Cosmic ray propagation and dark matter in light of the latest AMS-02 data

The AMS-02 experiment is measuring the high energy charged cosmic rays with unprecedented accuracy. We explore the possibility of determining the cosmic-ray propagation models using the AMS-02 data $alone$. A global Bayesian analysis of the constraints on the cosmic-ray propagation models from the latest AMS-02 data on the Boron to Carbon nuclei flux ratio and proton flux is performed, with the assumption that the primary nucleon source is a broken power law in rigidity. The ratio of the diffusion coefficient $D_{0}$ to the diffusive halo height $Z_{h}$ is found to be determined with high accuracy $D_{0}/Z_{h}\simeq 2.00\pm0.07\text{cm}^{2}\text{s}^{-1}\text{kpc}^{-1}$. The best-fit value of the halo width is $Z_{h}\simeq 3.3$ kpc with uncertainty less than $50\$. As a consequence, the typical uncertainties in the positron fraction is within a factor of two, and that in the antiproton flux is within an order of magnitude. Both of them are significantly smaller than that from the analyses prior to AMS-02. Taking into account all the uncertainties and correlations in the propagation parameters we derive conservative upper limits on the cross sections for DM annihilating into various standard model final states from the current PAMELA antiproton data. We also investigate the reconstruction capability of the future AMS-02 antiproton data on the DM properties. The result shows that for DM particles lighter than 100 GeV and with typical thermal annihilation cross section, the cross section can be well reconstructed with uncertainties about a factor of two for the AMS-02 three-year data taking.

Boron to Carbon nuclei flux ratio and proton flux is performed, with the assumption that the primary nucleon source is a broken power law in rigidity. The ratio of the diffusion coefficient D 0 to the diffusive halo height Z h is determined with high accuracy D 0 /Z h ≃ 2.00± 0.07 cm 2 s −1 kpc −1 , and the value of the halo width is found to be Z h ≃ 3.3 kpc with uncertainty less than 50%. As a consequence, the typical uncertainties in the positron fraction predicted from dark matter (DM) annihilation is reduced to a factor of two, and that in the antiproton flux is about an order of magnitude. Both of them are significantly smaller than that from the analyses prior to AMS-02. Taking into account the uncertainties and correlations in the propagation parameters, we derive conservative upper limits on the cross sections for DM annihilating into various standard model final states from the current PAMELA antiproton data. We also investigate the reconstruction capability of the future high precision AMS-02 antiproton data on the DM properties. The results show that for DM particles lighter than ∼ 100 GeV and with typical thermal annihilation cross section, the cross section can be well reconstructed with uncertainties about a factor of two for the AMS-02 three-year data taking.

Introduction
Although compelling evidence from astronomical observations has indicated that dark matter (DM) contributes to 26.8% of the total energy density of the Universe [1], the particle nature of DM remains largely unknown. If DM particles in the galactic halo can annihilate or decay into the standard model (SM) final states, they may contribute to primary sources of cosmic-ray particles, which can be probed by precision DM indirect detection experiments.
Recently, the Alpha Magnetic Spectrometer (AMS-02) collaboration has updated its measurement of the cosmic-ray positron fraction, i.e., the ratio between cosmic-ray positron flux and the total flux of electrons and positrons in the energy range of 0.5-500 GeV [2]. The high precision data indicate that the positron fraction increases with energy in the energy range  GeV, consistent with the previouse measurements by PAMELA [3,4] and Fermi-LAT [5] but with much higher accuracy. For the first time, it was shown that the positron fraction ceases to increase at the energy ∼ 270 GeV. The rise and the existence of a maximum in the positron fraction is unexpected from the conventional astrophysics in which the majority of positrons are believed to be from the collisions of primary cosmicray nuclei with interstellar gas. Besides astrophysical explanations, an exciting possibility is that the observed positron fraction excess is due to DM annihilation or decay in the galactic halo.
In DM interpretations, through analysing the cosmic-ray positron anomaly, the properties of DM particle such as its mass and annihilation cross section or decay life-time can be inferred, and different DM models can be distinguished or even excluded ( for recent global analyses on AMS-02 data, see e.g. Refs [6][7][8][9][10][11][12][13][14][15][16][17][18][19] ). However, the conclusions are in general sensitive to the choice of cosmic-ray propagation model, cosmic-ray background as well as the profile of DM halo density distribution. The main source of the uncertainty is related to that in the propagation models. Analyses based on the data prior to AMS-02 have shown that the uncertainties of this type can reach O(10) in the prediction for positron flux [20] and O(100) for anti-proton flux for DM annihilation [21]. Note that the backgrounds of primary and secondary cosmic-ray particles which are of crucial importance in identifying the DM signals also depend on the propagation models.
