Some of the more notable theories beyond general relativity or applying general relativity in non-conventional (and not widely accepted) ways that are gravity based models to explain dark matter and/or dark energy phenomena which are not MOND or a close elaboration of it include the following theories:
* Moffat's MOG, see, e.g., J.W. Moffat (2015); MOG also explains the cosmic microwave background power spectrum. J.W. Moffat (2001); John W. Moffat, "Wide Binaries and Modified Gravity (MOG)" arXiv:2311.17130 (November 28, 2023), Mahmood Roshan, "Stellar Bar evolution in the absence of dark matter halo" (January 25, 2018);
* Verlinde's entropic/emergent gravity, see e.g., Erik P. Verlinde, "Emergent Gravity and the Dark Universe" arXiv:1611.02269 (November 7, 2016), Jungjai Lee, Hyun Seok Yang, "Dark Energy and Dark Matter in Emergent Gravity" arXiv:1709.04914 (September 14, 2017, last revised November 1, 2022) (published at 81(9) Journal of the Korean Physical Society 910-920 (2022)); A. Schlatter, R. E. Kastner, "Gravity from Transactions: A Review of Recent Developments" arXiv:2209.04025 (September 8, 2022);
* Deur's work, see, e.g., A. Deur, "Effect of the field self-interaction of General Relativity on the Cosmic Microwave Background Anisotropies" arXiv:2203.02350 (March 4, 2022)), with a somewhat similar approach from Yuta Ito, "Gravitational Amplification of Test-Mass Potential in the Self-gravitating Isothermal Gaseous Systems" arXiv:2303.02631 (March 5, 2023);
* f(R) gravity, Eddington-inspired-Born-Infeld (EiBI) and general relativity with renormalization group effects (RGGR) (three different modified gravity theories discussed in one paper) see Alejandro Hernandez-Arboleda, Davi C. Rodrigues, Aneta Wojnar, "Normalized additional velocity distribution: a fast sample analysis for dark matter or modified gravity models" arXiv:2204.03762 (April 7, 2022);
* f(R) gravity, see, e.g., Vesna Borka Jovanović, Predrag Jovanović, Duško Borka, Salvatore Capozziello, "Fundamental plane of elliptical galaxies in f(R) gravity: the role of luminosity" (December 28, 2018);
* f(Q) gravity, see, e.g., Gaurav N. Gadbail, "Cosmological dynamics of interacting dark energy and dark matter in f(Q) gravity" arXiv:2406.02026 (June 4, 2024);
* f(T) gravity, see, e.g., A. R. P. Moreira, "Geometrically contracted structure in teleparallel f(T) gravity" arXiv:2212.08948 (December 17, 2022);
* scalar-tensor gravity theories, see, e.g., Thomas P. Sotiriou, Valerio Faraoni, "Modified gravity with R-matter couplings and (non-)geodesic motion" arXiv:0805.1249 (September 27, 2008);
* long range quantum gravity, see, e.g., Matteo Tuveri, Mariano Cadoni "Galactic dynamics and long-range quantum gravity" arXiv:1904.11835 (April 26, 2019);
* modified general relativity, see, e.g., Gary Nash, "Modified general relativity" arXiv:1904.10803 (April 22, 2019);
* metric skew tensor gravity, see, e.g., W.M. Stuckey, Timothy McDevitt, A.K. Sten, Michael Silberstein, "The Missing Mass Problem as a Manifestation of GR Contextuality" 27(14) International Journal of Modern Physics D 1847018 (2018). DOI: 10.1142/S0218271818470181;
* curvature and topology based model of gravity, see, e.g., Valeri P. Frolov, "Limiting curvature models of gravity" arXiv:2111.14318 (November 29, 2021); Richard Lieu, arXiv:2406.04355 "The binding of cosmological structures by massless topological defects" (May 16, 2024);
