THE SPLASHBACK RADIUS AS A PHYSICAL HALO BOUNDARY AND THE GROWTH OF HALO MASS

In order to elucidate the connection between the collapse of a halo and its boundary, we wish to consider a simple, analytical model. The simplest such model describes the collapse of a top-hat perturbation. However, as laid out in the introduction, peaks in a Gaussian density field have density profiles that decrease with radius, significantly different from a constant density spherical top-hat. While a top-hat perturbation collapses at a well-defined moment in time, the radial shells associated with realistic peaks collapse at different times, resulting in a halo formation history that is extended in time. Thus, we expect better guidance from collapse models that describe the extended collapse of matter.

One such model considers the secondary infall of matter onto a pre-existing overdensity in an ${{\rm{\Omega }}}_{{\rm{m}}}=1$ universe (Fillmore & Goldreich 1984; Bertschinger 1985). In this spherical collapse model, all the mass in the universe is bound to the pre-existing overdensity. It is instructive to consider the dynamics of infinitesimally thin mass shells around the overdensity. Each shell will initially expand with the Hubble flow, decelerate, eventually turn around and start contracting. At some point, the shell will cross previously collapsed shells that are now oscillating in the perturbation potential, thereby entering the multi-stream region of the halo. The matter of the shell will eventually pass through the pericenter of its orbit in the inner region of the perturbation and expand to the apocenter of its first orbit. Each successive shell collapses onto a deeper potential well than the preceding shell, and thus acquires a higher energy and a larger orbit apocenter.

In this picture, material piles up near the apocenter due to its small radial velocity in this region of the orbit, creating a density enhancement or caustic which is extremely sharp in the case of spherical symmetry (see, for example, the detailed discussion in Mohayaee & Shandarin 2006). This caustic occurs at radii $r\approx 0.1-0.4$ times the turnaround radius of the material at apocenter (see, e.g., Figure 4 of Vogelsberger et al. 2011), depending on the slope of the density profile of the initial perturbation which determines the mass accretion rate of the collapsing halo. Even in the case of ellipsoidal collapse, the caustic region is marked by a sharp jump in the density profile (Adhikari et al. 2014).

We note that the outermost caustic corresponds to the apocenter of matter on its first orbit, i.e., the splashback radius of the newly accreted matter, which we denote as ${R}_{\mathrm{sp}}$. The splashback radius cleanly separates the multi-stream region of the perturbation at $r\lt {R}_{\mathrm{sp}}$ from the infall region at $r\gt {R}_{\mathrm{sp}}$, where successive shells of matter have not yet crossed. The spherical collapse model thus motivates ${R}_{\mathrm{sp}}$ as a natural definition of the halo boundary, and the halo mass as the mass within this radius, ${M}_{\mathrm{sp}}\equiv M(\lt {R}_{\mathrm{sp}})$. By definition, the increase in ${M}_{\mathrm{sp}}$ between two epochs ${z}_{{\rm{i}}}$ and ${z}_{{\rm{f}}}$ is entirely due to the mass shells that have entered the multi-stream region in the interval ${\rm{\Delta }}z={z}_{{\rm{i}}}-{z}_{{\rm{f}}}$. Therefore, ${{dM}}_{\mathrm{sp}}/{dt}$ is the true halo mass growth rate in this model. In contrast, if we had chosen a smaller radius ${R}_{{\rm{\Delta }}}\lt {R}_{\mathrm{sp}}$ as the halo boundary, the halo mass growth rate would be due to both the accretion of new matter during ${\rm{\Delta }}z$ and matter previously accreted at $z\gt {z}_{{\rm{i}}}$, as we discuss in detail in the Appendix.

Although the collapse of realistic CDM halos is considerably more complicated than the collapse of a single peak in the secondary infall model, we can still use this model to guide our choices of the halo boundary and mass definitions. As shown by Diemer & Kravtsov (2014, see also Adhikari et al. 2014; Lau et al. 2014), halos that accrete mass at a sufficiently high rate exhibit a sharp steepening of the density profile in the outer regions, which is due to the caustic formed by recently accreted matter. The radius at which the profile achieves its steepest slope depends on the halo accretion rate and varies from $\approx 0.8-1.5{R}_{200{\rm{m}}}$ (Diemer & Kravtsov 2014). Adhikari et al. (2014) have confirmed this result and showed that the location of the steepest slope can be reproduced using the radius of the outermost caustic in the simple model of spherical collapse discussed in Section 2.1. In particular, they demonstrated that the overdensity corresponding to the splashback radius depends on both the mass growth rate and cosmological parameters such as ${{\rm{\Omega }}}_{{\rm{m}}}(z)$.

We operationally define the mass accretion rate the same way as in Diemer & Kravtsov (2014),

$$\begin{eqnarray}&&{\rm{\Gamma }}\equiv {\rm{\Delta }}\mathrm{log}({M}_{\mathrm{vir}})/{\rm{\Delta }}\mathrm{log}(a).\end{eqnarray} \tag{ 2 }$$

Figures 1 and 2 illustrate the correspondence between density profiles and ${R}_{\mathrm{sp}}$ using the example of two individual, cluster-sized halos with similar masses but very different mass accretion rates, representative of the slow (${\rm{\Gamma }}\lt 1$) and fast accreting (${\rm{\Gamma }}\gt 2$) sub-populations. Figure 1 shows the density distribution in a slice through the halo center, while Figure 2 shows the spherically averaged density profiles and their logarithmic slope. Both figures contrast ${R}_{\mathrm{sp}}$ (dashed lines) with ${R}_{200{\rm{m}}}$ (dot–dashed lines) and the "virial" radius ${R}_{\mathrm{vir}}$ (solid lines). The ${R}_{\mathrm{sp}}$ radii shown in the figures were predicted using the median relation given by Equation (5) below and the Γ of the specific halos as determined from the halo assembly histories. Figure 1 clearly demonstrates that the density fields exhibit a sharp jump at $R\approx {R}_{\mathrm{sp}}$, and that this radius occurs at a smaller multiple of ${R}_{200{\rm{m}}}$ for the faster accreting halo. We confirm these impressions by considering the spherically averaged density profiles of the same halos in Figure 2 which highlights how steep the density profile can get around ${R}_{\mathrm{sp}}$ (a logarithmic slope of −7).

