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.