In a recent paper, Chamseddine and Mukhanov have proposed a modification of general relativity that features a limiting curvature. The model is in the framework of their earlier work on "mimetic gravity" and coupled general relativity to a scalar field $\phi$ that is constrained as

\[

g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi = 1

\]

in the mostly minus signature convention. In addition to this constraint, the action contains the term $f(\Box \phi)$. In the homogeneous and isotropic sector, $\Box \phi$ is proportional to the Hubble rate $\dot a / a$.

The main input that is needed to fully define this model is the function $f$, which can a priori be chosen arbitrarily up to a consistency requirement with general relativity at low curvatures. In their paper, Chamseddine and Mukhanov make the proposal

\[

f(\chi) = - \chi_m^2 \, g\left( \sqrt{\frac{2}{3}} \frac{\chi}{\chi_m} \right), ~~~ g(y) = -1 -\frac{y^2}{2} + y \arcsin y + \sqrt{1-y^2}

\]

which leads in the homogeneous and isotropic sector to a very simple correction to the Friedmann equation of the form

\[

\frac{\dot a}{a} \propto \rho \left(1- \frac{\rho}{\rho_\text{crit}}\right)

\]

where $\rho$ is the matter energy density and $\rho_\text{crit}$ a free parameter. This equation leads to a bouncing universe at the critical matter energy density $\rho_\text{crit}$.

In the homogeneous but non-isotropic sector, one can again solve the equations of motion to find singularity resolution, however with somewhat more complicated details. An analogous modification for Schwarzschild black holes is discussed in a second paper.

It is interesting to note that the corrected Friedmann equation above is already known from loop quantum gravity, more precisely from the effective equations of loop quantum cosmology. This already suggests that at least in the homogeneous and isotropic sector, the two theories should agree.

In fact, the choice for $f$ can be read off from the effective action of loop quantum cosmology, as computed first in this paper, equation 7. A more detailed investigation in based on the Hamiltonian formulation of the model proposed by Chamseddine and Mukhanov confirms this. The details have been written up in this paper, see also this paper for an intendedly simultaneous publication of related results.

In addition, a quantum mechanical argument is given there why this choice of $f$ is the simplest one consistent with quantum mechanics: Mukhanov and Chamseddine motivate the structure of their model with non-commutative geometry, in particular the quantisation of three-volume. In a canonical quantisation setting, this leads one directly to the simplest version of loop quantum cosmology. The choice for $f$ then follows by a Legendre-transform.

If one goes beyond the homogeneous and isotropic setting, the effective loop quantum cosmology deviates from the model of Chamseddine and Mukhanov in the details. Still, the qualitative behavior is the same for homogeneous but non-isotropic metrics. Beyond this, a comparison is hard to make because the effective LQG dynamics are in general not known. This on the other hand suggests to simply use the new model as a toy model for loop quantum gravity, i.e. as a toy model effective action that is completely known. The main application of this would be to obtain an understanding for the effective geometries that can emerge from the quantum corrections build into loop quantum gravity beyond the cosmological setting.

# relatively quantum

A blog around my research in quantum gravity

## Monday, April 3, 2017

## Friday, December 23, 2016

### Holographic signatures of resolved cosmological singularities

After some longer silence partly due to moving to a new location (LMU Munich) and teaching my first regular lecture (Theoretical mechanics for lyceum teachers and computational science at Regensburg University), I hope to write more regularly again in the future.

As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared here. Together with Andreas SchÃ¤fer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is resolved in a manner inspired by results from loop quantum gravity.

Building (specifically) on recent work by Engelhardt, Hertog, and Horowitz (as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to some definite statement of how exactly they should be done.

As a start, a new paper on using loop quantum gravity in the context of AdS/CFT has finally appeared here. Together with Andreas SchÃ¤fer and John Schliemann from Regensburg University, we asked the question of what happens in the dual CFT if you assume that the singularity on the gravity side is resolved in a manner inspired by results from loop quantum gravity.