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The correlation between ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$ and the accretion rate mimics the correlation between the ratio of the last caustic and turnaround radii and the slope of the initial perturbation profile in the spherical collapse model (see Vogelsberger et al. 2011, and Section 2.1). By further analogy with the spherical collapse model, the splashback radius of CDM halos should include approximately all of the mass ever accreted by a halo. Thus, changes in ${M}_{\mathrm{sp}}$ should always correspond to the current accretion of new mass, implying that ${M}_{\mathrm{sp}}$ is largely unaffected by pseudo-evolution. In Sections 3.1 and 4.6 we discuss how ${R}_{\mathrm{sp}}$ can be measured in cosmological simulations and observations, and how ${M}_{\mathrm{sp}}$ and its evolution relate to conventional spherical overdensity masses.

As shown by Diemer & Kravtsov (2014), ${R}_{\mathrm{sp}}$ can be measured from the halo density profile as the radius where the density profile steepens sharply beyond what is expected from the Navarro–Frenk–White density profile (Navarro et al. 1997, hereafter NFW) or the Einasto profile (Einasto 1965; Navarro et al. 2004; Graham et al. 2006). We use the same simulation suite and fit the median density profiles of halo samples with a range of different masses, accretion rates, and redshifts using the fitting function in Equation (4) of Diemer & Kravtsov (2014), which we reproduce here for completeness,

$$\begin{eqnarray}\rho (r) & = & {f}_{\mathrm{trans}}\;{\rho }_{\mathrm{Einasto}}+{\rho }_{\mathrm{outer}}\\ {\rho }_{\mathrm{Einasto}} & = & {\rho }_{{\rm{s}}}\mathrm{exp}(-\displaystyle \frac{2}{\alpha }[{(\displaystyle \frac{r}{{r}_{{\rm{s}}}})}^{\alpha }-1])\\ {f}_{\mathrm{trans}} & = & {[1+{(\displaystyle \frac{r}{{r}_{{\rm{t}}}})}^{\beta }]}^{-\gamma /\beta }\\ {\rho }_{\mathrm{outer}} & = & {\rho }_{{\rm{m}}}[{b}_{{\rm{e}}}{(\displaystyle \frac{r}{5{R}_{200{\rm{m}}}})}^{-{s}_{{\rm{e}}}}+1].\end{eqnarray} \tag{ 3 }$$

For consistency, we fix some of the parameters in the fitting function in all fits (regardless of whether the samples were selected by mass or accretion rate), namely $\beta =6$, $\gamma =4$. Similar to Diemer & Kravtsov (2014), we also fix α according to the relation with the peak height ν as calibrated by Gao et al. (2008),⁶

$$\begin{eqnarray}&&\alpha (\nu )=0.155+0.0095{\nu }^{2}.\end{eqnarray} \tag{ 4 }$$

The other parameters, namely ${r}_{{\rm{s}}}$, ${r}_{{\rm{t}}}$, ${b}_{{\rm{e}}}$, and ${s}_{{\rm{e}}}$, are determined from a least-squares fit.

Diemer & Kravtsov (2014) showed that, at z = 0, the turnover radius, ${r}_{{\rm{t}}}$, depends on the mass accretion rate Γ. In Figure 3, we extend this analysis to higher redshifts and use ${R}_{\mathrm{sp}}$, defined as the radius where the median profile of halos reaches the steepest slope, instead of ${r}_{{\rm{t}}}$. The redshift intervals over which Γ are measured are the same as the redshifts listed in Figure 3, i.e., for z = 0, Γ is measured between $z=0.5$ and z = 0, for $z=0.5$ between z = 1 and $z=0.5$, and for z = 4 between z = 6 and z = 4. The choice of the redshift intervals defining Γ is somewhat arbitrary, but corresponds reasonably closely to the expected crossing time through the full extent of the halo, $2R$ (for example, at z = 0, ${\rm{\Delta }}z$ corresponds to about 5 Gyr whereas the crossing time is about 4 Gyr).

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We bin halos both by ν and by Γ, and only use halo samples with $\nu \gt 1$ ($M\gt 3\times {10}^{12}\;{h}^{-1}{M}_{\odot }$ at z = 0, $M\gt {10}^{11}\;{h}^{-1}{M}_{\odot }$ at z = 1) for this analysis, as the density jump associated with the splashback radius is difficult to measure robustly from spherically averaged profiles in halos with low ν and low Γ. This issue is apparent in Figure 10 in Diemer & Kravtsov (2014): for profiles with low ν and low Γ, the 2-halo term begins to dominate at radii smaller than ${R}_{\mathrm{sp}}$, thus concealing the steepening in the density profile. This does not mean that low-mass halos do not exhibit a steepening in their density profile; however, they are strongly influenced by their environment, making it difficult to discern the location of ${R}_{\mathrm{sp}}$.

In Figure 3, the similarity of the colored symbols with different shades demonstrates that, at fixed Γ, ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$ does not depend on ν, but does depend on z. In particular, Adhikari et al. (2014) showed that the overdensity associated with the splashback radius depends on ${{\rm{\Omega }}}_{{\rm{m}}}(z)$, where ${{\rm{\Omega }}}_{{\rm{m}}}(z)\equiv {\rho }_{{\rm{m}}}(z)/{\rho }_{\mathrm{crit}}(z)$. Thus, we parameterize the dependence of ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$ on Γ and z with the fitting function

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{\mathrm{sp}}}{{R}_{200{\rm{m}}}}=0.54\;[1+0.53{{\rm{\Omega }}}_{{\rm{m}}}(z)]\;(1+1.36{e}^{-{\rm{\Gamma }}/3.04}),\end{eqnarray} \tag{ 5 }$$

shown with solid lines in Figure 3. Given this function, we could now compute the median ${M}_{\mathrm{sp}}$ from ${R}_{\mathrm{sp}}$ by assuming a particular form of the density profile. However, we get a more accurate fit by directly calibrating the median ratio of ${M}_{\mathrm{sp}}$ and ${M}_{200{\rm{m}}}$ using the simulated density profiles,

