There’s a lot of compelling evidence for the Big Bang, but what preceded it? The most accepted model is called Inflation, but it’s not the kind of inflation that Alan Greenspan needs to fear.

Physicists define the boundaries of physics by trying to describe them theoretically and then testing that description against observation. Our observed expanding Universe is very well described by flat space, with critical density supplied mainly by dark matter and a cosmological constant that should expand forever.

When the scale factor a(t) was very small, radiation energy density was much larger than the matter and vacuum energy densities. The temperature gets smaller as the scale factor rises:

The experimental understanding of particle physics starts to poop out after energies above the electroweak unification scale, around 1TeV. At a very small scale factor, or a very high temperature, Grand Unified Theories, supersymmetry, and string theory have to be taken into account in the cosmological modeling.

This exploration is guided by three outstanding problems with the Big Bang cosmological model:

1. The flatness problem

2. The horizon problem

3. The magnetic monopole problem

Contents

## Flatness problem

The Einstein equation predicts that any deviation from flatness in an expanding Universe filled with matter or radiation tends to grow larger as the Universe expands. The ratio of the matter density to the curvature term in the Einstein equation

shows that tiny deviations from flatness at a much earlier time would grow linearly with scale factor as the Universe grows and comes to dominate the evolution of spacetime. This is consistent with the fact that matter attracts matter through the gravitational force. Small lumps are going to get bigger when gravity does its thing.

If the deviations from flatness are observed to be very small today, then extrapolating back to when the Universe was much smaller, the deviations from flatness must have been immeasurably small.

So why did the Big Bang start off with the deviations from flat spatial geometry being immeasurably small? This is called the **flatness problem** of Big Bang cosmology.

## Horizon problem

The cosmic microwave background is the cooled remains of the radiation from the radiation-dominated phase of the Big Bang. Observations of the cosmic microwave background show that it is highly **isotropic** thermal radiation. The temperature of this thermal radiation is 2.73° Kelvin. The variations observed in this temperature across the night sky are very tiny.

If the cosmic microwave background is at such a uniform temperature, it should mean that the photons have been thermalized through repeated particle collisions. But this presents a problem with causality in an expanding universe. Using the Robertson-Walker metric with k=0, assuming that a(t) ~ t^{m}, the distance a photon could have traveled since the beginning of the Big Bang at t=0 to some other time t_{0} is given by the horizon size r_{H}(t_{0})

The power m is set by the equation of state for the energy source under consideration, so that

For a matter or radiation dominated Universe, m=2/3 or 1/2, respectively. Therefore the horizon size is finite, because the integral converges as t -> 0 for m<1, and it is much smaller than necessary to account for the isotropy observed in the cosmic microwave background. To make the horizon integral diverge or grow extremely large would require a Universe that expanded more rapidly than is possible using matter or radiation in the Einstein equations.

The horizon size predicted by the existing Big Bang model is too small to account for the observed isotropy in the cosmic microwave background to have evolved naturally by thermalization. So that’s the horizon problem.

## Magnetic monopole problem

A **magnetic monopole** would be a magnet with only one pole. In other words, it would have net magnetic charge. But magnetic monopoles have never been observed or created experimentally. When a magnet with a north and south pole is cut in half, it becomes two magnets, each with its own north and south poles. There doesn’t seem to be a way to create a magnet with only one pole. Yet particle theories like Grand Unified Theories and superstring theory predict magnetic monopoles should exist.

In particle theory, a magnetic monopole arises from a topological glitch in the vacuum configuration of gauge fields in a Grand Unified Theory or other gauge unification scenario. The length scale over which this special vacuum configuration exists is called the correlation length of the system. A correlation length cannot be larger than causality would allow, therefore the correlation length for making magnetic monopoles must be at least as big as the horizon size determined by the metric of the expanding Universe.

According to that logic, there should be at least one magnetic monopole per horizon volume as it was when the symmetry breaking took place.

This creates a problem, because it predicts that the monopole density today should be 10^{11} times the critical density of our Universe, according to the Big Bang model.

But so far, physicists have been unable to find even one.

## Inflationary universe?

Matter and radiation are gravitationally attractive, so in a maximally symmetric spacetime filled with matter, the gravitational force will inevitably cause any lumpiness in the matter to grow and condense. That’s how hydrogen gas turned into galaxies and stars. But vacuum energy comes with a high vacuum pressure, and that high vacuum pressure resists gravitational collapse as a kind of repulsive gravitational force. The pressure of the vacuum energy flattens out the lumpiness, and makes space get flatter, not lumpier, as it expands.

