Asymmetric capture of Dirac dark matter by the Sun
AAsymmetric capture of Dirac dark matter by the Sun
Mattias Blennow ∗ and Stefan Clementz † Department of Theoretical Physics,School of Engineering Sciences, KTH Royal Institute of Technology,Albanova University Center, 106 91, Stockholm, Sweden
Abstract
Current problems with the solar model may be alleviated if a significant amount of dark matterfrom the galactic halo is captured in the Sun. We discuss the capture process in the case where thedark matter is a Dirac fermion and the background halo consists of equal amounts of dark matterand anti-dark matter. By considering the case where dark matter and anti-dark matter havedifferent cross sections on solar nuclei as well as the case where the capture process is considered tobe a Poisson process, we find that a significant asymmetry between the captured dark particles andanti-particles is possible even for an annihilation cross section in the range expected for thermalrelic dark matter. Since the captured number of particles are competitive with asymmetric darkmatter models in a large range of parameter space, one may expect solar physics to be alteredby the capture of Dirac dark matter. It is thus possible that solutions to the solar compositionproblem may be searched for in these type of models. ∗ Electronic address: [email protected] † Electronic address: [email protected] a r X i v : . [ h e p - ph ] A ug . INTRODUCTION Over the past few decades, a large amount of evidence in support of the existence of darkmatter (DM) has been assembled, see e.g., refs [1, 2]. To describe the properties of DMhas become one of the main issues not only in cosmology but also in particle physics, sincethe Standard Model leaves no room for its existence, which means extending it is necessary.Some of the most studied particle candidates of DM are weakly interacting massive particlessuch as the lightest neutral supersymmetric particle, the neutralino (an in-depth review canbe found in ref [3]).If DM interacts with regular matter, it may scatter in astrophysical bodies such as theSun and become gravitationally bound. Over time, this can lead to a large accumulation ofDM, the effect of which used to be one of the proposed explanations of the solar neutrinoproblem [4]. This idea was later discarded in favour of neutrino flavour conversion by theneutral current phase of the SNO experiment [5]. Since the downward revision of heavyelements in the standard solar model [6], theoretical predictions and observations of helio-seismology do not match and the Sun now faces the solar composition problem [7]. Again, aproposed solution is DM trapped in the Sun due to its interaction with regular matter. Theidea is that DM particles collide with nuclei in the Sun, lose enough energy to become grav-itationally bound and to eventually settle in the solar core after additional scattering. Oncein the core, DM can scatter off of the thermal distribution of nuclei and gain some energywhich is then lost by scattering in the outer regions of the Sun, the result of which is a lowercore temperature [8]. This would ultimately affect helioseismology and possibly provide acure for the solar composition problem. The shift in temperature of the solar core would alsochange the solar neutrino fluxes that are observed by experiments, in particular the B fluxwhich varies as T [9]. The effects of DM captured by the Sun has been extensively studiedfor various DM models [10–16] and also in other stars [17–21]. Annihilating DM can alsoprovide a channel for indirect detection of DM as a flux of high energy neutrinos from theseannihilations might be detectable on Earth. Such signals are being searched for by neutrinotelescopes [22, 23]. In the case where the DM has self-interactions, DM particles alreadycaptured by the Sun provide a possibility of DM self-capture, which could lead to higherconcentrations of DM inside the Sun. If DM annihilations produce high-energy neutrinos,2his would increase the number of annihilations and thus the signal in neutrino telescopescould be enhanced [24]. Energy injection into the Sun by the DM annihilation products hasbeen studied, but requires the halo density to be many orders of magnitude higher than thelocal density for the star to be affected [25].The models which achieve the highest numbers of captured DM particles in the Sun aremodels in which annihilation is suppressed or does not occur at all, the latter of which is thecase in asymmetric DM (ADM) models, reviews of which can be found in [26, 27]. It wasproposed in [12] that self-interacting ADM may alleviate the solar composition problem butit was shown in [13, 14] that ADM models are incapable of doing so with the cross-sectionsallowed by direct detection experiments. Recently, there