EXTRACTING THE SIZE OF THE COSMIC ELECTRON–POSITRON ANOMALY

The calculation of the posterior probability distributions $\mathcal {P}(p_i|D)$ requires us to numerically integrate over the parameter region where the (cumulative) likelihood function is non-negligible. The CPU demand to reliably sample the posterior density depends on the number of free theoretical parameters N and the speed of the numerical implementation. In the case of the diffusion model encoded in GalProp the number of free input parameters is around a hundred, and for a given set of parameters the code runs for several minutes on a single CPU. This makes it unfeasible to attempt the calculation of the posterior without simplifications.

Fortunately, both the number of relevant free parameters and the running time can be substantially reduced. To reduce the dimension of the parameter space, we tested the robustness of the electron–positron flux against the variation of nearly all individual parameters and found that it is mostly sensitive to the following propagation parameters:

$$\begin{equation} P = \lbrace \gamma ^{e^-}, \gamma ^{{\rm nucleus}}, \delta _1, \delta _2, D_{0xx} \rbrace . \end{equation} \tag{ 15 }$$

Here, $\gamma ^{e^-}$ and γ^nucleus are the primary electron and nucleus injection indices parameterizing an injection spectrum without a break, δ₁ and δ₂ are spatial diffusion coefficients below and above a reference rigidity ρ₀, and D_0xx determines the normalization of the spatial diffusion coefficient.

We found that the electron–positron spectra are fairly insensitive to the rest of the parameters. We also found that the electron–positron spectrum is sensitive to not only the power-law dependence of the spatial diffusion coefficient D_xx, but the presence of a kink therein. So following Strong et al. (2004), we introduce two coefficients δ₁ and δ₂ that parameterize this power-law dependence in Equation (2) below and above a reference rigidity. We fix this reference rigidity to 4 GeV as in Strong et al. (2004).

Our calculations also confirmed the findings of a recent study by Cotta et al. (2011) that the electron–positron flux is sensitive to the change of the Galactic plane height L. Indeed, Seo & Ptuskin (1994) have shown that there is a connection between L and D_0xx:

$$\begin{equation} D_{0xx} = \frac{2 c (1-\delta) L^{1-\delta }}{3 \pi w \delta (\delta + 2)}. \end{equation} \tag{ 16 }$$

Thus, varying the cylinder height amounts to the redefinition of D_0xx as also noticed by Di Bernardo et al. (2011b). In the light of this, we fix L to 4 kpc and use D_0xx as a free parameter.

We treat the normalizations of the e⁻, e⁺, $\bar{p}$/p, B/C, (SC+Ti+V)/Fe, and Be-10/Be-9 fluxes as theoretical nuisance parameters.

$$\begin{eqnarray} &&P_{{\rm nuisance}} = \big\lbrace \Phi _{e^-}^0, \Phi _{e^+}^0, \Phi _{{\bar{p}}/p}^0, \Phi _{{\rm B/C}}^0, \Phi _{({\rm SC+Ti+V)/Fe}}^0, \Phi _{{\rm Be-10/Be-9}}^0 \big\rbrace .\nonumber\\ \end{eqnarray} \tag{ 17 }$$

They are kept free because the electron–positron flux is either directly or indirectly sensitive to these parameters. On the other hand, prior information is available for these parameters, enabling us to reduce them to the nuisance level. Since GalProp calculates normalizations based on local cosmic-ray measurements, the results of this calculation can be used as a guideline to the central values of the nuisance parameters. The uncertainties of the normalizations can be reliably estimated by an initial scan over the full parameter space.³

Varying the parameters listed in Equations (15) and (17), we confirmed the result of Trotta et al. (2011) that the electron+positron flux of Fermi-LAT can be well reproduced by the theoretical calculation. We also found that by changing these parameters the theory can match well the latest PAMELA electron spectrum (Adriani et al. 2011) and the latest PAMELA positron fraction data (Adriani et al. 2010a). (We defer the discussion of the quantitative details to the results section.) This demonstrates that varying the selected parameters gives us enough flexibility to fit all the observed features of the electron–positron spectra.

