Issue 
A&A
Volume 553, May 2013



Article Number  A101  
Number of page(s)  4  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201220781  
Published online  16 May 2013 
Dark energy and the structure of the Coma cluster of galaxies
^{1} Tuorla Observatory, Department of Physics and Astronomy, University of Turku, 21500 Piikkiö, Finland
email: pekkatee@utu.fi
^{2} Sternberg Astronomical Institute, Moscow University, 119899 Moscow, Russia
^{3} Space Research Institute, Russian Academy of Sciences, 117997 Moscow, Russia
^{4} University of Alabama, Tuscaloosa, AL 354870324, USA
^{5} Department of Physics, University of Rome “La Sapienza”, 00185 Rome, Italy
Received: 23 November 2012
Accepted: 18 March 2013
Context. We consider the Coma cluster of galaxies as a gravitationally bound physical system embedded in the perfectly uniform static dark energy background as implied by ΛCDM cosmology.
Aims. We ask if the density of dark energy is high enough to affect the structure of a large and rich cluster of galaxies.
Methods. We base our work on recent observational data on the Coma cluster, and apply our theory of local dynamical effects of dark energy, including the zerogravity radius R_{ZG} of the local force field as the key parameter.
Results. 1) Three masses are defined that characterize the structure of a regular cluster: the matter mass M_{M}, the darkenergy effective mass M_{DE} (<0), and the gravitating mass M_{G} (=M_{M} + M_{DE}). 2) A new matterdensity profile is suggested that reproduces the observational data well for the Coma cluster in the radius range from 1.4 Mpc to 14 Mpc and takes the dark energy background into account. 3) Using this profile, we calculate upper limits for the total size of the Coma cluster, R ≤ R_{ZG} ≈ 20 Mpc, and its total matter mass, M_{M} ≲ M_{M}(R_{ZG}) = 6.2 × 10^{15} M_{⊙}.
Conclusions. The dark energy antigravity affects the structure of the Coma cluster strongly at large radii R ≳ 14 Mpc and should be considered when its total mass is derived.
Key words: galaxies: clusters: individual: Coma / dark matter / dark energy
© ESO, 2013
1. Introduction
The Coma cluster of galaxies (A1656) is the most massive wellstudied regular gravitationally bound aggregation of matter in the observable Universe. In his classic work Zwicky (1933, 1937) applied the virial theorem to the cluster and showed that dark matter dominates the system. Zwicky estimated its mass as 3 × 10^{14} M_{⊙}, when normalized to the Hubble constant h = 0.71 used hereafter.
Decades later, The & White (1986) found an order of magnitude higher value, 2 × 10^{15} M_{⊙}, with a generalization of the virial theorem that must be used when the observed sample does not include the entire system. Hughes (1989, 1998) obtained a similar value (1 − 2) × 10^{15} M_{⊙} with Xray data under the assumption that the cluster’s hot intergalactic gas is in hydrostatic equilibrium. In a similar way, Colles (2006) reports the mass 4.4 × 10^{14} M_{⊙} inside the radius of 1.4 Mpc. A weaklensing analysis gives 2.6 × 10^{15} M_{⊙} (Kubo et al. 2007) within 4.8 Mpc radius. Geller et al. (1999, 2011) extended mass estimates to the outskirts of the cluster using the caustic technique (Diaferio & Geller 1997; Diaferio 1999) and find the mass 2.4 × 10^{15} M_{⊙} within the 14 Mpc radius. Here the 2σ error is 1.2 × 10^{15} M_{⊙}, so it does not contradict the apparently higher mass within 4.8 Mpc.
In this paper, we reexamine the mass estimation of the Coma cluster using the data above and a new theory model that describes the cluster as a bound spherical system embedded in the cosmic background of dark energy. We find that dark energy strongly affects the cluster structure at large distances R ≥ 14 Mpc from the cluster center and must be taken into account in the mass estimate.
Basic theory is outlined in Sect. 2, three characteristic masses of a regular cluster are introduced in Sect. 3, a new matter mass profile is defined in Sect. 4, the upper bound on the total size of the Coma cluster is calculated and discussed in Sect. 5, and the results are summarized in Sect. 6.
