aa r X i v : . [ g r- q c ] M a r EPJ manuscript No. (will be inserted by the editor)
Hubble tension vs two flows
V.G. Gurzadyan , and A. Stepanian Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University, Yerevan, Armenia SIA, Sapienza University of Rome, Rome, Italythe date of receipt and acceptance should be inserted later
Abstract.
The Hubble tension is shown to be solvable, without any free parameter, conceptually andquantitatively, within the approach of modified weak-field General Relativity involving the cosmologicalconstant Λ . That approach enables one to describe in a unified picture both the dynamics of dark mattercontaining galaxies and the accelerated expansion of the Universe, thus defining a local Hubble constant ofa local flow and the global one. The data on the dark matter content of peculiar galaxy samples are shownto be compatible to that unified picture. Future more refined surveys of galaxy distribution, hierarchicaldynamics and flows within the vicinity of the Local group and the Virgo supercluster can be decisive inrevealing the possible common nature of the dark sector.
PACS.
XX.XX.XX No PACS code given
The Hubble tension as the claimed discrepancy between the values of the Hubble parameter associated to the early andlate Universe treatments, is attracting much attention [1,2,3,4,5,6,7,8,9]. The key issue is whether it is a signature ofprincipally new physical concepts or outlines the need of more accurate observational data analysis and interpretationwithin the existing concepts. At the same time, the dark sector continues to remain one of key problems of cosmologyand fundamental physics and its possible link with the Hubble tension is a natural quest.Among the models proposed to explain the available observational data on the dark matter (DM) is the one basedon a modification of General Relativity (GR) with the cosmological constant Λ entering its weak-field limit [10,11,12]. That approach follows from the Newton theorem on the equivalency of sphere’s gravity and that of a point masssituated in its center. Within that approach both the dark matter and dark energy are determined by cosmologicalconstant Λ which acts as a second fundamental constant of gravity along with G and the DM is defined by weak-fieldlimit of GR [11,12,13,14]. That Λ -gravity approach enables one to explain the dynamical properties of groups andclusters of galaxies [10,12]. Preliminary, the Λ -gravity vs the H -tension was considered in [14], and now more refineddata are available which are analysed below. These data enable one to define local and global Hubble constants, to revealtheir quantitative difference over the distance ladder and to show, without any free parameter, their correspondence tothe cosmological parameters. Along with that, we show that recently studied galaxy samples [15,16,17], either claimedwith no DM or as made up mostly (98%) of DM [18], are also compatible to the Λ -gravity.Among other approaches regarding the dynamics of the Local group surroundings we mention e.g. [19,20,21,22,23,24,25,26,27] and the references therein, based on various assumptions or gravity modifications.Within the Λ -gravity approach discussed below, the Hubble parameter defining the cosmological model (the earlyUniverse) and those obtained at distance ladder studies (late Universe), are explained naturally, without an extraparameter, as a consequence of the common nature of dark energy and dark matter. While the observational dataindicate the robust value of the Hubble parameter for the local volume galactic system dynamics [28,29,30,31], weshow the compatibility of Λ -gravity to flow dynamics at several scales around Local Group.Importantly, the Hubble tension, in view of the indications revealed in the analysis below, thus can act as anindependent test for the modified weak-field General Relativity, complementing the possibilities of gravity lensing[32,33], celestial mechanics [34], galaxy cluster dynamics and cosmic voids [35,36,37,38], cosmological perturbationevolution [39] and dedicated GR experimental programs [40,41]. V.G. Gurzadyan, A. Stepanian: Hubble tension vs two flows
Table 1.
