The Dark Side of the Halo Occupation Distribution

TL;DR

This work demonstrates that the distribution of subhalos within larger halos in dissipationless ΛCDM simulations can be captured by a simple, physically motivated halo occupation distribution (HOD) framework. By separating central and satellite populations, the authors show that centrals follow a near-step function $P_c(M)$ while satellites follow a Poisson distribution with a mean $\langle N_s \rangle=(M/M_1)^{\beta}$, with $\beta\approx 1$ and $M_1$ scaling with $M_{\min}$ and redshift. The resulting HOD yields a halo two-point correlation function that matches observed galaxy clustering at $z=0$ and reveals departures from a single power-law at higher redshift due to the evolving 1-halo term, implying potential biases in high-$z$ clustering analyses if a power law is assumed. Overall, the study provides a compact, three-parameter model for the galaxy-halo connection that aligns with previous semi-analytic and hydrodynamic results and clarifies the roles of central versus satellite populations in shaping small- and large-scale clustering.

Abstract

We analyze the halo occupation distribution (HOD), the probability for a halo of mass M to host a number of subhalos N, and two-point correlation function of galaxy-size dark matter halos using high-resolution dissipationless simulations of the concordance flat LCDM model. The halo samples include both the host halos and the subhalos, distinct gravitationally-bound halos within the virialized regions of larger host systems. We find that the first moment of the HOD, <N>(M), has a complicated shape consisting of a step, a shoulder, and a power law high-mass tail. The HOD can be described by a Poisson statistics at high halo masses but becomes sub-Poisson for <N><4. We show that the HOD can be understood as a combination of the probability for a halo of mass M to host a central galaxy and the probability to host a given number Ns of satellite galaxies. The former can be approximated by a step-like function, while the latter can be well approximated by a Poisson distribution, fully specified by its first moment <Ns>(M). We find that <Ns>~M^b with b~1 for a wide range of number densities, redshifts, and different power spectrum normalizations. This formulation provides a simple but accurate model for the halo occupation distribution found in simulations. At z=0, the two-point correlation function (CF) of galactic halos can be well fit by a power law down to ~100/h kpc with an amplitude and slope similar to those of observed galaxies. At redshifts z>~1, we find significant departures from the power-law shape of the CF at small scales. If the deviations are as strong as indicated by our results, the assumption of the single power law often used in observational analyses of high-redshift clustering is likely to bias the estimates of the correlation length and slope of the correlation function.

Figures (12)

• Figure 1: Distribution of dark matter particles (points) and dark matter halos (circles) identified by our halo finding algorithm centered on the most massive halo in the $\Lambda$CDM$_{80}$ simulation at $z=3$ (left) and $z=0$ (right). The radius of the largest circle indicates the actual virial radius, $R_{180}$, of the most massive halo ($R_{180}=0.67h^{-1}$ comoving Mpc at $z=3$ and $R_{180}=2.1h^{-1}$ Mpc at $z=0$); the radii of all other halos are scaled using the halo' maximum circular velocities ($r_{\rm h}=0.65V_{\rm max}$ kpc with $V_{\rm max}$ in $\rm km\ s^{-1}$).
• Figure 2: The maximum circular velocity of halos in the $60h^{-1}{\ }{\rm Mpc}$ ( dashed line) and $80h^{-1}{\ }{\rm Mpc}$ ( solid line) $\Lambda$CDM simulations vs. the $r$-band absolute magnitude of the SDSS galaxies of the same number density. The curves are obtained by matching the cumulative velocity functions $n(>V_{\rm max})$ (at $z=0$) to the SDSS luminosity function $n(<M_r)$. The dashed lines show the power-law luminosity--circular velocity relation, $L_r\propto V_{\rm max}^{a}$, for $a=7$ and $a=3$.
• Figure 3: Bottom panels: cumulative mass functions of the halo samples (left panel: $\Lambda$CDM$_{80}$, right panel: $\Lambda$CDM$_{60}$) used in our analysis at different redshifts. Note that the number density here includes all the centers of the isolated host halos and the subhalos located within the hosts (see eq. \ref{['eq:nhalo']} in § \ref{['sec:hodmodel']}). The horizontal dotted lines indicate the number density thresholds adopted in our analysis. The curves were plotted down to the minimum halo mass of 50 particles. Top panels: the fraction of halos with masses larger than $M$ classifed as subhalos: $f_{\rm sub}=(n - n_{\rm host})/n$.
• Figure 4: Bottom panel: the first moment of the halo occupation distribution, as a function of host mass for the halo sample with number density $n=5.86\times 10^{-2}h^3\rm Mpc^{-3}$ in the $\Lambda$CDM$_{80}$ simulation at $z=0$. The solid line shows the mean total number of halos including the hosts, while the long-dashed line shows the mean number of satellite halos. The error bars show the uncertainty in the mean. The dotted line shows the step function corresponding to the mean number of "central" halos. Note that by definition, the solid line is the sum of the dotted and long-dashed lines. The two short-dashed lines indicate the dependencies $\propto M_{\rm h}$ and $M_{\rm h}^{0.8}$. Upper panel: the parameter $\alpha\equiv \langle N(N-1)\rangle^{1/2}/\langle N\rangle$ for the full HOD ( solid points) and the HOD of satellite halos ( open points). The dotted line at $\alpha=1$ shows the case of a Poisson distribution. Note that the HOD becomes sub-Poisson at small host masses. However, the HOD of satellites remains close to Poisson down to masses an order of magnitude smaller than for the full HOD. Indeed, if the satellite HOD is Poisson, $\alpha=(1-1/\langle N\rangle^2)^{1/2}$ for the full HOD [see eq. (\ref{['eq:alphahod']})]. This expression is shown by the dot-dashed line, which describes the points very well. The full HOD at small $M_{\rm h}$ is also well described by the nearest integer distribution [see eqs. (\ref{['eq:honint']}) and (\ref{['eq:xi2xi3']})] shown by the dashed line.
• Figure 5: The mean number of subhalos $\langle N_s\rangle$ as a function of host mass for the halo samples of different number densities (symbols of different type) at different redshifts. The error bars show the uncertainty in the mean. The mean is plotted as a function of host mass in units of the minimum mass of the sample (see Fig. \ref{['fig:vmf']}; note that, by definition, $\langle N_s\rangle=0$ at $M_{\rm h}/M_{\mathrm{min}}=1$ and non-zero points shown at or below $M_{\rm h}/M_{\mathrm{min}} =1$ are caused by binning). The mass $M_{\rm h}$ in the $x$-axis is the mass within the radius corresponding to overdensity 180 with respect to the mean density of the universe. The number densities in units of $h^3\rm Mpc^{-3}$ are indicated in the legend. The solid line in each panel shows the linear relation $\langle N_s\rangle\propto M_h$. The figure shows that the mean number of subhalos at different number densities is remarkably similar and shows only a mild evolution. The mass dependence is approximately linear $\langle N_s\rangle\propto M_{\rm h}$ for masses $M_{\rm h}/M_{\rm min}\gtrsim 5$ (or $\langle N_s\rangle\gtrsim 0.2).$