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{sp}}}{{M}_{200{\rm{m}}}}=0.59\;[1+0.35{{\rm{\Omega }}}_{{\rm{m}}}(z)]\;(1+0.92{e}^{-{\rm{\Gamma }}/4.54}).\end{eqnarray} \tag{ 6 }$$

These formulae were calibrated using a flat ΛCDM cosmology where ${{\rm{\Omega }}}_{{\rm{m}}}=1-{{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.27$, $h=100/{H}_{0}=0.7$, ${\sigma }_{8}=0.82$, ${n}_{{\rm{s}}}=0.95$. These calibrations can be used to compute the dependence of the overdensity, Δ, of halos on Γ and ${{\rm{\Omega }}}_{{\rm{m}}}(z)$. Since we do not find a large dependence of ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$ on ν, we will extrapolate the relations calibrated above even for $\nu \lt 1$ halos in the subsequent sections while discussing our results. We also note that Adhikari et al. (2014) have presented a calibration for Δ as a function of the instantaneous mass accretion rate, s and ${{\rm{\Omega }}}_{{\rm{m}}}(z)$. Once the differences between s and Γ are accounted for, our calibration is largely consistent with theirs.

In observations, however, the accretion rate or the exact density profile of a halo are not readily available. Thus, we also quantify the dependence of ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ on the conventionally defined, observable ${M}_{{\rm{\Delta }}}$, or rather peak height, $\nu \equiv {\delta }_{c}/\sigma ({M}_{{\rm{\Delta }}})/D(z)$, in Figure 4.⁷ This dependence arises because halos of higher peak height exhibit, on average, higher accretion rates (see, e.g., Figure 8 in Diemer & Kravtsov 2014). In order to translate Equations (5) and (6) into functions of ν rather than Γ, we use the model of Zhao et al. (2009) to calculate halo mass growth histories. For each redshift along an accretion history we compute the accretion rate Γ across the same redshift intervals as in Diemer & Kravtsov (2014) and calculate ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ using Equations (5) and (6). Figure 4 shows the results as a functionof peak height. The relations are more or less independent of redshift (within 10%), and well fitted by the approximations

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{\mathrm{sp}}}{{R}_{200{\rm{m}}}}=0.81\;(1+0.97{e}^{-\nu /2.44})\end{eqnarray} \tag{ 7 }$$

and

$$\begin{eqnarray}&&\displaystyle \frac{{M}_{\mathrm{sp}}}{{M}_{200{\rm{m}}}}=0.82\;(1+0.63{e}^{-\nu /3.52}),\end{eqnarray} \tag{ 8 }$$

shown with dashed lines in Figure 4. Thus, in the concordance cosmological model, the dependence of ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$, ${M}_{\mathrm{sp}}/{M}_{200{\rm{m}}}$ or the overdensity of halos on ${{\rm{\Omega }}}_{{\rm{m}}}(z)$ is approximately cancelled by the dependence of ${\rm{\Gamma }}(\nu )$ on redshift based on the median mass accretion histories of halos. For rare, massive halos that accrete mass at a fast rate (on average), the splashback radius is close to ${R}_{200{\rm{m}}}$, while for low-ν halos it extends to a considerably larger radius of ${R}_{\mathrm{sp}}\sim 1.5{R}_{200{\rm{m}}}$.

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The ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ calibrations presented in Equations (7) and (8) were obtained using the median profiles of halos of a given Γ or ν. In observations, however, stacking would result in an average of the density profile, not the median ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$. We have checked that the ν–${R}_{\mathrm{sp}}$ and ν–${M}_{\mathrm{sp}}$ relations obtained from averaged profiles are almost identical to Equations (7) and (8).

Given that real CDM halos are not spherical, the density jump associated with the splashback radius occurs at different radii in different directions from the halo center. The corresponding feature in the spherically averaged density profile is thus not a sharp jump but rather a steepening of the profile that spans a range of radii. The finite radial extent of the steepening creates a certain ambiguity in the choice of the splashback radius definition. For instance, an alternative definition of the splashback radius could be the radius where the average radial velocity in a shell is most negative, ${R}_{\mathrm{infall}}$. This radius is more likely to include most of the accreted mass, although the majority of the mass between ${R}_{\mathrm{sp}}$ and ${R}_{\mathrm{infall}}$ is infalling for the first time. We find that ${R}_{\mathrm{infall}}\approx 1.4{R}_{\mathrm{sp}}$ and ${M}_{\mathrm{infall}}\approx 1.2{M}_{\mathrm{sp}}$ at all redshifts and halo masses. Thus, the fitting formulae in Equations (7) and (8) can easily be modified to return ${R}_{\mathrm{infall}}$ and ${M}_{\mathrm{infall}}$.

Figure 1 above shows both ${R}_{\mathrm{sp}}$ and ${R}_{\mathrm{infall}}\approx 1.4{R}_{\mathrm{sp}}$ (dotted line). In contrast to ${R}_{\mathrm{sp}}$, ${R}_{\mathrm{infall}}$ clearly extends into the filamentary regions and is not associated with the collapsed halo matter. Figure 2 confirms this impression, as ${R}_{\mathrm{infall}}$ does not correspond to a particular feature in the density profiles. Thus, the splashback radius definition based on the steepest density profile slope is preferable, and will be used for the remainder of this paper.

The results presented in Section 3.1 demonstrate that the splashback radius of halos is quite large. For some purposes, it may be instructive to consider the mass evolution in the inner regions of halos. In principle, one could characterize the inner regions simply by using ${M}_{{\rm{\Delta }}}$ with a high value of Δ. However, given that any spherical overdensity radius ${R}_{{\rm{\Delta }}}\lt {R}_{\mathrm{sp}}$ is subject to pseudo-evolution, we would prefer to define the inner mass using a radius that is not tied to any cosmological reference density.

