Bayesian Analysis of the DAMPE Lepton Spectra and Two Simple Model Interpretations

Bayesian Analysis of the DAMPE Lepton Spectra and Two Simple Model Interpretations

Jia-Shu Niu jsniu@itp.ac.cn CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China    Tianjun Li tli@itp.ac.cn CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, No. 19A Yuquan Road, Beijing 100049, China    Ran Ding Center for High-Energy Physics, Peking University, Beijing, 100871, P. R. China    Bin Zhu Department of Physics, Yantai University, Yantai 264005, P. R. China    Hui-Fang Xue Astronomy Department, Beijing Normal University, Beijing 100875, P.R.China    Yang Wang School of Mathematical Sciences, Shanxi University, Shanxi 030006, P.R. China.
July 23, 2019
Abstract

Recently, DAMPE has released its first results on the high-energy cosmic-ray electrons and positrons (CREs) from about to , which directly detect a break at . This result gives us an excellent opportunity to study the source of the CREs excess. In this work, we used the data fo proton and helium flux (from AMS-02 and CREAM), ratio (from AMS-02), positron flux (from AMS-02) and CREs flux (from DAMPE without the peak signal point at ) to do global fitting simultaneously, which can account the influence from the propagation model, the nuclei and electron primary source injection and the secondary lepton production precisely. For extra source to interpret the excess in lepton spectrum, we consider two separate scenarios (pulsar and dark matter annihilation via leptonic channels) to construct the bump () and the break at . The result shows: (i) in pulsar scenario, the spectral index of the injection should be and the cut-off should be ; (ii) in dark matter scenario, the dark matter particle’s mass is and the cross section is . Moreover, in the dark matter scenario, the annihilation channel is highly suppressed.

Furthermore, we present two simple dark matter models to explain the DAMPE results by introducing an SM singlet scalar as dark matter particle. In one model, we introduce a doublet scalar as a mediator, while in the other model we introduce a pair of vector-like fermions . In these models, the real scalar dark matter can annihilate dominantly into the charged lepton pairs.

I Introduction

Recently, DAMPE (DArk Matter Particle Explorer) (Chang, 2014; Chang et al., 2017) Satellite, which has been launched on December 17, 2015, has released its first data on high-energy cosmic-ray electrons and positrons (CREs) (Ambrosi et al., 2017). DAMPE has measured the CREs (i.e., ) spectrum in the range of with unprecedented energy resolution (better than ). The results shows a bumps at about which is consistent with previous results (Adriani et al., 2009; PAMELA Collaboration et al., 2010; AMS Collaboration et al., 2014a; Fermi-LAT Collaboration et al., 2012; Collaboration et al., 2017a, b). More interesting, a break at and a peak signal at have been detected. All of these features cannot be described by a single power law and provide us an opportunity to study the source of high-energy CREs.

The peak signal at has been studied by many works which employed nearby pulsars wind, supernova remnants (SNRs) and dark matter (DM) sub-structures (Malyshev et al., 2009; Kuhlen and Malyshev, 2009; Brun et al., 2009; Gendelev et al., 2010; Profumo, 2012; Panov, 2013; Fang et al., 2017; Yuan et al., 2017a; Athron et al., 2017; Fan et al., 2017; Duan et al., 2017; Gu and He, 2017; Liu and Liu, 2017; Cao et al., 2017; Profumo et al., 2017). At the same time, considering the statistical confidence level of this signal is about which needs more counts in future, we exclude the peak signal and do a global fitting on the left points in DAMPE CREs spectrum in this work. As a result, if we refer to the DAMPE CREs flux in this work, the peak point is excluded except special emphasis.

In cosmic ray (CR) theory, the CR electrons are expected to be accelerated during the acceleration of CR nuclei at the sources, e.g. SNRs. But the CR positrons are produced as secondary particles from CR nuclei interaction with the interstellar medium (ISM) (Adriani et al., 2009; AMS Collaboration et al., 2013a; Barwick et al., 1997; AMS-01 Collaboration, 2007). From the results of the flux of positrons and electrons (AMS Collaboration et al., 2013b, 2014b, 2014c, 2014a), we can infer that there should be some extra sources producing electron-positron pairs. This can be interpreted both by the astrophysical sources’ injection (Shen, 1970; Zhang and Cheng, 2001; Yüksel et al., 2009; Hooper et al., 2009; Profumo, 2012; Blasi, 2009; Hu et al., 2009; Fujita et al., 2009) and DM annihilation or decay (Bergström et al., 2008; Barger et al., 2009; Cirelli et al., 2009; Zhang et al., 2009; Bergström et al., 2009; Yin et al., 2013; Dev et al., 2014).