So one possible solution to the flatness problem would be if our Universe went through a phase where the only energy density present was a uniform vacuum energy. The maximally symmetric solution to the Einstein equation under those conditions is called de Sitter space and the metric can be written

In de Sitter cosmology, the Hubble parameter H is constant and related to the cosmological constant as shown.

The vacuum energy density is uniform in space and time, so the ratio of the curvature of space to the energy density will decrease exponentially as space expands in time:

Any deviations from flatness will be exponentially suppressed by the exponential expansion of the scale factor, and the flatness problem is solved.

Both the de Sitter spacetime and the Robertson-Walker spacetime start expanding from a(t) close to zero. But for a spacetime with matter or radiation, a(t) goes to zero when the time t goes to zero, because a(t) goes like a power of t. When the scale factor depends exponentially on time, the scale factor goes to zero when time t goes to minus infinity. Therefore the horizon distance integral can blow up instead of neatly converge

and solve the horizon problem.

## But how does Inflation work?

The vacuum energy that drives the rapid expansion in an inflationary cosmology comes from a scalar field that is part of the spontaneous symmetry breaking dynamics of some unified theory particle theory, say, a Grand Unified Theory or string theory.

This scalar field is sometimes called the **inflaton**. The equation of motion for this field in the de Sitter metric above is

and the Einstein equation with a scalar field density becomes

The conditions for inflationary behavior require that the scalar field time derivatives are small compared to the potential, so that most of the energy of the scalar field is in potential energy and not kinetic energy

These are called the slow roll conditions because the scalar field evolves slowly when these conditions are satisfied.

Another crucial element in an inflationary model is the thermal behavior of the scalar field effective potential V_{eff}(f). The effective potential includes quantum corrections from particle scattering. The shape of the potential can change with temperature, allowing for phase transitions. At very high temperatures, higher than some critical temperature T_{crit}, the minimum of the effective potential is at zero, in the symmetric phase of the theory.

As the temperature drops to T=T_{crit}, a second minimum forms in the potential at some value f_{0} and the vacuum with f=0 becomes metastable. At temperature T<T_{crit}, the new minimum f=f_{0} becomes the energetically favorable vacuum configuration. (The scale usually assumed for T_{crit} is the GUT scale of about 10^{14} GeV.)

In an inflation model, rather than making a uniform transition to the new vacuum, the field stays in the old vacuum, now called the false vacuum. (When steam does this in the gas-to-liquid phase transition of water at T_{crit}=373°K, it is called supercooling.) The vacuum energy of the supercooled false vacuum drives a de Sitter expansion of the Universe (or the part of it that becomes our Universe) which is called the period of inflation, with cosmological constant L given by

where V(0) is the value of the scalar potential in the false vacuum.

Eventually bubbles form of the true vacuum in the broken symmetric phase with f=f_{0}. The slow roll parameters grow large and the inflationary phase comes to an end. If the false vacuum bubble has expanded by at least 60 e-folds, the horizon and flatness problems are no more, because the radiation-dominated expansion that follows comes out of one extraordinarily flat causally connected domain.

## A testable prediction?

It’s always good to have testable predictions from a theory of physics, and the inflation theory has a distinct prediction about the density variations in the cosmic microwave background. A bubble of inflation consists of accelerating vacuums. In this accelerating vacuum, a scalar field will have very small thermal fluctuations that are nearly the same at every scale, and the fluctuations will have a Gaussian distribution. This prediction fits current observations and will be tested with greater precision by future measurements of the cosmic microwave background.

## So are all the problems solved?

Despite the prediction above, inflation as described above is far from an ideal theory. It’s too hard to stop the inflationary phase, and the monopole problem has other ways of resurfacing in physics. Many of the assumptions that go into the model, such as an initial high temperature phase and a single inflating bubble have been questioned and alternative models have been developed.

Today’s inflation models have evolved beyond the original assumption of a single inflation event giving birth to a single Universe, and feature scenarios where universes nucleate and inflate out of other universes in the process called **eternal inflation**.

There is also another attempt to solve the problems of Big Bang cosmology using a scalar field that never goes through an inflationary period at all, but evolves very slowly so that we observe it as being constant during our own era. This model is called **quintessence**, after the ancient spiritual belief in the Quinta Essentia, the spiritual matter from which the four forms of physical matter are made.

Another currently unsolved problem is how to accommodate for Inflation in string cosmology and M-theory cosmology. There are dimensions to compactify, branes to wrap, hierarchies to set, geometry to resolve, supersymmetry to break — a laundry list of processes and transitions that have to be described within a string theory cosmology.

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