Probing Dark Matter Self-Interaction in the Sun with IceCube-PINGU

We study the capture, annihilation and evaporation of dark matter (DM) inside the Sun. It has been shown that the DM self-interaction can increase the DM number inside the Sun. We demonstrate that this enhancement becomes more significant in the regime of small DM mass, given a fixed DM self-interaction cross section. This leads to the enhancement of neutrino flux from DM annihilation. On the other hand, for DM mass as low as a few GeVs, not only the DM-nuclei scatterings can cause the DM evaporation, DM self-interaction also provides non-negligible contributions to this effect. Consequently, the critical mass for DM evaporation (typically 3 $\sim$ 4 GeV without the DM self-interaction) can be slightly increased. We discuss the prospect of detecting DM self-interaction in IceCube-PINGU using the annihilation channels $\chi\chi \rightarrow \tau^{+}\tau^{-}, \nu\bar{\nu}$ as examples. The PINGU sensitivities to DM self-interaction cross section $\sigma_{\chi\chi}$ are estimated for track and cascade events.


I. INTRODUCTION
Convincing observational evidences indicate that roughly 80% of the matter in our universe is dark matter (DM). On the other hand, the understanding of the DM nature is still at the budding stage. If the form of DM is any kind of elementary particle χ, theoretical predictions for its mass lie in a wide range, from sub-eV scale to grand unified scale. Even though, more and more experimental searches, including direct and indirect detection with terrestrial or satellite instruments, are looking for the DM signals. The knowledge on DM is accumulating rapidly. If DM interacts with ordinary matters weakly, it may leave tracks in the detector and be caught directly. The possibility to see such signals depends on DM mass and its interaction cross section with the ordinary matter in the detector. The situation is similar for the indirect searches where one looks for the flux excess in cosmic rays, gamma rays, and neutrinos. The flux excess for the above particles are due to DM annihilation or DM decays. Different search strategies would have sensitivities to different parameter ranges and all together cover quite a broad range for DM mass and DM interaction cross section, respectively.
In this paper, we study the general framework of DM capture, annihilation and evaporation in the Sun. The capture of galactic DM by the Sun through DM-nuclei collisions was first proposed and calculated in Refs. [1][2][3][4][5][6]. It was then observed that the assumption of DM thermal distribution according to the average temperature of the Sun is a good approximation for capture and annihilation processes, but the correction to the evaporation mass can reach to 8% in the true distribution calculation [7]. The abundance of DM inside the Sun hence results from the balancing among DM capture, annihilation and evaporation processes. Updated calculations on these processes are given in Refs. [8][9][10]. In our study, we particularly note that for both collisionless cold DM and warm DM there exists a socalled core/cusp problem [11] which addresses the discrepancy between the computational structure simulation and the actual observation [12][13][14][15][16][17]. DM self-interaction has been introduced to resolve this inconsistency [18]. The type of analyses put constraints on the ratio of DM self-interaction cross section to the DM mass, 0.1 < σ χχ /m χ < 1.0 (cm 2 /g), from observations of various galactic structures [19][20][21][22]. It is worth mentioning that the authors in Ref. [23] used the IceCube data to constrain the magnitude of σ χχ for m χ in the range of O(10) GeV to O(1) TeV (also see the study of high energy neutrino flux from DM annihila-tion within the Sun with the inclusion of DM self-interaction in Ref. [24]). In their study the evaporation effect can be neglected for the considered DM mass region and the detectability of σ χχ by the IceCube through observing the final state neutrino flux produced by the DM annihilation is analyzed. In this paper we however concentrate on the low mass region of O(1) GeV DM mass. In this case one has to consider evaporation processes for determining the DM abundance inside the Sun. This mass region is favored by the so-called Asymmetric Dark Matter paradigm [25] (also see the recent review [26] and references therein) where the simple relation ρ DM /ρ B ∼ 5 is utilized, and one has m χ ∼ 5m p ≈ 5 GeV. We shall demonstrate that DM evaporation can arise both from DM-nuclei scatterings and DM selfinteractions. In fact the inclusion of DM self-interaction can both increase the trapped DM number inside the Sun and raise the critical mass for DM evaporation inside the Sun. Since we are interested in the scenario of low DM mass, we only consider DM annihilation into τ + τ − and νν, respectively. The final-state neutrinos can be detected by terrestrial neutrino telescopes such as IceCube-PINGU [27]. This paper is organized as follows. In Section II, we calculate the DM number accumulated inside the Sun via four processes. We then present the parameter space in which the thermal equilibrium condition holds. In Section III, we present the numerical results of our analysis. The sensitivity of IceCube-PINGU to DM self-interaction inside the Sun is presented in Section IV. Finally, we conclude in Section V.

