A halo-independent lower bound on the dark matter capture rate in the Sun from a direct detection signal

We show that a positive signal in a dark matter (DM) direct detection experiment can be used to place a lower bound on the DM capture rate in the Sun, independent of the DM halo. For a given particle physics model and DM mass we obtain a lower bound on the capture rate independent of the local DM density, velocity distribution, galactic escape velocity, as well as the scattering cross section. We illustrate this lower bound on the capture rate by assuming that upcoming direct detection experiments will soon obtain a significant signal. When comparing the lower bound on the capture rate with limits on the high-energy neutrino flux from the Sun from neutrino telescopes, we can place upper limits on the branching fraction of DM annihilation channels leading to neutrinos. With current data from IceCube and Super-Kamiokande non-trivial limits can be obtained for spin-dependent interactions and direct annihilations into neutrinos. In some cases also annihilations into $\tau\tau$ or $b\bar b$ start getting constrained. For spin-independent interactions current constraints are weak, but they may become interesting for data from future neutrino telescopes.


Introduction
From gravitational effects we know that dark matter (DM) constitutes a significant fraction of the energy density of the Universe. Among the most promising ways to search for non-gravitational manifestations of DM particles are direct detection (DD) experiments, which are looking for the scattering of DM particles from the galactic halo in underground detectors [1][2][3][4][5][6][7][8], and neutrino telescopes looking for high-energy neutrinos from the annihilations of DM particles in the Sun [9][10][11]. The latter signal emerges from the capture of galactic DM particles in the gravitational potential of the Sun after loosing enough energy in a DM-nucleus scattering event in the Sun. Hence, the capture rate of DM in the Sun is determined by the scattering cross section of the DM particle on nuclei, the same process which provides the signal in DD experiments. The resulting neutrino flux from the Sun will also depend on the annihilation channels of the DM particle. Therefore, viable information on DM properties can in principle be obtained by comparing the two signals.
However, the DD signal and the DM capture in the Sun depend on different parts of the DM velocity distribution. While DD experiments are sensitive to DM particles with velocity larger than a certain minimal velocity (which depends on the mass of the nuclei in the detector and its energy threshold as well as on the DM mass), the DM capture in the Sun is sensitive to values below a certain maximum velocity, above which capture of DM particles is kinematically forbidden. Therefore, in order to explore the complementarity of the two signals, it is common to adopt specific DM velocity distributions, for instance the so-called Standard Halo Model (SHM) consisting of a truncated Maxwellian distribution, see refs. [12][13][14][15][16][17][18][19] for a very incomplete list for examples for that approach. However, the properties of the DM velocity distribution as well as the local DM density are plagued with large uncertainties and halo-independent methods are desirable to draw robust conclusions about possible signals. The impact of variations of halo properties on the neutrino signal have been studied for instance in refs. [20,21] and in the context of the neutrino/DD comparison in ref. [22]. The authors of ref. [23] investigated the potential to extract DM parameters from a combination of data from DD experiments and from a neutrino signal based on a polynomial parameterization of the DM velocity distribution, whose parameters are fitted together with the DM parameters.
In the present paper we develop a completely halo-independent method to compare a signal from a DD experiment with a neutrino signal from the Sun. We show that from a precise measurement of the DD nuclear recoil spectrum a halo-independent lower bound on the capture rate in the Sun can be derived. It is based on the overlap region in velocity space and therefore does not require any assumptions about the halo properties. Our bound extends the halo-independent methods developed in the context of DD [24,25] to the capture rate in the Sun.
The remainder of this paper is organized as follows: We review the phenomenology of DD in section 2 and of the DM induced neutrino signal from the Sun in section 3. We then discuss the relation of direct detection to the capture rate in the Sun in section 4, where the central result of the paper (the lower bound on the capture rate) is given in section 4.2. In section 5, we apply the bound to mock data from future direct detection experiments and compare them to the upper bounds from the IceCube and Super-Kamiokande neutrino telescopes. We summarize and give our concluding remarks in section 6. In appendix A, we discuss how to use an annual modulation signal in a DD experiment to provide the lower bound on the capture rate, and apply our results to the DAMA signal. In appendix B we show how to use our results in the case of more general scattering cross sections, beyond contact interactions.

