Alternativas Propuestas para el Abordaje de los Elementos Relevantes de la

We will first have a closer look at how the primordial density fluctuations seeded by the inflationary epoch grow in general. This growth is mainly driven by the cold dark matter density component ρCDM which only interacts gravitationally. In the early Universe, how-

ever, the strong coupling of photons and baryons (the photon-baryon fluid) has important consequences on the structure formation. The homogeneous Dark Energy density ρDE, as

the fourth player of perturbation growth, has an additional indirect influence by governing the dynamics of the expanding background cosmology.

The hydrodynamic equations are the starting point for the description of the gravita- tional instability of density fluctuations. Besides Euler’s Equation (momentum conserva- tion), the continuity equation (mass conservation) and Poissions’s Equation are needed to close the system. These equations are transformed to comoving coordinates and the matter density field ρ(x, t) = ¯ρm(t)·[1 +δ(x, t)] is expressed through the dimensionless

density contrastδ(x, t)

δ(x, t) = ρ(x, t)−ρ¯m(t) ¯

ρm(t)

. (3.29)

The system can then be linearized for small fluctuation amplitudes |δ ¿1| on top of the mean matter density field ¯ρmand separated into the homogeneous background solution for

the Hubble expansion (Equ. 3.4) and the inhomogeneous solutions for the evolution of the density contrast δ. The result of the perturbation analysis is a second order differential equation that governs the gravitational amplification of the fractional density contrast δ, which can be expressed in its general form as a damped wave equation

¨ δ+ 2a˙ aδ˙ =δ " 4πGρ¯m− c2 sk2 a2 # . (3.30)

The left hand side contains the acceleration and the damping term, where the Hubble expansion H = ˙a/a acts as a friction force or a Hubble drag. The right hand side of Equ. 3.30 shows the gravitational force due to density fluctuation and an additional term originating from Euler’s Equation when considering fluids with pressure, as is the case for the photon-baryon fluid of the early Universe. The speed of sound for this relativistic fluid with equation-of-state according to Equ. 3.5 is then given by c2

s=∂p/∂ρ'c2/3. For

sufficiently large perturbation wave numbers k= 2π/L, corresponding to small wavelength scales L, the right hand side is negative implying an oscillating solution.

At this point, we will focus on the growing solution, which is obtained for the large scale perturbations withk¿(a/cs)

√

4πGρ¯min the early Universe or as the general perturbation

growth mode after matter-radiation decoupling, when the baryonic pressure vanishes. For a negligible oscillation term on the right hand side, Equ. 3.30 does not contain any spatial derivatives implying that the time evolution for the perturbation growth is separable from the initial spatial perturbation field δi+(x, ti) according to

δ(x, t) = D+(t)·δi+(x, ti) +D−(t)·δi−(x, ti). (3.31)

The decaying solution D−(t) is of little physical interest and will not be considered. The general expression for the growing perturbation solution D+(t) (e.g. Borgani, 2006 ) in

linear approximation as a function of redshift z is given by D+(z) = 5 2ΩmE(z)· Z _∞ z 1 +z0 E(z0₎3dz 0 _{. (3.32)}

In an Einstein-de Sitter Universe (Ωm= 1) the perturbation amplitudes grow proportional

to the scale factor D+∝a∝(1+z)−1∝(t/ti)2/3. For an empty Universe (Ωtot≡0), (virtual)

perturbations are frozen and do not grow at all. In a low density Universe (e.g. Ωm∼0.3,

ΩDE= 0) the driving gravitational force on the right hand side of Equ. 3.30 is decreased

resulting in a slower growth rate at low redshifts. At early epochs, however, the matter density of such models is still close to critical, i.e. to the Einstein-de Sitter case, with an analogous fast early growth phase which slows once the matter density drops significantly

below the critical value. When looking backwards in time from the present perspective, the effect of this slowed structure evolution is an increased number of objects, e.g. clusters, at high redshift relative to a critical matter density Universe. For a concordance cosmological model (Ωm ∼ 0.3, ΩDE = 0.7), the addition of a Dark Energy component results in an

intermediate degree of evolution, since the Hubble expansion must have been slower in the past, relative to an open Universe, leading to a smaller friction term in Equ. 3.30. Note that the ‘Hubble drag’term depends on the cosmological parameters, which implicitly influence the perturbation growth in addition to the explicit dependance on the mean matter density

¯ ρm.

