Observational constraints on interacting vacuum energy with linear interactions

We explore the bounds that can be placed on interactions between cold dark matter and vacuum energy, with equation of state $w=-1$, using state-of-the-art cosmological observations. We consider linear perturbations about a simple background model where the energy transfer per Hubble time, $Q/H$, is a general linear function of the dark matter density, $\rho_c$, and vacuum energy, $V$. We explain the parameter degeneracies found when fitting cosmic microwave background (CMB) anisotropies alone, and show how these are broken by the addition of supernovae data, baryon acoustic oscillations (BAO) and redshift-space distortions (RSD). In particular, care must be taken when relating redshift-space distortions to the growth of structure in the presence of non-zero energy transfer. Interactions in the dark sector can alleviate the tensions between low-redshift measurements of the Hubble parameter, $H_0$, or weak-lensing, $S_8$, and the values inferred from CMB data. However these tensions return when we include constraints from supernova and BAO-RSD datasets. In the general linear interaction model we show that, while it is possible to relax both the Hubble and weak-lensing tensions simultaneously, the reduction in these tensions is modest (reduced to less slightly than $4\sigma$ and $2\sigma$ respectively).