Mass determination of groups of galaxies: Effects of the cosmological constant

Abstract

The spherical infall model first developed by Lemaıˆtre and Tolman was modified in order to include the effects of a dark energy term. The resulting velocity–distance relation was evaluated numerically. This equation, when fitted to actual data, permits the simultaneous evaluation of the central mass and of the Hubble parameter. Application of this relation to the Local Group, when the dark energy is modeled by a cosmological constant, yields a total mass for the M31-Milky Way pair of (2.5 ± 0.7) × 10¹²M_⊙, a Hubble parameter H₀ = 74 ± 4 km s⁻¹ Mpc⁻¹ and a 1-D velocity dispersion for the flow of about 39 km s⁻¹. The zero-velocity and the marginally bound surfaces of the Local Group are at about 1.0 and 2.3 Mpc, respectively, from the center of mass. A similar analysis for the Virgo cluster yields a mass of (1.10 ± 0.12) × 10¹⁵M_⊙ and H₀ = 65 ± 9 km s⁻¹ Mpc⁻¹. The zero-velocity is located at a distance of 8.6 ± 0.8 Mpc from the center of the cluster. The predicted peculiar velocity of the Local Group towards Virgo is about 190 km s⁻¹, in agreement with other estimates. Slightly lower masses are derived if the dark energy is represented by a fluid with an equation of state P = wϵ with w = −2/3.

Introduction

New and high quality data on galaxies belonging to nearby groups have improved considerably estimates of masses and mass-to-light ratios (M/L) of these systems. Searches on the POSS II and ESO/SERC plates (Karachentseva and Karachentsev, 1998, Karachentseva and Karachentsev, 2000) as well as “blind” HI surveys (Kilborn et al., 2002) lead to the discovery of new dwarf galaxies, increasing substantially their known population in the local universe. Moreover, in the past years, using HST observations, distances to individual members of nearby groups have been derived from magnitudes of the tip of the red giant branch by Karachentsev and collaborators (Karachentsev, 2005 and references therein), which have permitted a better membership assignment and a more trustful dynamical analysis.

Previous estimates of M/L ratios for nearby groups were around 170M_⊙/L_B,⊙ (Huchra and Geller, 1982). However, virial masses derived from the aforementioned data are significantly smaller, yielding M/L ratios around 10–30M_⊙/L_B,⊙. If these values are correct, the local matter density derived from nearby groups would be only a fraction of the global matter density (Karachentsev, 2005). However, for several groups the crossing time is comparable or even greater than the Hubble time and another approach is necessary to evaluate their masses, since dynamical equilibrium is not yet attained in these cases.

Lynden-Bell, 1981, Sandage, 1986 proposed an alternative method to the virial relation in order to estimate the mass of the Local Group, which can be extended to other systems dominated either by one or a pair of galaxies. Their analysis is essentially based on the spherical infall model. If the motion of bound satellites is supposed to be radial, the resulting parametric equations describe a cycloid. Initially, the radius of a given shell embedding a total mass M expands, attains a maximum value and then collapses. At maximum, when the turnaround radius R₀ is reached, the radial velocity with respect to the center of mass is zero. For a given group, if the velocity field close to the main body, probed by satellites, allows the determination of R₀, then the mass can be calculated straightforwardly from the relation

$$M=\left(\frac{{\mathrm{\pi }}^2{R}_0^3}{8{\mathit{GT}}_0^2}\right)\text{,}$$

where T₀ is the age of the universe and G is the gravitational constant.

Data on the angular power spectrum of temperature fluctuations of the cosmic microwave background radiation derived by WMAP (Spergel et al., 2003) and on the luminosity-distance of type Ia supernovae (Riess et al., 1998, Perlmutter et al., 1999), lead to the so-called “concordant” model, e.g., a flat cosmological model in which Ω_m = 0.3 and Ω_v = 0.7. The later density parameter corresponds to the present contribution of a cosmological constant term or a fluid with negative pressure, dubbed “quintessence” or dark energy. The radial motion leading to the aforementioned M = M(R₀,T₀) relation neglects the effect of such a term, which acts as a “repulsive” force. This repulsive force is proportional to the distance and its effect can be neglected if the zero-velocity surface is close to the center of mass. Turnaround radii of groups are typically of the order of 1 Mpc (Karachentsev, 2005), while the characteristic radius at which gravitation is comparable to the repulsion force is $R_{\ast }=1.1M_{12}^{1/3}\mathrm{Mpc}$, where M₁₂ = M/(10¹²M_⊙) and the Hubble parameter H₀ was taken equal to 70 km s⁻¹Mpc⁻¹. This simple argument suggests that the effect of the cosmological term can not be neglected when deriving masses from the M = M(R₀,T₀) relationship.

In this paper, we revisit the velocity–distance relationship when a dark energy term is included in the dynamical equations and calculate the resulting M = M(R₀,H₀) relation. The presence of the dark energy term has been also invoked as a possible explanation for the smoothness of the local Hubble flow (Chernin, 2001, Teerikorpi et al., 2005) being a further reason to investigate its effects on the M = M(R₀,T₀) relation. For a cosmological density parameter associated to the “vacuum” energy Ω_v = 0.7 (our preferred solution), numerical computations indicate that the “zero-energy” surface, beyond which galaxies will never collapse onto the core, is located at about 2.3R₀. In order to illustrate our results, some applications are made to the Local Group and the Virgo cluster. As we shall see, values of the Hubble parameter resulting from fits of the actual data to the velocity–distance relation including a cosmological term, are in better agreement with recent estimates than those derived from the relation obtained either by Lynden-Bell, 1981, Sandage, 1986. In Section 2, the relevant equations are introduced, in Sections 3 The local group, 4 The Virgo cluster the results are applied to the Local Group and the Virgo cluster and finally, in Section 5 the concluding remarks are given.