Possible Evidence for the Stochastic Acceleration of
Secondary Antiprotons by Supernova Remnants
The antiproton-to-proton ratio in the cosmic-ray spectrum is a sensitive probe of new physics. Using recent measurements of the cosmic-ray antiproton and proton fluxes in the energy range of 1 – 1000 GeV, we study the contribution to the ratio from secondary antiprotons that are produced and subsequently accelerated within individual supernova remnants. We consider several well-motivated models for cosmic-ray propagation in the interstellar medium and marginalize our results over the uncertainties related to the antiproton production cross section and the time-, charge-, and energy-dependent effects of solar modulation. We find that the increase in the ratio observed at rigidities above 100 GV cannot be accounted for within the context of conventional cosmic-ray propagation models, but is consistent with scenarios in which cosmic-ray antiprotons are produced and subsequently accelerated by shocks within a given supernova remnant. In light of this, the acceleration of secondary cosmic rays in supernova remnants is predicted to substantially contribute to the cosmic-ray positron spectrum, accounting for a significant fraction of the observed positron excess.
The ratio of cosmic-ray (CR) antimatter to matter is a powerful probe of new physics, in particular of dark matter annihilation or decay Bergstrom et al. (1999); Hooper et al. (2004); Bertone et al. (2005); Profumo and Ullio (2004); Bringmann and Salati (2007). However, antimatter can also be produced astrophysically through the interactions of CR protons with gas. As the astrophysical flux of CR antimatter depends on CR propagation, its measurement depends on diffusion in the interstellar medium (ISM) Pato et al. (2010); Simet and Hooper (2009); Di Bernardo et al. (2010); Strong et al. (2007). Exotic contributions to the antimatter flux can be differentiated from conventional astrophysics through the observation of secondary CRs, such as boron, that are produced via spallations, but not from dark matter.
Over the past decade, an intriguing rise with energy in the CR positron fraction () has been observed by both PAMELA Adriani et al. (2010) and AMS-02 Aguilar et al. (2013). The dark matter interpretation of this excess has received significant attention (Bergstrom et al., 2008; Cirelli and Strumia, 2008; Cholis et al., 2009a; Cirelli et al., 2009; Nelson and Spitzer, 2010; Arkani-Hamed et al., 2009; Cholis et al., 2009b, c; Harnik and Kribs, 2009; Fox and Poppitz, 2009; Pospelov and Ritz, 2009; March-Russell and West, 2009; Chang and Goodenough, 2011). Dark matter models that explain the positron fraction typically invoke particles with masses of 1 – 3 TeV annihilating or decaying to ee pairs through intermediate two- or three-body decays (Cholis and Hooper, 2013; Cirelli et al., 2009; Dienes et al., 2013; Finkbeiner and Weiner, 2007). Alternatively, the positron excess could very plausibly be generated by nearby pulsars, with ages of – years Hooper et al. (2009); Yuksel et al. (2009); Profumo (2011); Malyshev et al. (2009); Grasso et al. (2009); Linden and Profumo (2013); Cholis and Hooper (2013).
A third explanation for the rising positron fraction includes a two-step process where positrons are first produced via hadronic interactions (followed by pion/muon decay) within supernova remnants (SNRs), and are then accelerated by shocks within those same remnants before escaping into the ISM Blasi (2009); Mertsch and Sarkar (2009); Ahlers et al. (2009); Blasi and Serpico (2009); Mertsch and Sarkar (2014). In contrast to dark matter or pulsar scenarios, this “stochastic acceleration” predicts a similar rise for all species of CR secondaries produced via hadronic interactions (see also Fujita et al. (2009); Kohri et al. (2016); Malkov et al. (2016)). In Ref. Cholis and Hooper (2014), the observed boron-to-carbon (B/C) ratio was utilized to show that stochastic acceleration could not account for the entirety of the positron excess, though this assumes that both CR protons and carbon nuclei are produced equally across the population of SNRs (see, however, Kachelriess et al. (2011); Kachelrieß and Ostapchenko (2013)).
