is simply N, the total number of those photons. At any moment every cubic centimetre of space is being traversed by about 300 relic photons – that’s how ubiquitous the photons of the Big Bang are. The Universe is about 84 billion light years across, which means it has a volume of about 5 × 1086 cubic centimetres.6 Consequently, the amount of information required to describe it is about 1089 bits.
In summary, the Universe started out containing only 1,000 bits of information but now contains 1089 bits. That is an increase of 1086, or 100 trillion trillion trillion trillion trillion trillion trillion, times. It may not be obvious that this extraordinary increase in information is a puzzle, but it is. To understand why, it is necessary to understand something about the laws of physics.
The well-known laws of physics such as Newton’s laws of motion are recipes for predicting the future with 100 per cent certainty. For instance, if we know the location of the Moon today, by applying Newton’s laws of motion and his law of gravity we can predict the location of the Moon tomorrow. Since knowing the location of the Moon yesterday is all that is needed to determine its location tomorrow, it follows that no new information is added. The location tomorrow is contained within the location today. And this is true of all the non-quantum, or ‘deterministic’, laws of physics. Since a single state of the system in the present completely determines a single state of the system in the future, no new information is created. In fact, deterministic laws are synonymous with the conservation of information.7
The relevance of this to the Universe is that its evolution is described by Einstein’s theory of gravity – the general theory of relativity – which is a deterministic theory. In other words, general relativity, when applied to the whole Universe, describes how a particular state of the Universe evolves into another state at a later time. There is no change in the information content of the Universe.
What, then, are we to make of the fact that the pre-inflation Universe contained only 1,000 bits of information and today’s Universe contains 1089? In other words, there were only 21,000 distinguishable ways the stuff of the Universe could have been arranged at the start yet there are a mind-numbing 2(10^89) ways it could be arranged now?8 Since a non-quantum, or ‘classical’, theory like general relativity permits 21,000 states at a particular time only to evolve into 21,000 states at a later time, it can only mean that the complexity of the Universe today cannot have been determined at the beginning of time. ‘Take the books on my bookshelf,’ says Hsu. ‘Why is my copy of The Feynman Lectures in Physics next to my copy of A Brief History of Time? The information argument is saying that the state of my bookshelf cannot be traced back to a unique state in the Big Bang. In fact, almost nothing in our Universe can be explained this way.’
Think of that leaf fluttering on a tree. What causes it to flutter? The wind, of course. But what causes the wind? Heat dumped into the atmosphere by sunlight. But what causes sunlight? Heat generated in the Sun’s core by nuclear reactions … It may seem that such a chain of cause and effect can be followed all the way back to the birth of the Universe. But the fact that there is vastly more information in today’s Universe than at the beginning is telling us that this is not true. Eventually, if the chain of cause and effect is followed back far enough, there will come an effect without a prior cause. Something which happened for no reason at all. An event which was utterly random.
And this is the clue to where all the information in today’s Universe has come from. Randomness is synonymous with information. This is far from obvious. In fact, at first sight, it appears counter-intuitive. However, imagine there is a 100-digit number whose digits are random. The only way you can communicate it to someone else is to send all 100 digits. Contrast this with a number that consists of a string of a hundred 3s. You can communicate this by exploiting the pattern and simply saying ‘3 repeated 100 times’. This shows that a random number contains a lot of information, whereas a non-random number contains very little. A lot is redundant.
So what processes have been responsible for injecting the information/randomness into the Universe since the beginning of inflation? Hsu is in no doubt. ‘Those processes can only be quantum processes,’ he says. Quantum processes are non-deterministic. A unique state of a system in the past does not lead to a unique state in the future. The laws of quantum physics are not a recipe for predicting the future with 100 per cent certainty. They are a recipe for predicting myriad possible futures, each of which may happen with a particular probability. ‘Things are happening in the world around us today because of countless bursts of randomness injected into the Universe since the Big Bang – because of the roll of a quantum dice,’ says Hsu.9
Hsu believes that the principal process that injected randomness into the Universe was inflation itself. No one knows what drove it, although physicists often talk about an ‘inflaton’ field, some kind of ‘stuff ’ which pervaded the Universe and whose repulsive gravity made the vacuum balloon enormously in size. During inflation, the Universe doubled in size, and doubled in size again more than 60 times over.10
At some point – and nobody knows why – inflation ran out of steam. The inflaton field decayed, leaving the vacuum as the normal vacuum we see around us today.11 The key characteristic of this decay was that it was quantum – that is to say, random. This means that the inflaton decayed at slightly different times in different locations and dumped a different amount of energy in different locations. Since energy can neither be created nor destroyed, merely changed from one form into another, the energy of the inflaton manifested itself in other forms. Those forms were the mass-energy of subatomic particles and their energy of motion. In short, the decay of the inflaton made the matter of the Universe and simultaneously heated it up to a blisteringly high temperature. It created the ‘hot’ Big Bang.
