# Inflation for beginners

Another old essay.  Still accurate as far as the history goes, although there have been further developments in the 21st century.

INFLATION has become a cosmological buzzword in the 1990s. No self-respecting theory of the Universe is complete without a reference to inflation — and at the same time there is now a bewildering variety of different versions of inflation to choose from. Clearly, what’s needed is a beginner’s guide to inflation, where newcomers to cosmology can find out just what this exciting development is all about. This is it — new readers start here.

The reason why something like inflation was needed in cosmology was highlighted by discussions of two key problems in the 1970s. The first of these is the horizon problem — the puzzle that the Universe looks the same on opposite sides of the sky (opposite horizons) even though there has not been time since the Big Bang for light (or anything else) to travel across the Universe and back. So how do the opposite horizons “know” how to keep in step with each other? The second puzzle is called the flatness problem This is the puzzle that the spacetime of the Universe is very nearly flat, which means that the Universe sits just on the dividing line between eternal expansion and eventual recollapse.

The flatness problem can be understood in terms of the density of the Universe. The density parameter is a measure of the amount of gravitating material in the Universe, usually denoted by the Greek letter omega (O), and also known as the flatness parameter. It is defined in such a way that if spacetime is exactly flat then O = 1. Before the development of the idea of inflation, one of the great puzzles in cosmology was the fact that the actual density of the Universe today is very close to this critical value — certainly within a factor of 10. This is curious because as the Universe expands away from the Big Bang the expansion will push the density parameter away from the critical value.

If the Universe starts out with the parameter less than one, O gets smaller as the Universe ages, while if it starts out bigger than one O gets bigger as the Universe ages. The fact that O is between 0.1 and 1 today means that in the first second of the Big Bang it was precisely 1 to within 1 part in 1060). This makes the value of the density parameter in the beginning one of the most precisely determined numbers in all of science, and the natural inference is that the value is, and always has been, exactly 1. One important implication of this is that there must be a large amount of dark matter in the Universe. Another is that the Universe was made flat by inflation.

Inflation is a general term for models of the very early Universe which involve a short period of extremely rapid (exponential) expansion, blowing the size of what is now the observable Universe up from a region far smaller than a proton to about the size of a grapefruit (or even bigger) in a small fraction of a second. This process would smooth out spacetime to make the Universe flat, and would also resolve the horizon problem by taking regions of space that were once close enough to have got to know each other well and spreading them far apart, on opposite sides of the visible Universe today.

Inflation became established as the standard model of the very early Universe in the 1980s. It achieved this success not only because it resolves many puzzles about the nature of the Universe, but because it did so using the grand unified theories (GUTs) and understanding of quantum theory developed by particle physicists completely independently of any cosmological studies. These theories of the particle world had been developed with no thought that they might be applied in cosmology (they were in no sense “designed” to tackle all the problems they turned out to solve), and their success in this area suggested to many people that they must be telling us something of fundamental importance about the Universe.

The marriage of particle physics (the study of the very small) and cosmology (the study of the very large) seems to provide an explanation of how the Universe began, and how it got to be the way it is. Inflation is therefore regarded as the most important development in cosmological thinking since the discovery that the Universe is expanding first suggested that it began in a Big Bang.

Taken at face value, the observed expansion of the Universe implies that it was born out of a singularity, a point of infinite density, some 15 billion years ago (cosmologists still disagree about exactly how old the Universe is, but the exact age doesn’t affect the argument). Quantum physics tells us that it is meaningless to talk in quite such extreme terms, and that instead we should consider the expansion as having started from a region no bigger across than the so-called Planck length (10-35m), when the density was not infinite but “only” some 1094 grams per cubic centimetre. These are the absolute limits on size and density allowed by quantum physics.

On that picture, the first puzzle is how anything that dense could ever expand — it would have an enormously strong gravitational field, turning it into a black hole and snuffing it out of existence (back into the singularity) as soon as it was born. But it turns out that inflation can prevent this happening, while quantum physics allows the entire Universe to appear, in this supercompact form, out of nothing at all, as a cosmic free lunch. The idea that the Universe may have appeared out of nothing at all, and contains zero energy overall, was developed by Edward Tryon, of the City University in New York, who suggested in the 1970s, that it might have appeared out of nothing as a so-called vacuum fluctuation, allowed by quantum theory.

