The splashback radius and the radial velocity profile of galaxy clusters in IllustrisTNG

We use 1697 clusters of galaxies from the Illustris TNG300-1 simulation (mass $M_{200c}>10^{14}$M$_\odot$ and redshift range $0.01\leq z \leq 1.04$) to explore the physics of the cluster infall region. We use the average radial velocity profile derived from simulated galaxies, ${\rm v_{rad}}(r)$, and the average velocity dispersion of galaxies at each redshift, ${\rm \sigma_v}(r)$, to explore cluster-centric dynamical radii that characterize the cluster infall region. We revisit the turnaround radius, the limiting outer radius of the infall region, and the radius where the infall velocity has a well-defined minimum. We also explore two new characteristic radii: (i) the point of inflection of ${\rm v_{rad}}(r)$ that lies within the velocity minimum, and (ii) the smallest radius where ${\rm \sigma_v}(r)$ = $|{\rm v_{rad}}(r)|$. These two, nearly coincident, radii mark the inner boundary of the infall region where radial infall ceases to dominate the cluster dynamics. Both of these galaxy velocity based radii lie within $1\sigma$ of the observable splashback radius. The minimum in the logarithmic slope of the galaxy number density is an observable proxy for the apocentric radius of the most recently accreted galaxies, the physical splashback radius. The two new dynamically derived radii relate the splashback radius to the inner boundary of the cluster infall region.


Introduction
Clusters of galaxies are massive self-gravitating systems of galaxies comprised of a dense central region in approximate virial equilibrium surrounded by an extended region where continuing infall dominates the dynamics.The infall region extends to scales 10 Mpc (Geller et al. 1999;Rines & Diaferio 2006;Rines et al. 2013;Umetsu et al. 2014Umetsu et al. , 2016Umetsu et al. , 2020)).Across this large volume, clusters present distinctive dynamical regimes.Characteristic dynamical radii mark the transition between the virial and infall regions.These radii inform comparisons with models of formation and evolution of cluster of galaxies (e.g., Press & Schechter 1974;White & Rees 1978;Bower 1991;Lacey & Cole 1993;Sheth & Tormen 2002;Zhang et al. 2008;Corasaniti & Achitouv 2011;De Simone et al. 2011;Achitouv et al. 2014;Musso et al. 2018).
In standard cosmology, the turnaround radius is the outer boundary of a cluster; it marks the radius where matter decouples from the Hubble flow.Within the turnaround, the gravitational potential of the cluster dominates the dynamics (Gunn et al. 1972;Silk 1974;Schechter 1980;Bertschinger 1985;Villumsen & Davis 1986;Cupani et al. 2008Cupani et al. , 2011)).Because the matter overdensity is small averaged over the volume within the turnaround radius, quasi-linear dynamics predicts the turnaround radius accurately.
Within the turnaround radius, clusters have an extended accretion region often referred to as the infall region.Here, there is a net radial infall of matter onto the cluster.The radial velocity in this region is negative, and the well-defined minimum of the radial velocity profile is a proxy for the radius where the infall is strongest (Cuesta et al. 2008;De Boni et al. 2016;Vallés-Pérez et al. 2020;Pizzardo et al. 2021Pizzardo et al. , 2023b;;Fong & Han 2021;Gao et al. 2023).Fong & Han (2021) and Gao et al. (2023) recently proposed two additional characteristic radii in the infall region.They define an inner depletion radius where the mass flow rate is maximum.This radius is similar to the radius where the radial velocity has a minimum.They also define a characteristic depletion radius at slightly larger clustercentric distance which traces the region where clustering around the halo is weakest relative to clustering around a random matter particle.
In the dense central regions of galaxy clusters, orbital motions dominate infall.The limiting radius of this approximately virialized region, the virial radius (e.g.Gunn et al. 1972;Peebles 1980;Lacey & Cole 1993), typically defines the cluster size.Proxies for the virial radius include R 200c and R 200 m , where R ∆ is the cluster-centric distance that encloses a mean density ∆ c times the critical density or ∆ m times the mean background density (Lacey & Cole 1993).
A more recently suggested boundary of the inner region of a cluster is the splashback radius, R spl , defined as the first apocenter of orbits of recently accreted material (Adhikari et al. 2014;Diemer & Kravtsov 2014;More et al. 2015;Diemer 2020).The splashback radius marks a natural boundary where the internal cluster dynamics dominates over systematic infall.Detailed studies use N-body simulations to explore the trajectories of individual dark matter particles as a theoretical proxy of R spl (e.g.Diemer & Kravtsov 2014;Diemer et al. 2017;Mansfield et al. 2017;Diemer 2018;Xhakaj et al. 2020).Usually, R spl is outside the virialized radius of a cluster, ∼1.5−2R 200c .At fixed redshift, R spl decreases with increasing halo mass accretion rate.At fixed mass accretion rate, R spl increases with increasing redshift.Adhikari et al. (2014) show that at ∼R spl particles generate caustics in phase-space density.They match this caustic to a sudden drop in the logarithmic derivative of the mass density profile of halos (Diemer & Kravtsov 2014).This feature in the mass (or number) density profile of halos, R ρ spl (R n spl ), is an observable proxy for R spl .In fact, a variety of observations detect the minimum in the logarithmic slope of cluster density associated with R spl (More et al. 2016;Baxter et al. 2017;Chang et al. 2018;Shin et al. 2019;Zürcher & More 2019;Murata et al. 2020;Adhikari et al. 2021;Bianconi et al. 2021;Gonzalez et al. 2021).
So far, the physical meaning of R spl relies solely on the dynamics of dark matter particles from pure N-body simulations.Studies based on hydrodynamical simulations investigate R spl through its proxies, R ρ spl or R n spl (Baxter et al. 2021;Deason et al. 2021;O'Neil et al. 2021O'Neil et al. , 2022;;Dacunha et al. 2022).Here we use the IllustrisTNG hydrodynamical simulations (Pillepich et al. 2018;Springel et al. 2018;Nelson et al. 2019) to interpret R spl based on the dynamics of simulated galaxies rather than dark matter particles.Connecting R spl directly to galaxy dynamics provides a way to unveil how galaxies as well as matter particles show clear signatures of the splashback physics.
The mean radial velocity profile of clusters in IllustrisTNG provides additional dimensions to the view of R spl .The R spl based on the matter density profile lies just within the radius where the radial velocity profile has a clear minimum.The radial velocity profile enables determination of two further characteristic radii: (1) the point of inflection inside the velocity minimum, and (2) the smallest radius where the local velocity dispersion exceeds the infall.From the turnaround radius inward, these two radii describe the radius where infall no longer dominates the dynamics.These two radii are essentially equal to the observable proxy of R spl .This agreement extends the meaning of R spl as the dynamical outer boundary of the region where orbital motions, as opposed to infall, dominate galaxy cluster dynamics.
Currently, these two galaxy radial velocity based radii are not directly observable.However, Odekon et al. (2022) show that data including independent distance measurements for galaxies around clusters and filaments allow extraction of the galaxy velocity field in the infall region.
Section 2.1 describes the IllustrisTNG simulations and the resulting cluster samples.Section 2.2 discusses the mass and velocity profiles.Section 3 outlines the definition of cluster dynamical radii based on the radial velocity profile.Sections 3.1 Power spectrum index 0.9667 and 3.2 summarize the main properties of the turnaround radius and the radial velocity minimum, respectively.In Sect.3.3 we identify radii based on the radial velocity profile that closely approximate the observable proxy of R spl from the galaxy number density profile.We discuss the impact of the mass distribution on the dynamical radii in Sect.4.1, compare the galaxy and total matter velocities in Sect.4.2, and conclude in Sect. 5. Table 1 defines the symbols used throughout the paper.