As a result, the CREs data contains the primary electrons, the secondary electrons, the secondary positrons and the extra source of electron-positron pairs. If we want to study the properties of the extra source, we should deduct the primary electrons and secondary electrons/positrons first. The primary electrons are always assumed to have a power-law form injection and the secondary electrons/positrons are determined dominatingly by the CR proton and helium particles interact with ISM. Consequently, we should do global fitting to these data simultaneously which can avoid the bias of choosing the lepton background parameters..

Considering the situations of high-dimentional parameter space of propagation model and precise data sets, we employ a Markov Chain Monte Carlo (MCMC (Lewis and Bridle, 2002)) method (embeded by DRAGON) to do global fitting and sample the parameter space of all the related parameters to reproduce the CREs spectrum (Liu et al., 2010; Lin et al., 2015; Yuan et al., 2017b; Niu and Li, 2017).

Moreover, because of the significant difference in the slopes of proton and helium, of about (Panov et al., 2006; Ahn et al., 2010; PAMELA Collaboration et al., 2011a; AMS Collaboration et al., 2015a, b), has been observed, we use separate primary source spectra settings for proton and helium. Note also that we consider propagation of nuclei only up to and neglect possible contributions from the fragmentation of nuclei, which should be a good approximation since their fluxes are much lower than the p and He fluxes (Korsmeier and Cuoco, 2016). In this condition, all the secondary particles (antiprotons and leptons) are produced from the interactions between proton, helium and ISM, which give us a self-consistent way to combine the nuclei and lepton data together.

Furthermore, we propose two simple models to explain the DAMPE results. In these models, we introduce an SM singlet scalar . We assume that is odd under a discrete symmetry and then a dark matter candidate. In Model I, we introduce a doublet scalar without a Vacuum Expectation Value (VEV) or with a very small VEV. and couples dominantly to the charged leptons. With the term where is the SM Higgs field, the dark matter particles can annihilate dominantly into charged leptons via channel after electroweak symmetry breaking. In Model II, we introduce a pair of vector-like fermions , which are odd under symmetry as well. Thus, the dark matter particles can annihilate into charged leptons via and channels.

This paper is organized as follows. We first introduce the setups of our work in Sec. II. The global fitting method and the chosen data sets and parameters is give in Sec. III. After present the fitting results and add some discussions in Sec. IV. We present our model in Sec. V, and summarize our results in Sec. VI.

Ii Setups

In this section, we just listed some of the most important setups in this work which is different from our previous work (Niu and Li, 2017). More detailed description can be found in Ref. (Niu and Li, 2017).

ii.1 Propagation model

In this work, we use the diffusion-reacceleration model which is widely used and can give a consistent fitting results to the AMS-02 nuclei data (see for e.g., (Niu and Li, 2017; Yuan et al., 2017b)). A uniform diffusion coefficient () is used in the whole propagation region.

At the same time, because high-energy CREs loss energy due to the process like inverse Compton scattering and synchrotron radiation, we parameterize the interstellar magnetic field in cylinder coordinates as

(1)

to calculate the energy loss rate. In Eq. 1, , ,  (Strong et al., 2000), and is the distance from the Sun to the galactic center.

ii.2 Primary Sources

In this work, considering the fine structure of spectral hardening for primary nuclei at (which was observed by ATIC-2 (Panov et al., 2006), CREAM (Ahn et al., 2010), PAMELA (PAMELA Collaboration et al., 2011a), and AMS-02 (AMS Collaboration et al., 2015a, b)) and the observed significant difference in the slopes of proton and helium (of about (PAMELA Collaboration et al., 2011b; AMS Collaboration et al., 2015a, b)), we use separate primary source spectra settings for proton and helium and each of them has 2 breaks at rigidity and . The corresponding slopes are (), () and (). For cosmic-ray electrons primary source, we followed the same configuration as proton and helium. But due to the DAMPE lepton data range (), we use 1 break for electron primary source, and the corresponding slopes are () and (()).

ii.3 Secondary sources

The secondary cosmic-ray particles are produced in collisions of primary cosmic-ray particles with ISM. The secondary antiprotons are generated dominantly from inelastic pp-collisions and pHe-collisions. At the same time, the secondary electrons and positrons are the final product of decay of charged pions and kaons which in turn mainly created in collisions of primary particles with gas. As a result, the corresponding source term of secondary particles can be expressed as

(2)

where is the number density of interstellar hydrogen (helium), is the differential production cross section, is the CR species density and is the total momentum of a particle.

