Observational constraints on running vacuum model
  We investigate the power spectra of the CMB temperature and matter density in
the running vacuum model (RVM) with the time-dependent cosmological constant of
$\Lambda = 3 \nu H^2 + \Lambda_0$, where $H$ is the Hubble parameter. In this
model, dark energy decreases in time and decays to both matter and radiation.
By using the Markov chain Monte Carlo method, we constrain the model parameter
$\nu$ as well as the cosmological observables. Explicitly, we obtain $\nu \leq
1.54 \times 10^{-4}$ (68\% confidence level) in the RVM with the best-fit
$\chi^2_{\mathrm{RVM}} = 13968.8$, which is slightly smaller than
$\chi^2_{\Lambda \mathrm{CDM}} = 13969.8$ in the $\Lambda$CDM model of $\nu=0$.