Dark matter direct detection
In this section we review the relevant expressions for DD of dark matter [26]. We focus on elastic scattering of DM particles χ with mass m χ off a nucleus with mass number A and mass m A , depositing the nuclear recoil energy E R . The differential rate (measured in events/keV/kg/day) for a single target detector is: 1 with ρ χ being the local DM mass density and v m is the minimal velocity of the DM particle required for a recoil energy E R : where µ χA is the reduced mass of the DM-nucleus system. The function f det ( v, t) describes the distribution of DM particle velocities in the detector rest frame, with the normalization and we define the angular averaged velocity distribution functionf bỹ where dΩ = d cos θ dφ. The velocity distributions in the rest frames of the detector, the Sun and the galaxy are related by where v e (t) is the velocity vector of the Earth relative to the Sun and v s is the velocity of the Sun relative to the galactic frame. The revolution of the Earth around the Sun encoded in v e (t) leads to an annual modulation of the DD signal [27,28].
To be specific, in the following we will concentrate on spin-independent (SI) and spin-independent (SD) scattering from a contact interaction. This implies that the differential scattering cross section dσ A (v)/dE R scales as 1/v 2 . Our results can also be generalized to other v dependences, as shown in appendix B. For SI contact interactions with equal DM couplings to neutrons and protons the cross section becomes where σ p χ is the total DM-proton scattering cross section at zero momentum transfer, µ χp is the DM-proton reduced mass, and F A (E R ) is a nuclear form factor. For SD interactions a similar formula applies with a different form factor and no A 2 enhancement.
Then the event rate eq. (2.1) becomes where v m is considered as a function of E R according to eq. (2.2) and we have defined Notice thatη depends only on the DM properties and the DM halo, but is independent of the characteristics of the experiment. For fixed DM mass, one can translate the event rate in E R space into v m space, andη(v m , t) then has to be the same for any experiment.
This is the basis of the halo-independent methods developed in refs. [24,25] and used extensively to compare results of different DD experiments, see, e.g., refs. [29][30][31][32][33][34][35][36][37]. For a specific detector the number of DM induced events in an energy range between E 1 and E 2 is given by where M and T are the detector mass and exposure time respectively, and is the detector response function describing the probability that a DM event with true recoil energy E R is reconstructed in the observed energy interval [E 1 , E 2 ], including energy resolution, energy dependent efficiencies, and possibly also quenching factors. In section 4.2 we will assume that a positive signal is observed in a direct detection experiment. In this case, the angular averaged distributionf (v) times the constant C can be extracted from the data (modulo experimental resolutions and uncertainties). Using eqs. (2.6) and (2.7), we find where E R is considered as a function of v = v m according to eq. (2.2), depending on the DM mass.