While the Galactic propagation of GeV or higher energy cosmic rays is relatively well understood, the propagation of a few GeV or lower energy electrons and positrons in the turbulent, magnetized ISM remains a formidable challenge (Prantzos et al. 2011). Local effects, such as solar modulation and the geomagnetic cutoff, significantly affect cosmic rays at lower energies (Pesce-Rollins & Fermi LAT Collaboration 2009). Since solar modulation effects, based on the force field model, are built into GalProp, we take these effects into account by varying the value of the modulation potential in the code. Following Gast & Schael (2009), we assume a charge-sign-dependent modulation, that is, positively and negatively charged cosmic rays are modulated differently by the Sun. Gast & Schael (2009) conclude that the effect of this charge-dependent modulation on (PAMELA) positrons is substantial. They also show that the modulation effect on the ${\bar{p}}/p$ ratio is comparable to the statistical uncertainties. As described in the next section, we absorb this effect in the systematic uncertainties of the ${\bar{p}}/p$ data. Heavier nuclei (B, C, Sc, Ti, V, Fe, and Be) can carry higher positive charges than that of the proton, but their charge-to-mass ratio is still lower. Since the modulation potential is proportional to the charge-to-mass ratio, the modulation effect on heavier nuclei is even milder. Considering that we use the ratio of their fluxes, most of the modulation effect cancels since they are positively charged. So the modulation effect on heavier nuclei can also be safely absorbed in the systematic uncertainties.

To be able to compare with experimental data, we set the positron (electron) modulation potential in GalProp to ϕ⁺ = 442 (2) MV. These values were determined by Gast & Schael (2009) for PAMELA. Usoskin et al. (2011) showed that the time dependence of the solar modulation potential is not substantial over the period of PAMELA's data taking, and about the same average values can be used for Fermi-LAT. We set the rest of the GalProp parameters to the values promoted by Strong et al. (2004).

In order to extract the most favored values of the propagation parameters, we have to calculate the posterior distribution $\mathcal {P}(p_i|D)$ using suitable Bayesian priors $\mathcal {P}(p_j)$. Assuming no prior knowledge justifies the use of uniform priors. Since we have previous knowledge about the order of magnitude of our parameters, we use uniform priors for the propagation parameters (rather than for some functions, such as log, of them). For the nuisance parameters prior knowledge is available in the form of a scan over GalProp predictions that are based on local measurements of cosmic-ray fluxes different from those listed in Table 1. Thus, for our nuisance parameters we use normally distributed priors.

When evaluating σ_i for the log likelihood in Equation (10), following Trotta et al. (2011), we ignore theoretical uncertainties and combine statistical and systematic experimental uncertainties in quadrature

$$\begin{eqnarray} &&\sigma _i^2 = \sigma _{i, {\rm statistical}}^2 + \sigma _{i, {\rm systematic}}^2. \end{eqnarray} \tag{ 18 }$$

This can be done for Fermi-LAT and the latest PAMELA e⁻ flux. Unfortunately, systematic uncertainties are not available for the rest of the cosmic-ray measurements. When this is the case, as an estimate of the systematics, we define σ_i as the rescaled statistical uncertainty

$$\begin{eqnarray} &&\sigma _i^2 = \sigma _{i, {\rm statistical}}^2/\tau _i. \end{eqnarray} \tag{ 19 }$$

For simplicity, in this study, we use the same scale factor τ_i for all data points where systematic uncertainty is not available. To remain mostly consistent with the work of Trotta et al. (2011), we set this common scale factor to a conservative value that they use: τ_i = 0.2. We checked that our conclusions only mildly depend on this choice.

We note that systematic errors in the data are not necessarily normally distributed point-to-point errors. In fact, typically systematic errors are correlated, such as a systematic shift in the energy scale, and could be described by various probability distributions other than a Gaussian. Unfortunately, these probability distributions are not provided by even those experimental collaborations that indicate a confidence interval for their systematic errors. In the lack of this information, we use the simplest ansatz that is adopted by most authors in the literature. This estimate of the systematic errors is a simplified approximation of a more complicated situation. Nevertheless, for astrophysical data it captures the essence of systematic uncertainties. After all, the simplest cosmic-ray flux is a falling power-law spectrum. For this case a systematic shift in the energy scale, for example, can be re-interpreted as a systematic normalization shift of the spectrum. Part of this shift is absorbed by our normalization nuisance parameters, and part of it is approximated as Gaussian error.⁴

Due to the simplicity of the posterior density (Trotta et al. 2011) and its relatively low dimensionality, we sample the parameter space P and P_nuisance according to a simple algorithm. We select random model points from the parameter space according to a uniform distribution for P and normally distributed for P_nuisance. While this sampling technique is less efficient than the Monte Carlo based ones, it enables us to trivially parallelize the numerical calculation. It also allows us to simply check the robustness of our results against the change of certain assumptions such as the prior, the scale factor τ, or the adequateness of the sampling.

