This content has been downloaded from IOPscience. Please scroll down to see the full text.
Download details:
IP Address: 140.113.38.11
This content was downloaded on 21/07/2015 at 09:44
Please note that terms and conditions apply.
Probing dark matter self-interaction in the Sun with IceCube-PINGU
View the table of contents for this issue, or go to the journal homepage for more JCAP10(2014)049
(http://iopscience.iop.org/1475-7516/2014/10/049)
JCAP10(2014)049
ournal of Cosmology and Astroparticle Physics
An IOP and SISSA journalJ
Probing dark matter self-interaction in
the Sun with IceCube-PINGU
Chian-Shu Chen,
a,cFei-Fan Lee,
bGuey-Lin Lin
band Yen-Hsun Lin
baPhysics Division, National Center for Theoretical Sciences,
Hsinchu 30010, Taiwan
bInstitute of Physics, National Chiao Tung University,
Hsinchu 30010, Taiwan
cDepartment of Physics, National Tsing Hua University,
Hsinchu 30010, Taiwan
E-mail: chianshu@phys.sinica.edu.tw,fflee@mail.nctu.edu.tw,glin@cc.nctu.edu.tw,
chris.py99g@g2.nctu.edu.tw
Received September 12, 2014 Accepted October 1, 2014 Published October 20, 2014
Abstract. 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 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 ∼ 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 χχ → τ+τ−, ν ¯ν as examples. The PINGU sensitivities to DM self-interaction cross section σχχ are estimated for track and cascade events.
Keywords: dark matter theory, neutrino detectors ArXiv ePrint: 1408.5471
JCAP10(2014)049
Contents
1 Introduction 1
2 DM accumulation in the Sun 2
2.1 DM evolution equation 2
2.2 Numerical results 5
3 Probing DM self-interaction at IceCube-PINGU 8
4 Conclusion 12
A DM self-interaction induced evaporation 12
1 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 evapo-ration in the Sun. The capture of galactic DM by the Sun through DM-nuclei collisions was first proposed and calculated in refs. [1–7]. It was then observed that the assumption of DM thermal distribution according to the average temperature of the Sun is a good approxima-tion for capture and annihilaapproxima-tion processes, but the correcapproxima-tion to the evaporaapproxima-tion mass can reach to 8% in the true distribution calculation [8]. 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. [9–11].
In our study, we particularly note that for both collisionless cold DM and warm DM there exists a so-called core/cusp problem [12] which addresses the discrepancy between the computational structure simulation and the actual observation [13–18]. DM self-interaction has been introduced to resolve this inconsistency [19]. This type of analyses put constraints on the ratio of DM self-interaction cross section to the DM mass, 0.1 < σχχ/mχ< 1.0 (cm2/g),
from observations of various galactic structures [20–23]. It is worth mentioning that the authors in ref. [24] used the IceCube data [25] to constrain the magnitude of σχχ for mχ in
JCAP10(2014)049
DM annihilation within the Sun with the inclusion of DM self-interaction in ref. [26]). In their work the evaporation effect can be neglected for the considered DM mass region and the detectability of σχχ by the IceCube through observing the final state neutrino flux produced
by the DM annihilation is analyzed. In this paper we shall concentrate on the low mass region of O(1) GeV DM mass since such a mass range has not been probed by the IceCube data mentioned above. Furthermore, this is also the mass range where the indirect search is crucial. In the case of spin-independent interaction, the sensitivity of DM direct search quickly turns poor for mχ less than 10 GeV [27]. Therefore the IceCube-PINGU [28] detector with a 1 GeV
threshold energy could be more sensitive than some of the direct detection experiments for mχ < 10 GeV. For spin-dependent interaction, the IceCube-PINGU sensitivity has been
estimated to be much better than constraints set by direct detection experiments [29]. For illustrative purpose, we shall only consider neutrino flux produced by DM annihi-lating into leptons such as χχ → τ+τ−and χχ → ν ¯ν, respectively. The final-state neutrinos
can be detected by terrestrial neutrino telescopes such as IceCube-PINGU [28]. We do not consider χχ → µ+µ− because muons will suffer severe energy losses in the Sun before they decay to neutrinos. The soft neutrino spectrum in this case is dominated by the atmospheric background. One also expects that the neutrino telescopes are less sensitive to heavy quark channels such as χχ → b¯b than their sensitivities to leptonic channels. This is caused by the relatively softer neutrino spectrum resulting from the b-hadron decays compared to the neutrino spectrum from τ decays [30]. For light quark channels χχ → q ¯q, the hadronic cas-cades produce pions or kaons in large multiplicities. These hadrons decay almost at rest and produce MeV neutrino fluxes, which from the observational point of view could be as promis-ing as the hard spectrum channels [10, 31]. However we would not discuss these signatures because the threshold energy of IceCube-PINGU is in the GeV range.
