Suppression of the neutralino relic density with supersymmetric CP violation

We study pair annihilations of the neutralino dark matter in the minimal supersymmetric standard model with CP violation. We consider the case that the higgsino mass and the trilinear scalar couplings have CP-violating phases of order unity, taking a scenario that the scalar fermions in the first two generations are much heavier than those in the third generation to avoid a severe constraint from experimental limits on electric dipole moments. It is found that, when the lightest neutralino ($\chi$) is bino-like, the cross sections of the $W$-boson pair production $\chi\chi$ $\to$ $W^+W^-$ and the lightest Higgs boson pair production $\chi\chi$ $\to$ $H^0_2 H^0_2$ for nonrelativistic neutralinos can be significantly enhanced by the phase of the higgsino mass. The relic density of the neutralino can be considerably suppressed by this effect. However, even this suppression is not enough to make bino-like dark matter consistent with a cosmological constraint. We also discuss the effect of CP violation on the positron flux from neutralino pair annihilations in the galactic halo.