An analytic derivation of the galaxy halo density profile

We examine some of the consequences of the exochronous (timeless) metric and the associated $\Sigma$GR cosmological model for the formation of galaxies, and in particular, their characteristic rotation curves. We show how the cumulative curvature from the multiple spatial hypersurfaces in this model leads to a modified version of the Poisson equation in which the gravitational potential is computed over 4D space.Using this new form of the Poisson equation, we derive an analytic expression for gravitational potential as a function of radial distance for a uniform gas cloud undergoing gravitational collapse. We show that this results in a radial velocity profile that provides an excellent fit with commonly observed galaxy rotation curves, and hence fully accounts for the effects previously ascribed to dark matter. An expression can be derived for the equivalent matter density profile corresponding to the $\Sigma$GR gravitational potential, from which it is evident that this is very similar in form to the well known Navarro-Frenk-White profile. As a further illustration of the consequences of adopting the exochronous metric, we show how the principle can readily be incorporated into particle-mesh N-body simulations of large scale structure evolution, using a relaxation solver for the solution to the Poisson equation and the evolution of the gravitational potential. Examples of the use of this simulation model are shown for the cases of: (a) initial evolution of large scale structure, and (b) galaxy formation from a gravitationally collapsing gas cloud. In both cases is is possible to directly visualise the build up of the gravitational potential in 3D space as the simulation evolves, and note how this corresponds to what is currently assumed to be dark matter.