arXiv:1006.0555v1 [astro-ph.CO] 3 Jun 2010 aetedsac oVrot e1 p n h Hubble the and Mpc 17 be to Virgo illustrate, to velocity To constant by cluster. distance peculiar (reviewed Virgo the the the Galaxy towards of take Group discovery our Local the the of to of motion led 1988) the Huchra of studies Early Introduction 1. Vroifl” iw h nvra xaso sexisting scales as cosmological” expansion ”truly universal on only the not views infall”) Virgo. ”Virgo towards km/s pe- observed our 220 reflects The about difference of km/s. The velocity 1224 km/s. culiar 1000 is about cluster is the velocity of velocity logical loo oa clsof scales local on also rmtecutrcne oaot200k/ ttedistances the at Mpc km/s 5-8 000 2 around about zero to center that nearly cluster from show the cluster. from range 2010) Virgo velocities Nasonova flow the & the from known (Karachentsev away is Galaxy data receeding (our what Recent galaxies are of of them) feature hundreds among particular flow: Virgocentric a the is as infall Virgo The flow Virgocentric The 1.1. (1977), 2000). (1999, Peebles al. et et 1977), Teerikorpi Ekholm (1986), and (1974, Sandage (1992), Silk infall (1982), al. Salpeter Virgo by & the e.g. Hoffman of studied aspects were theoretical and Observational .D Chernin D. A. srnm Astrophysics & Astronomy oebr3 2018 3, November hsetmt ftertre xaso teso-called (the expansion retarded the of estimate This rudi,det h olna rvt-nirvt interp words. gravity-antigravity Key nonlinear the quasi-st to a clusters: due and it, groups around for zero-gravit universal the be may 10; diagra pattern about velocity-distance of the factor Conclusions. In scaling a outflow. distances with its structure at and rate Group fl Hubble Virgocentric Local global the the controls and antigravity linearity the while cluster, 1 2 ocaiyti usin efcsnwo h ig lse an Cluster Virgo the on now focus we Methods. question, this clarify to akeeg akrud ersne yEnti’ cosmolo Einstein’s Aims. by represented background, energy dark 3 ehnc.Tekyprmtri h eogaiyrdu,th radius, zero-gravity the is whi Results. parameter model key analytical nonlinear The a mechanics. using environment, its and h ieaia tutr ftesse n otosisevo its controls and system the of structure kinematical the Context. Accepted / Received 4 uraOsraoy eateto hsc n srnm,U Astronomy, and Physics of Department Observatory, Tuorla trbr srnmclIsiue ocwUiest,Mos University, Moscow Institute, Astronomical Sternberg pca srpyia bevtr,NzniAky,36916 Arkhys, Nizhnii Observatory, Astrophysical Special nvriyo lbm,Tsaos,A 58-34 USA 35487-0324, AL Tuscaloosa, Alabama, of University H a akeeg aesrn yaia ffcso ml cosmic small on effects dynamical strong have energy dark Can 0 .Teitrlybtentegaiyo h lse n h an the and cluster the of gravity the between interplay The 1. h tnadΛD omlgclmdlipista l celes all that implies model cosmological ΛCDM standard The 2k//p,te h xetdcosmo- expected the then km/s/Mpc, 72 = eitrrtteHbl iga,fo e aaaeo veloc of database new a from diagram, Hubble the interpret We akeeg oiaini h igcnrcflow Virgocentric the in domination energy Dark aais oa ru,Vrocutr omlg:dr matt dark Cosmology: cluster, Virgo Group, Local galaxies: h hs n yaia iiaiyo h ytm ntescal the on systems the of similarity dynamical and phase The 1 , 2 .D Karachentsev D. I. , ∼ 0Mcol ekydisturbed. weakly only Mpc 10 aucitn.Virgo˙AA˙revised no. manuscript .M Domozhilova M. L. ∼ 3 ..Nasonova O.G. , 00Mc but Mpc, 1000 ∼ > 5Mc .Tecutradteflwfr ytmsmlrt the to similar system a form flow the and cluster The 3. Mpc. 15 ABSTRACT a ihtedr nrydmntn nteflwcomponent. flow the in dominating energy dark the with lay w rnigodradrglrt oteflw hc reaches which flow, the to regularity and order bringing ow, ia osat n xeineisrplieatgaiya antigravity repulsive its experience and constant, gical toaybudcnrlcmoetada xadn outflow expanding an and component central bound ationary aisfrtecutrsse sas 0tmslarger. times 10 also is system cluster the for radius y uin .Tegaiydmntsteqaisainr boun quasi-stationary the dominates gravity The 2. lution. 2 hicroae h nirvt oc ntrso Newtonian of terms in force antigravity the incorporates ch itnea hc rvt n nirvt r nbalance. in are antigravity and gravity which at distance e ,Russia 7, 3 n .G Byrd G. G. and , faot3 p.For Mpc. 30 about of o e ntelmyevrneta distances should at it environment where lumpy the Hubble in by be: discovered not cosmolog- was linear expansion the ical But smooth, a medium. of self-gravitating expansion uniform the Originally, described 2005; constant 2005). theory Karachentsev Friedmann Whiting Hubble 2003; the expansion 2003b; al. global 2002, of et al. the et Thim rate proceeds Karachentsev to 2001; identical, al. expansion the et not Moreover, local (Ekholm way. if the similar, regular that saw is quite fact He a 1999). the (1986, in in Sandage by mystery realized a paradox Mpc. a 30 of say, ple of, distance the cluster to the up around nonuniform distribution highly is matter the 3) overdensity; rpryo h nvrebyn h omc”elo unifor- of ”cell cosmic the ( beyond mity” universe flow, the