These constraints are mitigated if individual SNRs produce varying relative abundances of different primary CR species. For example, if nearby SNRs are efficient accelerators of secondaries, but have low abundances of intermediate mass nuclei, then the connection between the B/C ratio and the positron fraction could be weakened Cholis and Hooper (2014). However, a direct comparison exists between the positron fraction and the antiproton-to-proton ratio (), since secondary antiprotons and positrons are both generated through proton-proton interactions. In this paper, we examine in stochastic acceleration models. We find evidence for an excess of high-energy antiprotons measured with great accuracy by Aguilar et al. (2016), that can be explained by stochastic acceleration and that cannot be accounted for by uncertainties in solar modulation, cosmic-ray propagation or the antiproton production cross section. Intriguingly, our results suggest that a significant fraction of the observed positron excess originates from the secondary acceleration of positrons in SNRs.
There are a number of systematic uncertainties that must be treated carefully to interpret the ratio measured by AMS-02. In particular, we consider uncertainties associated with CR propagation in the ISM, the antiproton production cross section, and the effects of solar modulation.
Most CR antiprotons are produced through the hadronic interactions of high-energy protons and nuclei with interstellar gas. To model the injection and propagation of CRs through the Galaxy, we utilize Galprop, which numerically solves the transport equation to calculate the local flux of primary and secondary CR species http://galprop.stanford.edu/. (); Strong (2015); version of GALPROP availabe at: http://sourceforge.net/projects/galprop (); Strong and Moskalenko (1998). The primary uncertainties in this calculation are the injected spectrum and distribution of primary CRs, and the timescales for diffusion through the Galactic medium. Convection and diffusive re-acceleration can also be relevant. Numerous measurements provided by AMS-02, PAMELA and Voyager 1 constrain the characteristics of CR propagation. In this work, we follow the procedure described in Ref. Cholis et al. (2016) from which we take two propagation models (models C and E). We additionally produce a new (thin disk) model (model F). Each model provides a good fit to the proton spectrum measured by Voyager 1 and PAMELA, and the B/C data from AMS-02 and PAMELA. We use these three models to envelope the uncertainties related to CR production and propagation. These models are summarized in Table 1 (see Ref. Cholis et al. (2016)), and their predictions for the ratio are shown as a blue band in Fig. 1 (labelled as “Inj. & ISM Unc.”). We also show the ratio predicted by model F (solid black line).
We emphasize that the decreasing at high energies is a generic feature of any leaky-box diffusion model. Since the diffusion coefficient has an energy dependence , the total grammage encountered by cosmic-ray primaries falls as , and this softened spectrum is inherited by cosmic-ray secondaries. These secondaries are themselves softened by diffusive escape, leading to a primary-to-secondary ratio which falls as . Even if as observed above 0.5 TeV the proton spectrum becomes harder by 0.1 in its power-law spectrum value, and with those protons collisions giving the antiprotons at 100 GeV in energy; a value of will still give a ratio that falls with increasing energy. The nearly energy-independent ratio observed by AMS-02 at energies above 100 GeV Aguilar et al. (2016), can only be accommodated in models with 0, which are strongly ruled out Trotta et al. (2011); Pato et al. (2010); Simet and Hooper (2009); Di Bernardo et al. (2010); Strong et al. (2007). Thus, the error band shown in Fig. 1 is generically applicable to any Galprop model consistent with observations.
The cross section for antiproton production in inelastic collisions has been carefully studied Tan and Ng (1982, 1983); Duperray et al. (2003), using data from Refs. Dekkers et al. (1965); Capiluppi et al. (1974); Allaby et al. (1970); Guettler et al. (1976); Johnson et al. (1977); Antreasyan et al. (1979). However, there remain significant uncertainties reagrding the production for for antiprotons in the collisions of CR protons and nuclei. While Galprop v54 handles the production of antiprotons Moskalenko et al. (2002), it does not include the most recent measurements of the antiproton production cross section Arsene et al. (2007); Anticic et al. (2010), nor does it account for the uncertainties in this quantity, which can be significant in the determination of the ratio Bringmann et al. (2014); Hooper et al. (2015); Cuoco et al. (2016).