2. Local dynamical effects of dark energy
We adopt the ΛCDM cosmology that identifies dark energy with Einstein’s cosmological constant Λ and treats it as a perfectly uniform vacuumlike fluid with the constant density ρ_{DE} = 0.71 × 10^{29} g cm^{3}. The dark energy background produces antigravity that is stronger than the matter gravity in the present Universe as a whole. This makes the cosmological expansion accelerate, as discovered by Riess et al. (1998) and Perlmutter et al. (1999).
The cosmic antigravity can be stronger than gravity not only globally, but also locally on scales of ~ 1–10 Mpc (Chernin et al. 2000, 2006; Chernin 2001; Byrd et al. 2007, 2012), as studied using the HST observations made by Karachentsev’s team (e.g., Chernin et al. 2010, 2012a).
The local weakfield dynamical effects of dark energy can be adequately described in terms of Newtonian mechanics (e.g., Chernin 2008). Such an approach borrows from general relativity the major result: the effective gravitating density of a uniform medium is given by the sum (1)where ρ and P are the fluid’s density and pressure (c = 1 hereafter). In the ΛCDM model, the dark energy equation of state is P_{DE} = −ρ_{DE}, and its effective gravitating density (2)is negative, producing antigravity. Einstein’s “law of universal antigravity” says that a point mass M within uniform dark energy generates an acceleration a(r) that includes, in addition to the Newtonian term a_{N}(r) = −GM/r^{2}, the antigravity effect of dark energy (3)Then a test particle at the distance R from the center of a spherical matter mass M_{M} (beyond the mass) has the radial acceleration in the reference frame of the mass center: (4)Equation (4) comes from the Schwarzschildde Sitter spacetime in the weak field approximation. The net acceleration a(R) is zero at the distance (Chernin et al. 2000; Chernin 2001, 2008) (5)Gravity dominates at distances R < R_{ZG}, while antigravity is stronger than gravity at R > R_{ZG}. A gravitationally bound system with the mass M_{M} can only exist inside its zerogravity sphere of the radius R_{ZG}.
3. Three masses of a regular cluster
The presence of dark energy in the spherical volume of a regular cluster like Coma may be quantified by its effective gravitating mass within a given clustrocentric radius R: (6)The matter (dark matter and baryons) content of the cluster is characterized by the mass M_{M}(R) inside radius R: (7)Here ρ(R) is the matter density within the cluster. The sum (8)is the total gravitating mass within the radius R. It is this mass that can be directly measured by the methods cited in Sect. 1, which are all related to gravitation and give the gravitating mass M_{G}(R), rather than the matter mass M_{M}(R), which can, however, be derived from the data of Sect. 1 using Eq. (8): M_{M} = M_{G} − M_{DE}. So one has for R = 1.4 Mpc: (9)For R = 4.8 Mpc, (10)And for R = 14 Mpc, (11)As we see, in the inner cluster at R = 1.4 and 4.8 Mpc, the dark energy contributes practically nothing compared to the gravitating mass, so M_{G} ≃ M_{M} here. But (curiously) the absolute value of the dark energy mass M_{DE} nearly equals the gravitating mass M_{G} at R = 14 Mpc; as a result, the matter mass M_{M} ≃ 2M_{G} ≃ 2  M_{DE}  at this radius. The difference between the Eq. (11) estimate and the observed value of M_{G} is at the level of 4σ (, Sect. 1).
In the outer regions of the Coma cluster, (12)where the antigravity effect is thus significant.
4. Matter mass profile
Our estimate of the Coma matter mass within R = 14 Mpc (Eq. (11)) may be compared with estimates following from traditional matter density profiles for dark halos.
4.1. NFW and Hernquist profiles
The popular NFW profile (Navarro et al. 2005) is (13)where R is again the distance from the cluster center, ρ_{s} = ρ(R_{s}), and R_{s} are constant parameters. At small radii, R ≪ R_{s}, the matter density goes to infinity, ρ ∝ 1/R as R goes to zero. At long distances, R ≫ R_{s}, the density slope is ρ ∝ 1/R^{3}. With this profile, the matter mass profile is (14)To find the parameters ρ_{s} and R_{s}, we use the smallradii data of Sect. 1: M_{1} = 4.4 × 10^{14} M_{⊙} at R_{1} = 1.4 Mpc, M_{2} = 2.6 × 10^{15} M_{⊙} at R_{2} = 4.8 Mpc. At these radii, the gravitating masses are nearly equal to the matter masses there (cf. Sect. 3). The values of M_{1}, R_{1} and M_{2}, R_{2}, together with Eq. (14) lead to two logarithmic equations for the two parameters of the profile, which can easily be solved: R_{s} = 4.7 Mpc,ρ_{s} = 1.8 × 10^{28} g/cm^{3}. Then we find the matter mass within R = 14 Mpc, (15)to be considerably larger (over 70%) than given by Eq. (11).