* conformal gravity a.k.a. Weyl Conformal gravity, see, e.g., Philip D. Mannheim, "Is dark matter fact or fantasy? -- clues from the data" (March 27, 2019), James G. O'Brien, et al., "Radial Acceleration and Tully-Fisher Relations in Conformal Gravity" (December 7, 2018), Philip D. Mannheim, "Making the Case for Conformal Gravity" (October 27, 2011), and Leonardo Modesto, Tian Zhou, Qiang Li, "Geometric origin of the galaxies' dark side" arXiv:2112.04116 (December 8, 2021);
* GR with torsion added, see, e.g., S. H. Pereira, et al., "Dark matter from torsion in Friedmann cosmology" arXiv:2202.01807 (February 3, 2022);
* negative mass models, see, e.g., Hector Socas-Navarro, "Can a negative-mass cosmology explain dark matter and dark energy?" arXiv:1902.08287 (February 21, 2019);
* the effort of Naman Kumar to explain the dark energy with an anti-matter mirror universe, see, e.g., Naman Kumar, "On the Accelerated Expansion of the Universe" 30 Gravitation and Cosmology 85-88 arxiv.org:2406.04392 (April 4, 2024);
* string theory/brane theory approaches, see, e.g., Naman Kumar, "Variable Brane Tension and Dark Energy" arXiv:2404.17941 (April 27, 2024);
* non-Verlinde efforts based on entropy and Mach's principle, see, e.g., Santanu Das, "Aspects of Machian Gravity (III): Testing Theory against Galaxy Cluster mass" arXiv:2312.06312 (December 11, 2023) and Kimet Jusufi, Ahmad Sheykhi, Salvatore Capozziello, "Apparent dark matter as a non-local manifestation of emergent gravity" arXiv:2303.14127 (March 23, 2023), Rubén Arjona, et al., "A GREAT model comparison against the cosmological constant" arXiv:2111.13083 (November 25, 2021). Report number: IFT-UAM/CSIC-2021-136m, and Andre Maeder "Dynamical Effects of the Scale Invariance of the Empty Space: The Fall of Dark Matter?" 849(2) The Astrophysical Journal 158 (November 10, 2017) (pre-print here);
* general co-variance breaking gravity, see, e.g., Alexander P. Sobolev, "Foundations of a Theory of Gravity with a Constraint and its Canonical Quantization" 52 Foundations of Physics Article number: 3 arXiv:2111.14612 (open access, pre-print November 25, 2021, publication date anticipated 2022) DOI: 10.1007/s10701-021-00521-1;
* Pascoli's K-model, see, e.g., Gianni Pascoli, "A comparative study of MOND and MOG theories versus the κ-model: An application to galaxy clusters" arXiv:2307.01555 (July 4, 2023);
* gravitomagnetic and other perturbative GR effect approaches, see, e.g., Kostas Glampedakis, David Ian Jones, "Pitfalls in applying gravitomagnetism to galactic rotation curve modelling" arXiv:2303.16679 (March 29, 2023), A. N. Lasenby, M. P. Hobson, W. E. V. Barker, "Gravitomagnetism and galaxy rotation curves: a cautionary tale" arXiv:2303.06115 (March 10, 2023), Yogendra Srivastava, Giorgio Immirzi, John Swain, Orland Panella, Simone Pacetti, "General Relativity versus Dark Matter for rotating galaxies" arXiv:2207.04279 (July 9, 2022), G. O. Ludwig, "Galactic rotation curve and dark matter according to gravitomagnetism" 81 The European Physical Journal C 186 (February 23, 2021) (open access), F.I. Cooperstock, S. Tieu, "Galactic dynamics via general relativity: a compilation and new developments." 22 Int. J. Mod. Phys. A 2293–2325 (2007). arXiv:astro-ph/0610370 see also follow up papers in 2007, in 2011, and 2015; H. Balasin, D. Grumiller, "Non-Newtonian behavior in weak field general relativity for extended rotating sources." 17 Int. J. Mod. Phys. D 475–488 (2008) (arXiv version here); M. Crosta, M. Giammaria, M.G. Lattanzi, E. Poggio, "On testing CDM and geometry-driven Milky Way rotation curve models with Gaia DR2." 