Quantum uncertainty allows the temporary creation of bubbles of energy, or pairs of particles (such as electron-positron pairs) out of nothing, provided that they disappear in a short time. The less energy is involved, the longer the bubble can exist. Curiously, the energy in a gravitational field is negative, while the energy locked up in matter is positive. If the Universe is exactly flat , then as Tryon pointed out the two numbers cancel out, and the overall energy of the Universe is precisely zero. In that case, the quantum rules allow it to last forever. If you find this mind-blowing, you are in good company. George Gamow told in his book My World Line (Viking, New York, reprinted 1970) how he was having a conversation with Albert Einstein while walking through Princeton in the 1940s. Gamow casually mentioned that one of his colleagues had pointed out to him that according to Einstein’s equations a star could be created out of nothing at all, because its negative gravitational energy precisely cancels out its positive mass energy. “Einstein stopped in his tracks,” says Gamow, “and, since we were crossing a street, several cars had to stop to avoid running us down”.

Unfortunately, if a quantum bubble (about as big as the Planck length) containing all the mass-energy of the Universe (or even a star) did appear out of nothing at all, its intense gravitational field would (unless something else intervened) snuff it out of existence immediately, crushing it into a singularity. So the free lunch Universe seemed at first like an irrelevant speculation — but, as with the problems involving the extreme flatness of spacetime, and its appearance of extreme homogeneity and isotropy (most clearly indicated by the uniformity of the background radiation), the development of the inflationary scenario showed how to remove this difficulty and allow such a quantum fluctuation to expand exponentially up to macroscopic size before gravity could crush it out of existence.. All of these problems would be resolved if something gave the Universe a violent outward push (in effect, acting like antigravity) when it was still about a Planck length in size. Such a small region of space would be too tiny, initially, to contain irregularities, so it would start off homogeneous and isotropic. There would have been plenty of time for signals travelling at the speed of light to have criss-crossed the ridiculously tiny volume, so there is no horizon problem — both sides of the embryonic universe are “aware” of each other. And spacetime itself gets flattened by the expansion, in much the same way that the wrinkly surface of a prune becomes a smooth, flat surface when the prune is placed in water and swells up. As in the standard Big Bang model, we can still think of the Universe as like the skin of an expanding balloon, but now we have to think of it as an absolutely enormous balloon that was hugely inflated during the first split second of its existence.

The reason why the GUTs created such a sensation when they were applied to cosmology is that they predict the existence of exactly the right kind of mechanisms to do this trick. They are called scalar fields, and they are associated with the splitting apart of the original grand unified force into the fundamental forces we know today, as the Universe began to expand and cool. Gravity itself would have split off at the Planck time, 10-43 of a second, and the strong nuclear force by about 10(exp-35) of a second. Within about 10-32 of a second, the scalar fields would have done their work, doubling the size of the Universe at least once every 10-34 of a second (some versions of inflation suggest even more rapid expansion than this).

This may sound modest, but it would mean that in 1032 of a second there were 100 doublings. This rapid expansion is enough to take a quantum fluctuation 1020 times smaller than a proton and inflate it to a sphere about 10 cm across in about 15 x 1033 seconds. At that point, the scalar field has done its work of kick-starting the Universe, and is settling down, giving up its energy and leaving a hot fireball expanding so rapidly that even though gravity can now begin to do its work of pulling everything back into a Big Crunch it will take hundreds of billions of years to first halt the expansion and then reverse it.

Curiously, this kind of exponential expansion of spacetime is exactly described by one of the first cosmological models developed using the general theory of relativity, by Willem de Sitter in 1917. For more than half a century, this de Sitter model seemed to be only a mathematical curiosity, of no relevance to the real Universe; but it is now one of the cornerstones of inflationary cosmology.

When the general theory of relativity was published in 1916, de Sitter reviewed the theory and developed his own ideas in a series of three papers which he sent to the Royal Astronomical Society in London. The third of these papers included discussion of possible cosmological models — both what turned out to be an expanding universe (the first model of this kind to be developed, although the implications were not fully appreciated in 1917) and an oscillating universe model.

