Multipole analysis of IceCube data to search for dark matter accumulated in the Galactic halo

Dark matter which is bound in the Galactic halo might self-annihilate and produce a flux of stable final state particles, e.g. high energy neutrinos. These neutrinos can be detected with IceCube, a cubic-kilometer sized Cherenkov detector. Given IceCube's large field of view, a characteristic anisotropy of the additional neutrino flux is expected. In this paper we describe a multipole method to search for such a large-scale anisotropy in IceCube data. This method uses the expansion coefficients of a multipole expansion of neutrino arrival directions and incorporates signal-specific weights for each expansion coefficient. We apply the technique to a high-purity muon neutrino sample from the Northern Hemisphere. The final result is compatible with the null-hypothesis. As no signal was observed, we present limits on the self-annihilation cross-section averaged over the relative velocity distribution $<\sigma v>$ down to $1.9\cdot 10^{-23}\,\mathrm{cm}^3\mathrm{s}^{-1}$ for a dark matter particle mass of $700\,\mathrm{GeV}$ to $1000\,\mathrm{GeV}$ and direct annihilation into $\nu\bar{\nu}$. The resulting exclusion limits come close to exclusion limits from $\gamma$-ray experiments, that focus on the outer Galactic halo, for high dark matter masses of a few TeV and hard annihilation channels.

Abstract Dark matter which is bound in the Galactic halo might self-annihilate and produce a flux of stable final state particles, e.g. high energy neutrinos. These neutrinos can be detected with IceCube, a cubic-kilometer sized Cherenkov detector. Given IceCube's large field of view, a characteristic anisotropy of the additional neutrino flux is expected. In this paper we describe a multipole method to search for such a large-scale anisotropy in IceCube data. This method uses the expansion coefficients of a multipole expansion of neutrino arrival directions and incorporates signal-specific weights for each expansion coefficient. We apply the technique to a high-purity muon neutrino sample from the Northern Hemisphere. The final result is compatible with the nullhypothesis. As no signal was observed, we present limits on the self-annihilation cross-section averaged over the relative velocity distribution σ A v down to 1.9 · 10 −23 cm 3 s −1 for a dark matter particle mass of 700 GeV to 1000 GeV and direct annihilation into νν. The resulting exclusion limits are competitive when compared to exclusion limits from γray experiments, that focus on the outer Galactic halo, for high dark matter masses of a few TeV and hard annihilation channels.

Introduction
There is compelling evidence for dark matter, e.g. from cosmic microwave background anisotropies, large-scale structure formation, galaxy rotation-curves, and other astrophysical observations [1][2][3]. Despite this evidence, DM can not be (fully) explained by standard model particles, and its nature remains unknown [3]. Many theories, e.g. supersymmetry or extra dimensions, provide suitable candidates for dark matter [3]. The generic candidate for dark matter is a weakly interacting massive particle (WIMP) with a mass of a few GeV up to several hundred TeV [4,5]. Assuming that WIMPs interact at the scale of the weak force and were produced in the early universe in thermal equilibrium, the freeze-out of WIMPs leads to an expected dark matter abundance that is compatible with current estimates [2].
The density of WIMPs gravitationally trapped as dark halos in galaxies can be high enough that their pair-wise annihilation rate is not negligible. The final-state products of the annihilations decay to stable standard model particles, i.e., photons, protons, electrons or neutrinos, and, therefore, an observable flux of these particles could provide indirect evidence for dark matter. While charged cosmic rays are deflected by magnetic fields and photons have a large astrophysical foreground, astrophysical neutrinos from dark matter annihilation do not interact with inter-stellar matter and would point back to their origin. In certain models, neutrinos can also be produced directly [6], giving a monochromatic neutrino signal that would be a golden channel for neutrino telescopes.
Observations of an excess in the positron to electron ratio by PAMELA [7], that was confirmed by FERMI [8] and AMS-02 [9], may hint to dark matter in the GeV-TeV region. The nature of the positron signal is extremely difficult to interpret due to the complex propagation of electrons and positrons in the Galactic magnetic fields. The observation can also be explained by nearby astrophysical sources like pulsars [10] or supernova remnants [11]. If the positron excess is interpreted to originate from dark matter annihilation, leptophilic dark matter [12,13] is favored and could produce a prominent flux of neutrinos.
As mentioned above, the annihilation rate is significantly enhanced in regions where DM might have been gravitationally accumulated, since the annihilation rate scales with the square of the density. In particular, massive bodies like the Sun [14], the Earth [15], the Galactic Center [16,17] or dwarf galaxies and galaxy clusters [18], are good candidates to search for a neutrino flux from DM annihilations. Furthermore, and due to the expected shape of the dark halo around the Milky Way, annihilations in the halo would produce a diffuse flux of neutrinos with a characteristic largescale structure [19], depending on the assumed DM density distribution. While searches for a neutrino flux from the annihilation of DM captured in massive bodies are sensitive to the spin-dependent and spin-independent DM-nucleon cross -section, the Galactic and extra-galactic flux depends on the self-annihilation cross section [3].
In this paper we present a multipole method to search for a characteristic anisotropic flux of neutrinos from dark matter annihilation in the Galactic halo. The method is based on a multipole expansion of the sky map of arrival directions and an optimized test statistic using the expansion coefficients. This kind of method is ideal to search for anisotropies and provides flexibility even if the expected halo shape is not known. Furthermore this method provides the opportunity to reduce the influence of systematic uncertainties in the result, which arise from systematic uncertainties on the zenith dependent acceptance and zenith dependent atmospheric neutrino flux. A large-scale anisotropy as seen by [20][21][22][23] in cosmic-rays, is expected in the atmospheric neutrino flux. However this anisotropy is very small so that it is just an effect of few percent compared to our sensitivity on neutrinos from dark matter annihilations. This paper is organized as follows: In Section 2 the IceCube Neutrino Observatory is introduced. Section 3 gives the theoretical expected flux from dark matter annihilation with neutrinos as final state. Section 4 gives a short overview of the data sample used and the simulation of pseudo-experiments. In Section 5 the multipole analysis technique is introduced. The sensitivity of this analysis is given in Section 6. Section 7 addresses systematic uncertainties. In Section 8 and Section 9 the experimental result is presented and discussed, while in Section 10 we present our conclusions.