Cluster samples, velocity, and mass profiles
We built a sample of galaxy clusters from the TNG300-1 run of the IllustrisTNG simulations Pillepich et al. (2018), Springel et al. (2018), Nelson et al. (2019).We derived mass and radial velocity profiles for these clusters.We briefly discuss the cluster sample in Sect.2.1 and the profiles in Sect.2.2.

Simulations and catalogs
The IllustrisTNG simulations (Pillepich et al. 2018;Springel et al. 2018;Nelson et al. 2019) are a set of gravo-magnetohydrodynamical simulations based on the ΛCDM model.Table 2 lists the cosmological parameters of the simulations.TNG300-1 is the baryonic run with the highest resolution among the runs with the largest simulated volumes.The simulation has a comoving box size of 302.6 Mpc.TNG300-1 contains 2500 3 dark matter particles with mass m DM = 5.88 × 10 7 M and the same number of gas cells with average mass m b = 1.10 × 10 7 M .As in Pizzardo et al. (2023a) we used group catalogues compiled by the IllustrisTNG Collaboration to extract all of the Friends-of-Friends (FoF) groups in TNG300-1 with M 200c > 10 14 M .There are 1697 clusters in the 11 redshift bins in the range 0.01 ≤ z ≤ 1.04.Table 3 summarizes the main properties of the 11 subsamples, including redshift, number of clusters, median, interquartile range, and the minimum and maximum of the mass M 200c in each bin.
The mass and redshift sampling we adopted is consistent with the selection in recent studies that use hydrodynamical simulations to explore the outer region of galaxy clusters (e.g Baxter et al. 2021;Deason et al. 2021;O'Neil et al. 2021O'Neil et al. , 2022;;Dacunha et al. 2022).This sampling mimics characteristics of observational approaches that focus on cluster masses for systems typically at z 1 (Tamura et al. 2016;Ivezić et al. 2019;Marshall et al. 2019;Cornwell et al. 2022Cornwell et al. , 2023)).
For each FoF halo, IllustrisTNG provides a list of subhalos from the Subfind algorithm (Springel et al. 2001).For clusters in Table 3, we extracted subhalos with stellar mass M > 10 8 M and within 10R 200c of the center of the cluster halo.We identified these subhalos as cluster member galaxies.
Blue, orange, green, and red points in Fig. 1 show the relation between M 200c and the comoving R 200c for clusters in four A82, page 2 of 9 Colored squares with error bars show the median and the interquartile range for each sample.Figure 1 and Table 3 show that, as the redshift increases from z = 0.01 to z = 1.04, the sample minimum M 200c is nearly constant.The maximum M 200c decreases by 71% with increasing redshift.The lower and upper extrema of the interquartile ranges of M 200c behave in a similar way.
The mass functions in each bin are skewed towards low masses, M 200c 3 × 10 14 M , even at low redshift.Thus even at low redshift the high-mass tails have small statistical weight.As the redshift increases from z = 0.01 to z = 1.04, the median M 200c of clusters decreases by only 10%.The comoving R 200c 's are proportional to M 1/3 200c (Fig. 1).