The neutrino signal from DM annihilations in the Sun
In this section we briefly review the relevant expressions for the neutrino signal from the Sun. Due to the scattering of DM particles with the nuclei in the Sun they may lose energy and become gravitationally bound to the Sun. Their annihilation products can produce neutrinos with energies comparable to the DM mass, detectable at Earth [38][39][40]. The capture rate of dark matter particles is given by (see, e.g., refs. [40,41]) where, for SI interactions, the sum over A goes over all element abundances in the Sun up to nickel, while for SD interactions only hydrogen is relevant. We use the short-hand notationf Sun ≡f and the angular averaged velocity distribution function is defined in eq. (2.4). The functionf (v) refers to the velocity distribution at infinity, whereas w is the DM velocity inside the gravitational potential of the Sun at the radial distance r from the solar center, given by w 2 = v 2 + u 2 esc (r), with u 2 esc (r) being the escape velocity from the Sun depending on the location r. The quantity Ω A (w, r) is the rate with which a DM particle with velocity w will be gravitationally captured by scattering on the nucleus A in a spherical shell at radius r with thickness dr: where ρ A (r) is the mass density of the element A in the Sun (ρ A /m A being the number density). For the numerical calculations we use the standard solar model from ref. [42].
Here, E R is the recoil energy of the nucleus after the scattering. For a given DM velocity w there is a minimal and maximal nuclear recoil energy, E min (w) and E max (w), such that the DM particle gets trapped, i.e., its velocity after the scattering is less than the local escape velocity. Writing them in terms of the velocity v at infinity one has where the former constraint is the requirement for the DM particle to be captured and the latter is based on the maximal energy transfer allowed by kinematics. In figure 1 we show E min and E max (r) versus velocity v for scattering on hydrogen for two different DM masses, m χ = 10 and 100 GeV, and for the two extreme values of the escape velocity in the Sun: 618 km s −1 at the surface and 1381 km s −1 in the center of the Sun.
One can see that E min and E max (r) cross, which defines the maximum velocity the DM particles can have in order to be trapped in a single interaction, v A cross (r). From eq.
Therefore, v cross (r) decreases with DM mass m χ , and the capture rates will decrease and eventually vanish for large DM masses. For SD interactions, where only hydrogen is relevant, we can use m p m χ and v p cross (r) ≈ 2 Typically, for heavier nuclei v A cross (r), which is relevant for SI interactions, is larger than v p cross (r), relevant for SD interactions. The cross section dσ A (w)/dE R in eq. (3.2) is the same as the one entering in the event rate for DD experiments. Again, restricting to contact interactions 2 , we use eq. (2.5) for SI interactions and an analogous expression for SD. For the nuclear form factors for the elements in the Sun we use the approximation [40]. Hence, we obtain for the capture rate in SI interactions. The coefficient C is defined in eq. (2.7) and contains the DM-nucleus scattering cross section. For SD interactions only hydrogen is relevant, i.e., the sum contains only one term with A = 1, and the form factor is trivial, F 2 H (E R ) = 1. If equilibrium between DM capture and annihilation in the Sun is reached, the final annihilation rate is independent of the annihilation cross section and is given by: Notice that we are neglecting evaporation, which is justified for m χ 3 GeV [39,40,43]. The neutrino flux at the Earth from the annihilation channel f with branching where d is the Sun-Earth distance and dN f ν /dE ν is the neutrino spectrum per annihilation of flavour f that reaches a distance of 1 AU, which needs to take into account flavour transitions (neutrino oscillations, MSW effect), absorption and regeneration, see, e.g., refs. [44,45].

Relating DM direct detection to the DM capture rate in the Sun
We now want to relate a DD signal to the DM capture rate in the Sun in a haloindependent way. We assume that the DM velocity distributions relevant in the two cases are the same. This implies two important consequences: 1. We neglect the small velocity of the Earth with respect to the Sun and adopt the approximation forf (defined in eq. (2.4)) Hence we use that v e ≈ 29 km/s v m for typical values of v m . This also implies that we are ignoring the (small) annual modulation signal in the direct detection rate (we will comment on this in appendix A).