The IceCube Neutrino Observatory
IceCube is a cubic-kilometer Cherenkov neutrino detector located at the geographic South Pole [24]. When a neutrino interacts with the clear Antarctic ice, secondary leptons and hadrons are produced. These relativistic secondary particles produce Cherenkov light which is detected by Digital Optical Modules (DOMs) that contain a photomultiplier tube. The IceCube array consists of 86 strings, each instrumented with 60 DOMs, which are located at depths from 1.45 km to 2.45 km below the surface. The strings are arranged in a hexagonal pattern with an inter-string spacing of about 125 m and a DOM-to-DOM distance along each string of 17 m. A more compact sub-array, called DeepCore, consisting of eight densely-instrumented strings, has been embedded in the center of IceCube in order to lower the energy threshold from about 100 GeV to about 10 GeV [25]. The detector construction was completed in December 2010, however data were already taken with partial configurations [26]. The footprint of IceCube in its 79-string configuration (IC79) is shown in Figure 1. This is the configuration used in this analysis. Due to its unique position at the geographic South Pole, the zenith angle in local coordinates is directly related to the declination and the right ascension for a given azimuth angle only depends on the time.
3 Neutrino flux from dark matter annihilation in the Galactic Halo N-body simulations [27][28][29] predict the mass distribution ρ DM (r) in galaxies as function of the distance r to the Galactic Center, assuming a spherically symmetric distribution. The resulting dark matter halo profile is parameterized by an extension of the Hernquist model [30] where (α, β , γ, δ ) are dimensionless parameters. r s is a scaling radius and ρ 0 is the normalization density. Both have to be determined for each galaxy.
In this paper the halo profile of Navarro, Frenk and White (NFW) [31,32] with (1, 3, 1, 0) is used as baseline. For the Milky Way r s = 16.1 +17.0 −7.8 kpc and ρ(r s ) = 0.47 +0.05 −0.06 GeV/cm 3 are used [33]. A currently favored model is the Burkert profile, that was obtained by the observation of dark matter dominated dwarf galaxies.  [33]. While for the central part of the galaxy the models differ by orders of magnitude, the outer profiles are rather similar.
The expected differential neutrino flux dφ ν /dE at Earth depends on the annihilation rate Γ A = σ A v ρ(r) 2 /2 along the line of sight l, the muon neutrino yield per annihilation dN ν /dE, and the self-annihilation cross-section of dark matter averaged over the velocity distribution σ A v . The flux is given by [35]: where m χ denotes the mass of the dark matter particle. J(ψ) is the dimensionless line of sight integral, that depends on the angular distance to the Galactic Center, ψ, and is defined by [35]: where ρ 2 DM is evaluated along the line of sight, that is parameterized by R 2 SC − 2lR SC cos ψ + l 2 and ρ SC is the local dark matter density at the distance R SC = 8.5 kpc of the Sun from the Galactic Center [35]. d max is the upper boundary of the integral and is sufficiently larger than the size of the galaxy. The dimensionless line of sight integral for different halo profiles is shown in Figure 2 for small angles ψ can be seen, while for the outer part a similar factor is expected for all models.
This analysis searches for an anisotropy in the neutrino arrival directions on the Northern Hemisphere. Here we expect a characteristic anisotropy, proportional to J(ψ) as shown in Figure 3.
The neutrino multiplicity per annihilation for the flavors e, µ, τ, are obtained with DarkSUSY, which is based on Pythia6 [19,36]. The muon neutrino multiplicity per annihilation at Earth, dN ν /dE, includes the oscillation probability into muon neutrinos in the long baseline limit. The effective oscillation probability was calculated by numerical averaging the oscillation probability over sufficient oscillation length using mixing angles and amplitudes from [37]. Since the nature of the DM particles, as well as the branching ratio for different annihilation channels, are unknown, a 100 % branching ratio to a few benchmark channels is assumed. Similar to previous analyses [18], we use the annihilation to bb as a soft channel, W + W − as a medium and µ + µ − as a hard channel. Furthermore we investigate direct annihilation to νν (assuming 1:1:1 neutrino flavor at source and then use the long-baseline approximation as for all other spectra), resulting in a line-spectrum, that is modeled by a uniform distribution within ±5 % of m χ . The different muon neutrino multiplicity per annihilation spectra  on event quality parameters and a selection by a Boosted Decision Tree (BDT) [38] the contamination of misreconstructed atmospheric muons that mimic up-going neutrinos was reduced to < 3 % [39]. The detailed selection is described in [40] as "Sample B" for IC79. After the rejection of atmospheric muons, the sample consists of 57281 upgoing muon events from the Northern Hemisphere, mostly atmospheric muon neutrinos, which are background for the search of neutrinos from self-annihilating dark matter in the Galactic halo. Unlike signal, the integrated atmospheric neutrino flux is nearly constant with right ascension [19]. The reconstructed arrival directions of all events in the final sample are shown in Figure 5. From full detector simulation it was found that 90 % of the events have a neutrino energy in the range from about 100 GeV to about 10 TeV, with a median of 613 GeV. The median angular resolution is < 1 • for energies above 100 GeV [40]. Further details on the sample properties can be found in [40].

