Constraints on particle dark matter from cosmic-ray antiprotons

Cosmic-ray antiprotons represent an important channel for dark matter indirect-detection studies. Current measurements of the antiproton flux at the top of the atmosphere and theoretical determinations of the secondary antiproton production in the Galaxy are in good agreement, with no manifest deviation which could point to an exotic contribution in this channel. Therefore, antiprotons can be used as a powerful tool for constraining particle dark matter properties. By using the spectrum of PAMELA data from 50 MV to 180 GV in rigidity, we derive bounds on the dark matter annihilation cross section (or decay rate, for decaying dark matter) for the whole spectrum of dark matter annihilation (decay) channels and under different hypotheses of cosmic-rays transport in the Galaxy and in the heliosphere. For typical models of galactic propagation, the constraints are significantly strong, setting a lower bound on the dark matter mass of a"thermal"relic at about 50-90 GeV for hadronic annihilation channels. These bounds are enhanced to about 150 GeV on the dark matter mass, when large cosmic-rays confinement volumes in the Galaxy are considered, and are reduced to 4-5 GeV for annihilation to light quarks (no bound for heavy-quark production) when the confinement volume is small. Bounds for dark matter lighter than few tens of GeV are due to the low energy part of the PAMELA spectrum, an energy region where solar modulation is relevant: to this aim, we have implemented a detailed solution of the transport equation in the heliosphere, which allowed us not only to extend bounds to light dark matter, but also to determine the uncertainty on the constraints arising from solar modulation modeling. Finally, we estimate the impact of soon-to-come AMS-02 data on the antiproton constraints.

detection studies. Current measurements of the antiproton flux at the top of the atmosphere and theoretical determinations of the secondary antiproton production in the Galaxy are in good agreement, with no manifest deviation which could point to an exotic contribution in this channel. Therefore, antiprotons can be used as a powerful tool for constraining particle dark matter properties. By using the spectrum of PAMELA data from 50 MV to 180 GV in rigidity, we derive bounds on the dark matter annihilation cross section (or decay rate, for decaying dark matter) for the whole spectrum of dark matter annihilation (decay) channels and under different hypotheses of cosmic-rays transport in the Galaxy and in the heliosphere. For typical models of galactic propagation, the constraints are significantly strong, setting a lower bound on the dark matter mass of a "thermal" relic at about 50 -90 GeV for hadronic annihilation channels. These bounds are enhanced to about 150 GeV on the dark matter mass, when large cosmic-rays confinement volumes in the Galaxy are considered, and are reduced to 4-5 GeV for annihilation to light quarks (no bound for heavy-quark production) when the confinement volume is small. Bounds for dark matter lighter than few tens of GeV are due to the low energy part of the PAMELA spectrum, an energy region where solar modulation is relevant: to this aim, we have implemented a detailed solution of the transport equation in the heliosphere, which allowed us not only to extend bounds to light dark matter, but also to determine the uncertainty on the constraints arising from solar modulation modeling. Finally, we estimate the impact of soon-to-come AMS-02 data on the antiproton constraints.

Introduction
Several astronomical observations confirm the fact that the vast majority of the matter content of the Universe is in the form of an unknown component called dark matter (DM) [1]. Among those DM candidates that are best motivated under a theoretical point of view, weakly interacting massive particles (WIMPs) play a special role: their weak interaction may allow them to possess the correct relic abundance to explain the observed amount of dark matter and, at the same time, lead to the possibility for WIMPs to produce observable astrophysical signals: gamma-rays, neutrinos, electrons/positrons, antiprotons, antideuterons [2] and further indirect electromagnetic signals, in the whole electromagnetic spectrum down to radio frequencies.
Among the various channels for DM indirect detection, antiprotons are known to represent one of the best options, since the flux of cosmic antiprotons has been measured in recent years by many experimental collaborations to a good level of precision: BESS [3,4], AMS [5], BESS-Polar [6] and PAMELA [7,8]. Novel data are expected from AMS-02. On the theoretical side, antiprotons have been suggested for the first time as a possible signature of DM in [9,10] and then they have been studied as a way to constrain the properties of annihilating or decaying DM particles in a huge variety of theoretical frameworks starting from supersymmetry [11][12][13][14][15][16][17][18][19][20][21][22][23][24] to Kaluza-Klein DM [25][26][27] but also in relation to minimal DM models [28] or, more recently, as a constraining signal for DM models with internal bremsstrahlung [29][30][31].