In the diffusion models of cosmic-ray propagation, the major propagation parameters involve the diffusion halo height Z h , the spatial diffusion coefficient D 0 , the convection velocity V c related to the galactic wind, the Alfvèn speed V a related to the reaccelaration, and the primary source terms, etc.. The propagation models and parameters can be constrained by a set of astrophysical observables. The ratio between the fluxes of cosmicray secondary and primary nuclei such as that of Boron to Carbon nuclei (B/C) and the ratio of the radioactive isotopes such as that of Beryllium nuclei 10 Be/ 9 Be are commonly used to determine these parameters without knowing the primary sources ( for recent global fits, see e.g. [22][23][24][25] ). The primary source terms can be determined separately by the fluxes of primary cosmic-ray nuclei such as that of cosmic-ray protons.
Recently, the AMS-02 collaboration has reported the measurement of the B/C ratio in the kinetic energy interval from 0.5 to 670 GeV/nucleon with an unprecedented accuracy [26]. The AMS-02 experiment also released the data of proton flux as a function of rigidity from 1 GV to 1.8 TV [27], which is consistent with the previous measurement made by PAMELA in the low rigidity range from 20 to 100 GV [28]. In the high rigidity region above ∼ 100 GV, the proton spectrum measured by AMS-02 is consistent with a single power law spectrum. Under the assumption that the primary source is a broken power law in rigidity, the two type of data can be used together to determine the cosmic-ray propagation parameters.
In light of the recent significant experimental progresses, it is of interest to revisit the constraints on the cosmic-ray propagation models and explore the potential of the AMS-02 experiment on the capability of DM discovery. In this work, we first determine the main propagation parameters through a global Bayesian analysis to the AMS-02 data. We follow the strategy of determining both the propagation parameters and the primary sources in the same framework, using the data of B/C ratio and the proton flux. We show that the combination of B/C ratio and proton flux can lift the degeneracy in Z h and D 0 , and both the parameters can be well determined by the AMS-02 data alone. We find that the ratio of the diffusion coefficient D 0 to the diffusive halo height Z h is determined with high accuracy D 0 /Z h ≃ 2.00 ± 0.07 cm 2 s −1 kpc −1 , and the best-fit value of the halo width is Z h ≃ 3.3 kpc with uncertainty within 50%. From the allowed regions of parameter space, we estimate the uncertainties in the positron fraction and antiproton fluxes predicted by DM annihilation. We show that the uncertainties in the predicted positron fraction is within a factor of two and that in the antiproton flux is within an order of magnitude, which are significantly smaller than that from the previous analyses prior to AMS-02 ( see e.g. [22,23]). We construct reference propagation models corresponding to the minimal, median and maximal antiproton fluxes from DM annihilation into b-quarks. Combined with the PAMELA antiproton data, we derive conservative upper limits on the cross sections of DM annihilating into typical SM final states. We further project the sensitivity of the forthcoming AMS-02 data on the antiproton flux. The results show that for DM particle lighter than ∼ 100 GeV with a typical thermal annihilation cross section, the cross section can be reconstructed with uncertainties within a factor of two for the AMS-02 three-year data taking. This paper is organized as follows. In Sec. 2, we outline the formulas describing the propagation of cosmic-ray particles. In Sec. 3, we briefly overview the method of Bayesian inference used in our analysis. In Sec. 4, we present results on constraining the propagation models from the AMS-02 data of cosmic-ray B/C ratio and proton flux. In Sec. 5, we discuss the uncertainties in the prediction for positron fraction from DM annihilation into typical leptonic final states. In Sec. 6, we select typical propagation models corresponding to the minimal, median and maximal antiproton fluxes from DM annihilation into bb. In Sec. 7, taking into account the uncertainties in the propagation parameters, we derive upper limits on the DM annihilation cross sections for typical annihilation channels from PAMELA antiproton data. The reconstruction capability for the future AMS-02 data on the DM mass and annihilation cross sections is discussed. The conclusions are given in Sec. 8.

Propagation of cosmic-ray charged particles
It has been recognized that the propagation of cosmic rays in the Galaxy can be effectively described as a process of diffusion [29]. In this section, we briefly overview the main features of the cosmic-ray diffusion within the Galaxy. Detailed reviews of the transportation of processes can be found in Ref. [30]. The Galactic halo within which the diffusion processes occur is parametrized by a cylinder with radius R h = 20 kpc and half-height Z h = 1 − 20 kpc. The diffusion equation for the cosmic-ray charged particles reads (see e.g. [31]) where ψ(r, p, t) is the number density per unit of total particle momentum, which is related to the phase space density f (r, p, t) as ψ(r, p, t) = 4πp 2 f (r, p, t). For steady-state diffusion, it is assumed that ∂ψ/∂t = 0. The number densities of cosmic-ray particles are vanishing at the boundary of the halo, i.e., ψ(R h , z, p) = ψ(R, ±Z h , p) = 0. The spatial diffusion coefficient D xx is energy dependent and can be parametrized as where ρ = p/(Ze) is the rigidity of the cosmic-ray particle with electric charge Ze. The the power spectral index δ can have different values δ = δ 1(2) when ρ is below (above) a reference rigidity ρ 0 . The coefficient D 0 is a normalization constant, and β = v/c is the velocity of the cosmic-ray particle with c the speed of light. The convection term in the diffusion equation is related to the drift of cosmic-ray particles from the Galactic disc due to the Galactic wind. The direction of the wind is assumed to be along the direction perpendicular to the galactic disc plane and have opposite sign above and below the disc. The diffusion in momentum space is described by the reacceleration parameter D pp which is related to the velocity of disturbances in the hydrodynamical plasma, the so called Alfvèn speed V a as follows [31] where w characterise the level of turbulence. We take w = 1 as only V 2 a /w is relevant in the calculation. In Eq. (1), the momentum loss rate is denoted byṗ which could be due to ionization in the interstellar medium neutral matter, Coulomb scattering off thermal electrons in ionized plasma, bremsstrahlung, synchrotron radiation, and inverse Compton scattering, etc.. The parameter τ f (τ r ) is the time scale for fragmentation (radioactive decay) of the cosmic-ray nuclei as they interact with interstellar hydrogen and helium.
