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A complete Leibniz-Mach cosmology: From the Leibniz Question to MOND

The claim that the large scale structure of the Universe is hierarchical has a very long history going back at least to Charlier's papers of the early 20th century. In recent years, the debate has centered largely on the works of Sylos Labini, Joyce, Pietronero and others, who have made the quantative claim that the large scale structure of the Universe is quasi-fractal with fractal dimension $D\approx 2$. There is now a concensus that this is the case on medium scales, with the main debate revolving around what happens on the scales of the largest available modern surveys. This paper, which is a realization of a worldview which is deeply rooted in the ideas of Leibniz and Mach shows, as a very special case of a general formalism, that such a fractal $D\approx 2$ world is necessarily a world of dynamical equilibrium. We use the cosmology to write down a simple galaxy model as an arbitrary spherically symmetric bounded perturbation of the $D\approx 2$ equilibrium environment. A necessary mass continuity condition on the perturbation boundary between galactic interior and the exterior environment immediately defines that boundary as a critical acceleration boundary with $a_0 \sim 4\pi G \Sigma_F$ where $a_0$ is the critical acceleration parameter and $\Sigma_F$ is the characteristic mass surface density of the $D \approx 2$ fractal environment. The MOND acceleration condition $V_0^2/R_0 = a_0$ for circular orbits on the critical boundary follows in a straightforward manner.

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