Recently, several groups have studied and quantified the uncertainties in the antiproton production cross section di Mauro et al. (2014); Kappl and Winkler (2014); Kachelriess et al. (2015). To fit the AMS-02 data Aguilar et al. (2016), we first calculate the antiproton spectrum for a given propagation model, and then renormalize those fluxes by the following continuous (in kinetic energy, prior to solar modulation, ) function:
We bound the values of , , and such that resides within the 3 uncertainties presented in di Mauro et al. (2014). We add uncertainty in Eq. 1 to account for the local galactic gas uncertainties. The impact of this uncertainty is shown in Fig. 1, by the orange band surrounding the central prediction of propagation model F (labeled “p-p cr. sec. Unc.”).
As CRs enter the Solar System, they experience heliospheric forces resulting in solar modulation. In treating solar modulation, we adopt the standard formula:
where is the kinetic energy of CRs at Earth, is their charge, the differential CR flux at Earth, and is the local ISM differential flux. is the modulation potential, for which we use the predictive time-, charge- and rigidity-dependent formula presented in Cholis et al. (2016):
where is the strength of the heliospheric magnetic field (HMF) measured at Earth, is its polarity, and the tilt angle of the heliospheric current sheet. , is the CR rigidity before entering the heliosphere (see Refs. Cholis et al. (2016); Potgieter (2013)). is during eras in which the HMF does not have a well-defined polarity. We adopt GV and marginalize over the solar modulation uncertainties described in Ref. Cholis et al. (2016), allowing GV and GV.
|ISM mod.||(from back only)|
Given that the data of Aguilar et al. (2016) utilized in this study, have been taken over several years, between May 2011 and May 2015, we must account for the time-evolving properties of the HMF. We note that the modulation potential, , depends on both and , and thus is not linear with time. To account for this, we break the model into six-month periods and use the time-averaged values of (from the ACE magnetometer http://www.srl.caltech.edu/ACE/ASC/ ()) and (calculated by the Wilcox Solar Observatory http://wso.stanford.edu/Tilts.html ()) for each interval (see Table 2). For periods where the HMF geometry was being re-configured, we adopt values of chosen to result in a smooth transition of the second term in Eq. Possible Evidence for the Stochastic Acceleration of Secondary Antiprotons by Supernova Remnants. We use these values of , and to calculate the modulated spectra for each individual six-month period, and then combine these eras to determine the total CR spectrum over the period observed by AMS-02. The impact of the uncertainties related to solar modulation is depicted in Fig. 1 by the green band (labelled “Sol. Mod. Unc.”).
To combine the uncertainties associated with propagation through the ISM, the antiproton production cross section, and solar modulation, we calculate our fit to the AMS-02 data for each of our three propagation models, marginalizing over uncertainties in the parameters , , , , , and . The best-fit parameters for each propagation model are shown in Table 3, while the range of the combined uncertainties is depicted by the red band in Fig. 1. At kinetic energies below 1 GeV, the largest source of uncertainty is solar modulation. Between 2 – 20 GeV the main uncertainty is the antiproton production cross section. Above 20 GeV, uncertainties in the antiproton production cross section and CR propagation are both important.
We now consider the stochastic acceleration of CR secondaries in SNRs. We assume that SNR shocks are supersonic, with a compression ratio of , where is the plasma down-stream velocity and is the plasma up-stream velocity, both defined in the frame of the shock front. As particles are accelerated inside the SNR, to a spectrum , they interact with the dense gas and spallate with a partial cross-section , to produce lighter species , or decay to them with a time-scale of Blasi (2009); Mertsch and Sarkar (2009); Cholis and Hooper (2014). For these lighter species, the source term is:
is the gas density where the spallations occur and is the kinetic energy per nucleon.
These secondaries then undergo further spallations and decays at a rate:
where and are the spallation cross section and decay lifetime of nuclei species, , respectively. Including to the above, advection, diffusion, and adiabatic energy losses, one gets the transport equation for species :
is the diffusion coefficient, the advection velocity, the phase space density of species and the relevant source term.