High energy electrons/positrons loss energy due to the processes like inverse Compton scattering and synchrotron radiation. The typical propagation length is around a few kpc for electron energy around 100 GeV. In the calculation of energy loss rate, the interstellar magnetic field in cylinder coordinates (R, z) is assumed to have the form where B 0 = 5 × 10 −10 Tesla, R B = 10 kpc, z B = 2 kpc [32], and r ⊙ ≈ 8.5 kpc is the distance from the Sun to the galactic center. The spectrum of a primary source term for a cosmic-ray nucleus A is assumed to have a broken power low behaviour with γ A = γ A1 (γ A2 ) for the nucleus rigidity ρ below (above) a reference rigidity ρ As . For cosmic-ray electrons, sometimes two breaks ρ es1 , ρ es2 are introduced with three power law indices γ e1 , γ e2 and γ e3 . The spatial distribution of the primary sources is assumed to have the following form [33] where η = 0.5, ξ = 1.0, and the normalization parameters q 0 is determined by the EGRET gamma-ray data. Secondary cosmic-ray particles are created in collisions of primary cosmic-ray particles with interstellar gas. The secondary antiprotons are created dominantly from inelastic ppand pHe-collisions. The corresponding source term reads q(p) = βcn i i=H,He dp ′ σ i (p, p ′ ) dp ′ n p (p ′ ) where n i is the number density of interstellar hydrogen (helium), n p is the number density of primary cosmic-ray proton per total momentum, and dσ i (p, p ′ )/dp ′ is the differential cross section for p + H(He) →p + X.
The primary source term of cosmic-ray particles from the annihilation of Majorana DM particles has the following form where σv is the velocity-averaged DM annihilation cross section multiplied by DM relative velocity (referred to as cross section) which is the quantity appears in the Boltzmann equation for calculating the evolution of DM number density. ρ(r) is the DM energy density distribution function, and dN (X) /dp is the injection energy spectrum of antiprotons from DM annihilating into SM final states through all possible intermediate states X with η X the corresponding branching fractions. The injection spectra dN (X) /dp from DM annihilation are calculated using the numerical package PYTHIA v8.175 [34], in which the long-lived particles such as neutron and K L are allowed to decay and the final state interaction are taken into account. Since PYTHIA v8.15 the polarization and correlation of final states in τ -decays has been taken into account [35]. The fluxes of cosmic-ray particles from DM annihilation depend also on the choice of DM halo profile. N-body simulations suggest a universal form of the DM profile where ρ ⊙ ≈ 0.43 GeV cm −3 is the local DM energy density [36]. The values of the parameters α, β, γ and r s for the Navarfro-Frenk-White (NFW) profile [37], the isothermal profile [38] and the Moore profile [39,40] are summarized in Tab Values of parameters α, β, γ and r s for three DM halo models, NFW [37], Isothermal [38], and Moore [39,40].
adopted DM profile is the Einasto profile [41] with α E ≈ 0.17 and r s ≈ 20 kpc. The interstellar flux of the cosmic-ray particle is related to its density function as For high energy nuclei v ≈ c. At the top of the atmosphere (TOA) of the Earth, the fluxes of cosmic-rays are affected by solar winds and the helioshperic magnetic field. This effect is taken into account using the force-field approximation [42]. In this approach, Φ TOA the cosmic-ray nuclei flux at the top of the atmosphere of the Earth which is measured by the experiments is related to the interstellar flux as follows where T TOA = T − φ F is the kinetic energy of the cosmic-ray nuclei at the top of the atmosphere of the Earth. It is known that some of the propagation parameters are strongly correlated. For instance, although both D 0 and Z h can change the comic ray flux, in the absence of spallation, the flux of a stable cosmic ray nuclei is sensitive only to the combination D 0 /Z h . In the re-acceleration term the Alfvèn speed V a scales as √ D xx . At high energies above 10 GeV, the approximate relation δ + γ A ≈ 2.7 holds very well. The B/C ratio as the ratio of secondary to primary can be used to determine the ratio of D 0 /Z h and other propagation parameters such as δ, V c and V a . The value of Z h can be determined by fitting both the B/C ratio and the ratio of the isotopes of Beryllium nuclei 10 Be/ 9 Be as 10 Be is radioactive and sensitive to Z h . Making use of the flux ratios, the propagation parameter can be determined without knowing the primary sources.
