On the validity of the continuity equation in a modified gravity framework with CMB, DES 3x2pt and tomographic ISW data

In this work we propose a phenomenological modification to the continuity equation at the linear perturbation level and test it using combinations of the CMB temperature, polarization and lensing potential angular spectrum, the ISW-galaxy cross power spectrum and the 3$\times$2pt lensing and galaxy clustering from DES survey. We investigate two parametrisations of this modification, both proportional to a new parameter $A_c$, which is assumed to be either constant in time, or proportional to the scale factor $a$, in order to be more relevant at late times. We find DES and ISW data to be consistent with the standard continuity equation when $A_c$ is constant, but 2--3$\sigma$ hints of a non-zero modification appear when Planck data is included. The model $A_c \propto a$ results in stronger tensions. We also test the effects of including the common extra parameters $\mu$ and $\eta$ that modify the Poisson equation and Weyl potential. Although $A_c$, $\mu$ and $\eta$ are correlated, we still find non-zero $A_c$ when Planck data is included or without Planck if $A_c \propto a$ and only either $\eta$ or $\mu$ are allowed to vary. We conclude that violations of the continuity equation should be considered with care when testing additional deviations from general relativity.