A Bound Violation on the Galaxy Group Scale: The Turn-Around Radius of NGC 5353/4
aa r X i v : . [ a s t r o - ph . C O ] D ec A BOUND VIOLATION ON THE GALAXY GROUP SCALE:THE TURN-AROUND RADIUS OF NGC 5353/4
Jounghun Lee , Suk Kim , and Soo-Chang Rey ABSTRACT
The first observational evidence for the violation of the maximum turn-aroundradius on the galaxy group scale is presented. The NGC 5353/4 group is chosenas an ideal target for our investigation of the bound-violation because of itsproximity, low-density environment, optimal mass scale, and existence of a nearbythin straight filament. Using the observational data on the line-of-sight velocitiesand three-dimensional distances of the filament galaxies located in the bound zoneof the NGC 5353/4 group, we construct their radial velocity profile as a functionof separation distance from the group center and then compare it to the analyticformula obtained empirically by Falco et al. (2014) to find the best-fit value ofan adjustable parameter with the help of the maximum likelihood method. Theturn-around radius of NGC 5353/4 is determined to be the separation distancewhere the adjusted analytic formula for the radial velocity profile yields zero.The estimated turn-around radius of NGC 5353/4 turns out to substantiallyexceed the upper limit predicted by the spherical model based on the ΛCDMcosmology. Even when the restrictive condition of spherical symmetry is released,the estimated value is found to be only marginally consistent with the ΛCDMexpectation.
Subject headings: cosmology — large scale structure of universe
1. INTRODUCTION
After the epoch of recombination, the growth of an overdense region in the primordialmatter density field would be driven by the competition between its self-gravity and the Astronomy Program, Department of Physics and Astronomy, Seoul National University, Seoul 151-742,Korea [email protected] Department of Astronomy and Space Science, Chungnam National University, Daejeon 305-764, Korea r t , max , on the turn-around radius that a bound object with mass M can havein a flat ΛCDM (cosmological constant Λ and Cold Dark Matter) universe: r t , max = A (cid:18) M G Λ c (cid:19) / , (1)where A is a proportionality factor whose value depends on the geometrical shape of an initialoverdense region from which the bound object originates. Only, provided that the shapeof an overdense region possesses a perfect spherical symmetry, the factor A equals unity,being independent of M . For the realistic case of a non-spherical region whose gravitationalgrowth tends to proceed quite anisotropically (e.g., Zel’dovich 1970; Bond & Myers 1996), A slightly exceeds unity (roughly 1 .
3) taking on the mass dependence (see Figure 3 in 3 –Pavlidou & Tomaras 2014). From here on, the spherical bound ( non-spherical bound ) refersto r t , max with A = 1 ( A >
1) in Equation (1).Pavlidou & Tomaras (2014) suggested that the ”zero-velocity surface” around a boundobject should be a good approximation of the true turn-around radius of the initial over-dense site at which the object formed. Here the zero-velocity surface around a bound ob-ject is defined as the radial distance from the object center at which the peculiar motionsgenerated by the object’s gravity become equal in magnitude to the Hubble expansion.Pavlidou & Tomaras (2014) claimed that a violation of the spherical (non-spherical) boundgiven in Equation (1) by any bound object observed in the universe would contest the stan-dard ΛCDM cosmology as a smoldering (smoking) gun counter-evidence, urging explorationof such bound-violating objects on the galaxy group and cluster scales. Unfortunately, how-ever, the observational estimate of the turn-around radius of a galaxy group/cluster requiresus to fulfill a tough mission: accurately measuring the peculiar velocities of the neighborgalaxies located in the bound-zone of the galaxy group/cluster (see Section 2 for the defini-tion of a bound-zone) .Here, we develop a novel methodology to estimate the turn-around radius of a galaxygroup/cluster without directly measuring the peculiar velocities. It utilizes the universalformula for the radial velocity profile empirically derived by Falco et al. (2014) to estimatethe turn-around distances where the total radial velocities of the bound-zone galaxies van-ish. We apply this new methodology to the NGC 5353/4 group, a galaxy group with mass3 × M ⊙ , at a distance of 34 . m = 0 . , Ω Λ = 0 . , h = 0 .
