Adopción del informe y observaciones finales

Astrophysical observations depend strongly on their associated physical processes. Therefore, in order to have a reliable simulation, gravity on dark matter particles is not sufficient, and a reliable representation of baryons is necessary. This matter component is modeled assuming it is an ideal fluid. The following three equations describe its behaviour:

The continuity equation dρ dt + ρ∇ · v = ∂ρ ∂t + ∇ · (ρv) = 0 (2.5) Euler’s equation: dv dt = ∂v ∂t + (v · ∇)v = − 1 ρ∇P − ∇Φ (2.6)

The Energy equation (first law of thermodynamics): du dt = ∂u ∂t + v · ∇u = − P ρ∇ · v (2.7)

Where P and u represent the pressure and internal energy per unit mass. For the hydrogen-dominated monoatomic gas at the beginning of the simulation, P = 2/3ρu is the equation of state that connects these two parameters. At this stage, no radiative process is taken into consideration.

These three equations have to be adapted to an expanding universe. After adding the scale factor, they must be solved in the following form:

∂v ∂t + 1 a(v · ∇) v ˙a av = − 1 aρ∇P − 1 a∇Φ (2.8) ∂ρ ∂t + 3 ˙a a ρ + 1 a∇ · (ρv) = 0 (2.9) ∂ (ρu) ∂t + 1 av · ∇ (ρu) = − (ρu + P ) 1 a∇ · v + 3 ˙a a  (2.10)

This system of equations can either be solved with respect to the particles or on the nodes of a grid. Codes which solve for particles are called Lagrangians, codes that solve for grids are Eulerian.

pared to non-collisional gravity alone. Evolved density structures induce motions often supersonic, characterized by shocks and discontinuities. Hydrodynamics traces multiple physical quantities (temperature, pressure, etc.) and has to do so on a wide range of scales and orders of magnitude.

2.3.1

Smoothed-Particle Hydrodynamics

Smoothed Particle Hydrodynamics (SPH) is, among Lagrangian methods, the most widely used and also the easiest to couple with gravity. In SPH as we’ve mentioned earlier in this chapter, the fluid is described by a finite set of mass elements, as opposed to volume elements for Eulerian methods. As the mass resolution is kept fixed, denser regions will have more particles and their average distance will be smaller, therefore SPH is an adaptive code in spatial resolution (Monaghan, 1992). To obtain a continuous fluid, it is necessary to define the kernel smoothing method:

hA(r)i =

Z

W (|r − r0| , h)A(r0) dr03 (2.11) Here the continuous fluid is described by the functionA(r) and the kernel function

W has to be normalized, i.e.:

Z

W (|r − r0| , h) dr3 = 1 (2.12)

h is the SPH smoothing length. It is a variable parameter (Silverman, 1986). When h approaches zero, the kernel collapses into a delta function. The most common

kernel is the B2-spline:

W (r, h) = σ hν          1 − 6(_hr)2+ 6(_hr)3 0 ≤ _hr < 0.5 2(1 − _hr)3 0.5 ≤ _hr < 1 0 1 ≤ r h (2.13)

ν is the dimensionality (3 in 3D, 2 in 2D, etc.). σ is a normalization which varies

depending on ν. In the 3D case, σ = 8/π.

A discrete representation of the continuous fluid quantity A may be represented by the discretized SPH mass elements:

hAii = hA(rj)i = X j mj ρj AjW (|ri− rj| , h) (2.14)

Using the identity

(ρ∇) · A = ∇ (ρ · A) − ρ · (∇A) (2.15) it is possible from the discretized representation of A to derive the fluid equation and hence describe the motion of the particle.

The SPH method has a series of advantages and disadvantages. It conserves by construction physical quantities such as mass, energy, momentum, and entropy. It performs very efficiently in dense environments, where its resolution increases due to the higher number of particles describing the dense region. However, for this very same high density advantage, SPH underperforms in sparse regions (for a review, e.g. Dolag et al., 2008). SPH also has trouble treating fluid instabilities and discontinuities such as shocks.

2.3.2

Eulerian methods

Eulerian methods, or methods that separate the fluid spatially with a grid, perform well both at high and low density regions, however they are limited in spatial resolu- tion, and they may lead to inaccurate predictions of the thermal energy (Ryu et al., 1993; Bryan et al., 1995). A first attempt at solving the 3 hydrodynamical equations on a mesh was the central difference scheme (e.g. Cen, 1992). In these methods, the relevant hydrodynamical quantities such as density, pressure, and velocity were traced at the center of the mesh, and the derivatives were computed with a central finite difference [f (x + 1₂h) − f (x − 1₂h)]/h. These methods however break down

at discontinuities, so artificial viscosities were introduce to treat shocks. Eulerian methods are only first-order accurate as their error is directly proportional to h. More recently, Eulerian methods have geared toward reconstruction approaches which evolve a grid cell taking into consideration several of the neighboring cells (e.g. Colella and Woodward, 1984; Harten, 1983). Reconstruction approaches do not approximate hydrodynamical variables at specific points like the center of the grid of central finite difference methods, rather they compute the cell average ˆun:

ˆ

un =

Z xn+0.5

xn−0.5

fn,u(x) dx (2.16)

Here, fn,u(x) represents the shape of the reconstructed hydrodynamical quantity u,

n identifies the cell. A useful practice in the field is that to solve the Riemann prob-

lem, i.e. to find the solution to an initial value problem containing a conservation equation, continuous boundaries, and a single discontinuity (e.g. a shock). The

solution has to be found for each coordinate direction independently.

2.3.2.1 Leading Cosmological Simulation Codes

GADGET (GAlaxies with Dark matter and Gas intEracT, Springel et al., 2001; Springel, 2005) and all its variations is among the most popular Lagrangian codes, and the first one to become publicly available in its first two versions. It is also the code of choice of the present work. GADGET is fully parallelized with the MPI message-passing library. Notable are the Millennium II and Millennium XXL runs. In its default mode, gadget uses TreePM to compute gravity, and it includes an SPH prescription (ours follows the scheme by Beck et al., 2016). For what concerns Eule- rian codes,ENZO (Bryan et al., 2014) is a popular approach which is written to run in parallel with MPI on supercomputers. Gravity is treated with a PM algorithm combined with a triangular-shaped-cell interpolation. The hydrodynamics is based on Colella and Woodward (1984). Another approach which is gaining traction is

AREPO (Springel, 2010). Its gravity is a standard TreePM, but the hydrodynamics

is computed on a moving mesh obtained by fully-adaptive Voronoi tessellation, and it is computed on a Galilean-invariant Eulerian code. It has the advantage of good treatment of shocks.