The virialized mass of dark matter haloes

Virial mass is used as an estimator for the mass of a dark matter halo. However, the commonly used constant overdensity criterion does not reflect the dynamical structure of haloes. Here, we analyse dark matter cosmological simulations in order to obtain properties of haloes of different masses focusing on the size of the region with zero mean radial velocity. Dark matter inside this region is stationary, and thus the mass of this region is a much better approximation for the virial mass. We call this mass the static mass to distinguish from the commonly used constant overdensity mass. We also study the relation of this static mass with the traditional virial mass, and we find that the matter inside galaxy-sized haloes (M ~ 10¹²M_solar) is underestimated by the virial mass by nearly a factor of 2. At z ~ 0, the virial mass is close to the static mass for cluster-sized haloes (M ~ 10¹⁴M_solar). The same pattern - large haloes having M_vir > M_static - exists at all redshifts, but the transition mass M₀ = M_vir = M_static decreases dramatically with increasing redshift: M₀(z) ~ 3 × 10¹⁵h^-1M_solar (1 + z)^-8.9. When rescaled to the same M₀ haloes clearly demonstrate a self-similar behaviour, which in a statistical sense gives a relation between the static and virial mass. To our surprise, we find that the abundance of haloes with a given static mass, i.e. the static mass function, is very accurately fitted by the Press & Schechter approximation at z = 0, but this approximation breaks at higher redshifts z ~= 1. Instead, the virial mass function is well fitted as usual by the Sheth & Tormen approximation even at z <~ 2. We find an explanation why the static radius can be two to three times larger as compared with the constant overdensity estimate. The traditional estimate is based on the top-hat model, which assumes a constant density and no rms velocities for the matter before it collapses into a halo. Those assumptions fail for small haloes, which find themselves in an environment where density is falling off well outside the virial radius and random velocities grow due to other haloes. Applying the non-stationary Jeans equation, we find that the role of the pressure gradients is significantly larger for small haloes. At some moment, it gets too large and stops the accretion.