Variable Modified Newtonian Mechanics IV: Non Rotating Galaxies

At it stands, the $\Lambda CDM$ model does not anticipate the early emergence of massive galaxies. Canonical Modified Newtonian Dynamics (MOND) seems to fail at late time solar system scale and Wide-Binary scales. To match data, a MOND variant needs a variable MOND acceleration $a_0$ which is strong at high redshift galactic scale and diminishes over redshift to far below Newtonian gravity at solar system scale at late time. We found such a candidate in a relativistic frame-work. In a previous work, a new single-metric solution of Einstein Gravity is found for a point mass residing in an expanding universe, which apart from the Newtonian acceleration, gives rise to an additional MOND-like acceleration in which the MOND acceleration $a_0$ is replaced by the cosmological acceleration $\frac{1}{2}H^2(z)r$. This cosmological acceleration is shown to be far below Newtonian acceleration in the solar system and therefore avoids the problem of MOND program. In this work, we study the monolithic evolution of a Milky Way mass protogalactic cloud at recombination in this model where the non-Newtonian acceleration is stronger than Newtonian gravity. To obtain a spherical galaxy we assume that a point on a mass shell at turnaround will pick up sufficient non-systematic angular momentum. Assuming a violent relaxation process similar to the simulation studies for MOND and Newtonian gravity, we find that the central core can form a time independent Quasi-Stationary-State (QSS) by z>7, which could explain the galaxy morphology stability observations for $z<6.5$. The virialised potential has a Newtonian acceleration dominant central region and a MOND-like acceleration dominant outer region. We evaluate the corresponding MOND acceleration $a_0^{VM}$ in a virialised potential for a Milky-Way mass elliptical galaxy and find that $a_0^{VM}\sim a_0$.