Bianchi space-times in loop quantum cosmology

Brajesh Gupt, LSU 

Title: Bianchi I LQC, Kasner transitions and inflation
PDF of the talk (800k) Audio [.wav 30MB] Audio [.aif 3MB]

by Edward Wilson-Ewing

The Bianchi space-times are a generalization of the simplest Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmological models.  While the FLRW space-times are assumed to be homogeneous (there are no preferred points in the space-time) and isotropic (there is no preferred direction), in the Bianchi models the isotropy requirement is removed.  One of the main consequences of this generalization is that in a Bianchi cosmology, the space-time is allowed to expand along the three spatial axes at different rates.  In other words, while there is only one Hubble rate in FLRW space-times, there are three Hubble rates in Bianchi cosmologies, one for each of the three spatial dimensions.

For example, the simplest Bianchi model is the Bianchi I space-time whose metric is given by

ds² = - dt² + a₁(t)² (dx₁)² +  a₂(t)² (dx₂)² +  a₃(t)² (dx₃)²,

where the a_i(t) are the three scale factors.  This is in contrast to the flat FLRW model where there is only one scale factor.

It is possible to determine the exact form of the scale factors by solving the Einstein equations.  In a vacuum, or with a massless scalar field, it turns out that the i^th scale factor is simply given by the time elevated to the power k_i: a_i(t) = t^ki, where these k_i are constant numbers and are called the Kasner exponents.  There are some relations between the Kasner exponents that must be satisfied, so that once the matter content has been chosen, one Kasner exponent between -1 and 1 may be chosen freely, and then the values of the other two Kasner exponents are determined by this initial choice.

In addition to allowing more degrees of freedom than the simpler FLRW models, the Bianchi space-times are important due to the central role they play in the Belinsky-Khalatnikov-Lifshitz (BKL) conjecture.  According to the BKL conjecture, as a generic space-like singularity is approached in general relativity time derivatives dominate over spatial derivatives (with the exception of some small number of “spikes” which we shall ignore here) and so spatial points decouple from each other.  In essence, as the spatial derivatives become negligible, the complicated partial differential equations of general relativity reduce to simpler ordinary differential equations close to a space-like singularity.  Although this conjecture has not been proven, there is a wealth of numerical studies that supports it.

If the BKL conjecture is correct, and the ordinary differential equations can be trusted, then the solution at each point is that of a homogeneous space-time.  Since the most general homogeneous space-times are given by the Bianchi space-times, it follows that as a space-like singularity is approached, the geometry at each point is well approximated by a Bianchi model.

This conjecture is extremely important from the point of view of quantum gravity, as quantum gravity effects are expected to become important precisely when the space-time curvature nears the Planck scale.  Therefore, we expect quantum gravity effects to become important near singularities.  What the BKL conjecture is telling us is that understanding quantum gravity effects in the Bianchi models, which are relatively simple space-times, can shed significant insight into the problem of singularities in gravitation.

What is more, studies of the BKL dynamics show that for long periods of time, the geometry at any point is given by the Bianchi I space-time and during this time the geometry is completely determined by the three Kasner exponents introduced above in the third paragraph.  Now, the Bianchi I solution does not hold at each point eternally, rather there are occasional transitions between different Bianchi I solutions called Kasner or Taub transitions.  During a Kasner transition, the three Kasner exponents rapidly change values before becoming constant for another long period of time.  Now, since the Bianchi I model provides an excellent approximation at each point for long periods of time, understanding the dynamics of the Bianchi I space-time, especially at high curvatures when quantum gravity effects cannot be neglected, may help us understand the behaviour of generic singularities when quantum gravity effects are included.

In loop quantum cosmology (LQC), for all of the space-times studied so far including Bianchi I, the big-bang singularity in cosmological space-times is resolved by quantum geometry effects.  The fact that the initial singularity in the Bianchi I model is resolved in loop quantum cosmology, in conjunction with the BKL conjecture, gives some hope that all space-like singularities may be resolved in loop quantum gravity.  While this result is encouraging, there remain open questions regarding the specifics of the evolution of the Bianchi I space-time in LQC when quantum geometry effects are important.

One of the main goals of Brajesh Gupt's talk is to address this precise question.  Using the effective equations, which provide an excellent approximation to the full quantum dynamics for the FLRW space-times in LQC and are expected to do the same for the Bianchi models, it is possible to study how the quantum gravity effects that arise in loop quantum cosmology modify the classical dynamics when the space-time curvature becomes large and replace the big-bang singularity by a bounce.  In particular, Brajesh Gupt describes the precise manner of how the Kasner exponents ---which are constant classically--- evolve deterministically as they go through the quantum bounce.  It turns out that there is some sort of a Kasner transition that occurs around the bounce, the details of which are given in the talk.

The second part of the talk considers inflation in Bianchi I loop cosmologies.  Inflation is a period of exponential expansion of the early universe which was initially introduced in order to resolve the so-called horizon and flatness problems.  One of the major results of inflation is that it generates small fluctuations that are of exactly the form that are observed in the small temperature variations in the cosmic microwave background today.  For more information about inflation in loop quantum cosmology, see the previous ILQGS talks by William Nelson, Ivan Agullo, Gianluca Calcagni and David Sloan, as well as the blog posts that accompany these presentations.

Although inflation is often considered in the context of isotropic space-times, it is important to remember that in the presence of matter fields such as radiation and cold dark matter, anisotropic space-times will become isotropic at late times.  Therefore, it is not because our universe appears to be isotropic today that it necessarily was some 13.8 billion years ago.  Because of this, it is necessary to understand how the dynamics of inflation change when anisotropies are present.  As mentioned at the beginning of this blog post, there is considerably more freedom in Bianchi models than in FLRW space-times, and so the expectations coming from studying inflation in isotropic cosmologies may be misleading for the more general situation.

There are several interesting issues that are worth considering in this context, and in this talk the focus is on two questions in particular.  First, is it easier or harder to obtain the initial conditions necessary for inflation? In other words, is more or less fine-tuning required in the initial conditions?  As it turns out, the presence of anisotropies actually makes it easier for a sufficient amount of inflation to occur.  The second problem is to determine how the quantum geometry effects from loop quantum cosmology change the results one would expect based on classical general relativity.  The main modification found here has to do with the relation between the amount of anisotropy present in the space-time (which can be quantified in a precise manner) and the amount of inflation that occurs.  While there was a monotonic relation between these two quantities in classical general relativity, this is no longer the case when loop quantum cosmology effects are taken into account.  Instead, there is now a specific amount of anisotropy which extremizes the amount of inflation that will occur, and there is a turn around after this point.  The details of these two results are given in the talk.