This paper studies how a version of gravity built from “fractional” operators changes the equations that govern the universe. The authors work with a classical, nonlocal theory in which the usual wave operator (the d’Alembertian) is raised to a non-integer power γ. Such fractional operators are self-adjoint, which makes the equations well behaved classically. The goal is to begin exploring the cosmology of this proposal, which the authors describe as the classical limit of a candidate ultraviolet-complete theory of quantum gravity.

The researchers first derive covariant, nonlocal equations of motion for an arbitrary fractional exponent γ. They then reduce those equations to the cosmological (Friedmann) equations on a homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker background. Two self-adjoint mathematical representations of the fractional d’Alembertian are used: the Balakrishnan–Komatsu form and a Fresnel-type form. Both give the same cosmological solutions, a result the authors interpret as a universality property of the fractional formulation.

From these modified Friedmann equations the authors find that a de Sitter solution—an exponentially expanding universe, like the standard model’s dark-energy epoch—is an exact and stable solution of the theory. They also look for non-singular bouncing cosmologies, where the universe contracts and then re-expands without a big-bang singularity. Such exact bouncing solutions do exist in their set-up, but they require exotic matter: either a phantom fluid with equation-of-state parameter w<−1 (pressure more negative than energy density) or a ghost fluid with negative energy density ρ<0. In the ghost case they identify a new type of finite-future singular behaviour in the barotropic index (the relation between pressure and density) that they call “cosmic emptiness.”