To partially take into account the uncertainties when calculating the secondary fluxes, we employ a parameter and to re-scale the calculated secondary flux to fit the data (Tan and Ng, 1983; Duperray et al., 2003; Kappl and Winkler, 2014; di Mauro et al., 2014; Lin et al., 2015). Note that the above mentioned uncertainties may not be simply represented with a constant factor, but most probably they are energy dependent (Delahaye et al., 2009; Mori, 2009). Here we expect that a constant factor is a simple assumption.

ii.4 Extra sources

In this work, 2 kind of extra lepton sources are considered. The pulsar scenario account the extra lepton source to the pulsar ensemble in our galaxy, which is able to generate high energy positron-electron pairs from their magnetosphere. The injection spectrum of the CREs in such configuration can be parameterized as a power law with an exponential cutoff:

(3)

where is the normalization factor, is the spectral index, is the cutoff rigidity. The spatial distribution of this pulsar ensemble which provide continuous and stable CREs injection obeys the form as Eq. (5) in Ref. (Niu and Li, 2017), with slightly different parameters and Lin et al. (2015).

The DM scenario ascribe the extra lepton source to the annihilation of Majorana DM particles distributed in our galaxy halo, whose source term always has the form:

(4)

where present the DM density distribution, is the velocity-averaged DM annihilation cross section multiplied by DM relative velocity, and is the injection energy spectrum of CREs from DM annihilating into standard model (SM) final states through all possible channels with (the corresponding branching fractions). In this work, we considered DM annihilation via leptonic channels, the corresponding branching fractions for , , and are , , and respectively (). We use the results from PPPC 4 DM ID (Cirelli et al., 2011), which includes the electroweak corrections (Ciafaloni et al., 2011), to calculate the electron (positron) spectrum from DM annihilation by different channels. At the same time, we use Einastro profile (Navarro et al., 2004; Merritt et al., 2006; Einasto, 2009; Navarro et al., 2010) to describe the DM spatial distribution in our galaxy, which has the form:

(5)

with , and is the local DM energy density Catena and Ullio (2010); Weber and de Boer (2010); Salucci et al. (2010); Pato et al. (2010); Iocco et al. (2011).

ii.5 Solar modulation

We adopt the force-field approximation (Gleeson and Axford, 1968) to describe the effects of solar wind and helioshperic magnetic field in the solar system, which contains only one parameter the so-called solar-modulation . Considering the charge-sign dependence solar modulation represented in the previous fitting (Niu and Li, 2017), we use for nuclei (proton and helium) data and for data to do the solar modulation. At the same time, we use to modulate the positron flux. Because the DAMPE lepton data , we did not consider the modulation effects on electrons (or leptons).

ii.6 Numerical tools

The public code DRAGON 111https://github.com/cosmicrays/DRAGON (Evoli et al., 2008) was used to solve the diffusion equation numerically, because its better performance on clusters. Some custom modifications are performed in the original code, such as the possibility to use specie-dependent injection spectra, which is not allowed by default in DRAGON.

In view of some discrepancies when fitting with the new data which use the default abundance in DRAGON (Jóhannesson et al., 2016), we use a factor to rescale the helium-4 abundance (which has a default value of ) which help us to get a global best fitting.

The radial and grid steps are chosen as , and . The grid in kinetic energy per nucleon is logarithmic between and with a step factor of . The free escape boundary conditions are used by imposing equal to zero outside the region sampled by the grid.

Iii Fitting Procedure

iii.1 Bayesian Inference

As our previous works (Niu and Li, 2017), we take the prior PDF as a uniform distribution and the likelihood function as a Gaussian form. The algorithms such as the one by Goodman and Weare (2010) instead of classical Metropolis-Hastings is used in this work for its excellent performance on clusters. The algorithm by Goodman and Weare (2010) was slightly altered and implemented as the Python module emcee222http://dan.iel.fm/emcee/ by Foreman-Mackey et al. (2013), which makes it easy to use by the advantages of Python. Moreover, emcee could distribute the sampling on the multiple nodes of modern cluster or cloud computing environments, and then increase the sampling efficiency observably.

iii.2 Data Sets and Parameters

In our work, the proton flux (from AMS-02 and CREAM (AMS Collaboration et al., 2015a; Ahn et al., 2010)), helium flux (from AMS-02 and CREAM (AMS Collaboration et al., 2015b; Ahn et al., 2010)) and ratio ( from AMS-02 (AMS Collaboration et al., 2016)) are added in the global fitting data set to determine not only the propagation parameters but also the primary source of nuclei injections which further produce the secondary leptons. The CREAM data was used as the supplement of the AMS-02 data because it is more compatible with the AMS-02 data when .

