The ultimate halo mass in a ΛCDM universe

In the far future of an accelerating ΛCDM cosmology, the cosmic web of large-scale structure consists of a set of increasingly isolated haloes in dynamical equilibrium. We examine the approach of collisionless dark matter to hydrostatic equilibrium using a large N-body simulation evolved to scale factor a= 100, well beyond the vacuum-matter equality epoch, a_eq= 0.75, and 53h^-1 Gyr into the future for a concordance model universe (Ω_m= 0.3, Ω_Λ= 0.7). The radial phase-space structure of haloes - characterized at a<~a_eq by a pair of zero-velocity surfaces that bracket a dynamically active accretion region - simplifies at a>~ 10a_eq when these surfaces merge to create a single zero-velocity surface, clearly defining the halo outer boundary, r_halo, and its enclosed mass, M_halo. This boundary approaches a fixed physical size encompassing a mean interior density ~5 times the critical density, similar to the turnaround value in a classical Einstein-de Sitter model. We relate M_halo to other scales currently used to define halo mass (M₂₀₀, M_vir, M_180b) and find that M₂₀₀ is approximately half of the total asymptotic cluster mass, while M_180b follows the evolution of the inner zero-velocity surface for a<~ 2 but becomes much larger than the total bound mass for a>~ 3. The radial density profile of all bound halo material is well fit by a truncated Hernquist profile. An NFW profile provides a somewhat better fit interior to r₂₀₀ but is much too shallow in the range r₂₀₀ < r < r_halo.