In this paper, our purpose is to derive updated constraints on the DM annihilation cross section (or lifetime in the case of decaying DM) from experimental measurements of the antiprotons flux at the top of the atmosphere in a completely model independent framework [32][33][34][35][36]. In addition, and following the path traced in [37], we wish to add to the analisys of antiproton bounds also a detailed modeling of solar modulation, which is a critical element for low antiproton energies, where most of the experimental data are available and which are the relevant energies to constrain light DM. In fact, for DM masses below 50 GeV the constraints come from antiprotons with kinetic energies below 10 GeV, which is where solar modulation mostly affects the predicted fluxes. Solid and meaningful constraints for light DM therefore require a detailed modeling of cosmic rays transport in the heliosphere. We will profile ρ(r, z)/ρ parameters Isothermal (1 + r 2 /r 2 s )/(1 + (r 2 + z 2 )/r 2 s ) r s = 5 Kpc NFW (r / √ r 2 + z 2 )(1 + r /r s ) 2 /(1 + √ r 2 + z 2 /r s ) 2 r s = 20 Kpc Einasto exp(−2[( √ r 2 + z 2 /r s ) α − (r /r s ) α ]/α) r s = 20 Kpc , α = 0.17 Table 1. Dark matter density profiles ρ(r, z) adopted in the present analysis.
therefore study in detail the way in which a charge dependent solar modulation can affect the antiproton fluxes and the ensuing bounds. This will also allow us to quantify the impact of the uncertainties arising from solar modulation modeling. The paper is organised as follows: Section 2 very briefly summarizes the method used to describe the propagation of the antiprotons in our Galaxy. Section 3 deals with the issue of solar modulation, by introducing the fully numerical method employed to model the transport of cosmic rays in the heliosphere. Section 4 provides details about the way in which we calculate the bounds on the DM annihilation cross section (or decay rate). The bounds obtained from the PAMELA data are reported in Section 5, while Section 6 shows the projected sensitivity for future experiments, namely AMS-02. Section 7 summarizes our main conclusions.

Antiprotons production and propagation in the Galaxy
Antiprotons can be produced in the Galaxy through two main mechanisms: a primary flux is produced by DM in pair annihilation or decay events, while a secondary flux, which represent the astrophysical background, is produced by the spallation of cosmic rays on the nuclei that populate the interstellar medium (ISM).
Primary antiprotons are initially released in the ISM with an injected spectrum dNp/dT (T is the antiproton kinetic energy) which we model by using the PYTHIA MonteCarlo event generator (specifically, we have used the version 8.160 [38]). After being produced, antiprotons propagate in the galactic environment and are subject to a number of physics processes (diffusion, energy losses, drifts, annihilations) that can be described in terms of a transport equation (here expressed in cylindrical coordinates: a radial coordinate r along the galactic disk, a vertical coordinate z perpendicular to the disk): ∂ ∂z np(r, z, T ) + 2hδ(z)Γ ann p np(r, z, T ) = qp(r, z, T ) (2.1) The first term describes spatial diffusion, expressed through a diffusion coefficient K(r, z, T ), the second term refers to convection away from the galactic plane (V c is the convection velocity), the third term describes the possibility that antiprotons annihilate on the gas present in the galactic disk (Γ ann is the annihilation rate). The source term appearing in the right-hand-side is given by: for annihilating DM, and:  Table 2. Set of parameters of the galactic propagation models for charged cosmic rays employed in the analysis [16,39].
for decaying DM. In the previous equations, σ ann v is the thermally averaged annihilation cross section, Γ dec is the DM decay rate (Γ dec = 1/τ with τ the DM lifetime), ρ(r, z) is the DM density profile (in our analysis we will use the profiles listed in Table 1 and we adopt a local DM density of 0.39 GeV cm −3 ). As already stressed in [37], with Eq. (2.1) we are working in a framework in which reacceleration and energy losses are not taken into account. We use the fully analytical formalism of the two-zone diffusion model, which has been widely described in literature [39][40][41] and to which we address the reader for additional informations.
For the values of the astrophysical parameters that enter Eq. (2.1), we adopt the three sets called MIN, MED and MAX [16], listed in Table 2.
For the secondary antiproton flux the source term takes into account the hadronic interactions of primary cosmic rays on the ISM: where φ i (T i ) the flux of the primary cosmic rays species i impinging on the ISM nucleus j with a kinetic energy T i , while T th represents the minimal kinetic energy necessary to the production of one antiproton. For the secondary background we rely to Ref. [33]. We will comments on this secondary component and its uncertainties in Section 4.
