Preprint Article Version 4 Preserved in Portico This version is not peer-reviewed

Extended General Relativity for a Curved Universe

Version 1 : Received: 15 October 2020 / Approved: 15 October 2020 / Online: 15 October 2020 (11:47:59 CEST)
Version 2 : Received: 18 October 2020 / Approved: 19 October 2020 / Online: 19 October 2020 (10:43:06 CEST)
Version 3 : Received: 5 November 2020 / Approved: 6 November 2020 / Online: 6 November 2020 (13:11:37 CET)
Version 4 : Received: 1 January 2021 / Approved: 4 January 2021 / Online: 4 January 2021 (11:21:08 CET)

How to cite: Al-Fadhli, M.B. Extended General Relativity for a Curved Universe. Preprints 2020, 2020100320 (doi: 10.20944/preprints202010.0320.v4). Al-Fadhli, M.B. Extended General Relativity for a Curved Universe. Preprints 2020, 2020100320 (doi: 10.20944/preprints202010.0320.v4).


The recent Planck Legacy release revealed the presence of an enhanced lensing amplitude in the cosmic microwave background (CMB). Notably, this amplitude is higher than that estimated by the lambda cold dark matter model (ΛCDM), which endorses the positive curvature of the early Universe with a confidence level greater than 99%. Although General Relativity (GR) performs accurately in the local/present Universe where spacetime is almost flat, its lost boundary term, incompatibility with quantum mechanics and the necessity of dark matter and dark energy might indicate its incompleteness. By utilising the Einstein–Hilbert action, this study presents extended field equations considering the pre-existing/background curvature and the boundary contribution. The extended field equations consist of Einstein field equations with a conformal transformation feature in addition to the boundary term, which could remove singularities from the theory and facilitate its quantisation. The extended equations have been utilised to derive the evolution of the Universe with reference to the scale factor of the early Universe and its radius of curvature.

Subject Areas

General Relativity; Curved Universe

Comments (1)

Comment 1
Received: 4 January 2021
Commenter: Mohammed Al-Fadhli
Commenter's Conflict of Interests: Author
Comment: Dear Editor, 

I hope you are doing very well

This version includes the utilisation of the extended field equations to model the Universe's evolution

A very happy new year with best wishes 

Kindest regards,
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