On the other hand, when the primary source is assumed to be a power or broken power law in rigidity as in Eq. (5), the spectrum of the primary cosmic-ray flux such as that of proton can impose constraints on both the propagation parameters and the primary sources. Since the proton flux is the most precisely measured quantity, it is expected that the constraints can be stringent.
We solve the diffusion equation of Eq. (1) using the publicly available numerical code GALPROP v54 [43][44][45][46][47] which utilizes realistic astronomical information on the distribution of interstellar gas and other data as input, and considers various kinds of data including primary and secondary nuclei, electrons and positrons, γ-rays, synchrotron radiation, etc. in a self-consistent way. Other approaches based on simplified assumptions on the Galactic gas distribution which allow for fast analytic solutions can be found in Refs. [48][49][50][51][52].

Bayesian inference
The Bayesian inference is based on calculating the posterior probability distribution function (PDF) of the unknown parameter set θ = {θ 1 , . . . , θ m } in a given model, which actually updates our state of belief from the prior PDF of θ after taking into account the information provided by the experimental data set D. The posterior PDF is related to the prior PDF by the Bayes's therom where L(D|θ) is the likelihood function, and π(θ) is the prior PDF which encompasses our state of knowledge on the values of the parameters before the observation of the data. The quantity p(D) is the Bayesian evidence which is obtained by integrating the product of the likelihood and the prior over the whole volume of the parameter space The evidence is an important quantity for Bayesian model comparison. It is straight forward to obtain the marginal PDFs of interested parameters {θ 1 , . . . , θ n }(n < m) by integrating out other nuisance parameters {θ n+1 , . . . , θ m } The marginal PDF is often used in visual presentation. If there is no preferred value of θ i in the allowed range (θ i,min , θ i,max ), the priors can be taken as a flat distribution The likelihood function is often assumed to be Gaussian where f th,i (θ) are the predicted i-th observable from the model which depends on the parameter set θ, and f exp,i are the ones measured by the experiment with uncertainty σ i . For experiments with only a few events observed, the form of the likelihood function can be taken as Poisson. When the form of the likelihood function is specified, the posterior PDF can be determined by sampling the distribution according to the prior PDF and the likelihood function using Markov Chain Monte Carlo (MCMC) methods. A commonly adopted algorithm is Metropolis-Hastings MCMC which is implemented in the numerical package CosmoMC [53]. Other advanced sampling methods such as the MultiNest algorithm are also commonly adopted [54,55]. The statistic mean value of a parameter θ can be obtained from the posterior PDF P (θ|D) in a straight forward manner. Using the MCMC sequence {θ } of the parameter θ i with N the length of the Markov chain, the mean (expectation) value θ i is given by The 1σ standard deviation of the parameter θ i is given by

Constraining propagation models using AMS-02 data
The propagation models can be constrained by cosmic-ray data. Since the statistics of the AMS-02 data on charged cosmic-ray particles are now much higher than that of other experiments and will continue to increase, it is of interest to consider constraining the propagation models using the AMS-02 data alone. One advantage of this strategy is that the complicities involving the combination of the systematics of different type of experiments can be avoided. Furthermore, all the current AMS-02 data are taken in the same period of solar activity, which makes it easier to estimate the effect of solar modulation consistently. The AMS-02 data of which we shall include in the analysis are the spectra of the cosmic-ray nuclei ratio B/C (18 data points) [26] and the proton flux (100 data points) [27], namely, the whole data set is The role of the proton data in constraining the propagation parameters was not emphasised in the literature, as calculating the proton flux requires additional input of the primary source term. Under the assumption that the primary source is a broken power law in rigidity, which is widely adopted in the literature, the proton flux actually can impose a constraint on the parameters V a and D 0 . This is because the value of V a has to be adjusted to accommodate the low energy ( 20 GeV) proton flux data which do not follow a single power law in rigidity. Since the ratio D 0 /Z h can be constrained by the B/C ratio, including the constraint on V a and D 0 from the proton flux can lead to a constraint on the halo height Z h . Since we are focusing on determining the propagation parameters, the AMS-02 data of positrons and electrons [56] are not considered for the moment, as it is known that they are unlikely to be fully consistent with the conventional backgrounds, which calls for exotic contributions either from nearby astrophysical sources or from DM interactions.
We adopt the conventional diffusive reaccelaration (DR) models in which V c ≃ 0. It has been shown that in the GALPROP approach a nonvanishing V c results in the predicted peak of B/C spectrum to be too wide in comparison with the data [33,57]. We consider the case where R = 20 kpc and δ 1 = δ 2 ≡ δ, thus there are 4 free parameters related to the cosmic-ray propagation: Z h , D 0 , δ and V a . Two additional parameters γ p1 and γ p2 are introduced for the power-law indices of the primary source terms. The break in rigidity of the primary source is fixed at ρ ps = 10 4 MV. In the GALPROP code, the primary nuclei source term is normalized in such a way that the proton flux N p at a reference kinetic energy E kin =100 GeV is reproduced. We find N p = 4.83 ± 0.02 cm −2 sr −1 s −1 MeV −1 from interpolating the AMS-02 proton flux data at 100 GeV. The solar modulation amplitude φ which strongly correlates with γ p1 is fixed at φ = 550 MV. Thus in total there are 6 free parameters, The priors of all the parameters are chosen to be uniform distributions according to Eq. (16) with the prior intervals shown in Tab. 2.