expansion not of distances, property linear smaller the at 1) exist devia- relation surprisingly: strong linear But making the expansion. from flow, Hubble tions the global in the seen resemble it be may relation velocity-distance elutre l 99 ngoa cls esuggested 2001, al. we et scales, Baryshev 2001, global Chernin 1998; on 2000, al. al. 1999) et et (Riess al. (Chernin energy dark et of discovery Perlmutter the after Soon energy dark of relevance local The 1.2. o,199,Russia 119899, cow, .Teerikorpi P. , ,tecutrflwsrcuerpoue h group-flow the reproduces structure cluster-flow the m, h o fepninaon it. around expansion of flow the d eie h aua eitos ehv eea exam- an here have we deviations, natural the Besides iest fTru 10 Piikki¨o, Finland 21500 Turku, of niversity irvt ftedr nrybcgon determines background energy dark the of tigravity ≥ clsa ela lbly otnigorefforts our Continuing globally? as well as scales ilbde r meddi efcl uniform perfectly a in embedded are bodies tial 0 p) )teVrocutri toglocal strong a is cluster Virgo the 2) Mpc); 300 r akenergy dark er, so -0Mcsget htatwo-component a that suggests Mpc 1-30 of es te n itne fglxe ntecluster the in galaxies of distances and ities 1 .J Valtonen J. M. , 4 > R 10 − 5Mc ogl linear roughly a Mpc, 15 1 .P Dolgachev P. V. , V
= c < HR S 2018 ESO ction. 0Mpc. 20 sthe is , d 2 , 2 A.D. Chernin et al.: Dark energy in the Virgocentric flow
Karachentsev et al. 2003a, Teerikorpi et al. 2005, Chernin et al. 2004) that dark energy with its omnipresent and uni- form density (represented by Einstein’s cosmological con- stant) may provide the dynamical background for a regular quiescent expansion on local scales, resolving the Hubble- Sandage paradox. Our key argument came from the fact that the antigravity produced by dark energy is stronger than the gravity of the Local Group at distances larger than about 1.5 Mpc from the group center.1 The present paper extends our work from groups to the scale of clusters, focusing on the Virgo cluster.
2. The phase space structure Recent observations of the Virgo cluster and its vicinity permit a better understanding of the physics behind the visible structure and kinematics of the system. Fig. 1. The Hubble diagram with Virgocentric velocities 2.1. The dataset and distances for 761 galaxies of the Virgo Cluster and The most complete list of data on the Virgo Cluster and the outflow (Karachentsev & Nasonova 2010). The two- the Virgocentric flow has been collected by Karachentsev & component pattern is outlined by bold dashed lines. The Nasonova (2010). These include distance moduli of galax- zero-gravity radius RZG ∼ 10 Mpc is located in the zone ies from the Catalogue of the Neighbouring Galaxies (= between the components. The broken line is the running CNG, Karachentsev et al. 2005) and also from the litera- median used by K & N to find the zero-velocity radius ture with the best measurements prefered. Distances from R0 = 6 − 7 Mpc. Two lines from the origo indicate the the Tip of the Red Giant Branch (TRGB) and the Cepheids Hubble ratios H = 72 km/s/Mpc and HV = 65 km/s/Mpc. are used from the CNG together with new TRGB dis- Two curves show special trajectories, corresponding to the tances (Karachentsev et al. 2006, Tully et al. 2006, Mei parabolic motion (E = 0; the upper one) and to the min- et al. 2007). For galaxy images in two or more photomet- imal escape velocity from the cluster potential well (the ric bands obtained with WFPC2 or ACS cameras at the lower one). The flow galaxies with the most accurate dis- HST, the TRGB method yields distances with an accuracy tances and velocities (big dots) occupy the area between of about 7% (Rizzi et al. 2007). The database includes also these curves. data on 300 E and S0 galaxies from the Surface Brightness Fluctuation (SBF) method by Tonry et al. (2000) with a typical distance error of 12%. The total sample contains the velocities and distances of 1371 galaxies within 30 Mpc so that R0/RVirgo ≈ 0.45 or R0 ≈ 7.4 Mpc. The work by from the Virgo cluster center. Especially interesting is the Karachentsev & Nasonova (2010) puts the zero-velocity ra- sample of 761 galaxies selected to avoid the effect of un- dius at R0 = 5.0 − 7.5 Mpc. For R < R0, positive and known tangential (to the line of sight) velocity components. negative velocities appear in practically equal numbers; for The velocity-distance diagram for this sample taken from R > R0, the velocities are positive with a few exceptions Karachentsev & Nasonova (2010) is given in Fig.1. likely due to errors in distances. The zero-velocity radius gives the upper limit of the size of the gravitationally bound cluster, and the diagram shows that the Virgocentric flow 2.2. The zero-velocity radius and the cluster mass starts at R ≥ R0 and extends at least up to 30 Mpc. The zero-velocity radius within the retarded expansion field The zero-velocity radius has been often used for estimat- around a point-like mass concentration means the distance ing the total mass M0 of a gravitationally bound system. where the radial velocity relative to the concentration is According to Lynden-Bell (1981) and Sandage (1986), the zero. In the ideal case of the mass concentration at rest spherical model with Λ = 0 leads to the estimator within the expanding Friedmann universe this is the dis- tance where the radial peculiar velocity towards the con- 2 −2 3 centration is equal to the Hubble velocity for the same dis- M0 = (π /8G)tU R0. (1) tance. Using Tully-Fisher distances in the Hubble diagram,
