Inflation

views updated

INFLATION

The standard hot Big Bang cosmology has developed into a remarkably precise, well-tested theory of the evolution of the universe from its primordial state into the complex cosmos of today. Measurements of high precision have been performed over the last decade, confirming the basic predictions of the theory including Hubble's expansion law, the thermal spectrum of the cosmic background radiation, and the primordial abundances of the elements. A concordance model has emerged that with only a few parameters succeeds in fitting a great array of astronomical data ranging from the abundance and distribution of galaxies to the temperature pattern of the radiation left over from the Big Bang.

Despite its extraordinary success, the standard cosmology is clearly incomplete. It fails to answer many basic questions: Why is the universe so large, so smooth, and so geometrically flat on large scales? What was the original driving force behind the expansion of the universe? What was the origin of the density inhomogeneities that eventually grew to form galaxies, stars, and planets?

In the standard theory, it is assumed that the universe began in a hot, dense state that was almost uniform over macroscopic scales and undergoing very rapid expansion. Such a state is very special. Gravity tends to amplify density variations because denser regions undergo gravitational collapse and less dense regions fly apart. The beginning of the Big Bang had to be carefully adjusted to be very uniform and flat on large scales, with small variations at a level of about one part in a hundred thousand, just sufficient to lead to the formation of the observed structures.

The near uniformity of the temperature of the cosmic microwave sky provides a dramatic illustration of how special the initial conditions for the hot Big Bang had to be. Radiation received from opposite points on the sky is found to be at very nearly the same temperature. However, at least naively, the two emitting regions could not have been in causal contact because light from both is only now reaching the Earth. The puzzle therefore is the apparently strong correlation in the state of the universe on scales so large that no communication has been possible over the entire age of the universe.

Cosmic inflation was proposed by Alan Guth in 1979, although it was prefigured in the prior work of Alexei Starobinsky and others. It provides a beguilingly simple resolution of these basic puzzles. The basic idea is that an exotic form of matter (the simplest being scalar field matter) can have repulsive gravitational fields. If such matter was the dominant component of the early universe, it would have caused the universe to expand exponentially, making it very large, homogeneous, and flat. The most remarkable side-effect is that microscopic short-wave-length quantum mechanical variations in the scalar field matter are exponentially stretched and amplified during inflation, leading to very large-scale density variations. This provides a beautifully economical mechanism for the formation of structure in the universe, within a smooth background. So compelling, in fact, that for two decades inflation has dominated theoretical cosmology and to a large extent even set the agenda for observations.

The basic idea for inflation dates back to the 1930s when cosmological solutions to the equations of Einstein's theory of general relativity were still being developed. It was realized by Willem de Sitter, among others, that these equations allowed one to introduce an arbitrary constant, which became known as the cosmological constant. If this constant is positive, it causes the scale factor of the universe to grow exponentially and without bound. The cosmological constant was later interpreted as describing the energy density of the vacuum, that is, empty space. This vacuum has to be invariant under the Lorentz transformations of special relativity, and it follows that it possesses a negative pressure P equal in magnitude to its density ? times the speed of light c squared. In equations, the stress energy tensor where g_?? is the space-time metric.

In general relativity, one equates 8?GT_?? to the Einstein tensor G_??, measuring the curvature of space-time. A homogeneous, isotropic universe only evolves via an overall scale factor R , which is a function of time. Einstein's equations read: written in "Newtonian" form: G is Newton's gravitational constant, and M (R ) is the mass in a sphere of radius R . Units have been chosen so that the speed of light c is unity. If the pressure is small, P ??, then one sees that for ordinary matter with positive energy, gravity is attractive. A universe that starts out static will tend to collapse. However, for matter in the form of a cosmological constant, equation (1), P = -?, and instead one finds gravity to be repulsive. The two solutions to (2) are easily obtained: as long as ? is constant. The physical consequence is that a universe dominated by positive vacuum energy density will generally expand exponentially, since the exponentially growing solution quickly dominates over the contracting one. Homogeneity is assumed here, but a nearly homogeneous region will also start to grow exponentially. And once exponential expansion begins, it will dilute the density of matter (which scales as R^-3) or radiation (which scales as R^-4), whereas the cosmological constant, or vacuum energy, remains constant. One finds in consequence that a positive cosmological constant causes expansion, which dilutes any matter or radiation initially present. It may also be shown that any curvature of space is likewise expanded away. Literally, the expansion of the universe blows the universe up, and if one assumes that the universe prior to inflation was smooth and flat on small scales, inflation caused it to become smooth and flat on large scales.

