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We investigate the running cosmological constant model with dark energylinearly proportional to the Hubble parameter, $\Lambda = \sigma H +\Lambda_0$, in which the $\Lambda$CDM limit is recovered by taking $\sigma=0$.We derive the linear perturbation equations of gravity under theFriedmann-Lema\"itre-Robertson-Walker cosmology, and show the power spectra ofthe CMB temperature and matter density distribution. By using the Markov chainMonte Carlo method, we fit the model to the current observational data and findthat $\sigma H_0/ \Lambda_0 \lesssim 2.63 \times 10^{-2}$ and $6.74 \times10^{-2}$ for $\Lambda(t)$ coupled to matter and radiation-matter, respectively,along with constraints on other cosmological parameters.

Running cosmological constant with observational tests