On the other hand, the AMS-02 positrons flux (AMS Collaboration et al., 2014c) is added to set calibration to the absolute positron flux in DAMPE CREs flux (Ambrosi et al., 2017). Although the electron energy range covered by AMS-02 is under and there are systematics between the AMS-02 and DAMPE CREs data, fittings to the AMS-02 leptonic data provide a self-consistent picture for the extra source models. As the extra sources accounting for the AMS-02 results may provide contribution to the scale, the AMS-02 data could also constrain the properties of the predicted spectrum above . Considering the degeneracy between the different lepton data, we use the positron flux from AMS-02 and CREs flux from DAMPE together to constraint the extra source properties. The systematics are dealt with by employing a re-scale factor on positron flux.

Altogether, the data set in our global fitting is

The parameter sets for pulsar scenario is

for DM scenario is

Note that, most of these 2 scenarios’ parameters in the set and is the same with each other except those who account the extra sources of lepton.

Iv Fitting Results and Discussion

The MCMC algorithm was used to determine the parameters in the 2 scenarios. When the Markov Chains have reached their equilibrium state, we take the samples of the parameters as their posterior PDFs. The best-fitting results and the corresponding residuals of the proton flux, helium flux and ratio for 2 scenarios are showed in Fig. 1, and the corresponding results of the positron and CREs flux are showed in Fig. 2. The best-fit values, statistical mean values, standard deviations and allowed intervals at CL for parameters in set and are shown in Table 1 and Table 2, respectively .

Figure 1: The global fitting results and the corresponding residuals to the proton flux, helium flux and ratio for 2 scenarios. The (deep red) and (light red) bound are also showed in the figures.
ID Prior Best-fit Posterior mean and Posterior 95%
range value Standard deviation range
[1, 20] 14.37 14.380.16 [13.95, 14.74]

[0.1, 1.0] 0.318 0.3170.003 [0.311, 0.326]

[0.5, 30.0] 25.08 25.130.22 [24.55, 25.69]

[0, 80] 41.34 41.340.38 [40.37, 42.32]

[1, 8] 4.46 4.460.01 [4.44, 4.49]

[1, 30] 25.88 25.780.20 [25.43, 26.41]

[60, 1000] 428.98 429.057.44 [409.86, 447.63]

[1.0, 4.0] 2.196 2.1980.006 [2.180, 2.209]

[1.0, 4.0] 2.465 2.4640.005 [2.453, 2.474]

[1.0, 4.0] 2.348 2.3490.008 [2.332, 2.368]

[1, 30] 12.07 12.090.15 [11.67, 12.50]

[60, 1000] 244.83 246.418.14 [220.09, 265.47]

[1.0, 4.0] 2.186 2.1880.007 [2.170, 2.199]

[1.0, 4.0] 2.422 2.4220.005 [2.411, 2.431]

[1.0, 4.0] 2.219 2.2190.012 [2.197, 2.241]

[0, 1.5] 0.73 0.730.01 [0.71, 0.76]
[0, 1.5] 0.28 0.280.01 [0.26, 0.30]

[0.1, 10.0] 3.93 3.890.11 [3.66, 4.22]
[0.1, 10.0] 1.37 1.370.02 [1.34, 1.41]


[-4, 0] -1.936 -1.9360.006 [-1.950, -1.926]

[0, 3] 1.64 1.640.03 [1.55, 1.75]

[1.0, 4.0] 2.56 2.570.02 [2.50, 2.61]

[1.0, 4.0] 2.39 2.390.01 [2.36, 2.42]

[-8, -4] -6.15 -6.150.02 [-6.19, -6.11]

[0, 3.0] 0.65 0.650.01 [0.61, 0.69]

[2, 5] 2.81 2.800.02 [2.78, 2.86]

