ON THE EVOLUTION OF CLUSTER SCALING RELATIONS

We use a dissipationless simulation of a 1 (Gpc/h)³ cubic volume in the ΛCDM model. The large box size ensures that the volume contains tens of thousands of cluster-sized halos. For consistency with the analysis of D13, we chose the same cosmological parameters as the Bolshoi simulation (Klypin et al. 2011), namely a flat ΛCDM model and $\Omega _{\rm m}=1-\Omega _\Lambda =0.27$, Ω_b = 0.0469, h = H₀/(100 km s⁻¹ Mpc⁻¹) = 0.7, σ₈ = 0.82 and n_s = 0.95. These parameters are compatible with measurements from WMAP7 (Jarosik et al. 2011), a combination of WMAP5, Baryon Acoustic Oscillations and Type Ia supernovae (Komatsu et al. 2011), X-Ray cluster studies (Vikhlinin et al. 2009b), and observations of the clustering of galaxies and galaxy–galaxy/cluster weak lensing (see, e.g., Tinker et al. 2012; Cacciato et al. 2013). The same cosmology was used for all analytical calculations in this paper. The initial conditions for the simulation were generated at redshift z = 49 using a second-order Lagrangian perturbation theory code (2LPTic; Crocce et al. 2006). The simulation was run using the publicly available code Gadget2 (Springel 2005), and followed 1024³ dark matter particles, corresponding to a mass resolution of 7 × 10¹⁰ h⁻¹ M_☉. The spline force softening was set to 1/30 of the inter-particle separation, 33 h⁻¹ kpc.

We used the phase-space halo finder Rockstar (Behroozi et al. 2013a) to extract all isolated halos and subhalos from the 100 snapshots of the simulation, and applied the merger tree code of Behroozi et al. (2013b) to these halo catalogs. Whenever we refer to the progenitor of a halo, we mean the halo along its most massive progenitor branch at each redshift. We focus on a sample of isolated cluster-sized halos with M_vir > 5 × 10¹³ h⁻¹ M_☉ at z = 0, consisting of ≈57, 000 halos, though we impose further mass cuts for some of the experiments below. For 99.9% of the halos a z = 1 progenitor could be identified, and for 98.6% of them a z = 2 progenitor was found. In the following analysis, only progenitors with M > 10¹³ h⁻¹ M_☉ (referring to whichever mass definition used) were considered, corresponding to about 140 particles in the smallest halos. Once this lower limit was reached, earlier progenitors were also removed from the analysis. For all halos in the sample, we extract spherically averaged density profiles in 80 logarithmically spaced bins between 0.05 R_vir and 10 R_vir.

For the fits to the M–σ relation in Section 2.3, we also follow E08 in excluding satellite halos, i.e., halos whose R_200c overlaps with a larger halo. This requirement excludes more halos than just subhalos that are required to have their centers lie inside R_200c of a larger halo. We find the satellite fraction to be a few percent depending on the mass range, somewhat lower than E08. The difference is presumably attributable to the different halo finders and cosmologies used.

Since we are interested in subtle differences between the M–σ relation of different halo samples, we need to measure the slope and offset of the power-law fits (i.e., linear fits in log-log space) robustly. For the M–σ data at hand, a simple least-squares fit is not appropriate because of the presence of intrinsic scatter (Hogg et al. 2010). Instead, we perform a Markov-Chain Monte Carlo analysis with 30,000 samples for each fit, sampling the likelihood of the offset, slope, and intrinsic scatter of the linear fit in the $\log M{-}\log \sigma$ plane.

The spherical overdensity mass definition given in Equation (1) is based either on the critical or mean matter density of the universe, both of which decrease with time, albeit at different rates. As a result, the threshold density ρ_Δ decreases, even for the virial mass definition where the evolution of Δ_vir and ρ_c partially cancel. Thus, the virial radius and mass of a halo increase, even if the halo does not evolve and its density profile remains constant. The amount of this pseudo-evolution of mass can be computed for a given density profile by numerically solving Equation (1) with a varying density threshold ρ_Δ(z). D13 found pseudo-evolution to account, on average, for a growth of about a factor of two in M_200m since z = 1, albeit with large scatter.