DM direct detection samples the DM distribution today, while for the DM capture
in the Sun the velocity distribution on time scales relevant for the equilibration of capture and annihilations is relevant. We will assume that the DM velocity distribution is constant on those time scales and that the samef (v) applies for the neutrino capture and direct detection. Similarly, we also assume that the energy density is constant on times scales relevant for equilibration and equal to the current one.

The overlap in v m
From inspection of the relations given in sec. 2 we see that direct detection is sensitive to high DM velocities. Let us denote the threshold energy of a given experiment by E thr . If we ignore the finite energy resolution of the experiment, this defines a threshold velocity via eq. (2.2): v thr ≡ v m (E thr ). 3 Hence, this experiment is sensitive to DM velocities v > v thr . In a realistic halo there will be a maximal velocity of DM particles, set by the escape velocity from the galactic gravitational potential, v esc . Since the precise value of this escape velocity is uncertain (see, e.g., ref. [46] and references therein) we want to keep the discussion as independent of this argument as possible.
In contrast to direct detection, eq. (3.6) shows that only DM velocities v < v A cross (r) contribute to the capture rate in the Sun. Hence, direct detection is sensitive to DM velocities above a certain value given by v thr , whereas for DM to be captured in the Sun velocities below the value set by v A cross (r) are relevant. In order to relate the two phenomena we need to consider the overlap regions in velocity space. If v A cross < v thr , there is no overlap and direct detection and DM capture decouple. In this case no statement can be made halo-independently and a connection can be established only by referring to some a-priori assumptions about the DM velocity distribution.
In figure 2 we show with blue curves the minimum velocity probed in direct detection experiments versus the DM mass, using Ge (left) and Xe (right) for different examples of threshold energies. Those curves can be compared with the maximal velocity relevant for the DM capture in the Sun, v cross , which is shown with red curves for scattering on hydrogen for the two extreme values of the solar escape velocity u esc corresponding to the values in the centre and at the surface of the Sun. From the figure we observe that for a large range of DM masses there is an overlap of the DM velocities probed by direct detection and solar capture, i.e., v thr < v cross , at least for scattering events which occur near the centre of the Sun. A similar figure can be found in ref. [23], where it is also shown that v cross is even larger for heavier nuclei relevant for SI interactions and therefore the overlap is also enhanced with respect to SD interactions.
In the following we are going to make use of the overlap region in velocity space in order to compare results from direct detection experiments to neutrino searches from the Sun without specifying any halo properties apart from the two assumptions stated at the beginning of this section. In particular, in section 4.2 below we are going to assume a positive signal in direct detection and derive a halo-independent lower bound on the capture rate in the Sun. We will use the fact that from a precise measurement of the nuclear recoil spectrum the angular averaged distributionf (v) times the constant C can be extracted from the data via eq. (2.9), and we can make use of the signal precisely in the overlap region in velocity space.
Before we proceed with this, let us briefly comment on the case of a positive signal from neutrinos. In this case no direct information can be obtained aboutf (v), and the signal can come from any DM velocity below v cross . For a positive neutrino signal no general lower bound for the direct detection rate can be established halo-independently, since the capture may happen entirely from velocities below the threshold of the DD experiment. If no signal is seen in direct detection at the relevant level, such a situation could point to a halo dominated by low velocity DM particles (in the solar frame), for instance a co-rotating dark disk [20]. Alternatively this may indicate more exotic particle physics such as self-interacting DM, which enhances the capture rate by DM-DM scattering, independently of the DM-nucleus scattering cross section [47].

