Generalizations of quasilinear MOND

I present a class of theories that generalize quasilinear modified Newtonian dynamics, MOND (QUMOND). These generalized-QUMOND (GQUMOND) theories are derived from an action, and, like QUMOND, they require solving only the linear Poisson equation (twice), and are thus amenable to relatively simple numerical solution. Unlike QUMOND, their Lagrangian depends on higher derivatives of the Newtonian potential. They thus dictate different “phantom” densities as virtual sources in the Poisson equation for the MOND potential. These theories are not necessarily more appealing than QUMOND itself from a fundamental viewpoint. But, they might open new avenues to more fundamental theories, and, in the least, they have much heuristic value. Indeed, I use them to demonstrate that even within limited classes of modified-gravity formulations of MOND, theories can differ substantially on lower-tier MOND predictions, showing that it would be imprudent, at present, to equate MOND itself with some specific effective theory, such as QUMOND. Such GQUMOND theories force, generically, the introduction of dimensioned constants other than the MOND acceleration,

a_{0}

, such as a length, a frequency, etc. As a result, some of these theories reduce to QUMOND itself only, e.g., on length scales (or, in other versions, dynamical times) larger than some critical value. But in smaller systems (or, alternatively, in ones with shorter dynamical times), MOND effects are screened, even if their internal accelerations are smaller than

a_{0}

. In such theories it is possible that MOND (expressed as QUMOND) applies on galactic scales, but its departures from Newtonian dynamics are substantially suppressed in some subgalactic systems—such as binary stars, and open, or globular star clusters. The same holds for the effect of the galactic field on dynamics in the inner Solar System, which can be greatly suppressed compared with what QUMOND predicts. Tidal effects of a galaxy on smaller subsystems are the same as in QUMOND, for the examples I consider. I also describe briefly versions that do not involve dimensioned constants other than

a_{0}

, and yet differ from QUMOND in important ways, because they revolve around more than one acceleration variable.

Note:

12 pages, minor changes, to match the version published in Phys. Rev. D

gravitation: potential
gravitation: model
star: cluster
star: binary
derivative: high
MOND
galaxy
acceleration
dark matter
Poisson equation

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