De Sitter’s solution to Einstein’s equations seemed to describe an empty, static Universe (empty spacetime). But in the early 1920s it was realised that if a tiny amount of matter was added to the model (in the form of particles scattered throughout the spacetime), they would recede from each other exponentially fast as the spacetime expanded. This means that the distance between two particles would double repeatedly on the same timescale, so they would be twice as far apart after one tick of some cosmic clock, four times as far apart after two ticks, eight times as far apart after three ticks, sixteen times as far apart after four ticks, and so on. It would be as if each step you took down the road took you twice as far as the previous step.

This seemed to be completely unrealistic; even when the expansion of the Universe was discovered, later in the 1920s, it turned out to be much more sedate. In the expanding Universe as we see it now, the distances between “particles” (clusters of galaxies) increase steadily — they take one step for each click of the cosmic clock, so the distance is increased by a total of two steps after two clicks, three steps after three clicks, and so on. In the 1980s, however, when the theory of inflation suggested that the Universe really did undergo a stage of exponential expansion during the first split-second after its birth, this inflationary exponential expansion turned out to be exactly described by the de Sitter model, the first successful cosmological solution to Einstein’s equations of the general theory of relativity.

One of the peculiarities of inflation is that it seems to take place faster than the speed of light. Even light takes 30 billionths of a second (3 x 10(exp-10) sec) to cross a single centimetre, and yet inflation expands the Universe from a size much smaller than a proton to 10 cm across in only 15 x 10(exp-33) sec. This is possible because it is spacetime itself that is expanding, carrying matter along for the ride; nothing is moving through spacetime faster than light, either during inflation or ever since. Indeed, it is just because the expansion takes place so quickly that matter has no time to move while it is going on and the process “freezes in” the original uniformity of the primordial quantum bubble that became our Universe.

The inflationary scenario has already gone through several stages of development during its short history. The first inflationary model was developed by Alexei Starobinsky, at the L. D. Landau Institute of Theoretical Physics in Moscow, at the end of the 1970s — but it was not then called “inflation”. It was a very complicated model based on a quantum theory of gravity, but it caused a sensation among cosmologists in what was then the Soviet Union, becoming known as the “Starobinsky model” of the Universe. Unfortunately, because of the difficulties Soviet scientists still had in travelling abroad or communicating with colleagues outside the Soviet sphere of influence at that time, the news did not spread outside their country.

In 1981, Alan Guth, then at MIT, published a different version of the inflationary scenario, not knowing anything of Starobinsky’s work. This version was more accessible in both senses of the word — it was easier to understand, and Guth was based in the US, able to discuss his ideas freely with colleagues around the world. And as a bonus, Guth came up with the catchy name “inflation” for the process he was describing. There were obvious flaws with the specific details of Guth’s original model (which he acknowledged at the time). In particular, Guth’s model left the Universe after inflation filled with a mess of bubbles, all expanding in their own way and colliding with one another. We see no evidence for these bubbles in the real Universe, so obviously the simplest model of inflation couldn’t be right. But it was this version of the idea that made every cosmologist aware of the power of inflation.

In October 1981, there was an international meeting in Moscow, where inflation was a major talking point. Stephen Hawking presented a paper claiming that inflation could not be made to work at all, but the Russian cosmologist Andrei Linde presented an improved version, called “new inflation”, which got around the difficulties with Guth’s model. Ironically, Linde was the official translator for Hawking’s talk, and had the embarrassing task of offering the audience the counter-argument to his own work! But after the formal presentations Hawking was persuaded that Linde was right, and inflation might be made to work after all. Within a few months, the new inflationary scenario was also published by Andreas Albrecht and Paul Steinhardt, of the University of Pennsylvania, and by the end of 1982 inflation was well established. Linde has been involved in most of the significant developments with the theory since then. The next step forward came with the realization that there need not be anything special about the Planck- sized region of spacetime that expanded to become our Universe. If that was part of some larger region of spacetime in which all kinds of scalar fields were at work, then only the regions in which those fields produced inflation could lead to the emergence of a large universe like our own. Linde called this “chaotic inflation”, because the scalar fields can have any value at different places in the early super-universe; it is the standard version of inflation today, and can be regarded as an example of the kind of reasoning associated with the anthropic principle (but note that this use of the term “chaos” is like the everyday meaning implying a complicated mess, and has nothing to do with the mathematical subject known as “chaos theory”).