Top-hat collapse

An important approach to tackle the non-linear structure evolution regime is provided by Birkhoff’s theorem, which states that a closed sphere within a homogeneous Universe evolves independent of its surroundings, i.e. as if no external forces are exerted on the sphere10_{. This implies that any overdense region of the Universe can be conceived as a}

homogeneous mini-Universe with the evolution driven by the local density parameters of the region under consideration.

The spherical top-hat model is the only system for which the collapse of an overdense region of the Universe can be treated analytically. For this we consider a spherical region with radius R and a homogeneous overdensity δTH embedded in a background field with

constant density ¯ρm. According to Birkhoff’s theorem the evolution of this region is inde-

pendent of the background field and can be described by the Newtonian approximation of the first Friedmann equation 3.1, i.e. with vanishing pressure and zero cosmological con- stant term. In analogy to a closed Universe, this finite region will expand up to a maximum radius Rmax at turn-around time tturn and then re-collapse in a time-symmetric fashion at

timetcol= 2·tturn. In practice, the overdense region will not collapse to a point but stabilize

in a bound dynamic equilibrium state at radius Rvir'Rmax/2 once the Virial condition

2·E¯kin+ ¯Epot= 0 is fulfilled. The gravitationally bound, virialized object with total energy11

Etot= ¯Epot/2 =−3GM2/(5·2Rvir) is now detached from the Hubble flow, i.e. it does not

take part in the overall expansion of the background cosmology.

In the case of an Einstein-de Sitter cosmology with Ωm= 1 some universal characteristics

of the collapse process can be specified, with only small variations for other cosmologies. The density contrast at turn-around time tturn, i.e. at the time of decoupling from the

Hubble flow, is aboutδturn'4.55. At collapse timetcolthe contrast has increased to its final

(universal) equilibrium value12 _{of δ}

vir'177, which gives rise to the widely used definition

of R200 as the cluster radius with an average density of 200 times the critical density ρcr.

In comparison, the linear extrapolation of the density contrast according to Equ. 3.32 at tcol yieldsδvir(linear)'1.69, and at turn-aroundδturn(linear)'1.06, emphasizing the onset

10_{In reality residual tidal forces are expected which can be neglected for this discussion.} 11_{A factor of 3/5 arises for the potential energy of a sphere with uniform density.}

12_{This density contrast is roughly obtained from the turn-around density by considering the compression}

of the non-linear regime and the break-down of the linear approximation during object collapse. Note that a virialized cluster with final radiusRvir'1.5h−701Mpc originates from

a collapsed region of size ∼6·Rvir'10h−701Mpc.

These considerations imply that any spherical overdensities reaching the critical density contrastδturndetach from the Hubble flow and form a virialized object attvir=tcol'2·tturn.

For the cold dark matter standard model with a scale invariant initial perturbation spec- trum, sub-galactic halos are the first to decouple from the Hubble flow, since perturbations on small length scales have greater amplitudes (see Equ. 3.36). Galaxy clusters on the other hand have formed relatively recently and hence appear late on the cosmic stage, marking the largest mass scales that have had sufficient time to virialize.

The top-hat collapse model enables important physical insights into the cluster forma- tion process. However, it rests upon many simplifications (spherical symmetry, homoge- neous mass distribution, no external tidal forces, only gravitational physics) which are at best only an approximate description of the cluster formation conditions. The Millennium Run simulation of Springel et al. (2005) provides the most detailed and realistic cluster formation scenarios currently available. Figure 3.4 shows simulation snap-shots of the for- mation of a rich galaxy cluster over the last 10 Giga years of cosmic time (2≥z≥0). Besides the detailed dark matter distribution in the left panels, these simulations also trace the intracluster medium gas density (center panels) and the ICM temperature (right panels), which are directly accessible through X-ray observations. Note that the ICM gas density is a very good tracer of the underlying dark matter distribution, and that the substructure and asymmetry increases drastically beyond z >_∼1.