So in the standard picture of cosmology, the Universe starts with just vacuum. The vacuum is in an unusually energetic state, which mushrooms in size wildly. The more vacuum that is created, the more vacuum energy there is. Inflation, as pointed out by many physicists, is the ‘ultimate free lunch’. Finally, inflation ends and the energy of all the newly created vacuum heats up the Universe and creates the fireball of the Big Bang. Because the energy dumped into every location of the Universe by the decay of the inflaton was different, the temperature in each place was different too. ‘The decay of the inflaton was like a random number generator, injecting a fantastic amount of randomness across the length and breadth of the Universe,’ says Hsu. ‘This quantum randomness is the reason my The Feynman Lectures in Physics is next to A Brief History of Time.’
The Universe has continued to expand since the end of inflation, creating more vacuum and more vacuum energy along with it, so you might think this would have injected more randomness into the Universe. However, the key difference between the expanding vacuum today and the inflationary vacuum is that the former has remained stable against decay. And it is only the decay of the vacuum – an inherently quantum process – that unleashes randomness into the wider Universe.12
Hsu does not believe, however, that the decay of the inflaton was the only process that injected randomness/information into the Universe. He believes that since the end of inflation, information has been continuously injected into the Universe by countless quantum events such as the random disintegration of atomic nuclei and the random emission of photons by atoms.
Recall, for instance, that the ‘spin’ of an electron may be clockwise or anticlockwise. Before it is recorded by some kind of detector, its spin is undefined. There is nothing to describe, no information. However, once the electron makes its mark on a detector – in the jargon, ‘decoheres’ – it is found to be either spinning clockwise or anticlockwise. There are two possible outcomes, which takes a single bit to describe. Where once there was no information, now there is some. Imagine ten spinning electrons in a row. If they impress themselves on some kind of detector, suddenly some are spinning clockwise and others anticlockwise. There are 210 possible outcomes open to the ten electrons – more than 1,000 possibilities. So a whole load of information has been injected into the Universe just by these ten electrons registering their presence.
Just imagine how much information can be injected into the Universe by the trillions upon trillions of subatomic particles registering their presence. ‘It’s a tremendously powerful way of injecting information into the Universe,’ says Hsu.
How is the increase of information in the Universe compatible with the remorseless increase of entropy in the Universe, as decreed by the second law of thermodynamics? Well, entropy and information turn out to be intimately connected: entropy = e(information). The nineteenth-century physicists recognised that entropy increased because the number of disordered states open to atoms and their like overwhelmingly outnumbered the ordered states available. ‘Although they did not know it, those states are in fact quantum states such as the state of an electron spinning in an atom,’ says Hsu. ‘Entropy increases because the disordered quantum states open to subatomic particles overwhelmingly outnumber the ordered ones. Everything fits.’
Einstein, as pointed out before, considered quantum processes – random and without cause – utterly abhorrent. However, according to Hsu, they are far from abhorrent. They are absolutely essential. We owe our existence here today to quantum unpredictability. Look around you – at a rose, a newborn baby, a plane riding a vapour trail across the blue sky. We live in a world of boundless complexity. But all the complexity you see is merely the result of a long sequence of quantum coin tosses since the end of inflation. Like it or not, we live in a random reality.
1. The complexity of the everyday world is indeed due to the fact that there are 92 types of atomic building blocks rather than just one, as pointed out in Chapter 3. However, as with many things in science, there is a deeper level of explanation. And that is what we are talking about here – the ultimate source of the complexity of the Universe.
2. Notice the switch from talking about the information needed to describe the Universe to the information contained in the Universe. The two statements are equivalent. They reflect the growing suspicion among physicists that information is a fundamental ‘thing’, underpinning all of physics.