The idea of chaotic inflation led to what is (so far) the ultimate development of the inflationary scenario. The great unanswered question in standard Big Bang cosmology is what came “before” the singularity. It is often said that the question is meaningless, since time itself began at the singularity. But chaotic inflation suggests that our Universe grew out of a quantum fluctuation in some pre-existing region of spacetime, and that exactly equivalent processes can create regions of inflation within our own Universe. In effect, new universes bud off from our Universe, and our Universe may itself have budded off from another universe, in a process which had no beginning and will have no end. A variation on this theme suggests that the “budding” process takes place through black holes, and that every time a black hole collapses into a singularity it “bounces” out into another set of spacetime dimensions, creating a new inflationary universe — this is called the baby universe scenario.

There are similarities between the idea of eternal inflation and a self-reproducing universe and the version of the Steady State hypothesis developed in England by Fred Hoyle and Jayant Narlikar, with their C-field playing the part of the scalar field that drives inflation. As Hoyle wryly pointed out at a meeting of the Royal Astronomical Society in London in December 1994, the relevant equations in inflation theory are exactly the same as in his version of the Steady State idea, but with the letter “C” replaced by the Greek “Ø”. “This,” said Hoyle (tongue firmly in cheek) “makes all the difference in the world”.

Modern proponents of the inflationary scenario arrived at these equations entirely independently of Hoyle’s approach, and are reluctant to accept this analogy, having cut their cosmological teeth on the Big Bang model. Indeed, when Guth was asked, in 1980, how the then new idea of inflation related to the Steady State theory, he is reported as replying “what is the Steady State theory?” But although inflation is generally regarded as a development of Big Bang cosmology, it is better seen as marrying the best features of both the Big Bang and the Steady State scenarios.

This might all seem like a philosophical debate as futile as the argument about how many angels can dance on the head of a pin, except for the fact that observations of the background radiation by COBE showed exactly the pattern of tiny irregularities that the inflationary scenario predicts. One of the first worries about the idea of inflation (long ago in 1981) was that it might be too good to be true. In particular, if the process was so efficient at smoothing out the Universe, how could irregularities as large as galaxies, clusters of galaxies and so on ever have arisen? But when the researchers looked more closely at the equations they realised that quantum fluctuations should still have been producing tiny ripples in the structure of the Universe even when our Universe was only something like 10(exp-25) of a centimetre across — a hundred million times bigger than the Planck length.

The theory said that inflation should have left behind an expanded version of these fluctuations, in the form of irregularities in the distribution of matter and energy in the Universe. These density perturbations would have left an imprint on the background radiation at the time matter and radiation decoupled (about 300,000 years after the Big Bang), producing exactly the kind of nonuniformity in the background radiation that has now been seen, initially by COBE and later by other instruments. After decoupling, the density fluctuations grew to become the large scale structure of the Universe revealed today by the distribution of galaxies. This means that the COBE observations are actually giving us information about what was happening in the Universe when it was less than 10-20 of a second old.

No other theory can explain both why the Universe is so uniform overall, and yet contains exactly the kind of “ripples” represented by the distribution of galaxies through space and by the variations in the background radiation. This does not prove that the inflationary scenario is correct, but it is worth remembering that had COBE found a different pattern of fluctuations (or no fluctuations at all) that would have proved the inflationary scenario wrong. In the best scientific tradition, the theory made a major and unambiguous prediction which did “come true”. Inflation also predicts that the primordial perturbations may have left a trace in the form of gravitational radiation with particular characteristics, and it is hoped that detectors sensitive enough to identify this characteristic radiation may be developed within the next ten or twenty years.

The clean simplicity of this simple picture of inflation has now, however, begun to be obscured by refinements, as inflationary cosmologists add bells and whistles to their models to make them match more closely the Universe we see about us. Some of the bells and whistles, it has to be said, are studied just for fun. Linde himself has taken great delight in pushing inflation to extremes, and offering entertaining new insights into how the Universe might be constructed. For example, could our Universe exist on the inside of a single magnetic monopole produced by cosmic inflation? According to Linde, it is at least possible, and may be likely. And in a delicious touch of irony, Linde, who now works at Stanford University, made this outrageous claim in a lecture at a workshop on the Birth of the Universe held recently in Rome, where the view of Creation is usually rather different. One of the reasons why theorists came up with the idea of inflation in the first place was precisely to get rid of magnetic monopoles — strange particles carrying isolated north or south magnetic fields, predicted by many Grand Unified Theories of physics but never found in nature. Standard models of inflation solve the “monopole problem” by arguing that the seed from which our entire visible Universe grew was a quantum fluctuation so small that it only contained one monopole. That monopole is still out there, somewhere in the Universe, but it is highly unlikely that it will ever pass our way.

