Fractional Derivative Cosmology

TL;DR

This paper explores the implications of using fractional derivatives in cosmology, modifying the Friedmann equations, and discusses the challenges of defining fractional derivative geometry.

Contribution

It introduces fractional derivative modifications to cosmological equations and examines the complexities of fractional geometry in a simplified two-dimensional model.

Findings

Fractional derivatives modify cosmological equations.

Defining fractional geometry is mathematically challenging.

Simple two-dimensional models reveal ambiguities in fractional derivatives.

Abstract

The degree by which a function can be differentiated need not be restricted to integer values. Usually most of the field equations of physics are taken to be second order, curiosity asks what happens if this is only approximately the case and the field equations are nearly second order. For Robertson-Walker cosmology there is a simple fractional modification of the Friedman and conservation equations. In general fractional gravitational equations similar to Einstein's are hard to define as this requires fractional derivative geometry. What fractional derivative geometry might entail is briefly looked at and it turns out that even asking very simple questions in two dimensions leads to ambiguous or intractable results. A two dimensional line element which depends on the Gamma-function is looked at.