Vacuum solutions for GRSign Spacetime Isometry group
Λ > Λ = 0 Minkowski (M) IO(1,3) Λ <
The Newton theorem on “sphere-point” equivalency enables one to arrive to the weak-field modification of GeneralRelativity given by the metric [11] ( c = 1) g = 1 − Gmr − Λr g rr = (cid:18) − Gmr − Λr (cid:19) − . (1)This metric was known before (Schwarzschild - de Sitter metric), however when deduced based on Newton theorem,it provides a description of astrophysical structures such as the galaxy clusters within the weak-field limit of GR [12].The general function for force F ( r ) satisfying Newton’s theorem has the form (see [42,10,11]) F ( r ) = (cid:18) − Ar + Br (cid:19) ˆ r . (2)The second term here leads to the cosmological term in the solutions of Einstein equations and the cosmologicalconstant Λ appears in weak-field GR [14].The appearance of Λ s both in Eq.(1) and Eq.(2) has a clear group-theoretical background. Namely, depending onthe sign of Λ - positive, negative or zero - one has three different vacuum solutions for Einstein equations correspondingto isometry groups, as shown in Table 1.These maximally symmetric Lorentzian 4D-geometries have Lorentz group O(1,3) as their isometry stabilizergroup. The group O(1,3) of orthogonal transformations in these Lorentzian geometries implies spherical symmetry(in Lorentzian sense) at each point of spacetime, so that for all these cases O(3) is the stabilizer group of spatialgeometry, that is each point (in spatial geometry) admits O(3) symmetry. This statement can be regarded as grouptheory formulation of Newton theorem [14].The next important fact is that, the force of Eq.(2) defines non-force-free field inside a spherical shell, thus drasti-cally contrasting with Newton’s gravity when the shell has force-free field in its interior. The non-force-free field agreeswith the observational indications that galactic halos do determine features of galactic disks [43]. The weak-field GRthus ensures that any matter, seen or unseen [44], at large galactic scales is interacting by the law of Eq.(1) and forwhich the virial theorem yields [10] Λ = 3 σ c R ≃ − ( σ kms − ) ( R kpc ) − m − , (3)where σ is the velocity dispersion at a given radius of halo.This relation in [11,13] was shown to be compatible with the observed dynamics of groups and clusters of galaxies,where the value of the cosmological constant was derived from the dynamics of those galactic systems, dependingon their degree of virialization. Further analysis of the compatibility of Eqs.(1,2) to the galactic systems, involving,importantly, extremal galaxies, i.e. claimed as DM-free or DM-rich, is performed in Section 4. Λ : local and global According to Eq.(1) (for details see [14]) the same cosmological constant enters both in the FLRW cosmologicalequations (as dark energy) and in the weak-field GR (as dark matter) to define galactic system structure and dynamics.Then, one has two different flows and two different Hubble constants respectively H local = 8 πGρ local Λc , (4) H global = 8 πGρ global Λc − kc a ( t ) , (5) .G. Gurzadyan, A. Stepanian: Hubble tension vs two flows 3 where k is the spatial curvature of FLRW metric and a ( t ) is the scale factor. Hence, local and global values of Hubbleconstant do arise. Namely, while in Eq.(4) we are dealing with the density of local universe, the measurement of H global in Eq.(5) becomes related to cosmological parameters and the geometric features of FLRW metric.Then, for the Λ -gravity it is possible to obtain a flow in the local universe. Namely, one can define a critical radius where the Λ -term in Eq.(1) becomes dominant r crit = 3 GMΛc . (6)Accordingly, due to repulsive nature of Λ -term, one can conclude that beyond r crit the gravitational field of the centralobject becomes repulsive and that can cause the local H-flow. Meantime, for Λ -gravity we find one more limit, besidesthe Newtonian limit (related to the Newtonian term in Eq.(1)), beyond which the second term’s contribution becomesimportant Λr << , r ≈ . Gpc. (7)It should be noticed that Eq.(4) should not be confused as the non-relativistic limit of FLRW equations. Indeed,both from conceptual and fundamental points of view Eq.(4) has nothing to do with the FLRW universe. It is obtainedbased on the McCrea-Milne model [45,46] and the consideration of Λ -gravity. Namely, we get the Eq.(4) since in theequations of McCrea-Milne model we use the gravitational potential energy according to Λ -gravity i.e. Φ = − GM mr − Λc r m . (8)In this sense, the local H-flow occurs in all non-relativistic limits due to the presence of Λ in the weak-field limit andnot as a result of residuals of the expansion of the Universe. Speaking in other words considering the Λ -gravity, wewill have the gravitational repulsion beyond the r crit and as a result of that, no matter our relativistic backgroundgeometry is dynamic, static or even collapsing we will get a local H-flow according to Eq.(4).The observations indicate that the Universe is flat, i.e. k = 0 in Eq.(5). Thus, although Eq.(4) and Eq.(5) aresimilar in their form, as mentioned above, their essense is rather different. The two currently indicative values of theHubble constant are those determined by P lanck [9] and
