Fundamentos para el desarrollo del modelo de evaluación de riesgos.

We have shown in sec. 1.3.3 the intuitive rule that temperature falls with density as the universe expands. Similarly, the rate for the any kind of interaction, Γint,x ≡nxhσint,x|vx|i,

16_{Alternatively, we can specify that we are stating our equations in this section in a comoving frame.} 17

A degenerate form of matter is the counterpart to the perfect fluid and it implies that all particles are in the lowest possible energy state.

which depends on the number density of the interacting particles, n, the interaction cross- section, σ and the particle speed, |v|, decreases with the expanding universe. This leads to three important thermodynamic events for any particle species (see alsoKolb&Turner,1994, ch. 3 and 5), assuming of course that it is completely stable, i.e. that it does not decay into other species:

Decoupling from the heat bath, when a species no longer is in local thermodynamic equilibrium. This means it no longer interacts with the other species in the heat bath and therefore does not receive the transference of entropy from other species. This means that its temperature is no longer related to the number of relativistic degrees of freedom, g∗, but rather redshifts as Tx ∝ 1/a. Specifically, in the adiabatic early

universe it can be shown that before decoupling, the entropy, S = g∗T3a3 = const.,

meaning thataT ∝g∗−1/3 (T being the temperature of the heat bath). Once decoupled,

by considering the equilibrium distribution we can show that: Tx =

TD,x(aD,x/a) ultra-relativistic, Ek∝E∝1/a

TD,x(aD,x/a)2 non-relativistic, Ek∝ |p|2 ∝1/a2 .

(1.55) The temperature, TD at the time of decoupling, tD therefore fully determines the tem-

perature of the species in the future and depends entirely on the number of relativistic degrees of freedom and the scale factor at decoupling via:

aDTD,x∝

1 g1∗/3(tD,x)

. (1.56)

Since g∗ decreases with time as more and more species in equilibrium become nonrela-

tivistic, an early decoupling means that the temperature of the species,Tx will be low.

Such is the assumption in ΛCDM; cold dark matter is defined by its low temperature and hence early decoupling. On the other hand massive neutrinos are one of the last species to decouple from the photon-baryon plasma. For this reason they can be consid- ered hot dark matter (HDM). Any species with properties between these two extremes is considered to be warm dark matter (WDM).

The phase space distribution function for a species thatdecoupled while relativistic and in equilibriumis given by eq. (1.51).

Freeze-out is characterised by the cessation of the processes that keep a matter species in thermal equilibrium with itself18. Similarly to the case of LTE, if a species is in equilibrium with itself when it becomes nonrelativistic, its abundance is exponentially suppressed. This reflects the fact that the species’ abundance, Ωx is determined by

its time of freeze-out. The important factors in calculating the freeze-out time are the particle mass, the cross-section for annihilation in the case of a so called cold relic. However in the case of ahotorwarm relic, these factors are irrelevant and the abundance can be calculated from the effective number of relativistic degrees of freedom at freeze- out, g∗,fo, the particle mass for the species,mx and the effective number of the degrees

of freedom for the species,gx (Kolb& Turner,1994, eq. 5.31):

Ωxh2 ≈70 gx g∗,fo mx keV . (1.57) 18

In other words, the species decouples from itself, but this process is not known as decoupling to avoid confusion.

We meet this equation again in sec. 2.3.3. Let us just note here that a mx ∼ keV

weakly interactive particle would match the measured abundance, if it froze out when g∗,fo ∼500gx.

In this thesis, we assume that we know this abundance to be as measured by the WMAP satellite (Dunkley et al.,2009), which we discuss a bit more in the next chapter (sec 2.2.1).

Becoming nonrelativistic means that the temperature of the species, Tx drops toTx

mx. A transfer of entropy that can heat up the heat bath occurs if a massive species be-

comes nonrelativistic while still in LTE and therefore its density becomes exponentially suppressed, n∝exp(−m/T). This can be calculated fromeq. (1.48)for nonrelativistic particles. We consider the effects of particles that become nonrelativistic later than regular CDM in sec. 2.3.3.

It is important to note that even though many simplifications can be made, in order to treat the process of decoupling and other non-equilibrium dynamics, one should use the full, covariant Boltzmann equation to describe the evolution of the phase space distribution function, f(Pµ, xµ):

ˆ

L[f] =C[f] , (1.58)

where ˆLis the Liouville operator in the nonrelativistic case becomes: ˆ Lnr = d dt +v·∇x+ F m ·∇c , (1.59)

and in the fully covariant, relativistic case: ˆ L=Pµ ∂ ∂xµ −Γ µ νκPνPκ ∂ ∂Pµ , (1.60)

where Γµνκ is now as previously the Christoffel symbol. We revisit the Boltzmann equation

insec. 2.2.4.

The matter distribution in our universe starts out almost completely uniform, but in the era of gravitation, gravitational instability amplifies initial tiny fluctuations into structure. This is the topic of the next two chapters.

CHAPTER

2

The Early, Linear Structure

In the previous chapter we considered the time-varying, spatially-flat, absolutely homogeneous and isotropic FLRW metric (eq 1.14) as a solution to the nonlinear Einstein field equations (eq 1.10). We found the equations for the dynamics of our spacetime (eq 1.44) by solving for the components of the spatially constant FLRW metric.

This is a relatively straightforward solution to the field equations, but it is far from most general. As in many systems of equations, the perturbative approach can be attempted to solve the field equations in the local case, where the components of the metric depend not only on time, but also vary with position. It is clear that such solutions are necessary in order to describe the real universe in which structure exists. In this chapter we will write down the result of using the simplest perturbative extension to the homogeneous FLRW metric. We will still make many assumptions to facilitate the calculations. In particularly, we will only consider the regime in which only thefirst order perturbations to the FLRW metric are considered. This is a reasonable assumption in some regimes, in particular at early times and on large scales. We will attempt to justify why in the following sections.

2.1

Perturbing the field equations

We have shown how the Einstein field equations can be solved fairly straightforwardly with a homogeneous and isotropic FLRW metric (eq 1.12) to describe the geometric LHS (eq 1.10), sourced by the perfect fluid EM tensor on the RHS (eq 1.17).

We now wish to drop the assumption that the field equations are constant in space and reintroduce spatial dependence of the EM and the Einstein tensors (eqns. 1.16 and 1.17). Finding solution in this case is very difficult and in fact has not yet been achieved without the use of approximating assumptions and numerical methods. The method we use in this chapter iscosmological perturbation theory, which makes small perturbations around the analytically obtained homogeneous solutions.