have been improvements of thestandard solar model using DM has been achieved with long range interactions [28] andmomentum-dependent cross sections [29]. Common for the models described above is thatthey contain only one species of DM particles. In this paper we will study the accumulationof DM in the Sun under the assumption that it is a self-interacting Dirac fermion and thatthe DM halo is composed of equal amounts of DM and anti-DM particles. This opens up apossibility for an asymmetry in the captured number of DM and anti-DM to occur, eitherdue to different capture cross sections for DM and anti-DM on solar nuclei or due to randomfluctuations in the capture process. Moreover, the more abundant species of captured DMcan not fully annihilate with its counterpart which allows for large number of capturedparticles even for an annihilation cross section expected for thermal relic DM. If the size ofthe population of captured DM of the models considered here can compete with the ADMmodels, DM of the standard freeze-out scenario may alter solar observables.The remainder of this paper is organised as follows: In section II, we present the frame-work that governs the amount of DM and anti-DM present in the Sun and discuss thebehaviour of the DM to anti-DM asymmetry in the limiting cases. Then, in section III, weapply our framework to the case of the Sun and in section IV we present our results. Finally,in section V, we summarise our results and present our conclusions.3 I. ACCUMULATION AND THE ASYMMETRY
We model the total number of DM particles in the Sun N and the total number ofanti-DM particles in the Sun ¯ N using the coupled system of first order differential equations˙ N = c + CN + ¯ C ¯ N − Γ N ¯ N , (1)˙¯ N = ¯ c + ¯ CN + C ¯ N − Γ N ¯ N . (2)Here, c and ¯ c are the capture rates on solar nuclei for DM and for anti-DM, respectively,while C and ¯ C will depend on the self-capture rates of DM and anti-DM and on the ejectionrates of DM and anti-DM that is already trapped by incoming particles from the DM halo.The DM–anti-DM annihilation rate is given by Γ. In principle, C and ¯ C also includeevaporation effects, but it has been shown that this is negligible for DM particles with amass (cid:38) N − ¯ N . It evolves according to ˙∆ = d + D ∆ , (3)where we have defined d = c − ¯ c and D = C − ¯ C . Assuming negligible amounts of both DMand anti-DM at the birth of the Sun, ∆(0) = 0, we find∆ = (cid:90) t de D ( t − τ ) dτ. (4)An important note is that the absolute value of ∆ also represents the minimum amount ofdark particles (DM or anti-DM) present in the Sun. The sum of DM and anti-DM, N + ¯ N ,will at any time always be larger or equal to the asymmetry.Another important note is that there is a geometric limit for the self-capture rates. When N and ¯ N are large enough, our approach is no longer valid. When DM and anti-DM istrapped inside the Sun, they will accumulate in the center roughly inside a sphere of radius r χ (derived in the next section). When σ χχ N + σ χ ¯ χ ¯ N is larger than the cross-sectional area ofthe sphere, πr χ , every incoming DM particle will scatter off either trapped DM or anti-DM.When the distributions of trapped DM and anti-DM are identical, the fraction of collisionsof DM on DM is f χχ = σ χχ N/ ( σ χχ N + σ χ ¯ χ ¯ N ) while the fraction of collisions on anti-DMis given by f χ ¯ χ = σ χ ¯ χ ¯ N / ( σ χχ N + σ χ ¯ χ ¯ N ). The same argument holds for incoming anti-DM4xcept the total cross section σ χ ¯ χ N + σ χχ ¯ N can not exceed πr χ . In this case, the fraction ofcollisions on DM is given by f χ ¯ χ with all N and ¯ N interchanged. Similarly, the fraction ofcollisions on anti-DM is given by f χχ , again with N and ¯ N interchanged. When this occurs,the equation for ∆ will fail since eqs. (1) and (2) must be corrected in order to take thegeometric limit into account. II.1. Intrinsic different capture rates
When the scattering cross section of DM and anti-DM on solar nuclei are different, c and¯ c are different and thus d (cid:54) = 0. We choose c to be larger than ¯ c (identifying anti-DM as thespecies with lower capture probability on solar matter). The solution for ∆ is∆ = dD ( e Dt − , (5)which has three interesting limits:∆ = dt if | Dt | (cid:28) − dD if | Dt | (cid:29) , D < dD e Dt if | Dt | (cid:29) , D > Dt is small, self-capture is negligible and the asymmetry will be proportional tothe difference in the capture rates. When D <
0, the system eventually stabilizes, sincethe additional capture of anti-DM by already captured DM balances the difference in thecapture rate on normal matter. For
D >
0, DM captures itself at a larger rate than anti-DM.Once this process becomes dominant, it leads to an exponential increase in the amount ofDM captured in the Sun.
