Prepared for submission to JCAP Impact of the Dark Matter Velocity Distribution on Capture Rates in the Sun K. Choia C. Rottb Y. Itowa,c aSolar-Terrestrial Environment Laboratory, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan bDepartment of Physics, Sungkyunkwan University, Suwon 440-746, Korea cKobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, 464-8601, Japan E-mail: [email protected], [email protected], [email protected] Abstract. Dark matter could be captured in the Sun and self-annihilate, giving rise to an observable neutrino flux. Indirect searches for dark matter looking for this signal with neutrino telescopes have resulted in tight constraints on the interaction cross-section of dark matter with ordinary matter. We investigate how robust limits are against astro-physical uncertainties. We study the effect of the velocity distribution of dark matter in our Galaxy on capture rates in the Sun. We investigate four sources of uncertainties: orbital speed of the Sun, escape velocity of dark matter from the halo, dark matter velocity distribution functions and existence of a dark disc. We find that even extreme cases currently discussed do not arXiv:1312.0273v3 [astro-ph.HE] 22 Apr 2014 decrease the sensitivity of indirect detection significantly because the capture is achieved over a broad range of the velocity distribution by integration over the velocity distribution. The effect of the uncertainty in the high-velocity tail of dark matter halo is very marginal as the capture process is rather inefficient at this region. The difference in capture rate in the Sun for various scenarios is compared to the expected change in event rates for direct detection. The possibility of co-rotating structure with the Sun can largely boost the signal and hence makes the interpretation of indirect detection conservative compared to direct detection. Keywords: dark matter theory, neutrino astronomy, dark matter experiments 1 Introduction Weakly interacting massive particles (WIMPs) are prominent candidates to explain the dark matter observed in the Universe. WIMPs (denoted χ) naturally arise in supersymmetry [1] and models with large extra dimensions [2]. They are predicted to have masses ranging from a few GeV to a few TeV [3]. A very promising way to identify the WIMP nature of dark matter is to search for an excess neutrino flux from the Sun generated by self- annihilations of WIMPs [4–7] inside the Sun. The high-energy neutrino flux originating from decay of annihilation products as well as a low-energy neutrino signal from hadronic particle showers in the Sun [8, 9] are expected to arrive at neutrino telescopes. Tight limits on the WIMP-induced neutrino flux from the Sun have been placed with neutrino telescopes such as Super-Kamiokande [10], IceCube [11], Baksan [12] and ANTARES [13]. Searches for these neutrinos have to consider various sources of uncertainties on the expected event rates at detectors which can be grouped into three uncorrelated categories: (1) uncertainties on annihilation in the Sun, (2) uncertainties on neutrino propagation from the Sun to the detector, (3) uncertainties related to detector response. The last two categories are detector- and location-specific and hence need to be studied by the corresponding experiments. The DarkSUSY [14] / WimpSim [15] packages can, for example, be utilised to obtain precise neutrino flux at the detector. Uncertainties on the annihilation rate in the Sun are common to searches at all neutrino detectors. For the commonly assumed case of equilibrium between the capture rate and the annihilation rate, the uncertainty on the annihilation is directly given by the uncertainty on the WIMP capture in the Sun. This capture mechanism is theoretically well understood so that one can study the impact of the various effects that contribute to its uncertainty, which is the subject of this work. Uncertainties on the capture of WIMPs in the Sun orig- inate from a variety of effects. These are related to the WIMP mass, the WIMP-nucleus elastic-scattering cross-section, the local dark matter density, the velocity distribution of dark matter and the composition of the Sun, etc.. Uncertainties coming from the form factor in the scattering cross-section is negligible for spin-dependent (SD) coupling since capture is dominated by hydrogen. For spin-independent (SI) coupling WIMPs, it is predicted to be a few to 20 % depending on the model used [16–18]. The uncertainty on the composition of the Sun is predicted to be negligible for SD coupling [17] and at the 20% level for SI coupling WIMPs [18]. Larger uncertainty comes from the