A. DM evolution equation
We assume that both the galactic DM and the nuclei inside the Sun follow the thermal distributions. If DM interacts with nuclei in the Sun then it can be captured by the Sun when its final velocity is smaller than the escape velocity from the Sun. Alternatively, DM trapped inside the Sun will be kicked out if its final velocity after the scattering with the nuclei is larger than the escape velocity. The inclusion of DM self-interaction will also have effects on the capture and evaporation of DM inside the Sun. We will come to details of this in the later discussion. The captured DM could have reached to an equilibrium state if the equilibrium time scale is less than the age of the Sun (t ≈ 10 17 s). The general DM evolution equation in the Sun is given by with N χ the DM number in the Sun, C c the rate at which DM are captured by the Sun, C s the rate at which DM are captured due to their scattering with DM that have already been trapped in the Sun, C e the the DM evaporation rate due to DM-nuclei interactions, C a the DM annihilation rate, and C se the evporation rate induced by the interaction between DM particles in the Sun. The coefficients C a,c,e,s,se are taken to be positive and time-independent.
We note that C c can be categorized by the type of interactions between DM particles and nucleons. For spin-dependent (SD) interactions, the capture rate is given by [5,6] C SD where ρ 0 is the local DM density,v is the velocity dispersion, σ SD H is the SD DM-hydrogen scattering cross section and m χ is the DM mass. The capture rate from spin-independent (SI) scattering is given by [5,6] C SI Here σ SI H and σ SI He are SI DM-hydrogen and -helium cross sections respectively. Taking the approximation m p ≈ m n , the DM-nucleus cross section σ i is related to DM-nucleon cross section σ χp by for SD interactions and for SI interactions, where A is the atomic number and m A is the mass of the nucleus.
The DM evaporation rate in the Sun, C e , has been well investigated in Refs. [4,7]. The evaporation rate is usually ignored in the DM evolution equation since it happens for a very low DM mass, m χ 3 GeV. A updated calculation in Ref. [10] has shown that, for where v esc (0) is the escape velocity from the core of the Sun, T χ is the DM temperature in the Sun, andr is average DM orbit radius which is the mean DM distance from the solar center. The quantity Σ evap is the sum of the scattering cross sections of all the nuclei within a radius r 95% , where the solar temperature has dropped to 95% of the DM temperature.
Although the approximate form of C e can be obtained as the above equation, we shall adopt the exact form of C e given in Ref. [7] for our subsequent numerical calculations.
As stated before, C s is the DM capture rate by scattering off the DM that have been captured inside the Sun. This kind of scattering may result in the target dark matter particles being ejected from the Sun upon recoil. However, because the escape speed from the Sun is sufficiently large, the effect of target DM ejection by recoil is only a small correction to the simple solar capture estimate. Hence the self-capture rate in the Sun can be approximated by [24] where φ χ 5.1 [28] is a dimensionless average solar potential experienced by the captured DM within the Sun, n χ is the local number density of halo DM, σ χχ is the elastic scattering cross section of DM with themselves, v esc (R ) is the Sun's escape velocity at the surface, and C a is the annihilation coefficients given by [4] where V j 6.5 × 10 28 cm 3 10 GeV jm χ is the DM effective volume inside the Sun and σv is the relative velocity averaged annihilation cross section.
C se is the self-interaction induced evaporation. Since the DM can interact among themselves, DM trapped in the solar core could scatter with other trapped DM and results in the evaporation. Essentially, one of the DM particles could have velocity greater than the escape velocity after the scattering. This process involves two DM particles just like annihilation.
We note that both processes lead to the DM dissipation in the Sun. On the other hand C se does not produce neutrino flus as C a does. Since the derivation of C se has not been given in the previous literature, we present some details of the derivation in Appendix A [29].
With N χ (0) = 0 as the initial condition, the general solution to Eq. (1) is with the time-scale for the DM number in the Sun to reach the equilibrium. If the equilibrium state is achieved, i.e., tanh(t/τ A ) ∼ 1, one has The DM annihilation rate in the Sun's core is given by By setting C s = C se = 0, we can recover the results in Refs. [4,[7][8][9][10] for the absence of DM self-interaction. By setting C e = C se = 0, we recover the result in Ref. [24], which includes the DM self-interaction while neglects the DM evaporation.

B. Numerical results
The coefficients C c,e,s have been worked out in Refs. [4,7,24], which we adopt for our numerical studies. The numerical result for C se is based on the analytic expression we have given in the Appendix. We first identify the equilibrium region on the σ χp -σ χχ plane.