2. THEORETICAL PREDICTION
The regions outside of the virial radius, r v , of an isolated massive object in the expandinguniverse can be divided into three distinct zones. The nearest to the object is the infall zonewhere the object’s gravity defeats the cosmic expansion. If a lower-mass object is locatedin this zone, it is expected to undergo infall into the potential well of the massive objectto eventually become its satellite (Zu et al. 2014). The farthest from the object is theHubble zone where the cosmic expansion wins over the object’s gravity. If a smaller objectis located in this zone, then its effective motion would be just the Hubble expansion sincethe gravitational influence from the massive object should be negligible. The in-between is 4 –the bound zone where the object’s gravity is not dominant but strong enough to slow downthe Hubble flow. The three zones (the infall, the bound, and the Hubble zones) correspondroughly to the following ranges of the separation distances, r , from the object center: r ≤ r v ,3 r v ≤ r < r v and r > r v , respectively (Pavlidou & Tomaras 2014).Falco et al. (2014) has discovered the existence of the following universal profile of thebound-zone peculiar velocities: v p · ˆ r = − V c (cid:18) rr v (cid:19) − n v . (2)Here v p is the peculiar velocity of a test particle at a separation distance of r from a massiveobject with virial mass M v , ˆ r is the unit vector in the radial direction from the center ofthe object to the position of a test particle, r v is the virial radius of the object related to itsvirial mass as M v = 4 π ∆ c r v / c is 93 . ρ crit , , and V c is thecentral velocity of a test particle at r v given as V c ≈ ( GM v /r v ) / . The negative sign in theright-hand side (RHS) of Equation (2) reflects that the object’s gravity is in the direction of − ˆ r . The universality of the above profile is manifested by the independence of the power-law index n v from the masses and redshifts of the objects. Analyzing the numerical dataaround the cluster halos identified from high-resolution N -body simulations for a flat ΛCDMcosmology, Falco et al. (2014) have demonstrated that Equation (2) with a constant power-law index of n v ≈ .
42 approximates the peculiar velocity profile of dark matter particlesin the bound zones well, no matter what masses and redshifts the cluster halos have. Theirnumerical result implied that the expectation value of the peculiar velocity at any point ofthe bound zone around a cluster could be theoretically evaluated.It has also been found by Falco et al. (2014) that the matter-to-halo bias does not alterthe functional form of Equation (2) by demonstrating that it validly approximates not onlythe peculiar velocity profile of dark matter particles, but also that of the galactic haloslocated in the bound zone. This numerical finding has elevated the practicality of Equation(2) since what is directly observable is not the positions of dark matter particles but thoseof the bound-zone galaxies. Here, we claim that the very existence of the universal peculiarvelocity profile in the bound zone of a massive object (like a galaxy group and cluster) shouldenable us to estimate its turn-around radius without directly measuring the peculiar velocityfield in the bound-zone.Note that the bound-zone of a galaxy cluster is similar to the linear regime when the ini-tial proto-cluster evolved until the turn-around moment. Accordingly, the peculiar velocitiesof the bound-zone galaxies predicted by Equation (2) can be treated as the linear perturba-tions to the zero mean and thus must be a good approximation to the peculiar velocity of the 5 –initial proto-cluster region. Now, extrapolating Equation (2) to the turn-around moment,we determine the value of r t as the separation distance at which the following equality holdstrue. H r t = V c (cid:18) r t r v (cid:19) − n v . (3)The left-hand side (LHS) of Equation (3) represents nothing but the Hubble expansion whilethe RHS is the peculiar velocity of an initial proto-cluster region in the radial direction atthe turn-around radius r t approximated by the predictable peculiar velocity of a bound-zonegalaxy in the radial direction at the same distance r t .To solve Equation (3) in practice, we consider the total radial velocity profile, v r ( r ),defined as the combination of the Hubble expansion with the peculiar velocity in the radialdirection: v r ( r ) = H r − V c (cid:18) rr v (cid:19) − n v , (4)and then look for the range of r where the function, v r ( r ), crosses the zero line. It shouldbe emphasized that finding a solution to v r ( r t ) = 0 with Equation (4) requires informationonly on the radial positions of the bound-zone galaxies but not on their peculiar velocities.We plot Equation (4) for the case of M v = 3 × M ⊙ (the virial mass of NGC 5353/4,see Tully (2015)) as the gray region in Figure 1. Given the numerical result of n v = 0 . ± . V c = (0 . ± . GM v /r v ) / obtained by Falco et al. (2014), we let the power-law index n v and the multiplicative constant a ≡ V c ( GM v /r v ) − / vary in the ranges of 2 . ≤ n v ≤ . . ≤ a ≤ .