For O(1) GeV DM mass, it is necessary to consider evaporation processes for determining the DM abundance inside the Sun. We shall demonstrate that DM evaporation can arise both from DM-nuclei scatterings and DM interactions. In fact the inclusion of DM self-interaction can raise the critical mass such that evaporation can take place for DM lighter than this mass scale. On the other hand, DM self-interaction also enhances total DM number trapped inside the Sun for mχ greater than the critical mass.
This paper is organized as follows. In section II, we calculate the DM number accumu-lated inside the Sun via four processes. We then present the parameter space in which the equilibrium condition holds. In section III, we present the numerical results of our analy-sis. The sensitivity of IceCube-PINGU to DM self-interaction inside the Sun is presented in section IV. Finally, we conclude in section V.
2 DM accumulation in the Sun 2.1 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 ≈ 1017s). The general DM
JCAP10(2014)049
evolution equation in the Sun is given by dNχ
dt = Cc+ (Cs− Ce)Nχ− (Ca+ Cse)Nχ
2 (2.1)
with Nχ the DM number in the Sun, Cc the rate at which DM are captured by the Sun, Cs
the rate at which DM are captured due to their scattering with DM that have already been trapped in the Sun, Ce the the DM evaporation rate due to DM-nuclei interactions, Ca the
DM annihilation rate, and Cse the evaporation rate induced by the interaction between DM
particles in the Sun. The coefficients Ca,c,e,s,se are taken to be positive and time-independent.
We note that Cc can be categorized by the type of interactions between DM particles
and nucleons. For spin-dependent (SD) interactions, the capture rate is given by [6,7] CcSD' 3.35 × 1024 s−1 ρ0 0.3 GeV/cm3 270 km/s ¯ v 3 GeV mχ 2 σSDH 10−6 pb , (2.2)
where ρ0 is the local DM density, ¯v is the velocity dispersion, σHSD is the SD DM-hydrogen
scattering cross section and mχ is the DM mass. The capture rate from spin-independent
(SI) scattering is given by [6,7] CcSI' 1.24 × 1024 s−1 ρ0 0.3 GeV/cm3 270 km/s ¯ v 3 GeV mχ 2 2.6σSI H + 0.175σHeSI 10−6 pb . (2.3) Here σHSI and σSIHe are SI DM-hydrogen and -helium cross sections respectively. Taking the approximation mp ≈ mn, the DM-nucleus cross section σi is related to DM-nucleon cross
section σχp by σSDi = A2 mχ+ mp mχ+ mA 2 4(Ji+ 1) 3Ji |hSp,ii + hSp,ii|2σχpSD (2.4)
for SD interactions and
σSIi = A2 mA mp 2 mχ+ mp mχ+ mA 2 σχpSI (2.5)
for SI interactions, where A is the atomic number, mA the mass of the nucleus, Ji the total
angular momentum of the nucleus and hSp,ii and hSn,ii the spin expectation values of proton
and of neutron averaged over the entire nucleus [32–35].
The DM evaporation rate in the Sun, Ce, has been well investigated in refs. [5, 8].
The evaporation rate is usually ignored in the DM evolution equation since it happens for a very low DM mass, mχ . 3 GeV. A updated calculation in ref. [11] has shown that, for
mχ/mA> 1, Ce' 8 π3 s 2mχ πTχ(¯r) v2esc(0) ¯ r3 exp −mχv 2 esc(0) 2Tχ(¯r) Σevap, (2.6)
where vesc(0) is the escape velocity from the core of the Sun, Tχis the DM temperature in the
Sun, and ¯r is average DM orbit radius which is the mean DM distance from the solar center. The quantity Σevapis the sum of the scattering cross sections of all the nuclei within a radius
r95%, where the solar temperature has dropped to 95% of the DM temperature. Although
the approximate form of Cecan be obtained as the above equation, we shall adopt the exact
JCAP10(2014)049
As stated before, Cs 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 [26] Cs = r 3 2nχσχχvesc(R ) vesc(R ) v D b φχ Eerf(η) η , (2.7) whereDφbχ E
' 5.1 [36] is a dimensionless average solar potential experienced by the captured DM within the Sun, nχ is the local number density of halo DM, σχχ is the elastic scattering
cross section of DM with themselves, vesc(R ) is the Sun’s escape velocity at the surface, and
η2 = 3(v /v)2/2 is the square of a dimensionless velocity of the Sun through the Galactic
halo with v = 220 km/s Sun’s velocity and v = 270 km/s the local velocity dispersion of
DM in the halo.
