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Distributional cosmological quantities solve the paradox of soft singularity crossing

Both dark energy models and modified gravity theories could lead to cosmological evolutions different from either the recollapse into a Big Crunch or exponential de Sitter expansion. The newly arising singularities may represent true endpoints of the evolution or allow for the extension of geodesics through them. In the latter case the components of the Riemann tensor representing tidal forces diverge. Sudden Future Singularities (SFS) occur at finite time, finite scale factor and finite Hubble parameter, only the deceleration parameter diverges. The energy density of a perfect fluid is regular and its pressure diverges at the SFS. A particular SFS, the Big Brake occurs when the energy density vanishes and the expansion arrives at a full stop at the singularity. Such scenarios are generated by either a particular scalar field (the tachyon field) or the anti-Chaplygin gas. By adding matter to these models, an unwanted feature appears: at the finite scale factor of the SFS the energy density of matter remains finite, implying (for a spatially flat universe) a finite Hubble parameter, hence finite expansion rate, rather then full stop. The universe would further expand through the singularity, this nevertheless seems forbidden as the energy density of the tachyonic field / anti-Chaplygin gas would become ill-defined. This paradox is relieved in the case of the anti-Chaplygin gas by redefining its energy density and pressure in terms of distributions peaked on the singularity. The regular cosmological quantities, continuous across the SFS are the energy density and the square of the Hubble parameter; those allowing for a jump at the SFS are the Hubble parameter and expansion rate (both being mirror-symmetric). The pressure and the decelaration parameter will contain Dirac delta-function contributions peaked on the SFS, however they anyhow would diverge at the singularity.