Zemp (2014) discusses several alternative halo mass and radius definitions that are not subject to pseudo-evolution. In particular, it was argued that the mass within a radius that encloses a fixed physical density independent of redshift, such as $200{\rho }_{{\rm{m}}}(z=0)$, could be used as an alternative measure of mass. Such a definition, however, has a number of drawbacks. First, if the threshold density is chosen to be too low, the corresponding radius will be much larger than the virialized region of halos at high z. If the density is chosen to be too high, the enclosed mass will correspond only to the inner region of the halo. Most importantly, the mass and radius defined in this way do not track the physical growth of a halo in its fast accretion regime, where the halo profile does exhibit a well-defined characteristic scale close to ${R}_{200{\rm{m}}}(z)$ (Diemer & Kravtsov 2014, see also Figure 5). Zemp (2014) has also discussed the possibility of using the scale radius (the radius where the density profile has a logarithmic slope of −2) as the halo boundary, and define the mass as $M(\lt {r}_{{\rm{s}}})$.

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Here we use a halo radius and mass definition where $R=4{r}_{{\rm{s}}}$ and ${M}_{\lt 4{r}_{{\rm{s}}}}\equiv M(\lt 4{r}_{{\rm{s}}})$. The multiple of 4 in this definition is motivated by the fact that the concentration, ${c}_{{\rm{\Delta }}}={R}_{{\rm{\Delta }}}/{r}_{{\rm{s}}}$, is approximately equal to four as long as the halo is in the fast accretion regime (Zhao et al. 2003, 2009).⁸ In this regime, where the contribution of pseudo-evolution to the mass growth is relatively small, ${M}_{\lt 4{r}_{{\rm{s}}}}$ approximately tracks ${M}_{{\rm{\Delta }}}$. Subsequently, as the mass growth and physical evolution of the inner region of the halo profile slow down, the scale radius approaches a constant (Bullock et al. 2001) and ${M}_{\lt 4{r}_{{\rm{s}}}}$ tracks the actual evolution of the inner halo mass due to real profile changes, unaffected by pseudo-evolution. Assuming an NFW density profile, ${M}_{\lt 4{r}_{{\rm{s}}}}$ is given by

$$\begin{eqnarray}&&{M}_{\lt 4{r}_{{\rm{s}}}}={M}_{{\rm{\Delta }}}\displaystyle \frac{\mu (4)}{\mu ({c}_{{\rm{\Delta }}})}.\end{eqnarray} \tag{ 9 }$$

where $\mu (x)=\mathrm{ln}(1+x)-x/(1+x)$ and the mass and concentration could correspond to any of the commonly used density contrast choices.

For example, for a Milky Way sized halo of mass ${M}_{\mathrm{vir}}={10}^{12}\;{h}^{-1}{M}_{\odot }$ (${R}_{\mathrm{vir}}=207\;{h}^{-1}\;\mathrm{kpc}$) and a typical concentration at that mass, ${c}_{\mathrm{vir}}\approx 9$ (e.g., Zhao et al. 2009; Diemer & Kravtsov 2015), $4{r}_{{\rm{s}}}\approx 92\;{h}^{-1}\;\mathrm{kpc}$, and ${M}_{\lt 4{r}_{{\rm{s}}}}\approx 5.8\times {10}^{11}\;{h}^{-1}{M}_{\odot }$. For comparison, the median ${R}_{\mathrm{sp}}$ for such a halo is $358\;{h}^{-1}\;\mathrm{kpc}\approx 511$ kpc and the median ${M}_{\mathrm{sp}}$ is $1.4\times {10}^{12}\;{h}^{-1}{M}_{\odot }$, about 2.4 times larger than ${M}_{\lt 4{r}_{{\rm{s}}}}$. We caution that these values are medians, and there is large scatter both in the concentration–mass relation and in ${R}_{\mathrm{sp}}$ at fixed mass.

We now contrast the redshift evolution of ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ with the evolution of the boundary and mass for the definitions ${M}_{\lt 4{r}_{{\rm{s}}}}$ and ${M}_{{\rm{\Delta }}}$, using some common choices of Δ. Once again, we use the model of Zhao et al. (2009) to calculate concentrations and halo mass growth histories. The concentrations are used to convert between the different choices of Δ, while the halo mass growth histories are also used to derive Γ (and thus ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ using the fitting formulae presented in Section 3.1).

The upper panels of Figure 5 show the evolution of ${M}_{\mathrm{sp}}$ and the traditional definitions ${M}_{200{\rm{c}}}$, ${M}_{\mathrm{vir}}$, and ${M}_{200{\rm{m}}}$. At high z (i.e., in the fast mass growth regime), ${M}_{\mathrm{sp}}\approx {M}_{200{\rm{c}}}\approx {M}_{\mathrm{vir}}\approx {M}_{200{\rm{m}}}$. At low z, ${M}_{\mathrm{sp}}$ evolves faster than ${M}_{\mathrm{vir}}$ for halos of all masses. In particular, the figure demonstrates that even dwarf and Milky Way-sized halos do accrete new mass at low redshifts. In the lower panels, we compare the evolution of the corresponding radii. While ${R}_{\mathrm{sp}}$ can be significantly larger than ${R}_{200{\rm{m}}}$ (see also Figures 3 and 4), only a relatively small fraction of the total halo mass resides at those radii, reducing ${M}_{\mathrm{sp}}/{M}_{200{\rm{m}}}$ compared to ${R}_{\mathrm{sp}}/{R}_{200{\rm{m}}}$.

The differences in the growth rates of ${M}_{\mathrm{sp}}$, ${M}_{200{\rm{c}}}$, ${M}_{\mathrm{vir}}$, and ${M}_{200{\rm{m}}}$ arise because the mass accreted within ${R}_{\mathrm{sp}}$ at late times is distributed with an approximately isothermal $\rho \propto {r}^{-2}$ profile and contributes significantly only at large radii, $r\gtrsim {R}_{\mathrm{vir}}$ (see Section 4.1). This highlights an important point: at low redshift, the halo mass distribution in the inner regions may be relatively stable and evolve slowly, while the outer regions may evolve fast.