In the GALPROP code, the diffusion equation is solved numerically on a spatial grid with widths ∆R = 1 kpc and ∆Z = 0.2 kpc. The momentum grid is on a logarithmic scale with a scale factor 1.4. For sampling the posterior distributions and calculating the marginal distributions, we use the numerical package CosmoMC [53] which implements the Metropolis-Hastings algorithm in the MCMC scan of the whole parameter space. We have built 18 parallel MCMC chains with ∼1500 samples in each chain after burn-in. These chains satisfy the convergence condition that the ratio of the inter-chain variance and intra-chain variance is less than 0.2 [58]. In total 2.6 × 10 4 samples were obtained from the MCMC scan. The results of the best-fit values, statistical mean values, standard deviations and allowed intervals at 95% confidence level (CL) for these parameters are summaried in Tab. 2. For a comparison, we also list the allowed ranges determined from a previous analysis in Ref. [23] which is based on the data prior to AMS-02 such as the B/C ratio from HEAO-3 [59], ATIC-2 [60] and CREAM-1 [61], the data of 10 B/ 9 Be from ACE [62], and the data of Carbon and Oxygen nuclei fluxes from ACE [63].
As it can be seen from the table, although the fitting strategy is quite different, the parameters determined by the AMS-02 data are similar to that in Ref. [23], but the uncertainties in the parameters are significantly smaller. For instance, the ratio D 0 /Z h is found to be D 0 Z h = (2.00 ± 0.07) cm 2 s −1 kpc −1 . TAB. 2: Constraints on the propagation models from the global Bayesian analyses to the AMS-02 data of B/C ratio and proton flux. The prior interval, best-fit value, statistic mean, standard deviation and the allowed range at 95% CL are listed for each propagation parameter. The parameter D 0 /Z h is in units of 10 28 cm 2 · s −1 kpc −1 . For a comparison, we also list the mean values and standard deviations of these parameters from a previous analysis in [23]. The value of D 0 /Z h in the parentheses is obtained from [23] using a naive combination of D 0 and Z h without considering the correlation.
The uncertainty is within 5%, which is mostly constrained by the B/C data. Note that a relatively small halo height is favoured by the AMS-02 data Compared with Z h = 5.4 ± 1.4 kpc obtained in Ref. [23], the value of Z h from this work is ∼ 40% lower with the uncertainty smaller by a factor of two. The determined power-law index in the diffusion term is δ = 0.29 ± 0.01, close to 1/3 from the Kolmogorov-type diffusion. The determined Alfvèn speed is V a = 44.6 ± 1.3 km s −1 which is about ∼ 16% larger than that from Ref. [23]. The power-law indices of the nuclei source term are found to be γ p1 = 1.78 ± 0.01 and γ p2 = 2.45 ± 0.01, which are mostly constrained by the data of proton flux. The index γ p2 is larger than 2.38 obtained from fitting the Carbon and Oxygen fluxes in Ref. [23], which indicates that the spectrum of proton flux is softer than that of heavier cosmic-ray nuclei at high energies. For an estimate of the goodness-of-fit, we evaluate the χ 2 function which is defined as χ 2 = −2 ln L. Using the best-fit parameters, we find that in total χ 2 = 49.0 in which the contribution from B/C is 6.1 and that from proton flux is 42.9. Thus χ 2 /dof = 49.0/112 which indicates a good agreement with the data. Based on the MCMC samples, the contours of allowed regions at 68% and 95% CL for a selection of propagation parameters are shown in Fig. 1. It can be seen that D 0 /Z h is positively (negatively) correlated with V a (δ), δ is negatively correlated with V a , and γ p1   and γ p2 are positively correlated. The approximate relation δ + γ p2 ≈ 2.72 holds very well. Less pronounced correlations are found between parameters V a and γ p1,p2 . The onedimensional marginal posterior PDFs for some of the parameters are shown in Fig. 2. In the figure, the best-fit values, mean values with standard deviations are also shown. The figure shows that all the one-dimensional PDFs are close to Gaussian. Fig. 3 shows the calculated spectra of the cosmic-ray B/C ratio, proton fluxes, antiproton fluxes and antiproton/proton ratio using the parameters allowed within 95% CL. One can see that the AMS-02 data on proton flux and B/C ratio are well reproduced by the GALPROP DR models. The predicted antiproton fluxes are consistent with the PAMELA data only for the kinetic energies above 10 GeV. At lower energies, the predicted antiproton flux is about 40% lower than the data of PAMELA and BESS-Polar II, which is a typical feature of the DR models in GALPROP [57]. The low energy antiproton spectrum can be correctly reproduced if one constructs sophisticated GALPROP models with a flattening of the diffusion coefficient together with a convection term and a break in the injection spectrum [57]. Another possibility is that the solar modulation may have a charge sign dependence, namely, the modulation for antiprotons is different from that of protons.