Pseudo-Experiments
The sensitivity of this analysis has been estimated and optimized by pseudo experiments with simulated sky maps of neutrino arrival directions. These sky maps contain background from atmospheric neutrinos and misreconstructed atmospheric muons and signal from dark matter annihilation.
Signal events are generated at a rate proportional to the line-of-sight integral. Furthermore, the arrival direction is smeared according to the angular resolution [19], which was determined with the full detector simulation. Moreover, the acceptance of each event is randomized according to the declination-dependent effective area. It is assumed that the acceptance is constant in RA, due to IceCube's special position at the South Pole and the daily rotation of the Earth Muon neutrino multiplicity per annihilation dN ν /d ln(E) as function of energy is shown for the investigated benchmark channels and m χ = 600 GeV. The oscillation probability into muon neutrinos, in the long baseline limit, is included in dN ν /d ln(E). Beside the neutrino line spectrum, the spectra were calculated with DarkSUSY [36]. The background generation is done by scrambling experimental data. Here, the declination of the experimental events are kept and the RA is uniformly randomized. The rescrambling of experimental data to generate background is justified by a negligible signal contamination in the experimental data. By this technique the background estimation is not affected by systematic uncertainties from Monte-Carlo simulation.

Galactic Center
The number of signal events in a sky map is fixed to N sig . The total number of events in a sky map N ν is fixed to the total number of events in the experimental sample, so that the sky maps are filled up with N ν − N sig background events.