Teerikorpi et al. (1992) could for the first time see the lo- With the age of the universe tU = 13.7 Gyr (Spergel et cation of the zero-velocity radius R0 for the Virgo system, al. 2007), Karachentsev & Nasonova (2010) find the Virgo 14 1 cluster mass M0 = (6.3 ± 2.0) × 10 M⊙. This result agrees The increasing observational evidence and theoretical con- 14 with the virial mass Mvir = 6 × 10 M⊙ estimated by siderations favoring this view have been discussed by Maccio 14 Hoffman & Salpeter (1982) and ∼ 4 × 10 M⊙ of Valtonen et al. (2005), Sandage (2006), Teerikorpi et al. (2006, 2008), Chernin et al. (2006, 2007a,b,c), Byrd et al. (2007), Valtonen et al. (1985) and Saarinen & Valtonen (1985). Teerikorpi et al. (2008), Balaguera-Antolinez et al. (2007), Bambi (2007), et al. (1992) and Ekholm et al. (1999, 2000) found that the Chernin (2008), Niemi & Valtonen (2009), Guo & Shan (2009). real cluster mass might be from 1 to 2 the virial mass, or 15 For some counter-arguments to this new approach see Hoffman (0.6−1.2)×10 M⊙. Tully & Mohayaee (2004) derived the 15 et al. (2008) and Martinez-Valquero et al. (2009). mass 1.2 × 10 M⊙ with the ”numerical action” method. A.D. Chernin et al.: Dark energy in the Virgocentric flow 3
2.3. The Virgo infall Figure 1 shows clearly the Virgo infall: at distances R > R0, the galaxies are located mostly below the Hubble line VH = HR with H = 72 km/s/Mpc (Spergel et al. 2007). The deviation of the median velocity of the flow Vm from the Hubble velocity is well seen in the distance range 7 – 13 Mpc. At larger distances the retardation effect gets weaker and gradually sinks below the measurement error level. For the distance and peculiar velocity of the Milky Way, we take figures from Karachentsev & Nasonova (2010): the distance to the Virgo cluster 17 Mpc, the recession veloc- ity 1004 ± 70 km/s; with the Hubble constant H = 72 km/s/Mpc the regular expansion velocity is 1224 km/s. Then the peculiar velocity directed to the cluster center δV = 220 ± 70 km/s (this agrees with the result based on TF distances and a different analysis by Theureau et al. 1997; a complete velocity field on the scale of 10–20 Mpc is given by Tully et al. 2008.) The effect is not very strong. It is much more instructive, however, that there are big non- Fig. 2. The Hubble diagram for 57 galaxies of the Local linear deviations from the linear velocity-distance law near Group and the local flow (Karachetsev et al. 2009). The the cluster: the infall effect is strongest there. thin line indicates the zero-velocity radius R0 = 0.7 − 0.9 Mpc. The two-component phase structure is outlined by the bold dashed line. Zoomed in about 10 times, the group- 2.4. Two-component phase structure scale structure reproduces well the cluster-scale structure (Fig.1). The zero-gravity radius R is about 10 times The Virgo system reveals an obvious two-component struc- ZG ture in Fig.1: the cluster and the flow, outlined roughly smaller for the group than the same key quantity of the with bold dashed lines. As we noted, the positive and neg- cluster system. The similarity supports a universal cosmic two-component grand-design for groups and clusters. ative velocities (from -2000 to +1700 km/s) are seen in the cluster area in equal numbers, so the component is rather symmetrical relative to the horizontal line V = 0. The bor- factor of 10, would roughly reproduce the phase structure der zone between the components, in the range 6
2.5. Similarity with the Local system 3.1. Antigravity in Newtonian description The two-component phase structure of the Virgo sys- According to the simplest, most straightforward and quite tem looks very similar to that of the Local system (the likely view adopted in the ΛCDM cosmology, dark energy is Local Group and the outflow). The diagram of Fig.2 represented by Einstein’s cosmological constant Λ, so that 2 (Karachentsev et al. 2009) is based on the most complete ρV = (c /8πG)Λ. If so, then dark energy is the energy of and accurate data obtained with the TRGB method using the cosmic vacuum (Gliner 1965) and is described macro- the HST. The velocities within the Local Group range from scopically as a perfectly uniform fluid with the equation −150 to +160 km/s, about 10 times less than the velocity of state pV = −ρV (here pV is the dark energy pressure; spread in the Virgo cluster. The zone between the two com- c = 1). This interpretation implies that dark energy exists ponents of the Local system is in the range 0.7 − 1.2 Mpc, everywhere in space with the same density and pressure. also 10 times less than in Fig.1. The local flow