Another widely used density profile (Hernquist 1990) is (16)Its smallradius behavior is the same as in the NFW profile: ρ → ∞, as R goes to zero. The slope at large radii is different: ρ ∝ 1/R^{4}. The corresponding mass profile is (17)The parameters M_{0} and α can be found from the same data as above on M_{1}, R_{1} and M_{2},R_{2}: M_{0} = 1.4 × 10^{16} M_{⊙},α = 6.4 Mpc, giving another value for the mass within 14 Mpc: (18)Now the difference from the figure of Eq. (11) is about 40%.
Fig. 1 Matter mass M vs. radius R for three profiles. 1) NFW mass profile of Eq. (14): – dashed line; 2) Hernquist profile of Eq. (17): – thin line; 3) New mass profile of Eq. (19): – thick line. Here R is in Mpc and M is in 10^{15} M_{⊙}. The values of the matter mass at radii R = 1.4,4.8,14 Mpc are also shown as given by Eqs. (9)–(11) with 1σ error bars. 
4.2. A modified mass profile
In a search for a more suitable mass profile for the Coma cluster, we try a modified version of Hernquist’s relation: (19)The power of three is used now instead of the power of two in Eq. (17). This mass profile comes from the density profile: (20)The density goes to a constant as R goes to zero; at large radii, ρ ∝ 1/R^{4}, as in Hernquist’s profile.
The parameters M_{∗} and R_{∗} are found again from the data for the radii of 1.4 and 4.8 Mpc: M_{∗} = 8.7 × 10^{15} M_{⊙},R_{∗} = 2.4 Mpc. The new profile leads to a lower matter mass at 14 Mpc: (21)which is equal to the Eq. (11) value within 15% accuracy. The three profiles can be found in Figs. 1 and 2.
Fig. 2 Matter density ρ vs. radius R. 1) NFW density profile (Eq. (13)): – dashed line; 2) Hernquist profile (Eq. (16)): – thin line; 3) New density profile (Eq. (20)): – thick line. 
5. Discussion
We now discuss some implications of the above results.
5.1. Upper limits to size and mass
The strong effect of dark energy at large radii puts an absolute upper limit on the total size of the cluster. The system can be gravitationally bound only if gravity dominates in its volume (as we mentioned in Sect. 2). In terms of the three different masses, this criterion may be given in the form (22)Both inequalities are met, if the system is not larger than its zerogravity radius (Eq. (5)): R ≤ R_{max} = R_{ZG}.
If the radius of a system with matter mass M_{M} is equal to the maximal radius R = R_{max}, its mean matter density (see BisnovatyiKogan & Chernin 2012) is (23)This relation and the new profile (Eq. (19)) now lead to R_{max} and the corresponding matter mass, M_{max} = M_{M}(R_{max}): (24)For comparison, the other profiles lead to
5.2. Close to the maximal size?
Our studies of nearby systems like the Local Group and the Virgo and Fornax clusters (e.g., Chernin et al. 2010, 2012a) suggest their sizes are not far from the zerogravity radii. Around them, flows of galaxies are seen (Karachentsev et al. 2009; Nasonova et al. 2011), the systems are located in the gravitydominated regions (R < R_{ZG}), and the outflows are at R > R_{ZG}. If the local systems have nearly maximal sizes, this may explain the apparent underdensity of the local universe (Chernin et al. 2012b).
It is tempting to ask if the matter distribution could extend to somewhere near the maximal distance of 20 Mpc in the Coma cluster as well. If so, its mass would be near the upper limit evaluated above, and still consistent with the theory of largescale structure formation that claims the range 2 × 10^{15} < M < 10^{16} M_{⊙} for the most massive bound objects in the Universe (Holz & Perlmutter 2012; Busha et al. 2005). Another implication is the predicted (Eq. (23)) mean matter density of the system = twice the dark energy density, which does not depend on the density profile assumed (Merafina et al. 2012; BisnovatyiKogan & Chernin 2012). Its observational confirmation would directly indicate the key role of dark energy for the structure of the system. Using the cosmological matter density parameter Ω_{m} = 0.27, its mean density contrast would be (27)Another general prediction (Chernin et al. 2006) is that at distances R > R_{ZG} any galaxy in the outflow should have a velocity higher than (28)Here depends on the dark energy density alone. Furthermore, within the simplified model of the EinsteinStraus vacuole, one expects the flow to reach the global Hubble rate at the edge of the vacuole ( ≈ 1.7 R_{ZG}, Teerikorpi & Chernin 2010; Hartwick 2011).