Multipole Expansion of Sky Maps
The sky maps of reconstructed neutrino arrival directions are expanded by spherical harmonics Y m [41]. Spherical harmonics are given by where θ is the declination and φ the RA. , m are integer numbers with 0 ≤ and − ≤ m ≤ . P m are the associated Legendre polynomials. Because spherical harmonics are a complete set of orthonormal functions, one can expand all square-integrable functions f (θ , φ ) on a full sphere Ω into spherical harmonics. The expansion is given by with expansion coefficients a m . Here, is the order of the expansion and corresponds to an angular scale of approximately 180 • / , while m corresponds to the orientation of the spherical harmonic. The expansion coefficients are given by The sky map of reconstructed arrival directions is represented by where (θ i , φ i ) are reconstructed coordinates of event i in equatorial coordinates. N ν is the total number of events in the data sample and δ D is the Dirac-delta-distribution. Since the median angular resolution of the events (< 1 • ) is much smaller than the anisotropy to search for, the usage of Diracdelta-distributions is justified. Coefficients with negative m do not provide additional information, because the sky map is described by a real function, leading to |a m | = |a −m |, and a fixed relation between arg(a m ) and arg(a −m ) [41].
The multipole expansion is linear and the expansion coefficients for signal and background follow the superposition principle. This can be seen from Equation 6, if one uses is the sky map for pure signal, f bgd (θ , φ ) is the sky map for pure background, and s is the relative signal strength.
In practice the expansion is stopped at some large max . Information on structures of an angular scale smaller 180 • / will be lost. Hence, the value of max should be sufficiently large to include all angular scales of interest.

Application of Multipole-Expansion to Pseudo-Experiments
For this analysis the calculation of the expansion coefficients is done with the software package HealPix [42,43]. Figure 6 and Figure 7 show the expansion coefficient corresponding to Y m with = 2 and m = 1 for a signal, as described in Section 3, and a uniform distributed background in RA, as described in Section 4.
For different signal strength s = N sig /N ν , 1000 pseudoexperiments were performed and a 1 2 was calculated. For pure background sky maps (s = 0 %) with no preferred direction in RA (uniform) the expansion coefficient shows no preferred phase and is almost normal distributed around the origin of the complex plane. For pure signal (s = 100 %) there is a clear separation from the origin. Also, a clear preferred phase can also be observed, which corresponds to the orientation of the expected anisotropy in the sky. This phase is the same as the preferred phase for the sky maps with partial signal (0 % < s < 100 %). Furthermore, a linear dependency between the signal strength s and the mean power |a 1 2 | can be seen.
In practice the number of events N ν in the map is limited. Therefore, the value of a m has a statistical error, which depends on the total number of events in the sky map N ν and weakly on the signal fraction s. For the value N ν > 57000 of this analysis the error can be well approximated as Gaussian. For the pure background case most of the power is contained in the coefficients a 0 , related to the pure zenith structure. All coefficients with m = 0 are at the level of 10 −3 − 10 −4 , which is the level of the statistical noise in the map. Because the background was assumed to be isotropic in RA all coefficients with m = 0 are at the noise level, corresponding e.g. to the width of the distribution, as shown in Figure 7 and Figure 6. From equation (4) one can see that spherical harmonics with m = 0 are independent of RA and thus purely depend on declination. The a 0 coefficients have an absolute value larger than the noise level that means they contain power. These coefficients describe the full declination structure, that is mainly influenced by the declinationdependent acceptance and the declination-dependent variation of the atmospheric neutrino flux. Furthermore it was found that there is no preferred phase in any coefficient for background.
For pure signal (Figure 8, right) there is also power in coefficients with m = 0, resulting from the characteristic anisotropy of the signal. It was found that all coefficients that have a power larger than the noise level also have a preferred phase. The characteristic checkered pattern in the coefficients results from the observation of just one hemisphere θ > 0, leading to a suppression of coefficients with even + m, that correspond to a symmetric spherical harmonic with respect to the equator.
From Figure 8 (right panel) it becomes obvious that coefficients with small and m carry most power. This is due to the large-scale anisotropy of the line-of-sight integral (see

Im
Re small signal-fraction pure background pure signal projection Fig. 9 Sketch to illustrate the projection of complex expansion coefficients, on the axis corresponding to the preferred direction. The large gray circles represent the central part of the Gaussian in the complex plane in a large ensemble limit for different signal strength. In contrast to that distributions the star corresponds to one specific value of the expansion coefficient for one measured sky map. The projected value of this specific expansion coefficient is given by the length of the thick red line.

