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\indent We study the impact of a modified expansion rate on the dark matter relic abundance and leptogenesis in two types of scalar-tensor theories. The first scenario is motivated from string theory constructions, which have conformal as well as disformally coupled matter to the scalar. The second scenario is called disformal D-brane coupling, which stems from D-brane models of cosmology. In both scenarios, the conformal and disformal couplings modify the expansion rate of the universe prior to  BBN. Furthermore, we investigate the effects of such couplings to the dark matter relic abundance for a wide range of initial conditions, masses, and cross-sections. It is found that the annihilation cross-section required to satisfy the dark matter content can differ from the thermal average cross-section in the standard cosmology, which can be tested in future experiments. In addition, we discuss how non-standard cosmologies can open new pathways for low scale leptogenesis. Within these scenarios direct tests of leptogenesis could also provide information on the very early Universe evolution, corresponding to temperatures in the TeV range.


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