Han Aung, Daisuke Nagai, Eduardo Rozo, Rafael García, The phase-space structure of dark matter haloes, Monthly Notices of the Royal Astronomical Society, Volume 502, Issue 1, March 2021, Pages 1041–1047, https://doi.org/10.1093/mnras/staa3994

The phase space structure of dark matter haloes can be used to measure the mass of the halo, infer mass accretion rates, and probe the effects of modified gravity. Previous studies showed that the splashback radius can be measured in position space using a sharp drop in the density profile. Using N-body simulations, we model the distribution of the kinematically distinct infalling and orbiting populations of subhaloes and haloes. We show that the two are mixed spatially all the way to redge, which extends past the splashback radius defined by the drop in the spherically averaged density profile. This edge radius can be interpreted as a radius that contains a fixed fraction of the apocentres of dark matter particles. Our results highlight the possibility of measuring the outer boundary of a dark matter halo using its phase space structure and provide a firm theoretical foundation to the satellite galaxy model adopted in the companion paper, where we analysed the phase space distribution of Sloan Digital Sky Survey redMaPPer clusters.

Over the past several decades, numerical simulations have provided significant insights into our understanding the structure and formation of dark matter haloes in the concordance Lambda cold dark matter model (Frenk & White 2012, for review). The density profile of a halo in N-body simulations is typically characterized by Navarro–Frenk–White profile (Navarro, Frenk & White 1996) or Einasto profile (Einasto 1965), with a shallow slope inside the cluster that gets progressively steeper with increasing radius. The velocity dispersion profile is related to the density profile of the halo through Jeans equation, and has been found to be increasing when the density slope is shallow, and decreasing outward when the density slope is steep (Cole & Lacey 1996; Taylor & Navarro 2001). Studies have shown that the density profile, and thus the velocity dispersion profile, reflects the initial density peaks and assembly history of the halo (Dalal, Lithwick & Kuhlen 2010; Ludlow et al. 2014). Recent simulations showed that haloes have a sharp drop in the slope of the density profile at large radii, where the precise location of this feature is dependent on the peak height and mass accretion rate of the halo (Diemer & Kravtsov 2014).

The simple spherical collapse model predicts that there exists the outermost physical caustic in the phase space structure of the halo (Bertschinger 1985). The splashback radius is defined by the apocentres of the recently accreted spherical shells of particles that are at their second turnaround, and the sharp jump in the slope of the spherically symmetric density profile coincides with the caustic in the phase space (Adhikari, Dalal & Chamberlain 2014). Even without perfect spherical symmetry, such a density drop can be detected in the spherically averaged density profile inN-body simulations, and has been regarded as a physical boundary of the halo that encompasses most of the bounded particles (Diemer & Kravtsov 2014; More, Diemer & Kravtsov 2015). In practice, the detailed analysis of individual particle trajectories in N-body simulations revealed a broad distribution in the apocentres of the splashback particle population (Diemer 2017), and the splashback surface where the density slope is minimal can be highly aspherical (Mansfield, Kravtsov & Diemer 2017). This splashback surface contains most haloes which have been inside the central halo, with only 1–2 per cent of flyby haloes outside of this surface (Mansfield & Kravtsov 2020). However, the volume-averaged radius of the surface encloses only 87 per cent of the apocentres of the particle trajectories, while the radius from the spherically averaged density profile only encompassed 75 per cent, regardless of the mass accretion rate and mass of the haloes (Diemer et al. 2017). Analysis of hydrodynamics simulations also reveals that some galaxies outside the splashback radius of a halo have been inside the halo before (Haggar et al. 2020). Consequently, haloes appear to extend at least somewhat past the ‘average’ splashback radius defined using the density profile.