0, respectively, to plot Equation (4). The section of the dotted line( v r = 0) inside the gray region represents the expected range of the turn-around radius ofa galaxy group with M v = 3 × M ⊙ for a flat ΛCDM cosmology. The blue and greensolid lines indicate the locations of the spherical and non-spherical bounds, respectively.The comparison of the estimated range of r t with the spherical and non-spherical bounds inFigure 1 leads us to answer the question of whether it is possible to find a bound-violatinggalaxy group with mass M v = 3 × M ⊙ in a flat ΛCDM universe. As can be seen, thecrossing between the blue (green) solid and the black dotted lines occurs inside (outside) thegray region. This result indicates that although it is not impossible for such a galaxy groupto violate the spherical bound in a flat ΛCDM model, the violation of the non-sphericalbound would rarely occur on that mass scale.To see whether this critical prediction depends on the mass scale, we vary the values of M v in Equation (4) and find the turn-around radius r t as a function of M v , which is shownin Figure 2 as gray region. The blue and green solid lines correspond to the spherical andnon-spherical bounds as a function of M v , respectively. As can be seen, the blue (green)solid line is inside (outside) the gray region at all mass scales. Thus, the ΛCDM prediction 6 –against the bound-violation is extended to all mass scales: it is quite unlikely to find abound object whose turn-around radius exceeds the non-spherical bound in the standardflat ΛCDM cosmology, no matter what mass scale the object has.
3. NGC 5353/4: A BOUND-VIOLATING STRUCTURE3.1. Radial Velocity Profile of the Filament Galaxies around NGC 5353/4
Pavlidou & Tomaras (2014) suggested that the optimal target for the investigation of thebound violation should be a ”nearby galaxy group located in the low-density environment.”The proximity condition is necessary to minimize the observational uncertainties. A low-density environment around a target is required to minimize the difference between the virialand the turn-around masses. Note that M in Equation (1) represents the turn-around massenclosed by the turn-around radius r t which is expected to be larger than the virial mass M v (Pavlidou & Tomaras 2014). The turn-around mass is often approximated by the virialmass since the former is unmeasurable while a variety of methods has been developed tomeasure the latter. This approximation, however, works well for the case that a given targetis located in the low-density environment.The reason for the preference of the groups to the clusters lies in the hierarchical natureof structure formation process: the galaxy groups are more relaxed systems than the galaxyclusters since the former must have formed earlier than the latter. Thus, the systematicerrors produced by the deviation of the true dynamical state from complete relaxation wouldcontaminate the measurements of the virial masses of the galaxy groups less severely thanthose of the galaxy clusters.Here, we require one more condition in addition to the above three for an optimaltarget: the presence of a thin straight filament in the bound-zone. According to Falco et al.(2014), the total radial velocities and three-dimensional positions of the galaxies could bereadily inferred from the observable line-of-sight velocities and the two-dimensional projectedpositions, if the bound-zone galaxies are located along one-dimensional filament. If thegalaxies are distributed along one-dimensional filaments, they exhibit coherent motions alongthe filaments, which in turn makes it easier to judge whether or not the observed galaxiesare located in the bound-zone of a target (see also S. Kim et al. 2015, in preparation). Inour previous work (Lee et al. 2015), which reconstructed the radial velocity profile of theVirgo cluster and compared it to Equation (4), it was confirmed that the presence of a thinstraight filament is a key ingredient for the accurate reconstruction of the radial velocityprofile of the bound-zone galaxies. 7 –We find that the NGC 5353/4 group meets all of the above four conditions. It isonly 34 . x c ) is measured by using available information on its equato-rial coordinates and comoving distance. To estimate the virial mass, M v , of NGC 5353/4,Tully & Trentham (2008) used two distinct methods: one was based on the measurements ofthe velocity dispersions of the NGC 5353/4 satellites and the other employed the projectedmass estimator given by Heisler et al. (1985). They took the average over the two estimatesto find M v = 2 . × M ⊙ . Recently, however, Tully (2015) has updated M v to a slightlyhigher value, 3 × M ⊙ . The corresponding virial radius of NGC 5353/4 is determinedto be r v = 1 . r v = [3 M v / (4 π ∆ c )] / . Adopting the conserva-tive definition of Falco et al. (2014), we confine the bound-zone of the NGC 5353/4 groupto the region enclosed by a spherical shell whose inner and outer radii equal 3 r v and 8 r v ,respectively.S. Kim et al. (2015, in preparation) recently detected around the NGC 5353/4 group athin straight filament, which is mainly composed of dwarf galaxies with B-band magnitudes12 . ≤ m B ≤ .
