Abstract
The phase-space structure of primordial dark matter halos is revisited using cosmological simulations with three sine waves and cold dark matter (CDM) initial conditions. The simulations are performed with the tessellation based Vlasov solver ColDICE and a particle-mesh (PM) N-body code. The analyses include projected density, phase-space diagrams, radial density ρ(r), and pseudo-phase space density: Q(r) = ρ(r)/σv(r)3 with σv the local velocity dispersion. Particular attention is paid to force and mass resolution. Because the phase-space sheet complexity, estimated in terms of total volume and simplex (tetrahedron) count, increases very quickly, ColDICE can follow only the early violent relaxation phase of halo formation. During the violent relaxation phase, agreement between ColDICE and PM simulations having one particle per cell or more is excellent and halos have a power-law density profile, ρ(r) ∝ r−α, α ∈ [1.5, 1.8]. This slope, measured prior to any merger, is slightly larger than in the literature. The phase-space diagrams evidence complex but coherent patterns with clear signatures of self-similarity in the sine wave simulations, while the CDM halos are somewhat scribbly. After additional mass resolution tests, the PM simulations are used to follow the next stages of evolution. The power law progressively breaks down with a convergence of the density profile to the well-known Navarro–Frenk–White universal attractor, irrespective of initial conditions, that is even in the three-sine-wave simulations. This demonstrates again that mergers do not represent a necessary condition for convergence to the dynamical attractor. Not surprisingly, the measured pseudo phase-space density is a power law Q(r) ∝ r−αQ, with αQ close to the prediction of secondary spherical infall model, αQ ≃ 1.875. However this property is also verified during the early relaxation phase, which is non-trivial.
Highlights
In our current understanding of large-scale structure formation, the main matter component of the Universe is cold dark matter (CDM; e.g., Peebles 1982, 1984; Blumenthal et al 1984), which can be modelled as a self-gravitating collisionless fluid obeying Vlasov-Poisson equations: ∂f ∂t + u.∇r f − ∇rφ.∇u f = 0, (1)∆rφ = 4π G ρ = 4π G f (r, u, t) d3u, (2)where f (r, u, t) is the phase-space density at physical position r and velocity u, φ the gravitational potential, and G the gravitational constant
Calculation of the density on the computational mesh in ColDICE corresponds in practice to nearest grid point (NGP) interpolation (e.g., Hockney & Eastwood 1988) in terms of convolution, so one can expect a loss of effective force resolution of the PM code compared to ColDICE when using the same value of ng, which in practice will turn out to be of about a factor 1.7 in the subsequent analyses
A non-exponential behaviour of the simplex count reflects a quiescent behaviour of the dynamics or, in other words, the absence of chaos. This is what we find for the early, monolithic, violent relaxation phase of halos growing from initial conditions with three sine waves, except possibly in the axisymmetric case, but this is a very degenerate configuration
Summary
In our current understanding of large-scale structure formation, the main matter component of the Universe is cold dark matter (CDM; e.g., Peebles 1982, 1984; Blumenthal et al 1984), which can be modelled as a self-gravitating collisionless fluid obeying Vlasov-Poisson equations:. It should be noted that the pre-collapse phase (i) of the evolution of the phase-space sheet is well described by perturbation theory; the early violent relaxation phase (ii) and the convergence to the universal dynamical attractor (iii) are still poorly understood from the theoretical point of view despite numerous investigations involving maximum of entropy methods (e.g., Lynden-Bell 1967; Hjorth & Williams 2010; Pontzen & Governato 2013; Carron & Szapudi 2013), the Jeans equation (e.g., Taylor & Navarro 2001; Dehnen & McLaughlin 2005; Ogiya & Hahn 2018), post-collapse perturbation theory (e.g., Colombi 2015; Taruya & Colombi 2017; Rampf et al 2019), as well as self-similar solutions and secondary infall models (e.g., Fillmore & Goldreich 1984; Bertschinger 1985; Henriksen & Widrow 1995; Sikivie et al 1997; Zukin & Bertschinger 2010a,b; Alard 2013; Sugiura et al 2020).
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