Ca is the annihilation coefficient given by [5]
Ca' hσvi V2 V2 1 , (2.8) where Vj ' 6.5 × 1028 cm3 10 GeV jmχ 3/2 (2.9) is the DM effective volume inside the Sun and hσvi is the relative velocity averaged annihi-lation cross section.
Cseis the self-interaction induced evaporation. Since the DM can interact among
them-selves, 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 Cse
does not produce neutrino flux as Ca does. Since the derivation of Csehas not been given in
the previous literature, we present some details of the derivation in appendix A.1 With Nχ(0) = 0 as the initial condition, the general solution to eq. (2.1) is
Nχ(t) = Cctanh(t/τA) τA−1− (Cs− Ce) tanh(t/τA)/2 , (2.10) with τA= 1 pCc(Ca+ Cse) + (Cs− Ce)2/4 (2.11) the time-scale for the DM number in the Sun to reach the equilibrium. If the equilibrium state is achieved, i.e., tanh(t/τA) ∼ 1, one has
Nχ,eq= Cs− Ce 2(Ca+ Cse) + s (Cs− Ce)2 4(Ca+ Cse)2 + Cc Ca+ Cse . (2.12)
1We thank S. Palomares-Ruiz for pointing out to us the importance of C se.
JCAP10(2014)049
Figure 1. The values of tanh(t/τA) over σχpSD− σχχ plane at the present day, t = t . The red-circled
area is the non-equilibrium region for Nχ.
Figure 2. The values of tanh(t/τA) over σSIχp− σχχplane at the present day, t = t . The blue-circled
area is the non-equilibrium region for Nχ. The vertical line at the right panel indicates the LUX
bound, σχpSI ≤ 10−45 cm2, for mχ = 20 GeV.
The DM annihilation rate in the Sun’s core is given by ΓA=
Ca
2 N
2
χ. (2.13)
By setting Cs= Cse= 0, we can recover the results in refs. [5,8–11] for the absence of DM
self-interaction. By setting Ce = Cse = 0, we recover the result in ref. [26], which includes
the DM self-interaction while neglects the DM evaporation. 2.2 Numerical results
The coefficients Cc,e,shave been worked out in refs. [5,8,26], which we adopt for our numerical
JCAP10(2014)049
ΣΧΧ= 510 -24cm2 ΣΧΧ= 10-24cm2 ΣΧΧ= 10 -25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1036 1037 1038 1039 1040 mΧ@GeVD NΧ ΣΧ p SD = 10-41cm2 ΣΧΧ= 510 -24cm2 ΣΧΧ= 10-24cm2 ΣΧΧ= 10 -25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1036 1037 1038 1039 1040 mΧ@GeVD NΧ ΣΧ p SD = 10-43cm2 ΣΧΧ= 510-24cm2 ΣΧΧ= 10 -24cm2 ΣΧΧ= 10-25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1036 1037 1038 1039 1040 mΧ@GeVD NΧ ΣΧ p SI = 10-44cm2 ΣΧΧ= 510-24cm2 ΣΧΧ= 10 -24cm2 ΣΧΧ= 10-25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1036 1037 1038 1039 1040 mΧ@GeVD NΧ ΣΧ p SI = 10-45cm2Figure 3. The number of DM particles trapped inside the Sun, Nχ, as a function of DM mass mχwith
and without DM self-interaction. Both SI and SD DM-nucleon couplings are considered. Different
colors represent different values for σχχ. The 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
since Nχ drops quickly for mχ smaller than this mass scale.
the appendix. We first identify the equilibrium region on the σχp-σχχ plane. The SD and
SI cases are presented in figure 1and figure2, respectively with two benchmark DM masses mχ= 5 GeV and 20 GeV. We have taken 10−45≤ σSDχp/cm2 ≤ 10−41and 10−47≤ σχpSI/cm2≤
10−43 for our studies. The range of SD cross section is below those bounds set by direct detection experiments, COUPP [37] and Simple [38], and the indirect search by IceCube [25] for mχ ≈ 20 GeV. This range of SI cross section is below the direct detection bound set by
LUX [39] at mχ = 5 GeV. For mχ = 20 GeV, the LUX bound on σχpSI is 10−45 cm2. We have
indicated this bound on the right panel of figure 2. The dark areas represent those regions with tanh(t/τA) . 1 at the present day whereas the light areas are the equilibrium regions.