3 Antiproton propagation in the heliosphere: solar modulation Before they are detected at Earth, CRs lose energy due to the solar wind while diffusing in the solar system [42]. This modulation effect depends, via drifts in the large scale gradients of the solar magnetic field (SMF), on the particle's charge including its sign [43]. Therefore, it depends on the polarity of the SMF, which changes periodically every ∼11 years [44]. Besides the 11 year reversals, the SMF has also opposite polarities in the northern and southern hemispheres: at the interface between opposite polarity regions, where the intensity of the SMF is null, a heliospheric current sheet (HCS) is formed (see e.g. [45]). The HCS swings then in a region whose angular extension is described phenomenologically by the tilt angle α. The magnitude of α depends on solar activity. Since particles crossing the HCS suffer from additional drifts because of the different orientation of the magnetic field lines, the intensity of the modulation depends on the extension of the HCS. This picture explains, at least qualitatively, the annual variability and the approximate periodicity of the fluctuations of CR spectra below a few GeV. The propagation of CRs in the heliosphere can be described by the following transport equation [46]: where f represents the CR phase space density, averaged over momentum directions, K represents the (symmetrized) diffusion tensor, V sw the velocity of the solar wind, v d the divergence-free velocity associated to drifts, P the CR momentum. The transport equation is solved in a generic 3D geometry within the heliosphere, with a boundary at 100 AU (see [47] and Refs. therein). The CR interstellar flux is given as a boundary condition and we assume that no sources are present within the solar system at the energies relevant to this work. A model for solar propagation is specified by fixing the solar system geometry, the properties of diffusion and those of winds and drifts. We describe the solar system diffusion tensor by K(ρ) = diag(K , K ⊥r , K ⊥θ )(ρ), where and ⊥ are set with respect to the direction of the local magnetic field. We assume no diffusion in the ⊥ ϕ directions and we describe as drifts the effect of possible antisymmetric components in K. For the parallel CR mean-freepath we take λ = λ 0 (ρ/1 GeV)(B/B ) −1 , with B = 5 nT the value of the magnetic field at the Earth position, according to [48,49]. For ρ < 0.1 GeV, λ does not depend on rigidity. We then compute K = λ v/3. Perpendicular diffusion is assumed to be isotropic. According to numerical simulations, we assume λ ⊥r,θ = 0.02λ [50].
For the SMF, we assume a Parker spiral, although more complex geometries might be more appropriate for periods of intense activity: where Ω is the solar differential rotation rate, θ is the colatitude, B 0 is a normalization constant such that B = 5 nT and A = ±H(θ − θ ) determines the MF polarity through the ± sign. The presence of a HCS is taken into account in the Heaviside function H(θ − θ ). The HCS angular extent is described by the function θ = π/2 + sin −1 (sin α sin(ϕ + Ωr/V SW )), where 0 < α < 90 • is the tilt angle. The drift processes, due to magnetic irregularities and to the HCS, are related to the antisymmetric part K A of the diffusion tensor as [51]: where K A = pv/3qB, r L is the particle's Larmor radius and q is its charge. We refer to [48,49] for more details on the implementation of the HCS and of drifts. Adiabatic energy losses due to the solar wind expanding radially at V SW ∼ 400 km/s are taken into account. As clear from Eq. (3.1), CRs lose energy adiabatically, due to the expansion of the solar wind, while propagating in the heliosphere. It is straightforward to notice that the larger their diffusion time (i.e. the shorter their mean-free-path) the more energy they lose in propagation. This fact is at the basis of the simplest modulation model used in the literature, the so called force-field model [42]. In this picture, the heliospheric propagation is assumed to be spherically symmetric, and energy losses are described by the modulation potential Φ ∝ |K|/V sw and Φ is to be fitted against data. However, this model completely neglects the effects of v d , which may significantly alter the propagation path. A and α are of particular importance in this respect. If q · A < 0, drifts force CRs to diffuse in the region close to the HCS, which enhances their effective propagation time and therefore energy losses, while if q · A > 0 drifts pull CRs outside the HCS, where they can diffuse faster [48,49]. As this is the only effect that depends on the charge-sign in this problem, and given that the force-field model does not account for it, the latter model cannot be used to describe CR spectra below a few GeV, where charge-sign effects are demonstrated to be relevant [43,47,[52][53][54][55].
We exploit then the recently developed numerical program HelioProp [55] for the 4D propagation of CRs in the solar system. The main effects of solar system propagation on antiprotons are demonstrated in Fig. 1, where we show how the TOA energy of these particles corresponds to the LIS energy of the same particle, for a sample of 10 4 particles generated at each E TOA in HelioProp. While at high energy E LIS = E TOA , because diffusion is so fast that no energy losses occur, at low energies, below a few GeV, E LIS > E TOA and the actual energy lost during propagation can vary significantly from particle to particle in our sample. This is due to the fact that energy losses are a function of the actual path, and the path is determined by a combination of drifts and random walks, being in fact a stochastic variable. Operationally, the flux observed at Earth at E TOA is determined as a proper weighted average of the LIS flux at the energies E LIS corresponding to that E TOA , as in Fig. 1.

Antiproton fluxes and determination of the bounds on DM properties
The most recent, accurate and statistically significant datasets on cosmic antiprotons are currently provided by the space-borne PAMELA detector [7,8] (in the interval between 90 MeV and 240 GeV) and by the balloon-borne BESS-Polar detector [6] (from 170 MeV to 3.5 GeV). The top-of-atmosphere (TOA) fluxes are reported in Fig. 2, together with the theoretical determination of the antiproton secondary production in the Galaxy obtained in Ref. [33]. The figure shows that secondary production is in good agreement with the data, and therefore additional (exotic) antiproton components, with dominant contribution in the 500 MeV to 50 GeV energy range, appear to be strongly constrained, unless significant modifications to the standard picture of cosmic rays production and propagation are invoked.