It is therefore also interesting to contrast the evolution of ${M}_{\mathrm{sp}}$ with ${M}_{\lt 4{r}_{{\rm{s}}}}$, which characterizes the mass distribution in the inner regions of halos at low z (see Section 3.2). This comparison is shown in Figure 6, both for the masses (top panel) and radii (bottom panel). The most massive halos are largely still in the fast accretion regime today, and the evolution of ${M}_{\mathrm{sp}}$ and ${M}_{\lt 4{r}_{{\rm{s}}}}$ are quite similar. Low-mass halos, on the other hand, are in the slow mass growth regime and their ${M}_{\mathrm{sp}}$ and ${M}_{\lt 4{r}_{{\rm{s}}}}$ evolve quite differently. For example, for halos of ${M}_{\mathrm{vir}}(z=0)={10}^{8}\;{h}^{-1}{M}_{\odot }$, ${M}_{\lt 4{r}_{{\rm{s}}}}$ evolves significantly slower than ${M}_{\mathrm{sp}}(z)$ at $z\lesssim 3.5$. At $z\lesssim 1$, ${M}_{\lt 4{r}_{{\rm{s}}}}$ is approximately constant, while ${M}_{\mathrm{sp}}$ for these halos changes by $\approx 30\%$ between z = 1 and z = 0. Finally, we compare the halo mass growth rates for the ${M}_{\mathrm{sp}}$, ${M}_{200{\rm{c}}}$ and ${M}_{\lt 4{r}_{{\rm{s}}}}$ definitions in Figure 7. The growth rate of ${M}_{\lt 4{r}_{{\rm{s}}}}$ is approximately an order of magnitude lower than that of ${M}_{\mathrm{sp}}$ at $z\lesssim 1$ for low-mass halos.

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In summary, Figures 5–7 show that during the fast accretion regime ${M}_{\mathrm{vir}}$, ${M}_{200{\rm{m}}}$, ${M}_{\mathrm{sp}}$, and ${M}_{\lt 4{r}_{{\rm{s}}}}$ are approximately equivalent. In the slow mass growth regime, however, the mass within the inner radii ($r\lt {r}_{4{r}_{{\rm{s}}}}$) for halos with ${M}_{\mathrm{vir}0}\lesssim {10}^{12}\;{h}^{-1}{M}_{\odot }$ stops growing, while ${M}_{\mathrm{sp}}$ keeps growing even faster than ${M}_{\mathrm{vir}}$.

Figures 5 and 6 show that ${M}_{\mathrm{sp}}$ grows significantly all the way to z = 0 for halos of all masses, while the mass in the inner regions (characterized by ${M}_{\lt 4{r}_{{\rm{s}}}}$) evolves considerably slower at $z\lesssim 1-2$, and not at all for galaxy-sized halos. The reason for the slow evolution of the inner regions is not that the overall accreted mass is small (galaxy-sized halos approximately double their ${M}_{\mathrm{sp}}$ between z = 1 and z = 0; see Figure 5), but that the newly accreted matter has a relatively shallow density profile (see Section 4.2 in Lithwick & Dalal 2011). The shallow profile arises because the radial profile of the potential over most of the halo volume is shallow, meaning that the velocity of the accreted matter does not vary strongly with radius. Hence, the time spent at each radius r is $\delta t(r)\sim r/v\propto r$. The time averaged mass profile of newly accreted mass is thus ${M}_{\mathrm{acc}}(\lt r)\propto \delta t(r)\propto r/v\propto {r}^{\alpha }$ with $\alpha \sim 1$, leading to a density profile which is close to isothermal, $\rho \propto {r}^{-2}$. This profile is shallower than the overall NFW-like profile of CDM halos, which implies that the previously accreted matter dominates in the inner regions of halos while newly accreted mass contributes significantly in the outer regions near the splashback radius.

This highlights the possibility that systems that appear quiescent in their interior regions may actually still actively grow, particularly when much of the mass growth is due to the accretion of diffuse mass and small halos rather than due to major mergers. The orbit of this material may take it to the interior regions, but it is predominantly deposited in the outskirts. Let us consider some implications for both massive, cluster-sized halos as well as galaxy-sized halos.

For cluster-sized halos, relaxed "cool-core" inner regions do not preclude the possibility of active accretion in the outer regions. Abell 133 (further discussed in Section 4.5) may be an example of such a system, as it combines the characteristics of a cool core system (albeit a disturbed one) and a system undergoing rapid mass accretion. We indicate this possibility in Figure 8, which contrasts the average evolution of mass (calculated using the Zhao et al. 2009 model) within two NFW scale radii, ${M}_{2{r}_{{\rm{s}}}}$, and mass within the splashback radius, ${M}_{\mathrm{sp}}$, for cluster-sized halos. We show the mass within $2{r}_{{\rm{s}}}\approx 2{R}_{200{\rm{c}}}/{c}_{200{\rm{c}}}\approx 0.5{R}_{200{\rm{c}}}\approx {R}_{500{\rm{c}}}$ because this radius is often used to measure halo masses in cluster studies. The figure shows that for ${10}^{14}\;{h}^{-1}{M}_{\odot }$ halos ${M}_{2{r}_{{\rm{s}}}}$ grows from z = 1 to z = 0 by a factor of two, while ${M}_{\mathrm{sp}}$ grows by a factor of four. For ${10}^{15}\;{h}^{-1}{M}_{\odot }$ halos which grow at a much faster rate, the growth of the two masses over the same redshift interval is comparable, which indicates a significant growth of both the inner and outer regions of the halo. We conclude that individual cluster-sized halos can exhibit very different types of mass evolutions: for some halos the difference between the growth of ${M}_{2{r}_{{\rm{s}}}}$ and ${M}_{\mathrm{sp}}$ is more than a factor of two, while for other, fast growing halos the difference may be small.