54 in the redshift range of 0 . ≤ z ≤ . .The three-dimensional positions, x , of the 17 filament galaxies are determined and theirseparation displacement vectors r from the group center are calculated as r ≡ x − x c . Then,we select only those among the 17 filament galaxies whose separation distances, r ≡ | r | ,satisfy the bound-zone condition of 3 ≤ r/r v ≤
8. From here on, the filament galaxies, whichhave their comoving distances measured and belong to the bound-zone of NGC 5353/4, arereferred to as the bound-zone filament galaxies of NGC 5353/4. A total of four bound-zonefilament galaxies is selected.The angle, β , at which the radial direction, ˆ r ≡ r /r , of each bound-zone filament galaxyis inclined to the line-of-sight direction of the group center, ˆ x c ≡ x c /x c , can be determinedas cos β = ˆ r · ˆ x c . Now, the radial velocity of each bound-zone filament galaxy in unita of km https://ned.ipac.caltech.edu − can be evaluated from the measurable inclination angle β and the redshift difference ∆ z between the center of the NGC 5353/4 and its bound-zone filament galaxies, v r ( r ) = c ∆ z cos β , (5)where c ∆ z is basically the line-of-sight velocity of a bound-zone filament galaxy relative tothe NGC 5353/4 center. Λ CDM Cosmology with NGC 5353/4
In Section 3.1 we have measured the radial velocities of the bound-zone filament galaxiesof NGC 5353/4 from observations. We are now ready to adjust Equation (4) to this observa-tional result and to eventually estimate the turn-around radius of NGC 5353/4 by equatingthe RHS of Equation (4) to zero. The adjustable parameter is nothing but the power-lawindex, n v , in the RHS of Equation (4). As mentioned in Section 2, Falco et al. (2014) found n v = 0 . ± .
16 from their numerical experiment for the standard flat ΛCDM cosmology.Without ruling out the possibility that the true universe deviates from the standard flatΛCDM cosmology, it is not unreasonable for us to suspect that the value of n v might devi-ate from the estimate of Falco et al. (2014). Moreover, our previous work has found that,although the functional form of Equation (4) itself describes well the reconstructed radialvelocity profile of the Virgo cluster from observational data, the best agreement is reachedwhen the power-law index n v has a lower (negative) value than 0 .
42 (Lee et al. 2015).Varying the power-law index of n v , we fit the RHS of Equation (4) to the measuredradial velocities of the bound-zone filament galaxies from information on β and c ∆ z (seeEq.(5)). Then, we search for the best-fit value of n v that maximizes the following likelihoodfunction: p ( n v | M v , r i , z i , β i ) ∝ exp (cid:18) − χ ν (cid:19) , (6)where χ is the reduced chi-square given as χ ν ( n v | M v , r i , z i , β i ) = 1 ν N g X i =1 (cid:2) v ob ( z i , β i ) − v th ( r i ; M v , n v ) (cid:3) . (7)Here N g denotes the number of the bound-zone filament galaxies, v ob ( r i ) ≡ c ∆ z i / cos β i rep-resents the observational radial velocity of the i th bound-zone filament galaxy with redshiftdifference ∆ z i and inclination angle β i estimated by Equation (5), and v th ( r i ; n v ) is the the-oretical radial velocity at a separation distance r i predicted by Equation (4). Note that thedegree of freedom ν equals N g − n v . 9 –Normalizing p ( n v ) to satisfy R dn v p ( n v ) = 1, we plot it as a black solid line in Figure3. As can be seen, the likelihood function p ( n v ) reaches its sharp peak at n v , p = − .
