Determining the neutrino mass hierarchy with cosmology

The combination of current large-scale structure and cosmic microwave background anisotropies data can place strong constraints on the sum of the neutrino masses. Here we show that future cosmic shear experiments, in combination with cosmic microwave background constraints, can provide the statistical accuracy required to answer questions about differences in the mass of individual neutrino species. Allowing for the possibility that masses are nondegenerate we combine Fisher matrix forecasts for a weak lensing survey like Euclid with those for the forthcoming Planck experiment. Under the assumption that neutrino mass splitting is described by a normal hierarchy we find that the combination Planck and Euclid will possibly reach enough sensitivity to put a constraint on the mass of a single species. Using a Bayesian evidence calculation we find that such future experiments could provide strong evidence for either a normal or an inverted neutrino hierarchy. Finally we show that if a particular neutrino hierarchy is assumed then this could bias cosmological parameter constraints, for example, the dark energy equation of state parameter, by $\gtrsim 1\sigma$, and the sum of masses by $2.3\sigma$. We finally discuss the impact of uncertainties on the theoretical modeling of nonlinearities. The results presented in this analysis are obtained under an approximation to the nonlinear power spectrum. This significant source of uncertainty needs to be addressed in future work.

Figure 1

68% and 95% probability contours (two-parameter) in the plane $\alpha$-$\sum _{}^{}{m}_{\nu }$ for $\mathrm{\text{Planck}}+\mathrm{\text{Euclid}}$ from our Fisher matrix calculation for the 10-parameter space described in the text. The grey region is the $3\sigma$ parameter space allowed by oscillations constraints on the hierarchy according to values of Eq. (1) (neglecting the difference $\Delta m_{21}^2$).

Figure 2

Relative error on $\alpha$ as a function of the target model for the combination $\mathrm{\text{Planck}}+\mathrm{\text{Euclid}}$. The fiducial value for the total mass is $\sum _{}^{}{m}_{\nu }=0.055\mathrm{eV}$.

Figure 3

68% 2-parameter probability contours in the plane $\sum _{}^{}{m}_{\nu }$-$\alpha$ for Planck (green, light ellipse), Euclid (blue, dark ellipse) and the combination (red, central ellipse) from our Fisher matrix calculations for the 8-parameter space described in the text. The plots are for fiducial values $\alpha =0.86$ (left), $\alpha =0.88$ (middle) and $\alpha =0.9$ (right).

Figure 4

Derivatives of matter power spectrum with respect to $\alpha$ for two target models. The derivative for $\alpha =1/3$ is multiplied for a factor of 10. See text for more details.

Figure 5

Constraints on $\alpha$ for several fiducial models and for different values of ${\ell }_{\max }$.

Figure 6

Solid line is the difference of matter power spectra calculated for normal hierarchy and degeneracy; dashed line is the difference between normal hierarchy and inverted hierarchy; dotted line is the numerical uncertainty (5%) on the nonlinear correction. Power spectra are calculated up to $k_{\max }={\ell }_{\max }/\chi (z)$ for ${\ell }_{\max }=1000$.

Figure 7

Same of previous figure but for ${\ell }_{\max }=2000$ and for an uncertainty of 1% according to future codes.

Figure 8

Absolute value of $\ln B$ as a function of $|\delta \alpha |$. The lines indicate the limits of the Jeffreys scale. On the right of the cusp is $B<1$, meaning evidence favors a more general parametrization of neutrino mass hierarchy.

Figure 9

Jeffreys scale contours of $\ln B$ as a function of $\delta \alpha$ and $\delta \sum _{}^{}{m}_{\nu }$. The inner part of the plot (from the innermost $\ln B=1$ contour) corresponds to values $B>1$ and hence evidence for the simpler model (see text). The cross indicates the assumption of degenerate masses fixing the total mass to the correct value; the square indicates a typical inverted hierarchy scenario.