If enough CRs of species are produced and accelerated in the SNR before spallating or decaying (1/ ), they can have a significant impact on the observed secondary-to-primary ratios. Following Refs. Blasi (2009); Mertsch and Sarkar (2009); Cholis and Hooper (2014), we assume Bohm diffusion for CRs around the shock front:
where is the Larmor radius, is the magnetic field, and and are the charge and energy of the CR. is a factor Mertsch and Sarkar (2009) scaling as Blasi (2009), allowing for faster diffusion of CRs around the shock front. Measurements of the B/C ratio were used in Ref. Cholis and Hooper (2014) to constrain at the 95 (99, 99.9) confidence level.
Starting with the heaviest isotopes, we calculate the spectrum of all secondaries down to positrons in each SNR, and then average over the Galactic Disk, assuming a rate of three SNRs per century (see Cholis and Hooper (2014)). The injected spectrum of CRs in the ISM, after integrating over the volume of the SNR is:
We take yr, cm s and is the phase space density of species down-stream.
Treating as a free parameter, we calculate the spectrum of accelerated secondary antiprotons and protons and compare this result to the ratio measured by AMS-02. The contribution from accelerated antiprotons is insignificant at low energies, but can increase the ratio significantly at energies above 10 – 100 GeV. After accounting for the uncertainties described above, we identify a statistical preference for stochastic acceleration. In Table 4, we provide, for each propagation model, the best-fit value of , along with the 95 confidence interval for this quantity (corresponding to ). Even the lower limits on are consistently positive, and the fit improves at a level of 10 – 21 when accelerated secondaries are included, corresponding to a statistical preference of 3.2 – 4.6 111In the fits we have added in quadrature the reported statistical and systematic errors. At the highest energies the magnet spectrometer resolution and elastic scatterings of protons inside the detector might lead to charge confusion..
In Fig. 2, we show the impact of accelerated secondary antiprotons on the spectrum. The best-fit model is propagation model C with . Given the uncertainties associated with this calculation we also provide a best-fit range (dark purple band) which covers 6.1 – 10.4, bracketing the values obtained for the three propagation models considered in this study (see Table 4). We also show a 95 confidence band (light purple) corresponding to 4.6 – 12.4. This suggests that on average inside SNRs is only a factor of few above 1 (between and 3.5) and is in agreement with constraints on ISM CR acceleration Cowsik (1980). We note that these ranges are consistent with the B/C ratio upper limits of Cholis and Hooper (2014), (, at 95, 99 CL), especially given that the efficiency of SNRs for acceleration of CR secondaries may vary between different environments.
In Fig. 3, we illustrate the impact of accelerated secondaries on the positron fraction, showing the result predicted without the acceleration of secondaries (red band) and including accelerated secondary positrons, using the same range of as shown in Fig. 2 (purple bands). We do not include contributions from primary positron sources, such as dark matter or pulsars. The shaded bands account for the combined uncertainties associated with the CR propagation and solar modulation parameters, as well as the local energy loss rate. For the range of values required to explain the rising measured by AMS-02 Aguilar et al. (2016), we predict that accelerated secondary positrons will also account for a significant fraction of the positron excess.
Although we have treated as a simple parameter in this study, this quantity may vary with rigidity. CR diffusion results from particles scattering with random magnetohydrodynamic waves and discontinuities, and thus depends on the spectrum of underlying magnetic perturbations. As such, scattering is only efficient for perturbations on length scales comparable to the Larmor radius of the particle. The spectrum of magnetic perturbations found in SNR environments and future AMS-02 data will determine the rigidity dependence of .
In this paper, we have used the CR spectrum, as presented by the AMS-02 Collaboration, to test scenarios where CR secondaries are produced and accelerated within individual SNRs. The spectrum Aguilar et al. (2016) exhibits a clear rise at energies above 100 GeV. We show that this feature cannot be accounted for by conventional CR sources, even after accounting for the uncertainties pertaining to their injection and propagation through the ISM, the antiproton production cross section, and the effects of solar modulation. Instead, we find that the observed rise is consistent with a contribution of antiprotons that are produced as secondaries and then further accelerated within SNRs. We quantify the range of parameters that can produce this observation, and note that for our best fit models, the acceleration of secondary positrons should contribute substantially to the CR positron flux, potentially accounting for a significant fraction of the observed positron excess.