understanding of the local dark matter phase space structure. The uncertainty in local dark matter density is expected to be a factor of two [19– 23], as are the uncertainties in capture rate and signal rate of direct searches, as scattering rate simply increases with the local density. However, the local velocity distribution of dark matter impacts direct and indirect searches very differently. As opposed to direct detection looking for energetic scattering of WIMPs, less energetic WIMPs are more easily captured by the Sun. Therefore direct detection is sensitive to WIMPs with high-velocity, while indirect detection is sensitive to the low part of the velocity distribution. In the literature there have been extensive discussions about the impact of uncertainties related to WIMP phase space on direct detection [24–38] and indirect detection for more specific scenarios [39–46]. In this work, we will give a comprehensive and general treatment of uncertainties from the dark matter velocity distribution on indirect solar WIMP searches. Previous works have only discussed specific effects, while this work quantifies effects as function of the WIMP mass, making it easy to incorporate in future experimental and theoretical works. This paper is – 2 – structured as follows: In section 2, we first review the WIMP capture process in the Sun to manifest how the dark matter velocity distribution factors into it. In section 3, we introduce a variety of velocity distributions, for which we then discuss the impact on the capture rate in section 4. In section 5, we relate our finding to velocity distribution dependencies for direct detection experiments. We conclude and summarise our results in the final section 6. 2 WIMP capture and annihilation in the Sun In this section we review the WIMP capture mechanism that leads to the accumulation of WIMPs in the Sun. The capture process has been extensively discussed in the literature [6, 7], which will be reviewed here to see in particular how the dark matter velocity distribution is involved. WIMPs which are abundant in the dark matter halo of the Milky Way can scatter off a nucleus in the Sun as it revolves around the Galactic centre. The WIMP may lose enough energy by elastically scattering off a nucleus to fall below the escape velocity of the Sun at the point of scatter to be gravitationally captured. The differential WIMP capture rate on nucleus i per unit shell volume dV at distance r from the centre takes the following form : ∞ dCi ρH fη(u) = du Ωi(Q), (2.1) dV Mχ Z0 u 3 where ρH is the local dark matter halo density set to be 0.3 GeV/cm [47, 48] and Mχ is the mass of the WIMP. fη(u) is the velocity distribution function (VDF) seen in the reference frame of the Sun. For isotropic VDF, fη(u) can be simply calculated from the original VDF in the Galactic frame, fo(v), as 1 2 2 fη(u)= fo( v + v⊙ + 2vv⊙cosθ) dcosθ, (2.2) Z−1 q where ~u = ~v + ~v⊙ and θ is the angle between ~v and ~v⊙. The bound velocity of a WIMP in the gravitational field of the Sun, w, and the escape velocity at scattered position, uesc(r) 2 2 2 are related via w = u + uesc(r) . A WIMP can be captured when w drops below uesc(r) after losing kinetic energy through scattering off a nuclei. The capture probability Ωi(Q) can be written as [6] Qmax Mi 2 Ωi(Q)= σini 2 F (Q)dQ, (2.3) 2µi ZQmin where σi is the WIMP-nucleus i elastic-scattering cross-section at zero-momentum transfer, ni is the number density of nucleus i in the Sun, Mi is the mass of nucleus i, µi is the reduced mass of nucleus i and the WIMP, Q is the recoil energy where 1 2 Qmin = M u (2.4) 2 χ is the minimum recoil energy required to be captured, 1 2 Qmax = β+M w , (2.5) 2 χ is the kinematically determined maximal recoil energy with 4MiMχ β± = (2.6) (M M )2 i ± χ – 3 – and F 2(Q) is a nuclear form factor taken to have exponential form from [6]. For SD WIMP capture we neglect the contributions from the heavier elements than hydrogen. For SI cou- pling case, we assume isospin conserving scattering and write the SI zero-momentum transfer SI cross-section to a nucleus σi with the mass number I as 2 SI SI 2 µi σi = σp I 2 (2.7) µp SI where σp is the SI WIMP-nucleon zero-momentum transfer scattering cross-section and µp is the WIMP-nucleon reduced mass. To calculate the total capture rate, the above expression has to be summed over all relevant nuclear species inside the Sun and integrated over the distance r from the centre as R⊙ 2 dCi CC = 4πr dr , (2.8) Z dV 0 Xi where R⊙ is the radius of the Sun. The maximum velocity of a WIMP to be captured by nucleus i at distance r from the centre of the Sun is given where Qmin (eq.