Inflation Theory

Contents

Inflation Theory
A Simple Mathematical Model
Magnetic Monopoles

Inflation Theory

inflation According to the latest cosmological model, the universe sprang into being about 14 billion years ago. At birth, the space was likely to have been curved and warped due to quantum effect within the tiny speck and time may be meaningless. After about 10-35 seconds, there began a brief period of exponentially fast expansion, known as inflation, that ironed out any curves or warps in space and made the universe flat (because it becomes so large). Inflation also predicts a much smaller initial region, which is required for smoothing out the distribution of matter and radiation, only leaving behind tiny density fluc-

Figure 01 Cosmic Inflation

 tuations that match the observed spatial variations in the cosmic microwave background radiation and provide the seeds for galaxy formation.
inflation The blue strip in Figure 01 shows the period of inflation from 10-35 sec to 10-32 sec after the initial expansion. Figure 02 shows the actual size of the universe after the inflation. Our observable unvierse is only part of the whole thing. The mechanism to drive the inflation is related to a "yet-to-be-discovered" inflaton field, which is thought to be similar to the Higgs fields responsible for the mass of the elementary particles. When the temperature fell below a certain value, a phase transition (similar to the transition of water to ice at 0oC with the release of latent heat) of the inflaton field occurred. The phase transition released energy, which was conversed to hot matter and radiation. It also developed repulsive force to drive the inflation. The inflation stopped when the inflaton field settled down into lower energy state.

Figure 02 Unobservable Universe
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Recent observations of the cosmic microwave background radiation in the post-inflationary universe support the inflation theory. But the precise inflation mechanism is still unclear. There are issues about the nature of the inflaton field, the events that led to the onset of inflation and then the nagging question about the beginning of the universe. It is hoped that advances in fundamental physics (such as superstring theory) will eventually address these issues.

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A Simple Mathematical Model

inflation 2 The concept of cosmic inflation can be illustrated by simple mathematics using only elementary calculus. Suppose the universe is uniform and isotropic as demonstrated by observations. This means that every point in the universe is similar to every other point and can be considered as the "centre" (Figure 03). Now consider a "test particle" of mass m and at distance R from the centre. Since only mass inside the sphere has a net effect on the particle, and if the total mass M insider the sphere of radius R is constant, then the potential energy of

Figure 03 Repulsion vs Attraction

 the particle is:

V(R) = -GMm/R ---------- (1)
where G is the gravitational constant.

The gravitational force on the particle is the negative derivative of V(R):

F(R) = -dV(R)/dR = -GMm/R2 ----------(2)
where the minus sign signifies attraction, which tends to pull the particle towards the centre.

Next consider the case where the total mass M insider the sphere of radius R is not constant, instead the density D is fixed. The the gravitational energy of the particle is:

V(R) = -GmD x (4/3) x R2 ---------- (3)

The force on the particle is now:

F(R) = -dV(R)/dR = 2GmD x (4/3) x R ---------- (4)
where the plus sign indicates that the force is repulsive.

The particles are now being pushed away, the whole universe will expand (inflate). The mass inside the sphere will increase with R like:

M = (4/3) x D x R3 ---------- (5)

Thus in this case, there is a repulsive force to drive the inflation; matter and radiation are being created according to
Eq.(5). Under normal circumstance, the density D would decrease with the expansion. It is thought that the decay of the "inflaton field" to the lower energy state is responsible for keeping the density D constant.

Inflation Figure 04 shows the variation of the energy density since the Big Bang. It maintained a constant value during the inflationary era as described in the simple mathematical model. The energy density of water (1021 ergs/cm3) and of an atomic nucleus (1036 ergs/cm3) are included in the graph for comparison.

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Figure 04 Energy Density

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Magnetic Monopoles

The inflation theory also offers an explanation for the absence of magnetic monopoles. It was back in the 19th century when scientists noticed the asymmetry between the electric and magnetic field sources. While electric field can be generated from monopole, dipole, and other multipoles, the magnetic monopole is missing in this world. In experiment, no one can isolate a "north" or a "south" magnetic pole - chopping up a magnet just produces smaller magnets, each with two poles. Correspondingly in theory, there is no magnetic charge in the electromagnetic field equations. Observationally, the magnetic field pervading the entire Milky Way implies that there is a lack of magnetic monopoles to cancel out (or short out) such field on the galactic scale. But the idea about monopoles was rather persistent. In 1931 Paul Dirac was trying to understand why electric charges were quantized. He devised an elegant explanation, which would work only if monopoles indeed existed. Monopoles are now perceived in a new guise as "knots" in the vacuum according to the Grand Unified Theories (GUT). It was realized that if GUT were correct, monopoles must have created only 10-36 seconds after the Big Bang, when the forces differentiated. These monopoles, would be very massive - about 1015 times heavier than ordinary particles - and would therefore be impossible to make in the lab. However, the number expected to have survived from the early universe seemed embarrassingly large: there would have been enough to short out the galactic magnetic field; even worse, their total mass would far exceed that of everything else in the universe (far too much, even, for the dark matter). For GUT physics, monopoles are extremely interesting objects: they have an onion-like structure, which contains the whole world of GUT (Figure 04):
Magnetic Monopole
  • Near the center ( about 10-29 cm ) there is a GUT symmetric vacuum.
  • At about 10-16 cm, its content is the electroweak gauge fields of the standard model.
  • At 10-15 cm, it is made up of photons and gluons.
  • At the edge to the distance of 10-13 cm, there are fermion-antifermion pairs.
  • Far beyond nuclear distances it behaves as a magnetically-charged pole of the Dirac type.

    • In effect, the sequence of events during the earliest moment of the universe had been fossilized inside the magnetic monopole.

Figure 04 Monopole Structure in GUT

One of the successes with the inflation theory was that it solved this so-called "monopole problem": monopoles would be exponentially diluted during the inflation, to such an extent that there would be little chance of even one in the Milky Way. By the way, skeptics about exotic physics might not be hugely impressed by a theoretical argument to explain the absence of particles that are themselves only hypothetic.

Meanwhile in 1995 the MACRO§ (Monopole, Astrophysics, and Cosmic Ray Observatory) detector located at the Gran Sasso National Laboratories in Italy has been designed to look for supermassive magnetic monopoles among other exotic particles. It had stopped collecting data by December 2000 and represents another failed attempt for the search of magnetic monopoles.

§A short introduction to MACRO can be found in: http://www.aas.org/publications/baas/v31n2/head99/246.htm