The secondary background flux is the critical element in the derivation of bounds on exotic components, including dark matter antiproton production. In Fig. 2 we show the central estimate for the MED set of propagation parameters, together with a (conservative) uncertainty band. Galactic propagation does not represent a major source of uncertainties for the secondary production: Ref. [33,56] showed that it accounts for about a 20-30% uncertainty when the propagation model is varied inside the MIN/MED/MAX models described in Sect. 2. This is at significant variance with the case of DM antiproton production, where different sizes of the confinement volume and of the corresponding diffusion coefficient induce a variation of the antiproton flux by a factor of about 10 up (for the MAX model) or down (for the MIN case), with some dependence on the antiproton energy [16]. A specific example, which can help in guiding the discussion of the next Sections on the DM bounds, is reported in Fig AU. For each set of lines, the upper/median/lower curve refers to the MAX/MED/MIN set of galactic propagation parameters. We notice that a change in solar modulation modeling has an impact which sizably differs depending on the interstellar flux, i.e. on the galactic transport model at hand. In the MED case, the uncertainty on the TOA fluxes due to solar modulation is maximal at lower kinetic energies, where it reaches the maximal size of 10% (15% for decaying DM) in the energy range below 10 GeV. This maximal effect occurs for larger values of the mean free path λ. In the case of the MIN model, the largest uncertainties are just around antiproton energies of 10 GeV, and they significantly decrease down to the few percent level at antiproton kinetic energies below 1 GeV. In the MAX model, the effect is instead enhanced, and can reach 20%-30% for very low kinetic energies, slowly decreasing to 10% at energies of 10 GeV. The origin of this different impact of solar modeling is traced back to the different energy behavior of the interstellar fluxes in the MIN/MED/MAX cases, as reported in Fig. 3: larger confinement volumes allow for steeper insterstellar fluxes in the 1-10 GeV kinetic-energy range (the range which is more relevant in the determination of the TOA fluxes after solar modulation occurs) and this therefore induces larger influence of solar modeling parameters in the the low-energy spectra at the Earth. In the MIN case, the lower confinement volume produces interstellar fluxes which are less steep in the few GeV range and this translates in less sensitivity of the TOA fluxes on variation of solar modeling. As stated, a similar behavior is found for different production channels.
While galactic transport (which has a relevant impact on the signal) is a minor source of uncertainty for the secondary antiproton production, the background component possesses some additional sources of uncertainty, which may arise from uncertainties in the knowledge of the primary proton and helium fluxes, as well as on the detailed mapping of the interstellar gas on which the primary protons impinge to produce the antiproton background. Moreover, the nuclear physics processes at the basis of the antiproton secondary production still suffer from indeterminacies, mostly related to the lack of updated data on the production cross sections at the center-of-mass energies relevant for low-energy cosmic rays studies. While variations due to the transport phenomena are consistently taken into account by performing both the DM signal and background calculation in the same model, nuclear uncertainties represent an irreducible element in the analysis, and we therefore assume a 40% theoretical error [16,56] on the secondary antiproton flux, which is represented by the shaded area in Fig. 2. In other recent analyses, like Ref. [36], the uncertainty on the secondary flux has been taken into account by allowing a free normalization and a free variation on the spectral index of the background flux: we instead assume the reference flux calculation of [16,33,56], obtained under physical assumptions, and allow for it a 40% uncertainty. The two approaches are both motivated, provided that the free normalization/spectral-index case does not require or allow for too-much arbitrary values in the normalization and spectral index (which actually is not the case in Ref. [36]). The approach of using a physical reference flux is well-founded on the fact that the background flux is calculated under the same physical assumptions used to determine the DM signal (same propagation model) and using a physical model based on data for the determination of the secondary production (primary proton and helium fluxes, gas distribution). We will adopt this approach consistently in our analysis and therefore consider the whole available antiproton energy spectrum, including the low-energy data below 10 GeV, which are relevant to constrain light dark matter.
For cosmic rays energies below 10 GeV, solar modulation effects are important, and for this we adopt the detailed techniques discussed in Sec. 3 for studying the antiproton transport in the heliosphere. While the most relevant source of variation on the bounds arises from galactic propagation, a goal of our analysis is in fact to determine the impact on the DM bounds arising from proper treatments of solar modulation. This is a source of uncertainty which is independent from the one arising from galactic propagation: improvements in the galactic transport modeling, hopefully coming from the new cosmic-rays measurements of the AMS detector, will still leave open the issue of solar modulation. It is therefore a relevant information to quantify these uncertainties. We will find that they can be as large as 50%, depending on the signal production mechanism (annihilation vs. decay) and they have quite different size and behavior depending on the interstellar flux at the edge of the heliosphere (which are in turn determined by the specific galactic transport model). The impact of solar modulation uncertainties on the bounds on DM is therefore correlated to the galactic transport modeling.