The simplicity of the likelihood function and the high number of data points used in this analysis also make convergence testing relatively simple. To test the validity of our results, we can evaluate an approximate value of the posterior means, variances, and the evidence adopting the procedure described by Tierney & Kadane (1986). To assure the adequacy of the sampling, we can simply increase the number of samples of the posterior density until the numerically calculated evidence is within 5% of the one obtained by the Laplace method. During this procedure, we found that to extract the posterior probabilities presented in this paper about one million samples of the posterior density were required over the parameter space in Equations (15) and (17). The gathering of this sample consumed about 2 × 10⁵ CPU hours.

We included 219 of the most recent experimental data points in our statistical analysis. These contained 114 electron–positron related and 105 boron/carbon, anti-proton/proton, (Sc+Ti+V)/Fe, and Be-10/Be-9 cosmic-ray flux measurements. As a number of experiments have energy ranges that overlap, we chose the most recent experimental data points in those energy ranges.

For e⁺ + e⁻ we used the most recent data from AMS by Aguilar et al. (2002), Fermi-LAT by Ackermann et al. (2010), and HESS by Aharonian et al. (2008, 2009). The energy ranges in which we use each experiment are listed in Table 1. The AMS experiment reported an excess in high-energy positrons for energies greater than 10 GeV. The Fermi-LAT collaboration reported a high-precision measurement of the e⁺ + e⁻ spectrum for energies from 7 GeV to 1 TeV using its Large Area Telescope (LAT). This spectrum extended their previously published electron–positron spectrum over an energy range of 20 GeV to 1 TeV (Abdo et al. 2009) and is flatter than results reported by earlier experiments. HESS's atmospheric Cerenkov telescope reported a significant steepening of the electron plus photon spectrum above 1 TeV.

The PAMELA collaboration measured the flux of the positron fraction e⁺/(e⁺ + e⁻), between 1.5 and 100 GeV (Adriani et al. 2009). They observed that this differential positron fraction falls slower than expected for energies above 10 GeV. This behavior is different from that of the background of secondary positrons produced during propagation of cosmic rays in the Galaxy. Recently PAMELA released the measurement of the e⁻ flux alone (Adriani et al. 2011), robustly confirming the e⁺ + e⁻ spectrum by Fermi-LAT.

Cosmic-ray anti-protons can be used to study the production of primary and secondary cosmic rays and their transport throughout the Galaxy. Detailed anti-proton spectra require a large number of measurements over a larger energy range, with good statistics. Previous balloon-borne experiments such as CAPRISE98 (Boezio et al. 2001) and HEAT (Beach et al. 2001) detected only a small number of anti-protons with limited statistics. The PAMELA satellite experiment (Adriani et al. 2010b) provided a comprehensive measurement of the anti-proton/proton flux ratio for an energy range of 1–100 GeV. PAMELA's spectrum follows the same trend as other recent anti-proton/proton ratio measurements. The energy range over which we use the PAMELA experiment for the anti-proton/proton ratio is listed in Table 1.

In comparison to primary/primary or secondary/secondary cosmic-ray ratios, stable secondary-to-primary cosmic-ray ratios, such as boron/carbon and (Sc+Ti+V)/Fe ratios, are the most sensitive to variation in the propagation of cosmic rays in the Galaxy. Their sensitivity arises from the fact that primary cosmic rays are generated by the original source while secondary cosmic rays are created by the interaction of their primaries with the ISM (Childers & Duvernois 2008). Primary/primary and secondary/secondary cosmic-ray ratios have a low sensitivity to variation in the propagation parameters as the denominator and numerator are produced by similar propagation mechanisms. The boron/carbon and (Sc+Ti+V)/Fe ratios provide an indication (over different energy ranges) of the amount of interstellar material that primary cosmic rays traverse as a function of energy (Childers & Duvernois 2008). The experiments used to define the B/C and (Sc+Ti+V)/Fe ratios for our analysis are found in Table 1.

