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Postcollapse perturbation theory in 1DWarning: this page is better viewed with Firefox or Safari Authors : S. Colombi & A. Taruya Articles:
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1. ContextIt is widely admitted that the matter content of Universe is dominated by a collisionless dark component, which can be assimilated to a collisionless, selfgravitating fluid following VlasovPoisson equations. Without taking into account the expansion of the universe, these equations read, in one dimension, where f(x,v,t) is the phasespace density at position x, velocity v and time t, Φ is the gravitational potential and G is the gravitational constant. The ``cold'' nature of dark matter implies a virtually null initial velocity dispersion, which can be expressed as follows: where ρ_{ini} corresponds to the initial projected density and v_{ini} to the initial velocity field. In one dimension, the dynamics of such a system corresponds to the gravitational interaction between parallel infinite elementary planes of collisionless matter moving along axis x. It is straightforwardly and exactly solved by the linear Lagrangian solution (section 2), as long as shellcrossing does not take place (see, e.g., Novikov 1969). Here, we focus on the multistream regime, which is more problematic. In Colombi (2015) and Taruya & Colombi (2017) we attempt to describe the multistream flow with a Lagrangian perturbative approach modelling the system shortly after collapse, by taking as an initial condition the state of the system at collapse time. While Colombi (2015) focuses on the dynamics of an isolated structure in empty space in a standard physical setup (section 3), Taruya & Colombi (2017) extend the calculations to the cosmological case in a periodic box (section 4). In the latter case, it means one still considers infinite planes interacting gravitationally with each other, but inside a threedimensional expanding universe following standard FriedmannLemaître dynamics. 2. Precollapse dynamicsIn Colombi (2015), we consider the evolution of a single isolated structure with the initial smooth but still rather generic following set up, Following the footsteps of e.g. Shandarin & Zeldovich (1989), we employ a Lagrangian approach to study the dynamics of this system, i.e. trace the phasespace distribution function with a curve [x(q,t),v(q,t)], where q is a Lagrangian coordinate along the curve corresponding to initial position of matter elements. The initial configuration is a horizontal line: Without loss of generality (after proper scaling of time, position and velocity) one can set In the one dimensional case we consider here, the equations of motion can be solved analytically in the single stream regime, i.e. in the regime where function x(q) remains monotonous. The solution is given by linear Lagrangian perturbation theory, often named Zel'dovich approximation (Zel'dovich 1970). For the particular case we consider, we have Collapse time t_{c} corresponds to At this time we have and the density present the wellknown singular behavior (e.g., Arnold, Shandarin & Zeldovich 1982) 3. Postcollapse dynamicsThe above calculations show that positions and velocities can be expressed as third order polynomials of the Lagrangian position q at collapse time: where a, b and c are positive numbers, here a=1, b=2, c=2. This property is in fact always true in the vicinity of the singularity (here, q=0) as long as the density field is smooth, non degenerate, and can be locally expanded in a form given by equation (1) above. In the ballistic approximation, one can extrapolate linearly the dynamics of the system slightly beyond collapse time, starting from initial condition (2) which are expressions linear in time and cubic in Lagrangian position q. The third order polynomial nature of position (and velocity) allows us to resolve the multiple value problem illustrated in figure 1: to compute the force on a point of coordinate x_{0} of the S shape, which is proportional to the mass interior to x_{0}, we need to resolve the equation x(q)=x(q')=x_{0}, which can be done simply if x is a third order polynomial in q. Figure 1: Schematic representation of the phasespace structure of the system studied in Colombi (2015) shortly after collapse time. It has a "S" shape, which can be approximated by third order polynomials in Lagrangian coordinate q for the position x(q) and velocity v(q). In this framework, the multiple value problem x(q_{1})=x(q_{0})=x(q_{2})=x_{0} has a simple solution, which allows one to compute easily the force exerted on point of coordinate x_{0}. This force is simply proportional to the interior mass to x_{0}, i.e. the total mass between the two red dashed vertical lines.
Using the ballistic approximation, that is assuming constant in time velocity from the initial condition given by equation (2), one can thus compute the force after collapse time, hence the correction to motion to leading order in time h=tt_{c} after collapse. In Colombi (2015), we furthermore propose to solve the problem for a more general setup, where the ballistic approximation is replaced with a harmonic oscillator due to a background homogeneous density approximating the contribution of a halo: in this case, the system is the addition of a central S part rotating in phasespace and a slowly growing halo. The central part of the S contracts, while its tails feed the halo. Although conceptually simple, the actual calculation of the correction to the motion is cumbersome, because one has to integrate the force during time and deal with various regimes of the motion. One interesting property of this postcollapse approximation is that in practice, it remains surprisingly accurate until next crossing time, allowing one to iterate the procedure, as proposed and tested in Colombi (2015) using the waterbag code 4. The cosmological caseExtension of one dimensional postcollapse perturbation theory to the cosmological case is straightforward and meaningful since it approximates the dynamics of pancakes in the expanding universe. The starting point is the Zel'dovich approximation which relates the comoving position x to the Lagrangian comoving coordinate q where D_{+} is the growing mode from linear perturbation theory. The normalized displacement field is related to the initial density contrast through where D_{+}(t_{ini}) δ_{L}(q) is the contrast of the initial fluctuations in the density field. Again, in our one dimensional setting, Zel'dovich solution remains exact in all places where the flow is single valued. However, collapse will take place at positions corresponding to local maxima of the initial density field. An expansion at third order of the displacement field in the vicinity of a local maximum q_{m} gives (Taruya & Colombi 2017) which is the analogue of equation (3) in the cosmological case. This is the setup we use as our "ballistic" approximation. Note that because this setup corresponds to Zel'dovich motion, it is not exactly ballistic because the velocity is not constant, but this does not affect the procedure: equation (4) is used to extrapolate the dynamics beyond shellcrossing and a correction at leading order to the motion is computed by solving again the multiple value problem illustrated on Figure 1, with some minor differences in the calculation of the force to account for cosmology. Again, because there are several regimes to consider, the actual calculation of postcollapse motion needs several steps involving additional approximations when integrating the force over time. Thus, the final expressions for position and velocities, given as functions of q and time, are slightly intricate. On the other hand, the motion in single valued regions is given, let us repeat it again, by Zel'dovich dynamics. Our formalism is only applicable until next crossing time, but, coupled with the proper adaptive smoothing procedure, can give account of the nonlinear powerspectrum of density fluctuations very accurately, and surprisingly enough, quite far in the nonlinear regime. To test how postcollapse perturbation theory performs compared to the exact solution, we wrote a Nbody code,
5. Bibliography