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For galaxy-sized halos, the slow evolution of the inner regions can be exploited to identify progenitor halos across redshifts at $z\lesssim 1$. In particular, Figure 6 shows that ${M}_{\lt 4{r}_{{\rm{s}}}}$ for halos of ${M}_{\mathrm{vir}}\lesssim {10}^{12}\;{M}_{\odot }$ evolves very little at $z\lt 1$. This suggests that changes in galaxy properties at fixed ${M}_{\lt 4{r}_{{\rm{s}}}}$ can be used to deduce the evolution of these properties along their progenitor histories.

As an example, we estimate the stellar mass growth in low-mass halos at $z\lt 1$. For this purpose, we assume the stellar mass–halo mass relation (SHMR) and its evolution as parameterized by Behroozi et al. (2013), and convert ${M}_{\mathrm{vir}}(z)$ to ${M}_{\lt 4{r}_{{\rm{s}}}}(z)$ to obtain the ${M}_{*}-{M}_{\lt 4{r}_{{\rm{s}}}}$ relation as a function of redshift. We can now read off the star formation history at fixed ${M}_{\lt 4{r}_{{\rm{s}}}}$, shown as solid lines in Figure 9. The shaded areas show the observed star formation histories as derived from the sophisticated modeling of Behroozi et al. (2013). They use a combination of halo merger trees and observational data on the redshift dependence of the stellar mass function and the specific star formation rate–stellar mass relation to infer the growth histories of stellar mass in halos. On the other hand, our simple inference at fixed ${M}_{\lt 4{r}_{{\rm{s}}}}$ only utilizes constraints on the SHMR at different redshifts, such as the ones that can be obtained using abundance matching. The stellar mass growth histories agree well with the detailed modeling results at low halo masses, and starts to diverge once the evolution of ${M}_{\lt 4{r}_{{\rm{s}}}}$ becomes significant. Thus inferences related to the stellar mass growth histories in studies of the evolution of the galaxy–halo connection (e.g., Leauthaud et al. 2012) at low halo masses can be simplified by casting those results as a function of ${M}_{\lt 4{r}_{{\rm{s}}}}$. However, as we will discuss later, tying the growth histories of galaxies to the baryon accretion rate in the central regions is considerably more complicated.

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As we saw in the previous section, most of matter accreted by halos is deposited at the outskirts, while their inner regions can be relatively quiet. This finding relates directly to the mass accretion rate in different mass definitions, and to the question whether this accretion is physical or due to pseudo-evolution.

First, we note that the halo masses in the standard definitions are within a factor of ∼1.5 of ${M}_{\mathrm{sp}}$ at all z (Figure 5), though higher density contrasts (such as ${M}_{500{\rm{c}}}$) would lead to smaller radii and thus larger differences with ${M}_{\mathrm{sp}}$. The mass accretion rate of ${M}_{\mathrm{sp}}$, on the other hand, differs significantly from the spherical overdensity definitions, and can be up to three times larger than the change in ${M}_{200{\rm{c}}}$ (Figure 7). This may seem surprising, since the mass growth in conventional definitions already suffers from pseudo-evolution, i.e., a growth in addition to the physical growth within the corresponding halo boundary.

However, there are two competing differences between ${{dM}}_{\mathrm{sp}}/{dt}$ and ${{dM}}_{200{\rm{c}}}/{dt}$. While ${{dM}}_{200{\rm{c}}}/{dt}$ is larger than the physical change of the mass within ${R}_{200{\rm{c}}}$ due to pseudo-evolution, it also misses mass that is added outside ${R}_{200{\rm{c}}}$ between two epochs, i.e., the physical growth in the outskirts discussed in Section 4.1. While it is not a priori clear which of the two effects should dominate, Figure 7 demonstrates that ${{dM}}_{\mathrm{sp}}/{dt}$ is larger than ${{dM}}_{200{\rm{c}}}/{dt}$ at all masses and redshifts shown, meaning that physical evolution in the outskirts dominates over the pseudo-evolution of ${M}_{200{\rm{c}}}$. However, for halos which are truly non-accreting (${{dM}}_{\mathrm{sp}}/{dt}\approx 0$), pseudo-evolution would still lead to a non-zero ${{dM}}_{200{\rm{c}}}/{dt}$.

Whether the evolution in ${M}_{{\rm{\Delta }}}$ under- or overestimates the true accretion rate, it does not always reflect the physical addition of matter. The relatively small difference between ${M}_{{\rm{\Delta }}}$ and ${M}_{\mathrm{sp}}$ does not reflect the physical correctness of ${M}_{{\rm{\Delta }}}$, but rather the shallow mass profile in the outer regions of halos which means that large changes in radius lead to small changes in enclosed mass.

The importance of taking pseudo-evolution into account when interpreting changes in halo mass can be illustrated by the results of Watson & Conroy (2013) who tested the common assumption that the same SHMR is valid for both host and satellite halos. They showed that the relations are somewhat different and interpreted the difference as a 10% growth in the stellar mass of satellite galaxies since the time of infall, possibly due to residual star formation before final quenching (Figure 10). They estimated the average time of infall to be around 4 Gyr ago, or $z\approx 0.5$. However, in their analysis (similar to all abundance matching studies) the halo masses of centrals are defined at z = 0 while those of satellites are defined at their infall epoch ($z=0.5$). Therefore, some of the difference in the host and satellite SHMRs will arise from the pseudo-evolution of mass between these redshifts. We estimated this effect and show it with the green dot–dashed line in Figure 10. Simply accounting for pseudo-evolution explains most of the difference in the SHMRs found by Watson & Conroy (2013), and does so better than the suggested blanket 10% increase in the stellar mass of satellites across all halo masses. This agreement suggests that satellite galaxies maintain levels of star formation similar to their central counterparts for a substantial period after infall. This increase in stellar mass would depend on a satellite's halo mass, a qualitatively different view from that presented by Watson & Conroy (2013).