Positron fraction from DM annihilation
Recently the measurement of the positron fraction was extended to the energy range up to 500 GeV by AMS-02 [2]. For the first time, it was shown that the positron fraction stops to increase with energy at ∼ 270 GeV. The spectral features of the positron fraction such as the rate of increase with energy, the energy beyond which it ceases to increase and the rate at which it falls beyond the turning point are of crucial importance in distinguishing the DM models. The uncertainties in the propagation parameters affects the prediction for the spectrum of positron fraction from DM interactions. Making use of the constraints on the propagation parameters Z h , D 0 /Z h , D 0 , δ, γ p1 and γ p2 obtained in the previous section, we investigate how the the backgrounds and DM signals change in reference DM models due to the uncertainties in these parameters. In Fig. 4, we show the predicted electron and positron fluxes and the positron fraction for the case of background only. We choose a reference electron primary source with 2  [27], PAMELA [28] and CREAM [64] are also shown. (Upper right) the fitted spectra of B/C ratio. The data of AMS-02 [26], ACE [65], CREAM [61] and HEAO-3 [59] are also shown. (Lower left) the prediction for the antiproton flux at 95% CL. The data of PAMELA [66] and BESS-Polar II [67] are shown. (Lower right) the prediction for the antiproton to proton flux ratio at 95% CL. The data of PAMELA [68] are shown. breaks at ρ e1 = 4 GV and ρ e2 = 86.8 GV and three power law indices between the breaks are γ e1 = 1.46, γ e2 = 2.72 and γ e3 = 2.49, respectively. The shaded bands in the figure correspond to the variation of the propagation parameters within 95% CL. One sees that the typical uncertainties in background positron flux and positron fraction can reach a factor of two. Fig. 5 shows the predicted positron fraction for four typical DM annihilation channels χχ → 2µ, 4µ, 2τ and 4τ with the DM profile being Einasto. In each case, the DM particle mass and annihilation cross section are taken as the best-fit values obtained in a previous analysis based on the AMS-02 data in the year 2013 [11] 2µ : m χ = 570 GeV, σv = 6.7 × 10 −24 cm 3 s −1 ,

Energy[GeV]
It can be seen from the figure that these best-fit values still describe the AMS-02 new data quite well. The uncertainties in the predictions for positron fraction are within a factor of two at low energies below ∼ 500 GeV. However, at higher energies, the uncertainties are significantly smaller, which indicates that the future AMS-02 data of high energy positrons will be very useful in studying the properties of DM and distinguishing different DM models.

Antiproton flux from DM annihilation
Compared with cosmic-ray electrons, which loss energy quickly due to the inverse Compton scattering and synchrotron radiation, the cosmic-ray protons lose much less energy in the propagation process. Thus they can travel across a longer distance in the galaxy before arriving at the detectors, which makes the proton/antiproton fluxes more sensitive to the propagation parameters.
In the previous section, we have shown that with the current AMS-02 data the important propagation parameters such as D 0 /Z h and Z h can be determined with better precisions, which is useful in improving the predictions for the cosmic-ray antiproton fluxes induced from DM interactions. In this section, we estimate the uncertainties in the prediction for antiproton flux from DM annihilation and construct reference propagation models which give rise to the typically minimal, median and maximal antiproton fluxes within 95% CL. Such reference models are useful for a quick estimation of the propagation uncertainties in future analyses. We shall focus only on the case of DM annihilation. It is In each plot the hatched band indicates the uncertainty of the background at 95% CL. The data of AMS-02 [2], PAMELA [4] and Fermi-LAT [5] are also shown.  [39,40] and Einasto (lower right) [41]. The mass of the DM particle is 130 GeV and the annihilation cross section is fixed at σv 0 = 3 × 10 −26 cm 3 s −1 .
the uncertainties due to the propagation parameters are within an order of magnitude.
In some previous analysis, the choice of benchmark models leads to an uncertainty of O(100) [21]. Such a significant improvement is related to the precision AMS-02 data on the B/C ratio. Fig. 6 also shows that the differences due to the DM profile are typically around a factor of two among the profiles of NFW, Isothermal and Einasto. In the Moore profile, the differences are bigger and can reach O(20).