In conjunction with stable secondary/primary ratios such as the boron/carbon ratio, unstable isotope ratios such as Beryllium-10/Beryllium-9 can be used to constrain the time it takes for cosmic rays to propagate through the Galaxy (Malinin 2004). In this work we use Be-10/Be-9 data from various experiments, such as ISOMAX98 (Hams et al. 2001), ACE-CRIS (Davis et al. 2000), ACE (Yanasak et al. 2001), and AMS-02 (Burger 2004).

We begin our results section by investigating the question whether the present cosmic-ray data can be used to justify the existence of an anomaly in the cosmic electron–positron spectrum. Both the reality of an anomaly in the PAMELA e⁺/(e⁺ + e⁻) flux and the absence of such in the anti-proton flux have been questioned by Katz et al. (2009) and Kane et al. (2009), respectively. Recently, Trotta et al. (2011) argued that the Fermi-LAT data can be well matched by the diffusion model, as encoded in GalProp, simply by adjusting the parameters of the propagation model. Their Figure 8 clearly shows that the Fermi-LAT data agree reasonably well with the propagation model that was their best fit to 76 cosmic-ray spectral data points. Trotta et al. (2011) also acknowledge that the "positron fraction, shown in the right panel of Figure 8, does not agree with the PAMELA data (Adriani et al. 2009), but this was expected since secondary positron production in the general ISM is not capable of producing an abundance that rises with energy." In other words, they conclude that PAMELA cannot be fitted by simply adjusting the propagation parameters. We take this as an important indication that the cosmic-ray anomaly is real and requires a detailed investigation rather than the adjustment of the propagation model to explain it.

The hypothesis that the adjustment of the propagation parameters does not solve the cosmic-ray anomaly is further supported by the fact that not all cosmic-ray data can be fitted well with a single set of these parameters. It is already evident from Figures 7 and 8 of Trotta et al. (2011) that the best fit of the propagation parameters to the rest of the cosmic-ray data does not fit well AMS, Fermi, and HESS simultaneously. This is exactly what we find in our analysis. Our best fit for all cosmic-ray data excluding AMS, Fermi, HESS, and PAMELA data with electron and/or positron fluxes gives a χ² per degree of freedom of 0.34. As a consequence, the best-fit curves all pass through the estimated systematic error bands, shown in gray, in Figure 1. When this fit is compared to the AMS, Fermi, HESS, and PAMELA electron and/or positron flux, the χ² per degree of freedom we obtain is 24, which signals considerable tension bordering exclusion. The converse also holds. By changing the propagation parameters, we can find an ideal fit for the electron–positron related fluxes with χ² per degree of freedom of 1.0. But for the rest of the cosmic-ray data the same fit results in a χ² per degree of freedom of 3.1, which is a significant pull for 105 degrees of freedom. These discrepancies signal a statistically significant tension between the electron–positron measurements and the rest of the cosmic-ray data.

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To further investigate the tension, we divide the cosmic-ray data into two groups: 114 measurements containing observations of electron and/or positron fluxes (AMS, Fermi, HESS, and PAMELA), and the remaining 105 data points (anti-proton/proton, boron/carbon, (Sc+Ti+V)/Fe, Be-10/Be-9). We perform a Bayesian analysis independently on these two sets of data, extracting the preferred values of the propagation parameters. Remarkably, we found that we can obtain information about the electron–positron related propagation parameters from the rest of the data because the propagation of the cosmic rays is entangled for several reasons. First, certain propagation parameters, most importantly for us D_0xx, are species independent. Second, the transport equation includes nuclear fragmentation and decay, which directly contributes to the secondary electron–positron fluxes. Third, since their energy density is comparable to the interstellar radiation and magnetic fields, various species of cosmic rays affect each other's dynamics.

