Dark energy in the two-body problem: The local group of galaxies

The two-body problem of classical mechanics can be extended in a natural way by introducing a universal dark-energy background, which acts as a third dynamical factor. For real systems of galaxies, the corresponding additional acceleration is described by general relativity in the Newtonian limit, in which deviations from the Minkowski metric are very small. It is shown that this acceleration has the same form in different inertial and non-inertial reference frames. The invariance of the acceleration produced by dark energy reflects the fact that, according to its mechanical properties, dark energy is a vacuum that is comoving with any motion. In this generalized formulation, as in the classical case, the two-body problem with a dark-energy background reduces to the motion of a single body in a central field. Two problems of this kind are considered for the Local Group of galaxies. The first, “internal,” problem concerns the binary system formed by the main bodies of the Local Group: our galaxy and M31. The subject of the second, “external,” problem is the binary system formed by the Local Group as a whole and its closest neighbor, the Virgo Cluster. In the internal problem, the effect of the dark energy is that the binary system is not bound if its mass does not exceed 3 × 10 12 M ⊙ , which is allowed by the observational data. The external problem demonstrates the possibility of an evolutionary scenario in which a group could initially be located in the volume of a cluster, but then leave it and, moving away with an acceleration created by dark energy, arrive at the observed distance from the cluster.