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A class of exact shock-wave solutions to the Einstein field equations for a perfect fluid will be constructed from matching an inner class of expanding wave solutions (dynamic expanding spherically symmetric self-similar solutions), to an outer class of TOV solutions (static spherically symmetric self-similar solutions) across a spherical shock surface. Such a matching will be required to satisfy the Rankine-Hugoniot jump conditions and this requirement will impose a constraint on the parameters of the inner and outer solutions. One of the parameters, known as the acceleration parameter, will specify the expansion rate of the inner solution. The applicability of these matched solutions stems from a paper by Smoller and Temple that demonstrates how expanding wave solutions may provide a mechanism for the expansion of the universe without the presence of a cosmological constant, and thus, without dark energy. The possibility then remains to find an inner expanding wave solution which when matched to a TOV solution, with both equations of state corresponding to pure radiation, will constrain the acceleration parameter to be close to the observed expansion rate of the universe in the radiation dominated epoch.