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A number of recent studies (Prada et al. 2006; Diemand et al. 2007; Cuesta et al. 2008; Diemer et al. 2013; Diemer & Kravtsov 2014; Zemp 2014; Correa et al. 2015) have pointed out that the inner density profiles of low-mass, galaxy-sized halos evolve very little at $z\lesssim 1$. The near-constant density profiles were sometimes interpreted as a lack of new mass accretion in these halos. As we showed in the previous section, ${M}_{\mathrm{sp}}$ does, on average, increase rapidly for halos of all masses, including dwarf-sized halos, implying the continued accretion of mass in all halos to low z. This newly accreted mass may briefly pass through the inner regions of halos, but is primarily deposited in the halo outskirts, with no substantial change in the inner regions.

In Section 4.1, we explained why the newly accreted mass contributes little to the inner density profile, but this argument applies only to collisionless dark matter. Baryons, on the other hand, experience additional pressure forces and possibly dissipation and interaction with feedback-driven winds (Nelson et al. 2015), so that their radial distribution and fate can be quite different (Faucher-Giguère et al. 2011; van de Voort et al. 2011; van de Voort & Schaye 2012; Wetzel & Nagai 2015; Feldmann & Mayer 2015). We note that the baryon accretion rate in the inner regions of halos could track the growth rate of halos (e.g., determined by ${M}_{\mathrm{sp}}$) if the trajectory of recently accreted material brings it closer to the central density peak. The accreted baryons could interact with intra-halo gas and feedback-driven winds and stay in the inner regions of halos. Such effects could conceivably explain the large baryon accretion rate and its consistency with ${f}_{{\rm{b}}}{{dM}}_{200{\rm{m}}}/{dt}$, as found by Wetzel & Nagai (2015). However, Woods et al. (2014) tracked particles in their hydrodynamical simulations and showed that the late time star formation in their simulated galaxies is a result of gas cooling from a reservoir which was accreted 2–8 Gyr earlier. Nelson et al. (2015), on the other hand, find that feedback-driven winds increase the time it takes the gas to reach the disk after it crosses ${R}_{\mathrm{vir}}$ by a factor of several, but does not introduce any significant mass dependence in the accretion rate.

Although the fate of baryons is not completely clear, our results show that galaxy-sized systems do have a continuing supply of fresh matter to z = 0.

We have shown that the splashback radius for galaxy-sized halos at low z can be more than a factor of two larger than ${R}_{200{\rm{c}}}$. For example, for Milky Way-sized halos, the splashback radius can be as large as 600 kpc, or two thirds of the distance to M31 (see, e.g., Figure 2 of Diemand & Kuhlen 2008 and Figure 5 above). The large ${R}_{\mathrm{sp}}$ may explain the presence of dwarf spheroidal galaxies, such as Cetus (e.g., Fraternali et al. 2009), at large distances, and highlights that the "zone of influence" of halos (i.e., their associated environmental effects) can extend to and beyond $2{R}_{200{\rm{c}}}$.

This effect was discussed previously in the context of "backsplash satellites" which were found in substantial numbers beyond the halo radii in the standard definitions (Balogh et al. 2000; Mamon et al. 2004; Gill et al. 2005, see also Wetzel et al. 2014 for in depth analysis of this issue and more recent references). Their occurrence at large radii is not surprising because satellite subhalos effectively trace the distribution of accreted matter. This also means that the radial number density profiles of subhalos in simulations and galaxies in observed clusters may exhibit the steepening at the splashback radius (see further discussion in Section 4.5). Considering the extent of the splashback radius, it is unnecessary to invoke special ejection processes to explain the presence of subhalos that have orbited their host at large radii and are deficient in gas or show suppressed star formation (e.g., Balogh et al. 2000; Solanes et al. 2002; Geha et al. 2012; Wetzel et al. 2012). Rather, these galaxies may reflect a population of halos that accreted onto their host halo late and are located close to the apocenter of their orbit at $\sim {R}_{\mathrm{sp}}$.

As ${R}_{\mathrm{sp}}$ corresponds to the physical boundary between the inner and infall regions of halos, ${R}_{\mathrm{sp}}$ may also be a natural boundary of the one-halo term in halo models of structure formation. Using such a boundary would result in the reclassification of some isolated halos as subhalos. In Figure 11, we explore the magnitude of this effect by comparing the fraction of density peaks which are subhalos (i.e., the satellite fraction) as a function of their peak circular velocity, ${V}_{\mathrm{peak}}$, in the standard virial definition to that in the splashback case.⁹ For low values of ${V}_{\mathrm{peak}}\sim 100\;\mathrm{km}\;{{\rm{s}}}^{-1}$, the satellite fraction based on ${R}_{\mathrm{sp}}$ shows an increase of 30%.

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The reclassification of isolated halos also has implications for assembly bias. It is well-known that the large-scale bias of halos depends upon their assembly history (Gao et al. 2005; Wechsler et al. 2006; Dalal et al. 2008; Li et al. 2008). At the low (high) mass end, the large scale bias of halos is (anti-) correlated with their concentrations. The low-mass halos that are classified as isolated in the virial definition but as subhalos in the splashback definition are not a random subsample of the parent population. They tend to be in the vicinity of more massive halos, have higher concentrations, and higher bias values. Thus, their reclassification as satellites can reduce the assembly bias of low-mass halos to some extent (see also Lacerna & Padilla 2011), but it does not explain the bulk of the effect (see also Ludlow et al. 2009; Wang et al. 2009).

In halo models, accurate modeling of the transition region between the one- and two-halo terms has remained a challenge. The use of ${R}_{\mathrm{sp}}$ as the boundary in such models may lead to some progress, but requires an accurate calibration of the halo mass function and bias as a function of ${M}_{\mathrm{sp}}$ using cosmological simulations. We could convert existing mass function and bias formulae for ${M}_{200{\rm{m}}}$ (e.g., Tinker et al. 2008, 2010) using the conversions in Equations (7) and (8), but given the large scatter in those relations the results would be approximate. Instead, we will require robust methods to measure ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ for individual halos in simulations, rather than using the mean or median density profiles of halo samples. Falck et al. (2012) recently introduced a halo finder, in which phase space sheet foldings were used to define structures such as sheets, filaments and halos. This definition is in the spirit of the splashback radius definition we advocate in this paper. However, the number of phase space sheet foldings does not easily map onto the splashback radius. This is because a particle that was accreted onto a halo could have been part of other structures in the past and thus exprience a number of foldings. Depending on the history of a particular particle this number can be quite different. The definition of the splashback radius specific to a given halo will thus require a different approach.