has a dis- Local dynamical effects of dark energy have been persion of about 30 km/s around the line V = HLR, where studied by Chernin et al. (2000, 2009), Chernin (2001), HL = HV ≃ 60 km/s/Mpc. This line defines the median for Baryshev et al. (2001), Karachentsev et al. (2003), Byrd et the local flow. The dispersion is about 10 times less than al. (2007), and Teerikorpi et al. (2008). For the Local Group in the Vigocentric flow for the subsample of the accurate and the outflow close to it, we showed that the antigrav- distances. The local phase diagram of Fig.2, if zoomed by a ity produced by the dark energy background exceeds the 4 A.D. Chernin et al.: Dark energy in the Virgocentric flow gravity of the group at distances beyond ≃ 1.5 Mpc from produced by the cluster at the distances of the flow is still the group barycenter. Similar results were obtained for the isotropic. The model remains also nonlinear: no restriction nearby M81 and Cen A groups (Chernin et al. 2007a,b). on the strength of the dynamical effects is assumed. Extending now our approach to the cluster scale, we The mass-point model may seem overly simple, as it consider the Virgo system (the cluster and the flow) em- is known that the Virgo cluster contributes only 20 % of bedded in the uniform dark energy background. We de- the bright galaxies within the volume up to the distance of scribe the system – in the first approximation – in a frame- the Local Group (Tully 1982). However, Tully & Mohayaee work of a spherical model (Silk 1974, 1977; Peebles 1976) (2004) conclude from their numerical action method that where the cluster is considered as a spherical gravitationally the mass-to-luminosity ratio of the E-galaxy rich Virgo clus- bound quasi-stationary system. The galaxy distribution in ter must be several times higher (they give M/L ∼ 900) the flow is represented by a continuous dust-like (pressure- than in the surrounding region (∼ 125) in order to pro- less) matter. The flow is expanding, and it is assumed that duce the infall velocities observed close to Virgo (a quali- the concentric shells of matter do not intersect each other tatively similar result was found by Teerikorpi et al. 1992). in their motion (the mass within each shell keeps constant). Therefore, the model with most of the mass in and close to Since the velocities in the cluster and the flow are much the Virgo cluster could be a sufficient first approximation. less than the speed of light, we may analyze the dynamics In this simplified/generalized version, the mass M(r)= of the cluster-flow system in terms of Newtonian forces. M is constant in Eq. 4, which shows that antigravity and A spherical shell with a Lagrangian coordinate r and a gravity balance each other at the distance R = RZG, where Eulerian distance R(r, t) from the cluster center is affected 3M 1/3 by the gravitation of the cluster and the flow galaxies within RZG = [ ] (6) the sphere of the radius R(r, t). According to the Newtonian 8πρΛ gravity law, the shell experiences an acceleration is the zero-gravity radius (Chernin et al. 2000; Chernin GM(r) 2001; Dolgachev et al. 2003,2004). Gravity dominates at F (r, t)= − , (2) R < R , and antigravity is stronger than gravity at N R(r, t)2 ZG R > RZG. The zero-gravity radius is the major quantitative in the reference frame related to the cluster center. Here factor in the gravity-antigravity interplay in the system. M(r) is the total mass within the radius R(r, t). According to various estimates (Sect.2), the total mass of the Virgo cluster is 1 – 2 times the virial mass: M = The flow is affected also by the repulsion produced 15 by the local dark energy. The resulting acceleration in (0.6 − 1.2) × 10 M⊙. Then the known global ρΛ gives Newtonian terms is given by the ”Einstein antigravity law”: RZG = (9 − 11) Mpc. (7) 4π R(r, t)3 8π Karachentsev & Nasonova (2010) argue in favour of one F (r, t)=+G 2ρ 3 =+ Gρ R(r, t) (3) E Λ R(r, t)2 3 Λ virial mass for the cluster mass; if so the lower limit for RZG in Eq.(7) may be more realistic. Here −2ρΛ = ρΛ +3pΛ is the effective (General Relativity) Eqs.6,7 indicate that the zero-gravity radius RZG of gravitating density of dark energy (for details see, e.g., the Virgo system is not far from the zero-velocity radius Chernin 2001). It is negative, and therefore the accelera- R0 = 5.0 − 7.5 Mpc. In fact, one expects that RZG is tion is positive, speeding up the shell away from the center. slightly larger than R0 (Teerikorpi et al. 2008) in the stan- dard cosmology. The zero-velocity radius constrains the size 3.2. Zero-gravity radius in the point mass approximation of the cluster (Sec.2), and there also gravity dominates (R < R0 ∼< RZG), the necessary condition for a gravitation- The equation of motion of an individual shell has the form ally bound cluster. On the other hand, the Virgocentric flow is located beyond the distance RZG ≃ R0, hence the flow is GM(r) 8π R¨(r, t)= F + F = − + Gρ R(r, t). (4) dominated dynamically by the dark energy antigravity. N E R(r, t)2 3 Λ The zero-gravity radius of the Virgo system is about 10 times larger than that of the Local system for which RZG = Its integration leads to the mechanical energy conservation 1 − 1.3 Mpc (Chernin 2001; Chernin et al. 2000, 2009). We law for each matter shell: see here the same factor 10 as for the phase structures of the Virgo and Local systems (Sec.2). Thus the phase similarity 1 2 GM(r) 4π 2 V (r, t) = + GρΛR(r, t) + E(r). (5) is complemented by the dynamical similarity. 2 R(r, t) 3