IC acknowledges support from NASA Grant NNX15AB18G and from the Simons Foundation. DH is supported by the US Department of Energy under contract DE-FG02-13ER41958. Fermilab is operated by Fermi Research Alliance, LLC, under Contract No. DE- AC02-07CH11359 with the US Department of Energy. TL acknowledges support from NSF Grant PHY-1404311. FERMILAB-PUB-17-010-A, data from the Advanced Composition Explorer (ACE) Science Center http://www.srl.caltech.edu/ACE/ASC/ () and the Wilcox Solar Observatory obtrained in http://wso.stanford.edu/Tilts.html (), courtesy of J. T. Hoeksema, were used in this study. The Wilcox Solar Observatory is currently supported by NASA.
- Bergstrom et al. (1999) L. Bergstrom, J. Edsjo, and P. Ullio, Astrophys. J. 526, 215 (1999), eprint astro-ph/9902012.
- Hooper et al. (2004) D. Hooper, J. E. Taylor, and J. Silk, Phys. Rev. D69, 103509 (2004), eprint hep-ph/0312076.
- Bertone et al. (2005) G. Bertone, D. Hooper, and J. Silk, Phys. Rept. 405, 279 (2005), eprint hep-ph/0404175.
- Profumo and Ullio (2004) S. Profumo and P. Ullio, JCAP 0407, 006 (2004), eprint hep-ph/0406018.
- Bringmann and Salati (2007) T. Bringmann and P. Salati, Phys. Rev. D75, 083006 (2007), eprint astro-ph/0612514.
- Pato et al. (2010) M. Pato, D. Hooper, and M. Simet, JCAP 1006, 022 (2010), eprint 1002.3341.
- Simet and Hooper (2009) M. Simet and D. Hooper, JCAP 0908, 003 (2009), eprint 0904.2398.
- Di Bernardo et al. (2010) G. Di Bernardo, C. Evoli, D. Gaggero, D. Grasso, and L. Maccione, Astropart. Phys. 34, 274 (2010), eprint 0909.4548.
- Strong et al. (2007) A. W. Strong, I. V. Moskalenko, and V. S. Ptuskin, Ann. Rev. Nucl. Part. Sci. 57, 285 (2007), eprint astro-ph/0701517.
- Adriani et al. (2010) O. Adriani et al. (PAMELA), Phys. Rev. Lett. 105, 121101 (2010), eprint 1007.0821.
- Aguilar et al. (2013) M. Aguilar et al. (AMS), Phys. Rev. Lett. 110, 141102 (2013).
- Bergstrom et al. (2008) L. Bergstrom, T. Bringmann, and J. Edsjo, Phys. Rev. D78, 103520 (2008), eprint 0808.3725.
- Cirelli and Strumia (2008) M. Cirelli and A. Strumia, PoS IDM2008, 089 (2008), eprint 0808.3867.
- Cholis et al. (2009a) I. Cholis, L. Goodenough, D. Hooper, M. Simet, and N. Weiner, Phys. Rev. D80, 123511 (2009a), eprint 0809.1683.
- Cirelli et al. (2009) M. Cirelli, M. Kadastik, M. Raidal, and A. Strumia, Nucl. Phys. B813, 1 (2009), [Addendum: Nucl. Phys.B873,530(2013)], eprint 0809.2409.
- Nelson and Spitzer (2010) A. E. Nelson and C. Spitzer, JHEP 10, 066 (2010), eprint 0810.5167.
- Arkani-Hamed et al. (2009) N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer, and N. Weiner, Phys. Rev. D79, 015014 (2009), eprint 0810.0713.
- Cholis et al. (2009b) I. Cholis, D. P. Finkbeiner, L. Goodenough, and N. Weiner, JCAP 0912, 007 (2009b), eprint 0810.5344.
- Cholis et al. (2009c) I. Cholis, G. Dobler, D. P. Finkbeiner, L. Goodenough, and N. Weiner, Phys. Rev. D80, 123518 (2009c), eprint 0811.3641.
- Harnik and Kribs (2009) R. Harnik and G. D. Kribs, Phys. Rev. D79, 095007 (2009), eprint 0810.5557.