Motivated by analysis in the companion paper (Tomooka et al. 2020), in this work we set out to determine whether a detailed study of the phase space structure of dark matter haloes can shed light on their bonafide outermost physical boundary. The phase space structure of a dark matter halo can be used to constrain cosmology through cluster mass measurements (Evrard et al. 2008; Munari et al. 2013; Bocquet et al. 2015; Hamabata, Oguri & Nishimichi 2019), to constrain modified gravity models (Schmidt 2010; Lam et al. 2012; Zu et al. 2014; Mitchell et al. 2018) and to understand astrophysical processes such as assembly bias (Hearin 2015; Xu & Zheng 2018; Mansfield & Kravtsov 2020). Detailed characterization of the phase space structure of dark matter haloes, however, reveals that near the splashback radius, the tracers of the potential well cannot be cleanly separated into infalling and orbiting matter, which gives rise to the velocity structure of the halo, using a simple radial cut. Throughout this work, we define the orbiting population to be subhaloes and haloes which have experienced their first pericentre event, which marks the end of the first radial infall. Instead, the spatial distribution usually exhibits a mix of these two types of tracers. Indeed, the infalling stream may penetrate all the way into the halo centre (Zu & Weinberg 2013, hereafter ZW13). For these reasons, halo models that split the density distribution into a one-halo term at small scales and a two-halo term at large scales usually break down near the edge of the halo, with differences in velocity dispersion as large as |$20{{\ \rm per\ cent}}$| (Lam et al. 2013). This difference is comparable to the changes in phase space which arise from assembly bias, and is much larger than the effects from modified gravity. Thus, proper understanding of the phase space structure of dark matter haloes is needed to make reliable testable predictions for cosmology and astrophysics using galaxy surveys.

In this paper, we analyse the phase space structure of dark matter haloes with the goal of understanding the transition from the orbiting to infalling region better. In particular, we identify the ‘edge radius’ beyond which one does not find any additional orbiting structures. Specifically, we (1) characterize the phase-space structure of dark matter haloes in and around the edge radius, (2) show how this radius differs from the ‘splashback radius’ defined by the steep feature of the slope of the density profile, and (3) relate this radius to the splashback radius, and interpret it as enclosing a certain percentile of splashback particles. To analyse the phase-space structure of the dark matter haloes, we use the dark matter haloes and subhaloes from the MDPL2 (Multi-Dark Planck) N-body simulation as tracers. Section  2 describes the simulations and mock catalogue. We present our results in Section  3. We summarize our findings in Section  4.

In this work, we analyse the MDPL2 dark matter-only N-body simulation performed with l-gadget -2 code, a version of the publicly available cosmological code gadget -2 (Springel 2005). The simulation has a box size of |$1 {\rm \, Gpc}\, h^{-1}$|⁠ , with a force resolution of |$5\!-\!13{\rm \, kpc}\, h^{-1}$|⁠ . The mass resolution for dark matter particle is |$1.51\times 10^9 \, \mathrm{M}_{\odot }\, h^{-1}$|⁠ , corresponding to 38403 particles. It assumes the Planck 2013 cosmology with Ωm = 0.307, ΩΛ = 0.693, σ8 = 0.823, and |$H_0 = 68 {\rm \, km(s\, Mpc)}^{-1}$|⁠ . More details of the simulation can be found in Klypin et al. ( 2016). The haloes and subhaloes are identified using the Rockstar 6D phase space halo finder (Behroozi, Wechsler & Wu 2013a), and the merger tree is built using the Consistent-Tree algorithm (Behroozi et al. 2013b). For this study, we treat the subhaloes and haloes around the main haloes equally and are selected with a peak mass cut |$M_p\gt 3\times 10^{11}\, \mathrm{M}_{\odot }\, h^{-1}$|⁠ , which corresponds to at least 200 particles before falling on to the haloes. The main central haloes are selected using a mass cut |$M_{200m}\gt 10^{14}\, \mathrm{M}_{\odot }\, h^{-1}$|⁠ . All analyses are performed using the stacked profiles of the central haloes.

To understand the phase space around haloes, we study the radial and tangential velocities using dark matter haloes as tracers. The velocity of the tracer with respect to the central halo is given by |${\boldsymbol{v}} = {\boldsymbol{v}}_{\rm tracer}-{\boldsymbol{v}}_{\rm cen}$|⁠ . The radial velocity is |$v_r = {\boldsymbol{v}}\cdot \hat{r}$| and the tangential velocity is |$v_{\rm tan} = \sqrt{v^2-v_r^2}$|⁠ . Thus, the radial velocity is directional, positive for outgoing, and negative for infalling, while the tangential component is only a magnitude.

