Figure 1

The upper panel shows the factor by which the neutrino fraction $f_{\nu }$ (dashed curve) suppresses the current fluctuation amplitude ${\sigma }_M$ (upper solid curve) and consequently the galaxy number density $n_G$ (lower solid curve). The lower panel shows the resulting probability distribution for the neutrino mass sum for priors $m_{\nu }^n$ with $n=0$, 1, 2, 3 and 4, peaking from left to right, respectively.Reuse & Permissions

Figure 2

The probability distribution for the neutrino mass sum for flat prior $(n=0)$. The dark/red tails contain 5% probability each. The dotted line shows the lower limit 0.05 eV from atmospheric neutrino oscillations [49, 50, 51, 52].Reuse & Permissions

Figure 3

Same as Fig. 2, but showing the robustness of the results to changing various assumptions. We have changed the baseline calculation from Fig. 2 (heavy black curve) by evaluating the galaxy density in the infinite future $x\to \infty$ (dotted red/ gray curve) and at redshift unity, $x=7/3(1+z{)}^3=7/24$ (dot-long-dashed blue/ gray curve), decreasing the density threshold to ${\delta }_c=1.5$ (short-dashed magenta/ gray curve), lowering the primordial fluctuation amplitude on the galactic scale by 25% (solid blue/ gray curve), including the baryon correction as per Eisenstein and Hu (long-dashed green curve) and additionally including the neutrino infall correction as per Eisenstein and Hu (dot-dashed cyan/ gray curve).Reuse & Permissions

Figure 4

The $\Lambda \mathrm{CDM}$ growth function $G_{\Lambda }$ is shown as a function of cosmic time, first growing as the scale factor $a\propto x^{1/3}$, then asymptoting to $1.44$ as dark energy halts the fluctuation growth at $x\gtrsim 1$. The middle panel shows the exact result of Eq. (a1) (solid curve), our approximation given by Eq. (7) (green thick dashed curve), the Carroll, Press and Turner approximation [63] (red dotted curve) and the power law approximation $G_{\Lambda }={\Omega }_{\mathrm{m}}^{0.21}$ [66] (blue long-dashed curve). In the bottom panel, the various approximations have been divided by the exact result, showing that Eq. (5) is accurate to better than 1.5% for all $x$. The top panel shows the collapse density threshold ${\delta }_c(x)$ dropping from $1.6865$ early on to $1.62978$ in the infinite future.Reuse & Permissions