Non-linear coupling in the dark sector as a running vacuum model
  In this work we study a phenomenological non-gravitational interaction between dark matter and dark energy. The scenario studied in this work extends the usual interaction model proportional to the derivative of the dark component density adding to the coupling a non-linear term of the form $Q = \rho'/3(\alpha + \beta \rho)$. This dark sector interaction model could be interpreted as a particular case of a running vacuum model of the type $\Lambda(H) = n_0 + n_1 H^2 + n_2 H^4$ in which the vacuum decays into dark matter. For a flat FRW Universe filled with dark energy, dark matter and decoupled baryonic matter and radiation we calculate the energy density evolution equations of the dark sector and solve them. The different sign combinations of the two parameters of the model show clear qualitative different cosmological scenarios, from basic cosmological insights we discard some of them. The linear scalar perturbation equations of the dark matter were calculated. Using the CAMB code we calculate the CMB and matter power spectra for some values of the parameters $\alpha$ and $\beta$ and compare it with $\Lambda$CDM. The model modify mainly the lower multipoles of the CMB power spectrum remaining almost the same the high ones. The matter power spectrum for low wave numbers is not modified by the interaction but after the maximum it is clearly different. Using observational data from Planck, and various galaxy surveys we obtain the constraints of the parameters, the best fit values obtained are the combinations $\alpha = (3.7 \pm 7 )\times 10^{-4} $, $-(1.5\times10^{-5} {\rm eV}^{-1})^{4} \ll \beta < (0.07 {\rm eV}^{-1})^4$. 
Josue De-Santiago, Iv\'an E. S\'anchez G., David Tamayo
astro-ph.CO gr-qc