Nonlocal metric formulations of modified Newtonian dynamics with sufficient lensing

Abstract

We demonstrate how to construct purely metric modifications of gravity which agree with general relativity in the weak field regime appropriate to the solar system, but which possess an ultra-weak field regime when the gravitational acceleration becomes comparable to a0 ∼ 10 −10 m/s 2 . In this ultra-weak field regime, the models reproduce the MOND force without dark matter and also give enough gravitational lensing to be consistent with existing data. Our models are nonlocal and might conceivably derive from quantum corrections to the effective field equations.

Nonlocal metric realizations of MOND

“…Requiring that the ultra-weak field limit of the B(r) equation should reproduce the Tully-Fisher relation, along with no change in the A(r) equation, restricts invariant, metric-based gravitational Lagrangians to take the form of general relativity plus [23],…”

“…MOND phenomenology only fixes the behavior of f (Z) for positive Z [23]. We are therefore free to adjust the negative Z branch so that the MOND additions to the Einstein tensor (50-51) make up for the absence of dark matter in determining the expansion history a(t).…”

“…It was finally possible to accommodate both requirements by involving a more complicated curvature [23]. (See also [26,27,28].)…”

Relativistic MOND from modified energetics

“…This approach, the modified inertia approach proposed in [38,39], in the non-relativistic limit, keeps gravitational laws unchanged yet lets in nonlinear kinetic terms. In this framework, it is found that the kinetic term of the point mass involves all derivatives of acceleration [38][39][40][41], yet it is stable and respects causality [15][16][17]34,35]. In the present work, we pursue this modified inertia viewpoint to generalize it to general-relativistic domain.…”

“…In this case, one is necessarily led to modified Newtonian gravity [18,19] or General Relativity (GR) extended by geometrical scalar and vector fields [20][21][22][23][24][25]. Besides, there are alternative approaches based on f (R) gravity [26][27][28][29], bimetric gravity [30,31], time foliation [32,33], nonlocal metric theories [34,35], Galileons [36], and Horava-Lifshitz gravity [37]. In general, modified gravity theories introduced to replace the DM necessarily lead to a MONDian structure.…”