- Fox and Poppitz (2009) P. J. Fox and E. Poppitz, Phys. Rev. D79, 083528 (2009), eprint 0811.0399.
- Pospelov and Ritz (2009) M. Pospelov and A. Ritz, Phys. Lett. B671, 391 (2009), eprint 0810.1502.
- March-Russell and West (2009) J. D. March-Russell and S. M. West, Phys. Lett. B676, 133 (2009), eprint 0812.0559.
- Chang and Goodenough (2011) S. Chang and L. Goodenough, Phys. Rev. D84, 023524 (2011), eprint 1105.3976.
- Cholis and Hooper (2013) I. Cholis and D. Hooper, Phys. Rev. D88, 023013 (2013), eprint 1304.1840.
- Dienes et al. (2013) K. R. Dienes, J. Kumar, and B. Thomas, Phys. Rev. D88, 103509 (2013), eprint 1306.2959.
- Finkbeiner and Weiner (2007) D. P. Finkbeiner and N. Weiner, Phys. Rev. D76, 083519 (2007), eprint astro-ph/0702587.
- Hooper et al. (2009) D. Hooper, P. Blasi, and P. D. Serpico, JCAP 0901, 025 (2009), eprint 0810.1527.
- Yuksel et al. (2009) H. Yuksel, M. D. Kistler, and T. Stanev, Phys. Rev. Lett. 103, 051101 (2009), eprint 0810.2784.
- Profumo (2011) S. Profumo, Central Eur. J. Phys. 10, 1 (2011), eprint 0812.4457.
- Malyshev et al. (2009) D. Malyshev, I. Cholis, and J. Gelfand, Phys. Rev. D80, 063005 (2009), eprint 0903.1310.
- Grasso et al. (2009) D. Grasso et al. (Fermi-LAT), Astropart. Phys. 32, 140 (2009), eprint 0905.0636.
- Linden and Profumo (2013) T. Linden and S. Profumo, Astrophys.J. 772, 18 (2013), eprint 1304.1791.
- Blasi (2009) P. Blasi, Phys. Rev. Lett. 103, 051104 (2009), eprint 0903.2794.
- Mertsch and Sarkar (2009) P. Mertsch and S. Sarkar, Phys. Rev. Lett. 103, 081104 (2009), eprint 0905.3152.
- Ahlers et al. (2009) M. Ahlers, P. Mertsch, and S. Sarkar, Phys. Rev. D80, 123017 (2009), eprint 0909.4060.
- Blasi and Serpico (2009) P. Blasi and P. D. Serpico, Phys. Rev. Lett. 103, 081103 (2009), eprint 0904.0871.
- Mertsch and Sarkar (2014) P. Mertsch and S. Sarkar, Phys. Rev. D90, 061301 (2014), eprint 1402.0855.
- Fujita et al. (2009) Y. Fujita, K. Kohri, R. Yamazaki, and K. Ioka, Phys. Rev. D80, 063003 (2009), eprint 0903.5298.
- Kohri et al. (2016) K. Kohri, K. Ioka, Y. Fujita, and R. Yamazaki, PTEP 2016, 021E01 (2016), eprint 1505.01236.
- Malkov et al. (2016) M. A. Malkov, P. H. Diamond, and R. Z. Sagdeev, Phys. Rev. D94, 063006 (2016), eprint 1607.01820.
- Cholis and Hooper (2014) I. Cholis and D. Hooper, Phys. Rev. D89, 043013 (2014), eprint 1312.2952.
- Kachelriess et al. (2011) M. Kachelriess, S. Ostapchenko, and R. Tomas, Astrophys. J. 733, 119 (2011), eprint 1103.5765.
- Kachelrieß and Ostapchenko (2013) M. Kachelrieß and S. Ostapchenko, Phys. Rev. D87, 047301 (2013), eprint 1211.1033.
- Aguilar et al. (2016) M. Aguilar et al. (AMS), Phys. Rev. Lett. 117, 091103 (2016).
- (46) http://galprop.stanford.edu/.
- Strong (2015) A. W. Strong (2015), eprint 1507.05020.
- (48) N. version of GALPROP availabe at: http://sourceforge.net/projects/galprop.