The SD and SI cases are presented in Fig. 1 and Fig. 2, respectively with two benchmark DM masses m χ = 5 GeV and 20 GeV. We have taken 10 −45 ≤ σ SD χp /cm 2 ≤ 10 −41 and 10 −47 ≤ σ SI χp /cm 2 ≤ 10 −43 for our studies. The range of SD cross section is below those bounds set by direct detection experiments, COUPP [30] and Simple [31], and the indirect search by IceCube [32] for m χ ≤ 20 GeV. This range of SI cross section is below the direct detection bound set by LUX [33] at m χ = 5 GeV. For m χ = 20 GeV, the LUX bound on σ SI χp is 10 −45 cm 2 . We have indicated this bound on the right panel of Fig. 2. The dark areas represent those regions with tanh(t/τ A ) 1 at the present day whereas the light areas are the equilibrium regions. In Fig. 3, we show the effect of DM self-interaction on the number of DM particles trapped inside the Sun. It is seen that N χ can be significantly enhanced for sufficiently large σ χχ .
The N χ peak for each parameter set is the maximal DM number trapped inside the Sun, and the corresponding DM mass can be viewed as the evaporation mass scale because N χ drops quickly for m χ smaller than this mass scale. We observe that the inclusion of DM self-interaction tends to lift the evaporation mass by about 1 GeV. This is because the evaporation due to the captured DM-DM self-interaction comes to operate.
To quantify the effects of DM self-interaction and evaporation (the one by C e ) on N χ , it is useful to define a dimensionless parameter R by such that and where one takes the positive sign for C s > C e and the negative sign for C e > C s . The expression for N χ in the non-equilibrium case can also be simplified in a similar way. Comparing C s with C e , DM self-interaction dominates in the high mass region whereas the evaporation process takes over in the low mass region. Here we have taken into account both effects and found that the transition between two effects occurs at O(1) GeV DM mass. In Fig. 4, we show the behavior of R as a function of m χ . The dip of R for each parameter set represents the narrow mass range where C s ≈ C e . On the right side of the dip, C s dominates over C e while the reverse is true on the left side of the dip.
It is seen that R in the evaporation dominant region is growing up in small m χ since the velocity of final state χ after the collision can easily be larger than the escape velocity from the Sun in this case. The parameter space for R > 1 over the σ χp − σ χχ plane is shown in region where DM self-interaction is relevant. It has been seen that DM self-interaction not only enhances N χ significantly for m χ < 10 GeV but also affects the evaporation mass scale.
Therefore in the next session we shall explore the possibility of probing DM self-interaction for m χ < 10 GeV by IceCube-PINGU detector.

III. PROBING DM SELF-INTERACTION AT ICECUBE-PINGU
The annihilation rate of the captured DM in the Sun is given by Eq. (13). It is worth mentioning that in the absence of both evaporation (the one due to C e ) and self-interaction, the annihilation rate Γ A with an equilibrium N χ is which only depends on the capture rate C c . However, with the presence of either C e or selfinteraction, Γ A depends on other coefficients as well even N χ has reached to the equilibrium.
We plot Γ A as a function of m χ with and without self-interaction in Fig. 6.
To probe DM self-interaction for small m χ , we consider DM annihilation channels, χχ → τ + τ − and νν, for producing neutrino final states to be detected by IceCube-PINGU [27].
The neutrino differential flux of flavor i, Φ ν i , from χχ → ff can be expressed as where R is the distance between the neutrino source and the detector, P ν j →ν i (E ν ) is the neutrino oscillation probability during the propagation, B f is the branching ratio corresponding to the channel χχ → ff , dN ν /dE ν is the neutrino spectrum at the source, and Γ A is the DM annihilation rate in the Sun. To compute dN ν /dE ν , we employed WimpSim [34] with a total of 50,000 Monte-Carlo generated events.
The neutrino event rate in the detector is given by where E th is the detector threshold energy, dΦ ν /dE ν is the neutrino flux from DM annihilation, A ν is the detector effective area, and Ω is the solid angle. We study both muon track events and cascade events induced by neutrinos. The PINGU module will be implanted inside the IceCube in the near future [27] and can be used to probe neutrino energy down to O(1) GeVs. We take ice as the detector medium, so that the IceCube-PINGU neutrino effective area is expressed as where V eff is the IceCube-PINGU effective volume, N A is the Avogadro constant, M ice is the mass of ice per mole, n p,n is the number of proton/neutron of an ice molecule and σ νp,n is the neutrino-proton/neutron cross section which can be approximated by [35] σ for 1 GeV ≤ E ν ≤ 10 GeV. As the neutrinos propagate from the source to the detector, they encounter high-density medium in the Sun, the vacuum in space, and the Earth medium.
The matter effect to the neutrino oscillation has been considered in P ν j →ν i in Eq. (18).
The atmospheric background event rate can also be calculated by Eq. (19) with dΦ ν /dE ν replaced by the atmospheric neutrino flux. Hence In our calculation, the atmospheric neutrino flux dΦ atm ν /dE ν is taken from Ref. [36,37]. We set E max = m χ in order to compare with the DM signal.