In figure 3, we show the effect of DM self-interaction on the number of DM particles trapped inside the Sun. It is seen that Nχ can be significantly enhanced for sufficiently large
σχχ. The Nχ 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 Nχ drops quickly for mχ smaller than this mass scale. We observe that the inclusion of
JCAP10(2014)049
HΣΧpSI,ΣΧΧL=H10-43,5´10-24Lcm2 HΣΧpSI,ΣΧΧL=H10-43,10-25Lcm2 HΣΧpSI,ΣΧΧL=H10-45,5´10-24Lcm2 HΣΧpSI,ΣΧΧL=H10-45,10-25Lcm2 1.0 1.5 2.0 3.0 5.0 7.0 10.0 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 mΧ@GeVD R Spin-independent ΣΧ p HΣSDΧp,ΣΧΧL=H10-41,5´10-24Lcm2 HΣSDΧp,ΣΧΧL=H10-41,10-25Lcm2 HΣSDΧp,ΣΧΧL=H10-43,5´10-24Lcm2 HΣSDΧp,ΣΧΧL=H10-43,10-25Lcm2 1.0 1.5 2.0 3.0 5.0 7.0 10.0 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 103 mΧ@GeVD R Spin-dependent ΣΧ pFigure 4. Ratio R versus DM mass mχ. The dip occurs when DM self-interaction and evaporation
effects cancel each other, Ce≈ Cs.
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 Ce) on Nχ,
it is useful to define a dimensionless parameter R by R ≡ (Cs− Ce) 2 Cc(Ca+ Cse) , (2.14) such that Nχ,eq= r Cc Ca+ Cse ± r R 4 + r R 4 + 1 ! , (2.15) and ΓA= 1 2 CcCa Ca+ Cse ± r R 4 + r R 4 + 1 !2 , (2.16)
where one takes the positive sign for Cs> Ceand the negative sign for Ce> Cs. The
expres-sion for Nχ in the non-equilibrium case can also be simplified in a similar way. Comparing
Cs with Ce, 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 figure 4, we show the behavior of R as a function of mχ. The dip of R for each parameter set represents
the narrow mass range where Cs≈ Ce. On the right side of the dip, Cs dominates over Ce
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 mχsince 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 σχp− σχχ plane is shown in
JCAP10(2014)049
Figure 5. Ratio R over the σχp− σχχ plane. The upper panel is for SI interaction and the lower
panel is for SD interaction. The red-circled region is for R > 1.
region where DM self-interaction is relevant. It has been seen that DM self-interaction not only enhances Nχ significantly for mχ < 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 mχ< 10 GeV by IceCube-PINGU detector.
3 Probing DM self-interaction at IceCube-PINGU
The annihilation rate of the captured DM in the Sun is given by eq. (2.13). It is worth mentioning that in the absence of both evaporation (the one due to Ce) and self-interaction,
the annihilation rate ΓA with an equilibrium Nχ is
ΓA= 1 2Ca× Cc Ca = Cc 2 , (3.1)
which only depends on the capture rate Cc. However, with the presence of either Ce or
self-interaction, ΓAdepends on other coefficients as well even Nχhas reached to the equilibrium.
JCAP10(2014)049
ΣΧΧ= 510-24cm 2 ΣΧΧ= 10-24cm 2 ΣΧΧ= 10-25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1019 1020 1021 1022 1023 1024 1025 mΧ@GeVD GA S I@s -1D ΣΧp SI =10-44cm2 ΣΧΧ= 510-24cm 2 ΣΧΧ= 10-24cm 2 ΣΧΧ= 10-25cm2 No self-interaction 1.0 1.5 2.0 3.0 5.0 7.0 10.0 1019 1020 1021 1022 1023 1024 1025 mΧ@GeVD GA S D @s -1D ΣΧp SD =10-41cm2Figure 6. The annihilation rate ΓA of the captured DM inside the Sun. The left panel assumes
DM-nuclei scattering is dominated by SI interaction while the right panel assumes such scattering is dominated by SD interaction.
To probe DM self-interaction for small mχ, we consider DM annihilation channels, χχ →
τ+τ−and ν ¯ν, for producing neutrino final states to be detected by IceCube-PINGU [28]. The neutrino differential flux of flavor i, Φνi, from χχ → f ¯f can be expressed as
dΦνi dEνi = Pνj→νi(Eν) ΓA 4πR 2 X f Bf dN νj dEνj f (3.2)
where R is the distance between the neutrino source and the detector, Pνj→νi(Eν) is the
neutrino oscillation probability during the propagation, Bf is the branching ratio
correspond-ing to the channel χχ → f ¯f , dNν/dEν is the neutrino spectrum at the source, and ΓA is
the DM annihilation rate in the Sun. To compute dNν/dEν, we employed WimpSim [40] with
a total of 50,000 Monte-Carlo generated events.