II.2. Stochastically induced difference
When the capture rates are equal ( c = ¯ c ), the amount of DM might at some point belarger than the amount of anti-DM simply due to random variations in the capture processwhich can be modelled by adding a white noise signal δ c to the capture rates c = c + δ c ( t ) , (cid:104) δ c ( t ) (cid:105) = 0 , (cid:104) δ c ( t ) δ c ( τ ) (cid:105) = sδ ( t − τ ) . (7)5he white noise is normalized such that the expected number of captured DM particles andits variation matches those of a Poisson distribution, i.e., (cid:104) n (cid:105) − (cid:104) n (cid:105) = (cid:104) n (cid:105) . We find that (cid:104) n (cid:105) = (cid:28)(cid:90) t c + δ c ( τ ) dτ (cid:29) = c t (8) (cid:104) n (cid:105) = (cid:42)(cid:18)(cid:90) t c + δ c ( τ ) dτ (cid:19) (cid:43) = c t + st (9)and hence s = c . Using the same argument for the capture rate of anti-DM with a whitenoise signal δ ¯ c , we find d = δ d ( t ) = δ c ( t ) − δ ¯ c ( t ) . (10)Since δ c and δ ¯ c are independent, δ d has the properties (cid:104) δ d ( t ) (cid:105) = 0 , (cid:104) δ d ( t ) δ d ( τ ) (cid:105) = 2 c δ ( t − τ ) (11)The expectation value of the asymmetry ∆ is zero, which should be expected since theprobability of having an over-abundance of DM to anti-DM must be the same as that ofhaving an over-abundance of anti-DM due to symmetry. To estimate the typical magnitudeof the asymmetry, we can study the standard deviation of the stochastic variable ∆, givenby ˜∆ = (cid:112) (cid:104) ∆ (cid:105) . We find that˜∆ = (cid:90) t (cid:90) t e D (2 t − τ − σ ) (cid:104) δ d ( τ ) δ d ( σ ) (cid:105) dτ dσ = c D ( e Dt − . (12)Thus, the limiting behaviour is similar to the case for intrinsic different capture rates˜∆ = √ c t if | Dt | (cid:28) (cid:112) − c D if | Dt | (cid:29) , D < (cid:112) c D e Dt if | Dt | (cid:29) , D > . (13)The major difference is that the short time limit | Dt | (cid:28) t rather than linearly and that the coefficients arerelated to c and D by a square root. For small Dt , we note that this result is preciselywhat would be expected from the difference between two Poisson distributions of expectationvalue c t while an equilibrium or an exponential growth occur for strong self-capture.6 II. DM AND ANTI-DM SELF-CAPTURE RATES
For a self-interacting model of DM, the total capture of DM in the Sun will be the sumof a capture rate due to interactions with solar nuclei, a term proportional to the alreadycaptured DM and a similar term proportional to the number of captured anti-DM particles.The capture of anti-DM is completely analogous to DM capture although the rates maydiffer depending on the various scattering cross sections. The specific formulas to computethe capture rates and ejection rates are presented in the appendix but the complexity ofthe capture rates C and ¯ C requires some discussion. In what follows, we assume that thetime it takes for a captured DM particle to fall into thermal equilibrium in the solar core isnegligible.Generally, the formula for the capture rates of DM by DM and anti-DM as well as theejection rates are given by: C ( σ ) = (cid:90) R (cid:12) πr (cid:90) ∞ f ( u ) u w Ω du dr. (14)The factor Ω is the rate at which a particle with velocity w will scatter at radius r andcontain information on the probability that the incoming particle is captured and whetherthe target particle is ejected or not.The rate of capture of an incoming particle without ejecting the target particle is givenby C s ( σ ) = (cid:90) R (cid:12) πr (cid:90) v esc f ( u ) u σn ( r )( v esc − u ) du dr. (15)Depending on the collision, the scattering cross section σ is either σ χχ or σ χ ¯ χ . The velocityof the particle before it falls into the gravitational potential is u and the velocity at radius r is then w = (cid:112) u + v esc where v esc is the escape velocity.The Knudsen number is a measure of the distance DM particles travel on average betweencollisions and is given by K = l (0) r χ , l ( r ) = (cid:32)(cid:88) i σ i n i ( r ) (cid:33) − . (16)The parameter r χ is a length scale that describes the size of the distribution. For K (cid:29) n ISO ( r ) = n ISO (0) e − φ ( r ) /kT , (17)7here n ISO (0) is the normalization constant, T is the temperature of the distribution and φ ( r ) is the gravitational potential energy at radius r from the core which is calculated from φ ( r ) = (cid:90) r Gm χ M ( r (cid:48) ) r (cid:48) dr (cid:48) . (18)Here, M ( r ) is the total mass inside the sphere of radius r around the solar core; M ( r ) = (cid:90) r πρ ( r (cid:48) ) r (cid:48) dr (cid:48) . (19)In the isothermal case, the normalization constant is n ISO (0) = π − r − χ N and the length scale r χ is defined as r χ = 3 kT c / πGρ c m χ assuming a constant temperature T ( r ) = T c and density ρ ( r ) = ρ c with ρ c and T c being the density and temperature in the solar center. Indeed, fora DM particle with mass m χ = 5 GeV, the length scale r χ ∼ .
