Monday, March 26, 2012

Bianchi models in loop quantum cosmology

by Edward Wilson-Ewing, Marseille.

Parampreet Singh, LSU
Title: Physics of Bianchi models in LQC
PDF of the talk (500KB)
Audio [.wav 40MB], Audio [.aif 4MB].

The word singularity, in physics, is often used to denote a prediction that some observable quantity should be singular, or infinite. One of the most famous examples in the history of physics appears in the Rayleigh-Jeans distribution which attempts to describe the thermal radiation of a black body in classical electromagnetic theory. While the Rayleigh-Jeans distribution describes black body radiation very well for long wavelengths, it does not agree with observations at short wavelengths. In fact, the Rayleigh-Jeans distribution becomes singular at very short wavelengths as it predicts that there should be an infinite amount of energy radiated in this part of the spectrum: this singularity -which did not agree with experiment- was called the ultraviolet catastrophe.

This singularity was later resolved by Planck when he discovered what is called Planck's law, which is now understood to come from quantum physics. In essence, the discreteness of the energy levels of the black body ensure that the black body radiation spectrum remains finite for all wavelengths. One of the lessons to be learnt from this example is that singularities are not physical: in the Rayleigh-Jeans law, the prediction that there should be an infinite amount of energy radiated at short wavelengths is incorrect and indicates that the theory that led to this prediction cannot be trusted to describe this phenomenon. In this case, it is the classical theory of electromagnetism that fails to describe black body radiation and it turns out that it is necessary to use quantum mechanics in order to obtain the correct result.

In the figure below, we see that for a black body at a temperature of 5000 degrees Kelvin, the Rayleigh-Jeans formula works very well for wavelengths greater than 3000 nanometers, but fails for shorter wavelengths. For these shorter wavelengths, it is necessary to use Planck's law where quantum effects have been included.

Picture credit.

There are also singularities in other theories. Some of the most striking examples of singularities in physics occur in general relativity where the curvature of space-time, which encodes the strength of the gravitational field, diverges and becomes infinite. Some of the best known examples are the big-bang singularity that occurs in cosmological models and the black hole singularity that is found inside the event horizon of every black hole. While some people have argued that the big-bang singularity represents the beginning of time and space, it seems more reasonable that the singularity indicates that the theory of general relativity cannot be trusted when the space-time curvature becomes very large and that quantum effects cannot be ignored: it is necessary to use a theory of quantum gravity in order to study the very early universe (where general relativity says the big bang occurs) and the center of black holes.

In loop quantum cosmology (LQC), simple models of the early universe are studied by using the techniques of the theory of loop quantum gravity. The simplest such model (and therefore the first to be studied) is called the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) space-time. This space-time is homogeneous (the universe looks the same no matter where you are in it), isotropic (the universe is expanding at the same rate in all directions), and spatially flat (the two other possibilities are closed and open models which have also been studied in LQC) and is considered to provide a good approximation to the large-scale dynamics of the universe we live in. In LQC, it has been possible to study how quantum geometry effects become important in the FLRW model when the space-time curvature becomes so large that is is comparable to one divided by the Planck length squared. A careful analysis shows that the quantum geometry effects provide a repulsive force that causes a “bounce” and ensures that the singularity predicted in general relativity does not occur in LQC. We will make this more precise in the next two paragraphs.

By measuring the rate of expansion of the universe today, it possible to use the FLRW model in order determine the size and the space-time curvature of the universe was in the past. Of course, these predictions will necessarily depend on the theory used: general relativity and LQC will not always give the same predictions. General relativity predicts that, as we go back further in time, the universe becomes smaller and smaller and the space-time curvature becomes larger and larger. This keeps on going until around 13.75 billion years ago the universe has zero volume and an infinite space-time curvature. This is called the big bang.

In LQC, the picture is not the same. So long as the space-time curvature is considerably smaller than one divided by the Planck length squared, it predicts the same as general relativity. Thus, as we go back further in time, the universe becomes smaller and the space-time curvature becomes larger. However, there are some important differences when the space-time curvature nears the critical value of one divided by the Planck length squared: in this regime there are major modifications to the evolution of the universe that come from quantum geometry effects. Instead of continuing to contract as in general relativity, the universe instead slows its contraction before starting to become bigger again. This is called the bounce. After the bounce, as we continue to go further back in time, the universe becomes bigger and bigger and the space-time curvature becomes smaller and smaller. Therefore, as the space-time curvature never diverges, there is no singularity.

Photo credit.

In loop quantum cosmology, we see that the big bang singularity in FLRW models is avoided due to quantum effects and this is analogous to what happened in the theory of black body radiation: the classical theory predicted a singularity which was resolved once quantum effects were included.

This observation raises an important question: does LQC resolve all of the singularities that appear in cosmological models in general relativity? This is a complicated question as there are many types of cosmological models and also many different types of singularities. In this talk, Parampreet Singh explains what happens to many different types of singularities in models, called the Bianchi models, that are homogeneous but anisotropic (each point in the universe is equivalent, but the universe may expand in different directions at different rates). The main result of the talk is that all “strong” singularities in the Bianchi models are resolved in LQC.