Pulsars versus dark matter interpretation of ATIC/PAMELA

In this paper, we study the flux of electrons and positrons injected by pulsars and by annihilating or decaying dark matter in the context of recent ATIC, PAMELA, Fermi, and HESS data. We review the flux from a single pulsar and derive the flux from a distribution of pulsars. We point out that the particle acceleration in the pulsar magnetosphere is insufficient to explain the observed excess of electrons and positrons with energy $E\sim 1\mathrm{TeV}$ and one has to take into account an additional acceleration of electrons at the termination shock between the pulsar and its wind nebula. We show that at energies less than a few hundred GeV, the expected flux from a continuous distribution of pulsars provides a good approximation to the expected flux from pulsars in the Australia Telescope National Facility catalog. At higher energies, we demonstrate that the electron/positron flux measured at the Earth will be dominated by a few young nearby pulsars, and therefore the spectrum would contain bumplike features. We argue that the presence of such features at high energies would strongly suggest a pulsar origin of the anomalous contribution to electron and positron fluxes. The absence of features either points to a dark matter origin or constrains pulsar models in such a way that the fluctuations are suppressed. Also we derive that the features can be partially smeared due to spatial variation of the energy losses during propagation.

Figure 1

Time evolution of $e^{+}{e}^{-}$ flux on the Earth from a pulsar at a distance of 1 kpc with $\eta W_0=3\times {10}^{49}\mathrm{erg}$, an injection index $n=1.6$, and an injection cutoff $M=10\mathrm{TeV}$. The diffusion and energy losses are described in Sec. 2a. We assume the delta-function approximation for the emission from the pulsar, $Q(\mathbf{x},E,t)=Q(E)\delta (\mathbf{x})\delta (t)$. The flux from a young pulsar (the 3 kyr curve on the right) has an exponential suppression because the electrons have not had enough time to diffuse from the pulsar to the Earth. The cutoff moves to the left due to cooling of electrons and becomes sharper. After reaching a maximal value, the flux decreases since the electrons diffuse over a large volume.

Figure 2

Electron and positron flux from a single pulsar together with a primary background $\sim E^{-3.3}$ and a secondary background $\sim E^{-3.6}$. The pulsar is at a distance of 0.3 kpc. It has $\eta W_0=2.2\times {10}^{49}\mathrm{erg}$, age of 200 kyr, and injection index and cutoff $n=1.6$ and $M=10\mathrm{TeV}$, respectively. The propagation parameters are described in Sec. 2a. The cutoff $M\gg 1\mathrm{TeV}$ results in a significant bump around 1 TeV which is consistent with the ATIC data. For a smaller injection cutoff $M\sim 1\mathrm{TeV}$, the flux from the pulsar takes the form of a power law with an exponential cutoff that can be used to fit the Fermi and PAMELA data (see, e.g., 56).

Figure 3

The expected spectrum from the continuous flux distribution and that from pulsars in the ATNF catalog pulsar [28]. The latter is calculated using $\eta =0.065$, $n=1.5$, and a pulsar time scale $\tau =1\mathrm{kyr}$ for each pulsar. This last fact, in conjunction with its spin-down age and current spin-down luminosity, is used to calculate each pulsar’s initial rotational energy through Eq. (29). We also use the value of the propagation parameters given in Sec. 2a. Several hundred pulsars contribute below 300 GeV and the continuous distribution provides a good approximation for these energies. Above 300 GeV, there are only $\sim 10$ contributing pulsars, and the observed flux in this energy range is strongly dependent on their individual properties. The reason for the significant discrepancy between these two curves above 2 TeV has to do with the actual local distribution of pulsars versus the averaged flux seen by many observers in the Galaxy, as discussed in Sec. 3b.

Figure 4

The predicted flux from pulsars in the ATNF catalog calculated using the same procedure as in Fig. 3 but accounting for spatial variations of energy losses as described in Appendix . The assumed backgrounds are the same as in Fig. 2.