A lower bound on the capture rate from a positive direct detection signal
We can derive a lower bound on the capture rate, eq. (3.6), by using that for any positive function where we defined with E min (v) and E max (v) given by eq. (3.3). Notice that E max (v), F A (v, r) and v A cross , eq. (3.4), depend on r via u esc (r).
Let us assume now that a significant direct detection signal has been observed. We can then use eq. (2.9) to obtainf (v) from the data and insert it in eq. (4.1): The direct detection signal has to be precise enough forη(v) to be extracted from the observed energy spectrum via eq. (2.9) with sufficient precision. Either the derivative or the functionη(v), including its value at the experimental threshold, have to be determined, including unfolding of experimental resolutions and backgrounds. This will require a significant number of events such that an accurate spectral analysis can be performed. If those conditions are met, eq. (4.3) provides a lower bound on the capture rate in the Sun without specifying the DM velocity distribution, the galactic escape velocity, the scattering cross section or the local DM density. This lower bound is the central result of this paper and we will illustrate it numerically in the following section for possible future signals in DD experiments.

Mock data for direct detection
The halo-independent bound on the DM capture rate in the Sun derived in the previous section will be useful once a clear and highly significant signal in a direct detection experiment has been observed. 4 In the following we will assume that the DM-nucleon scattering cross section is just below the current limits [5,8], which will allow upcoming 4 The CDMS collaboration reports 3 candidate events from their data with a silicon target, with a 0.19% probability for the known-background-only hypothesis when tested against the alternative DM+background hypothesis [7]. In order to apply our bounds a detailed spectral measurement is required and not enough information can be extracted from the 3 observed events in CDMS-Si. However, motivated by this potential signal, we take DM parameter values similar to those preferred by CDMS-Si events for our example of mock data in germanium. The annual modulation signal reported by the DAMA/LIBRA collaboration [1] will be discuss in appendix A. We emphasize that both signals are either excluded or in strong tension with several other experiments [2][3][4][5][6]8] (see for instance refs. [33,37] for halo-independent analyses). experiments to obtain a significant signal. As representative examples we will consider a future xenon based experiment [48][49][50] as well as a germanium detector [51,52]. For calculating mock data we adopt the conventional Maxwellian velocity distribution (SHM) withv = 220 km/s, truncated at the escape velocity of v esc = 544 km/s, and we assume a local DM density ρ χ = 0.3 GeV/cm 3 . The velocity of the Sun in galactic coordinates is (10, 233, 7) km/s.
For the xenon experiment we adopt a threshold of 5 keV (we also show the effect of a reduced threshold of 3 keV below) and we take natural abundances of the isotopes with spin 129 Xe (26.4 %) and 131 Xe (21.2 %). We use the values of 10 −45 cm 2 for the SI cross section and 2 · 10 −40 cm 2 for the SD cross section, with equal couplings to protons and neutrons in both cases. Assuming m χ = 100 GeV, for an exposure of 1 ton yr at 100% efficiency about 154 (267) events would be observed in the energy range 5 − 45 keV for SI (SD) case.
We also generate mock data for a future germanium experiment, with a threshold of 1 keV, focusing on low DM masses. Assuming a DM mass m χ = 6 GeV and cross sections of σ SI = 5 · 10 −42 cm 2 and σ SD = 2 · 10 −40 cm 2 (with equal couplings to protons and neutrons) we would obtain about 1.5 × 10 4 (2-3) events for SI (SD) interactions in the energy range 1-10 keV for an exposure of 100 kg yr with energy resolution of 30%. While the SI case will allow for a high statistics reconstruction of the event spectrum, this will not be possible for the SD case with the assumed parameters. This follows from the missing A 2 enhancement with respect to SI interactions, as well as the small abundance of only 7% of the relevant Ge isotope with spin. Nevertheless we are going to include SD interactions in the following discussion, keeping always in mind that this will require larger exposures or a different target nucleus with better sensitivity to SD interactions. In addition, the bounds are stronger for larger DM masses, but we want to provide an illustrative example for the small mass region.
In the following analysis we neglect the energy resolution, effects of binning the data, possible contamination with background, and experimental errors. Hence, we assume thatη(v m ) can be extracted from the observed nuclear recoil spectrum (for a specified m χ ). This idealized analysis suffices to illustrate the power of our bounds. Once applied to real data an appropriate statistical analysis will have to be performed.
In figure 3 we show the lower bound on the capture rate for the mock data of future DD xenon (left) and germanium (right) experiments as described above, both for SD and SI interactions and compare them to the true capture rate assuming the SHM. In the case of Ge we focus on the low DM mass region, i.e., m χ 30 GeV, keeping in mind that the bounds are stronger (i.e., closer to the true capture rate) for m χ 20 GeV. The bounds as well as the capture rate are shown for the "true" DM mass, which has been used to calculate the mock data. We see that our bounds are strong in a large portion of parameter space for the xenon experiment: 20 m χ 1000 GeV for SD, and for m χ 50 GeV for SI.

Limits from neutrino telescopes
In order to compare the lower limit on the DM capture rate in the Sun from a DM direct detection signal with the upper bounds on the neutrino rate from neutrino telescopes Figure 3. Lower bounds on the DM capture rate in the Sun for xenon (left) and germanium (right) DD mock data as described in the text, both for SI (red curves) and SD (blue curves) interactions. For illustrative purposes, in the Xe SD case we show the effect of changing the default threshold energy of E thr = 5 keV (solid) to 3 keV (dashed). The dotted curves correspond to the actual capture rate. To calculate the DD mock data as well as the capture rate we assume the standard halo model.
we proceed as follows. We assume equilibrium between capture and annihilations and the neutrino-induced muon rate in neutrino telescopes will then depend on the specific annihilation channel, which depends on the particle physics model for DM. In particular cases, the high-energy neutrino rate may be strongly suppressed by the available annihilation channels, for instance for annihilations into e ± , µ ± or u, d, s quarks (see, however, ref. [53] for higher order effects). Hence, translating a given upper bound on the neutrino-induced muon rate into a bound on the capture rate depends on the DM annihilation channel, see section 3.
Below we are going to compare the lower bounds on C Sun to the limits from Ice-Cube (IC) and Super-Kamiokande (SK). For IC we use the upper bounds of ref. [10], where results are given directly as upper limit on the capture rate for various annihilation channels as a function of the DM mass. We will show results for two cases, namely annihilations into bb + τ τ and into W W + τ τ . For SK there are no limits on the capture rate available. Therefore, we are using the limits on the capture rate calculated in ref. [54] (Tab. II) based on SK data from ref. [9]. We show results for the three annihilation channels bb, τ τ and-as an extreme case-direct annihilation into neutrinos, ν µνµ . The quoted bounds at the 90% CL from IC and SK apply assuming that annihilations proceed with 100% branching ratio into the indicated channels.