Fig.  1 shows the phase space structure of dark matter haloes, illustrated as the 2D histograms of radial and tangential velocities in 4 representative radial bins. Note that the velocities are normalized by the circular velocity at r200m of the halo, |$v_c = \sqrt{GM_{\rm 200m}/r_{\rm 200m}}$|⁠ . All other radial bins are qualitatively similar to one of the four bins shown below.

The 2D histograms of the radial and tangential velocity distribution at 4 representative radii. The top-left panel shows the inner region (r/r200m = [0.5–0.55]) which consists of orbiting and infalling populations. The blue dashed line for vr < 0 separates the two populations. The velocity structure of the splashback stream outside the blue line mirrors that of the infall stream. The top-right panel also shows a mix of orbiting and infalling haloes at r/r200m = [0.8 − 0.85]. The bottom-left panel shows similar structure outside halo, but with orbiting population less prominent at r/r200m = [1.1–1.15]. The bottom-right panel at r/r200m = [1.95–2] shows an infalling region. A small panel inside each histogram shows the radial position of the histogram along with average radial velocity.

The top-left panel shows the distributions of haloes in the vr–vtan plane for the radial bin, r/r200m = [0.5–0.55]. The phase-space structure at this radius is typical of haloes, with approximately zero mean radial velocity. However, we can see a faint split between low and high total velocities for negative radial velocity component. The blue-dashed line, determined as the local minimum in the distribution P(v|vr < 0), denotes the valley between low and high total velocity (or kinetic energy) components. The average infall time of the high-energy haloes is less than a dynamical time, 1 indicating that these are haloes that have recently fallen into the central halo. Turning to the distribution of haloes with positive radial velocities, we can see a large population of haloes with large kinetic energy, similar to those in the infall stream outside blue dashed line. 2 As we move radially outward in the top-right panel to r/r200m = [0.8 − 0.85], the splitting between the low- and high-velocity components for the negative radial velocity becomes more apparent. However, the ‘arc’ of outgoing material with large velocities becomes less distinct with slightly less kinetic energy than the infall stream. These outgoing subhaloes form the splashback stream, which recently fell into the central halo.

The bottom-left panel shows the result at r/r200m = [1.1–1.15] and exhibits features that are quite similar to those found in the previous radial bin at r/r200m = [0.8 − 0.85], despite the fact that this bin is past the r200m radius of the central halo. In both cases, there are two kinematically distinct populations. The first one has a slightly positive average radial velocity, indicating structures similar to the orbiting populations within the central halo. The second population has a negative radial velocity on average, corresponding to infalling haloes. In addition, there is also a small population with the total velocity larger than the blue dashed line and positive radial velocity, associated with the splashback stream. Traditionally, the haloes with zero radial velocity found at r = [1.1–1.15]r200m are not considered subhaloes of the central halo, because they lie outside most halo radius definitions (such as r200m or r200c). However, it is clear that these haloes are kinematically distinct from the infalling population, and are better thought of as subhaloes associated with the central halo.

We can see in these three panels that in general the infall streams have the largest total velocity, followed by the splashback stream, and then the rest of the orbiting haloes. The difference between infall and splashback streams is most pronounced at large radii (r/r200m = [0.8−0.85] and r/r200m = [1.1–1.15] in Fig.  1), because the splashback population was accreted earlier when the halo was less massive and is also affected by dynamical friction longer compared to the infalling population. The infall and splashback streams have almost symmetric velocity distributions with respect to vr = 0 in the inner part of the halo (r/r200m = [0.5–0.55] in Fig.  1), because the difference in the infall time between the two populations becomes small. Orbiting haloes, which fell in even earlier, have even lower kinetic energy than the splashback haloes. After the first apocentric passage, orbiting subhaloes form multiple caustic-like phase space structures whose kinetic energy depends on the number of pericentric passages (Sugiura et al. 2019).

Finally, the bottom-right panel of Fig.  1 shows that the orbiting populations have disappeared by r/r200m = [1.95–2], leaving behind only the infalling component. As we move further away from the central halo, the average velocity of the infalling component becomes less negative, being eventually overtaken by the Hubble flow at the turnaround radius rta. Beyond this radius, the distance between haloes increases due to the expansion of the Universe.