05 R (cid:12) which shows that thevast majority of captured DM particles are concentrated in a very small volume in the centerof the Sun. In the case of large scattering cross sections, K (cid:28)
1, the particles scatter so oftenthat they will be in local thermal equilibrium with the surrounding nuclei. The distributionfor this case was derived in [8] which introduces a dependence on the temperature gradient.However, for the cross sections considered in this paper ( σ SD ≤ − cm ), K (cid:38)
90 andso we will work with the isothermal distribution. Defining (cid:15) ( r ) = n ( r ) /N , the distributioncan be written n ( r ) = (cid:15) ( r ) N . For anti-DM, the radial distribution is taken to be the sameexcept N → ¯ N .The ejection rate of DM captured by in the Sun by collisions with DM or anti-DM fromthe halo is given by C eject ( σ ) = (cid:90) R (cid:12) πr (cid:90) ∞ f ( u ) u u σN (cid:15) ( r ) dudr. (20)We must also take into account that while a dark matter particle is being ejected from theSun, it is also likely that the particle from the halo is captured by the same process. The ratefor this exchange occurring is C exch = C eject − C eject 2 , where C eject 2 is the rate of ejections inwhich both the incoming halo particle and the particle from the Sun are ejected, given by C eject 2 = (cid:90) R (cid:12) πr (cid:90) ∞ v esc f ( u ) u ( u − v ) σN (cid:15) ( r ) dudr. (21)The possible cases depending on velocity for capture and ejection are shown schematicallyin Fig. 1 along with the velocity distribution of the standard halo model.8 IG. 1. The behaviour of the functions w Ω for different capture scenarios. The horizontal linerepresents the escape velocity v esc in the center of the Sun, where most of the DM resides. Thevertical size of the colored regions represents the quantity which needs to be integrated along withthe distribution f ( u ) /u in order to yield the capture rates. The black curve shows shape of f ( u ) /u at v (cid:12) = 220 km/s for reference. As can be seen from this figure, the escape velocity will generally be so large that weexpect that the self-capture will be dominant with some contribution from halo particlesbeing captured while ejecting the target particle.Summarizing this, the capture rates C and ¯ C relevant for the evolution of the dark matterand anti-dark matter numbers in the Sun can be written as C = C s ( σ χχ ) − C eject ( σ χ ¯ χ ) − C eject 2 ( σ χχ ) , (22)¯ C = C s ( σ χ ¯ χ ) + C exch ( σ χ ¯ χ ) . (23)Here, the single self-ejection events do not appear in C , as the net change in the DM numberin the Sun is zero for these events, but the full ejection induced by the opposite species fromthe halo must be taken into account as the capture of the halo particle does not compensatefor the ejected one. For the capture of DM on anti-DM, the relevant quantities are thecapture without ejection and the ejection of the target particle while capturing the haloparticle. 9he annihilation rate is computed as [32]Γ = (cid:104) σv (cid:105) (cid:90) R (cid:12) πr (cid:15) ( r ) dr, (24)where (cid:104) σv (cid:105) is the thermally averaged annihilation cross section. As long as R (cid:12) (cid:29) r χ , theupper limit of the integral can be set to ∞ rather than R (cid:12) and the annihilation rate evaluatesto Γ = (cid:104) σv (cid:105) π ) / r χ . (25) IV. RESULTS
In the following, we will make some explicit assumptions in order to estimate the effectsdescribed in the previous sections. The velocity distribution f ( u ) of the halo is assumed tobe a standard Maxwell-Boltzmann distribution shifted to the solar frame moving throughthe halo at v (cid:12) = 220 km/s. It can be expressed as [4] f χ ( u ) = n χ u √ πv (cid:12) (cid:32) e − ( u − v (cid:12) )2 v (cid:12) − e − ( u + v (cid:12) )2 v (cid:12) (cid:33) . (26)It is assumed that the DM and anti-DM components in the halo are identical and that thedensity of each are equal. They will then each have a density ρ χ = ρ ¯ χ = 0 .