HST [1] data, namely
P lanck : H global = 67 . ± . kms − M pc − (9) HST : H local = 74 . ± . kms − M pc − . (10)Currently, after the publication of [14], new results on accurate measurements of H have been reported. In [47], theauthors have obtained H = 69 . ± . kms − M pc − . The importance of their reported value lies on the fact that theyhave analyzed the data of HST which has been generally used to obtain the “local H ”. However, according to authorsthey have used the calibration of the Tip of the Red Giant Branch (TRGB) applied to Type Ia supernovae (SNe Ia)to measure the H . Considering the nature of measurement and the fact that SNe Ia is used to make measurements in0 . < z < . M pc ) about 132 < r < H ” in [9]. Again, while the nature of the global flow is related to the FLRWequations, the local flow i.e. the recession of galaxies from the center of a gravitationally bound object would occurdue to the presence of Λ in the weak-field limit of GR. Since the mentioned distance scales are far larger than of anygravitationally bound structure’s, it is expected to have a value for H closer to ”global” rather than the ”local” one.Let us mention also the recent paper by H0LiCOW team [48], where the reported value of Hubble parameter iscloser to the local H and is in tension with the global flow. This case drastically differs from the above mentioned one,since it is based on the gravitational lensing which itself is a local effect. As a result, the time delay in [48] is relatedto “time-delay distance” which itself is sensitive to H (nevertheless, the authors have mentioned that ”although thereis a weak dependence on other parameters” ), and hence does not give an independent information about the ”global”structure of spacetime geometry; for details see the analysis of lensing effect for modified gravity [33].The next remarkable reported data are of [2], where the value H = 73 . ± . kms − M pc − is obtained, viameasuring the distance to M106 galaxy using its supermassive black hole. In this case, considering the nature ofmeasurement, the reported value should be in agreement with H local in Eq.(9). Note, that another analysis relatedto the measurement of “local H” is [49], with reported values of H (in kms − M pc − ) shown in Table2, which are inagreement with the value of H in [2].Thus, within our approach the discrepancy between the reported values of H is a result of measuring of twoparameters defining two different dynamical effects, i.e. local and global H-flows. Quantitatively, as follows from Eq.(4)and Eq.(5), that discrepancy is related to the discrepancy between the definitions of the local and global densities i.e. ρ local and ρ global = 8 . × − Kg m − [9].By considering Eq.(4) and the reported values of H [2,28,29,30,31] we obtain the local density ρ local . Then, forthree hierarchical systems we find the distances (in M pc ) with respect to the central object where the local H-flow
V.G. Gurzadyan, A. Stepanian: Hubble tension vs two flows
Table 2.
Reported value for Hubble constant[49]
Gravitational lens system H HE0435-1223 82 . +9 . − . PG1115+080 70 . +4 . − . RXJ1131-1231 77 . +4 . − . The joint adaptive optics(AO) re-sults 75 . +3 . − . The joint adaptive optics(AO) +
HST results 76 . +2 . − . occurs. Our analysis shows that, on the one hand, the results are in an exact agreement with observations and, on theother hand, with the theoretical principles. Finally, we also obtain an estimation of mass for Laniakea Supercluster.Namely, according to Eq.(4) the estimations for ρ local yield (in Kg m − ): ρ local = 4 . . . × − [2]; (11) ρ local = 4 . . . × − [28]; (12) ρ local = 4 . . . × − [29]; (13) ρ local = 4 . . . × − [30]; (14) ρ local = 4 . . . × − [31] . (15) Mass = 2 . × M ⊙ [50], Radius = 1 . M pc , r crit = 1 . M pc .Applying Eqs.(4),(6) to the observer at the center of the Local Group (LG) the galaxies in a certain vicinity of theLG will be repelled and it will cause a local H-flow; see also [21,22]. Namely, we can define the distance at which theflow will be exactly equal to the reported value (in
M pc )2 . < r < .
