Implications of an extended dark energy cosmology with massive neutrinos for cosmological tensions

We perform a comprehensive analysis of the most common early- and late-universe solutions to the $H_0$, $\mathrm{Ly}\text{−}\alpha$, and $S_8$ discrepancies. When considered on their own, massive neutrinos provide a natural solution to the $S_8$ discrepancy at the expense of increasing the $H_0$ tension. If all extensions are considered simultaneously, the best-fit solution has a neutrino mass sum of $\sim 0.4\mathrm{eV}$, a dark energy equation of state close to that of a cosmological constant, and no additional relativistic degrees of freedom (d.o.f). However, the $H_0$ tension, while weakened, remains unresolved. Motivated by this result, we perform a nonparametric reconstruction of the evolution of the dark energy fluid density (allowing for negative energy densities), together with massive neutrinos. When all data sets are included, there exists a residual $\sim 1.9\sigma$ tension with $H_0$. If this residual tension remains in the future, it will indicate that it is not possible to solve the $H_0$ tension solely with a modification of the late-universe dynamics within standard general relativity. However, we do find that it is possible to resolve the tension if either galaxy baryon acoustic oscillation (BAO) or joint light-curve analysis supernovae data are omitted. We find that negative dark energy densities are favored near redshift $z\sim 2.35$ when including the $\mathrm{Ly}\text{−}\alpha$ BAO measurement (at $\sim 2\sigma$). This behavior may point to a negative curvature, but it is most likely indicative of systematics or at least an underestimated covariance matrix. Quite remarkably, we find that in the extended cosmologies considered in this work, the neutrino mass sum is always close to 0.4 eV regardless of the choice of external data sets, as long as the $H_0$ tension is solved or significantly decreased.

Figure 1

The posterior distribution of $\{H_0,{\sigma }_8,{\mathrm{\Omega }}_m,\sum m_{\nu },w_0,w_a,\mathrm{\Delta }{N}_{\text{fluid}}\}$ when fitting to all data sets considered in this work, compared to the $\mathrm{\Lambda }\mathrm{CDM}$ fit of the same data set.

Figure 2

Reconstructed ExDE energy density and Hubble expansion rate (compared to the $\mathrm{\Lambda }\mathrm{CDM}$ prediction from Planck TT, TE, $\mathrm{EE}+\mathrm{SIMlow}$, black line) with $\sum m_{\nu }=0.06\mathrm{eV}$ (left panel) or $\sum m_{\nu }$ left as a free parameter (right panel), when including all data sets considered in this work and for different choices of prior on ${\mathrm{\Omega }}_{\text{ExDE}}$ (see text). The thick solid lines show the best fit spline in each case, while the thin lines show samples from the 68% confidence region. The vertical arrows show the positions of the knots. The orange band indicates the uncertainty on the Hubble parameter as measured by SH0ES (strictly speaking it is only valid at $z=0$).

Figure 3

Left panel: A comparison between the 1D and 2D posterior distributions of (${\sigma }_8$, ${\mathrm{\Omega }}_m$, $H_0$, $\sum m_{\nu }$) obtained in various models when using all data sets considered in this work. The grey band shows the R16 measurement, the purple band is the Planck SZ determination of $S_8$. Right panel: Reconstructed DE energy density and Hubble expansion rate (compared to the $\mathrm{\Lambda }\mathrm{CDM}$ prediction from Planck TT, TE, $\mathrm{EE}+\mathrm{SIMlow}$, black line) with $\sum m_{\nu }$ left as a free parameter. We include either the BAO (red) or JLA (blue) data. The thick solid lines show the best fit spline in each case, while the thin lines show draws from the 68% most likely fits. The red arrows pointing upwards show the locations of the BAO knots, while the blue arrows pointing downwards show the positions of the JLA knots. The orange band indicates the uncertainty on the Hubble parameter as measured by SH0ES (strictly speaking it is only valid at $z=0$).

Figure 4

The 1D and 2D posterior distributions of (${\sigma }_8$, ${\mathrm{\Omega }}_m$, $H_0$, $\sum m_{\nu }$) with a fixed neutrino mass sum (left panel) and a free neutrino mass sum (right panel) when using SDSS DR7 CFHTLens, SH0ES, CMB, $\mathrm{Ly}\text{−}\alpha$ BAO DR11, and either galaxy BAO DR12 (red curves) or JLA (blue curves). The grey band shows the SH0ES measurement, while the purple band is the Planck SZ determination of $S_8$.