Mass and velocity profiles
Pizzardo et al. (2023a) computed a cumulative mass profile for each cluster, M(<r), based on the 3D distribution of matter extracted from raw snapshots.These profiles include all matter species: dark matter, gas, stars, and black holes.For each cluster, Pizzardo et al. (2023a) computed M(<r) in 200 logarithmically spaced bins covering the radial range (0.1−10)R 3D 200c .These profiles define R 200c and M 200c for each cluster; they allow straightforward computation of cumulative and shell density profiles for each cluster.Pizzardo et al. (2023b) computed a single average galaxy radial velocity profile at each redshift by averaging over all the individual galaxy radial velocity profiles for the clusters in the redshift bin.They used a subsample of the cluster sample we used here, because they included only the 78% of systems allowing application of the caustic technique (Diaferio & Geller 1997;Diaferio 1999;Serra et al. 2011), an observational method for estimating the cluster mass profile.We included the entire set of IllustrisTNG clusters to maximize the sample size and to minimize any potential biases in the mass function and resulting radial velocity profile.
We computed the galaxy radial velocity profile of individual clusters in each subsample.Based on the comoving position of simulated galaxies with respect to the cluster center, r c,i , and the galaxy peculiar velocity, u p,i , we computed the radial velocity of each galaxy: , where H(z s ) and a(z s ) are the Hubble function and the scale factor at the red- 0.0 0.5 1.0 1.5 Frequency shift z s of the snapshot.We computed the mean radial velocity profile of the cluster by averaging over the galaxy v rad (r)'s within 100 linearly spaced radial bins covering the range (0, 10)R 3D 200c .At each redshift, we averaged over all of the individual radial velocity profiles for the snapshot.We obtained a single mean galaxy radial velocity profile along with the dispersion around it.
The solid curves in Fig. 2 show the average galaxy radial velocity profiles for clusters at three redshifts z = 0.01, 0.62, and 1.04, from left to right, respectively.The dashed curves show Savitzky-Golay (Savitzky & Golay 1964) smoothed profiles derived with an 11 bin window and a fourth order polynomial interpolation.Shaded bands show the error in the smoothed profiles.Fluctuations and errors increase with increasing redshift because of the decreasing size of the cluster samples (see second column of Table 3).

Dynamical radii from radial velocity profiles
The galaxy radial velocity profile provides direct measures of both the turnaround radius and the infall velocity minimum for a cluster of galaxies.The radial velocity profile also provides a complementary view of the splashback radius R spl as the inner boundary of the region where infalling galaxies dominate the cluster dynamics.We derive the turnaround radius and the point of minimum radial velocity in Sects.3.1 and 3.2, respectively.Section 3.3, shows how these dynamically determined radii provide an interpretation of R spl based on the radial velocity profile.We compare the dynamically derived radii with R n g spl , an observational proxy for R spl (e.g.Adhikari et al. 2014;Diemer & Kravtsov 2014;More et al. 2015;Diemer 2017Diemer , 2018;;Diemer et al. 2017;O'Neil et al. 2021).

