Relic Density

The relic abundance of a typical weakly interacting massive particle (WIMP) is determined by its interaction with other particles in early universe. When WIMP interacts effectively with the cosmic plasma such that it experiences thermal equilibrium with the plasma, its number density is simply determined by its mass through Boltzmann factor e^−mT. However, as the interaction rate become less effective than the Hubble expansion rate, the WIMP enters the freeze-out regime. In order to study the WIMP abundance during the freeze-out, one needs to solve Boltzmann equation as explained in the previous chapter. The relevant quantity of interest is the thermally averaged cross section⟨σv⟩ which encodes the WIMP interaction that controls the WIMP number density.

The thermally averaged cross section⟨σv⟩ contains all possible processes that affect the existence of dark matter species. These include dark matter self annihilation as well as dark

Fig. 5.1 The Feynman diagrams of dark matter annihilation into W⁺W⁻pair. The h_iand Z_i denote the three Higgs boson and three neutral gauge boson mediators. The t on the bottom left panel stands for t-channel diagram.

matter interaction with heavier Z₂odd particles or coannihilation. In case of complex scalar dark matter within G2HDM model, the relevant dark matter self annihilations includes all possible annihilations into SM and non-SM final states. However, in this study, there are only four final states that dominates the dark matter self annihilation: W⁺W⁻, h₁h₁, ZZ, and ¯f f. The h₁is the usual SM Higgs while ¯f f refers to SM fermion pair ( ¯cc, ¯τ τ , ¯bb, ¯tt).

These four final states control the relic abundance of the complex scalar dark matter D. This applies to all possible complex scalar dark matter candidate in this study, starting from inert doublet-like, triplet-like, or Goldstone boson-like dark matter.

The first dominant annihilation cross section is given by W⁺W⁻ final state. The contri-bution coming from this final state is more than 50%. It occurs when the dark matter mass is above 130 GeV as will be explained more detail below. The relevant Feynman diagrams are listed in Fig. 5.1. The second important final state is given by SM Higgs pair h₁h₁. This contribution is typically of order 25% or more and it is relevant for heavy dark matter mass.

The next important final state for heavy dark matter regime is ZZ pair. This will take of order 20% portion of annihilation cross section. For fermion pair, ¯cc, ¯τ τ , and ¯bbcontribute significantly for dark matter mass below 70 GeV. As for ¯tt pair, the contribution is important in the mass range between 170 GeV to 1 TeV. The percentage of this final states vary from

10% to 20% in this mass range. The relevant diagrams for these final states are shown in Figs. 5.2, 5.3 and 5.4.

Fig. 5.2 The Feynman diagrams of dark matter annihilation into h₁h₁pair. The h_idenote the three Higgs boson mediators. The t (u) represents t (u)-channel diagram respectively.

Fig. 5.3 The Feynman diagrams of dark matter annihilation into ZZ pair. The h_idenote the three Higgs boson mediators. The t (u) represents t (u)-channel diagram.

The coannihilation of dark matter with other Z₂odd particles typically occurs when the mass difference between them is less than 10%. In this case, the number density of the dark matter is comparable with those of coannihilating partners. Therefore, one needs to include coannihilation in determining the abundance of dark matter. In this study, the dark matter candidate D can coannihilate with heavier complex scalar e∆, charged Higgs H^±, new W^′ gauge boson, and heavy fermions f^H. However, in the case of heavy fermion coannihilation, we restrict ourselves to exclude this possibility. This can be realized via the following set up

Fig. 5.4 The diagrams relevant for ¯f f final states. The h_i, Z_i, and f_i^H denote the three Higgs boson, three neutral gauge boson as well as three heavy fermion mediators.

for the heavy fermion mass

m_fH = max[1.5 TeV, 1.2m_D] (5.2)

where m_fH denotes heavy fermion mass. This means that the heavy fermion mass is always bigger than 1.5 TeV and when it is bigger than this value, its mass is set to be 1.2 dark matter mass in order to avoid the coannihilation. The 1.5 TeV coming from recent search of new fermion at the LHC [44]. Note that even though the mass difference determine the abundance of coannihilating particles, the coannihilation contribution to dark matter abundance is controlled by its cross section. Therefore, if the coannihilating particle has very small interaction with the dark matter, it will not affect the dark matter abundance.

As we use the micrOMEGAs code [45] in our relic density calculation, the criteria we use in determining the initial abundance of the coannihilating particles is given by the Boltzmann suppression factor B as

B=K₁ (m_i+ m_j)/T

K₁(2m_D/T ) ≈ e^−Xf

(mi+m j−2mD)

mD > B_ε, (5.3)

where K₁is the modified Bessel function of the first kind. It is a function of the coannihilating particle masses m_iand m_j or the dark matter mass m_D. The X_f is the usual freeze-out param-eter which is defined as ^m_T^D. The B_ε is the minimum threshold to account for coannihilation.

Its default value is 10⁻⁶. However in our calculation, we set B_ε to be greater than 10⁻⁴in order to optimize the result.

Finally, to compare the relic density calculation against the experimental value, we will consider the latest result from the PLANCK collaboration [46] for the relic density, Ω_χh²= 0.120± 0.001. In particular, we will require that the model predicts this measured value within 2σ accuracy.