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Interactions between cosmic fluids may appear in many cosmological scenariosthat go far beyond the usually studied energy exchange in the dark sector. Inthe absence of known microscopic interaction mechanisms, phenomenologicalansatzes are usually proposed in order to describe such models. In this paper,we derive a generalization of one of the most frequently used of suchansatzes:the one based on a initial proposal of Shapiro, Sol\`a, Espa\~na-Bonetand Ruiz-Lapuente who described a time-dependent cosmological "constant" whosevariation arises from quantum effects near the Planck scale [I. L. Shapiro, J.Sol\`a, C. Espa\~na-Bonet, and P. Ruiz-Lapuente, (2003). This physicallymotivated model was based on a single free parameter $\nu$, and wassubsequently studied by Wang and Meng (2005), under the pure phenomenologicalreasoning that the vacuum decay would slightly modify the exponent describinghow the energy density of dark matter decreases with the scale factor. Thismodification is described by a single parameter $\varepsilon$ (= 3 $\nu$ of theformer paper). The generalization derived in the present article requires twofree parameters ($\varepsilon_{1}$, $\varepsilon_{2}$). We show that thisextension is dynamically equivalent to the ansatz proposed by Barrow andClifton (2006) which deals with the transfer of energy between any two fluidsusing a two-parameter scheme ($\alpha_{1}$, $\alpha_{2}$) and that has theadvantage of exhibiting explicitly the form of the interaction factor appearingin the continuity equation of each fluid.

Equivalence between generalized phenomenological schemes for the  interaction of cosmological fluids: applications to arbitrary linear  barotropic fluids and vacuum decay