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.

Figure 1

CMB power spectra for the cases with $N_{\nu }=1$ (blue dotted line), 3 (red solid line), and 5 (green dashed line). Other cosmological parameters are taken as the mean value from the WMAP5-alone analysis for a power-law flat $\Lambda \mathrm{CDM}$ model.

Figure 2

The 1D posterior distributions of $N_{\nu }$. The red solid line uses WMAP5 alone (with $Y_p=0.24$ fixed) and the other lines use $\mathrm{WMAP}5+\mathrm{ACBAR}+\mathrm{BOOMERANG}+\mathrm{CBI}$ with different assumptions on $Y_p$. The green dashed line fixes it to be $Y_p=0.24$, the blue dotted line uses the BBN relation to fix $Y_p$ from ${\omega }_b$ and $N_{\nu }$, and the magenta dot-dashed line treats $Y_p$ as a free parameter. For the analysis with WMAP5 alone, we assumed the prior on the cosmic age as $10\mathrm{Gyr}<t_0<20\mathrm{Gyr}$.

Figure 3

The 68% and 95% allowed regions in the plane of $N_{\nu }$ vs other parameters when WMAP5 alone is used with $Y_p=0.24$ (red solid line), $\mathrm{WMAP}5+\mathrm{ACBAR}+\mathrm{BOOMERANG}+\mathrm{CBI}$ are used with $Y_p=0.24$ (green dashed line), $Y_p$ is fixed by the BBN relation (blue dotted line), and $Y_p$ is treated as a free parameter (orange and yellow shaded regions).

Figure 4

An illustration of the degeneracy of $N_{\nu }$ with other cosmological parameters. Here the values of the effective number of neutrinos are assumed as $N_{\nu }=1$ (blue dotted line), 3 (red solid line), and 5 (green dashed line), and other cosmological parameters are chosen such that the CMB spectra become the same as that with the fiducial parameters.