Probing the effective number of neutrino species with the cosmic microwave background

We discuss how much we can probe the effective number of neutrino species $N_{\nu }$ with the cosmic microwave background alone. Using the data of the WMAP, ACBAR, CBI, and BOOMERANG experiments, we obtain a constraint on the effective number of neutrino species as $0.96<N_{\nu }<7.94$ at 95% C.L. for a power-law $\Lambda \mathrm{CDM}$ flat universe model. The limit is improved to be $1.39<N_{\nu }<6.38$ at 95% C.L. if we assume that the baryon density, $N_{\nu }$, and the helium abundance are related by the big bang nucleosynthesis theory. We also provide a forecast for the Planck experiment using a Markov chain Monte Carlo approach. In addition to constraining $N_{\nu }$, we investigate how the big bang nucleosynthesis relation affects the estimation for these parameters and other cosmological parameters.