Here V (r, t) = R˙ (r, t) is the Eulerian radial velocity of 3.3. Antigravity push the shell, and E(r) is its total mechanical energy (per unit mass). This is the basic equation which describes the out- Now we return to the Milky Way and estimate the net flow in our model. It is the Newtonian analogue of the LTB gravity-antigravity effect at its distance RMW = 17 Mpc spacetime of General Relativity (e.g., Gromov et al. 2001). from the cluster center. With RZG = 9 − 11 Mpc, we find A simpler model results, if we can assume that the clus- that here the antigravity force is stronger than the gravity: ter mass M0 makes the majority of the mass M(r), and FE/|FN|≃ 4 − 7, R = RMW. (8) the flow galaxies can be treated as test particles. Curiously enough, this assumption makes the model more general in As we see, the extra gravity produced by the Virgo cluster the sense that no spherical symmetry is required now for mass is actually overbalanced by the antigravity of the dark the galaxy distribution in the flow (not necessarily concen- energy located within the sphere of the radius RMW around tric shells around the cluster). However, the gravity force the cluster. Thus the Milky Way experiences not a weak A.D. Chernin et al.: Dark energy in the Virgocentric flow 5 extra pull (cf. Secs.1,2), but rather a strong push (in the of Sec.3 works well. Beyond the distance RES ≃ RMW, the Virgocentric reference frame) from the Virgo direction. model needs modification, but remains qualitatively correct The dark energy domination in the flow changes the up to the distances of 25-30 Mpc. Anyway since 2ρΛ > current dynamical situation drastically in comparison with ρM (t0), the antigravity of dark energy dominates in the the earlier models by Silk (1974, 1977) and Peebles (1976). entire area of the Virgocentric flow in Fig.1. At the periphery of the Virgocentric flow (at about 30 Mpc) the force ratio is impressively large: F /|F |≃ 20 − 37. E N 4.2. Two special trajectories
3.4. Antigravity domination Eq.5 shows that the flow has zero velocity at R = RZG ≃ R0, if the total energy of the shell or an individual galaxy The basic trend of the flow evolution is seen from Eq.5. With M(r)= M = const, it takes at large times the form: 3 GM E(r)= Emin = − . (12) 2 RZG 8π 1/2 V (r, t = ∞) = [ GρΛ] R. (9) 3 It is easy to see that Emin is the minimal energy needed for a particle to escape from the potential well of the cluster. In this limit, F /|F | → ∞, and the gravity of the cluster E N Eq.5 with E(r) = Emin gives the corresponding velocity is practically negligible compared to the dark energy anti- Vmin(R) of the flow at different distances in the flow: gravity. As a result, the flow acquires asymptotically the linear velocity-distance relation with the constant Hubble R R V = H R[1 + 2( ZG )3 − 3( ZG )2]1/2. (13) factor expressed via the dark energy density only: min Λ R R 8πG 1/2 At large distances (R >> RZG), the trajectories with HΛ = [ ρΛ] = 63 − 65 km/s/Mpc, (10) 3 E(r)= Emin approach the asymptotic given by Eq.11. Another special value of the mechanical energy is At a given moment of time (that of observation) t = t0 Eq.5 presents the spatial structure of the flow. At large distances E(r) = 0, corresponding to parabolic motion. In this case,
RZG V (r = ∞,t0) → HΛR(r, t0). (11) V = H R[1 + 2( )3]1/2. (14) par Λ R The flow trajectories converge to the asymptotic of Eq.11 independently, in general, of the ”boundary/initial condi- The two special trajectories with E(r)= Emin and E(r)= tions” near the cluster (see figure 2 in Teerikorpi & Chernin 0 (see Fig.1) converge to the asymptotic of Eq.11 at large 2010). In particular, strong perturbations at small distances distances – in agreement with the results of Sec.3. in the flow are compatible with its regularity at larger dis- It is quite interesting that the most accurate data on the tances. No tuning of the model parameters are required for flow (from the TRGB and Cepheids; dots in Fig.1) show a this – the only significant quantity is the zero-gravity radius low dispersion around the line V = HVR, and the cor- responding trajectories occupy the area just between the RZG determined by the cluster mass and the DE density. Thus, dark energy dominates the dynamics of the curves Vmin(R) and Vpar(R). The rather wide spread of the Virgocentric flow on its entire observed extension from 12 other data in Fig.1 is mostly due to observational errors to at least 30 Mpc. Because of the strong antigravity, the (Karachentsev & Nasonova 2010 – especially their Fig.7b). flow proves to be rather regular acquiring the linear Hubble