Dark matter and pulsar origins of the rising cosmic ray positron fraction in light of new data from the AMS

The rise of the cosmic ray positron fraction with energy, as first observed with high confidence by PAMELA, implies that a large flux of high energy positrons has been recently (or is being currently) injected into the local volume of the Milky Way. With the new and much more precise measurement of the positron fraction recently provided by the Alpha Magnetic Spectrometer (AMS), we revisit the question of the origin of these high energy positrons. We find that while some dark matter models (annihilating directly to electrons or muons) no longer appear to be capable of accommodating these data, other models in which $\sim 1–3\mathrm{TeV}$ dark matter particles annihilate to unstable intermediate states could still be responsible for the observed signal. Nearby pulsars also remain capable of explaining the observed positron fraction. Future measurements of the positron fraction by the AMS Collaboration (using a larger data set) combined with their anticipated measurements of various cosmic ray secondary-to-primary ratios may enable us to further discriminate between these remaining scenarios.

Figure 1

The predicted cosmic ray positron fraction (left) and $\mathrm{\text{electron}}+\mathrm{\text{positron}}$ spectrum (right) in dark matter models annihilating to $e^{+}{e}^{-}$ (top) and to ${\mu }^{+}{\mu }^{-}$ (bottom). The error bars shown represent the positron fraction as measured by the AMS (black, left) and PAMELA (red, left), and the $\mathrm{\text{electron}}+\mathrm{\text{positron}}$ spectrum as measured by Fermi and AMS-01 (black, right). In the Fermi error bars we do not include the overall shift from the energy resolution uncertainty. In each case, we have adopted a propagation model that provides a good fit to the various secondary-to-primary ratios as described in the text and with a diffusion zone half-width of $L=4\mathrm{kpc}$. The expected backgrounds are shown as black dotted lines. For a given mass and channel we fit the annihilation cross section to the AMS positron fraction ratio data and also show the equivalent result for the total lepton flux. For each annihilation channel, we show results for two masses. For annihilations to $e^{+}{e}^{-}$ and a mass of 350 GeV (900 GeV), we have used a thermally averaged annihilation cross section of $⟨\sigma v⟩=4.0\times {10}^{-25}{\mathrm{cm}}^3/\mathrm{s}$ ($2.2\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$). Our ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ is 15.3(10.6) from the AMS data and 6.5(9.2) from the Fermi data. For annihilations to ${\mu }^{+}{\mu }^{-}$ and a mass of 600 GeV (1.5 TeV), we have used a thermally averaged annihilation cross section of $1.6\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($8.5\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$), resulting in a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 9.3(14.6) to the AMS and 7.6(0.84) to the Fermi data. These models can not provide a consistent picture to the combined AMS and Fermi lepton flux measurements.

Figure 2

As in Fig. 1 but for dark matter which annihilates into a pair of intermediate states, $\varphi$, which proceed to decay to $e^{+}{e}^{-}$ (first row), to ${\mu }^{+}{\mu }^{-}$ (second row), to ${\pi }^{+}{\pi }^{-}$ (third row), and to a $1:1:2$ ratio of $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, and ${\pi }^{+}{\pi }^{-}$ (fourth row). For annihilations to $2e^{+}2e^{-}$ and a mass of 400 GeV (1.2 TeV), we have used a thermally averaged annihilation cross section of $⟨\sigma v⟩=6.6\times {10}^{-25}{\mathrm{cm}}^3/\mathrm{s}$ ($5.4\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$) resulting in a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 10.3(15.0) to the AMS and 8.8(2.0) to the Fermi data. For annihilations to $2{\mu }^{+}2{\mu }^{-}$ and a mass of 800 GeV (2.5 TeV), we have used a thermally averaged annihilation cross section of $3.4\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($2.6\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) with a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of 5.1(14.0) and 12.6(0.64) to the AMS and Fermi data, respectively. For annihilations to $2{\pi }^{+}2{\pi }^{-}$ and a mass of 1.0 TeV (3.0 TeV), we have used a thermally averaged annihilation cross section of $5.8\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($4.1\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) giving ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fits of 3.7(11.7) to the AMS and 19.4(0.61) to the Fermi data. Finally for annihilations to a $1:1:2$ ratio of $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, and ${\pi }^{+}{\pi }^{-}$ final states with a mass of 500 GeV (1.6 TeV), we have used a thermally averaged annihilation cross section of $1.7\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($1.2\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) which have ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fits of 2.3(9.5) to the AMS and 11.3(1.34) to the Fermi data. While there is a preference for DM models with softer annihilation $e^{\pm }$ spectra, for the given propagation/background assumptions all these models are excluded.