13. Theevaluation of the upper and lower errors (say σ u and σ l ) associated with the best-fit value n v , p as Z n v , p + σ u n v , p dn ′ v p ( n ′ v | M v , r i , z i , β i ) = Z n v , p n v , p − σ l dn ′ v p ( n ′ v | M v , r i , z i , β i ) = 0 .
34 (8)yields σ u = 0 .
14 and σ l = 0 .
16, respectively.Now that the best-fit power-law index, n v , p , for Equation (4) has been determined, thecorresponding turn-around radius r t can be calculated by equating Equation (4) to zero.Figure 4 plots how the turn-around radius varies with the power-law index n v as a red solidline. The location of the best-fit value, n v , p , and the amount of the associated errors, σ u and σ l , obtained by means of the maximum likelihood method are shown as the black dashedline and gray regions, respectively. The blue and the green solid lines indicate the locationsof the spherical and nonspherical bounds, respectively. Projecting the section of the redsolid line overlapped with the gray region onto the vertical axis in Figure 4 allows us todetermine the range of the turn-around radius of NGC 5353/4. As can be seen, the best-fitvalue r t is one σ higher than the upper bound predicted by the ΛCDM model. Althoughthe signal of the bound-violation is not so statistically significant less than 2 σ , we believethat the NGC 5353/4 group is a strong candidate for the bound-violation on the group scalesince its best-fit turn-around radius turns out to exceed not only the spherical, but also thenon-spherical, upper bound.We also examine how robust the fitting result is against the change of the bound-zonerange from 3 ≤ r/r v ≤ ≤ r/r v ≤ ≤ r/r v ≤
10. The likelihood functions, p ( n v ) for these two cases of the bound zone range are shown as red dashed and blue dottedlines, respectively, in Figure 3. As can be seen, the change of the bound-zone range producesa thicker tail of p ( n v ) in the high n v section and moves the location of the peak, n v , p , toa slightly higher value. Figures 5 and 6 plot the same as Figure 4 but for the cases of3 ≤ r/r v ≤ ≤ r/r v ≤
10, respectively, which demonstrate that the change of thebound-zone range enlarges the errors of r t , but does not alleviate the bound-violation by theNGC 5353/4 group much. Table 1 lists the numbers of the bound-zone filament galaxies, thecorresponding best-fit power-law index and the resulting ranges of the turn-around radius r t for three different cases of the bound-zone limit. 10 –
4. SUMMARY AND DISCUSSION
Our work was inspired and urged by the seminal paper of Pavlidou & Tomaras (2014)which theoretically proved the existence of the maximum turn-around radii, r t , max , of massiveobjects and proposed it as an independent test of the ΛCDM model by showing its sensitivedependence on the density and equation of state of dark energy (see also Pavlidou et al.2014). In the current work, we have taken a step toward realizing the ingenious idea ofPavlidou & Tomaras (2014) by accomplishing the following. First, we have devised a newmethodology to infer the turn-around radii of galaxy groups/clusters without informationon the peculiar velocity field. Based on the numerical finding of Falco et al. (2014) that theradial velocity profile of dark matter particles in the bound zone of massive objects displaysuniversality, this new methodology employs the extrapolation of the universal profile to theturn-around moment at which the radial velocities would vanish.Second, we have applied this new methodology to the nearby galaxy group, NGC 5353/4,around which a thin straight filament composed mainly of the dwarf galaxies has beendetected (S. Kim et al. 2015, in preparation). Assuming that the radial velocity profileof the filament galaxies in the bound zone of NGC 5353/4 follows the universal form well,we have shown that having information on the spatial positions of the filament galaxiessuffices to determine the best-fit value of the characteristic parameter of the profile. Then,we have constrained the ranges of the turn-around radius, r t , of NGC 5353/4 by locating theseparation distance at which the radial velocity profile with the best-fit parameter crosses thezero line. The comparison of the constrained range of r t with the theoretical upper bound, r t , max , derived by Pavlidou & Tomaras (2014) has led us to detect a 2 σ signal of the violationof the spherical bound and a 1 . σ signal of the violation of the non-spherical bound. Weakas the signal may appear at first sight, our result is significant because it reports the firstcase of possible violation of the non-spherical bound on the galaxy group scale.All of the previously reported cases of the bound-violation occurred on the cluster orsupercluster scales but none on the group scale (see Figure 1 in Pavlidou & Tomaras 2014).Lack of evidence for the bound-violating cases on the galaxy group scale had been interpretedas an indication that the signals of the bound-violation detected on the cluster scale mightbe spurious ones generated by large uncertainties in the measurements of the virial massesof the clusters as well as the peculiar velocities of their bound-zone galaxies (Nasonova et al.2011; Karachentsev et al. 2014). Our result provides the first observational counter-evidenceagainst the ΛCDM prediction for r t , max on the galaxy group scale, which has two crucialimplications. First, the observed signals of the bound-violation on the cluster scale shouldnot be interpreted just as false ones but deserve thorough reinvestigation by making effortsto estimate the turn-around radii more accurately. 