INFLATIONARY UNIVERSE

Adapted from P. Coles, 1999, The Routledge Critical Dictionary of the New Cosmology, Routledge Inc., New York. Reprinted with the author's permission. To order this book click here: http://www.routledge-ny.com/books.cfm?isbn=0415923549

A modern variation of the standard Big Bang theory that includes a finite period of accelerated expansion (inflation) in the early stages of its evolution. Inflation is the mechanism by which various outstanding problems of the standard cosmological models might be addressed, providing a possible resolution of the horizon problem and the flatness problem, as well as generating the primordial density fluctuations that are required for structure formation to occur and which appear to have produced the famous ripples in the cosmic microwave background radiation.

Assuming that we accept that an epoch of inflation is desirable for these reasons, how can we achieve the accelerated expansion physically. when the standard Friedmann models are always decelerating? The idea that lies at the foundation of most models of inflation is that there was an epoch in the early stages of the evolution of the Universe in which the energy density of the vacuum state of a scalar field , perhaps associated with one of the fundamental interactions, provided the dominant contribution to the energy density. In the ensuing phase the scale factor a(t) describing the expansion of the Universe grows in an accelerated fashion, and is in fact very nearly an exponential function of time if the energy density of the scalar field is somehow held constant. In the inflationary epoch that follows, a microscopically small region, perhaps no larger than the Planck length, can grow - or inflate - to such a size that it exceeds the size of our present observable Universe (see horizon).

There exist many different versions of the inflationary universe. The first was formulated by Alan Guth in 1981, although many of his ideas had been presented by Alexei Starobinsky in 1979. In Guth's model inflation is assumed to occur while the Universe undergoes a first-order phase transition, which is predicted to occur in some grand unified theories. The next generation of inflationary models shared the characteristics of a model called the new inflationary universe, which was suggested in 1982 independently by Andrei Linde, and by Andreas Albrecht and Paul Steinhardt. In models of this type, inflation occurs during a phase in which the region that will grow to include our observable patch evolves more gradually from a `false vacuum' to a `true vacuum'. It was later realised that this kind of inflation could also be achieved in many different contexts, not necessarily requiring the existence of a phase transition or a spontaneous symmetry-breaking. This of parameters of a particular grand unified theory which, in the absence of any other experimental evidence, appears a little arbitrary. This problem also arises in other inflationary models based on theories like supersymmetry or string theories, which are yet to receive any experimental confirmation or, indeed, are likely to in the foreseeable future. It is fair to say that the inflationary model has become a sort of paradigm for resolving some of the difficulties with the standard model, but no particular version of it has so far received any strong physical support from particle physics theories.

Let us concentrate for a while on the physics of generic inflationary models involving symmetry-breaking during a phase transition. In general, gauge theories of elementary particle interactions involve an order parameter which we can identify with the scalar field determining the breaking of the symmetry. The behaviour of the scalar field is controlled by a quantity called its Lagrangian action, which has two components: one (denoted by U) concerns time-derivatives of and is therefore called the kinetic term, and the other (V) describes the interactions of the scalar field and is called the potential term. A scalar field behaves roughly like a strange form of matter with a total energy density given by U + V and a pressure given by U - V. Note that if V is much larger than U, then the density and pressure are equal but of opposite sign. This is what is needed to generate inflation, but the way in which the potential comes to dominate is quite complicated.

The potential function V changes with the temperature of the Universe, and it is this that induces the phase transition, as it becomes energetically favourable for the state of the field to change when the Universe cools sufficiently. In the language of thermodynamics, the potential V() plays the role of the free energy of the system. A graph of V() will typically have a minimum somewhere, and that minimum value determines the value of which is stable at a given temperature. Imagine an inverted parabola with its minimum value at = 0; the configuration of the field can be represented as the position of a ball rolling on this curve. In the stable configuration it nestles in the bottom of the potential well at = 0. This might represent the potential V() at very high temperatures, way above the phase transition. The vacuum is then in its most symmetrical state. What happens as the phase transition proceeds is that the shape of V() changes so that it develops additional minima. Initially these `false' minima may be at higher values of V() than the original, but as the temperature continues to fall and the shape of the curve changes further, the new minima can be at lower values of V than the original one. This happens at a critical temperature T_c at which the vacuum state of the Universe begins to prefer one of the alternative minima to the original one.

