Running cosmological constant with observational tests

We investigate the running cosmological constant model with dark energy linearly proportional to the Hubble parameter, Λ=σH+Λ0Λ=σH+Λ0, in which the ΛCDMΛCDM limit is recovered by taking σ=0σ=0. We derive the linear perturbation equations of gravity under the Friedmann–Lemaïtre–Robertson–Walker cosmology, and show the power spectra of the CMB temperature and matter density distribution. By using the Markov chain Monte Carlo method, we fit the model to the current observational data and find that σH0/Λ0≲2.63×10−2σH0/Λ0≲2.63×10−2 and 6.74×10−26.74×10−2 for Λ(t)Λ(t) coupled to matter and radiation-matter, respectively, along with constraints on other cosmological parameters.