Constraining neutrino masses by CMB experiments alone

It is shown that a subelectronvolt upper limit can be derived on the neutrino mass from the cosmic microwave background (CMB) data alone in the $\Lambda \mathrm{C}\mathrm{D}\mathrm{M}$ model with the power-law adiabatic perturbations, without the aid of any other cosmological data. Assuming the flatness of the Universe, the constraint we can derive from the current Wilkinson Microwave Anisotropy Probe observations is $\sum _{}^{}{m}_{\nu }<2.0\mathrm{e}\mathrm{V}$ at the 95% confidence level for the sum over three species of neutrinos ($m_{\nu }<0.66\mathrm{e}\mathrm{V}$ for the degenerate neutrinos) by maximizing the likelihood over six other cosmological parameters. This constraint modifies little even if we abandon the flatness assumption for the spatial curvature. We argue that it would be difficult to improve the limit much beyond $\sum _{}^{}{m}_{\nu }\lesssim 1.5\mathrm{e}\mathrm{V}$ using only the CMB data, even if their statistics are substantially improved. However, a significant improvement of the limit is possible if an external input is introduced that constrains the Hubble constant from below. The parameter correlation and the mechanism of CMB perturbations that give rise to the limit on the neutrino mass are also elucidated.

Figure 1

Minimum ${\chi }^2$ as a function of the neutrino energy density ${\omega }_{\nu }$. The solid curve is for the flat universe. The dotted and the dashed curves show the cases for a negative and a positive curvature universe, respectively.

Figure 2

The cosmological parameters for the solutions that give minimum ${\chi }^2$ as a function of ${\omega }_{\nu }$. The two line segments shown in panel (c) are the cases for a negative (dotted line) and a positive (dashed line) curvature universe.

Figure 3

The values of reduced CMB observables for the solutions that give minimum ${\chi }^2$ as a function of ${\omega }_{\nu }$.

Figure 4

Constraints on the four reduced CMB observables. Local ${\chi }^2$ is computed using restricted sets of multipoles as explained in the text and is measured with ${\chi }_{\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{a}\mathrm{l}}^2$ in the relevant range indicated in the left ordinate. The ${\chi }^2$ of the global solution is measured for the value with respect to the entire data set as measured in the right ordinate. The relative normalization is fixed so that the global solution that gives ${\chi }^2$ minimum gives the local ${\chi }^2$ value measured in the left ordinate. Dotted curves are the envelopes for ${\omega }_{\nu }=0.01$, 0.02, and 0.03 in the order of increasing minimum ${\chi }^2$. The horizontal dashed line segments show the position of ${\chi }^2-{\chi }_{\mathrm{m}\mathrm{i}\mathrm{n}}^2=4$.

Figure 5

Response of the four reduced CMB observables to the variation of ${\omega }_{\nu }$. The isolated points show the values at ${\omega }_{\nu }=0$, which do not connect to the ${\omega }_{\nu }\ne 0$ values smoothly.

Figure 6

Dependence of ${\ell }_1$ on ${\omega }_{\nu }$ calculated from Eq. (30) (dashed line), as compared with the accurate numerical solution (solid line). The dotted curve shows the case when the effect of massive neutrinos on the early Sachs-Wolfe effect is ignored.

Figure 7

Multipoles corresponding to the neutrino free-streaming scale.

Figure 8

Effective parameters of the massless neutrino theory that are required to mock the massless neutrino world.

Figure 9

Dependence of $H_1$ on ${\omega }_{\nu }$ predicted in the mock massless neutrino theory (dashed line), as compared with the true theory with massive neutrinos (solid line). For illustration, the results with the theories, where only the early integrated Sachs-Wolfe effect is mocked by changing ${\omega }_m$ and $N_{\nu }$ (dotted line) and only the late Sachs-Wolfe effect is mocked by changing $h$, are also shown (dot-dashed line).

Figure 10

Dependence of (a) $H_2$ and (b) $H_3$ on ${\omega }_{\nu }$ predicted in the mock massless neutrino theory (dashed line), as compared with the true theory with massive neutrinos (solid line).

Figure 11

Likelihood functions (${\omega }_{\nu }=0$) for $n_s$ estimated from our ${\chi }^2$ statistics (solid line), as compared with those from MCMC given by the WMAP group (data points with errors) and Tegmark et al. (dashed curve). The maximum is normalized to unity.

Figure 12

An example of the likelihood function for ${\omega }_m$ with $n_s$ fixed to 0.98 and ${\omega }_{\nu }$ to zero. The ten data points are fitted with a Gaussian function. The maximum is normalized to unity.