The turnaround radius
The turnaround radius R turn of a galaxy cluster is defined as the cluster-centric distance where galaxies depart from the Hubble flow (Gunn et al. 1972;Silk 1974;Schechter 1980).Hence R turn is the radius where the smoothed galaxy radial A82, page 3 of 9 velocity v rad (R turn ) = 0.The red vertical lines in Fig. 2 show the turnaround radius at redshifts z = 0.01, 0.62, and 1.04, respectively from left to right.The red line in Fig. 3 shows R turn in units of R 200c as a function of redshift.The red shadowed band shows the uncertainty in R turn , based on bootstrapping 1000 samples at each redshift.The radius R turn is in the range (4.67−4.94)R200c , and it is independent of redshift with a typical value R turn = (4.81± 0.10)R 200c .
We compared the R turn values derived from the average radial velocity profile with the analytic predictions of Meiksin (1985;priv. comm. to Villumsen & Davis 1986), R Meik turn .In this approach, the density contrast is δ(r) = 3M(r)/4πρ bkg r 3 − 1, where M(r) is the mass of the cluster within a distance r from the cluster center and ρ bkg = Ω m ρ c is the background matter density.In the spherical collapse model, the radial velocity induced by the perturbation is v rad /Hr ≈ Ω 0.6 m P(δ) (Gunn et al. 1972;Silk 1974;Schechter 1980;Regos & Geller 1989).Meiksin's approach approximates the function P(δ) with a non-polynomial; the overdensity within the turnaround radius where v rad /Hr = 1, is At each redshift, we computed the set of R Meik turn 's from the mass profiles of individual clusters in Table 3.We adopted the Lahav et al. (1991) approximation for the growth factor.At each redshift, we computed the median value RMeik turn and the interquartile range.
The radius R turn derived from the average radial velocity profile is consistent with the analytic predictions of the Meiksin approximation.On average R turn exceeds RMeik turn by 3.9%.At each redshift, R turn is within the interquartile range of the set of individual R Meik turn 's, and R turn and RMeik turn agree to within ∼1.5σ.The radius RMeik turn increases by ∼4% as the redshift increases from z = 0.01 to z = 1.04; this increase is not ruled out by the Illus-trisTNG results.

The minimum radial velocity
The point of minimum galaxy radial velocity, R v min , is a characteristic feature of the v rad (r) profiles.Previous evaluations of R v min include computations based on dark matter particles within stacked halo samples in N-body simulations (Cuesta et al. 2008;De Boni et al. 2016;Pizzardo et  The blue vertical lines in Fig. 2 show R v min from the smoothed average galaxy radial velocity profiles at the three redshifts z = 0.01, 0.62, and 1.04, from left to right, respectively.The blue curve in Fig. 3 shows R v min as a function of redshift.R v min decreases with increasing redshift by ∼41%.The redshift dependence occurs because at fixed mass, clusters at low redshift accrete from relatively less dense surroundings than clusters at higher redshift.The cores of high redshift clusters are also more dense relative to their surroundings, bringing the maximum infall velocity closer to the cluster center.The splashback radius that A82, page 4 of 9 we consider next must lie within the radius where the radial velocity has its minimum.