All of this is interesting but not useful. After all, the universe has large amounts of matter and radiation today, and these must have dominated in the early universe if the theory of the abundances of the elements is to work. What Guth realized is that a certain form of matter postulated in unified theories of high-energy physics could provide a temporary cosmological constant in the early universe. The density ? can be nearly constant for some time but then decay into other forms of matter. When it does so, the exponential expansion ceases, and the standard hot Big Bang evolution can begin.

The form of matter in question is known as scalar field matter. This was developed in the early days of quantum field theory as a description of elementary particles (pions) before it was realized that pions are actually made out of quarks. Later, Peter Higgs realized that scalar fields could be used to give other fields a mass in particle physics, and Steven Weinberg and Abdus Salam built a detailed model of the weak interactions incorporating this mechanism. Particle physics experiments continue to decisively probe for the existence of the Higgs particle. If it is found, it will be the first proof that scalar field matter of the type needed for inflation actually exists. (The Higgs scalar field does not drive inflation because its potential V (?) [see below] is not sufficiently flat).

For simplicity, consider a spatially uniform scalar field ?. In an expanding universe, it evolves in time according to the equation where V (?) is an arbitrary function of ?, known as its potential. This equation is just for a ball moving in a one-dimensional potential, along with a damping term that depends on the rate of expansion of the universe. To describe cosmology with scalar field matter, the density and pressure must be determined: From the last equation here and equation (2), it follows that if the kinetic energy of the field is small, the scalar field causes R to accelerate, leading to exponential expansion.

A potential of the form shown in Figure 1 (e.g., a simple quadratic, V ??²) allows one to start the universe with a temporary cosmological constant as follows. The slope of V (?) is chosen to be shallow, so that it is possible for ? to roll slowly downhill. Then, one chooses the initial conditions for the universe so that ? starts out uphill, moving slowly. Finally, the initial conditions for the scale factor of the universe R are chosen so that it is expanding. If these conditions are fulfilled, then ? will gradually roll downhill, with its motion damped by the second term in (4). As it rolls slowly downhill, the scale factor R will drive an epoch of exponential expansion, making the universe very large, very smooth, and very flat. When the scalar field nears the minimum of the potential V (?), then its kinetic energy ?² begins to overwhelm V (?), and it is seen from (5) that

FIGURE 1

? + 3P becomes positive so the acceleration of the scale factor ceases.

When the field ? is incorporated into a unified theory of particle physics and forces, it automatically couples to all the particles of the Standard Model. This coupling provides a natural mechanism through which the energy stored in ? during inflation is eventually released into the radiation and matter needed to start the hot Big Bang. When ? reaches its potential minimum, it starts to oscillate, and these oscillations cause the creation of particles to which ? couples, particles that comprise the radiation and matter required for the hot Big Bang. The energy initially stored in ? is thus redistributed amongst all the particle species present. Generally, it is easy to arrange that sufficient energy is transferred to heat the universe to temperatures well above those needed for the production of the primordial elements in the Big Bang.