Test Statistic
The test statistic (T S) to separate signal from background combines the phase information and the power of a complex coefficient into one value. A projection of the complex coefficient onto the axis, corresponding to the preferred phase, is introduced [44]. This projection is illustrated in Figure 9 and is given by where arg a m is the argument of a m and arg(a m ,sig ) is the mean expected phase of the a m of pure signal pseudoexperiments.
This projected expansion coefficient has the following advantages. First A m is proportional to the power of the expansion coefficient. Second, the most sensitive direction is the axis of the preferred phase, and the value of the projection gets smaller, the more the phase differs. This results in negative values for A m , if the phase differs more than π/2, indicating that the anisotropy is in the opposite direction of the expectation.
Using these projected expansion coefficients the T S is defined as where sig(x) gives the sign of x and A m ,bgd and σ (A m ,bgd ) are the mean and standard deviation of an ensemble of A m estimated from pseudo-experiments of pure background [44]. w m are individual weights for each coefficient. The definition of the test statistic is motivated by a weighted χ 2 -function. The weights are chosen with respect to the separation power of the different coefficients and are defined below. Because the sign of the deviation is lost in the squared term, the sign is included as an extra factor. Coefficients with no power, especially in the background case, have randomly positive or negative sign. In average they add up to zero, however for signal always positive values contribute to the sum.
The weights are given by where A m ,sig is the expected projected expansion coefficient for pure signal, that can be calculated by averaging over the A m of an ensemble of pseudo-experiments for pure signal. Because A m is proportional to a m and, thus, to the signal strength, a smaller signal expectation would just lead to a different normalization of the weight, which is absorbed in the factor 1/ ∑ w m . Therefore the relative strength of coefficients in this test statistic does not depend on the signal strength s. The weights represent the power to separate signal from background for each coefficient. Insensitive expansion coefficients get assigned a small weight and do not contribute.

Test statistic application to the search for dark matter
To determine arg(a m ,sig ) , w m , A m ,bgd and σ (A m ,bgd ), 1000 pseudo-experiments were used in each case. The weights for an NFW profile for all coefficients with , m ≤ 100 are shown in Figure 10.
The weights range over orders of magnitude and are proportional to the values shown in Figure 8 (right panel) reflecting the separation power of the coefficients.
For IceCube, the coefficients with m = 0 contain the declination dependence and, due to the detector location at the geographic South Pole, this translates directly into the zenith dependence of the detector acceptance. In order to avoid introducing a zenith-dependent systematic uncertainty, the coefficients with m = 0 are omitted in this analysis and are not included in equation (9). Since spherical harmonics are orthonormal functions, no additional systematic is introduced by this choice. Possible systematic uncertainties in azimuth average out due to the daily rotation of Earth, and thus the detector. Because the anisotropy introduced by the flux from dark matter annihilation in the halo is a large scale anisotropy a maximal expansion order of max = 100 was chosen. In general the coefficients become less sensitive with lager . Due to this generic suppression of insensitive coefficients in the test statistic max does not need to be optimized.
Since the differences in weights for different halo profiles are found to be small, weights from NFW profiles are used for all model tests to avoid trial factors. Differences with respect to the halo profiles will be discussed below. Figure 11 shows the resulting test statistic for pseudoexperiments of pure background (N sig = 0, s = 0 %) and pseudo-experiments with signal contribution of N sig = 1000, 5000 (signal strength s = 1.7 %, 8.7 %) assuming a NFW profile.

Generalization of the method
In the previous sections the assumed signal was the characteristic anisotropy of the flux from dark matter annihilation. However, the method described here can be generalized to any other anisotropy of preferred direction.
If there is no preferred direction in the signal expectation, i.e. a characteristic event correlation structure which is distributed isotropically on the sky, the phase is also random in the signal case. This is the case e.g. in a search for many point-like sources that are too weak to be detected individually, but which lead to a clustering of events on specific angular scales. Even in these cases it is possible to define a test statistic analogously. Here one can use the averaged power on a characteristic scale C eff , that is given by  Note that also here the power coefficients are defined without the a 0 coefficients, resulting in coefficients that are not affected by systematic uncertainties in the declination acceptance. In the test statistic (equation (9)) one has to replace all A m by C eff and w m by w and remove the sum over m. Furthermore the sig(A m )-term now has to be written as sig(C eff − C eff ,bgd ). The weight w can be defined similar to equation (10) by replacing A m by C eff . An example where such a test statistic has been used is [45,46].

