tokens: (Planck Collaboration I 2016 ), is implemented in order to constrain the Λ( H ) models. Up to now, for these models only the set of the CMB shift parameters have been used in order to test the dynamical vacuum scenarios close to recombination. Then, we combine CMB in an overall likelihood analysis with other cosmological probes (SNe type Ia, BAOs, H 0 ) in order to extract the probability distribution (via Monte Carlo MCMC method) of the solutions for a large set of cosmological parameters, as well as, to search for deviations from the concordance ΛCDM model for which Λ( H ) = const. The structure of our paper is as follows: in Section 2 we provide a brief introduction of the running vacuum cosmology and we present the most popular Λ( H ) models that have appeared in the literature. In Section 3 , we discuss the methodology and the cosmological data that we utilize and we perform a detailed statistical analysis aiming to provide the corresponding best-fitting values and contour plots for the current Λ( H ) models. Finally, we discuss our results in the conclusions. 2 BACKGROUND EXPANSION IN RUNNING VACUUM MODELS In this section, we briefly describe the main features of the dynamical vacuum models for which Λ is not constant but evolves with cosmic time. This is perfectly allowed by the cosmological principle embedded in the FLRW metric (Perico et al. 2013 ). In general, if we model the expanding Universe as a mixture of perfect fluids N = 1, 2, ... then the total energy momentum tensor is given by where is the four-velocity field. In this case the components of are written as where the quantities p T and ρ T are the total pressure and energy density in the comoving frame , respectively. The next step is to apply the covariant local conservation law for the mixture, namely ∇ μ T μν = 0. Inserting equation ( 1 ) into the latter expression and with the aid of the relation (which comes from the fact that for any four-velocity vector, we have ) we arrive at (Grande, Pelinson Sola 2009 ): For a Friedmann–Lemaître–Robertson–Walker (FLRW) metric, it is easy to check that for a comoving frame (), one obtains: and thus equation ( 3 ) reduces to where is the Hubble parameter and a ( t ) is the scale factor of the Universe normalized to unity at the present epoch. For the rest of the paper we focus on the spatially flat FLRW metric. In the aforementioned discussion, we did not address the physics of the fluids involved. The total density ρ T receives contributions from non-relativistic matter (cold dark matter and baryons) ρ m = ρ b + ρ dm ( p m = 0), radiation ρ r ( p r = ρ r /3) and vacuum ρ Λ ( p Λ = −ρ Λ ). In this context, the index N specifies the specific components of the cosmic fluid, namely {dm, b, r , Λ}. Assuming that baryons and radiation are self-conserved, namely the corresponding densities evolve in the nominal way ρ b = ρ b0 a −3 and ρ r = ρ r0 a −4 the overall conservation law ( 5 ) becomes or Note that Q is the interaction term
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