Anyway, the restriction Emin ≤ E ≤ 0 puts an upper limit on the value |E| in Eq.5, and because of this, the asymptotic relation at distances ∼> 15 Mpc from the cluster center. This major conclusion from the model also resolves the Hubble- of Eq.11 may be reached more easily in the flow. Sandage paradox on the scale of ∼ 10 Mpc, in harmony It is also suggestive that the Local Group outflow with our general approach to the old puzzle (Sec.1). has the same feature: the galaxies prefer the area be- tween Vmin and Vpar in the Hubble diagram of Fig.2 (c.f. Teerikorpi et al. 2008). This is another similarity aspect 4. Discussion between the Virgo system and the Local system (Sec.2). Mathematically, the similarity reflects the fact that Eqs.13 We discuss here some specific features and implications of and 14 have exactly the same view for both systems in the the dark energy domination in the Virgocentric flow. dimensionless form y(x) = 0 with y = V/(HΛRZG), x = R/RZG; the scale factor is again about 10. 4.1. The Einstein-Straus radius This similarity suggests that the condition Emin ≤ E ≤ 0 has more general significance; it may reflect the kinemat- The mass of dark matter and baryons in the Virgo Cluster, 15 ical state of the flow at the early formation epoch. For ex- M = (0.6 − 1.2) × 10 M⊙, was initially collected from the ample, V > Vmin follows, if the galaxies of the flow escaped volume which at the present epoch has the radius RES = 3M 1/3 initially from the cluster (Chernin et al. 2004, 2007c). [ 4πρM (t0) ] = 15 − 19 Mpc, where ρM (t0) is the mean matter density in the Universe. The quantity RES is known as the Einstein-Straus radius in the vacuole model. In the 4.3. Comparison with cosmology presence of dark energy (Chernin et al. 2005; Teerikorpi The zero-gravity radius RZG is a local spatial counterpart of 2ρΛ 1/3 et al. 2008), RES = [ ρM (t0) ] RZG = 1.7RZG. Within the the redshift zΛ ≃ 0.7 at which the gravity and the antigrav- E-S vacuole, the spherical layer between the radii R0 and ity balance each other in the Universe as a whole. Globally, RES does not contain any mass. Therefore in this layer, the balance takes place only at one moment t(zΛ) in the the approximation of test particles adopted in the model entire co-moving space, while the local gravity-antigravity 6 A.D. Chernin et al.: Dark energy in the Virgocentric flow balance exists during all life-time of the system since its 4.4. Probing local dark energy formation, but only on the sphere of the radius RZG. The Virgocentric flow should not be identified with the The global density of dark energy can be derived, if the global cosmological expansion. The Virgo system lies deeply redshift zΛ is found at which the gravity of matter is ex- inside the cosmic cell of uniformity and does not ”know” actly balanced by the dark energy antigravity, and also the about the universe of the horizon scales. In the early epoch present matter density is known. The same logic works in of weak protogalactic perturbations, the dark matter and local studies: just find the zero-gravity radius RZG from the baryons of the system participated in the cosmological ex- outflow observations and the mass of the local system. pansion. But later, at the epoch of nonlinear perturba- We may now restrict the value of RZG using the diagram tions, the matter separated from the expansion and under- of Fig.1. Since the zero-gravity surface lies outside the clus- went violent evolution. Contrary to that, the unperturbed ter volume, it should be that RZG > R0 ≃ 6 − 7 Mpc. On flow at horizon-scale distances was not similarly affected the other hand, the linear velocity-distance relation, with and up to now the initial isotropy of its expansive motion the Hubble ratio close to H, is clearly seen from a distance and the uniformity of its matter distribution are conserved. of about 15 Mpc. This suggests that RZG < 15 Mpc. This Therefore, it is mysterious that the global Hubble constant argument that the zero-gravity surface is located below the 72 km/s/Mpc is so close to the local expansion rate 65 point where the local flow reaches the global expansion km/s/Mpc in the Virgocentric flow (two beams in Fig.1). rate, gained support from the calculations by Teerikorpi This is one aspect of the Hubble-Sandage paradox (Sec.1). & Chernin (2010). As for the cluster mass, we adopt the 15 Trying to find an explanation, we note that the range M = (0.6 − 1.2) × 10 M⊙ (Sec.2). Then Eq.6 in the Friedmann equation for the scale factor a(t) has much the 8π 3 form ρx = M/( 3 RZG) directly leads to robust upper and same form as the local energy conservation equation (5): lower limits to the unknown local dark energy density ρx: 1 GC 4π a˙(t)2 = + Gρ a(t)2 + B. (15) 2 a(t) 3 Λ −29 3 4π 3 0.2 <ρx < 5 × 10 g/cm . (19) The constant C = 3 ρM (t)a(t) has the dimension of mass, and ρM (t) is the density of matter (dark and baryonic); the energy constant B = 0 in the standard flat cosmology. The In fact, the upper limit obtained from the cluster size (RZG) cosmological Hubble factor comes from this equation: is conservative. We note that the theoretical minimum ve- locity curve in Fig.1 would be shifted up to the velocity- a˙ (t) 8πG 1/2 ρm 1/2 H(t)= = [ (ρM (t)+ρΛ)] = HΛ[1+ ] .