137 [2]; (16)2 . < r < .
168 [28]; (17)2 . < r < .
147 [29]; (18)2 . < r < .
153 [30]; (19)1 . < r < .
218 [31] . (20) Mass = 1 . × M ⊙ [23], Radius = 2 . M pc , r crit = 11 . M pc .In this case, it should be noticed that although r crit is outside the cluster, it is smaller than the distance betweenthe cluster and LG. This is due to the fact, that the center of the cluster is located in 16 . M pc from us. Again, onecan obtain the distance on which a gravitational system centered in the center of Virgo cluster can cause a H-flow (in
M pc ) 16 . < r < .
20 [2]; (21)16 . < r < .
45 [28]; (22)16 . < r < .
29 [29]; (23)16 . < r < . , [30]; (24)14 . < r < .
72 [31] . (25)A remarkable consequence of this result is the following. By comparing the limits of the above relation with thedistance between Virgo cluster and LG, one can state that the gravitational repulsion produced by Virgo cluster can .G. Gurzadyan, A. Stepanian: Hubble tension vs two flows 5 repel the whole LG in an exact accordance with the reported value of H . Inversely, we, as the observers located withinLG, can observe the Virgo cluster as moving away from us exactly according to local H-flow.A new estimation for the virial mass of Virgo cluster is obtained in [51], equal to (6 . ± . × M ⊙ . Withinthe Λ -gravity, from the appeared additional “effective mass” Λc r G we get the following upper limit for ΛΛ ≤ G E( M vir ) c r = 4 . × − m − . (26)On the other hand, the virial parameters also can be used (with obvious precautions regarding the degree ofvirialization) to find an upper limit for Λ , namely,(1 − E( σ ) σ ) ≤ − Λc r GM vir = 6 . × − m − . (27)Note that, although these limits are considerably small, neither contradicts the reported value for Λ of Planck [9].
Mass = 1 . × M ⊙ [52], Radius = 16 . M pc , r crit = 12 . M pc .This case means that one has to consider both LG and Virgo cluster within a larger scale structure, i.e. one gets(
M pc ) 17 . < r < .
45 [2]; (28)17 . < r < .
72 [28]; (29)17 . < r < .
54 [29]; (30)17 . < r < .
59 [30]; (31)15 . < r < .
00 [31] . (32)Note that, while the centers of Virgo Supercluster and Virgo cluster are considered to be identical, the LG is in theoutskirts of the supercluster. Then, the center where the local H-flow caused by Virgo Supercluster repels the LG islocated at r = 1 . M pc away from the center of the LG. In this sense, by comparing the reported data of LG, itturns out that this radius is smaller than the radius of LG and even smaller than r crit , which can be regarded as anindicator of gravitational boundness for a system. The mass of the revealed Laniakea supercluster [53] is evaluated of the order of 10 M ⊙ . Here, we can perform aninverse analysis, to obtain an estimation of the mass according to Eq.(6) and Eq.(4). Note that, although theses twoequations seem similar, the nature of their analyses is totally different. Namely, Eq.(6) is obtained via a dynamicalanalysis, while Eq.(4) is related to the gravitational energy. In this sense, by taking M = 10 M ⊙ , r crit will be 51 M pc . Now, considering the reported radius of Laniakea and the fact that it is located in 77
M pc from us, we can useEq.(4) to obtain the limits for the mass. Namely, considering the reported values of H [2,28,29,30,31], one can findan (average) estimation 1 . × < M/M ⊙ < . × . (33)The main consequence of this result is that, by considering the reported local value of H and Eq.(4), we are able tohave a mass estimation which agrees with the reported data.Thus, the Eqs.(1) and (2), via Eq.(4) and (5), enable to define local and global Hubble constants and, hence, self-consistently - without any free parameter - describe the observational data on the galaxy distribution and their flowsin the vicinity of the Local Group and Virgo supercluster.It should be stressed that, the obtained limit in Eq.(7) and the corresponding mass and radius for all the aboveanalyzed cases, guarantee that we are in the weak-field limit regime. Thus, we are justified to analyze the dynamicsof the said objects based on the weak-field of Λ -gravity and compare them with the results of [2,28,29,30,31]. Theresults of our analysis shows that for all above (hierarchical) systems, we will have the exact amount of density whichcan cause the recession of objects according to the measured H flow in [2,28,29,30,31]. Namely, the distance wherethe gravitational repulsion of the central object according to Λ -gravity forces the galaxies to move away is in completeagreement with the reported value of local H and its corresponding density.Thus, for the considered hierarchical systems of different scales, the galaxies move away from their centers startingfrom a critical distance, in accord to the presence of Λ in Eq.(1). This is what we call the ”local H-flow”. Accordingly,in order to show the difference of this flow with the relativistic flow of objects at cosmic scales, we have obtained forall systems, the relevant distance at which the objects start to flow. We have found that for all five reported values of H , the corresponding distance is in agreement with the observations. V.G. Gurzadyan, A. Stepanian: Hubble tension vs two flows