Dark matter properties from current and future antiproton data
Taking into account the uncertainties of all the propagation parameters, one can derive conservative constraints on the properties of DM particles from the current PAMELA data and make projections for the sensitivity of the upcoming AMS-02 antiproton measurement. Some previous analyses based on simplified assumptions of fixed background or allowing part of the propagation parameters to vary can be found in Refs. [71][72][73][74]. In the Bayesian approach, it is straightforward to consider the uncertainties and correlations of the propagation parameters consistently, as the posterior PDFs of the propagation parameters obtained in Sec. 4 can be used as the prior PDFs in the subsequent Bayesian analysis. The inclusion of the new data will also update the "degree of believe" of these parameters, as well as constrain the new parameters related to the properties of DM particles. In the case of DM annihilation, the new parameter set related to DM annihilation is θ ′ = { σv , m χ }. The new data set of cosmic-ray antiproton is p/p ) stands for the data of antiproton flux (antiproton to proton flux ratio) from PAMELA. The posterior PDF for the parameter set θ ′ can be written as whereπ(θ) is the prior PDF of the propagation parameter set θ defined in Eq. (20), which has been updated from uniform distributions after considering the constraints from the AMS-02 data set D in Eq. (19), i.e.,π(θ) = P (θ|D), where P (θ|D) is calculated using the Bayes's theorem in Eq. (13).

Constraints on DM properties from PAMELA antiproton data
We consider several reference DM annihilation channelsχχ → X where X = bb, tt, W + W − , Z 0 Z 0 and hh. The energy spectra of these channels are all similar at high energies. The main difference is in the average number of total antiprotons N X per DM annihilation of each channel. For a DM particle mass m χ = 500 GeV, the values of N X for typical final states are N qq = 2.97 (q = u, d), N bb = 2.66, N tt = 3.20, N W W = 1.42, N ZZ = 1.48, and N hh = 2.18, respectively. Note that some of them are related. For instance, N hh ≈ 2N bb · Br 2 (h → bb).
We include the data of antiproton flux and antiproton-to-proton flux ratio from the current PAMELA experiment [66,68]. To avoid the complicities involved in modelling the effect of solar modulation, we only include the data points with antiproton kinetic energy E > 10 GeV. In total 8 (7) data points from antiproton flux (antiproton-to-proton flux ratio) are included in the analysis. The DM profile is chosen to be Einasto profile. Fig. 7 shows the results of upper limits on the annihilation cross sections at 95% CL. When the uncertainties in the propagation parameters are included, the upper limits obtained are always above the typical thermal cross section σv 0 for the mass range m χ ≈ 10GeV − 1TeV For bb final state, the most stringent limit is σv 10 −25 cm 3 s −1 at m χ ≈ 70 GeV. For TeV scale DM particle, the upper limits are around 10 −24 cm 3 s −1 for all the channels. For a comparison, in Fig. 7 we also show the upper limits on the bb and W + W − channels obtained from the Fermi-LAT gamma-ray data of dwarf satellite galaxies of the Milky Way [75]. One can see from the figure that when the uncertainties in the propagation parameters are considered, the upper limits from the PAMELA bb data are slightly more stringent that from the gamma-ray data.

Projected AMS-02 sensitivity
The forthcoming AMS-02 data on the antiproton flux is eagerly awaited. The AMS-02 detector has a high rejection power to distinguish antiprotons from protons, which is extremely helpful in identifying small excesses in the antiproton fluxes. In this section, we investigate the prospect for AMS-02 on reconstructing the property of DM particle in the case where an excess in the cosmic-ray antiproton flux over the conventional astrophysical background is identified in the forthcoming AMS-02 antiproton data.
We generate mock data of antiproton flux according to the specifications of the AMS-02 detector for the case of an astrophysical background plus a contribution from DM annihilation into bb final states. The binning of the kinetic energy spectrum of the antiproton flux is based on the rigidity resolution of the AMS-02 detector which is obtained through fitting to the Fig. 2 of Ref. [76] ∆R This value is for the observed event tracks hitting on both layer-1 and layer-9 of the AMS-02 silicon tracker. The rigidity resolution reaches 100% for R ≈ 1.9 TV, which roughly sets the upper limit on the proton/antiproton rigidity that can be measured by the AMS-02 detector. The relation between the resolution of the kinetic energy T and that of the rigidity reads ∆T where m p is the proton mass. The expected number of antiprotons N in the i-th kinetic energy bin with kinetic energy T i for an exposure time ∆t is given by where ǫ is the efficiency of the detector, a(T i ) is the acceptance for antiproton at kinetic energy T i , φ(T i ) is the expected antiproton flux, and ∆T i is the width of the i-th kinetic energy bin. From Ref. [77], the acceptance is a(T ) ≈ 0.147 m 2 for 1 GeV ≤ T ≤ 11GeV and a(T ) ≈ 0.03 m 2 for 11 GeV ≤ T ≤ 150 GeV. For T ≥ 150 GeV, the acceptance drops very quickly with increasing kinetic energy. In numerical calculations, we interpolate the values of a(T ) from Fig. 8 of Ref. [77]. The efficiency is assumed to be a constant ǫ = 0.9 in this work. Due to the geomagnetic effects, the value of ǫ becomes very low at kinetic energies below 1 GeV [78], we thus only consider the mock data above 1 GeV. Under the assumption that the distribution of the observed antiproton events is Poissonian, the statistic uncertainty in N observed events is ∆N = √ N . Thus the statistic uncertainty in the flux φ(T i ) is The systematic uncertainties may have various sources, such as the misidentification of background protons and electrons as antiprotons. The AMS-02 detector has a rejection power of p :p ∼ 10 5 − 10 6 for protons and e − :p ∼ 10 3 − 10 4 for electrons. At multi-GeV energy region, the flux ratios of p/p and e − /p are ∼ 10 4 and ∼ 10 2 respectively. Thus the systematic uncertainty can reach ∼ 1 − 10%. In this work, we take the systematic uncertainty to be ∆φ sys = 8%. The total uncertainty is ∆φ(T i ) = ∆φ(T i ) 2 sta + ∆φ 2 sys . In Fig. 8, we show the mock data of the projected AMS-02 antiproton flux with 3-year data taking. The antiproton background is generated according to the best-fit propagation parameters listed in Tab. 2. We assume that the DM particles annihilate into bb final states with a typical thermal cross section σv 0 = 3 × 10 −26 cm 3 s −1 for different masse m χ = 10, 100, 250 and 500 GeV, respectively, and the cases of large cross sections σv = 1 and 3 × 10 −25 cm 3 s −1 for a large m χ = 500 GeV. The halo DM profile is assumed to be Einasto. As can be seen from the figure, only in the cases where a light 10 GeV DM particle with typical thermal cross section or a heavy 500 GeV DM particle with a large cross section, the DM contribution can lead to a visible change in the antiproton flux. However, it is still possible that a tiny change in the spectrum of antiproton flux can be identified by the AMS-02 experiment.
