Positive charge prevalence in cosmic rays: Room for dark matter in the positron spectrum

The unexpected energy spectrum of the positron/electron ratio is interpreted astrophysically, with a possible exception of the 100–300 GeV range. The data indicate that this ratio, after a decline between 0.5 and 8 GeV, rises steadily with a trend towards saturation at 200–400 GeV. These observations (except for the trend) appear to be in conflict with the diffusive shock acceleration (DSA) mechanism, operating in a single supernova remnant (SNR) shock. We argue that $e^{+}/e^{-}$ ratio can still be explained by the diffusive shock acceleration if positrons are accelerated in a subset of SNR shocks which (i) propagate in clumpy gas media and (ii) are modified by accelerated cosmic ray protons. The protons penetrate into the dense gas clumps upstream to produce positrons and charge the clumps positively. The induced electric field expels positrons into the upstream plasma where they are shock accelerated. Since the shock is modified, these positrons develop a harder spectrum than that of the cosmic ray electrons accelerated in other SNRs. Mixing these populations explains the increase in the $e^{+}/e^{-}$ ratio at $E>8\mathrm{GeV}$. It decreases at $E<8\mathrm{GeV}$ because of a subshock weakening which also results from the shock modification. Contrary to the expelled positrons, most of the antiprotons, electrons, and heavier nuclei, are left unaccelerated inside the clumps. Scenarios for the 100–300 GeV AMS-02 fraction exceeding the model prediction, including, but not limited to, possible dark matter contribution, are also discussed.

Figure 1

SNR shock propagating into ISM with MC upstream.

Figure 2

Flow profile near a CR modified shock. The line “MC” shows a MC trajectory on the phase plane. The speed of MC is considerably higher than that of the flow because of its inertia, resulting in a weaker slow down by the CR pressure than that for the main plasma. The drag from the plasma is also assumed to be not sufficient to slow down the MC significantly.

Figure 3

Matching of low (squared red points) and high (round blue points) momentum solutions of Eq. (20) given by Eq. (35). An overlap region at $s\gtrsim 1$ ensures a smooth transition between the two asymptotics. They deviate from the actual solution at high/low momenta. The matched asymptotic (compound), uniformly valid solution is shown by the solid line. The matching parameters in Eq. (35) are $\beta =0.95$, $B=0.05$, and $A=0.9785$. The subshock compression, $r_s=3$, so $q_s=4.5$.

Figure 4

Positron fraction, represented as a ratio of the $e^{+}$ spectrum to the sum of $e^{+}$ and $e^{-}$, as given in Eq. (37). A fit of the AMS-02 data to the solution of Eq. (20) given by Eq. (35) with $\beta =0.95$, $p_0=6.33$, $A=0.9785$, $B=0.05$ and the mix of species represented by Eq. (37) is shown for the two sets of normalization and $e^{\pm }$ mixing constants, $C$ and $\zeta$. They correspond to a high and low electron contribution to the mix, with $\zeta =9$ and $\zeta =5$, respectively. To comply with the AMS-02 at the spectrum minimum, we fixed the normalization constant at $C=1.45$ and $C=1.25$ for these two cases.

Figure 5

The same as Fig. 4 but plotted for an averaged shock modification: instead of the specified shock modification parameter $p_0=6.33$ used in Fig. 4, two different values of $p_0$ are chosen, $p_{0,1}=5.2$ and $p_{0,2}=8.0$. Shown is the positron fraction, obtained as an average between those obtained for $p_0=p_{0,1}$ and $p_{0,2}$ (solid line). The dashed line extends this solution to higher energies, using a simplified calculation with a fixed value of $p_0\approx (p_{0,1}+p_{0,2})/2$, because the effect of $p_0$ dispersion is not significant at high energies. Parameters in Eq. (37) are fixed at $C=0.061$ and $\zeta =0.35$, so that the saturation level predicted by Eq. (36) is $\approx 0.17$, shown in the upper right corner. AMS-02 error bars added, where they are significant ($E>30\mathrm{GeV}$).

Figure 6

Time evolution of the ion velocity profile, starting from $V_i(x,0)\equiv 0$, as described by Eqs. (7) and (8). The CR source term is prescribed according to Eq. (3), $n_{CR}\propto 1/(t_0-t)$. The ion density remains constant in $x$ at all times. Here $x=0$ corresponds to the mid point of the MC, while $x=1$—to its edge.