The weak-field GR given by Eqs.(1) and (2) has been applied to describe the dynamics of galactic halos, galaxy groupsand clusters [10,13] by means of the virial theorem for the gravitational potential containing besides the Newtonianterm also the one with the cosmological constant Λ . So, as in [13] at comparison with observational data, the currentnumerical value of the cosmological constant Λ has to be smaller than the error of velocity dispersion. We will nowextend such an analysis to two categories of extremal cases i.e. to galaxies with no DM and galaxies made up of DMonly. In all these cases, the reported data are in accordance with Newtonian dynamics. Namely, the measured velocitydispersion σ is related to dynamical mass M dyn via the following relation σ = GM dyn R , (34)where R is the typical radius of galaxy. Clearly, the above dynamical equation is obtained by considering the Newtoniangravity. However, by replacing the Newtonian gravitational force with Λ -gravity according to Eq.(1) we will get σ = GM dyn R − Λc R . (35)Comparing Eqs.(34, 35), it turns out that observed value of σ in the context of Λ -gravity should be smaller thanNewtonian case. Thus, in order to be a self consistent theory, the theoretically obtained value of σ in the context Λ -gravity should be larger than observed value of σ which is based on Newtonian gravity. Consequently, as a methodto check the validity of Λ -gravity theory, we can find the upper limit for the numerical value of Λ as follows( σ − E( σ ) σ ) ≤ − Λ c R GM dyn , (36)where E( σ ) is the error limit of velocity dispersion reported by observations. In this sense, it is expected that theobtained upper limits for Λ must be larger than numerical value of Λ = 1 . × − m − which have been reportedby Planck satellite.Analysis of such extreme cases can pose constraints over various theories of gravity and even rule them out [54].Indeed, by assuming Newtonian dynamics, the recent study [55] proposes two estimations for dynamical mass ofNGC 1052-DF2. Consequently, the upper limit over the Λ , for intrinsic and nominal velocity dispersions i.e. σ int and σ DF ⋆ will be σ int : Λ ≤ . × − , σ DF ⋆ : Λ ≤ . × − . (37)For the second DM-missing galaxy NGC 1052-DF4, we also obtain the upper limits as follows: σ int : Λ ≤ . × − , σ stars : Λ ≤ . × − , (38)where σ stars refers to velocity dispersion obtained by considering the stars alone.For the other extreme category we check the structure of Dragonfly 44 as one of best known ultra diffuse galaxies(UDG) [18]. Here by considering the total dynamical mass M dyn within the half-light radius i.e. r = 4 . kpc equal to0 . +0 . − . × M ⊙ we have Λ ≤ . × − . (39)Thus by considering the results of both categories of objects - galaxies lacking DM and the one made almostentirely of DM - it turns out that the modification of gravity according to Eq.(1) not only is able to describe thesestructures, but fits the considered weak-field GR with the numerical value of Λ not contradicting the observationaldata on these extremal astrophysical structures.Besides the above two categories of galaxies, a new group denoted as DM deficient dwarf galaxies has been studiedin [56]. For them it has been reported that the matter content consists mainly of baryons. We start our discussion bychecking the velocity of galaxies according to Eq.(1) i.e. V cir = GM dyn r − Λc r , (40)where M dyn is the total dynamical mass. Thus, by taking the reported values of these galaxies we find the error limitsof Λ . The results are shown in Table 3. The w
20 denotes the 20% of the HI line width which has been considered asindicator of the gas velocity. Considering the results of Table 3, it becomes clear that again there is no contradictionbetween Λ -modified gravity and the observed parameters of the galaxies. .G. Gurzadyan, A. Stepanian: Hubble tension vs two flows 7 Table 3.