496 Mon. Not. R. Astron. Soc. 2107–2122 (2020) (open access); Missing Mass Problem as a Manifestation of GR Contextuality" 27(14) International Journal of Modern Physics D 1847018 (2018). DOI: 10.1142/S0218271818470181, Federico Re, "Fake dark matter from retarded distortions" (May 30, 2020), Felipe J. Llanes-Estrada, "Elongated Gravity Sources as an Analytical Limit for Flat Galaxy Rotation Curves" 7(9) Universe 346 arXiv:2109.08505 (September 16, 2021) DOI: 10.3390/universe7090346, P. Tremblin, et al., "Non-ideal self-gravity and cosmology: the importance of correlations in the dynamics of the large-scale structures of the Universe" arXiv:2109.09087 (September 19, 2021) (submitted to A&A, original version submitted in 2019), Priidik Gallagher, Tomi Koivisto, "The Λ and the CDM as integration constants" arXiv (March 9, 2021), and Johan Hansson, et al., Nonlinear Effects of Gravity in Cosmology arXiv:1805.11043 (2016);
* non-local gravity theories, see, e.g., Ivan Kolář, Tomáš Málek, Anupam Mazumdar, "Exact solutions of non-local gravity in class of almost universal spacetimes" arXiv: 2103.08555; Reza Pirmoradian, Mohammad Reza Tanhayi, "Non-local Probes of Entanglement in the Scale Invariant Gravity" arXiv: 2103.02998, J. R. Nascimento, A. Yu. Petrov, P. J. Porfírio, "On the causality properties in non-local gravity theories" arXiv: 2102.01600, Salvatore Capozziello, Maurizio Capriolo, Shin'ichi Nojiri, "Considerations on gravitational waves in higher-order local and non-local gravity" arXiv: 2009.12777, Jens Boos, "Effects of Non-locality in Gravity and Quantum Theory" arXiv: 2009.10856, Jens Boos, Jose Pinedo Soto, Valeri P. Frolov, "Ultrarelativistic spinning objects in non-local ghost-free gravity" arXiv: 2004.07420;
* massive graviton theories, see, e.g., Kimet Jusufi, Genly Leon, Alfredo D. Millano, "Dark Universe Phenomenology from Yukawa Potential?" arXiv:2304.11492 (May 14, 2024) (Phys. Dark Univ. 42 (2023), 101318), Kimet Jusufi, et al., "Modified gravity/entropic gravity correspondence due to graviton mass" arXiv:2405.05269 (April 25, 2024), and O. Costa de Beauregard, "Massless or massive graviton?" 3 Foundations of Physics Letters 81-85 (1990));
* varying G and running gravitational coupling constant approaches, see, e.g., Hikaru Kawai, Nobuyoshi Ohta, "An Observation on the Beta Functions in Quadratic Gravity" arXiv:2405.05706 (May 9, 2024), Dimitris M. Christodoulou, Demosthenes Kazanas, "Gravitational Potential and Nonrelativistic Lagrangian in Modified Gravity with Varying G" (November 21, 2018);
* fifth force models, see, e.g., Marcus Högås, Edvard Mörtsell, "The Hubble tension and fifth forces: a cosmic screenplay" arXiv:2309.01744 (September 4, 2023); and
* Einstein-Aether theories, see, e.g., Vincenzo F. Cardone, Ninfa Radicella, "Can MONDian vector theories explain the cosmic speed up?" arXiv:0908.0095 (August 1, 2009) ("Generalized Einstein - Aether vector field models have been shown to provide, in the weak field regime, modifications to gravity which can be reconciled with the successful MOND proposal.").
* Quantized inertia, see, e.g., McCulloch, Mike E., "Testing Quantised inertia on Proxima Centauri." 352(1) MNRAS L67-69 (2024).
See also McGaugh's tree of gravity based theories (with regrettably very small print):
Another summary from a conference paper illustration is:
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