For definiteness, we will present the bounds obtained from the PAMELA dataset [7,8], since it covers a wider energy range. Since PAMELA reports slightly larger fluxes in the low-energy range, as compared to BESS-Polar, the derived bounds will be slightly more conservative. We will use the PAMELA data in the rigidity range from 50 MV up to 180 GV, for which a statistically relevant measurement of the antiproton flux is available (the highestrigidity bin, which reaches 350 GV currently provides only an upper limit on the antiproton flux).

Statistical analysis
The bounds on the DM properties are reported as upper limits on the annihilation cross section σ ann v (or lower limits in the case of the decay lifetime τ ) as a function of the DM mass m DM , for the different annihilation/decay channels which can produce antiprotons, and by assuming that the particle DM under study accounts for the whole DM in the Galaxy, regardless of the actual value of its annihilation cross section σ ann v or decay lifetime τ (as it is customary). We adopt a rastering technique, where we determine bounds on σ ann v (or τ ) at fixed values of the DM mass m DM . As a test statistic we employ a log-likelihood ratio R defined as: where L bg = i f (E i ) bg is the joint pdf of the background-only hypothesis (i runs on the energy bins E i ) and L(θ) bg+DM = i f (E i , θ) bg+DM , where θ denotes either σ ann v or τ . By assuming independent energy bins and gaussian pdfs, the test statistics is a chi-squared distribution with 1 degree of freedom, and we can set the bounds on the parameter θ by requiring that: where ∆χ 2 = χ 2 bg+DM − χ 2 bg , with: Let us comment that, as a consequence of experimental data being very well compatible with the background-only hypothesis, we have χ 2 bg ≈ χ 2 best fit . We conservatively determine upper [lower] bounds on σ ann v [τ ] at a one-sided confidence level of 3σ (i.e., CL = 99.86%), which corresponds to n = 10.21.
As discussed above, we allow theoretical uncertainties on the secondary background calculation at the level of 40%. The method we will adopt in the analysis is to assume the errors σ i,tot to be composed by two sources, which we add in quadrature: where σ i,theo = 0.4 × φ bg i , as stated, and where the experimental errors σ i,exp contain both the statistical and systematic uncertainties, which we add linearly: σ i,exp = σ i,stat + σ i,sys 1 . While this is a practical way of including the theoretical uncertainties, a more proper and statistically correct way is to generate a large sample of realizations of the background flux, normally distributed around the background reference flux [33] and with a standard deviation of 40%: for each background realization, a bound is derived by using only σ i,exp , and the ensuing distribution of the derived bounds on σ ann v (or τ ) can be analyzed. This has been done in one specific annihilation channel, in order to check the validity and the limitations of the method discussed above (which will be then adopted throughout). The left panel of Fig. 5 shows the statistical distribution of the 3σ upper bounds on σ ann v obtained with 10 5 statistical realizations of the background flux. The reference annihilation channel is bb and the bounds refer to a DM mass of 50 GeV. The mean value of the bounds is 1.1 × 10 −26 cm 3 s −1 (which corresponds to the upper limit obtained with the reference background flux), with a relatively broad distribution. This means that nuclear uncertainties in the background calculation represent a critical element in the ability to determine bounds on the DM properties (and on the possibility to detect a signal as well: with the upcoming AMS measurements, the dominant source of uncertainty will be in fact the theoretical one). The upper bound obtained with the technique discussed above is marked by the rightmost (red) vertical line, which corresponds to the 98% coverage of the cumulative distribution of the bounds found in our Monte Carlo analysis, as is clear from the right panel of Fig. 5, where the cumulative distribution function is reported. This shows that by adding the theoretical uncertainty to the experimental errors, as done in Eq. (4.5), well (and conservatively) intercepts the actual fluctuations on the background calculations due to nuclear uncertainties.   6 shows that the bounds arising from antiproton measurements are actually quite stringent: for light quarks, a thermal cross section is excluded for DM lighter than about 90 GeV, while for heavier quarks (which produce smaller antiproton multiplicities) the bound for thermal cross section is around 50 GeV. Light DM, below 10 GeV, is severely bounded, both in the annihilating and decaying case. These bounds, obtained for the central value of the allowed galactic-transport parameters set (the MED case) are actually competitive, if not better, than the limits obtained from gamma-rays measurements obtained with the Fermi-LAT detector, both from observations related to the extragalactic gamma-rays background and from observations of Milky Way satellites [57][58][59][60][61][62][63][64].