The splashback radius
The splashback radius R spl , a proxy for the physical boundary of a galaxy cluster, is the average location of the first apocenter of matter recently accreted by the cluster halo.Therefore R spl marks the boundary where the internal cluster dynamics dominates systematic infall (e.g.spl , is the minimum of the logarithmic slope of the local matter density ρ, γ = d log ρ(r) d log r (Adhikari et al. 2014).We used simulated galaxies in Illus-trisTNG to compute the average galaxy local number density n g (r) at each redshift; we then derived the logarithmic slope of n n g (r), γ n g (r), and smoothed it with a Savitzky-Golay filter (Savitzky & Golay 1964) with an 11 bin window and a fourth order polynomial.The minimum of the resulting function is R n g spl .We checked the stability of the results by considering filters with polynomials of order two to six and fixed 11 bin windows.All of the corresponding R n g spl s are within ∼5% of R n g spl s resulting from the fourth order polynomial; furthermore, χ 2 yields very similar results for all filter choices.
Figure 4 shows γ n g (r) based on the average galaxy local number density profile of the 282 TNG300-1 clusters with M 200c > 10 14 M at z = 0.01.The orange vertical line shows the minimum of γ n g (r), R n g spl , an observationally accessible proxy of R spl .The black dotted curve in the upper panel of Fig. 5 shows the dependence of R n g spl on redshift.The gray shadowed area shows the standard deviation based on bootstrap resampling.Over the redshift range we sample, R n g spl decreases by ∼38%.At fixed mass R spl should decrease with redshift (see Sect. 1).The uncertainty increases with increasing redshift.At higher redshift, the size of the cluster sample is small (see Table 3); hence the γ n g (r) profile is very noisy.Locating the minima of these noisy γ n g (r)'s is challenging and the uncertainty is correspondingly large.
The R   2021) at z ≤ 0.62; they are ∼20−50% larger at higher redshifts.However, the samples we extracted are not fully comparable with theirs: we defined galaxies as subhalos with stellar mass M > 10 8 M (see Sect. 2.1); they defined galaxies by selecting a total subhalo mass >10 9 M and nonzero M .Thus, the physical properties of the two samples may differ; the choice we made reduces the number of subhalos labelled as galaxies leading to poorer statistics 1 .The redshift dependence of R n g spl s we obtained is consistent with previous results based on N-body simulations (More et al. 2015;Diemer 2020;O'Neil et al. 2021), although with a higher normalization.
The average galaxy radial velocity profile based on Illus-trisTNG provides a route to define two new radii which closely approximate R dr profiles for the four redshifts z = 0.01, 0.31, 0.62, and 1.04, respectively.The vertical lines show the minimum R infl correspondingly to each colored curve.galaxy dynamics at R spl .We first explored a radius based on the inflection point in the radial velocity profile, R infl .We then turned to a radius based on a comparison between the velocity dispersion and the mean radial velocity, R σ v .In both cases these radii mark the limiting cluster-centric radius where infall no longer dominates the cluster dynamics.
The radial velocity profiles (Fig. 2) turn from concave d 2 v rad dr 2 < 0 to convex d 2 v rad dr 2 > 0 as the cluster-centric radius increases.The inflection point where the change occurs, R infl , lies between R 200c and R v min .The radius of the inflection point is in the range ∼(1−2)R 200c depending on redshift (see Sect. 3.2) and is evident at every redshift.Statistically, galaxies within the inflection point cannot escape to larger radii.Thus the inflection point of v rad (r) is a dynamically motivated radius that should approximate R spl .
Computationally, the radius corresponding to the inflection is the minimum of dv rad dr .Here v rad (r) < 0. The radial velocity also increases in absolute value as the cluster-centric distance increases.Thus, at the inflection point, the local spatial change of v rad (r) is a maximum in absolute value.
To measure R infl , we first computed dv rad dr at each redshift from the smoothed average galaxy radial velocity profile.The colored curves in Fig. 6 show the resulting dv rad dr profiles for the four redshifts listed in legend.At each redshift, the minimum of the corresponding profile locates R infl .The dotted vertical lines in Fig. 6 indicate the minimum R infl for the correspondingly coloured curves.
For comparison with other dynamical radii, the orange vertical lines in Fig. 2 show the inflection point R infl at the three redshifts z = 0.01, 0.62, and 1.04, from left to right, respectively.The orange curve in Fig. 3 summarizes the behavior of R infl as a function of redshift.The shadowed area indicates the standard deviation based on bootstrap resampling.From z = 0.01 to z = 1.04,R infl decreases from ∼2.1R 200c to ∼1.3R 200c (∼38%).On average, R infl is ∼28% smaller than R v min .
A second radius that improves understanding of the physics within the clusters' splashback region is R σ v , which we define from the average ratio between the 3D cluster velocity dispersion and the infall velocity, σ v (r)/v rad (r). Figure 2 shows that from R turn to R v min , v rad increases in absolute value from ∼0 to ∼250−500 km s −1 depending on the redshift.Then, from R v min to R 200c , v rad decreases in absolute value from its maximum to ∼0. Figure 7 shows that σ v (r) increases monotonically from R turn to R 200c ; σ v (r) increases from ∼250−350 km s −1 to ∼400−500 km s −1 .At R v min , the ratio between σ v (r) and v rad is always greater than −1: from z = 0.01 to z = 0.52 the ratio increases from ∼−0.97 to ∼−0.74; at higher redshifts, the ratio at R v min is ∼ − 0.72.The two profiles σ v (r) and v rad always have two intersections.When σ v (r) > |v rad (r)|, randomly oriented motions exceed motions aligned along the radial direction; radial motions dominate when σ v (r) < |v rad (r)|.We compute R σ v by identifying the smallest cluster-centric radius where σ v (r)/v rad (r) = −1.Within this radius galaxies orbiting in the cluster potential dominate the dynamics.
At each redshift, we computed the average galaxy velocity dispersion profile σ v (r) with an approach similar to the computation of the average v rad (r) profiles (Sect.2.2).We considered 100 shells with linearly spaced boundaries covering the range (0−10)R 200c .For each cluster, we computed the standard deviation of the velocities of all the galaxies inside that shell, the estimator of the velocity dispersion for the shell.In bin n for the shell with radial range (r n − r n+1 ), the velocity dispersion is the square root of where k runs over the three spatial components, v(n) k is the mean of the k spatial component of the velocity of galaxies in bin n, and N is the number of galaxies.The single average galaxy σ v (r) profile at each redshift is the mean of the velocity dispersions for the individual clusters in each bin.
The curves in Fig. 7 show the average σ v (r) for clusters at the four redshifts listed in the legend within the radial range (1 − 4)R 200c .Generally, σ v (r) deceases with increasing clustercentric radius.The σ v (r) profile at fixed cluster-centric radius increases with redshift by ∼40% from z = 0.01 to z = 1.04.Figure 8 shows the ratio σ v (r)/v rad (r) for the four redshifts listed in the legend.As in Fig. 7, we limit the radial range to Fig. 8. Average ratio between the galaxy velocity dispersion and the galaxy radial velocity, as a function of cluster-centric radius.The blue, orange, green, and red solid curves show the ratio at four redshifts z = 0.01, 0.31, 0.62, and 1.04, respectively.The dashed curves show the smoothed correspondingly colored solid curves.The horizontal dashdotted line shows σ v /v rad = −1.The vertical dotted lines show R σv corresponding to σ v (r)/v rad (r) = −1 from smoothed profiles.
At cluster-centric distances R 200c , where the cluster is in approximate dynamical equilibrium, σ v (r)/v rad (r) is ill-defined because v rad (r) ∼ 0 (Fig. 2).At radii ∼(1 − 2)R 200c , σ v (r)/v rad (r) increases but remains <−1.This behavior occurs at smaller radii with increasing redshift.In this radial range there is a net radial motion toward the cluster center, but the infall does not dominate over the internal cluster dynamics.
In the region where σ v (r)/v rad (r) > −1 (Fig. 8), radial infall dominates the dynamics.The center of this range is ≈R v min as expected (Sect.3.2).Statistically, galaxies in this region have radial velocities directed toward the cluster center (Fig. 2); the radial velocities exceed the orbital velocities.At smaller cluster-centric radii, where σ v (r)/v rad (r) < −1, orbital motions dominate.Once galaxies migrate from the infall region where σ v (r)/v rad (r) > −1 into the inner cluster region where σ v (r)/v rad (r) < −1, they can generally no longer reach radii where σ v (r)/v rad (r) > −1.Thus the radius where σ v (r)/v rad (r) = −1, R σ v , provides another complementary physical view of the dynamics near the splashback radius.
R σ v is the innermost cluster-centric radius where σ v (r)/v rad (r) = −1.To locate R σ v , we used the ratio between the smoothed σ v (r) and v rad (r) profiles.The dotted vertical lines in Fig. 8 show R σ v for the correspondingly colored ratios.The green dash-dotted curve in the upper panel of Fig. 5 shows that R σ v decreases with redshift.The green shadowed area indicates the error based on bootstrap resampling.From z = 0.01 to z = 1.04,R σ v decreases from ∼2.2R 200c to ∼1.3R 200c (∼40%).
Figure 5 compares the observable proxy for the splashback radius, R n g spl , with the two radii R infl and R σ v .These radii, derived from the mean radial velocity profile, mark the transition from the infall region to the approximately virialized region where orbital motions dominate.
In the upper panel of Fig. 5, the solid orange, dash-dotted green, and dotted black lines show R infl (same as the orange line in Fig. 3), R σ v , and R n g spl , respectively, as a function of redshift.The shadowed bands show the bootstrapped error.In the bottom panel, the solid orange and dash-dotted green lines show the ratios R infl /R n g spl and R σ v /R n g spl , respectively.The orange, green, and grey shadowed areas show the 1σ confidence levels for R infl , R σ v , and R n g spl , respectively.Averaged over redshift, R infl /R n g spl = 1.00 ± 0.12; R σ v /R n g spl = 1.006 ± 0.069 (bottom panel, Fig. 5).The redshift dependence of R n g spl agrees with that of R infl and R σ v over the whole redshift range.
The radii from the velocity profile, R infl and R σ v , have nearly the same value as R n g spl .The dynamically derived radii are unbiased and within ∼1σ of R n g spl .Although R infl and R σ v are not directly observable, they establish a connection between the splashback physics and the infall dynamics of clusters.The near equality of R infl , R σ v and R n g spl suggests that R spl corresponds to the boundary between the region where galaxies orbiting in the cluster potential dominate the dynamics and the region where infall of galaxies dominates.