- Strong and Moskalenko (1998) A. W. Strong and I. V. Moskalenko, Astrophys. J. 509, 212 (1998), eprint astro-ph/9807150.
- Cholis et al. (2016) I. Cholis, D. Hooper, and T. Linden, Phys. Rev. D93, 043016 (2016), eprint 1511.01507.
- Trotta et al. (2011) R. Trotta, G. Johannesson, I. Moskalenko, T. Porter, R. R. de Austri, et al., Astrophys.J. 729, 106 (2011), eprint 1011.0037.
- Tan and Ng (1982) L. C. Tan and L. K. Ng, Phys. Rev. D26, 1179 (1982).
- Tan and Ng (1983) L. C. Tan and L. K. Ng, J. Phys. G9, 227 (1983).
- Duperray et al. (2003) R. P. Duperray, C. Y. Huang, K. V. Protasov, and M. Buenerd, Phys. Rev. D68, 094017 (2003), eprint astro-ph/0305274.
- Dekkers et al. (1965) D. Dekkers, J. A. Geibel, R. Mermod, G. Weber, T. R. Willitts, K. Winter, B. Jordan, M. Vivargent, N. M. King, and E. J. N. Wilson, Phys. Rev. 137, B962 (1965).
- Capiluppi et al. (1974) P. Capiluppi, G. Giacomelli, A. M. Rossi, G. Vannini, A. Bertin, A. Bussiere, and R. J. Ellis, Nucl. Phys. B79, 189 (1974).
- Allaby et al. (1970) J. V. Allaby, F. G. Binon, A. N. Diddens, P. Duteil, A. Klovning, and R. Meunier (1970).
- Guettler et al. (1976) K. Guettler et al. (British-Scandinavian-MIT), Nucl. Phys. B116, 77 (1976).
- Johnson et al. (1977) J. R. Johnson, R. Kammerud, T. Ohsugi, D. J. Ritchie, R. Shafer, D. Theriot, J. K. Walker, and F. E. Taylor, Phys. Rev. Lett. 39, 1173 (1977).
- Antreasyan et al. (1979) D. Antreasyan, J. W. Cronin, H. J. Frisch, M. J. Shochet, L. Kluberg, P. A. Piroue, and R. L. Sumner, Phys. Rev. D19, 764 (1979).
- Moskalenko et al. (2002) I. V. Moskalenko, A. W. Strong, J. F. Ormes, and M. S. Potgieter, Astrophys. J. 565, 280 (2002), eprint astro-ph/0106567.
- Arsene et al. (2007) I. Arsene et al. (BRAHMS), Phys. Rev. Lett. 98, 252001 (2007), eprint hep-ex/0701041.
- Anticic et al. (2010) T. Anticic et al. (NA49), Eur. Phys. J. C65, 9 (2010), eprint 0904.2708.
- Bringmann et al. (2014) T. Bringmann, M. Vollmann, and C. Weniger, Phys. Rev. D90, 123001 (2014), eprint 1406.6027.
- Hooper et al. (2015) D. Hooper, T. Linden, and P. Mertsch, JCAP 1503, 021 (2015), eprint 1410.1527.
- Cuoco et al. (2016) A. Cuoco, M. Krämer, and M. Korsmeier (2016), eprint 1610.03071.
- di Mauro et al. (2014) M. di Mauro, F. Donato, A. Goudelis, and P. D. Serpico, Phys. Rev. D90, 085017 (2014), eprint 1408.0288.
- Kappl and Winkler (2014) R. Kappl and M. W. Winkler, JCAP 1409, 051 (2014), eprint 1408.0299.
- Kachelriess et al. (2015) M. Kachelriess, I. V. Moskalenko, and S. S. Ostapchenko, Astrophys. J. 803, 54 (2015), eprint 1502.04158.
- Potgieter (2013) M. Potgieter, Living Rev. Solar Phys. 10, 3 (2013), eprint 1306.4421.
- (71) http://www.srl.caltech.edu/ACE/ASC/.
- (72) http://wso.stanford.edu/Tilts.html.
- Cowsik (1980) R. Cowsik, Astrophys. J. 241, 1195 (1980).