Fig.  2 illustrates the haloes from the bottom-left panel of Fig.  1 separated into two categories: (1) top panel: haloes that have been in the central halo (the radial position of the halo is less than r200m of the central halo) at least once in the last 2 Gyr (approximately 1 dynamical time at z = 0, or 1.5 dynamical time at z = 0.36); (2) bottom panel: haloes that have not been in the central halo in the last 2 Gyr. The top panel shows that the haloes that have been in the central halo are the ones responsible for creating the orbiting components of the velocity distribution. These haloes have at least one pericentric passage with respect to the central halo. The escape velocity at these radii is |$\approx \sqrt{2}v_c$|⁠ , which means that most of these haloes are bounded to central haloes with highly elliptical orbits. In the bottom panel, the haloes that have never been in the central halo clearly correspond to the infalling population, and have not had a pericentric passage in their history. Our results are consistent with the findings in Haggar et al. ( 2020), which showed that backsplash galaxies can exist outside r200m.

A more detailed look at the haloes outside r200m, the third panel of Fig.  1 but with slightly larger radial bin. The haloes are now distinguished into haloes that have been inside the central halo in the past 2 Gyr and haloes that have never been in the central halo. The former constitutes a population of haloes around 〈vr〉 > 0, indicating that these haloes are orbiting, while the latter constitutes infalling haloes with largely negative radial velocity.

Fig.  1 presents a simple yet compelling way of describing the phase-space structure of orbiting dark matter structures around a central halo. At small radii, dark matter haloes in a halo belong to one of three categories: (1) haloes in approximate virial equilibrium with the central halo; (2) an infalling stream of haloes; and (3) an outgoing population of splashback haloes. As we move towards larger radii, the orbiting populations disappear, eventually leaving only a stream of infalling structures. In this work, we want to identify the radius redge which defines the transition from a mix of infall and orbiting populations to an infall only region based on the kinematics of haloes.

In a previous study of the phase-space structure of dark matter haloes, ZW13 defined the virial extent of a halo by modelling the distribution of galaxies near a halo as a mixture of orbiting 3 and infalling galaxies. The infall stream was modelled using a skewed t-distribution, whereas the orbiting structures are modelled as a Gaussian distribution with mean of 0. However, the model fit produced a decreasing orbiting fraction in the inner part of the halo. This is in contrast to the phase space structure in the Fig.  1, which shows the orbiting population increases towards the halo centre as expected. The degree of freedom of the t-distribution also hits the upper bound, turning the t-distribution into a Gaussian. In fitting the ZW13 model to our data, we find that these peculiarities arise because the t-distribution shifts to smaller median so that it ends up describing the wide-peaked orbiting population, rather than the infalling stream in the interior of the halo (see Appendix  A for details).

Fig.  3 shows the fraction of orbiting haloes as a function of radius recovered by our model. Following Fig.  2, we defined the ‘true’ fraction of orbiting haloes as those which have had their first pericentric passages. Our model recovers the fraction of orbiting structures correctly at all radial bins. We can see that the fraction starts out at 0 at large radii, and constantly rises after r ≲ 1.7−1.8r200m. It then asymptotically approaches towards but not equal to unity as we move towards smaller radii. forb(r) is well fit by a slight modification to the original function used in ZW13, namely |$f_{\rm orb}(r) = a\, {\rm exp}(-(r/r_0)^\gamma)$|⁠ , where a = 0.986 is the asymptotic fraction as it approaches centre, and r0 = 1.27r200m is the radius where the fraction reaches 1/e. The decreasing slope is fitted to γ = 3.5. Since the fraction of orbiting population approaches 0 as we move outward, we define the edge radius as the radius where the fraction reaches 0.01, which results in redge/r200m = 1.96.

Fraction of orbiting haloes, forb, as a function of radius. The result of fitting the equation ( 1) in different radial bins agrees with the fraction of haloes which have had their first pericentric passages (offset slightly in x-axis for clarity) and describes the evolution of infalling stream versus orbiting populations.

