El paradigma de los nuevos movimientos sociales o de la identidad

1.2.1.1. Standard Big Bang cosmology

The standard model of cosmology is not a xed term like the standard model of particle physics [80, 81]. Nevertheless, it contains some basic ingredients: everything started about 1010 _years

ago with the Big Bang. Since then, the universe expanded and cooled down. This expansion was in the rst 10−35 _{- 10}−33 _{seconds exponential and this period is called the ination. Ination is}

the reason, why the universe looks today to a very high level homogeneous and isotropic. While the early universe was very hot and consisted of a soup of matter and energy, more and more particles left the thermal equilibrium with decreasing temperature in the universe. During that time, an asymmetry between matter and anti-matter must have appeared, which prevented a complete annihilation of all matter. This so called baryogenesis is not yet completely understood. When the universe was cooled down to 1016 _{K after 10}−12 _{s, the electroweak interaction splits}

into electromagnetism and weak interaction. After about 10 s the protons and neutrons started to fuse to the light elements deuterium, helium and lithium. This process is called Big Bang nucleosynthesis (BBN). The universe cooled further down and free nuclei and electrons combined to atoms. This process is known as recombination and it is the rst time that photons could travel over large distances. This happened when the universe was about 400.000 years old. The quantitative description of this scenario is based on Einstein's eld equation [82]

Rµν−1

2gµνR =− 8πGN

c4 Tµν+ Λgµν . (1.55)

Here, Rµν and R are the Ricci tensor and scalar which incorporate the geometry of the space

time. gµν is the metric tensor and Tµν the energy-momentum tensor. GN is Newton's constant

and Λ is the cosmological constant which can be used to describe the accelerated expansion of the universe. Eq. (1.55) is a set of second order dierential equations which can analytically only be solved if some symmetries are assumed. Because of the isotropy and homogeneity of the space time, the line element can be approximated by the Friedman-Robertson-Walker metric as [8385] ds2=−c2dt2+ a(t)2  dr2 1_{− kr}2 + r 2_dΩ2  . (1.56)

1.2. DARK MATTER a(t)is the so called scale factor and k describes the curvature of the space time. Solving eq. (1.55) using this metric leads to the Friedman equation [83]

 ˙a a 2 + k a2 = 8πGN 3 ρtot. (1.57)

ρtot is the total averaged energy density of the universe. Obviously, this equation leads to a at

universe (k = 0) if the total density is equal to the critical density dened as ρcrit=

3H2 8πGN

. (1.58)

Here we have introduced the Hubble parameter H(t) = ˙a(t)

a(t). The measured value of the Hubble

parameter is H0= (70.4± 1.4) km_{s Mpc [86]. It is common to normalize the values for the matter}

and energy in the universe with respect to the critical density. This denes the quantity Ωi for

dierent species i of matter and energy as Ωi =

ρi

ρcrit

. (1.59)

1.2.1.2. The freeze out of species

Weakly interacting massive particles (WIMPs) χ with mass mχwere kept in thermal equilibrium

in the hot, early universe [87]. As long as they stayed in equilibrium, their number density neq

is exponentially suppressed with decreasing temperature. When the interaction rate became smaller than the expansion of the universe, i.e.

Γ < H , (1.60)

the particle left thermal equilibrium. After freeze out, the density of the WIMPs stayed approx- imatively constant in a comoving volume while the photons were diluted and redshifted. We can calculate the current WIMP density normalized to today's temperature T0 = 2.7 K and photon

density ρ0 ' 422photonscm3 of the CMB by calculating the decoupling temperature Tf between

WIMPs and the plasma dened by eq. (1.60).

The starting point for the calculation of the relic density is the Boltzmann equation

L[f ] = C[f ] . (1.61)

Here, L is the Liouville operator describing the evolution of the universe and C is the collision operator describing the interactions of particles. The way to rewrite this equation to the better known Lee-Weinberg formula [88]

˙n + 3Hn =−hσ|v|i n2− n2eq

(1.62) is shown in [89]. n is the density of the WIMP, hσ|v|i is the thermal averaged cross section and H is the Hubble parameter. By introducing the yield Y = n_s and freeze out parameter x = mχ

1 5 10 50 100 500 1000 10- 25 10- 20 10- 15 10- 10 10- 5 1 x = m T Y (x )/ Y (x = 1)

Y_∞ for small_hσvi Y∞ for largehσvi

Yeq xf1

Figure 1.5.: Dependence of the yield after freeze out Y∞ on the cross section hσvi. As long as the

particles are in thermal equilibrium, their yield Yeq is exponentially decreasing with temperature T , but

stays constant after freeze out.

we derive nally the Riccati equation dY dx =− xs H(T = m)hσvi Y 2_{− Y}2 eq
. (1.63)

This equation can often be solved by a non relativistic expansion in x: hσvi ' a +b

x. Hence, the

yield Y∞long after the freeze out is given by [90]

Y_∞−1= πg∗ 45 MPmχx −1  a + 3b x  . (1.64)

Since the present density of χ is given by ρχ= mχs0Y∞, the relic density can be easily calculated

by eq. (1.64). A good rst order estimation for the relic density of a WIMP is often Ωχh2'

0.1pb

hσvi . (1.65)

Here, we have inserted the today's entropy s0 = 2970cm−3, the critical density ρc= 1.05410−5 GeVcm3,

the freeze out parameter xf ' 20, the number of degrees of freedom √g∗ ' 10 and the Hubble

parameter h = H0

100 s Mpc. The general dependence between the cross section and the yield iskm

depicted in Fig. 1.5.

If we use the approximation eq. (1.65) for the relic density together with the bounds from CMB, we get an upper mass bound for a WIMP of

mχ≤ 120 TeV . (1.66)

As we will see below, this calculation doesn't hold necessarily for all realistic scenarios: especially the gravitino is so weakly interacting that it probably has never been in thermal equilibrium as discussed in sec. 1.3.2. Furthermore, the eects of coannihilation and resonances have been

1.2. DARK MATTER neglected so far. Both will be discussed for the case of a neutralino as dark matter in chapter 4 and chapter 5. The decays of other particles after the freeze out can change the relic density of a WIMP, too, as we will see now.

1.2.1.3. Late time decays

A particle Ψ with mass m and life time τ decays at time t ∼ H−1 _{∼ τ and at temperature}

TD. Here, we have neglected the freeze out time τF O, hence, the following discussion is valid for

τF O  τ. We assume that the energy content of the universe is dominated by Ψ, i.e. the energy

density of the universe is given by ρ ∼ ρΨ = sYim. The decay temperature TD and life time τ

of Ψ are related by [89] H2(TD)∼ YiTD3

m M_P2 ∼ τ

−2 _{. (1.67)}

If the particles decay into relativistic particles, they rapidly thermalize and yield a post-decay radiation density ρR of

ρR∼ g?TRH4 . (1.68)

By energy conservation this radiation must be equal to the energy density of Ψ just before its demise: H2

DMP2. Hence, the ratio of the entropy density before and after the decay is given by

s_after s_before = g?a3T_RH3 g?a3T_D3 ∼ g 1/4Yim√τ √ MP . (1.69)

This increment of the entropy dilutes the relic density of already frozen out particles by a factor ∆ = 1 + 4

3 M Y

TD

. (1.70)

Finally, the connection between the decay temperature TD of a particle and its decay width Γ is

given by TD =  g_∗π 2 90 1 4 p ΓMP . (1.71)

This dilution is very important for us when we try to solve the cosmological gravitino problem in chapter 3.