Cosmology with massive neutrinos III: the halo mass function and an application to galaxy clusters

TL;DR

This study investigates the impact of massive neutrinos on the halo mass function (HMF) using N-body simulations that include neutrinos as particles. It demonstrates that for SO haloes, the HMF is best described by the Tinker et al. fit when the CDM density ρ_{cdm} and the CDM power spectrum P_{cdm}(k,z) are used to compute σ(M,z), rather than total matter quantities. Applying this calibration to Planck-like cluster counts reveals up to ~30% differences in predicted counts for Σmν ≈ 0.4 eV and shows how neutrino treatment shifts the σ_8–Ω_m degeneracy, quantified by Δγ ≈ 0.05–0.14 and Δσ_8 ≈ 0.01–0.02 depending on neutrino split. These corrections tend to increase tension between cluster abundances and Planck CMB measurements in massive-neutrino cosmologies. The work provides a practical HMF calibration for neutrino cosmologies and underscores the importance of using CDM-based prescriptions in interpreting cluster data.

Abstract

We use a suite of N-body simulations that incorporate massive neutrinos as an extra-set of particles to investigate their effect on the halo mass function. We show that for cosmologies with massive neutrinos the mass function of dark matter haloes selected using the spherical overdensity (SO) criterion is well reproduced by the fitting formula of Tinker et al. (2008) once the cold dark matter power spectrum is considered instead of the total matter power, as it is usually done. The differences between the two implementations, i.e. using $P_{\rm cdm}(k)$ instead of $P_{\rm m}(k)$, are more pronounced for large values of the neutrino masses and in the high end of the halo mass function: in particular, the number of massive haloes is higher when $P_{\rm cdm}(k)$ is considered rather than $P_{\rm m}(k)$. As a quantitative application of our findings we consider a Planck-like SZ-clusters survey and show that the differences in predicted number counts can be as large as $30\%$ for $\sum m_ν= 0.4$ eV. Finally, we use the Planck-SZ clusters sample, with an approximate likelihood calculation, to derive Planck-like constraints on cosmological parameters. We find that, in a massive neutrino cosmology, our correction to the halo mass function produces a shift in the $σ_8(Ω_{\rm m}/0.27)^γ$ relation which can be quantified as $Δγ\sim 0.05$ and $Δγ\sim 0.14$ assuming one ($N_ν=1$) or three ($N_ν=3$) degenerate massive neutrino, respectively. The shift results in a lower mean value of $σ_8$ with $Δσ_8 = 0.01$ for $N_ν=1$ and $Δσ_8 = 0.02$ for $N_ν=3$, respectively. Such difference, in a cosmology with massive neutrinos, would increase the tension between cluster abundance and Planck CMB measurements.

Figures (5)

• Figure 1: Mass function of dark matter haloes identified using the SO criteria for different cosmological models at redshifts $z=0$ (upper-left), $z=0.5$ (upper-right) and $z=1$ (bottom). The points show the halo mass function obtained from the N-body simulations with massless neutrinos (red) and with neutrinos with masses $\Sigma_i m_{\nu_i}~$ = 0.3 eV (green) and $\Sigma_i m_{\nu_i}~$ = 0.6 eV (blue). The error bars represent the dispersion around the mean value obtained from the eight independent realizations for each cosmological model. The results of using the Tinker fitting formula Tinker2008 along with the matter and cold dark matter prescriptions (see text for details) are displayed with dashed and solid lines, respectively.
• Figure 2: Ratio between the halo mass function for cosmologies with massive neutrinos to the halo mass function for the cosmological model with massless neutrinos (reference model). The points represent the results from the N-body simulations whereas the solid and dashed lines correspond to the ratios between the halo mass functions computed using the Tinker fitting formula together with the cold dark matter prescription and the matter prescription for cosmologies with massive neutrinos, respectively.
• Figure 3: Ratio of the number counts obtained using the CDM over the matter prescription for different combinations of ($\sum m_\nu - \sigma_8$) values (colour-coded) and different neutrino mass splitting: one massive neutrino (left panel) and three degenerate massive neutrinos (right panel). For a given cosmology, the cold dark matter prescription predicts a larger number of clusters, especially for high neutrino mass and cosmology with three massive neutrinos. Black curves trace constant values of the ratio; from the left to the right: $1.05, 1.10,1.15,1.20$ (left panel); $1.1, 1.2,1.3,1.4$ (right panel).
• Figure 4: Curves of constant number counts ($N=600$, 200, 100 and 50, top to bottom) in the plane $\sum m_\nu - 10^9 \cdot A_s$ (left panel) and in the plane $\sum m_\nu$ - $\sigma_8$ (right panel), for the two prescriptions for the halo mass function, matter (black) and cold dark matter (red) and for two neutrino mass splitting schemes, single massive neutrino (solid lines) and three degenerate massive neutrinos (dashed lines). The different slope of the black and red curves shows the different degeneracy direction between parameters in the prescriptions.
• Figure 5: Comparison of the $68\%$ and $95\%$ C.L. contours in the $\Omega_{\rm m}-\sigma_8$ plane and degeneracy curves obtained using the matter (blue) and cold dark matter (green) prescription. We show the results when the sum of the neutrino masses is split between one massive neutrino family (left panel) and three degenerate neutrino families (right panel). Also shown in the right panel in orange the contours from PlanckCMB+BAO datasets for a $\Lambda$CDM+$\sum m_\nu$ model.