Constraints from PAMELA on the DM properties
The stringent bounds for DM lighter than about 50 GeV are mostly due to antiprotons arriving at the top-of-atmosphere with energies below 10 GeV. Data at low kinetic energies therefore represent an important tool to probe DM: however, this is also the energy range where solar modulation is operative and therefore a proper treatment of cosmic-rays transport in the heliosphere is important to determine the actual impact of antiproton measurements in this DM mass sector. To this aim we have carefully modeled solar modulation transport with the techniques described in Sec. 3, and we have adopted different models compatible with the PAMELA data-taking period in order to quantify uncertainties on the bounds arising from solar modulation treatment. Results for the representative case of thebb channel are shown in Fig. 7 For illustrative purposes, the annihilating case refers to thebb production channel (representative of heavy quark production), the decaying caseūu (representative of light quark production). Fig. 9 shows instead the case of DM annihilating (left panel) or decaying (right panel) into gauge bosons, specifically W + W − .
From Fig. 8 we can see that, for galactic propagation set at the MED case, the largest variation of the bounds occurs, as expected, for light DM and is of the order of 20% for annihilating DM and 40% for decaying DM. This maximal variation occurs for solar models with larger mean-free paths λ and is more relevant for light DM since in this case the bounds are mostly induced by the lower energy bins of the PAMELA measurements. For DM masses around 100 GeV, the variation in the bounds due to solar modulation modeling is still at the level of 10-15%, and decreases at a modest 5% level when the DM mass approaches 1 TeV. Variation of the annihilation channel in terms of quark production produces similar results. Fig. 9 shows the fractional variation R bounds in the case of the W + W − channel. Results are similar to the case of thebb channel: for DM masses of 100 GeV solar modulation modeling brings an uncertainty of the order of 20% , which steadily decreases to the few percent level for larger DM masses. In the case of gauge bosons production, the decrease in the uncertainty with the DM mass is steeper than in the case of quark production: this is due to the fact that the gauge-boson channel is harder than the quark channel, and this implies that the bounds on DM are coming from relatively larger energies, where solar modulation effects are smaller.
Finally, Fig. 7 also brings the information that in the MED annihilating case, solar modulation modeling introduces an uncertainty of 50% in the lower bound on the DM mass for thermal cross sections: it moves from 30 GeV for λ = 0.15 AU to 45 GeV for λ = 0.25 AU. We can therefore conclude that, in the case of the interstellar fluxes obtained with the MED galactic propagation, solar modulation modeling has an impact on the determination of antiproton bounds, especially for DM masses reaching up to 100 GeV, where the uncertainties can be seized to be of the order of 20-40%.
Coming back to the variation of the galactic transport modeling, this modifies the bounds as shown in Fig. 10 for the MIN set of propagation parameters and in Fig. 11 for the MAX set. Due to the significant variation of the absolute fluxes, as discussed above, the corresponding bounds are increased (decreased) by about an order of magnitude for the MAX (MIN) set of propagation parameters, as compared to the MED case. In the MIN case, thermal cross sections are excluded for DM masses below 4-5 GeV when annihilation occurs into light quarks, while they are not constrained when DM annihilates into heavy quarks. In the case of the MAX set of parameters, very stringent bounds are present: for thermal cross sections, all DM masses below 150 GeV are excluded. Summarizing the results in connection to galactic propagation modeling, the most probable set of transport parameters (MED) produces a lower limit on the DM mass around 50-90 GeV (depending on the annihilation channel). This may be considered as the most likely bound. Variation of galactic propagation modeling can sizably alter the bounds, setting the limit in a range between few GeV to 150 GeV. Refinements on the galactic propagation parameters, in the light of the new AMS-02 measurements on cosmicrays nuclei, will hopefully allow to reduce this source of uncertainty. Concerning decaying DM, antiprotons set a lower bound on the lifetime of the DM particle at about 10 28 s, which increases up to 10 29 s for DM masses of a few GeV and light-quarks production. These bounds are increased/decreased by about an order of magnitude for the MAX and MIN case.
Solar modulation modeling has an impact on the derived bounds which is more stable than what would be expected by the corresponding impact on the absolute fluxes, shown in Fig. 4. Fig. 8, representative for the quark production channels, and Fig. 9, representative for the gauge-bosons production channels, show that the impact of a variation of solar modulation modeling remains around 20-30% for light annihilating DM and can reach 30-50% for light decaying DM, regardless of the galactic transport model. The uncertainty is still of the same order of several tens of percent for DM with a mass around 10 GeV, and decreases to the few percent level at 1 TeV. We notice that in the case of the MAX galactic propagation, solar modulation uncertainties is always in excess of 10% even for DM masses of 1 TeV, when the production channel is in terms of quarks.
While these variations due to solar modulation modeling are not as large as those due to galactic transport modeling, nevertheless they have a size that can influence the ability to set bounds on the DM mass of annihilating DM which can reach 50%, once a galactic transport model is adopted, as it has been discussed above in connection to Fig. 7.