Precisely what was gained from this early epoch of exponential expansion? The puzzle is to explain why the universe is so smooth today, where the natural expectation from "random" initial conditions for the universe would be a lumpy universe becoming ever more lumpy under the influence of gravitational collapse. Exponential expansion improves this situation. A characteristic length scale in gravitational physics is the Planck scale By random initial conditions, one might mean a universe composed of many Planck-sized domains of differing energy densities and expansion rates. Compare this picture with the picture obtained if the current observed universe is traced all the way back to the Planck time, L_Planck/c or 10^-43 seconds. The observed universe is now some 13 to 15 billion light-years across. Following this scale back in time using the known densities of matter and radiation, the size of the region it occupied at the Planck time can be calculated. It turns out to be just a millimeter, which, even though small, is a huge scale in Planck units. The problem of the initial conditions needed for the hot Big Bang cosmology is that the universe had to have started in a state almost perfectly uniform on a scale of 10³² times the natural length scale.

Inflation greatly improves this situation. Take instead one Planck-sized region, 10^-33 centimeters across, within which the conditions required for inflation happen to be satisfied, with the scalar field ? large enough to ensure that inflation lasted for 100 expansion times within it. The size of such a region would have grown by e¹⁰⁰ ? 10⁴³ during inflation, so that by the end of inflation it would be approximately a kilometer across. Following inflation, this region would then undergo standard hot Big Bang expansion, during which a millimeter-sized subregion would grow to the size of the visible cosmos today. Thus, given just one Planck-sized region with the right inflationary initial conditions, one can account for the present vast cosmos of great uniformity and flatness (see Figure 2).

This argument is plausible but not necessarily compelling. After all, the field needed to drive inflation has been added by hand, and the discussion of the likelihood of obtaining inflation is hardly rigorous. What happened to all those regions that did not inflate? Is inflation still occurring somewhere else in the universe? Scientists have struggled with these questions, but no convincing answer has yet emerged.

What made inflationary theory much more convincing was the fact that it was able to explain the density inhomogeneities needed for cosmic structure formation in a quite magical way. This became apparent soon after inflation was invented and was a surprising, and for some a compelling, success.

FIGURE 2

Relativity and quantum mechanics lie at the heart of the inflationary mechanism for generating inhomogeneities. If ? is a field, then according to relativity, if it can vary in time, it must also be allowed to vary in space. But according to quantum mechanics (the Heisenberg uncertainty principle), there must be a minimal level of fluctuations in such a field so that even in empty space, it is constantly fluctuating about its minimal value.

In the context of inflation, the quantum "jitter" in the inflation field ? becomes stretched and amplified into large-scale density homogeneities. First, as the universe is blown up by inflation, the scale of any fluctuation in ? grows exponentially in time. When the scale of the fluctuation is short, it oscillates like a sound or light wave.

But when the scale of the fluctuation is stretched beyond a certain point, different regions of the fluctuation no longer communicate, and the fluctuation becomes "frozen," thereafter simply undergoing a continuous stretching until the end of inflation. The remarkable consequence of this mechanism is that the final spectrum of fluctuations is "scale-free." That is to say, over an exponentially large range of scales, at the end of inflation one finds that the density of the universe fluctuates with the same amplitude on all length scales.

It was realized long before inflation that scale-free primordial fluctuations could plausibly explain the observed distribution of galaxies in the universe. This spectrum became known as the Harrison-Z'eldovich spectrum, named after the physicists who first postulated it. Therefore, the realization that inflation, invented for very different purposes, automatically produced the Harrison-Z'eldovich spectrum was quite spectacular and convinced many physicists that inflation must have actually occurred.

The Harrison-Z'eldovich hypothesis can be understood as follows. The early universe is filled with many different particles: nucleons such as protons and neutrons, electrons, photons, and neutrinos, and some form of dark matter needed to explain the structure of galaxies as observed today. The simplest possibility is that the overall density of the universe varied from place to place, but the relative abundances of the different particle species were the same everywhere. This possibility is realized in the simplest inflation models. Again, the simplest possibility is that the density variations take the form of a Gaussian random field. That is to say, the amplitudes of plane waves of each wavelength are chosen at random from a Gaussian probability distribution, and there is no correlation between modes of different wavelengths. More prosaically, the density variations are like small ripples on the surface of the sea, with random locations and no special features.