11 –Second, the turn-around radii of those galaxy groups, which were estimated and foundto match the ΛCDM prediction well in previous studies (Karachentsev & Kashibadze 2006;Karachentsev et al. 2007) may need to be reestimated without placing too much confidenceon the measurements of the peculiar velocities of their bound-zone galaxies. Since the galaxygroups have weaker self-gravity and a smaller number of bound-zone galaxies than the galaxyclusters, it is in fact delicately more difficult to infer their turn-around radii by directlymeasuring the peculiar velocities of the bound-zone galaxies. Our methodology has allowedus to overcome this difficulty, being capable of producing a robust result that is expected to beless severely contaminated by observational errors. Yet, as mentioned in Pavlidou & Tomaras(2014), given the inherent stochastic nature of the structure formation process, it requiresmore counter-evidence on the galaxy group scale to contest the ΛCDM cosmology with the r t values estimated by our methodology.We admit that our result may suffer from the poor-number statistics. Only four bound-zone filament galaxies have been used in our analysis to determine the best-fit value ofthe power-law index of the radial velocity profile around NGC 5353/4, simply because it isonly those four galaxies that already have their comoving distances independently measured. It would definitely be desirable to measure the comoving distances of more bound-zonefilament galaxies of NGC 5353/4 and then to reestimate the best-fit value of the power-law index of the radial velocity profile by using them all, which is, however, beyond thescope of this paper. We also admit that not all possible errors including hidden systematicshave been taken into account to derive our results. For instance, if the uncertainties in themeasurements of the virial mass of NGC 5353/4 and the separation distances to the bound-zone filament galaxies were known and accounted for, the errors in the final estimate of r t would become larger, which might in turn alleviate the tension with the ΛCDM predictionfor r t , max . Furthermore, although the NGC 5353/4 group appears to be located in the low-density environment (Tully 2015), non-negligible difference between the virial and the turn-around masses of NGC 5353/4 may exist and could produce other uncertainties regardingthe estimate of its turn-around radius.Recently, Faraoni (2015) studied how the deviation of gravitational law from the Gen-eral Relativity would affect the value of the maximum turn-around radii and derived ageneral expression for r t , max in modified gravity (MG) models from the first principles. Ourmethodology will be useful to put an independent constraint on the MG models by effi-ciently estimating the turn-around radii of several nearby galaxy groups and comparing theestimates with the MG predictions given by Faraoni (2015). However, some numerical workhas to precede such a test with our methodology. Strictly speaking, the functional formof the radial velocity profile (Eq.(4)) empirically drawn by Falco et al. (2014) from N -bodysimulations is applicable only for a flat ΛCDM cosmology and it is not guaranteed that the 12 –same functional form can be applied for the cases of MG models. It will be necessary toinvestigate if and how not only the value of the power-law index but also the functional formitself of the radial velocity profile of the bound-zone galaxies differs from Equation (4) inMG models. We plan to work on this project as well as on testing the robustness of thecurrent result by using more improved data sets. We hope to report the result elsewhere inthe near future.This work was supported by a research grant from the National Research Foundation(NRF) of Korea to the Center for Galaxy Evolution Research (NO. 2010-0027910). J.L. alsoacknowledges the financial support of the Basic Science Research Program through the NRFof Korea funded by the Ministry of Education (NO. 2013004372). S.C.R. acknowledges thesupport of the Basic Science Research Program through the NRF funded by the Ministryof Education, Science, and Technology (NRF-2015R1A2A2A01006828). S.K. acknowledgessupport from the National Junior Research Fellowship of NRF (No. 2011-0012618). 13 – REFERENCES
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This preprint was prepared with the AAS L A TEX macros v5.2.
