Halo bias in mixed dark matter cosmologies

The large-scale distribution of cold dark matter halos is generally assumed to trace the large-scale distribution of matter. In a universe with multiple types of matter fluctuations, as is the case with massive neutrinos, the relation between the halo field and the matter fluctuations may be more complicated. We develop a method for calculating the linear bias factor relating fluctuations in the halo number density to fluctuations in the mass density in the presence of multiple fluctuating components of the energy density. In the presence of massive neutrinos we find a small but pronounced feature in the halo bias near the neutrino free-streaming scale. The neutrino feature is a small step with amplitude that increases with halo mass and neutrino mass density. The scale-dependent halo bias lessens the suppression of the small-scale halo power spectrum and should therefore weaken constraints on neutrino mass from the galaxy autopower spectrum and correlation function. On the other hand, the feature in the bias is itself a novel signature of massive neutrinos that can be studied independently.

Figure 1

Left: The scale-dependent changes to the matter power spectrum in a cosmology with a single massive neutrino $m_{\nu }=0.1\mathrm{eV}$. Right: The ratio of the CDM power spectrum and CDM-matter cross-power spectrum to the matter power spectrum in a cosmology with a single massive neutrino $m_{\nu }=0.1\mathrm{eV}$. Both quantities are plotted at $z=0$.

Figure 2

Left: The scale dependence of ${\dot{\delta }}_c/{\delta }_c$ at $z_i$, plotted for a number of different neutrino mass hierarchies. Also shown is the horizon scale at $z_i=200$ and the matter radiation equality scale. Right: The scale dependence of the linear evolution of CDM and baryon perturbations between $z=200$ and $z=0$. In both panels ${\mathrm{\Omega }}_c$ is fixed so varying ${\mathrm{\Omega }}_{\nu }$ changes the total matter density ${\mathrm{\Omega }}_m$.

Figure 3

Plotted is the relationship between the initial perturbation values ${\delta }_{c,iS}$ and ${\delta }_{c,iL}(k)$ for halos of $M={10}^{13}{M}_{\odot }$ that collapse at the same time $z_{\text{collapse}}=0.5$ for a range of $k$, the wave number of the long-wavelength mode. Left: $m_{\nu 1}=m_{\nu 2}=m_{\nu 3}=0\mathrm{eV}$, Right: $m_{\nu 1}=0.05\mathrm{eV}$, $m_{\nu 2}=m_{\nu 3}=0\mathrm{eV}$.

Figure 4

Plotted is the relationship between ${\delta }_{c,S}(z_{\text{collapse}})$ and ${\delta }_{c,iL}(k,z_{\text{collapse}})$ (the initial perturbation amplitudes linearly extrapolated to the collapse time) for halos of $M=10^{13}{M}_{\odot }$ that collapse at the same time $z_{\text{collapse}}=0.5$ for a range of values of $k$. Left: $m_{\nu 1}=m_{\nu 2}=m_{\nu 3}=0\mathrm{eV}$, Right: $m_{\nu 1}=0.05\mathrm{eV}$, $m_{\nu 2}=m_{\nu 3}=0\mathrm{eV}$.

Figure 5

The slopes of the relationships plotted in Fig. (4)—that is, the slope of the line relating ${\delta }_{c,S}$ to ${\delta }_{c,L}$. The left panel plots the relationship between the initial values of the two quantities. The right panel plots it with ${\delta }_{c,S}$ and ${\delta }_{c,L}$ evaluated at $z_{\text{collapse}}$ (using the scale-dependent linear growth functions). Each curve has a fixed ${\mathrm{\Omega }}_m$, but varying ${\mathrm{\Omega }}_c$ and ${\mathrm{\Omega }}_{\nu }\approx \sum _i{m}_{\nu i}$ where the neutrino masses are listed in the plot legend.

Figure 6

The shift in the Eulerian bias (left column) and Lagrangian bias with respect to the CDM (right column) relative to the values of the bias factors at very large scales. Precisely, the plotted quantity is $b(k)/b(k={10}^{-4}{\mathrm{Mpc}}^{-1})$. The top row is $b(M)$ for $M={10}^{13}{M}_{\odot }$ halos and the bottom row shows the shift in the bias for $M={10}^{14}{M}_{\odot }$ halos. In all plots the value of ${\mathrm{\Omega }}_m$ is fixed, but ${\mathrm{\Omega }}_c$ and ${\mathrm{\Omega }}_{\nu }$ vary. The neutrino free-streaming scale for each hierarchy, Eq. (16), is shown by the vertical dotted lines of the same color. In both panels the order of the legend matches the order of the curves.

Figure 7

The amplitude of the step feature in the halo bias seen in Fig. 6 defined here as $b(k=1{\mathrm{Mpc}}^{-1})/b(k={10}^{-4}{\mathrm{Mpc}}^{-1})-1$. Plotted is the size of the step in the Eulerian bias (solid lines) and Lagrangian bias with respect to the CDM fluctuations (dashed lines) for a range of neutrino mass hierarchies compatible with oscillation data.

Figure 8

The suppression in the halo power spectrum in the presence of massive neutrinos, relative to cosmologies without massive neutrinos. Precisely, the plotted quantity is the fractional difference between $P_{nn}(k|m_{\nu })/P_{nn}(k={10}^{-4}{\mathrm{Mpc}}^{-1}|m_{\nu })$ and $P_{nn}(k|m_{\nu }=0)/P_{nn}(k={10}^{-4}{\mathrm{Mpc}}^{-1}|m_{\nu }=0)$. The solid lines include the scale-dependent bias calculated in this paper; the dotted lines use the standard prediction of a constant value of $b$ defined by $b=1+b_c^{\text{Lagrangian}}$ where $b_c^{\text{Lagrangian}}$ is calculated assuming $d{\delta }_{\text{crit}}/d{\delta }_{c,L}=-1$.