Figure 3

The impact of the diffusion zone half-width, $L$, on the positron fraction in a representative DM model. In each case, we have chosen the diffusion coefficient to fit the nonleptonic background cosmic ray measurements. The dotted, dashed, dot-dashed, and solid lines correspond to $L=1$, 2, 4, and 8 kpc, respectively. The model used in this figure consists of a 1 TeV dark matter particle that annihilates to a pair of intermediate states, each decaying to ${\pi }^{+}{\pi }^{-}$, with a cross section given by 16, 9.5, 5.7, and $3.5\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$ and a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ of 16.6, 9.9, 3.7, and 1.18 for $L=1$, 2, 4, and 8 kpc, respectively.

Figure 4

The same as in Figs. 1, 2 but for a diffusion zone half-width of $L=2\mathrm{kpc}$. For annihilations to $2e^{+}2e^{-}$ and a mass of 400 GeV (1200 GeV), we have used a thermally averaged annihilation cross section of $⟨\sigma v⟩=1.2\times {10}^{-25}{\mathrm{cm}}^3/\mathrm{s}$ ($1.2\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$), with a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 23(33) to the AMS and 6.6(32) to the Fermi data. For annihilations to $2{\mu }^{+}2{\mu }^{-}$ and a mass of 800 GeV (2.5 TeV), we have used a thermally averaged annihilation cross section of $5.6\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($5.1\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) resulting in a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 13.1(30) to the AMS and 9.5(7.3) to the Fermi data. For annihilations to $2{\pi }^{+}2{\pi }^{-}$ and a mass of 1.0 TeV (3.0 TeV), we have used a thermally averaged annihilation cross section of $9.8\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($7.8\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$) with a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 9.9(25) to the AMS and 6.8(4.2) to the Fermi data. For annihilations to a $1:1:2$ ratio of $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, and ${\pi }^{+}{\pi }^{-}$ final states with a mass of 500 GeV (1.6 TeV), we have used a thermally averaged annihilation cross section of $2.7\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($2.3\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) resulting in a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 7.3(22) to the AMS and 8.9(0.70) to the Fermi data.

Figure 5

The same as in Figs. 1, 2, 4 but for a diffusion zone half-width of $L=8\mathrm{kpc}$. For annihilations to $2{\mu }^{+}2{\mu }^{-}$ and a mass of 1.0 TeV (2.5 TeV), we have used a thermally averaged annihilation cross section of $2.9\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($1.5\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) providing a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 1.18(4.4) to the AMS and 15.4(5.1) to the Fermi data. For annihilations to $2{\pi }^{+}2{\pi }^{-}$ and a mass of 1.0 TeV (3.0 TeV), we have used a thermally averaged annihilation cross section of $3.5\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($2.3\times {10}^{-23}{\mathrm{cm}}^3/\mathrm{s}$) giving a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 0.81(3.9) to the AMS and 15.2(3.8) to the Fermi data. For annihilations to a $1:1:2$ ratio of $e^{+}{e}^{-}$, ${\mu }^{+}{\mu }^{-}$, and ${\pi }^{+}{\pi }^{-}$ final states with a mass of 700 GeV (1.6 TeV), we have used a thermally averaged annihilation cross section of $1.6\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$ ($6.5\times {10}^{-24}{\mathrm{cm}}^3/\mathrm{s}$) with a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 0.83(3.0) to the AMS and 13.4(7.6) to the Fermi data.