(16) distance relation defined by the TRGB and Cepheid dis- a 3 2ρΛ −29 3 tances, if one takes ρx =1.4 × 10 g/cm . This would be When time goes to infinity, the matter density drops to a more realistic upper limit, and near the limit similarly ob- zero, the dark energy becomes dominant, and the cosmo- tained in Teerikorpi et al. (2008) using the Local Group and logical Hubble factor becomes time-independent: the M81 group. Thus the local dark energy density is near −29 8πG 1/2 the global value or may be the same; ρΛ = 0.73 × 10 H(t) → H = [ ρ ] = 63 − 65 km/s/Mpc. (17) 3 Λ 3 Λ g/cm lies comfortably in the range of Eq.19. Similar es- At the present epoch, t = 13.7 Gyr, the Hubble rate timates for the dark energy come from studies of the U Local Group, M81 group and Cen A group (Chernin et Ω H = H [1 + M ]1/2 ≃ 72 km/s/Mpc, (18) al. 2007a,b, 2009). Λ Ω Λ We may now argue the other way round: let us assume which is equal to the empirical (WMAP) value. that the density of dark energy is known from cosmological As we see, the Virgocentric flow and the global cosmo- observations and the zero-gravity radius RZG is between 6 logical expansion have a common asymptotic in the limit and 15 Mpc. Then we may estimate independently the mass 15 of dark energy domination: for the local flow, this is the of the Virgo Cluster: M = (0.3 − 4) × 10 M⊙. The value asymptotic in space (large distances from the cluster cen- of the cluster mass adopted earlier is within this interval. ter), and for the global flow, this is the asymptotic in time (a future of the Universe). If the asymptotic is approached now both globally and locally, we would have H ≃ HΛ on 4.5. The origin of the local flows? the largest scales and HV ≃ HΛ in the Virgocentric flow. In their present states, both flows are near the asymp- The model of Sec.3 describes the present structure of the totic, though do not quite reach it. That is why H, HV , and Virgocentric flow and also predicts its future evolution. HΛ have similar values. The flows are not related to each However, it says almost nothing about the initial state of other directly; however, they ”feel” the same uniform dark the flow. Our discussion suggests that the Virgo cluster energy background, and in this way both are controlled by and the outflow form a system with a common origin and the antigravity force which is equally effective near the cos- evolutionary history. Its formation was most probably due mic horizon and at the periphery of the Virgocentric flow. to complex linear and nonlinear processes (collisions and As a result, the local flow at its large distances looks much merging of galaxies and protogalactic units, violent relax- like a ”part” of the global horizon-distance flow. ation of the system in its self-gravity field, etc.). Similar pro- The near coincidence of the values H, HV, and HΛ ex- cesses are expected in the early history of the Local Group plained by our model clarifies the Hubble-Sandage paradox and other systems with the two-component structure. Their (Sec.1) for the Virgocentric flow: asymptotically, the flow basic features might be recognized from big N-body simu- reaches not only the linear Hubble law (Sec.3), but also ac- lations, like the Millennium Simulation (Li & White 2008). quires the expansion rate HV which approaches the global We leave it for a later study to clarify the way in which the rate when the measuring accuracy is improved. galaxies of local flows gain their initial expansion velocities. A.D. Chernin et al.: Dark energy in the Virgocentric flow 7
5. Conclusions Chernin A.D., Teerikorpi P., Baryshev Yu.V. (2000) [astro-ph/0012021] Adv. Space Res., 31, 459, 2003 The recently published systematic and most accurate data Chernin A.D., Karachentsev I.D., Valtonen M.J, et al. (2004) A&A on the velocities and distances of galaxies in the Virgo clus- 415, 19 ter and its environment (Karachentsev & Nasonova 2010) Chernin A.D., Karachentsev I.D., Nasonova O.G., et al. (2007a), Astrophys. 50, 405 shed light on the physics behind the observed properties of Chernin A.D., Karachentsev I.D., Makarov D.I., et al. (2007b), the cluster and the flow. We find that: A&AT, 26, 275 1. The cluster and the flow can be treated together as a Chernin A.D., Karachentsev I.D., Valtonen M.J.,et al. (2007c), A&A physical system embedded in the uniform dark energy back- 467, 933 Chernin A.D., Teerikorpi P., Valtonen M.J., Dolgachev V.P., ground; this is adopted in our analytical nonlinear model. Domozhilova,L.M. Byrd G.G. (2009) A&A 507, 1271 2. The nonlinear interplay between the gravity produced Dolgachev V.P., Domozhilova L.M., Chernin A.D. (2003) Astr.Rep. by the cluster mass (dark matter + baryons) and the anti- 47, 728 gravity of the dark energy is the major dynamical factor Dolgachev V.P., Domozhilova L.M., Chernin A.D. (2004) Astr.Rep. 48, 787 determining the kinematic structure of the system and con- Ekholm T. et al. (1999) A&A 351, 827 trolling its evolution; our model enables one to describe this Ekholm T. et al. (2000) A&A 355, 835 in a simple (exact or approximate) quantitative way. Ekholm T. et al. (2001) A&A 368, L17 3. The key physical parameter of the system is its zero- Gliner E.B. (1965) JETP 49, 542 (Sov.Phys. JETP 22, 376, 1966) Gromov, A., Baryshev, Yu., Suson, D., Teerikorpi, P. (2001) gravity radius, RZG = 9 − 11 Mpc, below which the grav- Gravitation & Cosmology, 7, 140 ity dominates the quasi-stationary bound cluster, while the Guo Q., Shan H.