Comparison of direct detection and neutrino data
In order to apply the lower bound on the capture rate derived in section 4.2 one has to specify the DM mass. In general, the dark matter mass cannot be extracted from a DD signal without referring to a specific DM halo model. Therefore, we do the analysis without assuming that the DM mass is known. We calculate mock data for a fixed "true" DM mass, but then we apply the lower bound on C Sun as a function of m χ (different from the "true" value). This procedure resembles the situation we would face in case of applying the method to real data. If the mass was known from some . Left: Lower bounds on the capture rate from a future xenon DD experiment for SI (red) and SD (blue) interactions compared to the 90% CL upper limits for the combined annihilation channels W W, τ τ and bb, τ τ from IceCube (IC, green curves) [10] and the channels ν µ ν µ , τ τ , bb from Super-Kamiokande (SK, black curves) [9,54]. Right: Upper bounds on the branching ratios versus dark matter mass for SD interactions. To calculate the DD mock data we assume a "true value" for the DM mass of m χ = 30, 100, 300 GeV in the top, middle, bottom panels, respectively, and values of 10 −45 cm 2 for the SI cross section and 2 · 10 −40 cm 2 for the SD cross section, with equal couplings to protons and neutrons in both cases. Assumptions about the mock data for the DD experiment are given in section 5.1.
other data (e.g., an observation at LHC or a γ line signal from indirect DM searches) one would of course perform the analysis only for that DM mass.
In figure 4 we show the lower bounds on the capture, assuming a true (but unknown) dark matter mass of m χ = 30 GeV (upper panels), 100 GeV (middle panels), Figure 5. Left: Lower bounds on the capture rate from a future germanium DD experiment for SI (red) and SD (blue) interactions compared to the 90% CL upper limits for the combined annihilation channels W W, τ τ from IceCube (IC, green curves) [10] and the channels ν µ ν µ , τ τ , bb from Super-Kamiokande (SK, black curves) [9,54]. Right: Upper bounds on the branching ratios versus dark matter mass. To calculate the DD mock data we assume a "true value" for the DM mass of m χ = 6 GeV and cross sections of σ SI = 5 · 10 −42 cm 2 and σ SD = 2 · 10 −40 cm 2 (equal couplings to protons and neutrons). Assumptions about the mock data for the DD experiment are given in section 5.1.
and 300 GeV (lower panels) for a xenon DD experiment and compare them to the limits from IC and SK. While for SI interactions the lower bound from DD would be consistent with the limits from IC and SK, we see that for SD interactions tension arises if DM annihilates into neutrinos or into τ τ /W W . In the right panels we show the ratio of the upper limit on C Sun to the lower bound. This can be interpreted as an upper bound on the branching ratio of the corresponding annihilation channel. This is conservative, since it assumes that there are no neutrinos from any other annihilation channels, which typically will not be the case. We see that direct annihilations into neutrinos would be constrained to branching ratios at the few % level, with some dependence on the DM mass. Annihilations into τ τ, W W could be constrained at the 10% level. However, note that for true m χ 100 GeV the bound is stronger when applied at a "wrong" DM mass, i.e., for values of m χ which are larger than the actual true value. When using the correct value of m χ the limits for the assumptions adopted here would be rather weak.
In figure 5 we show the lower bounds on the capture (left) and the upper bounds on the branching ratios for different channels (right), assuming the low threshold germanium experiment. Since this configuration is most sensitive at low DM masses we here take a "true" DM mass of 6 GeV. In this case SD and SI interactions lead to similar lower bounds which potentially can constrain annihilations into channels leading to neutrinos. Again we note the feature that bounds get stronger when applied for the "wrong" DM mass, in which annihilations into τ τ become also constraining.
We note that our comparison of DD and neutrino data is not fully consistent, in the sense that we are comparing a possible signal in a future DD experiment with current limits from neutrino telescopes. When the potential signals from DD will be available, limits from IC and/or SK may have improved and upgraded and/or new neutrino telescopes may be in place [55][56][57]. In this sense our results are conservative, since the comparison may become more stringent than what is shown here.
Let us comment briefly on the annual modulation of the DD signal. We expect that using information on the modulation in addition to the unmodulated rate does not provide significantly stronger bounds on the capture rate. If a future experiment only measures the modulation amplitude but cannot distinguish the unmodulated DM rate from background (similar to the DAMA experiment [1]) one can use the results of refs. [32,33] to derive a lower bound on C Sum based on the modulation amplitude. Those bounds require some additional (modest) assumptions on the halo and they can be found in appendix A, where we also apply those bounds for the DAMA signal [1], which however is incompatible with other DD limits halo-independently [33].

