Unveiling $\nu$ secrets with cosmological data: Neutrino masses and mass hierarchy

Using some of the latest cosmological data sets publicly available, we derive the strongest bounds in the literature on the sum of the three active neutrino masses, $M_{\nu }$, within the assumption of a background flat $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. In the most conservative scheme, combining Planck cosmic microwave background temperature anisotropies and baryon acoustic oscillations (BAO) data, as well as the up-to-date constraint on the optical depth to reionization ($\tau$), the tightest 95% confidence level upper bound we find is $M_{\nu }<0.151\mathrm{eV}$. The addition of Planck high-$\ell$ polarization data, which, however, might still be contaminated by systematics, further tightens the bound to $M_{\nu }<0.118\mathrm{eV}$. A proper model comparison treatment shows that the two aforementioned combinations disfavor the inverted hierarchy at $\sim 64\%\mathrm{C}.\mathrm{L}.$ and $\sim 71\%\mathrm{C}.\mathrm{L}.$, respectively. In addition, we compare the constraining power of measurements of the full-shape galaxy power spectrum versus the BAO signature, from the BOSS survey. Even though the latest BOSS full-shape measurements cover a larger volume and benefit from smaller error bars compared to previous similar measurements, the analysis method commonly adopted results in their constraining power still being less powerful than that of the extracted BAO signal. Our work uses only cosmological data; imposing the constraint $M_{\nu }>0.06\mathrm{eV}$ from oscillations data would raise the quoted upper bounds by $\mathcal{O}(0.1\sigma )$ and would not affect our conclusions.

Figure 1

Top: Nonlinear galaxy power spectrum computed using the Halofit method with the camb code [124] (red line) and the Coyote emulator (blue line) [120, 121, 122] at $z=0.57$ for the $\mathrm{\Lambda }\mathrm{CDM}$ best-fit parameters from Planck TT 2015 data and $M_{\nu }=0\mathrm{eV}$ (given that the emulator does not fully implement corrections due to nonzero neutrino masses on small scales). Green triangle data points are the clustering measurements from the BOSS DR12 CMASS sample. The error bars are computed from the diagonal elements $C_{ii}$ of the covariance matrix. For comparison with previous work [23], purple circles represent clustering measurements from the BOSS DR9 CMASS sample. A very slight suppression in power on small scales (large $k$) of the DR12 sample compared to the DR9 sample is visible. Note that the binning strategies adopted in DR9 and DR12 are different. Bottom: Residuals with respect to the nonlinear model with Halofit. The orange horizontal line indicates the $k$ range used in our analysis. As is visually clear, the $k$ range we choose is safe from large nonlinear corrections.

Figure 2

Posteriors of $M_{\nu }$ obtained with baseline data sets basePK and baseBAO, in combination with additional external data sets. This allows for a comparison of the constraining power of shape information in the form of the full-shape galaxy power spectrum and geometrical information in the form of BAO measurements, when CMB full temperature and low-$\ell$ polarization data are used. To compare the relative constraining power of shape and geometrical information, compare the solid and dashed lines for a given color: red (basePK against baseBAO), blue ($basePK+\tau 0p055$ against $baseBAO+\tau 0p055$), and black ($basePK+H073p02+\tau 0p055$ against $baseBAO+H073p02+\tau 0p055$). The dotted line at $M_{\nu }=0.0986\mathrm{eV}$ denotes the minimal allowed mass in the IH scenario. It can be clearly seen that with our current analyses methods geometrical information supersedes shape information in constraining power.

Figure 3

As Fig. 2, but with the addition of high-$\ell$ polarization anisotropy data. Hence, the data sets considered are the baseline data sets basePK and baseBAO and combinations with external data sets. Once more, it can be clearly seen that with our current analyses methods geometrical information supersedes shape information in constraining power.

Figure 4

68% C.L. (dark blue) and 95% C.L. (light blue) joint posterior distributions in the $M_{\nu }-w$ plane, along with their marginalized posterior distributions, for the baseBAO data combination (see the caption of Table 6 for further details). Ticks on the $w$ axis of the upper left plot are the same as those for the lower left plot.

Figure 5

68% C.L. (dark blue) and 95% C.L. (light blue) joint posterior distributions in the $M_{\nu }\text{−}{\mathrm{\Omega }}_k$ plane, along with their marginalized posterior distributions, for the baseBAO data combination (see the caption of Table 6 for further details). Ticks on the ${\mathrm{\Omega }}_k$ axis of the upper left plot are the same as those for the lower left plot.

Figure 6

Figures of merit ${\zeta }_{ij}$, defined in Eq. (a2) and which quantify the goodness of the 3 deg approximation, as a function of the total neutrino mass $M_{\nu }$. ${\zeta }_{ij}=0$ (not displayed in this plot due to the logarithmic scale on the $y$ axis) corresponds to the unphysical case in which the 3 deg approximation is exact. The red lines correspond to $i$, $j=1$, 2 [that is, $\zeta =3(m_2-m_1)/M_{\nu }$], whereas the blue lines correspond to $i$, $j=1$, 3 [that is, $\zeta =3|m_3-m_1|/M_{\nu }$], with solid and dashed lines corresponding to the NH and IH scenarios, respectively. The solid vertical line at $M_{\nu }=0.15\mathrm{eV}$ represents the indicative upper limit on $M_{\nu }$ of 0.15 eV obtained in our analysis.

Figure 7

As Fig. 6, but with the figures of merit plotted against the mass of the lightest mass eigenstate $m_0=m_1(m_3)$ for NH (IH). The solid and dashed vertical lines at $\simeq 0.03\mathrm{eV}$ and $\simeq 0.04\mathrm{eV}$, respectively, represent the masses of $m_0$ corresponding to the indicative upper limit on $M_{\nu }$ of 0.15 eV obtained in our analysis.