SEVERAL simplified expositions of cosmology have appeared in the past few years in response to the increased interest in the subject1–7. In these expositions the dimensionless scale factor R(t) at a universal cosmic time t plays an important part. It governs the universal expansion of all linear dimensions of the model, and is introduced through the equationwhere i labels a typical galaxy (or “particle” in the model), ri, is the distance of this galaxy from a galaxy i=0 which acts as origin, and t1 is a standard time at which the galaxies are labelled. The differential equation for R is thenwhere Ṙ=dR/dt, and B, D and E are constants. The argument, in its Newtonian framework, is based on a homogeneous and isotropic model universe in which the galaxies are assumed to be smeared out to give a mass density p(t) which is constant in space at any one time. Equation (2) is then obtained as an energy equation for a spherical distribution. There then arise some awkward problems concerning the significance of the origin, the significance of the surface of the sphere considered and the effect of the possibly infinite amount of matter outside the sphere. But authors usually manage to make it plausible that none of these effects change equation (2).
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LANDSBERG, P. Derivation of the Differential Equation for the Simpler Cosmological Models. Nature Physical Science 242, 104 (1973). https://doi.org/10.1038/physci242104a0