Recent observations of the cosmic background radiation have provided spectacular confirmation of the Harrison-Z'eldovich spectrum, combined with the simplest form of primordial density variations. The Boomerang and Maxima experiments used balloon-borne telescopes to map the cosmic background radiation over hundreds of square degrees on the sky, to a level of tens of microKelvin. They measured the amplitude of the temperature fluctuations as a function of angular scale. What was found was that as one goes from large to smaller scales, the amplitude grows and then oscillates. So far there is evidence for three peaks, on scales of a degree, half a degree, and one-third of a degree. This is in astonishingly close accord with the expectations from theory, under the assumption of the simplest form of perturbations. These measurements have also allowed a measurement of the spatial geometry of the universe on the largest visible scales. Again, the measurements are in accord with the simplest models of inflation, according to which there was a lot of inflation and the universe became spatially flat with exponential accuracy.

These measurements are a considerable success for inflation. However, one should not infer that inflation has been proven. The observations really confirm something much simpler: scale invariance and the "simplest" type of fluctuations. There is no direct evidence for the existence of the inflation field ?. Also, apart from the qualitative successes of inflation— the observed flatness of the universe and the scale invariance of the inhomogeneities—one does not have a quantitative prediction which provides specific evidence that inflation occurred. On the contrary, inflationary theory can only be reconciled with the observed amplitude of the primordial fluctuations if certain parameters in the theory are adjusted to very small values, to fit the data. It is possible that another physical mechanism could make the universe large, flat, and smooth and produce density variations of the observed form.

What would convince skeptics that inflation did, in fact, occur? One of the most distinctive inflationary predictions is that during the period of exponential expansion, gravitational waves would have been amplified and stretched to large scales just as the fluctuations in ? were. This leads again to a scale-invariant spectrum of gravitational waves, which would in principle be detectable in today's universe. The most powerful way to search for these waves employs the polarization of the cosmic microwave sky. If gravitational waves are present, they lead to a pattern of polarization that is impossible to obtain from ordinary density inhomogeneities. Observation of this pattern would be a much more direct confirmation of the inflationary mechanism. The Planck satellite, scheduled to be flown by the European Space Agency in 2007, should be able to see this signal, at least for the simplest inflation models.

Even if inflation did provide the mechanism for the Big Bang and the density variations in the universe, many theoretical questions are unresolved. The biggest problem in physics is that of quantizing the gravitational field. However, inflationary fluctuations involve both quantum mechanics and the effects of gravity. How can they be treated consistently, when there is no consistent theory? Current calculations of inflationary fluctuations are performed in an approximation (linear theory) in which the theoretical inconsistencies do not yet appear. Calculations to the next level of accuracy produce meaningless infinities and ambiguities that are still not resolved. Thus within its current framework, inflation can only be viewed as a provisional theory, recognized to be inconsistent at a deep level. More consistent theories of quantum gravity, including string theory and supergravity, so far do not seem to produce inflationary models of the form needed to match observation.

It is also important to emphasize that inflation does not solve many of the most fundamental puzzles in cosmology: Did the universe begin? And if so, how? Recent observations indicate the presence of a positive vacuum energy (or cosmological constant). This discovery was entirely unexpected within inflationary cosmology.

Why is there a cosmological constant today? The current cosmological constant is smaller than that needed for inflation by at least 100 orders of magnitude. So, inflationary cosmology requires two cosmological constants, differing by 10¹⁰⁰ in magnitude. Since the goal of inflation was to avoid "fine tuning," the actual need for it is quite disturbing. If the expansion of the universe is accelerating today, as observations indicate, where will that expansion lead? What is the future of the universe?

These questions illustrate that inflation is only a theory of the early universe, one that neither addresses how the universe began nor its current state nor its future direction. The successes of inflation are considerable, but they may yet be reproduced by other theories that are more complete.