15 GeV cm − ,which is equal to half the total local DM density of 0 . − [33, 34]. The numberdensity of DM and anti-DM is therefore n χ = ρ χ /m χ .The dark matter is assumed to scatter with regular matter with velocity independent spin-independent (SI) and/or spin-dependent (SD) cross sections through effective operators.For the case of SD capture in the Sun, we are interested in the bounds on the SD DM-proton cross section. Limits on these cross sections have been set in various direct detectionexperiments [35–40]. In the DM mass range 10 − σ SI (cid:46) − cm [35]. For smaller DM masses, the limits on the SI cross section weakenssignificantly. For a 5 GeV DM particle, σ SI (cid:46) − cm . The limits on the SD cross sectionin the mass range 10 − σ SD (cid:46) − cm [37]. For a 5 GeV particle, the boundis slightly reduced to σ SD (cid:46) − cm [40].Limits on the self-interaction of DM comes from astrophysical sources. When galaxyclusters collide, drag forces acting on the gas while the DM passes through unhindered10 −10 −5 m χ [GeV] C / σ [ s − m − ] C eject 2 / σ C exch / σ C s / σ FIG. 2. The self-capture rate C s /σ (black line), the exchange rate C exch /σ (red line) and theejection rate C eject 2 /σ (blue line) as a function of mass. would produce an offset in the mass and gas distribution of the clusters, the size of whichcan be used to put upper limits on the self-interaction of DM. In [41], one such collisionwas analysed and and set an upper limit on the self-interacting cross section of σ χχ /m χ (cid:46) · − cm /GeV.The relic abundance of DM has been precisely derived from WMAP [42] and Planck [43]experimental data. The thermally averaged annihilation cross section can be related to thisrelic abundance by solving the Boltzmann equation which is done in e.g., ref [3] and is heretaken to be 3 · − cm /s.As a model of the Sun, the AGSS09 solar model [44] is chosen. It contains the mass andradial distribution of elements up to Ni. The solar age is taken to be t (cid:12) = 4 . m χ = 5 GeV is calculated to be at most10 s − for a SD cross section of 10 − cm and 2 . · s − for a SI cross section of10 − cm . For a DM particle of mass 10 GeV, the bounds push SI capture down by fourorders of magnitude even though for a fixed SI cross section, the capture rate is only reducedat the percent level. Thus, the SD cross section allows for higher capture rates for all massesin the range 5-1000 GeV.Fig. 2 shows the values of C s , C exch and C eject 2 as a function of DM masses between11 and 1000 GeV. While C exch is roughly 20 times smaller than C s , C eject 2 is almost 15orders of magnitude lower. This is not surprising as the escape velocity is very large wherethe DM resides and particles with a large velocity in the halo are exponentially suppressed(cf. fig. 1). In the case of the Sun, C eject 2 may therefore be neglected, since C s and/or C exch will be completely dominant depending on the relative sizes of σ χχ and σ χ ¯ χ . The capturerates can now be written as C (cid:39) C s ( σ χχ ) − C exch ( σ χ ¯ χ ) , (27)¯ C (cid:39) C s ( σ χ ¯ χ ) + C exch ( σ χ ¯ χ ) , (28)by using that C eject ( σ ) (cid:39) C exch ( σ ) and one finds that D takes the form D = C − ¯ C = C s ( σ χχ ) − C s ( σ χ ¯ χ ) − C exch ( σ χ ¯ χ ) . (29)Note that even if the scattering cross sections σ χχ and σ χ ¯ χ are equal, D will be non-zero.This is due to the fact that ejection of the more dominant species occurs at a larger rate. IV.1. Asymmetric capture and ∆ Figure 3 shows the size of ∆ over time for a DM mass of 5 GeV, a capture rate of DMon solar nuclei at c = 10 s − and a capture rate of anti-DM on solar nuclei of ¯ c = 0 s − ,and various different self-scattering cross sections. It can be seen that, when the capture ofanti-DM occurs primarily by DM (negative D ), the asymmetry is smaller than if there wouldbe no self-capture at all or the difference between σ χχ and σ χ ¯ χ is such that D is small. Onthe other hand, the exponential growth is apparent when σ χχ is larger than σ χ ¯ χ as to make D positive. Since ∆ is definitely smaller or equal to N , the geometric limit of self-capturehas definitely been reached once σ χχ ∆ > πr χ . In the case of fig. 3, a redefinition of C and¯ C would have already been necessary for the two cases with D > c = c/