In the simple case of a static density profile, the amount of pseudo-evolution can be determined exactly if the density profile is known at any epoch. We sometimes refer to this case as "pure pseudo-evolution." In the more realistic case where a halo grows through a mixture of physical accretion and pseudo-evolution, the exact amount of pseudo-evolution can only be estimated, unless the density profile is known at all redshifts. In this paper, we use the minimum estimator of pseudo-evolution defined in D13, which corresponds to the mass between the virial radii at some initial redshift, R(z_i), and at a final redshift, R(z_f), which is already in place at z_i. This material is bound to contribute to the halo's pseudo-evolution as the virial radius moves out (see D13 for details on rare exceptions). As the density between the two virial radii increases after z_i, the actual amount of pseudo-evolution is larger. Thus, this estimator represents the minimum pseudo-evolution we deduce from the density profiles at z_i and z_f. We define the fraction of mass growth due to pseudo-evolution, f_PE, as the minimum pseudo-evolved mass divided by the actual difference in spherical overdensity mass between the main progenitor at z_i and its descendant at z_f (see D13 for an exact mathematical definition).

Most galaxy-sized halos are predominantly pseudo-evolving after z ≈ 1 (Cuesta et al. 2008). Moreover, the mass growth is dominated by pseudo-evolution for a non-negligible fraction of cluster-sized halos at z < 1 (Wu et al. 2013, D13). In this paper, we use z_i = 0.5 and z_f = 0 as the interval during which we estimate the amount of pseudo-evolution. This interval corresponds to about 5 Gyr, or a couple of dynamical times of clusters.

We first examine the relation between M_200c and σ_200c, as E08 found that the M–σ relation is particularly tight for this mass definition. We define σ as the averaged, one-dimensional, velocity dispersion inside a radius r (E08),

$$\begin{equation} \sigma ({<}r)\equiv \left[ \frac{1}{3 N_{\rm p}(r)} \sum _{i=1}^{N_{\rm p}(r)} \sum _{j=1}^{3} | v_{\rm i,j}-\bar{v}_{\rm j}|^2\right]^{\frac{1}{2}}, \end{equation} \tag{ 4 }$$

where N_p(< r) is the number of particles within r, v_{i, j} the jth component of the velocity of particle i, and $\bar{v}_{\rm j}$ the average jth component of all particles inside r.

We consider all isolated halos with M_200c > 10¹⁴ h⁻¹ M_☉ at z = 0, a sample of ≈13, 400 halos. We split those halos into six samples with different f_PE, the fraction of their mass evolution since z = 0.5 estimated to be due to pseudo-evolution. Table 1 lists the fraction of halos in each sample; for the halos with f_PE > 1, the density profile decreases at certain radii. For a small fraction of halos, M_200c decreases with time, and thus f_PE < 0. In the following, we refer to halos with 0.75 < f_PE < 1 as pseudo-evolving (PE), and those with 0 < f_PE < 0.25 as fast accreting (FA).

Figure 1 shows the M–σ evolution of halos in the four samples spanning the range 0 < f_PE < 1. The lines show the best-fit power-law relation to the entire sample at z = 0 and z = 0.5 (solid red lines), as well as to the halos in the respective f_PE sample at z = 0 (dashed green lines). The orange arrows indicate the average evolution of halos in three mass bins at z = 0. Table 1 lists the parameters of the best-fit power laws for the respective halo samples, some of which are visualized in Figure 2.

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In general, the M–σ relation of all halos in our simulation at z = 0 roughly agrees with the results of E08, and matches the redshift scaling predicted by Equation (3). There are, however, some subtle disagreements (Figure 2). First, our best-fit slope α ≡ d ln  σ/d ln  M = 0.3436 ± 0.0007 is steeper than the slope α = 0.3361 ± 0.0026 found by E08. It is interesting to note that the E08 slope is consistent with the expected value of 1/3, whereas our slope differs from this value by 14σ. One might suspect that the sharp mass cut-off at 10¹⁴ h⁻¹ M_☉ introduces an artificial steepening, but the progenitor halos at z = 0.5 are not subject to such a cut, and their M–σ relation exhibits the same slope as at z = 0, within the statistical uncertainties (Figure 1). Furthermore, the mass resolution of our simulation is sufficiently high to measure the slope of the M–σ relation without bias (Figure 6 of E08; our mass resolution corresponds to 14,000 particles in a 10¹⁵ h⁻¹ M_☉ halo). In Section 3.2, we present a physical explanation for the steepening of the slope, as well as for the trends in normalization and slope with f_PE. A further, minor difference appears in the intrinsic scatter about the best-fit relation that we find to be 0.0165 dex, while E08 reported a somewhat larger scatter of 0.0185 dex.

