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} $,
$-(1.5\times10^{-5} {\rm eV}^{-1})^{4} \ll \beta < (0.07 {\rm eV}^{-1})^4$.