Prospects for AMS-02
In this Section we derive prospects for a 13 years data-taking period of the Alpha Magnetic Spectrometer (AMS-02), which was deployed on the International Space Station in May 2011. AMS-02 is an experiment designed to give precision measurements of a wide number of cosmic-rays species, including antiprotons. This will allow possible improvements in the determination of antiproton bounds on DM: larger statistics and reduced systematics on the antiproton spectrum; improved data on the primary flux, which could help in reducing the uncertainty on the theoretical determination of the secondary antiproton background; improved data on cosmic rays nuclei, which could be instrumental to reduce the galactic transport uncertainties; large statistics data over a long exposure time on a large number of cosmic rays species (hadronic and leptonic), which could help in better shaping transport modeling in the heliosphere. On the other hand, the extension of latitudes covered by the International Space Station trajectory will limit the minimal accessible energies, due to the geomagnetic cutoff.
We perform the analysis of the prospects for AMS-02 by generating mock data according to the AMS-02 specifications and by adopting on the mock data the same analysis technique described in Sec. 4, and used in Sec. 5 for the analysis of the PAMELA data. The mock data are generated under the hypothesis of the presence of background only, for which we adopt the theoretical estimate of Ref. [33], i.e. the median curve of Fig. 2.
Concerning solar modulation, since the AMS-02 operational period will likely be very long (we consider a duration from 2011 to 2024) and will cover more than one solar cycle, we subdivide the data-taking period in three phases, for which we adopt the following solar modeling: We determine the energy binning of the mock data by first determining the AMS-02 resolution in the energy range of interest (which is here below 500 GeV). This is directly derived from the rigidity resolution which, following Ref., [36] can be parametrized as: From the rigidity resolution, the energy resolution is directly obtained as: Then, we require that mock-data bins are comparable in size to the energy resolution: in agreement with Ref. [23], we adopt 10 bins per energy decade. In the energy bin with a central energy value T i and a width ∆T i , the number of expected antiproton events is then given by: where denotes the efficiency (we assume = 1, for definiteness), ∆t is the length of the data taking period, a(T i ) denotes the energy-dependent acceptance, which we assume as in Ref. [36]: for T < 11 GeV we assume a(T ) = 0.147 m 2 sr, for larger kinetic energies we derive an energy dependence by fitting the curve in Fig. 8 of Ref. [65]. Finally, we assume that the statistical error of the mock data in each energy bin is poissonian, and we allow for a 5% systematic uncertainty. The generated AMS mock data, together with the theoretical uncertainty bands of 40%, 20% and 5% sizes, are reported in Fig. 13. Due to geomagnetic effects, the efficiency will drop starting from energies of about 30 GeV, down to sub-GeV energies where the detection efficiencies (or, alternatively, the effective area) will be reduced to few percent of its nominal value [66]. For this reason, we include in the analysis of AMS mock data only the energy range above T min = 1 GeV.
Results are shown in Fig. 14 for theūu production channel, in Fig. 15 for thebb channel, and in Fig. 16 for the W + W − channel. The plots show the projected sensitivity for AMS-02, for annihilating (left panel) and decaying (right panel) DM, compared to the current bounds from PAMELA. The representative case reported in Fig. 14, 15 and 16 refers to an Einasto density profile and the MED set of propagation parameters in the Galaxy. Each set of curves (in the left panel the "upper" blue band refers to PAMELA, the "lower" red band refers to AMS-02; the reverse occurs in the right panel: the ''lower" blue band refers to PAMELA, the "upper" red band refers to AMS-02) show the current PAMELA bound or the projected AMS-02 sensitivity, under three different assumptions on the size of the theoretical uncertainties on the secondary antiproton production: solid, dashed and dot-dashed lines refer to 40%, 20% and 5%, respectively. The solid lines for PAMELA reproduce the bounds reported in Fig. 6. The horizontal (green) line in the left panel denotes the "thermal" value σ ann v = 3 × 10 −26 First of all, we notice that the theoretical uncertainty on the background flux can represent a dominant and limiting factor in the ability to improve the bounds on DM. By comparing the current PAMELA limits and the AMS projected sensitivity obtained with a 40% uncertainty on the background flux (solid lines in Fig. 14, 15 and 16) we see that AMS-02 will improve the bounds in the whole mass range and for all antiproton production channels, but for DM masses below 100 GeV the improvement will likely not be large. Only for DM masses above 100 GeV the bounds can be significantly improved, mostly due to the fact that AMS-02 will have access to antiproton energies larger than those covered by PAMELA. For very light DM, which produces antiprotons at low kinetic energies, the geomagnetic cutoff can instead be a limiting factor: Fig. 14 shows that for DM lighter than a few GeV (which is a case relevant only for annihilation/decay into light quarks) AMS-02 sensitivity drops.
In the case theoretical uncertainties in the background flux can be reduced, both PAMELA bounds and AMS-02 projected sensitivities would improve. In this case, the larger statistics of AMS-02 could be more throughly exploited, and the expected reach significantly extended. This is manifest in Figs. 14, 15 and 16, especially for a reduction of the theoretical uncertainties where both a 20% level and a more ambitious level of 5% are reported, in which case an improvement of up to an order of magnitude can be obtained, depending on the antiproton production channel and DM mass range.