Due to the correlations pointed out above, certain parameters of the electron–positron propagation are constrained even if no electron–positron related data are used in our fit. Unfortunately, the injection indices remain virtually unconstrained. In order to fix those parameters, we resorted to using a minimal amount of information from the electron–positron related fluxes. We decided to use data points from the e⁻ + e⁺ spectrum because (1) they span the widest energy range, and the end points of the e⁻ + e⁺ spectrum, partially due to their high uncertainty, appeared to agree with the theoretical predictions even before we set out to find the most optimal parameter values; (2) in the low-energy region they are relatively insensitive for solar modulation effects; and (3) in the mid-energy range the e⁻ + e⁺ theoretical prediction develops an insensitivity to the values of the propagation parameters (cf. the distinct bow-tie shape of the theory uncertainty band).

With this in mind, we included four e^± related data points in the analysis together with the $\bar{p}$/p, B/C, (Sc+Ti+V)/Fe, and Be data. These were the lowest energy point of AMS, the highest energy point of HESS, and the 19.40 GeV and 29.20 GeV data points of Fermi-LAT. We have checked that our result is robust against this choice and does not bias the final conclusion.

Figure 2 clearly shows that the two subsets of cosmic-ray data are inconsistent with the hypothesis that the cosmic-ray propagation model and/or sources implemented in GalProp provide a good theoretical description. The five frames display the marginalized posterior probability densities of our selected propagation parameters. Dashed blue curves show results with likelihood functions containing only electron–positron related flux data (AMS, Fermi, HESS, and PAMELA), while the likelihood functions for the solid red curves contain only the rest of the cosmic rays (anti-proton/proton, boron/carbon, (Sc+Ti+V)/Fe, Be-10/Be-9). Shaded areas show the 68% credibility regions for the parameters. Table 2 shows the numerical values of the best fits and the 68% credibility ranges for the propagation parameters.

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Table 2. Best-fit Values of the Propagation Parameters and Their 68% Credibility Ranges

Parameter	Fit for the e± Related Data		Fit for the Rest of the Data
	Best-fit Value	68% Cr Range	Best-fit Value	68% Cr Range
$\gamma ^{e^-}$	2.55	{2.45, 2.60}	2.71	{2.54, 2.92}
γnucleus	1.60	{1.51, 1.69}	2.10	{1.88, 2.92}
δ1	0.24	{0.23, 0.26}	0.06	{0.04, 0.08}
δ2	0.10	{0.08, 0.12}	0.35	{0.32, 0.39}
D0xx [× 1028]	2.17	{1.85, 2.19}	11.49	{8.86,13.48}

Notes. Numerical values are shown for both fits: including the electron–positron related cosmic-ray data only and including the rest of the data.

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In the first two frames, showing the posterior densities of the electron and nucleus injection indices $\gamma ^{e^-}$ and γ^nucleus, there is a mild but tolerable tension between the electron–positron related and the rest of the cosmic-ray data. The last three frames, on the other hand, indicate statistically significant tension between the e⁺–e⁻ and the rest of the data. The 68% credibility regions for the spatial diffusion coefficients δ₁ and δ₂ and that of D_0xx fall far away from each other when determined using the two different cosmic-ray data sets. Although it is not shown, it is easily inferred that not even the 99% credibility regions overlap. It appears that by adjusting the cosmic-ray parameters we can indeed achieve a good fit to either the electron–positron related fluxes or the rest of the data but not both simultaneously.

Our interpretation of the tension between the electron–positron fluxes and the rest of the cosmic-ray data is the following. The measurements of PAMELA and Fermi-LAT are affected by new physics that is unaccounted for by the propagation model and/or cosmic-ray sources included in our calculation. We base this hypothesis partly on the earlier quoted statement of Trotta et al. (2011) that "secondary positron production in the general ISM is not capable of producing an abundance that rises with energy." The behavior of the PAMELA e⁺/(e⁺ + e⁻) data is unexpected based on general theoretical principles, and when it is fit by adjusting the propagation parameters, it leads to a bad fit to the rest of the data. An anomaly in PAMELA e⁺/(e⁺ + e⁻) is also expected to produce an anomaly in the Fermi-LAT e⁺ + e⁻ and the PAMELA e⁻ spectra.