The cluster mass distribution
Table 3 shows that the median cluster mass generally decreases as the redshift increases because very massive systems are progressively less abundant at higher redshift.The changing distribution of cluster masses could affect the resulting dynamical radii.We explored this issue by selecting subsamples that are homogeneous.
For each redshift, we constructed homogeneous samples that include clusters with mass >M 200c in the range (1.0−5.0)× 10 14 M .The last column of Table 3 shows that this mass range is sampled at every redshift.A Kolmogorov-Smirnov test demonstrates that the clipped samples have indistinguishable mass distributions; the p-values are in the range (0.32 − 1.00).For each redshift, we computed the average galaxy radial velocity and the average galaxy velocity dispersion profiles for the clipped sample following Sects.2.2 and 3.3, respectively.
The clipped R turn are on average ∼0.1% smaller than the full sample R turn , and they are always within 0.5% of the full sample values.The difference between clipped and full sample R turn as a function of redshift is unbiased.The clipped R v min 's are equal to the full samples R v min 's at every redshift except z = 0.73 where the clipped radius exceeds the full sample radius by ∼5%.
The clipped R infl 's are equal to the full sample R infl 's at every redshift except z = 0.62 where the clipped radius exceeds the full sample radius by ∼8%.The clipped R σ v 's are on average ∼0.2% below the full sample R σ v 's and they lie within 0.8% of the full sample R σ v 's.Differences between the clipped and full sample R σ v 's are unbiased.The dynamical radii are all insensitive to differences among the distribution of cluster masses at different redshifts in the full samples.