We now compare the edge radius we have identified based on the halo kinematics to the splashback radius defined using the SPARTA algorithm, calculated using the fitting formula in Diemer ( 2017) for the median mass (and thus peak height) and mass accretion rate of the haloes in each bin using COLOSSUS (Diemer 2018). SPARTA identifies the splashback radius of individual particles by tracking their trajectories. The splashback radius of a particle is defined as the apocentre of the orbit at the second turnaround. The splashback radius of the halo is defined as the radius within which a specified percentile of the particle apocentres lie. The splashback radius identified using the slope of the spherically averaged dark matter density profile corresponds to 75–87 percentile of particle apocentres (Xhakaj et al. 2019), while the splashback radius defined by line-of-sight density slopes corresponds to 87 percentile (Mansfield et al. 2017). In other words, at least 13 per cent of the particles in a halo lie outside the splashback radius identified using density profile.

Fig.  4 illustrates the mass and redshift dependence of the ratio of redge and |$r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ . We see that this ratio (⁠|$r_{\rm edge}/r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ ) is approximately constant throughout the entire mass and redshift range we sampled. Since the peak-height is a function of mass and redshift, the ratio also stays constant as a function of the peak-height as well. Thus, we interpret the edge radius as a splashback radius containing specific percentiles of the apocentres of orbiting haloes. Specifically, we can see that the edge radius (redge) extends further out than the radius encompassing 87 percentile of the dark matter particles. Beyond the 87 percentile, the splashback radius defined using particle apocentres diverges quickly (Diemer 2017). We conclude that |$r_{\rm edge}= 1.6r_{\rm sp,87{{\ \rm per\ cent}}}$| provides a better definition of the boundary of halo as we can infer from our fitting function that roughly |$40{{\ \rm per\ cent}}$| of haloes at |$r_{\rm sp,87{{\ \rm per\ cent}}}$| are still orbiting haloes. Mansfield & Kravtsov ( 2020) argued that the outlying haloes which were originally inside the central halo are contained within an aspherical splashback surface. We note that these splashback haloes should disappear after redge, likely coinciding with the maximum radius of the splashback surface.

The ratio of redge and |$r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ , the splashback radius containing 87 percentile of particles from SPARTA, demonstrating that the edge radius has the same mass and redshift dependence as |$r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ .

Fig.  5 also shows the ratio (⁠|$r_{\rm edge}/r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ ) as a function of the mass accretion rate (Γ), where the mass accretion rate is defined as Γ = dlog M/dlog a evaluated in the a = [0.600–0.733] range which spans one dynamical time. This figure further demonstrates the constancy of the ratio (⁠|$r_{\rm edge}/r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ ). It has the same mass accretion rate dependence as the splashback radius, and is again roughly a fixed multiple of |$r_{\rm sp,87{{\ \rm per\ cent}}}$|⁠ .

The ratio of redge and |$r_{\rm sp,87{{\ \rm per\ cent}}}$| as a function of mass accretion rate. The two have similar mass accretion rate dependence and the ratio remains roughly constant except at very low accretion regime. The three dashed lines indicate redge computed using haloes within different Mp bins. Higher mass haloes have smaller redge due to dynamical friction similar to splashback radius.

Analysis of the relative change of redge using different halo mass cuts also shows splashback-like behaviour as seen in Fig.  5. redge serves as the furthest splashback radius for all matter orbiting around the halo. When working with haloes, this radius is expected to be sensitive to the effects of dynamical friction. Dynamical friction tends to increase with the mass squared, so the higher the mass of the orbiting halo, the more kinetic energy the halo will lose and the smaller the splashback radius will be (Adhikari, Dalal & Clampitt 2016). Thus, redge decreases for a halo sample of larger Mp.

Our findings demonstrate that the average edge radius for haloes generally lies around 2r200m, consistent with the extent to which backsplash or ejected haloes and galaxies are found within clusters and high-mass haloes (Li et al. 2013; Haggar et al. 2020; Knebe et al. 2020). However, studies focusing on the low-mass haloes with |$M\approx 10^{12}\, \mathrm{M}_{\odot }\, h^{-1}$| show that ejected haloes may extend past 3r200c ≈ 2r200m, although the fraction of ejected haloes outside of this range is less than |$15{{\ \rm per\ cent}}$| (Ludlow et al. 2009; Wang, Mo & Jing 2009). This radius depends on mass accretion rate in addition to mass. In particular, redge is related to the 87 per cent splashback radius defined using SPARTA by a constant factor of ≈1.6, where the ratio of these two radii is independent of the mass accretion rate. As such, the steep slope of the spherically average density profile at the splashback radius, for example, occurs at a constant radius when normalized using redge. Notably, the spatial extent of the 1-halo term extends significantly beyond the traditionally defined splashback radius, and must be taken into account when modelling the structures of dark matter haloes.