Discussion and conclusions
We have established a halo-independent framework to relate a signal in a DM direct detection experiment to the neutrino rate in neutrino telescopes from DM annihilations in the Sun. Assuming that the DM velocity distribution and the DM density are constant on time scales relevant for equilibration in the Sun, we have derived a lower bound for the DM capture rate in the Sun in terms of a positive signal in a direct detection experiment, see eq. (4.3). If DM capture and annihilation in the Sun are in equilibrium we obtain (conservative) upper bounds on branching fractions for annihilations in channels involving neutrinos from the comparison of the lower bound on the capture rate from a direct detection signal with the upper limits from neutrino telescopes.
The lower bounds are based on the part in DM velocity space which contributes to both the capture in the Sun as well as the scattering in direct detection experiments. We find that for typical threshold energies of direct detection experiments a significant overlap region exists to apply our bounds. Hence, the lower bounds are independent of the velocity distribution, the escape velocity, the cross section or the local DM density, although we implicitly assume that these values are such that a direct detection signal can be measured. With some additional modest assumptions on the halo properties the bound can also be used for an annual modulation signal, as explained in appendix A.
To illustrate the power of the bounds, we have applied them to mock data from future xenon and germanium experiments, assuming that the true DM cross section is not too far from the current upper bounds. In such a case the halo-independent comparison to present limits from Super-Kamiokande and IceCube leads to non-trivial bounds on the branching fraction for direct annihilations into neutrinos, if the scattering is spin-dependent. In some cases also annihilations into τ τ start getting constrained. For spin-independent interactions current constraints are weak. In general we note that bounds are stronger for DM masses in the range between 100 to 500 GeV. For lower DM masses the lower bound from direct detection as well as the limits from neutrino telescopes become weaker. Those results can be found in figures 4 and 5. We note that this method is expected to become more powerful when data from future neutrino telescopes such as IceCube-Pingu, Hyper-Kamiokande, or KM3NET become available.
The halo-independent comparison has to be done for a specific model for the DMnucleon interaction (and as a function of the DM mass). In our work we assumed either SI or SD elastic scattering with equal couplings to neutrons and protons. Generalization to other types of couplings is straightforward. This may be relevant especially in the case of spin-dependent interactions, since then only scattering on free protons is relevant for the capture in the Sun, whereas for direct detection the sensitivity is governed by the spin-composition of the target nucleus. One may imagine a situation where the spin of the nucleus is dominated by neutrons, in which case the neutrino and direct detection signals largely decouple. In order to exclude such a case it will be essential that data from target nuclei with spin carried by protons is available.
In our work we always assumed that the differential scattering cross section dσ/dE R is proportional to 1/v 2 , which is true for DM-nucleon contact interactions, but in general this may not be the case for more exotic types of interactions (see for instance ref. [35] for generalizing halo-independent methods for direct detection to such cases). In appendix B we show that it is straightforward to generalize our lower bound on the capture rate to all cases where the velocity and nuclear recoil energy dependence of the differential cross section factorizes as dσ/dE R = g(v)h(E R ). Considering more complicated models is beyond the scope of this paper and is left for future work. Another particle physics variation for which our results do not apply directly is inelastic scattering of the type χ + N → χ * + N , where the mass difference between χ and χ * is of the order of the DM kinetic energy. This will change the kinematics of the scattering and generically for m χ * > m χ the capture rate is increased [58][59][60], see ref. [36] for halo-independent considerations in the context of DD. We leave also the generalization of the lower bound on the capture rate to this case for future work.
Note however, that the lower bounds on the capture rate in eq. (4.3) require a lower bound either on (−dη/dv m ) or on η(v m ), which cannot be obtained in terms of A η (v m ) alone from eq. (A.1). In principle eq. (A.1) can be re-written as a lower bound on (−dη/dv m ) involving both, A η (v m ) and η(v m ). Hence, this would require an experiment able to determine the modulation amplitude as well as the unmodulated rate. Moreover we do not expect a significant stronger lower bound on C Sun from such a procedure for modulation amplitudes which fulfill eq. (A.1) (as they must to be consistent with a DM signal).
Adopting some modest additional assumptions on the halo another bound on the modulation amplitude can be derived. If there is only one preferred direction in the DM velocity distribution one finds (see ref. [32] for details): where α is the angle between the preferred DM direction and the direction perpendicular to the Earth's orbit. Assuming an observed signal for A η (v m ), this provides us with a lower bound on (−dη/dv m ), which can be directly plugged in eq. (4.3) and we get: where in the last line we usedÃ η (v) ≡ C A η (v). In the following we will adopt the most conservative option sin α halo 1, but notice that in cases where the preferred DM direction is set by the direction of the Sun relative to the DM halo (e.g., for a halo with a dark disk) one has sin α halo 0.5 and the bounds scale accordingly.
Notice that observing annual modulation with typical exposures (and allowed cross section values) is extremely hard (see for instance ref. [61]). The DAMA/LIBRA experiment reports an annual modulation of the signal in their NaI scintillator detector, with a period of one year and a maximum around June 2nd with very high statistical significance [1]. This modulation is strongly disfavoured by other experiments haloindependently, both for elastic SI and SD interactions [33] and for inelastic scattering [36]. Despite these problems of the DM interpretation of the DAMA modulation signal we will use it in the following to illustrate the lower bounds on the capture rate eq. (A.3) based on a modulation signal.