We note that the recently released PAMELA e⁻ flux (Adriani et al. 2011), which is included among our electron–positron related data, considerably increases this tension. We checked that without the inclusion of the PAMELA e⁻ flux the tension is noticeably milder. This, as well as the effect of the extra data that we use, probably explains why this tension was not detected by Trotta et al. (2011).

We attempt to extract the size of the new physics signal, after arriving to the conclusion that new physics is buried in the electron–positron fluxes. Based on our findings, our working hypothesis is that the new physics is affecting the electron–positron fluxes but hardly influences the rest of the cosmic rays. Under this hypothesis the cosmic-ray propagation parameters can be determined from the unbiased data: anti-proton/proton, boron/carbon, (Sc+Ti+V)/Fe, Be-10/Be-9. This means that we can use the central values and credibility regions of the parameters determined using these data to calculate a background prediction for all cosmic-ray data including the electron–positron fluxes. Once we quantify the background, we can subtract it from the electron–positron data to see whether there is a statistically significant signal that can be extracted.

In the first step, we use the central values of the propagation parameters determined earlier using $\bar{p}$/p, B/C, (Sc+Ti+V)/Fe, Be-10/Be-9 to calculate a central value prediction for the PAMELA and Fermi-LAT electron–positron fluxes. Then we use all the scanned points in the parameter space lying within the 68% credibility region of all the five scanned parameters to establish a 1σ band around this central value. We will refer to this band as the 1σ uncertainty of the background. We overlay this uncertainty band on the Fermi-LAT electron+positron and the PAMELA electron and positron fraction fluxes.

Figure 3 shows the comparison of the measured electron–positron fluxes and their backgrounds. Statistical and systematic uncertainties combined in quadrature are shown for Fermi-LAT and PAMELA e⁻, while (τ = 0.2) scaled statistical uncertainties are shown for PAMELA e⁺/(e⁺ + e⁻) as gray bands. Our background prediction is overlaid as magenta bands. The central value and the 1σ uncertainty of the calculated anomaly are displayed as green dashed lines and bands. As the first frame shows, the Fermi-LAT measurements deviate from the predicted background both below 10 GeV and above 100 GeV.

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As we shall discuss later, the low-energy deviation might be due to the inadequacies of the propagation model, so here we concentrate on the deviation between the background and the measurements above 100 GeV. In our interpretation this is a weak but statistically significant signal of the presence of new physics in the electron+positron flux. Based on the difference between the central values of the data and the background, a similar conclusion can be drawn from PAMELA. Unfortunately, the sizable uncertainties for the PAMELA measurements prevent us from claiming a statistically significant deviation.⁵

After having determined the background for the electron–positron fluxes, we can subtract the background from the measured flux to obtain the size of the new physics signal. We determine the central value of the signal by subtracting the central value of the background from the central value of the data. The 1σ uncertainty of the signal is the quadratically combined experimental and background uncertainty. Results for the electron–positron anomaly are also shown in Figure 3. As expected based on the background predictions, a non-vanishing anomaly can be established for the Fermi-LAT e⁺ + e⁻ flux, while no anomaly with statistical significance can be claimed for PAMELA due to the large uncertainties.

Based on the available evidence, we can only speculate about the origin of the discrepancy between the data and predictions of the cosmic electron–positron spectra. Since the publication of the first PAMELA positron fraction measurement by Adriani et al. (2009), speculation has been abundant. The first obvious assumption is that some aspect of the propagation model used in the present calculation is insufficient for the proper description of the electron–positron fluxes arriving at Earth (Stawarz et al. 2010; Donato & Serpico 2011; Arakida & Kuramata 2011; Tawfik & Saleh 2010). In this case there exists no anomaly in the data. One such plausible effect, which is missed by the two-dimensional calculation in GalProp, is the spectral hardening of cosmic rays caused by non-steady sources (Cowsik & Lee 1979; Pohl & Esposito 1998; Pohl et al. 2003). It would be an interesting exercise to repeat our analysis using a different calculation, such as DRAGON by Maccione et al. (2010) (Evoli et al. 2008; Di Bernardo et al. 2010), USINE by Maurin et al. (2011), PPPC4DMID by Cirelli et al. (2011), or the code of Buesching et al. (2003), to confirm these possibilities.⁶