Building (specifically) on recent work by Engelhardt, Hertog, and Horowitz (as well as many others before them) using classical gravity, we found that a finite distance pole in the two-point-correlator of the dual CFT gets resolved if you resolve the singularity in the gravity theory. Several caveats apply to this computation, which are detailed in the papers. We view this result therefore as a proof of principle that such computations are possible, as opposed to some definite statement of how exactly they should be done.

## Wednesday, September 21, 2016

### Coarse graining and state refinements

Coarse graining has become an increasingly important topic in loop quantum gravity with several researchers working on it. As usual in physics, one is interested in integrating out microscopic degrees of freedom and doing computations on an effective coarse level. How exactly the states, observables, and dynamics of the theory should change under such a renormalisation group flow is only poorly understood at the moment.

Recently, I have written a paper about this topic in a simplified context which is tailored to reproduce loop quantum cosmology from loop quantum gravity. Here, one can explicitly coarse grain the relevant observables, the scalar field momentum and the volume of the spatial slice, and check that their dynamics remains unchanged under such a coarse graining.

The reason that this works is rooted in an exact solution to loop quantum cosmology which can be imported in this full theory setting. In particular, the form of the dynamics of this solution is independent of the volume of the universe. It then follows that if we concentrate the volume of some set of vertices at a single one, the evolution of this coarse grained volume will agree with the evolution of the sum of the individual volumes.

The content of the paper is sketched in the brief talk linked above.

## Wednesday, July 27, 2016

### Heuristics for Lorentz violations in LQG

Predictions for possible Lorentz violations are a key area of interesting phenomenology in quantum gravity. Within loop quantum gravity, it has not been possible so far to reliably extract predictions for Lorentz violations. Existing claims were based on simplified toy models inspired by LQG, but not implied. This situation hasn’t changed so far, but there is an interesting development which hints at which order we might expect such Lorentz violations to be found.

## Tuesday, July 19, 2016

### Elements of loop quantum gravity

The lectures start with a general introduction to quantum gravity, including a theoretical motivation, possible experimental tests, and the previously posted list on approaches to the subject. There is also an improved (as compared to here) estimate on the local Lorentz invariance violation based on anomaly freedom of effective constraints. I am planning to write about it in more detail in the future.

Next, an introduction to loop quantum cosmology is given, a draft of which has appeared here before. The new version features some improvements in the presentation and some simplifications in the derivation.

The remaining part of the lectures introduces full loop quantum gravity with minimal technical details. The derivation of geometric operators is sketched and different approaches to the dynamics are discussed. Promising lines of current research are mentioned and evaluated. Exercises are included at the end of each section.

The present lecture notes are somewhat complementary to several other sets of lecture notes existing in the literature in that they refrain from technical details and give a broad overview of the subject, including motivations and current trends. If someone spots mistakes or has suggestions for a better presentation, I would be happy to hear about it.

## Wednesday, June 8, 2016

### Strings meet Loops via AdS/CFT (Helsinki Workshop on Quantum Gravity 2016)

This rerecorded talk was originally given at the Helsinki Workshop on Quantum Gravity on June 2, 2016. I was invited by the organisers to talk about a recent paper, which was intended as an invitation for people to become interested in the subject, as opposed to giving concrete and detailed results.

In particular, I am very much looking forward to discussions about this topic and criticism of the ideas, in particular from experts in string theory. In the long run, much can be gained in my point of view from intensifying the exchange between researchers in loop quantum gravity and string theory.

In this context, it is certainly worth pointing out a recent article by Sabine Hossenfelder for Quanta Magazine, as well as a blogpost of hers on the paper I wrote.

## Wednesday, June 1, 2016

### Graph superspositions and improved regularisations

An open issue in this context has been to properly derive the so called “improved “ dynamics of loop quantum cosmology, which are consistent with observation and do not feature some unphysical properties of the original formulation. This is somewhat tricky if one uses standard connection variables along with a gauge group like SU(2) or U(1) for the following reason:

Subscribe to:
Posts (Atom)