2, the total number of captured particles in an ADM model, N Asym , is at most an order of magnitude larger than the number of captured particles in aDirac DM model in a large region of the σ χχ − σ χ ¯ χ plane. Note that in the case that ¯ c = 012 .1 110 ∆ Time [Gyrs]
FIG. 3. The evolution of the asymmetry ∆ over time for a 5 GeV DM particle with a solar capturerate of c = 10 s − and ¯ c = 0 and various σ χχ and σ χ ¯ χ . Blue line: σ χχ = 0 and σ χ ¯ χ = 3 · − cm ,red line: σ χχ = 0 and σ χ ¯ χ = 0, solid black line: σ χχ = 3 · − cm and σ χ ¯ χ = 0, dashed blackline: σ χχ = 4 · − cm and σ χ ¯ χ = 0. The purple line shows the geometric self-capture limit for σ χχ = 2 · − cm and σ χ ¯ χ = 0 σ χχ [cm ] σ χ ¯ χ [ c m ] ¯ c = 0 −25 −24 −23 −25 −24 −23 N D M / N A s y m σ χχ [cm ] σ χ ¯ χ [ c m ] ¯ c = c / −25 −24 −23 −25 −24 −23 N D M / N A s y m FIG. 4. The total number of captured DM and anti-DM particles of mass m χ = 5 GeV with across section on solar nuclei σ SD = 10 − cm in the σ χχ − σ χ ¯ χ plane normalized to the numbercaptured by an ADM model with the same DM mass and cross section and twice the halo density( ρ χ = 0 . ). The cross section of anti-DM on solar nuclei is 0 in the left plane and σ SD / .1 110 ˜ ∆ Time [Gyrs]
FIG. 5. The evolution of the stochastic asymmetry ˜∆ over the lifetime of the Sun for various σ χχ and σ χ ¯ χ using a 5 GeV DM mass and a capture rate at 10 s − . Blue line: σ χ ¯ χ = 10 − cm and σ χχ = 0, red line: σ χχ = σ χ ¯ χ = 0, solid black line: σ χχ = 2 · − cm and σ χ ¯ χ = 10 − cm ,dashed black line: σ χχ = 2 · − cm and σ χ ¯ χ = 0. and σ χ ¯ χ is small, N Asym is slightly larger than half that of the captured numbers in a Diracmodel. This is due to the fact that self-capture, CN , is limited to a much smaller rate than c . The capture of anti-DM is so small that annihilation barely occur and the capture isdescribed approximately by an ADM model in which the background density of DM ρ χ ishalved. IV.2. Symmetric capture and ˜∆ The case of c = ¯ c implies a simple solution to the steady state of eqs. (1) and (2). Whenthe capture rates are equal, the symmetry of the equations implies that N = ¯ N . Both thecapture of DM and anti-DM will then be given by˙ N = c + ( C + ¯ C ) N − Γ N . (30)14n the steady state, the capture rate is equal to the annihilation rate so that ˙ N = 0 whichplugged into the above gives N ∞ = C + ¯ C
2Γ + (cid:112) ( C + ¯ C ) + 4 c Γ2Γ . (31)The total number of captured particles will then be 2 N ∞ .Fig. 5 shows the evolution of ˜∆ over time for various D with a DM mass of 5 GeVand a capture rate of 10 s − . If Dt is small, the stochastic asymmetry at this point intime would be ˜∆ = 1 . · while increasing D on the negative side will always lead to astochastic asymmetry that is smaller. Considering the case of positive D , the asymmetrywill increase exponentially with D . With the same DM mass and solar capture rate butturning on self-capture using σ χχ = 2 · − cm and σ χ ¯ χ = 10 − cm , the number ofparticles at which equilibrium occurs is N ∞ = 2 . · while the size of ˜∆ is 4 . · .With the same self-scattering cross section and setting σ χ ¯ χ = 0, the stochastic asymmetrybecomes 5 orders of magnitude larger than the expected number of particles which slightlydecrease to N ∞ = 2 . · . Since ˜∆ is defined as the standard deviation of ∆, we may inthis case expect the asymmetry to be large. However, if the actual asymmetry was of thissize, the geometric limit has kicked in and ˜∆ is no longer given by eq. (12). Still, this is anindication that self-capture combined with a stochastically induced asymmetry has lead toa significant accumulation of DM in the Sun. V. SUMMARY AND DISCUSSION
In this paper, we have considered the capture of DM in the Sun under the assumptionthat it is a self-interacting Dirac particle and that the galactic background density consistsof equal amounts of DM and anti-DM. This opens up the possibility that a large differencein the captured amount of each type might occur so that the total number of particles in theSun may continue to grow even though the annihilation cross section is that expected fromstandard thermal relic DM. The initial asymmetry between the number of captured DMand anti-DM particles can occur either due to different scattering cross sections for DM andanti-DM on solar nuclei or due to stochastic fluctuations in the capture process. Any suchasymmetry may then be amplified by self-capture or counter-acted by capture of anti-DM15y DM. The size of the asymmetry is independent of the annihilation rate and an analyticalexpression for its size was derived.When the capture rates of DM and anti-DM are different ( c (cid:54) = ¯ c ), we have the case ofasymmetric capture. If the capture rates of DM by DM is C and the capture rate of DMby anti-DM is ¯ C , we define the difference in these rates as D = C − ¯ C . When D <