Sensitivity
The median sensitivity at a 90 % confidence level (C.L.), N 90 , is given by that number of signal events, where 50 % of the signal T S distribution is larger than the 90 % upper quantile of the background distribution. To estimate this median sensitivity for different signal contributions 25000 pseudoexperiments were generated for different numbers of signal events N sig . In Figure 12 the median of each T S distribution is shown versus the signal strength N sig for the different halo profiles. Further, the 90 %-quantile of the background distribution is shown. The resulting N 90 for the different halo profiles are shown in Table 1. It can be seen that differences in the value of N 90 are smaller than 10 %. Note that the N 90 value does not depend on the overall normalization but only on the different shape of the profiles.
The sensitivity on the number of signal events in the data set, and thus the flux, can be interpreted in terms of the self- annihilation cross-section of the dark matter. Using equation (2) the self-annihilation cross-section is given by Here, A eff is the effective area, which is shown for the chosen data set, averaged over the Northern Hemisphere, in Figure 13. The resulting sensitivity on the self-annihilation cross-section depends on the assumed annihilation channel and WIMP mass. The sensitivities will be presented together with the results in Section 8. In N-body simulations of structure formation using DM, self-similar substructures are found. These structures lead to an enhanced annihilation probability, because the gain of flux from denser regions is larger than the loss in dilute regions. The increase of the annihilation rate can be described by a boost factor B(r), which modifies the line of sight integral. An example of such a boost factor is given in [47] and has been discussed in [19]. For this analysis the modification of the line-of-sight integral as described in [19] results in a median sensitivity, N 90 , which is about 50 % worse, because the shape of the line-of-sight integral and thus the anisotropy becomes flatter. However, due to the larger total expected flux the sensitivity on the self-annihilation crosssection is 20 % more stringent. To be conservative, the results presented here do not take substructures into account.
As a cross-check, the sensitivity on the number of signal events of a cut-and-count based method as described in [19] was determined. For the same significance with respect to background 24 % more signal events are required for the cross-check method, resulting in a slightly less sensitive analysis.
Compared to a likelihood analysis the sensitivity of the analysis method presented here is similar. However a likelihood analysis is more model specific, potentially resulting Table 2 Systematic uncertainties resulting from pre-existing anisotropies in the experimental sky map. σ A v base denotes σ A v assuming no pre-existing anisotropy and σ A v syst assuming the N 90 changed by the systematic effects. in several trials, which would reduce the discovery potential.
Even if the sensitivity of a likelihood analysis is optimal, a likelihood method can be affected by systematic uncertainties. The multipole analysis has the advantage that the sensitivity does not depend strongly on the optimized model. Moreover here it is possible to strongly reduce systematic uncertainties simply by omitting the strongly affected coefficients a 0 . A similar strategy is more difficult to introduce in a likelihood formalism.