It is remarkable that although halos with different amounts of pseudo-evolution evolve very differently on the M–σ plane, the M–σ relations of the PE and FA samples evolve nearly identically. The slopes of the relation for the individual samples agree with the slope for the entire sample to better than 2%, with the largest disagreement in the FA sample (about 3σ). The slope of the best-fit relation to the PE sample agrees with the overall sample within statistical errors. The difference in the normalization between the FA and PE sets is roughly 2% as well. Thus, we conclude that pseudo-evolution does not have a significant impact on the M–σ relation for M_200c. The main difference between the samples is that the scatter around the M–σ relation is noticeably smaller for the PE sample, likely because the PE halos are the most relaxed sub-population.

Figure 1 clearly shows that both FA and PE halos in simulations (with masses defined within R_200c) follow a power-law M–σ relation,

$$\begin{equation} \sigma = \sigma _M^* S(z) \left(\frac{M}{M_*} \right)^{\alpha }. \end{equation} \tag{ 5 }$$

The slope of the best-fit power law is close to α = 1/3, and the evolution of the relation is consistent with $S(z) = \rho _{\Delta }^{1/6}$, the scaling expected from the simple estimate described in Section 1. We first consider the origin of this relation for FA halos.

It is well known that the density and mass profiles of CDM halos are universal as a function of radius scaled by the scale radius r_s,

$$\begin{equation} M({<}r) = M_{\Delta } \frac{\mu (x)}{\mu (c)}, \end{equation} \tag{ 6 }$$

where x = r/r_s, r_s is the radius where the density profile has a logarithmic slope of −2, and c = R_Δ/r_s is a dimensionless concentration parameter. In addition to the universal mass profile, CDM halos exhibit a power-law pseudo phase space density profile within the radius of the first shell crossing,

$$\begin{equation} Q(r) \equiv \frac{\rho (r)}{\sigma ^3(r)} = Q_* x^{\beta }, \end{equation} \tag{ 7 }$$

with β ≈ 1.9 (Taylor & Navarro 2001; Ascasibar et al. 2004; Rasia et al. 2004; Ludlow et al. 2011). The existence of universal density and Q profiles implies that σ(< r) also follows a universal profile,

$$\begin{equation} \sigma ({<}r)= \sigma _\ast s(x), \end{equation} \tag{ 8 }$$

where s(x) is another dimensionless function of the re-scaled radius. By dimensional considerations⁵

$$\begin{equation} \sigma _\ast = \sqrt{\frac{{GM}({<} r)}{r}}, \end{equation} \tag{ 9 }$$

where r is a control radius of the problem. Given that r_s appears to be the control radius in the density structure of halos, r can be readily identified with r_s. Thus,

$$\begin{eqnarray} \sigma ({<}r) &=& \sqrt{\frac{{GM}({<} r_s)}{r_s}}\, s(x) \nonumber \\ &=&\sqrt{\frac{{GM}_{\Delta }}{R_{\Delta }}}\, s(x)\left[\frac{\mu (c)}{c\,\mu (1)}\right]^{-1/2}, \end{eqnarray} \tag{ 10 }$$

which allows us to express σ(< r) in terms of the scaling of Equation (3),

$$\begin{equation} \sigma ({<}R_{\Delta }) \propto \rho _{\Delta }^{1/6}(z) M_{\Delta }^{1/3} \times s(c) \, \left[\frac{\mu (c)}{c\,\mu (1)}\right]^{-1/2}. \end{equation} \tag{ 11 }$$

For FA halos, c_vir = R_vir/r_s is approximately constant, c_vir ≈ 4 (Zhao et al. 2003a, 2009), and thus $\sigma ({<}R_{\rm vir}) \propto \rho _{\Delta }^{1/6}(z) M_{\rm vir}^{1/3}$ according to Equation (11). Note that this result does not depend on the actual shape of μ(x) and s(x), but only on their universality.