The situation is complicated by the fact that the Coma cluster is not isolated, but lies within the CfA Great Wall.
5.3. Dark energy estimator
We have assumed that the dark energy density inside the Coma cluster is equal to its global value. One can reverse the argument and consider the local dark energy density as an unknown constant ρ_{x}. Its value may be independently estimated using our concept of three cluster masses.
The data (Geller et al. 1999, 2011) give the gravitating mass M_{G} within R = 14 Mpc. The mass M_{M} at the same radius may be found by extrapolating the data from R = 1.4 Mpc and R = 4.8 Mpc using a reasonable matter density profile (Sect. 4). Then with the masses M_{G} and M_{M} known for R = 14 Mpc, we may find the dark energy mass, M_{DE}(R) = M_{G} − M_{M}, at the same radius. Finally, the dark energy density inside the cluster is estimated from (29)With the density profiles of Eqs. (13) and (16) in Sect. 4, Eq. (29) gives for the local dark energy density the values (30)equal to the global value ρ_{DE} within an orderofmagnitude accuracy. (To avoid a circular argument, we do not use the profile of Eq. (20) here, as was partly suggested from considerations related to the global value itself.)
6. Conclusions
Three masses that characterize the structure of a regular cluster (like Coma) are defined as functions of the radius R: the matter (dark matter and baryons) mass M_{M}(R), the darkenergy effective gravitating mass M_{DE} (negative), and the total gravitating mass M_{G}(R) = M_{M} + M_{DE}. Of these masses, only the gravitating mass M_{G} reveals itself directly in observations at various distances from the cluster center. The dark energy mass M_{DE} may be derived using the known global value of the dark energy density. The mass 2.4 × 10^{15} M_{⊙} measured at R = 14 Mpc by Geller et al. (1999, 2011) is the gravitating mass M_{G} inside this radius. The corresponding matter mass is M_{M} ≃ 2 M_{G}.
At small radii, R ≪ 14 Mpc, dark energy effects are almost negligible,  M_{DE}  ≪ M_{M}, and the gravitating mass M_{G} is practically equal to the matter mass M_{M}. At large radii R ≥ 14 Mpc, the antigravity effects are strong and  M_{DE}  ≥ M_{G}.
A new mattermass profile for the Coma cluster reproduces the observational data well and accounts for the dark energy effects in the radius range from 1.4 to 14 Mpc and beyond: , where the constants M_{∗} and R_{∗} can be found from the data for small radii.
The available observational data and the new mass profile give upper limits for the Coma cluster total size, R ≲ 20 Mpc, and total matter mass, M_{M} ≲ 6.2 × 10^{15} M_{⊙}.
Acknowledgments
We thank Yu. N. Efremov, I. D. Karachentsev, D. I. Makarov, O. G. Nasonova, and A. V. Zasov for many useful discussions. A.C. appreciates partial support from the RFBR grant 130200137. The work of G.B.K. was partly supported by RFBR grant 110200602, the RAN Program “Formation and evolution of stars and galaxies”, and Russian Federation President Grant for Support of Leading Scientific Schools NSh3458.2010.2. We also thank the anonymous referee for useful comments.
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All Figures
Fig. 1 Matter mass M vs. radius R for three profiles. 1) NFW mass profile of Eq. (14): – dashed line; 2) Hernquist profile of Eq. (17): – thin line; 3) New mass profile of Eq. (19): – thick line. Here R is in Mpc and M is in 10^{15} M_{⊙}. The values of the matter mass at radii R = 1.4,4.8,14 Mpc are also shown as given by Eqs. (9)–(11) with 1σ error bars. 

In the text 
Fig. 2 Matter density ρ vs. radius R. 1) NFW density profile (Eq. (13)): – dashed line; 2) Hernquist profile (Eq. (16)): – thin line; 3) New density profile (Eq. (20)): – thick line. 

In the text 
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