Constraints on Λ for DM deficient dwarf galaxies Galaxy log M dyn ( M ⊙ ) w w er (km/s) Λ ( m − ) ≤ AGC 6438 9.444 80.36 2.03 9 . × − AGC 6980 9.592 56.63 1.54 6 . × − AGC 7817 9.061 82.37 4.45 1 . × − AGC 7920 8.981 79.03 2.6 9 . × − AGC 7983 9.046 46.12 0.83 1 . × − AGC 9500 9.092 39.08 0.31 2 . × − AGC 191707 9.08 49.27 1.21 2 . × − AGC 205215 9.706 72.5 4.41 3 . × − AGC 213086 9.8 78.35 4.33 3 . × − AGC 220901 8.864 45.38 0.74 2 . × − AGC 241266 9.547 52.82 1.98 7 . × − AGC 242440 9.467 42.47 1.18 2 . × − AGC 258421 10.124 87.79 8.53 2 . × − AGC 321435 9.204 56.83 4.41 1 . × − AGC 331776 8.503 29.59 2.9 6 . × − AGC 733302 9.042 48.36 0.99 2 . × − AGC 749244 9.778 70.87 4.91 2 . × − AGC 749445 9.264 54.51 3.06 4 . × − AGC 749457 9.445 58.68 5.49 5 . × − Table 4.
Constraints on Λ for 24 dwarf galaxies surrounding the Milky Way Galaxy σ ( km/s ) r (pc) Λ ( m − ) ≤ Aquarius2 5 . ± . . ± . . × − Bootes1 2 . ± . . ± .
039 9 . × − Carina 6 . ± . . ± .
952 1 . × − Coma 4 . ± . . ± .
615 1 . × − CraterII 2 . ± . ±
86 1 . × − CVenI 7 . ± . . ± .
59 2 . × − CVenII 4 . ± . . ± .
22 1 . × − Draco 9 . ± . . ± .
079 3 . × − Draco2 2 . ± . . ± .
639 2 . × − Fornax 11 . ± . . ± .
837 2 . × − Hercules 3 . ± . . ± . . × − LeoI 9 . ± . . ± .
133 2 . × − LeoII 6 . ± . . ± .
926 2 . × − LeoIV 3 . ± . . ± .
03 7 . × − LeoV 2 . ± . . ± .
15 6 . × − Sagittarius 11 . ± . . ± .
78 4 . × − Sculptor 9 . ± . . ± . . × − Segue1 3 . ± . . ± .
79 8 . × − Sextans 7 . ± . . ± .
993 9 . × − TucanaII 8 . ± . . ± .
68 2 . × − UMaI 7 . ± . . ± .
01 2 . × − UMaII 6 . ± . . ± .