Figure 5

Statistical cutoff as a function of the diffusion index and the birth rate of pulsars in the Galaxy. The cutoff in $e^{+}{e}^{-}$ flux from pulsars is determined by the age of the youngest pulsar within the diffusion distance from the Earth. The average such cutoff is a universal quantity that depends on the properties of ISM (the energy losses and the diffusion coefficient) and on the pulsar birth rate, but it is insensitive to the properties of the injection spectrum from the pulsars. We assume $D_0=100{\mathrm{pc}}^2{\mathrm{kyr}}^{-1}$ and $b_0=5\times {10}^{-6}{\mathrm{GeV}}^{-1}{\mathrm{kyr}}^{-1}$.

Figure 6

The flux from a continuous distribution of pulsars. The parameters are chosen to fit the Fermi and PAMELA data points, $\eta W_0=6.5\times {10}^{48}\mathrm{erg}$ and $n=1.5$. In this plot, instead of the injection cutoff $M=10\mathrm{TeV}$, we use the statistical cutoff $M_{\mathrm{stat}}=1\mathrm{TeV}$. The backgrounds are the same as in Fig. 2. The propagation parameters are described in Sec. 2a.

Figure 7

Flux from DM model in 10 with the annihilation chain $\chi +\chi \to \varphi +\varphi \to 2e^{+}+2e^{-}$, $M_{\mathrm{DM}}=1\mathrm{TeV}$ and a boost factor $\mathrm{BF}\approx 500$. The primary background is $\sim E^{-3.3}$ and the secondary background is $\sim E^{-3.6}$.

Figure 8

The fluxes from annihilating dark matter, from a single pulsar, and from a continuous distribution of pulsars can be made similar, depending on the parameters of the models. The flux from a collection of pulsars may have significant deviations from a continuous curve. This property can be used to distinguish the pulsars from the sources producing a featureless spectrum. The flux from pulsars in the ATNF catalog is the same as in Fig. 4. The single pulsar has the age $t=100\mathrm{kyr}$, distance 0.3 kpc, $\eta W_0=9.2\times {10}^{48}\mathrm{erg}$, and $n=1.6$. The continuous pulsar distribution has $N_{\mathrm{b}}=1.8{\mathrm{kyr}}^{-1}$, $\eta W_0=6.5\times {10}^{48}\mathrm{erg}$, $n=1.5$, and $M_{\mathrm{stat}}=2\mathrm{TeV}$. The dark matter model is the same as in Sec. 4 but with $M_{\mathrm{DM}}=2\mathrm{TeV}$ and $\mathrm{BF}=2000$. The ISM properties are the same as in Sec. 2a.

Figure 9

The left panel has 41 pulsars within 4 kpc from the Earth and younger than 300 kyr. Assuming the braking index 3 and the pulsar time scale $\tau =1\mathrm{kyr}$ we find the average power $p={\mathrm{Log}}_{10}(W_0/\mathrm{erg})=49.1\pm 1.1$ with ${\chi }^2/\mathrm{dof}=1.2$. For the right panel we select the pulsars from the ATNF catalog that have braking index $2<k<10$ (there are 20 such pulsars). Assuming the pulsar time scale $\tau =1\mathrm{kyr}$, these pulsars have $p=48.9\pm 1.1$ with ${\chi }^2/\mathrm{dof}=0.3$. For the pulsar time scale $\tau =10\mathrm{kyr}$, the left (right) selection of pulsars would have $p=48.2\pm 1.1$ ($48.1\pm 1.2$). Thus, for $\tau =1\mathrm{kyr}$ (10 kyr) the average initial rotational energy is ${\overline{W}}_0\approx {10}^{50}\mathrm{erg}$ (${10}^{49}\mathrm{erg}$).

Figure 10

The fits of the continuous distribution flux to the ATIC and PAMELA data. The best-fit parameters are $\delta =0.48$, $b_0=1.05\times {10}^{-16}{\mathrm{GeV}}^{-1}{\mathrm{s}}^{-1}$, $\eta W_0=4.9\times {10}^{48}\mathrm{erg}$, $n=1.7$, and $M_{\mathrm{stat}}=1.0\mathrm{TeV}$. The contours show the confidence levels relative to the best fit, ${\sigma }^2=({\chi }^2-{\chi }_{*}^2)/\mathrm{dof}$, where ${\chi }_{*}^2$ is the chi squared for the best-fit parameters.