Neutrino clustering around spherical dark matter halos

TL;DR

This paper tackles how massive relic neutrinos cluster around spherical CDM halos within a $\ u\Lambda CDM$ background. It combines a Newtonian treatment of neutrino dynamics with three complementary methods—the BKT Boltzmann approach, an absorbing-barrier accretion model, and a full Boltzmann calculation—to determine the neutrino halo boundary $r_*$, the total neutrino mass inside the halo, and the bound-neutrino mass that persists into the late-time, $\ m \Lambda$-dominated era. The authors provide fitting formulae for the neutrino mass contribution as a function of halo mass and neutrino masses across several hierarchies, revealing that the neutrino halo is significantly more extended than the CDM virial radius and that, for cosmologically allowed neutrino masses, the bound mass fraction is small but non-negligible for heavier degenerate masses. The work illustrates how semi-analytic, halo-scale neutrino clustering can be incorporated into spherical-collapse treatments and benchmarks the limits of approximations like BKT against a full Boltzmann solution, with potential implications for weak-lensing detection of neutrino halos.

Abstract

Cold dark matter halos form within a smoothly distributed background of relic neutrinos -- at least some of which are massive and non-relativistic at late times. We calculate the accumulation of massive neutrinos around spherically collapsing cold dark matter halos in a cosmological background. We identify the physical extent of the "neutrino halo" in the spherical collapse model, which is large in comparison with the virial radius of the dark matter, and conditions under which neutrinos reaching the cold dark matter halo will remain bound to the halo at late times. We calculate the total neutrino mass and bound neutrino mass associated with isolated spherical halos for several neutrino mass hierarchies and provide fitting formulae for these quantities in terms of the cold dark matter halo mass and the masses of the individual neutrino species.

Figures (9)

• Figure 1: Comparison of the peculiar velocity of neutrinos to the depth of the gravitational potential well for different halo masses. Left: Thick solid curves are $\Delta\Psi \equiv \frac{3}{2} G\delta M/R_c$ for spherical-top hat halos with $M = 10^{13}M_\odot$, $= 10^{14}M_\odot$, $= 10^{15}M_\odot$ that virialize at roughly the same time (indicated by the vertical line). For each neutrino mass we plot $\frac{1}{2}u^2$ where $u = 3.151 T_\nu/m_\nu$ -- the average magnitude of the peculiar velocity for neutrinos drawn from a Fermi-Dirac distribution with temperature $T_\nu = 1.95 K$. Right: The distribution of neutrino velocities today for different neutrino masses (in the non-relativistic limit $p = m_\nu u$). Black vertical lines show the depth the potential wells $\sqrt{2\Delta\Psi}$.
• Figure 2: Left: Solid colored lines are trajectories of $m_\nu= 0.05 eV$ neutrinos on radial paths through a collapsing halo of mass $M=10^{14} M_\odot$. The different colors indicate neutrinos with different velocities at the time that they reach $R_\nu$. The radius of the collapsing halo is shown in the thick black line (after $t\sim 6$ Gyrs the halo is assumed to have constant proper size $R_{virial}$). The time at which $\dot{H}+H^2\rightarrow 0$ is indicated by the vertical dotted line. The dashed brown line (that diverges when $\ddot{a}/a\rightarrow 0$) is $R_\nu(t)$, the scale at which the gravitational force of the halo dominates over the background (see Eq. (\ref{['eq:Rnu']})) and another scale $r_*(t)$ is shown in the dashed gray line. Right: The total gravitational potential (the sum of $\Psi_H$ in Eq. (\ref{['eq:PsiHubble']}) and $\Psi_{pec}$ in Eq. (\ref{['eq:Psipec']})) plotted at several different times. $R_\nu(t)$ at each time is indicated by the dashed vertical line of the same color.
• Figure 3: Left: The density profile of neutrino mass from a single neutrino species calculated using the BKT approximation in Eq. (\ref{['eq:Mnuf1']}). Here, the CDM halo has $M = 10^{14} M_\odot$ and collapses around $t_{collapse} \sim$ 8.5 Gyrs, or $z_{collapse} \sim 0.5$. We show several different neutrino masses (the curves roughly have increasing neutrino mass from top to bottom) and scale the results by $m_{\nu}^{3/2}$. The extent of the "neutrino halo" is large compared to the CDM virial radius (solid vertical line) and $r_*$ as defined in Eq. (\ref{['eq:rstar']}) (dotted vertical lines) appears to be a good characterization of the scale. Right: The neutrinos that contribute to $\delta M_\nu$ originate at a range of distances far from the CDM halo. Plotted is $r d\delta M_\nu/dr(<r_*, t_{collapse})$ scaled by $m_{\nu}^{5/2}$. The colored vertical lines indicate particle horizon for a neutrino with average momentum $p = 3.151T_\nu$ and corresponding $m_\nu$ (the curves have increasing $m_\nu$ from left to right).
• Figure 4: Left: The neutrino mass fluctuation from a single neutrino species within radius $r_*$ at the collapse time as a function of CDM halo mass calculated using the BKT approximation in Eq. (\ref{['eq:Mnuf1']}). The value of $\delta M_\nu$ depends on the halo collapse time and for each ($m_\nu$, $M$) we plot points with a range of $z_{collapse}$ values; they are $z_{collapse} = 0$ (solid), $0.5$ (dashed), $1$ (dot-dashed), and $1.5$ (dotted). Roughly, $\delta M_\nu(<r_*, t_{collapse}) \propto M^{3/2}m_\nu^{5/2}$. Right: The fluctuation in neutrino mass within the CDM halo radius $R_c$ at the collapse time as a function of CDM halo mass, as we show in § \ref{['ssec:full']}, the BKT approximation significantly underestimates the mass interior $R_c$ for $m_\nu \mathrel{\hbox{$\sim$} \hbox{$>$}} 0.1eV$. In both figures the plotted values of $m_\nu$ increase from the bottom curve to the top.
• Figure 5: Left: The accreted bound neutrino mass (from a single neutrino species) within radius $r_*$ at the collapse time as a function of CDM halo mass calculated using Eq. (\ref{['eq:dMdt']}), Eq. (\ref{['eq:urho']}), and Eq. (\ref{['eq:deltarho']}). The value of $\delta M_\nu$ depends on the halo collapse time and for each ($m_\nu$, $M$) we plot points with a range of $z_{collapse}$ values; they are $z_{collapse} = 0$ (solid), $0.5$ (dashed), $1$ (dot-dashed), and $1.5$ (dotted). For $m_\nu \mathrel{\hbox{$\sim$} \hbox{$<$}} 0.2 eV$, the bound neutrino mass scales roughly as $\delta M_\nu(<r_*, t_{collapse}) \propto M^{2} m_\nu^{4}$ whereas for higher neutrino masses the scaling is closer to $\delta M_\nu(<r_*, t_{collapse}) \propto M^{3/2} m_\nu^{3}$. Right: A subset of the trajectories of $m_{\nu} = 0.05eV$ neutrinos captured by a $M= 10^{14} M_\odot$ halo. Also plotted are the radius of the CDM halo $R_{vir}$ today (solid) and our definition of the boundary of the neutrino halo $r_*$ (dotted) at $z = 0$.