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Extensions of the gravitational framework of Brans-Dicke (BD) are studied by considering two different scenarios: i) `BD-$\Lambda$CDM', in which a rigid cosmological constant, $\Lambda$, is included, thus constituting a BD version of the vanilla concordance $\Lambda$CDM model (the current standard model of cosmology with flat three-dimensional geometry), and ii) `BD-RVM', a generalization of i) in which the vacuum energy density (VED), $\rho_{\textrm{vac}}$, is a running quantity evolving with the square of the Hubble rate: $\delta\rho_{\textrm{vac}}(H)\propto \nu\, m^2_{\textrm{Pl}} (H^2-H_0^2)$ (with $|\nu|\ll 1$).
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This is not an undue limitation of our analysis. The Planck 2018 TT+lowE data are perfectly sound to test new models; and, as we indicated previously, for the sake of a fairer comparison of the current analysis with the aforementioned works it is perfectly sound to use such a valuable CMB data. In addition, let us note that the polarization data is still under analysis and is not completely free from moderate controversies in the literature, see e.g. [ 146 , 147 ] . While we do not judge at all these discussions we would like that our models are tested against the cleanest and nevertheless equally robust CMB data with which we already had tested similar models before
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