Often, however, the M–σ relation is defined within some radius R_Δ ≠ R_vir, e.g., R_500c. In such cases, c generally evolves with redshift (particularly at z ≲ 1), and we would expect some evolution in the normalization of σ in addition to $\rho _{\Delta }^{1/6}(z)$. Nevertheless, Figure 1 shows no such evolution, which means that the s(x)/[μ(x)/x]^1/2 term in Equation (11) must be approximately constant for different x ∼ R_Δ. We assume that the σ(< r) and M(< r) profiles can be approximated by power laws around R_Δ,

$$\begin{equation} \mu (x) = \tilde{\mu }x^{\gamma }; \quad s(x) = \tilde{s} x^{\xi }. \end{equation} \tag{ 12 }$$

The condition that s(x)/[μ(x)/x]^1/2 ≈ constant is satisfied if

$$\begin{equation} \xi -\frac{\gamma -1}{2}\approx 0; \quad \tilde{\mu }/\tilde{s}^{1/2} \approx {\rm constant}. \end{equation} \tag{ 13 }$$

Thus, we expect $\sigma \propto \rho _{\Delta }^{1/6}(z)M^{1/3}_{\Delta }$ for any R_Δ as long as the M and σ profiles satisfy the conditions expressed in Equation (13). We note that these conditions are valid for both PE halos and for halos that evolve through a combination of physical accretion and pseudo-evolution. Their σ_* stops increasing, and their R_Δ increases as a result of pseudo-evolution; but as long as the M and σ profiles obey the relations in Equation (13), the PE halos move along the same M–σ relation as the FA halos.

We now investigate whether the M and σ profiles of CDM halos in simulations obey Equation (13). Figure 3 shows the mass and velocity dispersion profiles of halos in a narrow mass bin with 10¹⁴ h⁻¹ M_☉ < M_vir < 1.1 × 10¹⁴ h⁻¹ M_☉, a sample of about 3000 halos. For simplicity, we focus on the profiles at z = 0. The profiles are plotted for r > 0.07 R_vir which corresponds to about twice the softening length of our simulation. We have examined the profiles of massive halos, M_vir > 10¹⁵ h⁻¹ M_☉, which are resolved with ten times more particles and have larger R_vir. We found very similar profile shapes in M and σ, which indicates that (1) the shapes are not strongly dependent on halo mass, and (2) the inner structure of the profiles for the 10¹⁴ h⁻¹ M_☉ halos does not suffer from significant resolution effects in the radial range shown.

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Figure 3 shows the median profiles for all halos, as well as the FA and PE sub-samples, along with the logarithmic slopes of the median profiles as a function of radius. The slopes were computed using a fourth-order Savitzky–Golay smoothing algorithm over the 15 nearest bins (Savitzky & Golay 1964) to smooth out random fluctuations, but without affecting the actual values of the slope.

In order to investigate deviations from the scaling of Equation (3), the bottom panel of Figure 3 shows the profile of the scaled velocity dispersion σ' ≡ σ(< r)/[GM(< r)/r]^1/2. For the cluster halos examined here, the median σ' is approximately constant from R_2500c to R_200m—i.e., the range of radii usually used to define M and σ. Thus, the first condition of Equation (13) is satisfied over these radii. For example, at R_200c, γ = 0.82 and ξ = −0.09, and thus ξ − (γ − 1)/2 ≈ 0. The second condition of Equation (3) demands that the normalization of the σ' profiles of halos with different masses and accretion rates is roughly constant. This condition is more or less satisfied, but we note two deviations. First, the FA halos (purple line in the bottom panel of Figure 3) have a slightly higher median σ' profile than the median and the PE halos. Second, higher-mass halos (M_vir > 10¹⁵ h⁻¹ M_☉, dashed dark blue line) follow the profile of the lower-mass FA halos almost exactly, which is not surprising since most high-mass halos are in the FA regime.

These slight deviations from the conditions of Equation (13) explain the deviations of the M–σ relation from a perfectly self-similar scaling that we observed in Section 2.3. First, the higher median σ' profiles of higher-mass halos mean that the slope of the M–σ relation of the entire sample is steeper than 1/3. The difference in the normalization appears to be larger at small radii such as R_2500c (Figure 3), and we measure the slope of the M–σ relation for M_2500c to be steeper, about 0.36. Second, the higher normalization of σ' of the FA compared to the PE halos explains why the M–σ relations for the FA samples have systematically higher normalizations than those of the PE samples (Figure 2). Finally, we note that the slopes measured within each sample also show a trend: For the FA sample, the slope is consistent with 1/3, but increases significantly toward the PE sample. This trend confirms that FA halos adhere to conditions of Equation (13) most accurately, as expected from the discussion in Section 3.1. The steeper slope within the PE sample could, for example, arise if the lowering of the σ' normalization happens gradually, and the higher-mass halos in a given PE sample have, on average, resided in the PE regime for less time than have their lower-mass counterparts.

