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Modified Newtonian Dynamics (MOND) is one of the successful theories to explain the dark matter problem in galaxies. However, the data from clusters and the cosmic microwave background (CMB) indicate some dark matter should exist in larger scales. In addition, recent dynamical studies of clusters show that the effect of dark energy should not be ignored in cluster scale. In this article, I will demonstrate how dark energy affects the cluster mass calculation by using MOND. Also, I will show that the calculated cluster mass is consistent with the total matter to baryonic matter ratio obtained by the CMB data.

The dark matter problem is one of the key issues in modern astrophysics. The existence of cold dark matter (CDM) particles is the generally accepted model to tackle the darkmatter problem. However, no such particles have been detected directly. In addition, the CDM model also encounter many well-known unresolved issues such as the cusp problem [1,2], the missing satellite problem [

It has recently been recognized that dark energy exists in our universe. Angus (2009) shows that if MOND theory is needed to satisfy the fitting in CMB spectrum, a large amount of dark energy is required. Therefore, both CDM and MOND theories should consider the effect of dark energy. The local dynamic effects of dark energy were first reported by Chernin, Teerikorpi and Baryshev (2003) [

The effective gravitational acceleration in MOND is given by [5,6]

when, where g_{n} is the Newtonian gravity and a_{0} = 1.2 × 10^{−8} cm·s^{−2}. If we assume that the hot gas with uniform temperature T in cluster is a pressure supported system, we have [

where m_{g} is the mass of a gas particle, is the density profile of the hot gas and g′ is the total gravity of the system. The global dark energy density is = 7 × 10^{−}^{30} g·cm^{−}^{3} [

Since the total gravity can be written as, by using Equations (1)-(3), we have

where M is the total mass of the cluster. By using the gas model in clusters for large r, [

where M_{m} ≈ 6 × 10^{12} M_{⊙}(T/1 keV)^{2} is the total cluster mass without dark energy in classical MOND [^{7} K and r = 1.5 Mpc, C_{2}/C_{1}_{ }≈ 0.17. Therefore, the total cluster mass is 1.17^{2} ≈ 1.4 times larger than the one calculated by the classical MOND. For larger clusters, the effect of dark energy will be much more significant. Therefore, the cluster mass probed from the hot gas by MOND is underestimated if we do not consider the dark energy. It means that more dark matter should exist in clusters.

In fact, observational data shows that the mass of hot gas can be fitted empirically by [

where M_{g} is the total mass of hot gas in a cluster. Therefore, the predicted cluster mass by the classical MOND is 3.5 times larger than the observed baryonic mass (≈ 3.5) [_{b}h^{2} = 0.0024 and Ω_{m}h^{2} = 0.117 respectively [^{2} ≈ 4.8, which is very closed to the ratio obtained by the cosmological density parameters. Therefore, the calculated ratio of total matter to baryonic matter in clusters by using MOND matches the result of the CMB if we include the effect of dark energy.

In this article, we consider the effect of dark energy in clusters. In the MOND regime, the contribution of the anti-gravity effect by dark energy density is significant to the total cluster mass calculation. The total cluster mass for a typical cluster can be 40% larger than the one calculated by the classical MOND. It represents a larger amount of dark matter should exist in clusters. Therefore, the existence of 2 eV active neutrinos in clusters is not enough to account for the missing mass. Since more massive active neutrinos (>2 eV) may violate the experimental bounds [

Since the CDM scenario encounters many fundamental problems including the cusp and the missing satellite problem, the MOND together with the existence of sterile neutrino hot dark matter (HDM) is the only theory which can retain in the recent challenges. Since neutrinos contain mass, there should exist right-handed neutrinos which may indeed be the massive sterile neutrinos [

To conclude, the existence of dark energy can affect the calculated cluster mass by MOND significantly. Also, the MOND + HDM scenario may be one of the best theories to explain the dark matter problem in the future.