Hence we consider neutrino events arriving from the solid angle range ∆Ω = 2π(1 − cos ψ) surrounding the Sun with ψ = 10 • . We present the IceCube-PINGU sensitivity to σ χχ in the DM mass region 3 GeV < m χ < 20 GeV for both SD and SI cases in Fig. 7 and Fig. 8, respectively. The sensitivities to σ χχ are taken to be 2σ significance for 5 years of data taking. The shadow areas in the figures represent those parameter spaces disfavored by the Bullet Cluster and halo shape analyses. Below the black solid line, the DM self-interaction is too weak to resolve the core/cusp problem of the structure formation. Two benchmark are used for our studies. We note that the latter value for σv does not contradict with the relic density, since DM annihilation inside the Sun occurs much later than the period of freeze-out.
We take σ SD χp = 10 −41 cm 2 and 10 −43 cm 2 for SD interaction, and take σ SI χp = 10 −44 cm 2 and 10 −45 cm 2 for SI interaction. We stress that σ SD χp = 10 −41 cm 2 is below the lowest value of IceCube bound σ SD χp ∼ 10 −40 cm 2 at m χ ∼ 300 GeV [32]. For SI interaction, σ SI χp = 10 −44 cm 2 is below the LUX bound for m χ < 8 GeV, while σ SI χp = 10 −45 cm 2 is below the LUX bound for m χ < 20 GeV [33]. We find that cascade events provide better sensitivities to DM self-interaction than track events do in all cases. One can also see that the sensitivity to σ χχ becomes better for smaller annihilation cross section σv for a fixed σ χp , as noted in earlier works [23,24] which neglect both C e and C se . This is evident from Eq. (16) since R increases as C a decreases. It is instructive to take the limit R 1 such that It is easily seen that Γ A is inversely proportional to C a (in the mass range that C se is negligible) and is independent of C c . In other words, only C s and C a determine the annihilation rate (we are in the region that C e is suppressed as compared to C s ). We also see that the sensitivity to σ χχ does become significantly worse as m χ → 4 GeV. This is the critical m χ below which the DM evaporations from the Sun is important.
We have studied the time evolution of DM number trapped inside the Sun with DM selfinteraction considered. We have focused on the low m χ range so that our analysis includes evaporation effects due to both DM-nuclei and DM-DM scatterings. The parameter region for the trapped DM inside the Sun to reach the equilibrium state is presented. We also found that the inclusion of DM self-interaction can increase the number of trapped DM and raise the evaporation mass scale. The parameter space on σ χχ − σ can be lowered down to 1 GeV. We considered cascade and track events resulting from neutrino flux induced by DM annihilation channels χχ → νν and χχ → τ + τ − inside the Sun. We found that cascade events always provide better sensitivity to σ χχ . The sensitivity to σ χχ is also improved with a smaller DM annihilation cross section σv .
where w is the DM velocity. The calculation of DM-DM scattering rate proceeds by choosing one of the DM as the incident particle and the other DM as one of the targets which satisfy Maxwell-Boltzmann distribution in their velocities. We then sum over the incident states with Maxwell-Boltzmann distribution as well. Let the velocity of the incident DM be w and the velocity of the faster DM in the final state as v, respectively. Since we are considering the DM evaporation due to their self interactions, we have v > w.
It is essential to take note on the symmetry factor for identical particle scattering (χχ scattering) as compared to the DM-nucleus scattering studied before. We note that the faster DM with velocity v can be either one of the DM particles in the final state. This generates an extra factor of 2 relative to DM-nucleus scattering. On the other hand, due to identical particles in the initial state, a factor 1/2 must be applied as we sum over initial states according to thermal distributions. Hence the DM-DM differential scattering rate with the velocity transition w → v can be inferred from DM-nucleus scattering with suitable parameter replacements, which is given by where we have summed up the target DM according to the thermal distribution f described above, Therefore the DM-DM scattering rate with v greater than the escape velocity v esc is given by the integral Carrying out the integral yields To calculate the evaporation rate per unit volume at the position r, we should sum up all possible states of the incident DM as follows: The DM number density inside the Sun is determined by where n 0 is the density at the solar core, and φ(r) is the solar gravitational potential with respect to the core so that with G the Newton gravitational constant, M (r) = 4π r 0 r 2 ρ (r )dr the solar mass enclosed within radius r, and ρ (r) the solar density. Thus, the integration in Eq. (A4) can be performed with E esc (r) the escape energy at the position r defined by E esc (r) = 1 2 m χ v 2 esc (r).
The escape velocity v esc (r) is related to the gravitational potential by v esc (r) ≡ 2[φ(∞) − φ(r)]. Finally, the self-interaction induced evaporation rate can be evaluated through the following: (A8)