content_image/372176.jpg content_image/372175.jpg content_image/372174.jpg
analytical formula). The total spectrum is their convolution, whose explicit form can be found in Eq. (B12). By the way, when we put Eq. (24) into Eq. (22), the τ$_{S}$ factor in the front of source term in Eq. (22) is cancelled by factor Γ$_{total}$ in Eq. (24). So, τ$_{eff}$ or Γ$_{eff}$ is still the free parameter in the calculations of positron propagation. 

Eq. (21) can be approximately solved analytically. The final positron flux observed at the Earth Φ$_{e}$ + ( r$_{⊙}$, E ) is factorized into the convolution of H $_{e}$( E$_{s}$, E ) which encodes the whole astrophysical information and the inject spectrum dN e $_{S}$/dE$_{s}$ 

<img src='content_image/372176.jpg'>

where β$_{e}$ + is the velocity of the observed positron and κ = ρ$_{⊙}$/ ( τ$_{eff}$ m$_{D}$ ). The concrete method to calculate the halo functions for electron H $_{e}$( E$_{s}$, E ) and for anti-proton H ¯ $_{p}$( E ) can be found in Ref. [8, 39]. 

To compare the data, we should consider the cosmic background. It is believed that the astrophysical e ± source is mainly due to supernova explosions (primary e $^{−}$) and the interactions between the CR nuclei and light atoms in interstellar medium (secondary e $^{±}$). They are usually parameterized in the following form [40] 

<img src='content_image/372175.jpg'>

where the unit is GeV − $^{1}$cm − $^{2}$s − $^{1}$sr − $^{1}$. Φ bk,pri e − , Φ bk,sec e − , and Φ bk,sec e + are the background fluxes for the primary electron, secondary electron, and secondary positron, respectively. Then the positron fraction observed by the PAMELA experiment and the normalized total elec- tron/positron fluxes observed by the FERMI and ATIC experiments are respectively given by 

<img src='content_image/372174.jpg'>

where κ ≤ 1 is a parameter which includes the uncertainties of the primary background electron production. 

The PAMELA positron fraction data and FERMI electron/positron data can be fitted very well by choosing proper τ$_{eff}$ and κ . The primary DM S three-body decays into muons, which subsequently decay to electron/positron, give the hard positron energy spectrum that is necessary to explain the steep rise in positron fraction observed by the PAMELA experiment [17]. As for the FERMI experiment, its data show a rather flat spectrum up to 