-Y. (2009) RAA 9, 151 antigravity dominates the expanding Virgocentric flow. Hoffman G.L., Salpeter E.E. (1980) ApJ 263, 485 4. The dark energy antigravity brings order and regu- Hoffman Y., et al. (2008) MNRAS 386, 390 larity to the Virgocentric flow and the flow acquires the lin- Huchra J.P. (1988) in ”The Extragalactic Distance Scale”, ASP, p.257 Karachentsev I.D. (2005) AJ 129, 178 ear velocity-distance relation at the Virgocentric distances Karachentsev I.D., Chernin A.D., Teerikorpi P. (2003a) Astrophysics R> 15 Mpc. Thus, the Hubble-Sandage paradox is under- 46, 491 stood on the cluster scale. Karachentsev I.D., Nasonova O.G. (2010) MNRAS (in press) 5. In the velocity-distance diagram, the zero-gravity ra- Karachentsev I.D., Sharina M.E., Dolphin A.E., et al. (2002a) A&A, 385, 21 dius is near to or somewhat larger than the zero-velocity Karachentsev I.D., Sharina M.E., Makarov D.I., et al. (2002b) A&A radius R0 =5.0 − 7.5 Mpc, as determined by Karachentsev 389, 812 & Nasonova (2010). Karachentsev I.D., Makarov D.I., Sharina M.E., et al. (2003b) A&A 6. With the value of R > R found from the di- 398, 479 ZG ∼ 0 Karachentsev I.D., et al. (2006) AJ 131, 1361 agram, one may estimate the mass of the Virgo cluster, Karachentsev I.D., et al. (2009) MNRAS 393, 1265 using the known global density of dark energy: M = Li Y.-S., White S.M.D. (2008) MNRAS 384, 1459 15 (0.3 − 4) × 10 M⊙. Lynden-Bell D. (1981) The Observatory 101, 111 Maccio, A.V. et al. (2005) MNRAS 259, 941 7. With the same value of RZG, one may estimate the Marinez-Vaquero L., et al. (2009) MNRAS 397, 2070 local density of the dark energy in the Virgo system, if the Mei S., Blakeslee J.P., Cˆote P., et al. 2007, ApJ 655, 144 mass of the cluster is considered known (from the virial or Niemi S.-M., Valtonen M.J. (2009) A&A 494, 857 −29 3 the zero-velocity method): ρΛ = (0.2 − 5) × 10 g/cm . Peebles P.J.E. (1976) ApJ 205, 318 8. The two-component Virgo system is similar to the Perlmuter S., Aldering G., Goldhaber G. et al. (1999) ApJ 517, 565 Riess A.G., Filippenko A.V., Challis P., et al. (1998) AJ 116, 1009 Local system formed by the Local Group together with the Rizzi L. et al. (2007) ApJ 661, 813 local outflow around it. In the velocity-distance diagram, Saarinen, S. & Valtonen, M.J. (1985) A&A 153, 130 the Virgo system reproduces the Local system with the Sandage A. (1986) ApJ 307, 1 scale factor of about 10. A dynamical similarity is seen in Sandage A. (1999) ApJ 527, 479 Sandage A., Tammann G., Saha A. 2006, ApJ, 653, 843 the fact that the zero-gravity radius for the Virgo system Silk J. (1974) ApJ 193, 525 is about 10 times larger than for the Local system. Silk J. (1977) A&A 59, 53 9. The similarity suggests that a universal two- Spergel D.N. et al. (2007) ApJ Suppl. 170, 377 component grand design may exist with a quasi-stationary Teerikorpi P. & Chernin A.D. (2010) A&A (in press) Teerikorpi P., Bottinelli, L., Gouguenheim L., Paturel G. (1992) A&A bound central component and an expanding outflow around 260, 17 it on the spatial scales of 1-30 Mpc. The phase and dynam- Teerikorpi P., Chernin A.D., Baryshev Yu.V. (2006) A&A 483, 383 ical structure of the real two-component systems reflects Teerikorpi P., Chernin A.D., Karachentsev I.D., et al. (2008) A&A the nonlinear gravity-antigravity interplay with dark en- 483, 383 ergy domination in the flow component. Theureau, G., Hanski M., Ekholm, T. et al. (1997), A&A 322, 730 Thim F., Tammann G., Saha A., et al. (2003) ApJ 590, 256 Tonry J.L., Blakeslee, J.P., Ajhar, E.A., & Dressler (2000) ApJ 530, Acknowledgements. A.C., V.D., and L.D. thank the RFBR for partial 625 support via the grant 10-02-00178. We also thank the anonymous Tully R.B. (1982) ApJ 257, 389 referee for useful comments. Tully R.B., Mohayaee R (2004) in ”Outskirts of Galaxy Clusters” (IAU Coll.195, A. Diaferio, ed.) IAU, p.205 Tully R.B., et al. (2006) AJ 132, 729 References Tully R.B., Shaya E.T., Karachentsev I.D. et al., 2008, ApJ 676, 184 Valtonen, M.J., Innanen, K.A., Huang, T.Y., & Saarinen, S. (1985) Balaguera-Antolinez A. et al. (2007) MNRAS 382, 621 A& 143, 182 Bambi C. (2007) PhRvD 75h3003 Valtonen M.J., Niemi S., Teerikorpi P. et al. (2008) DDA 39.0103 Baryshev Yu.V., Chernin A.D., Teerikorpi P. (2001) A&A 378, 729 Whiting A.B. (2003) ApJ 587, 186 Byrd, Chernin and Valtonen (2007) Cosmology: Foundations and Frontiers Moscow, URSS Chernin A.D. (2001) Physics-Uspekhi 44, 1099 Chernin A.D. (2008) Physics-Uspekhi 51, 267