The transition does not occur instantaneously. How it proceeds depends on the shape of the potential, and this in turn determines whether the transition is first or second order. If the phase transition is second order it moves rather smoothly, and fairly large `domains' of the new phase are generated (much like the Weiss domains in a ferromagnet). One such region (bubble or domain) eventually ends up including our local patch of the Universe. If the potential is such that the transition is first order, the new phase appears as bubbles nucleating within the false vacuum background; these then grow and coalesce so as to fill space with the new phase when the transition is complete.

Inflation arises when the potential term V greatly exceeds the kinetic term U in the action of the scalar field. In a phase transition this usually means that the vacuum must move relatively slowly from its original state into the final state. In fact, the equations governing the evolution of are mathematically identical to those describing a ball moving under the action of the force -dV / d, just as in standard Newtonian dynamics. But there is also a frictional force, caused by the expansion of the Universe, that tends to slow down the rolling of from one state into another. This provides a natural self-regulation of the speed of the transition. As long as the Universe is expanding at the start, the kinetic term U is quickly reduced by the action of this friction (or viscosity); the motion of the field then resembles the behaviour of particles during sedimentation.

In order to have inflation we must assume that, at sometime, the Universe contains some expanding regions in thermal equilibrium at a temperature T > T_c which can eventually cool below T_c before they re-collapse. Let us assume that such a vacuum phase, is sufficiently homogeneous and isotropic to be described by a Robertson-Walker metric. In this case the evolution of the patch is described by a Friedmann model, except that the density of the Universe is not the density of matter, but the effective density of the scalar field, i.e. the sum U + V mentioned above. If the field is evolving slowly then the U component is negligibly small. The Friedmann equation then looks exactly like the equation describing a Friedmann model incorporating a cosmological constant term but containing no matter. The cosmological model that results is the well-known de Sitter solution in which the scale factor a(t) exp (Ht), with H (the Hubble parameter) roughly constant at a value given by H² = 8 G V / 3. Since does not change very much as the transition proceeds, V is roughly constant.

The de Sitter solution is the same as that used in the steady state theory, except that the scalar field in that theory is the so-called C-field responsible for the continuous creation of matter. In the inflationary Universe, however, the expansion timeseale is much more rapid than in the steady state. The inverse of the Hubble expansion parameter, 1/H, is about 10^-34 seconds in inflation. This quantity has to be fixed at the inverse of the present-day value of the Hubble parameter (i.e. at the reciprocal of the Hubble constant, 1/H₀) in the steady state theory, which is around 10¹⁷ seconds.

This rapid expansion is a way of solving some riddles which are not explained in the standard Big Bang theory. For example, a region which is the same order of size as the horizon before inflation, and which might therefore be reasonably assumed to be smooth, would then become enormously large, encompassing the entire observable Universe today. Any inhomogeneity and anisotropy present at the initial time will be smoothed out so that the region loses all memory of its initial structure. Inflation therefore provides a mechanism for avoiding the horizon problem. This effect is, in fact, a general property of inflationary universes and it is described by the so-called cosmic no-hair theorem (see also black hole).

Another interesting outcome of the inflationary Universe is that the characteristic scale of the curvature of spacetime, which is intimately related to the value of the density parameter , is expected to become enormously large. A balloon is perceptibly curved because its radius of curvature is only a few centimetres, but it would appear very flat if it were blown up to the radius of the Earth. The same happens in inflation: the curvature scale may be very small indeed initially, but it ends up greater than the size of our observable Universe. The important consequence of this is that the density parameter should be very close to 1. More precisely, the total energy density of the Universe (including the matter and any vacuum energy associated with a cosmological constant) should be very close to the critical density required to make a flat universe.

Because of the large expansion, a small initial patch also becomes practically devoid of matter in the form of elementary particles. This also solves problems associated with the formation of monopoles and other topological defects in the early Universe, because any defects formed during the transition will be drastically diluted as the Universe expands, so that their present density will be negligible.