In this work, we analysed the phase-space structure of dark matter haloes using dark matter subhaloes and nearby haloes as tracers. Our main findings are summarized as follows:

The phase-space structure inside dark matter haloes can be modelled as a mixture of haloes on their first infall, a splashback stream of haloes that are on their way to their first apocentric passage, and haloes that have orbited the main halo at least once. We refer to the latter two-halo populations as ‘orbiting’, in that they are in an orbit around the central halo, bounded or unbounded.

The edge of the halo can be defined by the radius (redge), beyond which little (⁠|$\lt 1{{\ \rm per\ cent}}$|⁠ ) orbiting populations exist. Inside the edge radius (r < redge), orbiting and infalling structures are mixed in physical space, but they are distinct in velocity space. Outside redge and up to the turnaround radius rta, the haloes are infalling to the central halo. Outside rta, the haloes are receding away from the central halo due to the Hubble flow.

The edge radius (redge) coincides with a fixed multiple of the splashback radius defined either using the slope of the density profile or the splashback radius containing 87-percentile of apocentres of dark matter particles. We reinterpret the edge radius redge, which has been previously found as part of the phase space analysis in ZW13, as the radius within which all apocentres of splashback tracers lie. This is supported by the fact that it has similar mass, redshift, and mass accretion rate dependence as the splashback radii.

Our results suggest a new way of defining the halo boundaries based on the phase-space structure of haloes around dark matter haloes. The edge radius (redge) is larger than the traditional splashback radius defined based on the slope of the dark matter density profile. Our finding is consistent with previous studies showing that the splashback radius defined based on the density slope does not encompass all the splashback particles. We show, however, that the edge radius redge is clearly defined in phase space, and encompasses more than |$99{{\ \rm per\ cent}}$| of all orbiting structures. That is, the edge radius (redge) defines a real kinematic boundary for a dark matter halo. In addition, we improved upon the previous characterization of a phase space model by ZW13, by enforcing that the t-distribution used in the model corresponds to the same physical population of structures at all radii (namely infalling structures).

The improved modelling and phase space and new definition of the halo boundary will allow us to use phase-space measurements of cluster galaxies for cosmology and astrophysics. In the companion paper (Tomooka et al. 2020), we present the first detection of the outer edge of galaxy clusters based on spectroscopic measurements of SDSS cluster galaxy kinematics. Our study presents the physical interpretation of the edge radius defined based on the halo kinematics and its connection to the splashback radius and its properties. In future work, we plan to investigate observational and systematic uncertainties in extracting the 3D phase space information from line-of-sight velocity measurements and test the robustness of the method used by Tomooka et al. ( 2020) to infer the cluster edge radius. Such work is particularly important for measuring the phase space structures of dark matter haloes accurately and precisely with the next generation spectroscopic galaxy surveys, e.g. DESI (DESI Collaboration 2016) and Subaru PFS (Takada et al. 2014).

We thank Benedikt Diemer, Keiichi Umetsu, and anonymous referee for comments and suggestions on the manuscript and Susmita Adhikari, Eric Baxter, Chihway Chang, Bhuvesh Jain for illuminating discussions during the early phase of this project. The CosmoSim database used in this paper is a service by the Leibniz-Institute for Astrophysics Potsdam (AIP). The MultiDark database was developed in cooperation with the Spanish MultiDark Consolider Project CSD2009-00064. This work was supported in part by NSF AST-1412768 and the facilities and staff of the Yale Center for Research Computing. DN and ER acknowledge funding from the Cottrell Scholar program of the Research Corporation for Science Advancement. ER was supported by DOE grant DE-SC0015975.

The data underlying this article are available at https://www.cosmosim.org/.