Comparison of galaxy and total-matter distribution radii
Galaxies may be biased tracers of the underlying distribution of matter in the Universe, mainly comprised of dark matter (e.g., Kaiser 1984;Davis et al. 1985;White et al. 1987) Fig. 9. Ratio between dynamical radii from radial velocity profiles based on galaxies and on the total matter.The red, blue, and orange lines show the ratio between the turnaround, the minimum radial velocity, and the inflection point of average radial velocity profiles of galaxies and total matter, as a function of redshift.The horizontal dotted lines show the median ratio of the corresponding curve for the 11 redshift bins.The shadowed band in each case shows the 1σ confidence range.
dynamical radii derived from the radial velocity profiles based on galaxies by comparing them with the values derived for the total matter distribution.
We computed a single average radial velocity for the total matter distribution at each redshift with the approach of Sect.2.2.The total matter content includes dark matter, gas, stars, and black holes, which are all included in IllustrisTNG (see Sect. 2.1).We took 200 logarithmically spaced bins (rather than the 100 we used for the velocity profile based on galaxies alone) in the radial range (0.1−10)R 200c .The total matter distribution enables the more finely spaced bins.
At each redshift, we located the turnaround, the minimum radial velocity, and the inflection point of the average total matter radial velocity profiles (Sect.3).The radii derived from the total mass distribution are R all turn , R all v min , and R all infl , respectively.The red, blue, and orange lines in Fig. 9 show the ratios between the galaxy based and total matter based radii: R turn /R all turn , R v min /R all v min , and R infl /R all infl as a function of redshift.The horizontal dotted lines show the corresponding median ratio for the 11 redshift bins.The shadowed band of corresponding color shows the 1σ confidence range for each of the ratios.
The radii R turn and R v min based on galaxy velocities are consistent with the radii based on the total matter velocities.Averaging over redshift, the median R turn /R all turn ≈ 0.997 (red dotted horizontal line); R turn is always within 4% of R all turn .On average, R v min /R all v min ≈ 0.997 (blue dotted horizontal line), and R v min overestimates (underestimates) R all v min by +10% (−8%) at most.The R infl 's based on galaxy velocities generally exceed the corresponding radii based on the total matter.Averaging over redshift, R infl /R all infl ≈ 1.082 (orange dotted horizontal line), and R infl overestimates (underestimates) R all infl by +14% (−12%).The results are consistent with equality at ∼1σ level.Figure 9 shows that the ratios are essentially independent of redshift.
The mild overestimation of R infl relative to R all infl is consistent with the behaviour of R spl found by O'Neil et al. (2021) who also analyze TNG300-1.They computed the proxy for R spl based on galaxies, R n g spl , and compared it with the analogous splashback radius based on all of the matter, R n all spl .For systems with masses 10 14 M , R n g spl /R n all spl ∼ 1.02−1.03,with a considerable dispersion that allows overestimations as large as ∼15% (see their Fig. 6).For the inflection point, the average ratio over the 11 redshift bins we sampled is R infl /R all infl = 1.082 ± 0.074.This value is consistent with O'Neil et al. ( 2021) within the uncertainty.We conclude that, as O'Neil et al. (2021) show for the splashback proxy, the bias between dynamical radii based on galaxies and all of the matter is small.