But Linde has discovered that, according to theory, the conditions that create inflation persist inside a magnetic monopole even after inflation has halted in the Universe at large. Such a monopole would look like a magnetically charged black hole, connecting our Universe through a wormhole in spacetime to another region of inflating spacetime. Within this region of inflation, quantum processes can produce monopole-antimonopole pairs, which then separate exponentially rapidly as a result of the inflation. Inflation then stops, leaving an expanding Universe rather like our own which may contain one or two monopoles, within each of which there are more regions of inflating spacetime.

The result is a never-ending fractal structure, with inflating universes embedded inside each other and connected through the magnetic monopole wormholes. Our Universe may be inside a monopole which is inside another universe which is inside another monopole, and so on indefinitely. What Linde calls “the continuous creation of exponentially expanding space” means that “monopoles by themselves can solve the monopole problem”. Although it seems bizarre, the idea is, he stresses, “so simple that it certainly deserves further investigation”.

That variation on the theme really is just for fun, and it is hard to see how it could ever be compared with observations of the real Universe. But most of the modifications to inflation now being made are in response to new observations, and in particular to the suggestion that spacetime may not be quite “flat” after all. In the mid-1990s, many studies (including observations made by the refurbished Hubble Space Telescope) began to suggest that there might not be quite enough matter in the Universe to make it perfectly flat — most of the observations suggest that there is only 20 per cent or 30 per cent as much matter around as the simplest versions of inflation require. The shortfall is embarrassing, because one of the most widely publicised predictions of simple inflation was the firm requirement of exactly 100 per cent of this critical density of matter. But there are ways around the difficulty; and here are two of them to be going on with.

The first suggestion is almost heretical, in the light of the way astronomy has developed since the time of Copernicus. Is it possible that we are living near the centre of the Universe? For centuries, the history of astronomy has seen humankind displaced from any special position. First the Earth was seen to revolve around the Sun, then the Sun was seen to be an insignificant member of the Milky Way Galaxy, then the Galaxy was seen to be an ordinary member of the cosmos. But now comes the suggestion that the “ordinary” place to find observers like us may be in the middle of a bubble in a much greater volume of expanding space.

The conventional version of inflation says that our entire visible Universe is just one of many bubbles of inflation, each doing their own thing somewhere out in an eternal sea of chaotic inflation, but that the process of rapid expansion forces spacetime in all the bubbles to be flat. A useful analogy is with the bubbles that form in a bottle of fizzy cola when the top is opened. But that suggestion, along with other cherished cosmological beliefs, has now been challenged by Linde, working with his son Dmitri Linde (of CalTech) and Arthur Mezhlumian (also of Stanford).

Linde and his colleagues point out that the Universe we live in is like a hole in a sea of superdense, exponentially expanding inflationary cosmic material, within which there are other holes. All kinds of bubble universes will exist, and it is possible to work out the statistical nature of their properties. In particular, the two Lindes and Mezhlumian have calculated the probability of finding yourself in a region of this super- Universe with a particular density — for example, the density of “our” Universe.

Because very dense regions blow up exponentially quickly (doubling in size every fraction of a second), it turns out that the volume of all regions of the super-Universe with twice any chosen density is 10 to the power of 10 million times greater than the volume of the super- Universe with the chosen density. For any chosen density, most of the matter at that density is near the middle of an expanding bubble, with a concentration of more dense material round the edge of the bubble. But even though some of the higher density material is round the edges of low density bubbles, there is even more (vastly more!) higher density material in the middle of higher density bubbles, and so on forever. The discovery of this variation on the theme of fractal structure surprised the researchers so much that they confirmed it by four independent methods before venturing to announce it to their colleagues. Because the density distribution is non-uniform on the appropriate distance scales, it means that not only may we be living near the middle of a bubble universe, but that the density of the region of space we can see may be less than the critical density, compensated for by extra density beyond our field of view.

