Cosmology with massive neutrinos II: on the universality of the halo mass function and bias

TL;DR

This paper shows that in cosmologies with nonzero neutrino masses, halo abundances and large-scale clustering are substantially more universal when expressed through the CDM-only variance $σ^2_{cc}(M,z)$ rather than the total matter variance $σ^2_m(M,z)$. Using CDM-based peak height $ν=δ_{cr}/σ_{cc}$ restores near-universality in the halo mass function across neutrino masses and redshifts, and halo bias becomes scale-independent on large scales when defined relative to the CDM power spectrum. The authors analyze extensive N-body simulations with neutrinos treated as CDM-like particles, demonstrating that neglecting the CDM-only perspective introduces sizable (and potentially bias-inducing) nonuniversality and scale dependence. These findings have direct implications for cosmological parameter inference from cluster counts and galaxy clustering in the presence of massive neutrinos, and they advocate CDM-based descriptions in both mass function and bias modeling. The work also clarifies degeneracies with σ8 and underscores the importance of the CDM component in interpreting nonlinear structure formation with massive neutrinos.

Abstract

We use a large suite of N-body simulations to study departures from universality in halo abundances and clustering in cosmologies with non-vanishing neutrino masses. To this end, we study how the halo mass function and halo bias factors depend on the scaling variable $σ^2(M,z)$, the variance of the initial matter fluctuation field, rather than on halo mass $M$ and redshift $z$ themselves. We show that using the variance of the cold dark matter rather than the total mass field, i.e., $σ^2_{cdm}(M,z)$ rather than $σ^2_{m}(M,z)$, yields more universal results. Analysis of halo bias yields similar conclusions: When large-scale halo bias is defined with respect to the cold dark matter power spectrum, the result is both more universal, and less scale- or $k$-dependent. These results are used extensively in Papers I and III of this series.

Figures (9)

• Figure 1: Halo mass function for the three models of set A at redshift $z=0$ ( left panels) and $z=1$ ( right panels). Top panels show the quantity $M^2\,n(M)$ as a function of mass, together with the predictions of the MICE fitting formula CrocceEtal2010 using $\sigma=\sigma_{mm}$ ( dotted curves) and $\sigma=\sigma_{cc}$ ( dashed curves). Black, blue and red data points correspond respectively to the $m_\nu=0$, $0.3$ and $0.6$ eV results. Lower panels show the residuals of measurements with respect to the MICE formula with $\sigma=\sigma_{cc}$. Here, as in the following figures, all symbols show the mean over the eight realizations while the error bars show the uncertainty on this mean.
• Figure 2: Ratio of the measured $\nu\,f(\nu)$ to the ST formula. Left panels use $\nu=\delta_{cr}/\sigma_{cc}$ while right panels use $\nu=\delta_{cr}/\sigma_{mm}$. Different panels, top to bottom, show the different redshifts $z=0$, 0.5 and 1 with all neutrino masses (distinguished by color) shown together. Dashed curves show the MICE fit of CrocceEtal2010.
• Figure 3: Left column: residuals of the measured $\nu f(\nu)$ w.r.t. to the best fit (assuming the ST form) to the $m_{\nu}=0$ data. Black, blue and red data points show the results respectively for the $m_{\nu}=0$, $0.3$ and $0.6$ eV models. Different panels correspond to $z=0$, $0.5$ and $1$, top to bottom. Right column: residuals of the measured mass function $f(\nu)$ w.r.t. to the best fit (assuming the ST form) to the $z=0$ data. Data points for the higher redshift outputs are shown by lighter shades of the given color. Different panels correspond to $m_{\nu}=0$, $0.3$ and $0.6$ eV, top to bottom.
• Figure 4: Degeneracy between $m_{\nu}$ and $\sigma_{8,mm}$, $\sigma_{8,cc}$: comparison of the mass functions as a function of the mass $M$ and at $z=0$ from $m_{\nu}=0.0$ eV and $m_{\nu}=0.6$ eV models, sharing the same value of $\sigma_{8,cc}$ or $\sigma_{8,mm}$. Left panels: comparison between model H6 (red data points) and models H0s8 (grey) and H0s8-CDM (light gray). The lower panel shows the ratio of H0s8 and H0s8-CDM to H6, with the black horizontal line showing the ratio $\Omega_c(\text{H0s8-CDM})/\Omega_c(\text{H6})$. Right panels: comparison between model H0 (black data points) and model H6s8 (orange). The lower panel shows the ratio of H6s8 to H0.
• Figure 5: Degeneracy between $m_{\nu}$ and $\sigma_{8,mm}$, $\sigma_{8,cc}$: ratio of the measured $f(\nu)$ for several models as a function of $\nu$ w.r.t. the best fit (assuming the ST functional form) to the massless neutrino model of set A, with $\sigma_{8,mm}=\sigma_{8,cc}=0.83$, taken as a reference (black data points). This is compared directly with a $m_\nu=0.6$ eV model with $\sigma_{8,mm}=0.83$ and $\sigma_{8,cc}=0.86$ (yellow points). In addition, the $m_\nu=0.6$ eV model with $\sigma_{8,mm}=0.67$ and $\sigma_{8,cc}=0.70$ from set A (red points) is compared with two massless neutrino models with both $\sigma_{8,mm}=\sigma_{8,cc}=0.67$ (gray points) and $\sigma_{8,mm}=\sigma_{8,cc}=0.70$ (lighter gray points). Left and right panels show the results at $z=0$ and $z=0.5$.