0, thecapture of anti-DM is more efficient than DM self-capture which implies that the asymmetry∆ will at some point find an equilibrium. On the other hand, if
D >
0, DM will captureitself more efficiently so that any initial asymmetry will grow exponentially. The asymmetryis found to become large enough to conclude that the geometric bound on the self-captureneeds to be taken into account for a wide range of solar capture rates. This is due to anexponential dependence on the size of the DM on DM and DM on anti-DM capture rateswhen
D > /t (cid:12) . This occurs for a 5 GeV DM particle with a capture rate of 10 s − onsolar nuclei, corresponding to a spin-dependent cross section of 10 − cm , for a D given by σ χχ = 2 · − cm and σ χ ¯ χ = 0, where σ χχ is the DM self-scattering cross section and σ χ ¯ χ isthe scattering cross section for DM on anti-DM. When the capture rate of anti-DM by DMand the capture rate of DM by DM makes D negative, the asymmetry will always be smallerthan in the case when there is no self-capture at all. In any case, the size of the asymmetryimplies that the total amount of DM captured may be large. Taking the geometric limit intoaccount and comparing the number of captured DM and anti-DM particles in the Sun bythose of an ADM model, it is found that in the region of parameter space where σ χχ (cid:38) σ χ ¯ χ ,a sizeable population of captured DM occurs where the numbers are within an order ofmagnitude than those of an ADM model with similar parameters but a twice as large halodensity. Since ADM models may have an impact on solar observables as demonstrated in[14], it is plausible that Dirac models may also have an impact on these. However, in regardto solving the solar composition problem and constraining Dirac DM models, numericalinvestigations such as those found in [13, 14, 29] needs to be performed.When the capture rates on solar nuclei are equal for DM and anti-DM ( c = ¯ c ), thestochastic asymmetry ˜∆, which estimates the typical magnitude of the actual asymmetryinduced by the stochastic variation of c and ¯ c , is always extremely small in comparison tothe total number of particles in the Sun when the self-scattering cross sections σ χχ and σ χ ¯ χ are such that D is small or negative. However, in the case when D is positive, the16xponential dependence of D may bring the stochastic asymmetry to a size several orders ofmagnitude larger than the expected total number of trapped particles at steady-state withno asymmetry. However, this case is an extreme since the scattering cross section σ χχ istaken to be right around the upper bound while σ χ ¯ χ = 0. Increasing σ χ ¯ χ to half that of σ χχ ,the stochastic asymmetry is reduced by over 10 orders of magnitude to a negligible levelcompared to the expected amount for symmetric capture. The window for the asymmetryinduced by stochastic variations for the Sun is very small and requires σ χχ (cid:29) σ χ ¯ χ so itmay not be expected that the solar asymmetry is large. However, the capture rates increaseproportionally to the background density of DM so that larger self-capture rates may beexpected for stars in regions where the background density is larger. Even if the likelihoodthat the Sun has a negligible asymmetry since the background density is small, stars in suchregions may have a D that is several orders of magnitude larger which would increase ˜∆significantly thus affecting the evolution of such stars.In this work, we have neglected the fact that a stochastic capture rate on solar nucleimay imply a stochastic variation of the self-capture rates as well. One such scenario wouldoccur if there are perturbations in the local background density. The investigation of thiscase is left for future work. ACKNOWLEDGMENTS
This work was supported by the G¨oran Gustafsson Foundation.