After the slow rolling phase is complete, the field falls rapidly into its chosen minimum and then undergoes oscillations about the minimum value. While this happens there is a rapid liberation of energy which was trapped in the potential term V while the transition was in progress. This energy is basically the latent heat of the transition. The oscillations are damped by the creation of elementary particles coupled to the scalar field, and the liberation of the latent heat raises the temperature again - a phenomenon called reheating. The small patch of the Universe we have been talking about thus acquires virtually all the energy and entropy that originally resided in the quantum vacuum by this process of particle creation. Once reheating has occurred, the evolution of the patch again takes on the character of the usual Friedmann models, but this time it has the normal form of matter. If V(₀) = 0, then the vacuum energy remaining after inflation is zero and there will be no remaining cosmological constant .

One of the problems left unsolved by inflation is that there is no real reason to suppose that the minimum of V is exactly at zero, so we would expect a nonzero -term to appear at the end. Attempts to calculate the size of the cosmological constant induced by phase transitions in this way produce enormously large values. It is important that the inflationary model should predict a reheating temperature sufficiently high that processes which violate the conservation of baryon number can take place so as to allow the creation of an asymmetry between matter and antimatter (see baryogenesis).

As far as its overall properties are concerned, our Universe was reborn into a new life after reheating. Even if before it was highly lumpy and curved, it was now highly homogeneous, and had negligible curvature. This latter prediction may be a problem because, as we have seen, there is little strong evidence that the density parameter is indeed close to 1, as required.

Another general property of inflationary models, which we shall not go into here, is that fluctuations in the quantum scalar field driving inflation can, in principle, generate a spectrum of primordial density fluctuations capable of initiating the process of cosmological structure formation. They may also produce an observable spectrum of primordial gravitational waves.

There are many versions of the basic inflationary model which are based on slightly different assumptions about the nature of the scalar field and the form of the phase transition. Some of the most important are described below.

Old inflation. The name now usually given to the first inflationary model, suggested by Guth in 1981. This model is based on a scalar field theory which undergoes a first-order phase that, being a first-order transition, it occurs by a process of bubble nucleation. It turns out that these bubbles would be too small to be identified with our observable Universe, and they would be carried apart by the expanding phase too quickly for them to coalesce and produce a large bubble which we could identify in this way. The end state of this model would therefore be a highly chaotic universe, quite the opposite of what is intended. This model was therefore abandoned soon after it was suggested.

New inflation. The successor to old inflation: again, a theory based on a scalar field, but this time the potential has no potential barrier, so the phase transition is second order. The process which accompanies a second-order phase transition, known as spinodal decomposition, usually leaves larger coherent domains, providing a natural way out of the problem of old inflation. The problem with new inflation is that it suffers from severe fine-tuning difficulties. One is that the potential must be very flat near the origin to produce enough inflation and to avoid excessive fluctuations due to the quantum field. Another is that the field is assumed to be in thermal equilibrium with the other matter fields before the onset of inflation; this requires to be coupled fairly strongly to the other fields that might exist at this time. But this coupling would induce corrections to the potential which would violate the previous constraint. It seems unlikely, therefore, that thermal equilibrium can be attained in a self-consistent way before inflation starts and under the conditions necessary for inflation to happen.

Chaotic inflation. One of the most popular inflationary models, devised by Linde in 1983; again, it is based on a scalar field but it does not require any phase transitions at all. The idea behind this model is that, whatever the detailed shape of the effective potential V, a patch of the Universe in which is large, uniform and relatively static will automatically lead to inflation. In chaotic inflation we simply assume that at some initial time, perhaps as early as the Planck time, the field varied from place to place in an arbitrary chaotic manner. If any region is uniform and static, it will inflate and eventually encompass our observable Universe. While the end result of chaotic inflation is locally flat and homogeneous in our observable patch, on scales larger than the horizon the Universe is highly curved and inhomogeneous. Chaotic inflation is therefore very different from both the old and new inflationary models. This difference is reinforced by the fact that no mention of supersymmetry or grand unified theories appears in the description. The field that describes chaotic inflation at the Planck time is completely decoupled from all other physics.

Stochastic inflation. A natural extension of Linde's chaotic inflation, sometimes called eternal inflation; as with chaotic inflation, the Universe is extremely inhomogeneous overall, but quantum fluctuations during the evolution of are taken into account. In stochastic inflation the universe will at any time contain regions which are just entering an inflationary phase. We can picture the Universe as a continuous `branching' process in which new `mini-universes' expand to produce locally smooth patches within a highly chaotic background Universe. This model is like a Big Bang on the scale of each mini-universe, but overall it is reminiscent of the steady state model. The continual birth and rebirth of these mini-universes is often called, rather poetically, the phoenix universe. This model has the interesting feature that the laws of physics may be different in different mini-universe, which brings the anthropic principle very much into play.