The neutrino event rate in the detector is given by Nν = Z mχ Eth dΦν dEν Aν(Eν)dEνdΩ (3.3)
where Eth is the detector threshold energy, dΦν/dEν is the neutrino flux from DM
anni-hilation, Aν 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 [28] 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
Aνeff(Eν) = Veff
NA
Mice
(npσνp(Eν) + nnσνn(Eν)), (3.4)
where Veff is the IceCube-PINGU effective volume, NA is the Avogadro constant, Miceis the
JCAP10(2014)049
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®ΝΝ, ΣΧ p SD = 10-41cm2Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®ΝΝ, ΣΧ p SD = 10-43cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®Τ+Τ-, Σ Χ p SD= 10-41cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®Τ+Τ-, Σ Χ p SD= 10-43cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Figure 7. The IceCube-PINGU sensitivities to DM self-interaction cross section σχχ as a function
of mχ. The DM-nucleus interaction inside the Sun is assumed to be dominated by SD interaction.
the neutrino-proton/neutron cross section which can be approximated by [41–44] σνN(Eν) Eν = 6.66 × 10−3 pb · GeV−1, (3.5a) σνN¯ (Eν) E¯ν = 3.25 × 10−3 pb · GeV−1, (3.5b) for 1 GeV ≤ Eν ≤ 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 Pνj→νi in eq. (3.2).
The atmospheric background event rate can also be calculated by eq. (3.3) with dΦν/dEν
replaced by the atmospheric neutrino flux. Hence
Natm= Z Emax Eth dΦatmν dEν Aν(Eν)dEνdΩ. (3.6)
In our calculation, the atmospheric neutrino flux dΦatmν /dEν is taken from ref. [45,46]. We
JCAP10(2014)049
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®ΝΝ, ΣΧ p SI = 10-44cm2Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®ΝΝ, ΣΧ p SI = 10-45cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®Τ+Τ-, Σ Χ p SI= 10-44cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Track Cascade XΣΥ\=310-26cm3s-1 XΣΥ\=310-27cm3s-1 10.0 5.0 20.0 3.0 7.0 15.0 10-25 10-24 10-23 10-22 mΧ@GeVD Σ Χ Χ @c m 2D ΧΧ®Τ+Τ-, Σ Χ p SI = 10-45cm2
Bullet Cluster excl. HRandall et al.L Halo shapes excl. HPeter et al.L
Figure 8. The IceCube-PINGU sensitivities to DM self-interaction cross section σχχ as a function
of mχ. The DM-nucleus interaction inside the Sun is assumed to be dominated by SI interaction.
The angular resolution for IceCube-PINGU detector at Eν = 5 GeV is roughly 10◦ [28].
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 σχχ in the
DM mass region 3 GeV < mχ < 20 GeV for both SD and SI cases in figure 7 and figure 8,
respectively. The sensitivities to σχχare 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 values of thermal average cross section, hσvi = 3 × 10−26 cm3s−1 and hσvi = 3 × 10−27 cm3s−1 are used for our studies. We note that the latter value for hσvi does not contradict with the relic density, since DM annihilation inside the Sun occurs much later than the period of freeze-out.
We take σχpSD= 10−41 cm2 and 10−43cm2 for SD interaction, and take σχpSI = 10−44cm2 and 10−45cm2for SI interaction. We stress that σSDχp = 10−41cm2 is below the lowest value of IceCube bound σχpSD∼ 10−40cm2at m
χ∼ 300 GeV [25]. For SI interaction, σSIχp= 10−44cm2
is below the LUX bound for mχ < 8 GeV, while σχpSI = 10−45 cm2 is below the LUX bound
for mχ < 20 GeV [39]. We find that cascade events provide better sensitivities to DM
JCAP10(2014)049
becomes better for smaller annihilation cross section hσvi for a fixed σχp, as noted in earlier
works [24,26] which neglect both Ceand Cse. This is evident from eq. (2.16) since R increases
as Cadecreases. It is instructive to take the limit R 1 such that ΓA→ (CcCa)R/2(Ca+Cse)
for Cs> Ce. It is easily seen that ΓA is inversely proportional to Ca (in the mass range that
Cse is negligible) and is independent of Cc. In other words, only Cs and Ca determine the
annihilation rate (we are in the region that Ce is suppressed as compared to Cs). We also
see that the sensitivity to σχχ does become significantly worse as mχ → 4 GeV. This is the
critical mχ below which the DM evaporations from the Sun is important.