[0, 1.5] 1.37 1.370.01 [1.36, 1.39]

[0.1, 10.0] 5.09 5.080.05 [5.03, 5.15]


Table 1: Constraints on the parameters in set . The prior interval, best-fit value, statistic mean, standard deviation and the allowed range at CL are listed for parameters. With for best fit result.
ID Prior Best-fit Posterior mean and Posterior 95%
range value Standard deviation range
[1, 20] 15.72 15.760.14 [15.47, 15.96]

[0.1, 1.0] 0.307 0.3070.004 [0.302, 0.313]

[0.5, 30.0] 28.59 28.390.22 [28.07, 28.78]

[0, 80] 42.46 42.600.48 [41.69, 43.32]

[1, 8] 4.50 4.480.02 [4.45, 4.51]

[1, 30] 23.18 23.190.20 [22.92, 23.60]

[60, 1000] 497.28 492.088.41 [480.08, 507.07]

[1.0, 4.0] 2.222 2.2260.009 [2.212, 2.239]

[1.0, 4.0] 2.477 2.4770.006 [2.468, 2.486]

[1.0, 4.0] 2.357 2.3520.009 [2.338, 2.368]

[1, 30] 11.06 11.230.17 [10.97, 11.57]

[60, 1000] 237.29 232.958.88 [219.91, 248.52]

[1.0, 4.0] 2.206 2.2070.008 [2.196, 2.221]

[1.0, 4.0] 2.435 2.4350.005 [2.426, 2.443]

[1.0, 4.0] 2.232 2.2320.013 [2.213, 2.257]

[0, 1.5] 0.77 0.780.01 [0.76, 0.80]
[0, 1.5] 0.25 0.260.01 [0.24, 0.27]

[0.1, 10.0] 3.68 3.560.11 [3.38, 3.74]
[0.1, 10.0] 1.47 1.470.02 [1.44, 1.50]


[-4, 0] -1.940 -1.9430.007 [-1.958, -1.928]

[0, 3] 1.62 1.630.04 [1.57, 1.74]

[1.0, 4.0] 2.55 2.540.03 [2.46, 2.60]

[1.0, 4.0] 2.37 2.370.01 [2.34, 2.40]

[1, 6] 3.082 3.0850.006 [3.076, 3.096]

 888In unit
[-28, -18] -22.83 -22.800.06 [-22.93, -22.70]

[0, 1] 0.484 0.4790.007 [0.466, 0.488]

[0, 1] 0.508 0.5080.008 [0.493, 0.518]

[0, 1] 0.008 0.0130.010 [0.001, 0.032]

[0, 1.5] 1.32 1.310.01 [1.296, 1.332]

[0.1, 10.0] 5.02 5.030.03 [4.97, 5.08]


Table 2: The same as Table. 1, but for the ones in set . With for best fit result.

In Fig. 1, we can see that the nuclei data is perfectly reproduced, which would provide a good precondition for the subsequent fitting on the lepton data. The proton and helium particles would produce the secondary particles (including anti-protons and positrons) in lower energy range. Although the CREAM proton and helium data in has a relative large uncertainties, the spectral hardening at is accounted and then its influence on secondary products is included.

The best-fitting results and the corresponding residuals of the lepton and positron spectra are showed in Fig. 2. The corresponding best-fit values, statistical mean values, standard deviations and allowed intervals at CL for these parameters are shown in Table 1 and Table 2.

In Fig. 2, the lepton data can be fitted within fitting uncertainties. Although we got smaller reduced from global fitting on pulsar scenario, if we consider the DAMPE CREs flux alone, the best fit results shows for pulsar scenario and for DM scenario.

In Table 1, the value is obviously lower than the usual pulsar models (Profumo, 2012; Panov, 2013). This should be noted for further research.

In Table 2, we can see that the DAMPE CREs flux constrain and effectively. It gives a dark matter particle’s mass and a cross section , which also need a suppress factor () to meet the value from cosmology (Jungman et al., 1996).

Another interesting property from the DM scenario is the leptonic branching channel of DM annihilation via is strongly suppressed, which is obviously different from the fitting results obtained from AMS-02 lepton data alone (Lin et al., 2015). Some of the lepton favored DM models has been constructed to interpret this results (see for e.g., (Chao et al., 2017)).

Figure 2: The global fitting results and the corresponding residuals to the AMS-02 positron flux and DAMPE lepton flux. The (deep red) and (light red) bound are also showed in the figures. The first column shows the fitting results of pulsar and the second shows the fitting results of DM. For DAMPE CREs flux only, we got for pulsar scenario and for DM scenario.