This is convenient, since those observations by the Hubble Space Telescope have suggested that cosmological models which require exactly the critical density of matter may be in trouble. But there is more. Those Hubble observations assume that the parameter which measures the rate at which the Universe is expanding, the Hubble Constant, really is a constant, the same everywhere in the observable Universe. If Linde’s team is right, however, the measured value of the “constant” may be different for galaxies at different distances from us, truly throwing the cat among the cosmological pigeons. We may seem to live in a low-density universe in which both the measured density and the value of the Hubble Constant will depend on which volume of the Universe these properties are measured over!

That would mean abandoning many cherished ideas about the Universe, and may be too much for many cosmologists to swallow. But there is a simpler solution to the density puzzle, one which involves tinkering only with the models of inflation, not with long-held and cherished cosmological beliefs. That may make it more acceptable to most cosmologists — and it’s so simple that it falls into the “why didn’t I think of that?” category of great ideas.

A double dose of inflation may be something to make the Government’s hair turn grey — but it could be just what cosmologists need to rescue their favourite theory of the origin of the Universe. By turning inflation on twice, they have found a way to have all the benefits of the inflationary scenario, while still leaving the Universe in an “open” state, so that it will expand forever.

In those simplest inflation models, remember, the big snag is that after inflation even the observable Universe is left like a mass of bubbles, each expanding in its own way. We see no sign of this structure, which has led to all the refinements of the basic model. Now, however, Martin Bucher and Neil Turok, of Princeton University, working with Alfred Goldhaber, of the State University of New York, have turned this difficulty to advantage.

They suggest that after the Universe had been homogenised by the original bout of inflation, a second burst of inflation could have occurred within one of the bubbles. As inflation begins (essentially at a point), the density is effectively “reset” to zero, and rises towards the critical density as inflation proceeds and energy from the inflation process is turned into mass. But because the Universe has already been homogenised, there is no need to require this bout of inflation to last until the density reaches the critical value. It can stop a little sooner, leaving an open bubble (what we see as our entire visible Universe) to carry on expanding at a more sedate rate. They actually use what looked like the fatal flaw in Guth’s model as the basis for their scenario. According to Bucher and his colleagues, an end product looking very much like the Universe we live in can arise naturally in this way, with no “fine tuning” of the inflationary parameters. All they have done is to use the very simplest possible version of inflation, going back to Alan Guth’s work, but to apply it twice. And you don’t have to stop there. Once any portion of expanding spacetime has been smoothed out by inflation, new inflationary bubbles arising inside that volume of spacetime will all be pre-smoothed and can end up with any amount of matter from zero to the critical density (but no more). This should be enough to make everybody happy. Indeed, the biggest problem now is that the vocabulary of cosmology doesn’t quite seem adequate to the task of describing all this activity.

The term Universe, with the capital “U”, is usually used for everything that we can ever have knowledge of, the entire span of space and time accessible to our instruments, now and in the future. This may seem like a fairly comprehensive definition, and in the past it has traditionally been regarded as synonymous with the entirety of everything that exists. But the development of ideas such as inflation suggests that there may be something else beyond the boundaries of the observable Universe — regions of space and time that are unobservable in principle, not just because light from them has not yet had time to reach us, or because our telescopes are not sensitive enough to detect their light.

This has led to some ambiguity in the use of the term “Universe”. Some people restrict it to the observable Universe, while others argue that it should be used to refer to all of space and time. If we use “Universe” as the name for our own expanding bubble of spacetime, everything that is in principle visible to our telescopes, then maybe the term “Cosmos” can be used to refer to the entirety of space and time, within which (if the inflationary scenario is correct) there may be an indefinitely large number of other expanding bubbles of spacetime, other universes with which we can never communicate. Cosmologists ought to be happy with the suggestion, since it makes their subject infinitely bigger and therefore infinitely more important!

Further reading: John Gribbin, Companion to the Cosmos, Weidenfeld & Nicolson, London, 1996.

## One comment on “Inflation for beginners”

1. Amr Khaled says:

Awesome article.. Really helped me out here