A.1 The DAMA modulation signal
In figure 6 we show the overlap in v m space between DAMA and the DM capture in the Sun assuming scattering either on Na or on I. Na dominates for DM masses m χ 20 GeV, whereas iodine is relevant for larger DM masses. For sodium we observe a large overlap region. Note that the DAMA threshold of 2 keVee corresponds to a recoil threshold of 6.7 keV (Na) and 22 keV (I) for usual quenching factors. This implies a relatively small overlap region in the case of iodine, especially for the SD case shown in figure 6.  . Lower bound on the capture rate from DAMA data assuming scattering on sodium (left) or iodine (right) for SI (red) and SD (blue) interactions, using the bound from eq. (A.3). Also shown are the 90% CL upper bounds from IceCube [10] for annihilations into W W + τ τ and W W + bb (green curves), and from Super-Kamiokande into bb, τ τ , and ν µ ν µ [9,54] (black curves).
To obtain a lower bound on the capture rate from DAMA data we consider a binned version of eq. (A.3) and use the observed values ofÃ η (v m ) corresponding to the energy bins reported by DAMA, see refs. [32,33,36] for details on this procedure. We assume either elastic SI or SD scattering on Na or on I with equal couplings to protons and neutrons. The resulting lower bounds on the capture rate are shown in figure 7 both for SI and SD, together with the 90% CL upper limits on the capture from neutrino telescopes.
From the left panel we see that for scattering on sodium there is tension between the lower and upper bounds implying that for SI DM annihilation into neutrinos, τ τ , bb are strongly constrained for m χ 5, 10, 30 GeV, respectively, while SD is excluded for annihilations into neutrinos and τ τ , and also into bb for m χ 8 GeV. For scattering on iodine shown in the right panel of figure 7 we note the strong dependence on the DM mass for SD scattering. This can be understood from figure 6, which shows that only for a small range of DM masses there is overlap in v m space. For SI interactions v cross is larger (not shown in the plot), leading to larger overlap in v m space, and we observe strong bounds on the annihilations shown in figure 7 for m χ 50 GeV (and even for m χ 10 GeV for the neutrino channel).

B Beyond contact interactions
In the main text of the paper we have always assumed contact interactions between DM and the nucleus, which leads to a 1/v 2 dependence of the differential scattering cross section. In more exotic models also other dependences on v and/or E R are possible, see for instance refs. [19,35,54] in the context of DD and the neutrino signal. Our bounds can be generalized in a straightforward way if the dependence of v and E R factorizes. Let us assume that the differential cross section on the nucleus with mass number A can be written in the form In the conventional case considered in the main text we have g A (v) ∝ 1/v 2 and h A (E R ) = F 2 A (E R ), see eq. (2.5). From the measured recoil spectrum in a DD experiment, R(E R ), we can then extract the DM velocity distribution bỹ .

(B.2)
Using this in the expression for the capture rate in the Sun (eqs. (3.1) and (3.2)) we find that the bound eq. (4.3) becomes where A DD indicates the mass number of the nucleus in the DD experiment, whereas the sum over A runs over the elements in the Sun, w 2 = v 2 + u 2 esc (r), and Eq. (B.3) corresponds to the lower bound on the capture rate if the scattering cross section can be factorized according to eq. (B.1). For certain models such a factorization may not be possible. Generalizing our bound to those cases is beyond the scope of this paper and we leave it for future work.