We first investigate the reconstruction capability for the cases where the DM annihilation cross section is fixed σv = σv 0 and the DM particle mass is allowed to vary in the range ∼ 10 − 500 GeV. In Fig. 9, we show the results of the reconstruction for m χ = 10, 30, 50, 100, 250 and 500 GeV. The figure shows that for m χ 100 GeV, the annihilation cross section can be reconstructed with uncertainties around a factor of two. For a fixed annihilation cross section, the reconstruction becomes difficult for heavier DM particle, as the source term is suppressed by m 2 χ . As shown in Fig. 9, when m χ > 250 GeV, only an upper limit is obtained from the mock data. We then consider the case where m χ is fixed at 500 GeV and σv differs significantly from σv 0 . For large annihilation cross sections σv = 1 × 10 −25 cm 2 and 3 × 10 −25 cm 2 , we find that the cross section can still be well reconstracted with uncertainty typically about a factor of two. In all the cases, we find that the DM particle mass can be well reconstructed with uncertainties less than ∼ 30%.

Conclusions
The AMS-02 experiment is measuring the spectra of cosmic-ray nuclei fluxes with unprecedented accuracies, which is of crucial importance in understanding the origin and propagation of the cosmic rays and searching for dark matter. We have performed a global Bayesian analysis of the constraints on the cosmic-ray propagation models from the recent AMS-02 data on the ratio of Boron to Carbon nuclei and proton flux with the assumption that the primary source is a broken power law in rigidity. The analysis is based on the method of MCMC sampling. The result has shown that the propagation parameters can be well determined by the AMS-02 data alone. For instance, the ratio of the diffusion coefficient to the diffusive halo height is found to be D 0 /Z h ≃ 2.0 cm 2 s −1 kpc −1 with uncertainty less than 5%. The best-fit value of the halo width is Z h ≃ 3.3 kpc with uncertainty less than 50%. Other parameters such as the Alfvèn speed and the power law indices of the primary sources have also been determined. Such results can be used to improve the prediction of the antiproton flux from DM interactions. Using the allowed regions of parameter space, we have estimate the uncertainties in the positron fraction and antiproton fluxes predicted by DM annihilation. We have shown that the uncertainty in the predicted positron fraction is within a factor of two and that in the antiproton flux is within an order of magnitude, which are much smaller than the estimations in the previous analyses prior to AMS-02. With all the uncertainties and correlations in the propagation parameters taken into account, we have derived conservative upper limits on the cross sections for DM annihilating into various standard model final states from the current PAMELA antiproton data. We have also investigated the reconstruction capability of the future AMS-02 antiproton data on the DM properties. The result have shown that if the DM particles are lighter than 100 GeV and the annihilation cross section is the typical thermal cross section, the annihilation cross section can be well reconstructed with uncertainties around a factor of two for the AMS-02 three-year data taking. data taking in the assumption of DM annihilating into bb final states with a typical thermal cross section σv 0 = 3 × 10 −26 cm 3 s −1 for DM particle masse m χ = 10, 100, 250, 500 GeV, respectively, and the cases of large cross sections σv = 1 and 3 × 10 −25 cm 3 s −1 for m χ = 500 GeV. In each plot, the dashed line represents the contribution from DM only, and the solid line represents the sum of background and DM contribution. The background is generated from the best-fit propagation parameters shown in Tab. 2. The halo DM profile is assumed to be Einasto. The mock data with kinetic energy below 1 GeV (shadowed region) is not used for the reconstruction of DM properties due to the geomagnetic cut off of the detection efficiency.