Conclusion
Simulated galaxies drawn from the IllustrisTNG300-1 simulations (Pillepich et al. 2018;Springel et al. 2018;Nelson et al. 2019) enable exploration of the infall region of 1697 galaxy clusters with M 200c > 10 14 M and redshift 0.01 ≤ z ≤ 1.04 (Pizzardo et al. 2023a,b).For these systems, we revisited the classical turnaround radius R turn that defines the outer boundary of a cluster (Gunn et al. 1972;Silk 1974;Schechter 1980) and the characteristic minimum infall velocity R v min (De Boni et al. 2016;Vallés-Pérez et al. 2020;Fong & Han 2021;Pizzardo et al. 2021Pizzardo et al. , 2023b)).Based on the galaxy radial velocity profile, v rad (r), we derived two new measures of the inner boundary of the infall region.Both of these radii lie within the radial velocity minimum and coincide with the splashback radius, R spl (Adhikari et al. 2014;Diemer & Kravtsov 2014;More et al. 2015).
The galaxy average radial velocity profile, v rad (r), enables identification of the turnaround radius, R turn , where galaxies decouple from the Hubble flow.The R turn 's lie in the range (4.67−4.94)R200c , are insensitive to redshift, and agree with the Meiksin analytic approximation (Meiksin 1985).
The galaxy average radial velocity profile, v rad (r), has a well-defined minimum, R v min .The value of R v min decreases from 2.8R 200c at z = 0.01 to 1.8R 200c at z = 1.04.The maximum infall velocity itself increases with redshift.
Inside R v min , we developed two new dynamical radii that mark the inner boundary of the infall region: (i) R infl is the inflection point of v rad (r), the cluster-centric radius where the derivative of v rad (r) is maximum in absolute value, and (ii) R σ v is the smallest cluster-centric radius where σ v (r) = |v rad (r)|.Outside these radii, infall dominates the cluster dynamics; within these radii orbital motions dominate over infall.
To within 1σ, the two dynamical radii, R infl and R σ v , coincide with R n g spl , the radius where the logarithmic derivative of the galaxy average number density profile has a minimum.This radius is an observable proxy for the splashback radius, R spl , often determined from N-body simulations (Diemer & Kravtsov 2014;Diemer et al. 2017;Mansfield et al. 2017;Diemer 2018;Xhakaj et al. 2020).Averaged over redshift, R infl /R n g spl = 1.00 ± 0.12, and R σ v /R n g spl = 1.006 ± 0.069.Galaxies may be biased tracers of the total matter content of a cluster.For the set of IllustrisTNG clusters, the ratios between galaxy and total matter R turn is 0.997 ± 0.014; the corresponding ratio for R v min is 0.997 ± 0.056.The radius R infl for galaxies exceeds the value for all matter by (8.2 ± 7.4)%.O'Neil et al. (2021) measure a similarly negligible bias between values of R spl for galaxies and the entire matter distribution.
The consistency between the dynamical radii, R infl and R σ v , and the splashback radius, R n g spl , provides an enhanced physical view of R n g spl as the inner boundary of the infall region where galaxies are infalling onto the cluster core.Currently, A82, page 8 of 9 R n g spl provides a direct observational route to this physical cluster boundary.In the future, observations that include accurate independent distance measures for galaxies around clusters and their filamentary structures will allow observational determination of the velocity field for infalling galaxies (Odekon et al. 2022).Some limited observations are available in the nearby Universe; they should become more broadly available at greater and greater depths in the Universe.Thus observational proxies for the dynamical radii R infl and R σ v should eventually complement determination of R spl from the cluster density profile.

Fig. 1 .
Fig. 1.Relation between M 200c and comoving R 200c for IllustrisTNG clusters.Left panel: Blue, orange, green, and red points show the M 200c − R 200c relation for clusters in four redshift bins z = 0.01, 0.31, 0.62, and z = 1.04, respectively.Colored squares with error bars show the median and interquartile range in each redshift bin.Right panel: Mass distribution histograms, with areas normalized to unity.

Fig. 2 .Fig. 3 .
Fig.2.Average radial velocity profiles and dynamical radii.Solid curves show the average galaxy radial velocity profiles of clusters at three redshifts: z = 0.01, 0.62, and 1.04, from left to right, respectively.Dashed curves show the Savitzky-Golay(Savitzky & Golay 1964) smoothed profiles.Shaded bands show the error in the smoothed profiles.In each panel, the red, blue, and orange lines show the turnaround, the minimum radial velocity, and the inflection point radii, respectively, derived from the average radial velocity profiles (see Sect. 3).
al. 2021; Fong & Han 2021; Gao et al. 2023), and derivations based on the intracluster gas from hydrodynamical simulations (Vallés-Pérez et al. 2020).Fong & Han (2021) and Gao et al. (2023) identify R v min with the inner depletion radius, the radius where the mass flow is maximum.Pizzardo et al. (2023b) also derive R v min from galaxy velocities of clusters from the IllustrisTNG simulations for redshifts 0.01 ≤ z ≤ 1.04.

Fig. 4 .
Fig. 4. Splashback radius from n g .The curve shows the smoothed logarithmic slope of the average local galaxy number density of the 282 clusters at z = 0.01.The orange vertical line shows the minimum, R ng spl .
n g spl we derived are consistent with recent results of O'Neil et al. (2021) who also computed R n g spl with TNG300-1.

Fig. 5 .
Fig. 5. Dynamical radii from cluster galaxy velocity in the splashback region.Upper panel: The solid orange, dash-dotted green, and dotted black lines show R infl , R σv , and R ng spl , respectively.The shadowed areas show the associated standard deviations.Bottom panel: The solid orange and dash-dotted green lines show the ratio between R infl and R ng spl , and between R σv and R ng spl , respectively.The dotted line shows equality.The orange, green, and grey shadowed areas show the 1σ confidence levels for each ratio with the corresponding color.

Fig. 6 .
Fig.6.Radial derivative of the smoothed v rad (r), and R infl .The blue, orange, green, and red curves show the smoothed dv rad dr profiles for the four redshifts z = 0.01, 0.31, 0.62, and 1.04, respectively.The vertical lines show the minimum R infl correspondingly to each colored curve.

Fig. 7 .
Fig.7.Average galaxy velocity dispersion as a function of clustercentric radius.The blue, orange, green, and red solid curves show the dispersion at four redshifts z = 0.01, 0.31, 0.62, and 1.04, respectively.The dashed curves show the smoothing of the correspondingly colored solid curves.
. IllustrisTNG enables measurement of the possible bias in the values of the A82