Extending Friedmann Equations Using Fractional Derivatives Using a Last Step Modification Technique: The Case of a Matter Dominated Accelerated Expanding Universe

We present a toy model for extending the Friedmann equations of relativistic cosmology using fractional derivatives. We do this by replacing the integer derivatives, in a few well-known cosmological results with fractional derivatives leaving their order as a free parameter. All this with the intention to explain the current observed acceleration of the Universe. We apply the Last Step Modification technique of fractional calculus to construct some useful fractional equations of cosmology. The fits of the unknown fractional derivative order and the fractional cosmographic parameters to SN Ia data shows that this simple construction can explain the current accelerated expansion of the Universe without the use of a dark energy component with a MOND-like behaviour using Milgrom's acceleration constant which sheds light into to the non-necessity of a dark matter component as well.

Automated summary

The study presents a toy model for extending the Friedmann equations of relativistic cosmology using fractional derivatives, with the aim to explain the current observed acceleration of the Universe without the need for a dark matter component. Using fractional calculus, the model shows that the accelerated expansion of the Universe can be explained without the use of dark energy component with a MOND-like behavior.