325 7 . × − UMi 9 . ± . . ± . . × − Willman1 4 . ± . . ± . . × − In addition to above extreme cases 62 dwarf spheroidals (dSphs) in the Local Group (LG) are considered as anothersample to analyze the validity of different modified theories of gravity and the paradigm of DM. Namely, the study ofdSphs surrounding the Milky Way has suggested those are DM-free structures [57]. Here, by considering Eq.(36) wehave obtained error limits of Λ for them. The results are exhibited in Table 4.Moreover, considering Eq.(1), the radial acceleration will be written as a ( r ) = GM tot r − Λc r , (41)where M tot is the total mass (both ordinary and DM) of the configuration according to [58]. Consequently, theconstrains over Λ will be obtained. For 20 of them these limits are shown in Table 5.It is worth noticing that although several parameters are taken into consideration in actual observations, they arebased on fundamental relations governing the dynamics of objects according to Newtonian gravity. What we havedone in this section is to modify the underlying dynamical equations in the context of Λ -gravity (i.e. Eqs.(40),(41)and analyze the error limits accordingly assuming that all the observational complications are the same.Thus, the galaxy samples which were previously used to test and/or reject certain dark matter models, here areshown to be in full compatibility with the Λ -gravity predictions. The Hubble tension problem, now attracting much attention, is shown to be resolvable conceptually and quantitativelyby the Λ -gravity, as modified weak-field General Relativity [10,11].We show that the suggested approach defines a ladder of distance scales for galaxy distribution hierarchy, from theLocal group to the Virgo and Laniakea superclusters, which links their local dynamics to the cosmological parameters. V.G. Gurzadyan, A. Stepanian: Hubble tension vs two flows
Table 5.
Constraints on Λ for 20 dwarf spheroidals of LG Galaxy log a ( r )( m/s ) r (pc) Λ ( m − ) ≤ Bootes I -11 . ± .
15 283 ± . × − Bootes II -9 . ± .
63 61 ±
24 2 . × − Canes Venatici I -11 . ± .
05 647 ±
27 1 . × − Canes Venatici II -10 . ± .
19 101 ± . × − Carina -10 . ± .
18 273 ±
45 2 . × − Coma Berenices -10 . ± .
16 79 ± . × − Draco -10 . ± .
12 244 ± . × − Fornax -10 . ± .
08 792 ±
58 3 . × − Hercules -11 . ± .
22 175 ±
22 1 . × − Hydra II -10 . ± .
12 88 ±
17 6 . × − Leo I -10 . ± .
06 298 ±
29 1 . × − Leo II -10 . ± .
14 219 ±
52 2 . × − Leo IV -11 . ± .
47 149 ±
47 3 . × − Leo V -11 . ± .
88 125 ±
47 3 . × − Leo T -10 . ± .
19 160 ±
10 8 . × − Sculptor -10 . ± .
13 311 ±
46 2 . × − Sextans -11 . ± .
15 748 ±
66 3 . × − Ursa Minor -10 . ± .
12 398 ±
44 1 . × − Ursa Major I -11 . ± .
88 125 ±
47 3 . × − Ursa Major II -10 . ± .
19 160 ±
10 8 . × − For those considered hierarchical systems of different scales we conclude that the galaxies move away from theircenters starting from a critical distance, in accord to the presence of Λ in Eq.(1) i.e. contribute to the ”local H-flow”.We show the difference of this flow with the relativistic flow at cosmological scales, i.e. we obtain for all systems therelevant distances at which the objects would start to participate the global flow and show their agreement with theobservations.Importantly, the Λ -gravity is also shown to agree with the data on extremal galaxies, i.e. those claimed as darkmatter rich galaxies and no dark matter ones. Several independent galaxy samples are considered, all shown withdata compatible to modified gravity constraints. Again, the principal point in our analysis and in comparison to theobservational data is the absence of any additional or free theoretical parameters.Future more refined observational surveys of galaxy distribution and dynamics in the vicinity of the Local groupand the Virgo supercluster can be decisive in testing the modified weak-field General Relativity, with direct impacton the nature of the dark sector. References