V Two Simple Models

We shall propose two simple models to explain the DAMPE results. In these models, we introduce an SM singlet scalar with quantum number under gauge symmetry. We assume that is odd under a discrete symmetry and then a dark matter candidate.

In Model I, we introduce a doublet scalar with quantum numbers . We assume that couples dominantly to the charged leptons, and does not have a VEV or has a very small VEV. The relevant new Lagrangian is given as follows

(6)

where , and , as well as , , , , are the left-handed quark doublets, right-handed up-type quarks, right-handed down-type quarks, left-handed lepton doublets, right-handed charged leptons, respectively. For simplicity, we assume that couples dominantly to the charged leptons, i.e., . After the electroweak symmetry breaking, we obtain , and then the dark matter particles can annihilate dominantly into charged leptons.

In Model II, we introduce a pair of vector-like fermions , which are odd under symmetry as well. The quantum numbers for and are and , respectively. The relevant new Lagrangian is given as follows

(7)

Thus, the dark matter particles can annihilate into charged leptons via and channels.

Vi Conclusion

In this work, we did Bayesian analysis on the newly released CREs flux (exclude the peak signal at ) from DAMPE to study the extra source properties in it. In order to deduct the primary electrons, secondary leptons in CREs flux consistently and precisely, we did a global fitting to reproduce the proton flux (from AMS-02 and CREAM), helium flux (from AMS-02 and CREAM), ratio (from AMS-02), positron flux (from AMS-02) and CREs flux (from DAMPE) simultaneously. Two independent extra source scenarios are considered, which account the excess of leptons to continuously distributed pulsars in the galaxy and dark matter annihilation (via leptonic channels) in the galactic halo. Both of these scenarios can fit the DAMPE CREs flux within the fitting uncertainties, while DM scenario gave a smaller and a obvious break at .

Additionally, in the DM scenario, the fitting result gives a dark matter particle’s mass and a cross section . This is benefited from the break at . In such situations, the cross section in this work still should have a suppress factor to meet the value . This discrepancy can be resolved by some proposed mechanisms like the non-thermal production of the DM (Jeannerot et al., 1999; Lin et al., 2001; Yuan et al., 2012), the Sommerfeld enhancement mechanism (Sommerfeld, 1931; Hisano et al., 2005; Arkani-Hamed et al., 2009), and Breit-Wigner type resonance of the annihilation interaction (Griest and Seckel, 1991; Gondolo and Gelmini, 1991). What’s more interesting, the constraints on the annihilation branching fraction shows the annihilation channel is strongly suppressed, while the and channels are almost equally weighted (, , and ).

Furthermore, we proposed two simple models to explain the DAMPE results. We introduce an SM singlet dark matter scalar in these models. In Model I, we introduced a doublet scalar without a VEV or with a very small VEV. And couples dominantly to the charged leptons. With the term , the dark matter particles can annihilate dominantly into charged leptons via channel after electroweak symmetry breaking. In Model II, we introduced a pair of vector-like fermions , and the dark matter particles can annihilate into charged leptons via and channels.

Note: In this work, we can see that the CREs spectrum from DAMPE without the peak can be reproduced by DM scenarios precisely. On the other hand, the spectrum with peak also can be reproduced by DM annihilation from a local DM sub-structure (Yuan et al., 2017a; Athron et al., 2017; Fan et al., 2017; Duan et al., 2017; Gu and He, 2017; Liu and Liu, 2017; Cao et al., 2017; Jin et al., 2017; Huang et al., 2017; Yang et al., 2017; Ge and He, 2017). Both of these situations call for DM particles with . Other experiment is needed to distinguish the excess in the CREs spectrum which can also be produced from some astrophysical sources (Fang et al., 2017; Yuan et al., 2017a; Cholis et al., 2017). Our recent works Niu et al. (2017) proposed a novel scenario to probe the interaction between DM particles and electrons with .

Acknowledgments

We would like to thank Maurin et al. (2014) to collect database and associated online tools for charged cosmic-ray measurements. This research was supported in part by the Projects 11475238 and 11647601 supported by National Science Foundation of China, and by Key Research Program of Frontier Sciences, CAS. The calculation in this paper are supported by HPC Cluster of SKLTP/ITP-CAS.

References

Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
114858
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description