4 Conclusion
We have studied the time evolution of DM number trapped inside the Sun with DM self-interaction considered. We have focused on the low mχ 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 σχχ− σSD (SI)χp plane for significant
enhancement on trapped DM number (R > 1) is identified. The parameter space for R > 1 becomes larger for smaller mχ. For Cs< Ce, the condition R > 1 leads to the suppression of
neutrino flux, since the first term on the right hand side of eq. (2.16) is negative. We have proposed to study σχχ with the future IceCube-PINGU detector where the energy threshold
can be lowered down to 1 GeV. We considered cascade and track events resulting from neutrino flux induced by DM annihilation channels χχ → ν ¯ν and χχ → τ+τ− inside the Sun. We found that cascade events always provide better sensitivity to σχχ. The sensitivity
to σχχ is also improved with a smaller DM annihilation cross section hσvi.
Acknowledgments
We thank S. Palomares-Ruiz for a very useful comment. CSC is supported by the National Center for Theoretical Sciences, Taiwan; FFL, GLL, and YHL are supported by Ministry of Science and Technology, Taiwan under Grant No. 102-2112-M-009-017.
A DM self-interaction induced evaporation
The derivation of DM self-interaction induced evaporation is similar to the usual nucleon induced evaporation [8–11]. One simply makes the parameter replacements
mN → mχ, TN → Tχ,
where Tχ is the DM temperature inside the Earth. The DM velocity distribution is
approxi-mated by Maxwell-Boltzmann distribution given by
f (w) = 4 √ π mχ 2Tχ 3/2 nχw2exp −mχw 2 2Tχ ,
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
JCAP10(2014)049
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
R+(w → v)dv = √2 πnχσχχ v we −κ2(v2−w2) χ(β−, β+)dv, (A.1)
where we have summed up the target DM according to the thermal distribution f described
above, β±= ±κw with κ = rm χ 2Tχ , and χ(a, b) ≡ Z b a due−u2 = √ π 2 [erf(b) − erf(a)].
Therefore the DM-DM scattering rate with v greater than the escape velocity vesc is given
by the integral
Ω+vesc(w) = Z ∞
vesc
R+(w → v0)dv0. (A.2)
Carrying out the integral yields Ω+vesc(w) = √2 π nχσχχ w Tχ mχ exp mχ(v 2 esc− w2) 2Tχ χ(β−, β+). (A.3)
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:
dCse
dV = Z vesc
0
f (w)Ω+vesc(w)dw. (A.4)
The DM number density inside the Sun is determined by nχ(r) = n0exp −mχφ(r) Tχ ,
where n0 is the density at the solar core, and φ(r) is the solar gravitational potential with
respect to the core so that
φ(r) = Z r 0 GM (r0) r02 dr 0 , with G the Newton gravitational constant, M (r) = 4π
Rr
0 r
02ρ
(r0)dr0 the solar mass
en-closed within radius r, and ρ (r) the solar density. Thus, the integration in eq. (A.4) can be
performed dCse dV = 4 √ π rm χ 2Tχ n20σχχ mχ exp −2mχφ(r) Tχ exp −Eesc(r) Tχ ˜ K(mχ) (A.5)
JCAP10(2014)049
where ˜ K(mχ) = r Eesc(r)Tχ π exp −Eesc(r) Tχ + Eesc(r) − Tχ 2 erf s Eesc(r) Tχ ! , (A.6)with Eesc(r) the escape energy at the position r defined by
Eesc(r) =
1 2mχv
2
esc(r). (A.7)
The escape velocity vesc(r) is related to the gravitational potential by vesc(r) ≡
p2[φ(∞) − φ(r)]. Finally, the self-interaction induced evaporation rate can be evaluated through the following:
Cse= R dCdVsed 3r R nχ(r)d3r 2. (A.8) References
[1] G. Steigman, C.L. Sarazin, H. Quintana and J. Faulkner, Dynamical interactions and
astrophysical effects of stable heavy neutrinos,Astron. J. 83 (1978) 1050[INSPIRE].
[2] D.N. Spergel and W.H. Press, Effect of hypothetical, weakly interacting, massive particles on
energy transport in the solar interior,Astrophys. J. 294 (1985) 663[INSPIRE].
[3] W.H. Press and D.N. Spergel, Capture by the sun of a galactic population of weakly interacting
massive particles,Astrophys. J. 296 (1985) 679[INSPIRE].
[4] J. Faulkner and R.L. Gilliland, Weakly interacting, massive particles and the solar neutrino
flux,Astrophys. J. 299 (1985) 994[INSPIRE].
[5] K. Griest and D. Seckel, Cosmic Asymmetry, Neutrinos and the Sun,Nucl. Phys. B 283
(1987) 681[Erratum ibid. B 296 (1988) 1034] [INSPIRE].
[6] G. Jungman, M. Kamionkowski and K. Griest, Supersymmetric dark matter,Phys. Rept. 267
(1996) 195[hep-ph/9506380] [INSPIRE].