Systematic Uncertainties
The relevant systematic uncertainties for the analysis can be categorized into three groups: -Systematic uncertainties on the background expectation, resulting from pre-existing anisotropies in the experimental sky map. -Systematic uncertainties on the signal detection efficiency. -Dependencies on the halo profile.
As the background expectation is generated from scrambled experimental events in RA, systematic effects can only be caused by pre-existing anisotropies. Such an anisotropy can arise from the zenith-dependent acceptance of the detector, the zenith-dependent variation of the atmospheric neutrino flux or the detector exposure. There is also the possibility of an anisotropy in the atmospheric neutrino flux, caused by the cosmic-ray anisotropy which has been measured by Milagro [20], TUNKA [21], ARGO-YBJ [23] and IceCube [22].
In order to study the influence of the zenith structure that arises from the acceptance of the detector and the variation of the atmospheric neutrino flux, pseudo-experiments were generated using events according to a histogram of experimental zenith values and not using the experimental data directly. To generate steeper and flatter zenith-spectra, the bincontents of the histogram are changed by raising the outer most left bin by 25 % and decreasing the outer most right bin by 25 %. The bins in between are raised or decreased according to a linear interpolation between +25 % and −25 %. The uncertainties on the zenith-spectrum are on the order of 5 %, however to study this effect and not be limited by statistics Table 3 Relative systematic uncertainties on σ A v resulting from uncertainties in the detection efficiency. Because the detection efficiency is energy-dependent, the uncertainties are given in dependence of annihilation channel and the mass of the DM particle m χ . σ A v base denotes σ A v assuming the baseline effective area and σ A v syst assuming the effective area changed by the systematic effects.
the slope of the zenith-spectrum was changed by ±50 % resulting in a large overestimation of the effect. Based on these pseudo-experiments the median sensitivity on σ A v was calculated and the resulting overestimated systematic uncertainty is shown in Table 2. The uncertainty is very small as a result of the fact that the zenith(declination)-dependent coefficients are ignored in the test statistic.
The up-time of the IceCube-detector is of the order of 98 %, however due to high quality criteria in the data selection the used data correspond to 91 % up-time. The geometry of the detector is almost symmetric in azimuth, and thus the exposure of each direction in the sky is nearly constant. However due to short down-times and a non flat azimuth acceptance an anisotropy of 0.02 % in the data sample (∼ 10 events) can occur. In the worst case this anisotropy can mimic a (anti-)signal and thus result in a small systematic effect on the median sensitivity σ A v (see Table 2).
Milagro, ARGO-YBJ and TUNKA have observed an anisotropy in cosmic rays at few hundred GeV to EeV energies of primary particles in the Northern Hemisphere [20,21,23]. Because the experimental sky map is dominated by atmospheric neutrinos, that were produced in air-showers initiated by cosmic rays an analogous anisotropy is expected in atmospheric neutrinos. Therefore, pseudo-experiments were generated that allow for an anisotropy as parameterized in [20]. The uncertainty on the median sensitivity on σ A v is given in Table 2.
Systematic uncertainties on the signal efficiency can be expressed by uncertainties in the effective area. Because the effective area depends on energy, the resulting systematic uncertainties on σ A v depend on the assumed energyspectrum and, thus, on the annihilation channel and the mass of the dark matter particle.
The main uncertainties of the detection efficiency arise from the optical efficiency of the DOMs and from the optical properties of the antarctic ice, described by the absorption and scattering length. The influence on the effective area of this effect was determined by a full detector simulation with ±10 % changed optical efficiency and ±10 % change in the absorption and scattering length. The uncertainties on the effective area were further propagated to uncertainties on σ A v , which depends on the dark matter particle mass m χ and the annihilation channel. The resulting uncertainties are listed in Table 3. They typically lie in the range 15 %-30 %, and they are the dominating uncertainties of this analysis.
The sensitivities as obtained from the different halo profiles using best-fit parameters differ by about 6 %. This is smaller than the uncertainty that arises from uncertainties on the profile fit values. The dominant contribution comes from the local dark matter density, and corresponds to an uncertainty on the sensitivity of up to 50 %. In the following the dependency of the assumed model is not treated as a systematic uncertainty, but as model uncertainty, and thus the experimental result will be interpreted for each of the different halo profiles, and benchmark annihilation channels, respectively.

Experimental Results
This analysis was performed blind, meaning it was developed by using pseudo-experiments only. After the analysis procedure was optimized and fixed, the data were unblinded. The experimental sky map has a test statistic value of T S exp = 0.23. The probability of a larger experimental value in the background-only case is 22 % and thus the result is compatible with the background-only hypothesis. The observation is an over-fluctuation corresponding to 0.8σ , where σ is the standard deviation of the background expectation of the test statistic. Note that the test statistic can not be approximated by a Gaussian due to larger tails. The experimental value and the background expectation of the test statistic are shown in Figure 14. Figure 15 shows the deviation of the experimental expansion coefficients from the background expectation, normalized to the standard deviation of the background coefficients. These values correspond to the last term in Equation 9 without the square. Also here no significant deviation can be seen.
As no signal was observed, upper limits on the number of signal events in the sample, N UL , were calculated at a 90 % C.L. following the approach of Feldman and Cousins [48]. In order to calculate the confidence belt, 25000 pseudo-experiments for different N sig were generated respectively. Due to limited computational resources the pseudo-experiments were not generated for each N sig , but for signal contributions N sim,i with a step-size of ∆ sim = 25 events. The test statistic distribution was interpolated for the remaining N sig , using a Gaussian, p gaus , with mean µ = N sig and standard deviation σ = N sig . The interpolated test statistic distributions are given by  where i runs over all generated test statistics. Furthermore the systematic uncertainties, as described in Section 7, are folded into the T S distributions. The resulting 90 % C.L. limits N UL for the different halo profiles are listed in Table 4.
By using equation (12) the limit on the signal events N UL can be interpreted in terms of a limit on the self-annihilation cross-section σ A v UL . The resulting limits are shown in Figure 16 as function of m χ and for the different benchmark annihilation channels. The limits are also listed in Table 5. Note that the sensitivity on σ A v of this analysis is better than the resulting exclusion limits by a factor of 0.5 for each halo profile, respectively.