1. Riess A.G. et al., ApJ , 85 (2019)2. Reid M. J., Pesce D. W., Riess A.G, ApJ, , L27 (2019)3. Verde L., Treu T., Riess A.G., Nature Astronomy, , 891 (2019)4. Pesce D. W., et al., ApJ, (2020) L15. Dhawan S., Brout D., Scolnic D., Goobar A., Riess A. G., Miranda V., ApJ, , 54 (2020)6. Riess A.G., Nature Review Physics, , 10 (2020)7. Riess A.G., et al., arXiv:2012.08534 (2020)8. Beenakker W., Venhoek D., arXiv:2101.01372 (2021)9. Planck Collaboration et al., arXiv:1807.06209 (2018)10. Gurzadyan V.G., Eur. Phys. J. Plus, , 98 (2019)11. Gurzadyan V.G., Stepanian A., Eur. Phys. J. C, , 632 (2018)12. Gurzadyan V.G., Stepanian A., Eur. Phys. J. C, , 169 (2019)13. Gurzadyan V.G. , Stepanian A., Eur. Phys. J. Plus, , 98 (2019)14. Gurzadyan V.G., Stepanian A., Eur. Phys. J. C, , 568 (2019)15. van Dokkum P., et al., Res. Notes AAS, , 54 (2018)16. van Dokkum P., et al., Nature, , 629 (2018)17. van Dokkum P., et al., ApJ Lett., , L5 (2019)18. van Dokkum P., et al., ApJ Lett., , L6 (2016)19. Rauzy, S. and Gurzadyan, V. G. MNRAS, , 114 (1998)20. Davis T. et al, Amer. J Phys., , 358 (2003)21. Karachentsev I.D., et al, MNRAS, , 1265 (2009)22. Chernin A.D., et al, A&A, , A104 (2010)23. Fouque P., et al., A & A , 3 (2001)24. Nandra R., Lasenby A.N., Hobson M.P., MNRAS, , 2931 (2012)25. Banik I., Zhao H, MNRAS, , 4033 (2018)26. McLeod M., Lahav O., arXiv:1903.10849 (2019)27. Christodoulou D.M., Kazanas D., MNRAS Lett. , L53 (2019)28. Riess A.G., et al., ApJ, , 56 (2016)29. Riess A.G., et al., ApJ, , 136 (2018).G. Gurzadyan, A. Stepanian: Hubble tension vs two flows 930. Riess A.G., et al., ApJ, , 126 (2018)31. de Jaeger T., et al., MNRAS , 3402 (2020)32. Collett T.E., et al, Science, , 1342 (2018)33. Gurzadyan V.G., Stepanian A., Eur. Phys. J. C, , 869 (2018)34. Kopeikin S., Efroimsky M. , Kaplan G. , Relativistic celestial mechanics of the solar system, (Wiley, New Jersey, 2001)35. Salzano V., et al, JCAP, , 033 (2016)36. Capozziello S., et al, MNRAS, , 2430 (2018)37. Gurzadyan V.G., Kocharyan A.A., Stepanian A., Eur. Phys. J. C, , 24 (2020)38. Gurzadyan V.G., Kocharyan A.A., A & A, , L61 (2009)39. Eingorn M., ApJ, , 84 (2016)40. Turyshev S., Phys. Uspekhi , 1 (2009)41. Ciufolini I., et al, Eur. Phys. J. C, , 872 (2019)42. Gurzadyan V.G., Observatory, , 42 (1985)43. Kravtsov A.V., ApJ Lett, , L31 (2013)44. Gurzadyan V.G., et al, A & A, 609, A131 (2018)45. McCrea W.H., Milne E.A., Q. J. Math. , 73 (1934)46. Milne E.A., Q. J. Math. , 64 (1934)47. Freedman W.L., et al., ApJ, , 34 (2019)48. Wong K.C. et al, arXiv:1907.04869 (2019)49. Chen G.C.-F., et al., MNRAS, , 2 (2019)50. Penarrubia J., et al., MNRAS, , 3 (2014)51. Kashibadze O.G., Karachentsev I.D. , Karachentseva V.E., A& A , A135 (2020)52. Einasto M., et al., A & A , 2 (2007)53. Tully R.B., et al., Nature, , 71 (2014)54. Islam T., Dutta K., Phys. Rev. D, , 104049 (2019)55. Emsellem E., et al, A & A , A76 (2019)56. Guo Q., et al., Nature Astronomy, 4, 246 (2019)57. Hammer F., et al., ApJ, , 171 (2019)58. Lelli F., et al., ApJ,836