A stratified framework for scalar-tensor theories of modified dynamics

Although the modified Newtonian dynamics (MOND) proposed by Milgrom successfully accounts for the systematics of galaxy rotation curves and cluster dynamics without invoking dark matter, the idea remains a largely ad hoc modification of Newtonian dynamics with no basis in deeper theory. Nonstandard scalar-tensor theories have been suggested as a theoretical basis for MOND; however, any such theory with the usual conformal relation between the Einstein and physical metrics fails to predict the degree of light deflection observed in distant clusters of galaxies. The prediction is that there should be no discrepancy between the detectable mass in stars and gas and the lensing mass, in sharp contradiction to the observations (Bekenstein & Sanders). In the present paper, I demonstrate that one can write down a framework for scalar-tensor theories that predict the MOND phenomenology for the low-velocity (upsilon much less than c) dynamics of galaxies and clusters of galaxies and are consistent with observations of extragalactic gravitational lenses, provided that one drops the requirement of the Lorentz invariance of gravitational. dynamics. This leads to ''preferred-frame'' theories characterized by a nonconformal relation between the two metrics. I describe a toy theory in which the local environment (the solar system, binary pulsars) is protected from detectable preferred-frame effects by the very same nonstandard (aquadratic) scalar Lagrangian that gives rise to the MOND phenomenology. Although this particular theory is also contrived, it represents a limiting case for two-field theories of MOND and is consistent with a wide range of gravitational phenomena. Moreover, it is a cosmological effective theory which may explain the near numerical coincidence between the MOND acceleration parameter and the present value of the Hubble parameter multiplied by the speed of light.