The dynamical time is the time-scale for haloes at r200m to fall into the centre of halo given a typical circular velocity, |$t_{\rm dyn} = r_{\rm 200m}/\sqrt{GM_{\rm 200m}/r_{\rm 200m}}$|⁠ ).

The line is reflected across Hubble velocity for the minimum in the distribution of P(v|vr < 0).

ZW13 use the term ‘virialized’ when referring to the orbiting population as defined in this paper.

ZW13 uses skewed t-distribution. However, the skewness disappears in the innermost radii, where the orbiting fraction of ZW13 and our model disagrees. Hence, a normal t-distribution suffices.

Adhikari   S. , Dalal   N. , Chamberlain   R. T. , 2014, J. Cosmol. Astropart. Phys., 11, 19  10.1088/1475-7516/2014/11/019

Adhikari   S. , Dalal   N. , Clampitt   J. , 2016, J. Cosmol. Astropart. Phys., 2016, 022  10.1088/1475-7516/2016/07/022

Behroozi   P. S. , Wechsler   R. H. , Wu   H.-Y. , 2013a, ApJ, 762, 109  10.1088/0004-637X/762/2/109

Behroozi   P. S. , Wechsler   R. H. , Wu   H.-Y. , Busha   M. T. , Klypin   A. A. , Primack   J. R. , 2013b, ApJ, 763, 18  10.1088/0004-637X/763/1/18

Bocquet   S.  et al.  , 2015, ApJ, 799, 214  10.1088/0004-637X/799/2/214

Cole   S. , Lacey   C. , 1996, MNRAS, 281, 716  10.1093/mnras/281.2.716

Dalal   N. , Lithwick   Y. , Kuhlen   M. , 2010, preprint (arXiv:1010.2539)

Diemer   B. , 2017, ApJS, 231, 5  10.3847/1538-4365/aa799c

Diemer   B. , 2018, ApJS, 239, 35  10.3847/1538-4365/aaee8c

Diemer   B. , Kravtsov   A. V. , 2014, ApJ, 789, 1  10.1088/0004-637X/789/1/1

Diemer   B. , Mansfield   P. , Kravtsov   A. V. , More   S. , 2017, ApJ, 843, 140  10.3847/1538-4357/aa79ab

Einasto   J. , 1965, Trudy Astrofizicheskogo Instituta Alma-Ata, 5, 87

Evrard   A. E.  et al.  , 2008, ApJ, 672, 122  10.1086/521616

Frenk   C. S. , White   S. D. M. , 2012, Ann. Phys., 524, 507  10.1002/andp.201200212

Haggar   R. , Gray   M. E. , Pearce   F. R. , Knebe   A. , Cui   W. , Mostoghiu   R. , Yepes   G. , 2020, MNRAS, 277  10.1093/mnras/staa273

Hamabata   A. , Oguri   M. , Nishimichi   T. , 2019, MNRAS, 489, 1344  10.1093/mnras/stz2227

Hearin   A. P. , 2015, MNRAS, 451, L45  10.1093/mnrasl/slv064

Klypin   A. , Yepes   G. , Gottlöber   S. , Prada   F. , Heß   S. , 2016, MNRAS, 457, 4340  10.1093/mnras/stw248

Knebe   A.  et al.  , 2020, MNRAS. 10.1093/mnras/staa1407

Lam   T. Y. , Nishimichi   T. , Schmidt   F. , Takada   M. , 2012, Phys. Rev. Lett., 109, 051301  10.1103/PhysRevLett.109.051301

Lam   T. Y. , Schmidt   F. , Nishimichi   T. , Takada   M. , 2013, Phys. Rev. D, 88, 023012  10.1103/PhysRevD.88.023012

Li   R. , Gao   L. , Xie   L. , Guo   Q. , 2013, MNRAS, 435, 3592  10.1093/mnras/stt1551

Ludlow   A. D. , Navarro   J. F. , Springel   V. , Jenkins   A. , Frenk   C. S. , Helmi   A. , 2009, ApJ, 692, 931  10.1088/0004-637X/692/1/931

Ludlow   A. D. , Navarro   J. F. , Angulo   R. E. , Boylan-Kolchin   M. , Springel   V. , Frenk   C. , White   S. D. M. , 2014, MNRAS, 441, 378  10.1093/mnras/stu483

