Characteristic Length Scale and Dynamics of χ 3/2 -MOND Cosmology Donniel Cruz 1,* and Emmanuel Rodulfo 2 1, 2 De La Salle University - Manila *Corresponding Author: donniel_c_cruz@dlsu.edu.ph / donniel_cruz@dlsu.edu.ph (Dated: July 7, 2023) Abstract: This work studies the cosmology of χ 3/2 -MOND gravity by Bernal et. al. (2011). This theory is a modification to Einstein’s General Relativity (GR) that uses a dimensionless curvature scalar χ by rescaling the Ricci scalar  by some characteristic length scale   , as well as a set of modified field equations that follows from a 3/2- power Lagrangian. The characteristic length scale is assumed to be built from the universal constants of the theory and the parameters of the system in question. In the weak field limit, this theory recovers Milgrom’s (1983a) Modified Newtonian Dynamics (MOND). MOND is a proposal that corrects Newtonian gravitational laws below an acceleration threshold  0 ≈ 1.2 × 10 −10 / 2 to explain the anomalous flattening of galactic rotation curves without imposing any dark matter components. In the cosmological case, this work asserts that the characteristic length scale is of the order  2 / 0 . This specific value is motivated in two ways: (1) it is shown that this scale defines a convergence of GR and MOND at some critical mass (with this as the corresponding length); (2) this length scale is shown to be an extremal value of   independent of the mass parameter. The established length scale is then used in the case of cosmology; the FLRW metric is plugged in into the modified field equations and the two modified Friedmann equations are derived incorporating the MOND effects by a manifest appearance of the constant  0 . Key Words: modified gravity; dark matter; dark energy; cosmology; general relativity 1. INTRODUCTION Modified Newtonian Dynamics (MOND) is a modification of the laws of (classical, non-relativistic) gravity first proposed by Milgrom (1983a, 1983c, 1983b) in order to explain the anomalous flat rotation curves of galaxies. The standard explanation is that Einstein’s General Relativity (GR) is a complete description of (classical) gravity and there must be some extra (invisible) “dark matter” in galaxies that explains the discrepancy in the measured gravitational strength (Rubin & Ford, 1970; Zwicky, 1933). But so far, no such dark matter particle has been directly detected. MOND asserts the opposite; no dark matter particles are needed but instead the laws of gravity must be modified for very weak gravitational strengths. This proposed correction can be stated as a modification of the Poisson equation for the gravitational scalar field ϕ due so some matter density ρ: ∇ ⋅ ( ( |∇ϕ|  0 ) ∇ϕ) = 4π (1) where  0 ≈ 1.2 × 10 −10 / 2 is the threshold gravitational acceleration below which MOND effects manifest (Milgrom, 2015); and () is some