Despite the small deviations discussed earlier, the profiles are close to a constant σ' with radius and mass. This property of the profiles is the main reason why the FA and PE halos evolve similarly, and why the cluster population exhibits a tight scaling relation of the form $\sigma \propto \rho _{\Delta }^{1/6}(z)M^{1/3}$ for all common definitions of M_Δ. There are, however, small deviations from a constant σ' at small and large radii such as R_2500c and R_200m. We examine the implications of these deviations for the M–σ relation in Section 3.4.

What is the origin of the remarkable constancy of σ' = σ(< r)/[GM(< r)/r]^1/2 with radius? We observe significant systematic differences in the shapes of the M(< r) and σ(< r) profiles of the FA and PE halos shown in Figure 3, and note that these differences are real, and not due to noise in the measurement. Nevertheless, σ'(r) is similar for both types of halos.

First, we note that the σ(< r) profile is well described by a simple prediction based on the Jeans equation. Specifically, we assume that halos follow the Navarro–Frenk–White (NFW) density profile,

$$\begin{equation} \rho _{\rm NFW} = \frac{\rho _0}{x(1+x)^2}, \end{equation} \tag{ 14 }$$

and thus

$$\begin{equation} M_{\rm NFW} = 4 \pi r_{\rm s}^3 \rho _0 f(x), \end{equation} \tag{ 15 }$$

where f(x) = ln (1 + x) − x/(1 + x). We assign the NFW profile the same virial radius as the median profile, and the mean concentration, c_vir = R_vir/r_s = 5.6, corresponding to the median mass, M_vir = 1.05 × 10¹⁴ h⁻¹ M_☉ of our halo sample. This mean concentration was derived from the concentration–mass relation of Zhao et al. (2009), which was calibrated with cosmological N-body simulations. The top panel of Figure 3 demonstrates that the NFW profile, shown with a dashed purple line, is a good approximation to the median mass profile of simulated halos at the radii of interest.

To compute the σ profile corresponding to an NFW density profile, we assume spherical symmetry, Jeans equilibrium,⁶ and that the orbits of dark matter particles in cluster halos are only mildly anisotropic across most radii within the virial radius (Eke et al. 1998; Colín et al. 2000; Cuesta et al. 2008; Lemze et al. 2012). We thus find (see, e.g., More et al. 2009)

$$\begin{equation} \sigma ^2({<}r) = \frac{c_{\rm vir} V_{\rm vir}^2}{f(c_{\rm vir}) f(x)} \int _0^{x} x^{\prime 2}\ dx^{\prime } \int _{x^{\prime }}^{\infty } \frac{f(x^{\prime \prime }) dx^{\prime \prime }}{x^{\prime \prime 3}(1+x^{\prime \prime })^2}. \end{equation} \tag{ 16 }$$

This predicted profile is a good approximation both to the median σ(< r) profile (center panel of Figure 3), and to the corresponding profile of the PE sample, even though most halos in this mass range are undergoing physical mass accretion. This equilibrium estimate offers a possible explanation why the FA and PE halos have similar σ' profiles, even though their M and σ profiles are noticeably different. This view is further supported by the fact that halos maintain power-law Q(r) profiles during different stages in their evolution. We can conclude, therefore, that halos maintain a structure close to equilibrium even during stages of rapid mass growth.

We now generalize the NFW prediction to any power-law density profile, ρ = ρ₀x^α where α ≠ −1. The mass enclosed within the scaled radius, x, is

$$\begin{equation} M({<}x) = \frac{4 \pi \rho _0 r_{\rm s}^3}{(\alpha +3)} x^{\alpha +3} = M_0 x^{\alpha +3}. \end{equation} \tag{ 17 }$$

The velocity dispersion profile at a given radius r, using the spherically symmetric Jeans equation, is given by

$$\begin{equation} \sigma ^2(x) = -\frac{{GM}_0}{r_{\rm s}} \frac{x^{\alpha +2}}{(2\alpha +2)}. \end{equation} \tag{ 18 }$$

Again assuming isotropic orbits, we obtain the mass-weighted velocity dispersion averaged within the scaled radius,

$$\begin{equation} \sigma ^2({<}x) = {-}\frac{{GM}_0}{r_{\rm s}}\frac{1}{(2\alpha +2)} \frac{\int x^{\prime \alpha +2} x^{\prime \alpha }x^{\prime 2}\ dx^{\prime } }{\int x^{\prime \alpha }x^{\prime 2}\ dx^{\prime }} \end{equation} \tag{ 19 }$$

$$\begin{equation} = {-}\frac{{GM}({<}x)}{r_{\rm s}x}\frac{(\alpha +3)}{(2\alpha +2)(2\alpha +5)}. \end{equation} \tag{ 20 }$$

The aforementioned calculations show that $\sigma ({<}r) \approx {\rm const}\times \sqrt{{GM}({<}r)/r}$ for both NFW profiles, and profiles that roughly follow a power law around the relevant radii. We conclude that a relation of the form $\sigma \propto \rho _{\Delta }^{1/6}(z) M^{1/3}$ can be generically expected for CDM halos, as long as they are in approximate Jeans equilibrium.