Modified gravity. There are versions of the inflationary Universe model that do not require a scalar field associated with the fundamental interaction to drive the expansion. We can, for example, obtain inflation by modifying the laws of gravity. Usually this is done by adding extra terms in the curvature of spacetime to the equations of general relativity. For certain kinds of modification, the resulting equations are mathematically equivalent to ordinary general relativity in the presence of a scalar field with some particular action. This effective scalar field can drive inflation in the same way as a genuine physical field can. An alternative way to modify gravity might be to adopt the Brans-Dicke theory of gravity. The crucial point with this kind of model is that the scalar field does not generate an exponential expansion, but one in which the expansion is some power of time: a(t) t^a. This modification even allows old inflation to succeed: the bubbles that nucleate the new phase can be made to merge and fill space if inflation proceeds as a power law in time rather than an exponential. Theories based on Brans-Dicke modified gravity are usually called extended inflation.

There are many other possibilities: models with more than one scalar field, models with modified gravity and a scalar field, models based on more complicated potentials, models based on supersymmetry, on grand unified theories, and so on. Indeed, inflation has led to an almost exponential increase in the number of cosmological models!

It should now be clear that the inflationary Universe model provides a conceptual explanation of the horizon problem and the flatness problem. It also may rescue grand unified theories which predict a large present-day abundance of monopoles or other topological defects. Inflationary models have gradually evolved to avoid problems with earlier versions. Some models are intrinsically flawed (e.g. old inflation) but can be salvaged in some modified form (e.g. extended inflation). The magnitude of the primordial density fluctuations and gravitational waves they produce may also be too high for some particular models. There are, however, much more serious problems associated with these scenarios. Perhaps the most important is intimately connected with one of the successes. Most inflationary models predict that the observable Universe at the present epoch should be almost flat. In the absence of a cosmological constant this means that 1. While this possibility is not excluded by observations, it cannot be said that there is compelling evidence for it and, if anything, observations of dark matter in galaxies and clusters of galaxies favour an open universe with a lower density than this. It is possible to produce a low-density universe after inflation, but it requires very particular models. To engineer an inflationary model that produces 0.2 at the present epoch requires a considerable degree of unpleasant fine-tuning of the conditions before inflation. On the other hand, we could reconcile a low-density universe with apparently more natural inflationary models by appealing to a relic cosmological constant: the requirement that space should be (almost) flat simply translates into ₀ + (c² / 3H₀²) 1. It has been suggested that this is a potentially successful model of structure formation, but recent developments in classical cosmology put pressure on this alternative.

The status of inflation as a physical theory is also of some concern. To what extent is inflation predictive? Is it testable? We could argue that inflation does predict that we live in a flat universe. This may be true, but a flat universe might emerge naturally at the very beginning if some process connected with quantum gravity can arrange it. Likewise, an open universe appears to be possible either with or without inflation. Inflationary models also produce primordial density fluctuations and gravitational waves. Observations showing that these phenomena had the correct properties may eventually constitute a test of inflation, but this is not possible at the present. All we can say is that the observed properties of fluctuations in the cosmic microwave background radiation indeed seem to be consistent with the usual inflationary models. At the moment, therefore, inflation has a status somewhere between a theory and a paradigm, but we are still far from sure that inflation ever took place.

FURTHER READING:

Guth, A.H., `Inflationary Universe: A possible solution to the horizon and flatness problems', Physical Review D, 1981, 23, 347. Albrecht, A. and Steinhardt, P.J., `Cosmology for grand unified theories with radiatively induced symmetry breaking', Physical Review Letters, 1982, 48, 1220. Linde, A.D., `Scalar field fluctuations in the expanding Universe and the new inflationary Universe scenario', Physics Letters B, 1982, 116, 335. Narlikar, J.V. and Padmanabhan, T., `Inflation for astronomers', Annual Reviews of Astronomy and Astrophysics, 1991, 29, 325.