3. Binary was invented by the seventeenth-century mathematician Gottfried Leibniz. It is a way of representing numbers as a string of 0s and 1s. Usually, we use decimal, or base 10. The right-hand digit represents the 1s, the next digit the 10s, the next the 10 × 10s, and so on. So, for instance, 9217 means 7 + 1 × 10 + 2 × (10 × 10) + 9 × (10 × 10 × 10). In binary, or base 2, the right-hand digit represents the 1s, the next digit the 2s, the next the 2 × 2s, and so on. So, for instance, 1101 means 1 + 0 × 2 + 1 × (2 × 2) + 1 × (2 × 2 × 2), which in decimal is 13.
4. Usually, the number of distinct arrangements of the subcomponents of a system is defined as e(information content) rather than 2(information content), with e, one of the most famous constants in maths, being 2.718281828 … The difference is not important since all the figures in this chapter are rough, ‘order of magnitude’ estimates.
5. Actually, it is a bit more complicated than this and the number of possible arrangements of N thermal photons is eN. But eN is approximately the same as 2N.
6. The reason our 13.7 billion-year-old Universe is 84 billion light years across and not 13.7 × 2 billion light years across, as might naively be expected, is that during inflation, it expanded faster than the speed of light. See Chapter 7.
7. The conservation of information is behind the black hole ‘information paradox’, highlighted by Stephen Hawking in 1976. A black hole, it turns out, is not completely black but shines with ‘Hawking radiation’, which allows it to ‘evaporate’ and eventually disappear. A paradox arises because the Hawking radiation can carry no information about the interior of the black hole, since, by definition, nothing can escape from one. So when the black hole has gone, there remains the puzzle of what happened to the information that described the dying star whose catastrophic shrinkage led to the creation of the black hole in the first place. The strong suspicion now is that it goes into creating myriad tiny bumps on the ‘event horizon’, the imaginary surface – or point of no return for in-falling matter – that surrounds the black hole. This means that the information that described the precursor star – a three-dimensional body – is encoded in the two-dimensional event horizon. The event horizon is like a hologram. The implications of this for the Universe are fascinating because it too is surrounded by a horizon – a horizon in time rather than space, but a horizon nonetheless. It appears that the three-dimensional Universe can contain no more information than can be impressed on the two-dimensional surface that surrounds it. Remarkably, it seems we are living in a giant cosmic hologram.
8.2(10^89) is 2 multiplied by itself (10 × 10 × 10 …) times, where there are 89 tens inside those brackets. Or, to put it another way, 2(10^89) is approximately (103)89 = 10267.
9. To some extent this was anticipated by the great British physicist Paul Dirac in 1939. Although he knew nothing of inflation and how ridiculously small the Universe had been at the outset, he nevertheless realised that if the Universe was expanding, as the observations of galaxies indicated, it would have been far smaller in the past, which meant it would have been too simple to seed the complexity we see around us today. At least it would be too simple if classical physics described the Universe. Dirac realised, however, that quantum theory might come to the rescue and that unpredictable quantum jumps in the early Universe might be the origin of the Universe’s complexity. By recognising the role of quantum theory in the origin of the Universe, Dirac anticipated by four decades the field of quantum cosmology. See The Strangest Man: The Hidden Life of Paul Dirac, Quantum Genius by Graham Farmelo (Faber & Faber, 2009).
10. See Chapter 8.
11. Bizarrely, the discovery of the ‘dark energy’ in 1998 shows that in the past few billion years, the normal vacuum has changed back into an inflationary-type vacuum, though with a tiny, tiny fraction of the energy that drove inflation. Dark energy, like the inflationary vacuum, is speeding up the expansion of the Universe. Nobody knows whether there is any connection between the inflationary phase and the current dark-energy-driven phase, though if there is, two mysteries would be reduced to one.
12. For a long time the vacuum energy was exactly zero, so it contained no energy to dump into other forms even if it were possible for it to decay. However, in the past few billion years, with the arrival on the cosmic stage of the dark energy, everything has changed. Since nobody knows why the dark energy switched on in the first place, it is always possible that one day it will switch off. However, since the dark energy is so small compared to the vacuum energy that drove inflation, the information injected into the Universe by such a decay will be correspondingly smaller.
Further reading:
‘Information, Information Processing and Gravity’ by Stephen Hsu (http://arxiv.org/abs/0704.1154).