Conclusions
In this paper we have presented the most updated analysis of the bounds on DM properties that can be obtained from antiprotons measurements. We have included in our analysis not only the uncertainties arising from galactic modeling (i.e. the DM density profile and, most relevant, the propagation parameters) which, as known, provide the largest variability in the derived bounds on DM properties, but we have also investigated the impact of solar modulation modeling, which we have shown to play a non negligible role, especially in the low DM mass range. To evaluate the importance of solar modulation, we have used a full numerical and charge-dependent solution of the equation that models cosmic rays transport in the heliosphere, tuned on data sensitive to solar activity. This detailed modeling has allowed us to quantify the impact of solar modulation on the derived bounds, once a galactic propagation model is adopted.
We have shown that the constraining power of the antiprotons measurements for DM particles that annihilate into quarks or gauge bosons is quite significant: bounds on the DM annihilation cross section (or lifetime, in the case of decaying DM) are very strong, similar or in some cases even stronger than those that arise from gamma-ray measurements. Considering the most probable set of galactic propagation parameters (the MED model) for annihilating DM and "thermal" cross section the whole DM mass range below 90 GeV is excluded when DM annihilates into light quarks; this bounds moved to 40 GeV when annihilation occurs into heavy quarks. In the case of decaying DM, the lower limit on the lifetime is set to 10 28 s for intermediate DM masses and can reach 10 29 s for very light DM particles annihilating into light quarks. Concerning solar modulation, variation of the modeling parameters, in particular the value of the mean free path λ, have an impact on the bounds that can be as large as 30-50% for the lightest DM particles and decreases as the DM particle mass grows. While these variations due to solar modulation modeling are not as large as those due to galactic transport modeling, nevertheless they have a size that can influence the ability to set bounds on the mass of annihilating DM: the quoted limit of 40 GeV for the mass of a DM particle annihilating into heavy quarks can be varied in a range of values which extends up to 60 GeV, when solar modulation modeling is taken into account.
In the last section of the paper, we have investigated the future perspectives for antiproton searches in the light of the AMS mission. We have shown that (and quantified how much) a high-precision experiment like AMS-02 will allow to set stronger bounds on DM properties, even if, as it has been stressed, effects such as the geomagnetic cutoff can play a non-negligible role, since they can limit the sensitivity in the lower DM masses region. However, in order to fully exploit the AMS increased sensitivity, a reduction of the theoretical errors (mostly related to nuclear uncertainties in the antiproton production processes and to the determination of the primary cosmic rays fluxes) in the determination of the astrophysical secondary antiproton background will be critically important.  Open circles (blue) data points refer to PAMELA measurements [7,8]. Open triangles (red) data points refer to BESS-Polar [6]. The solid line shows the antiproton secondary production, propagated in the Galaxy with the MED set of transport parameters [33] and further propagated in the heliosphere with a charge-dependent solar modulation with propagation parameters α = 20, λ = 0.15 AU and negative polarity. The band shows a (conservative) 40% theoretical uncertainties on the background calculation, mainly ascribable to nuclear-physics uncertainties in the production cross section and to uncertainties in the primary proton flux.         Figure 9. The same as in Fig. 4, for theW + W − production channel.   Figure 11. The same as in Fig. 6, for the MAX set of galactic propagation parameters.   Figure 13. Mock data for the AMS mission, used in the analysis for the AMS projected sensitivity. The mock data are generated from the central value of the antiproton theoretical background of Fig.  2. The three shaded bands around the mock data refer to a 40%, 20% and 5% uncertainty around the theoretical expectation. The vertical band for T < 1 GeV denotes the energy range not used in the analysis, because of the impact of the geomagnetic cutoff.  Figure 14. Projected sensitivity for AMS-02, for annihilating (left panel) and decaying (right panel) DM, compared to the current bounds from PAMELA. The representative case reported here refers to DM annihilation/decay into uū, an Einasto density profile and the MED set of propagation parameters in the Galaxy. In the derivation of these bounds, it has been assumed a low-energy threshold (due to the geomagnetic cut-off) for AMS-02 of T min p = 1 GeV. Each set of curves (in the left panel the "upper" blue band refers to PAMELA, the "lower" red band refers to AMS-02; the reverse occurs in the right panel: the ''lower" blue band refers to PAMELA, the "upper" red band refers to AMS-02) show the current PAMELA bound or the projected AMS-02 sensitivity, under three different assumptions on the size of the theoretical uncertainties on the secondary antiproton production: solid, dashed and dot-dashed lines refer to 40%, 20% and 5%, respectively. The solid lines for PAMELA reproduce the bounds reported in Fig. 6. The horizontal (green) line in the left panel denotes the "thermal" value σ ann v = 3 × 10 −26 cm 3 s −1 .