Discussion
The presented exclusion limits of this analysis are about a factor two less stringent compared to the sensitivity. Although the over-fluctuation is just 0.8σ the non-Gaussian tails of the test statistic results in worse exclusion limits than expected in a Gaussian test statistic case. Table 5 Limit on the self-annihilation cross-section σ A v for different annihilation channels, halo profiles and DM-particle masses m χ .
4.2 · 10 −19 7.6 · 10 −21 2.6 · 10 −21 1.9 · 10 −22 Compared to the predecessor analysis of IC22 data using a cut-and-count based method [19], the effective area increases by more than an order of magnitude in the low energy region (∼ 100 GeV) but just a factor of about 3 at high energies (∼ 10 TeV) in the relevant zenith region. The larger gain in effective area for low dark matter masses is caused by DeepCore, the low-energy extension of IceCube, which was not implemented in IC22, but was already in operation in IC79. The lager gain in the effective area at low energies causes an increase in sensitivity of more than an order of magnitude at these energies. However due to a much larger sample size, caused by the large increase in the num-  ber of low energy events, and just a slight increase in effective area at high energies, there is just a small gain in sensitivity for high dark matter masses. As this analysis measures an over-fluctuation and the IC22 analysis has measured a under-fluctuation the exclusion limits of the IC22 analysis are more stringent for high dark matter masses of a few TeV. However the limits in the low-mass region of 100 GeV are still an order of magnitude more stringent due to the larger increase in sensitivity. For comparison the exclusion limits (90 % C.L.) of the predecessor analysis are shown in Figure 17. Note that in the predecessor analysis a NFW profile was assumed, but with a local density of 0.3 GeV/cm 3 , while here 0.47 GeV/cm 3 is assumed. For reasons of comparison the limits in Figure 17 have been rescaled to the local density used in this analysis.
Furthermore, exclusion limits of an outer Galactic halo analysis by Fermi-LAT [49] are also shown in Figure 17 for annihilation into bb and µ + µ − . This analysis has measured the γ-ray emission along the Galactic plane in a window of ±15 • , whereas the central ±5 • are excluded. In [49] a NFW profile was assumed, but with a local density of 0.43 GeV/cm 3 , while here 0.47 GeV/cm 3 is assumed. For reasons of comparison the limits in Figure 17 have been rescaled to the local density used in this analysis. It can be The baseline limit curves are calculated for the NFW profile (markers). The model-dependence (bands) has been estimated from the Burkert profile. The gray band describes the natural scale if all dark matter consists of WIMPs and the gray upper region is excluded by the unitarity bound [5].
seen that for hard neutrino channels (µ + µ − ) the exclusion limits of this analysis are competitive to outer Galactic halo limits of Fermi.
The most stringent exclusion limits from γ-ray telescopes in the energy-range of this analysis are set by H.E.S.S [51], with an analysis focusing on the Galactic Center, and Fermi [52] with an analysis focusing at dwarf galaxies. These limits are about two orders of magnitude more stringent. However it is important to note, that the systematic uncertainties for γ-ray and neutrino telescopes are of very different nature. Also γ telescopes have the highest sensitivity for the soft channel and vice versa, smallest sensitivity for the hard channels.
The halo profile dependencies in this analysis are very small compared to the Galactic Center analysis of IceCube. This can be seen by comparing this analysis with a Galactic Center search, that focuses on the central part of the galaxy. The sensitivity for the W + W − annihilation channel of the IceCube-79 Galactic Center analysis described in [50] and the sensitivity of this analysis are compared in Figure 18. The bands represent the model uncertainties determined from Burkert and NFW profile, whereas NFW is used as baseline. It is clearly visible that the halo profile uncertainties are much smaller for a halo analysis, while the overall sensitivity of the two approaches are remarkably similar.

Conclusions
We have presented a competitive analysis technique to search for characteristic anisotropies by using a multipole expansion of the neutrino arrival direction sky map. It was found that the multipole analysis is a sensitive and robust analysis method, that has the feature to reduce systematic uncertainties in an easy way.
We applied the analysis to one year of data taken with the IceCube detector in its the nearly completed detector configuration. The search for a neutrino flux, resulting from dark matter annihilation, has found no significant deviation from the background expectation. Exclusion limits on the selfannihilation cross-section σ A v were calculated approaching 1.9 · 10 −23 cm 3 s −1 . The resulting exclusion limits are more stringent than the predecessor analysis of IC22 data in a wide parameter range. Furthermore the extracted limits are competitive to limits from γ-experiments, that also focus on the outer Galactic halo, for hard annihilation channels and large dark matter masses. The presented analysis, focusing on the Galactic halo, is very robust against halo profile uncertainties compared to analyses targeting the Galactic Center or dwarf spheroidal galaxies.