[7] G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints,
Phys. Rept. 405 (2005) 279[hep-ph/0404175] [INSPIRE].
[8] A. Gould, WIMP Distribution in and Evaporation From the Sun,Astrophys. J. 321 (1987) 560
[INSPIRE].
[9] R. Kappl and M.W. Winkler, New Limits on Dark Matter from Super-Kamiokande,Nucl.
Phys. B 850 (2011) 505[arXiv:1104.0679] [INSPIRE].
[10] N. Bernal, J. Mart´ın-Albo and S. Palomares-Ruiz, A novel way of constraining WIMPs
annihilations in the Sun: MeV neutrinos,JCAP 08 (2013) 011 [arXiv:1208.0834] [INSPIRE].
[11] G. Busoni, A. De Simone and W.-C. Huang, On the Minimum Dark Matter Mass Testable by
Neutrinos from the Sun, JCAP 07 (2013) 010[arXiv:1305.1817] [INSPIRE].
[12] W.J.G. de Blok, The Core-Cusp Problem,Adv. Astron. 2010 (2010) 789293
[arXiv:0910.3538] [INSPIRE].
[13] B. Moore, Evidence against dissipationless dark matter from observations of galaxy haloes,
Nature 370 (1994) 629[INSPIRE].
[14] R.A. Flores and J.R. Primack, Observational and theoretical constraints on singular dark
JCAP10(2014)049
[15] J.F. Navarro, C.S. Frenk and S.D.M. White, A Universal density profile from hierarchical
clustering,Astrophys. J. 490 (1997) 493[astro-ph/9611107] [INSPIRE].
[16] M.G. Walker and J. Penarrubia, A Method for Measuring (Slopes of ) the Mass Profiles of
Dwarf Spheroidal Galaxies,Astrophys. J. 742 (2011) 20[arXiv:1108.2404] [INSPIRE].
[17] S.-H. Oh, W.J.G. de Blok, E. Brinks, F. Walter and R.C. Kennicutt Jr., Dark and luminous
matter in THINGS dwarf galaxies,Astron. J. 141 (2011) 193[arXiv:1011.0899] [INSPIRE].
[18] A.V. Maccio, S. Paduroiu, D. Anderhalden, A. Schneider and B. Moore, Cores in warm dark
matter haloes: a Catch 22 problem,arXiv:1202.1282[INSPIRE].
[19] D.N. Spergel and P.J. Steinhardt, Observational evidence for selfinteracting cold dark matter,
Phys. Rev. Lett. 84 (2000) 3760[astro-ph/9909386] [INSPIRE].
[20] S.W. Randall, M. Markevitch, D. Clowe, A.H. Gonzalez and M. Bradac, Constraints on the Self-Interaction Cross-Section of Dark Matter from Numerical Simulations of the Merging
Galaxy Cluster 1E 0657-56,Astrophys. J. 679 (2008) 1173[arXiv:0704.0261] [INSPIRE].
[21] M. Rocha, A.H.G. Peter, J.S. Bullock, M. Kaplinghat, S. Garrison-Kimmel et al., Cosmological Simulations with Self-Interacting Dark Matter I: Constant Density Cores and Substructure,
Mon. Not. Roy. Astron. Soc. 430 (2013) 81[arXiv:1208.3025] [INSPIRE].
[22] A.H.G. Peter, M. Rocha, J.S. Bullock and M. Kaplinghat, Cosmological Simulations with
Self-Interacting Dark Matter II: Halo Shapes vs. Observations,arXiv:1208.3026[INSPIRE].
[23] J. Zavala, M. Vogelsberger and M.G. Walker, Constraining Self-Interacting Dark Matter with
the Milky Way’s dwarf spheroidals,Mon. Not. Roy. Astron. Soc. Lett. 431 (2013) L20
[arXiv:1211.6426] [INSPIRE].
[24] I.F.M. Albuquerque, C. P´erez de Los Heros and D.S. Robertson, Constraints on self interacting
dark matter from IceCube results,JCAP 02 (2014) 047[arXiv:1312.0797] [INSPIRE].
[25] IceCube collaboration, M.G. Aartsen et al., Search for dark matter annihilations in the Sun
with the 79-string IceCube detector,Phys. Rev. Lett. 110 (2013) 131302[arXiv:1212.4097]
[INSPIRE].
[26] A.R. Zentner, High-Energy Neutrinos From Dark Matter Particle Self-Capture Within the Sun,
Phys. Rev. D 80 (2009) 063501[arXiv:0907.3448] [INSPIRE].
