Impact of assuming flatness in the determination of neutrino properties from cosmological data

Cosmological data have provided new constraints on the number of neutrino species and the neutrino mass. However, these constraints depend on assumptions related to the underlying cosmology. Since a correlation is expected between the number of effective neutrinos $N_{\mathrm{eff}}$, the neutrino mass $\sum _{}^{}{m}_{\nu }$, and the curvature of the universe ${\Omega }_k$, it is useful to investigate the current constraints in the framework of a nonflat universe. In this paper we update the constraints on neutrino parameters by making use of the latest cosmic microwave background data from the Atacama Cosmology Telescope and South Pole Telescope experiments and consider the possibility of a universe with nonzero curvature. We first place new constraints on $N_{\mathrm{eff}}$ and ${\Omega }_k$, with $N_{\mathrm{eff}}=4.03\pm 0.45$ and ${10}^3{\Omega }_k=-4.46\pm 5.24$. Thus, even when ${\Omega }_k$ is allowed to vary, $N_{\mathrm{eff}}=3$ is still disfavored with 95% confidence. We then investigate the correlation between neutrino mass and curvature that shifts the 95% upper limit of $\sum _{}^{}{m}_{\nu }<0.45\mathrm{eV}$ to $\sum _{}^{}{m}_{\nu }<0.95\mathrm{eV}$. Thus, the impact of assuming flatness in neutrino cosmology is significant and an essential consideration with future experiments.

Figure 1

A demonstration of how the sound horizon $r_s$ changes with the effective neutrino number $N_{\mathrm{eff}}$ under the matter dominated approximation given by Eq. (5).

Figure 2

Correlation between ${\Omega }_k$ and $N_{\mathrm{eff}}$. The credible intervals are given at the 68% and 95% confidence levels and the markers indicate the locations of the marginalized values. $\mathrm{WMAP}+\mathrm{BAO}+H_0$ is shown in red while $\mathrm{WMAP}+\mathrm{ACT}+\mathrm{SPT}+\mathrm{BAO}+H_0$ is in blue. Note that the effect of adding additional data sets is significant.

Figure 3

Correlation between $H_0$ and $N_{\mathrm{eff}}$. The credible intervals are given at the 68% and 95% confidence levels and the markers indicate the locations of the marginalized values. $\mathrm{WMAP}+\mathrm{BAO}+H_0$ is shown in red while $\mathrm{WMAP}+\mathrm{ACT}+\mathrm{SPT}+\mathrm{BAO}+H_0$ is in blue. In this case the effect of adding additional data sets is also significant.

Figure 4

Comparison of the correlation between ${\Omega }_k$ and $\sum _{}^{}{m}_{\nu }$ under the two $\Delta N_{\mathrm{eff}}$ models. The model with three massive neutrinos is shown in blue while the model with additional relativistic species is in red. Intervals are given at the 68% and 95% confidence levels and markers indicate the locations of the marginalized values. Data sets include $\mathrm{WMAP}7+\mathrm{ACT}+\mathrm{SPT}+\mathrm{BAO}+H_0$. The addition of curvature allows $\sum _{}^{}{m}_{\nu }$ to be more than twice the previous constraint.