The impact of massive neutrinos on the abundance of massive clusters

TL;DR

This work develops a multi-component spherical top-hat collapse model that includes radiation, baryons, CDM, and massive neutrinos to quantify how neutrino masses in the range $m_{ u,tot}\sim 0.05$–$0.1$ eV affect massive-halo formation. It solves superhorizon evolution exactly and treats subhorizon CDM and baryon growth while modeling neutrino perturbations with the linearized Boltzmann hierarchy fed by nonlinear CDM/baryon potentials. The key findings are that massive neutrinos slow CDM collapse but do not fully catch up to nonlinear CDM growth, with the linear-theory overdensity $\delta^{\rm L}_{cb}(z_{coll};R)$ remaining a good clock; for halos around $M\sim 10^{15} h^{-1}M_\odot$ at $z\sim 1$, the halo abundance can decrease by up to a factor of ~2 for $m_{\nu,tot}=0.1$ eV, even though $\sigma_8$ shifts by only a few percent. This framework provides a practical analytic tool to interpret cluster-count data and reveals how neutrino physics can degenerate with cosmological expansion parameters in shaping the observed halo distribution.

Abstract

We study the spherical, top-hat collapse model for a mixed dark matter model including cold dark matter (CDM) and massive neutrinos of mass scales ranging from m_nu= 0.05 to a few 0.1eV, the range of lower- and upper-bounds implied from the neutrino oscillation experiments and the cosmological constraints. To develop this model, we properly take into account relative differences between the density perturbation amplitudes of different components (radiation, baryon, CDM and neutrinos) around the top-hat CDM overdensity region assuming the adiabatic initial conditions. Furthermore, we solve the linearized Boltzmann hierarchy equations to obtain time evolution of the lineariezed neutrino perturbations, yet including the effect of nonlinear gravitational potential due to the nonlinear CDM and baryon overdensities in the late stage. We find that the presence of massive neutrinos slows down the collapse of CDM (plus baryon) overdensity, however, that the neutrinos cannot fully catch up with the the nonlinear CDM perturbation due to its large free-streaming velocity for the ranges of neutrino masses and halo masses we consider. We find that, just like CDM models, the collapse time of CDM overdensity is well monitored by the linear-theory extrapolated overdensity of CDM plus baryon perturbation, smoothed with a given halo mass scale, if taking into account the suppression effect of the massive neutrinos on the linear growth rate. Using these findings, we argue that the presence of massive neutrinos of mass scales 0.05 or 0.1eV may cause a significant decrease in the abundance of massive halos compared to the model without the massive neutrinos; e.g., by 25% or factor 2, respectively, for halos with 10^15Ms and at z=1.

Figures (11)

• Figure 1: Radial profile of density perturbation for each component (CDM, baryon and massive neutrino) in the CDM top-hat overdensity region assuming the adiabatic initial conditions to determine relative amplitudes of different components. We plot the profiles in units of comoving scale, ${\rm Mpc}/h$. We assumed $m_\nu=0.05~{\rm eV}$ for neutrino mass scale, $R=6.89~h^{-1}{\rm Mpc}$ for radius of the top-hat region, which corresponds to halo mass scale $M=10^{14}~h^{-1 }M_\odot$ for our fiducial cosmological model, and determined the initial density amplitude so that the top-hat region collapses around $z=0.5$. Note that, for the baryon perturbation, we show its density contrast within the CDM top-hat region for illustrative clarity (the baryon top-hat perturbation computed is spatially more extended, and holds the mass conservation within its own top-hat region). The different panels show the profiles at different redshifts as indicated. At sufficiently early redshift such as $z=570$, after the decoupling epoch, $\delta_{\rm b},~\delta_{\rm \nu}\ll \delta_{\rm c}$, because baryon was coupled with radiation until the decoupling epoch ($z\simeq 1100$) and neutrino was free-streaming out of CDM potential well. As time goes by, the baryon perturbation eventually catches up with the CDM perturbation. Then at redshifts lower than $z\simeq 30$ for this case, the top-hat radius starts to shrink and the top-hat dynamics deviates from the linear theory and enters into the nonlinear regime. As for neutrinos, the large velocity dispersion of neutrino particles prevents from catching up with the nonlinear collapse and then becomes to have a more spatially-extended profile than the CDM and baryon top-hat region. Even when CDM and baryon collapse ($z\simeq 0.5$), the neutrino perturbation stays in the quasi nonlinear regime as $\delta_\nu \lesssim 1$.
• Figure 2: Time evolution of the density contrast of each component, averaged within the CDM top-hat region. We used the same initial conditions and model parameters as in Fig. \ref{['fig:sp_collapse']}. For this particular case, the neutrino perturbation averaged within the CDM top-hat region $\delta_\nu\simeq 0.19$ at the collapse redshift, and therefore is still in the quasi nonlinear regime.
• Figure 3: Time evolution of the CDM top-hat overdensity for different models. The dashed curve shows the result for our fiducial cosmological model without massive neutrino. We again assumed the halo mass scale $M=10^{14}~h^{-1}M_\odot$ and determined the initial CDM perturbation so that the top-hat region collapses at $z\simeq 0.5$. The solid curve shows the result when ignoring the baryon perturbation, where the CDM perturbation amplitude is set so as to match that of the dashed curve at the decoupling epoch $z\simeq 1100$. Note that the model of the solid curve leads the linear-theory extrapolated overdensity to be $\delta^{\rm L}=1.686$ at redshifts before dark energy domination in the cosmic expansion (the result shown here is affected by dark energy domination). Comparing the solid and dashed curves manifests that the presence of baryon perturbation, which has a smaller amplitude at earlier redshifts as implied from Figs. \ref{['fig:sp_collapse']} and \ref{['fig:density']}, delays the spherical collapse. The solid and dotted curves show the results when further including massive neutrino for a fixed dark matter density $\Omega_{\rm c0}+\Omega_{\nu0}$. The neutrino perturbation is smoother than that of CDM perturbation, and delays the spherical collapse.
• Figure 4: The linear-theory extrapolated density contrast of CDM plus baryon perturbation at the collapse redshift -- the so-called critical density that can be used to infer the collapse redshift based on the linear theory. The left and right panels show the results for halo mass scales $M=10^{14}$ and $10^{15}~h^{-1}M_\odot$, respectively. The different curves are the results without and with massive neutrino contribution assuming different neutrino mass scales. Note that the density contrast shown here is not for CDM perturbation alone, and the corresponding critical density of CDM perturbation is greater than shown in this plot. The overall change in the critical density from the Einstein de-Sitter result $\delta^{\rm L}=1.686$ arises from the effect of baryon perturbation for higher redshifts, while the change at lower redshift $z\hbox{$\buildrel < \@@over \sim$} 1$ is due to dark energy domination in the cosmic expansion. The effect of massive neutrino is in the range of the different curves. The curves show non-trivial dependence on neutrino mass scale (see the next figure).
• Figure 5: Similarly to the previous figure, but we here ignored the effect of neutrino perturbation on the spherical collapse and on the linear density calculation; i.e. we set $\delta_\nu=0$ and $\delta^{\rm L}_\nu=0$. The critical overdensity becomes smaller at each redshift with increasing the neutrino mass scale. Therefore, comparing this figure with Fig. \ref{['fig:dlc']} clarifies that the non-trivial dependence of $\delta^{\rm L}_{cb}$ on neutrino mass scale is due to the effect of neutrino perturbation (see text for details).