[27] P. Cushman, C. Galbiati, D.N. McKinsey, H. Robertson, T.M.P. Tait et al., Working Group
Report: WIMP Dark Matter Direct Detection,arXiv:1310.8327[INSPIRE].
[28] IceCube-PINGU collaboration, M.G. Aartsen et al., Letter of Intent: The Precision IceCube
Next Generation Upgrade (PINGU),arXiv:1401.2046[INSPIRE].
[29] J.L. Feng, S. Ritz, J.J. Beatty, J. Buckley, D.F. Cowen et al., Planning the Future of U.S.
Particle Physics (Snowmass 2013): Chapter 4: Cosmic Frontier,arXiv:1401.6085[INSPIRE].
[30] M. Cirelli, G. Corcella, A. Hektor, G. Hutsi, M. Kadastik et al., PPPC 4 DM ID: A Poor
Particle Physicist Cookbook for Dark Matter Indirect Detection,JCAP 03 (2011) 051[Erratum
ibid. 1210 (2012) E01] [arXiv:1012.4515] [INSPIRE].
[31] C. Rott, J. Siegal-Gaskins and J.F. Beacom, New Sensitivity to Solar WIMP Annihilation
using Low-Energy Neutrinos,Phys. Rev. D 88 (2013) 055005[arXiv:1208.0827] [INSPIRE].
[32] J.R. Ellis and R.A. Flores, Realistic Predictions for the Detection of Supersymmetric Dark
Matter,Nucl. Phys. B 307 (1988) 883[INSPIRE].
[33] J. Engel and P. Vogel, Spin dependent cross-sections of weakly interacting massive particles on
nuclei,Phys. Rev. D 40 (1989) 3132[INSPIRE].
[34] J. Engel, S. Pittel and P. Vogel, Nuclear physics of dark matter detection,Int. J. Mod. Phys. E
JCAP10(2014)049
[35] P.C. Divari, T.S. Kosmas, J.D. Vergados and L.D. Skouras, Shell model calculations for light
supersymmetric particle scattering off light nuclei, Phys. Rev. C 61 (2000) 054612[INSPIRE].
[36] A. Gould, Cosmological density of WIMPs from solar and terrestrial annihilations,Astrophys.
J. 388 (1992) 338[INSPIRE].
[37] COUPP collaboration, E. Behnke et al., First Dark Matter Search Results from a 4-kg CF3I
Bubble Chamber Operated in a Deep Underground Site,Phys. Rev. D 86 (2012) 052001
[arXiv:1204.3094] [INSPIRE].
[38] M. Felizardo, T.A. Girard, T. Morlat, A.C. Fernandes, A.R. Ramos et al., Final Analysis and
Results of the Phase II SIMPLE Dark Matter Search,Phys. Rev. Lett. 108 (2012) 201302
[arXiv:1106.3014] [INSPIRE].
[39] LUX collaboration, D.S. Akerib et al., First results from the LUX dark matter experiment at
the Sanford Underground Research Facility,Phys. Rev. Lett. 112 (2014) 091303
[arXiv:1310.8214] [INSPIRE].
[40] M. Blennow, J. Edsjo and T. Ohlsson, Neutrinos from WIMP annihilations using a full
three-flavor Monte Carlo,JCAP 01 (2008) 021[arXiv:0709.3898] [INSPIRE].
[41] R. Gandhi, C. Quigg, M.H. Reno and I. Sarcevic, Ultrahigh-energy neutrino interactions,
Astropart. Phys. 5 (1996) 81[hep-ph/9512364] [INSPIRE].
[42] J. Hisano, K. Nakayama and M.J.S. Yang, Upward muon signals at neutrino detectors as a
probe of dark matter properties,Phys. Lett. B 678 (2009) 101[arXiv:0905.2075] [INSPIRE].
[43] A.E. Erkoca, M.H. Reno and I. Sarcevic, Muon Fluxes From Dark Matter Annihilation,Phys.
Rev. D 80 (2009) 043514[arXiv:0906.4364] [INSPIRE].
[44] J. Kumar, J.G. Learned, M. Sakai and S. Smith, Dark Matter Detection With Electron
Neutrinos in Liquid Scintillation Detectors,Phys. Rev. D 84 (2011) 036007
[arXiv:1103.3270] [INSPIRE].
[45] IceCube collaboration, M.G. Aartsen et al., Measurement of the Atmospheric νe flux in
IceCube,Phys. Rev. Lett. 110 (2013) 151105[arXiv:1212.4760] [INSPIRE].
[46] M. Honda, T. Kajita, K. Kasahara, S. Midorikawa and T. Sanuki, Calculation of atmospheric
neutrino flux using the interaction model calibrated with atmospheric muon data,Phys. Rev. D
