Eternal Expansion or an End to Time

In the Friedmann cosmologies of the GTR, a finite space implies an end to proper time in the future, but this is not required in some nonstandard models. Assuming the Robertson-Walker form of the metric for a homogeneous and isotropic space-time, it is convenient to discuss the future evolution of any such expanding universe in terms of a dimensionless deceleration parameter, defined as  

Applications of the virial theorem of Newtonian mechanics to galactic rotation and the dynamics of clusters indicate that masses often exceed those inferred from visible light by about an order of magnitude. Such results push at the upper bound of the baryon density inferred from nucleosynthesis but are still far short of supporting. However, if only one-tenth of the ordinary matter in galaxies and clusters may be visible to us, is it not possible that there exists mass-energy, in as yet undetected forms or places, of sufficient quantity to produce deceleration exceeding the critical value Primordial black holes. 

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