15 –Fig. 1.— Radial velocity profile of the galaxies ( v r ) located in the bound zone of a galaxygroup with virial mass M v = 3 × M ⊙ (gray region) determined by Equation (4). Theturn-around radius ( r t ) of the galaxy group is in the range of the separation distances betweenthe bound-zone galaxies and the group center, r , where the gray region touches the dotted-line ( v r = 0). The blue and green solid lines indicate the locations of the spherical and thenon-spherical bounds predicted by the standard ΛCDM cosmology (Pavlidou & Tomaras2014). 16 –Fig. 2.— Variation of the turn-around radius ( r t ) with the virial mass of a bound object (grayregion) estimated by solving Equation (3). The spherical and non-spherical upper bounds, r t , max predicted by the ΛCDM model are plotted as blue and green solid lines, respectively. 17 –Fig. 3.— Posterior probability density distributions of the power-law index n v in the ana-lytic formula of Falco et al. (2014) fitted to the observed radial velocities of the bound-zonefilament galaxies around NGC 5353/4. The blue dotted, black solid, and red dashed linescorrespond to the cases in which the separation distances r of the bound-zone filament galax-ies from NGC 5353/4 are in the range of 3 ≤ r/r v ≤
7, 3 ≤ r/r v ≤
8, and 3 ≤ r/r v ≤ r v is the virial radius of NGC 5353/4. 18 –Fig. 4.— Variation of the turn-around radius with the power-law index n v of the radialvelocity profile in the bound zone of a galaxy group with mass M v = 3 × M ⊙ (red solidline). The dashed line and the gray region indicate the location of the best-fit n v and thewidth of the associated 1 σ scatter around the best-fit n v , respectively, which are estimatedby fitting the analytic formula of Falco et al. (2014) to the observed radial velocities of thebound-zone filament galaxies around NGC 5353/4 where the bound-zone range is given as3 ≤ r/r v ≤
8. The blue and green solid lines indicate the locations of the spherical andnon-spherical bounds (i.e., the upper limit of the turn-around radius, r t , max , predicted bythe standard ΛCDM model, respectively. 19 –Fig. 5.— Same as Figure 4, but for the case in which the bound-zone range is given as3 ≤ r/r v ≤
7. 20 –Fig. 6.— Same as Figure 4, but for the case in which the bound-zone range is given as3 ≤ r/r v ≤
10. 21 –Table 1. Ratios of the separation distances ( r ) of the bound-zone filament galaxies to thevirial radius ( r v ) of NGC 5353/4, the number of the bound-zone filament galaxies ( N g ),and the best-fit value of the power-law index of the radial velocity profile ( n v ) and theestimated range of the turn-around radius of NGC 5353/4 ( r t ). r/r v N g n v r t (Mpc)[3 ,
7] 3 − . +0 . − . [4 . , . ,
8] 4 − . +0 . − . [4 . , . ,
10] 5 − . +0 . − . [4 . , ..