Mansfield   P. , Kravtsov   A. V. , 2020, MNRAS, 493, 4763  10.1093/mnras/staa430

Mansfield   P. , Kravtsov   A. V. , Diemer   B. , 2017, ApJ, 841, 34  10.3847/1538-4357/aa7047

Mitchell   M. A. , He   J.-h. , Arnold   C. , Li   B. , 2018, MNRAS, 477, 1133  10.1093/mnras/sty636

More   S. , Diemer   B. , Kravtsov   A. V. , 2015, ApJ, 810, 36  10.1088/0004-637X/810/1/36

Munari   E. , Biviano   A. , Borgani   S. , Murante   G. , Fabjan   D. , 2013, MNRAS, 430, 2638  10.1093/mnras/stt049

Navarro   J. F. , Frenk   C. S. , White   S. D. M. , 1996, ApJ, 462, 563  10.1086/177173

Schmidt   F. , 2010, Phys. Rev. D, 81, 103002  10.1103/PhysRevD.81.103002

Springel   V. , 2005, MNRAS, 364, 1105  10.1111/j.1365-2966.2005.09655.x

Sugiura   H. , Nishimichi   T. , Rasera   Y. , Taruya   A. , 2020, MNRAS, 493, 2765  10.1093/mnras/staa413

Takada   M.  et al.  , 2014, PASJ, 66, R1  10.1093/pasj/pst019

Taylor   J. E. , Navarro   J. F. , 2001, ApJ, 563, 483  10.1086/324031

Tomooka   P. , Rozo   E. , Wagoner   E. L. , Aung   H. , Nagai   D. , Safonova   S. , 2020, MNRAS, 499, 1291  10.1093/mnras/staa2841

Wang   H. , Mo   H. J. , Jing   Y. P. , 2009, MNRAS, 396, 2249  10.1111/j.1365-2966.2009.14884.x

Xhakaj   E. , Diemer   B. , Leauthaud   A. , Wasserman   A. , Huang   S. , Luo   Y. , Adhikari   S. , Singh   S. , 2020, MNRAS, 499, 3534  10.1093/mnras/staa3046

Xu   X. , Zheng   Z. , 2018, MNRAS, 479, 1579  10.1093/mnras/sty1547

Zu   Y. , Weinberg   D. H. , 2013, MNRAS, 431, 3319  (ZW13)  10.1093/mnras/stt411

Zu   Y. , Weinberg   D. H. , Jennings   E. , Li   B. , Wyman   M. , 2014, MNRAS, 445, 1885  10.1093/mnras/stu1739

The distribution of radial velocities of all subhaloes for the radial bin of r = [0.6−0.7]r200m and of infalling subhaloes, which have not had a pericentric passage. Our best-fitting model based on the equation ( 1) is indicated with the red-dashed curve. The vertical lines indicate 3 means of the distributions, with the leftmost line indicating the infalling stream, while the other two indicate the means of orbiting Gaussian components. Employing ZW13 model with varying mean for Gaussian fails to capture the infall stream using the t-distribution.

Fig.  A2 shows the radial dependence of the various parameters obtained by fitting our new model described in equation ( 1) to the radial velocity distribution of subhaloes. The orbiting parameters also vary monotonically with radius. Specifically, the difference between the mean of two distributions disappears as it approaches the edge radius. The median of the infalling distribution decreases monotonically as enforced, while the scale parameter is approximately constant. We found that the fitted degree of freedom ν ≈ 2.2 is approximately the same outside the halo, but starts to gradually increase at r ≲ 1.5r200m and hits the upper bound of the prior at r ≲ 0.5r200m, i.e. the distribution is closer to Gaussian. Our results agree with ZW13 outside the halo, which is expected as the only difference in our model is the distribution of the orbiting population. However, inside the halo, the degree of freedom ν approaches ∞, and the distribution becomes Gaussian at smaller radii than the radius ZW13 model predicts. Our model accurately captures the phase space structure associated with the infalling stream.

The radial dependence of the fitted parameters based on the two Gaussians (top panel) and the t-distribution (bottom panel). The parameter σ2 is skipped for clarity in the figure and is approximately the same as σ1. The error band is the standard deviation of the MCMC posterior.

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