Figure 6

The same as in Figs. 1, 2, 4, 5 but for a diffusion zone half-width of $L=8\mathrm{kpc}$, and for broken power-law spectrum of electrons injected from cosmic ray sources ($d{N}_{e^{-}}/d{E}_{e^{-}}\propto E_e^{-2.65}$ below 85 GeV and $d{N}_{e^{-}}/d{E}_{e^{-}}\propto E_e^{-2.3}$ above 85 GeV). The cross sections are the same as those given in the caption of Fig. 5. With this cosmic ray background, we show the dark matter models compared to the measurements of the cosmic ray positron fraction and the overall leptonic spectrum. Even with the presence of a break, there is a preference towards models with softer injection $e^{\pm }$ spectra, with the 1.6 TeV to the $e^{\pm }$, ${\mu }^{\pm }$, ${\pi }^{\pm }$ case providing the best ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit to the AMS (Fermi) lepton data of 0.82(0.51). The 2.5 TeV to $2{\mu }^{+}$ $2{\mu }^{-}$ gives a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit of 1.32(1.07) and the 3.0 TeV to $2{\pi }^{+}$ $2{\pi }^{-}$ a fit of 1.00(1.03). We remind that in the Fermi error bars we do not include an overall shift from the energy resolution uncertainty.

Figure 7

The predicted cosmic ray positron fraction (left) and $\mathrm{\text{electron}}+\mathrm{\text{positron}}$ spectrum (right) from the sum of all pulsars throughout the Milky Way, for an injected spectrum of $d{N}_{e^{\pm }}/d{E}_{e^{\pm }}\propto E_{e^{\pm }}^{-1.55}\exp (-E_{e^{\pm }}/600\mathrm{GeV})$ and a diffusion zone half-width of $L=4\mathrm{kpc}$ (top), and for an injected spectrum of $d{N}_{e^{\pm }}/d{E}_{e^{\pm }}\propto E_{e^{\pm }}^{-1.65}\exp (-E_{e^{\pm }}/600\mathrm{GeV})$ and a diffusion zone half-width of $L=8\mathrm{kpc}$ (bottom). For normalization, we have assumed that 16% of the pulsars’ total energy goes into high energy electron-positron pairs. The error bars shown represent the positron fraction as measured by the AMS (black, left) and PAMELA (red, left), and the $\mathrm{\text{electron}}+\mathrm{\text{positron}}$ spectrum as measured by Fermi and AMS-01 (black, right). In each case, we have adopted a propagation model that provides a good fit to the various secondary-to-primary ratios as described in the text and a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit to the AMS data of 1.69 (top) and 1.11 (bottom), and 6.9 (top) and 8.7 (bottom) to the Fermi data. The expected backgrounds are shown as black dotted lines.

Figure 8

The same as in Fig. 7 but for a broken power-law spectrum of electrons injected from cosmic ray sources ($d{N}_{e^{-}}/d{E}_{e^{-}}\propto E_e^{-2.65}$ below 100 GeV and $d{N}_{e^{-}}/d{E}_{e^{-}}\propto E_e^{-2.3}$ above 100 GeV), and for slightly different pulsar spectral indices ($\alpha =1.6$ and 1.5 in the upper and lower frames, respectively). With this cosmic ray background, the pulsar models shown can simultaneously accommodate the measurements of the cosmic ray positron fraction and the overall leptonic spectrum giving a ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ fit to the AMS data of 0.85 (top), 0.88 (bottom), and 0.37 (top) 0.37 (bottom) to the Fermi data. By comparing to the results of Fig. 7 the presence of a break at $\simeq 100\mathrm{GeV}$ is preferred from both individual data sets and from their combination.