Interestingly, the steepening of the density profile associated with the splashback radius may already have been detected in several galaxy clusters. Rines et al. (2013) used the velocity caustic method to derive mass profiles of a number of clusters to large radii. Figure 11 in their paper shows that the radial density profiles of massive clusters sharply transition to slopes steeper than the value of −3 expected from the NFW profile. The steepening occurs at radii where $\rho (r)\sim 10-20{\rho }_{{\rm{m}}}$, roughly where the splashback radius is expected to be located for typical halos with ${\rm{\Gamma }}\sim 1-2$ (see Figure 2 above and Figure 2 in Diemer & Kravtsov 2014).

Furthermore, a deep Chandra observation of Abell 133 (A. Vikhlinin et al. 2015, in preparation) shows a strong steepening of the gas density profile just beyond ${R}_{200{\rm{c}}}$. Although the steepening is observed in gas rather than in the total mass profile, hydrodynamic simulations show that the gas profile exhibits a steepening at the same radius as the dark matter profile (Lau et al. 2014). According to the calibration of Equation (5), the proximity of the splashback radius to ${R}_{200{\rm{c}}}$ would correspond to a very high mass accretion rate onto this relatively relaxed, "cool core" cluster. In the future, the density profile steepening associated with splashback may be independently verified with mass profile measurements using weak lensing or via radial profiles of number density of galaxies.

Finally, Tully (2015) recently argued that a radius qualitatively similar to the splashback radius which the author called "the second turnaround radius" should be used to define the halo extent in analyses of observational samples. Tully (2015) presents evidence for a sharp drop in the number density and velocity dispersion profiles of galaxies in a number of systems from cluster-sized to the Local Group. We note that for the Milky Way and M31, Tully (2015) derives a splashback radius of $\approx 290$ kpc, considerably smaller than what we would expect for systems of this mass on average: ${R}_{\mathrm{sp}}\approx 1.4-1.5{R}_{200{\rm{m}}}\approx 400-600$ kpc (see Figure 4). Overall, Tully (2015) find a scaling of ${R}_{\mathrm{sp}}\approx 1.33{R}_{200{\rm{c}}}$, significantly smaller than our calibration. This can be traced to some of the simplifying assumptions made in his calculation of the radius.

In the standard spherical overdensity definition, the radius ${R}_{{\rm{\Delta }}}$ enclosing a given overdensity Δ depends on the amount of mass within ${R}_{{\rm{\Delta }}}$, but not on the distribution of this mass. In contrast, estimating ${M}_{\lt 4{r}_{{\rm{s}}}}$ demands knowledge of the scale radius (or concentration; Equation (9)), and estimating ${M}_{\mathrm{sp}}$ requires a measurement of the density profile at the outermost radii of a halo, or knowledge of its accretion rate (Equation (5)). In simulations, computing ${M}_{\lt 4{r}_{{\rm{s}}}}$ presents no problem, since the scale radius is routinely computed by halo finders. The splashback radius can be found by finding the radius of steepest slope in halo density profiles, or by fitting those profiles with the fitting function of Diemer & Kravtsov (2014). It may be possible to develop more robust methods not relying on spherically averaged profiles. For example, a clearer picture of the density caustic emerges from the distribution of halo particles on the phase-space plane (${v}_{{\rm{r}}}-r$, where ${v}_{{\rm{r}}}$ is the radial velocity with respect to the halo center; Adhikari et al. 2014).

Observationally, measurements of ${r}_{{\rm{s}}}$, ${R}_{\mathrm{sp}}$, or the accretion rate are challenging, but ${M}_{\lt 4{r}_{{\rm{s}}}}$ and ${M}_{\mathrm{sp}}$ can be estimated using their average relations with conventionally defined masses. For ${M}_{\lt 4{r}_{{\rm{s}}}}$, all we need is a well-calibrated model for halo concentrations to estimate ${r}_{{\rm{s}}}$ (e.g., Diemer & Kravtsov 2015). For ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$, we have quantified ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ as a function of ${R}_{200{\rm{m}}}$ and ${M}_{200{\rm{m}}}$ and the mass accretion rate, given in Equations (5) and (6). For the case where the mass accretion rate of a halo is not known, we have given ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$ as a function of peak height (or mass) in Equations (7) and (8). However, we note that there is one easily accessible quantity that contains information beyond halo mass: the concentration. Since the concentration is intimately related to the mass accretion history of a halo (e.g., Ludlow et al. 2013), we expect there to be a relation between Γ and concentration. We find that, in principle, this relation depends on redshift and mass in a non-trivial manner. However, at z = 0 we find a simple, mass-independent relation,

$$\begin{eqnarray}&&{\rm{\Gamma }}\approx 3.43-2.74{\mathrm{log}}_{10}({c}_{\mathrm{vir}}).\end{eqnarray} \tag{ 10 }$$

We note that this relation was only calibrated for the cosmology used in this paper, and for our particular definition of Γ. Nevertheless, the relation can be used to estimate the mass accretion rate, and thus ${R}_{\mathrm{sp}}$ and ${M}_{\mathrm{sp}}$, at z = 0.

For convenience, we have implemented all the calibrations of ${R}_{\mathrm{sp}}$, ${M}_{\mathrm{sp}}$, and ${M}_{\lt 4{r}_{{\rm{s}}}}$ given in this paper, as a function of mass accretion rate, peak height, and mass, in the public python code Colossus (Diemer 2015). This stand-alone module can be downloaded at www.benediktdiemer.com/code.