The results presented in this section illustrate the origin of tight scaling relations, such as the M–σ relation: they reflect equilibrium relations between cluster profiles. The scaling relations hold at any radius where such equilibrium relations apply.

We now return to the small deviations from a constant σ' observed in Figure 3, which can cause a difference in the evolution of the M–σ relation for FA and PE halos, depending on the chosen mass definition. To evaluate these deviations, we use the NFW model and the σ' profiles calculated using Jeans equilibrium and the assumption of isotropic orbits, which proved to be a good approximation to the median σ' profile (Section 3.3). We assign concentrations appropriate for a halo of a given mass using the model of Zhao et al. (2009). Figure 4 shows the evolution of σ' from z = 0 to z = 1 for four different mass definitions, normalized to z = 0. Each color corresponds to a certain M_vir (regardless of the mass definitions used in the respective panels). The resulting physical density and σ profiles are assumed to be constant over time. The gray shaded region highlights the mass range 10¹⁴ h⁻¹ M_☉ < M_vir < 10¹⁵ h⁻¹ M_☉ in which most cluster halos reside. An M–σ relation that is perfectly independent of radius, and thus of pseudo-evolution, corresponds to the dashed horizontal line indicating σ' = const.

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The results shown in Figure 4 are a reflection of the slope of the σ' profile shown in Figure 3, and the positions of the respective virial radii. The top left panel demonstrates that σ' is particularly constant for large halo masses and an M_200c mass definition. As expected, the deviations from a perfect scaling are small, Δσ' ≲ 10% at z = 1. The figure illustrates that there are two main factors that influence the impact of pseudo-evolution on σ', namely (1) the chosen definition of the virial radius, and (2) the concentration of the NFW profile, and thus the halo mass. For mass definitions with a low threshold density (M_200m, M_vir), the σ' profile around the virial radius has a positive slope and σ' increases between z = 1 and z = 0 for pseudo-evolving halos. Conversely, for mass definitions with high threshold densities (M_500c, M_2500c), the slope is negative and σ' decreases with time. We note that, for most mass definitions σ' is generally less constant for smaller mass halos because they have larger concentrations and their virial radii shift further along the σ' profile. For most mass definitions this shift leads to larger deviations from a flat slope, but R_2500c moves into a more favorable part of the profile as the concentration increases. Thus, we conclude that it is somewhat fortuitous that the M–σ relation for cluster halos is close to the simple scaling of Equation (3), as lower-mass halos should evolve to deviate from the self-similar prediction for the normalization by ≈10%, depending on the mass definition.

Figure 5 shows the evolution of the PE halos on the M–σ plane for the M_200m and M_2500c mass definitions. As M_2500c is significantly smaller than M_200c, the cut-off mass for the M_2500c sample was lowered to M_2500c ⩾ 2 × 10¹³ h⁻¹ M_☉. As expected from Figure 4, halos evolve in different directions in σ in the top and bottom panels. The fits to the PE sample (green dashed lines) are almost indistinguishable from the fits to all halos (solid red lines), as for the M_200c mass definition in Figure 1. However, the M–σ relation deviates somewhat from the expected redshift scaling; the dashed orange lines show the fit to the z = 0.5 sample re-scaled to z = 0 according to Equation (3). For the M_200m mass definition, this scaling underpredicts the measured σ at fixed mass. We expect this deviation, since the σ' profile increases with radius around R_200m and σ at z = 0 will thus be larger than we would expect from a constant σ' scaling. Conversely, the σ' profile decreases with radius around R_2500c, meaning that Equation (3) overpredicts the measured σ for this mass definition. These findings underscore that, for certain mass definitions, pseudo-evolution can have an impact on the M–σ relation of the entire population, not only the PE halos.

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We conclude that the M–σ relation exhibits an evolution close to that expected from the scaling relation of Equation (3) only for definitions of the virial radius that lie in the flat range of the σ' profile. Other definitions of masses and radii will result in small deviations from such a scaling.