Assuming that the propagation model satisfactorily describes physics over the Galaxy, the next reasonable thing is to suspect local effects modifying the electron–positron distribution (Pesce-Rollins & Fermi LAT Collaboration 2009). Further suspicion falls on the lack of sources included in the calculation (Di Bernardo et al. 2011b; de Vega et al. 2010; Blum 2011; Frandsen et al. 2011). Possible new sources of cosmic rays to account for the anomaly have been proposed in two major categories. The first category is known astrophysical objects with unknown or uncertain parameters (Lavalle 2011). These could be supernova remnants, pulsars, various objects in the Galactic center, etc. (Kawanaka et al. 2011; Kashiyama et al. 2011; Pato et al. 2010; Yuan et al. 2011; Guo et al. 2011; Di Bernardo et al. 2011a). Finally, more exotic explanations call for new astronomical and/or particle physics phenomena, such as dark matter (Pieri et al. 2011; Abidin et al. 2010; Josan & Green 2010; Cheng et al. 2011; Ko & Omura 2011; Cirelli et al. 2011; Cholis & Goodenough 2010; Palomares-Ruiz & Siegal-Gaskins 2010; Anderson 2010; Zaharijas et al. 2010; Yang 2010; Borriello et al. 2012; Kajiyama & Okada 2011; Finkbeiner et al. 2011; Buckley et al. 2011; Kyae 2010; Logan 2011; Hutsi et al. 2010; Feldman et al. 2010; Arina et al. 2010; Cholis 2011; Chen et al. 2011; Lineros 2010; Dugger et al. 2010; Vincent et al. 2010; Mohanty et al. 2010; Bell et al. 2011b; Kang et al. 2011; Haba et al. 2011; Cline 2010; Ishiwata et al. 2010; Barger et al. 2010; Kang & Li 2011; Carone et al. 2010; Cirelli & Cline 2010; Masina & Sannino 2011; Porter et al. 2011; Hutsi et al. 2011; Sanchez & Holdom 2011; Zavala et al. 2011; Ke et al. 2011; Zhu 2011; Bell et al. 2011a).

Possible deviations from the predicted background can occur for energies above 100 GeV, as electron propagation is limited by energy losses via inverse Compton scattering of interstellar dust and CMB light and synchrotron radiation due to the Galactic magnetic field. This results in a relatively short lifetime and a rapidly decreasing intensity of the cosmic ray, as energy increases. Hence, a large fraction of the detected electrons and positrons above 100 GeV are hypothesized to come from individual nearby sources that are within a few kiloparsecs of the Earth (Delahaye et al. 2009a; Grasso et al. 2009). Random fluctuations in the injection spectrum and the spatial distribution of those nearby sources produce significant differences in the most energetic part of the observed electron and positron spectrum. This can be the indication of new physics from either an astrophysical object(s) or dark matter.

Whatever the source of the anomaly is, if the size of the anomaly can be isolated, then the source will have to match that size. Figure 4 compares our extracted signal to a few attempts to match the anomaly that we randomly selected from the recent literature. The first frame shows the prediction of Ahlers et al. (2009) for unaccounted energetic electrons and positrons produced by supernova remnants. The top right frame features contributions from additional electron–positron primary sources (nearby pulsars or particle dark matter annihilation) calculated by Grasso et al. (2009). The bottom left frame contains predictions of Bergstrom et al. (2009) for anomalous electron–positron sources from dark matter annihilations. Similar to this, dark matter annihilation contributions suggested by Cholis et al. (2009) are shown in the last frame.

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The contributions of various new sources typically come with their own (theoretical) uncertainties. In some of the cases this uncertainty is unknown; thus, it is hard to draw any conclusion by comparing these speculations to our isolated signal. In the cases where the theoretical uncertainty is known, currently it tends to be large enough to prevent us from judging the validity of the given explanation. Nevertheless, based on the present amount of information, we can already select a few scenarios that are more favored than some others. By adding more data to our analysis, it is possible to shrink the uncertainty of the signal. Similarly, in most cases, the theoretical model of a given new source can be constrained further, producing a narrower prediction. With time, more data, and more precise calculations, the various suggestions of the cosmic electron–positron anomaly can be ruled out or confirmed.