Appendix A: Capture rates of DM
The capture of dark matter in celestial bodies is a standard calculation, first done by Pressand Spergel [4], later improved and corrected by Gould [45]. Given the velocity distributionof halo dark matter, f ( u ), in the frame of the Sun where u is the velocity very far awaywhere the gravitational potential of the Sun is negligible. The capture rate is given by c = (cid:90) R (cid:12) πr (cid:90) ∞ f ( u ) u w Ω du dr (A1)where w Ω is the rate at which a particle with velocity w at radius r will scatter and loseenough energy to be captured. 17 olar element capture For the SI cross section, Gould found that w Ω i = σ i n i ( r ) 2 E m χ µ µ (cid:34) e − mχu E − e − µµ mχw E (cid:35) θ ( µµ − u w ) (A2)while for the SD cross section w Ω = σ p n H ( r )( w − µ µ u ) θ ( µµ − u w ) . (A3) n i ( r ) is the radial distribution of element i in the Sun. The mass of and scattering crosssection on element i is m i and σ i , respectively. The mass of the DM particle is m χ , µ = m χ m i and µ + = µ . The SI scattering cross section scale as σ i = σ p A i µ i µ p (A4)where A i is the number of nucleons in element i and µ i the reduced mass of element i andthe DM. σ p is the proton scattering cross section and µ p the reduced mass of the DM andthe proton. The more complex formula for SI scattering is due to the form factor whichtakes the nuclear structure of the target into account for larger energy transfers given by | f (∆ E ) | = e − ∆ E/E i (A5)where E i = 3 (cid:126) M i R i , R i = (cid:32) . (cid:18) M i GeV (cid:19) + 0 . (cid:33) fm (A6)and M i is the mass of nuclei i . For hydrogen, this form factor is set to unity. The totalcapture rate for a SI capture rate is then the sum of the capture rate by each individualelement. For SD capture, hydrogen is the only element of importance since there is no A enhancement of the cross sections on other elements and the fraction of elements with spinis completely negligible compared to hydrogen. Self-capture
If DM has a non-zero scattering cross section on other DM and anti-DM particles, itmay also be captured by colliding with other DM and anti-DM particles. A derivation of18elf-capture is given in ref. [24] and we review the result and add three cases for which thetarget particle is ejected. For self-capture, Ω is broken down toΩ = σn ( r ) wP cap . (A7)Here, σ is the DM self-scattering cross section, n ( r ) the radial distribution of already cap-tured DM and P cap the probability that the particle is captured in a collision while not givingthe target particle enough energy to escape the Sun. The projectile and target particles aregravitationally unbound when their kinetic energy is greater than m χ v esc /
2. This meansthat, for capture of a particle without ejecting the target, the energy transfer ∆ E must bein the interval u w < ∆ EE < v esc w . (A8)The energy transfer distribution is assumed uniform on the interval which gives Ω( r, w ) as w Ω = σn ( r )( v esc − u ) θ ( v esc − u ) (A9)and the self-capture rate is given by C s = (cid:90) R (cid:12) πr (cid:90) v esc f ( u ) u σn ( r )( v esc − u ) du dr (A10) Ejection
When the transferred energy in a collision involving DM and anti-DM is greater than m χ v esc /
2, the particle that is hit will be gravitationally unbound and escape the Sun. Thisejection rate is calculated using the same formula as self-capture but with a different Ω. Wecan divide ejection into two regions, one in which u < v esc and one in which u > v esc .If u < v esc and the incoming particle is trapped after a collision, the target particle mayor may not be trapped. If the target is trapped, we have the case of self-capture that isdescribed above. If the target particle is ejected, the transferred energy is in the range v w < ∆ EE < w Ω is given by w Ω = σn ( r ) u Θ( v esc − u ) . (A12)19f u > v esc , entrapment of the incoming particle will always result in the ejection of thetarget particle. In the case that the target particle is still trapped, ∆ E falls in the interval u w < ∆ EE < . (A13)The factor w Ω is then w Ω = σn ( r ) v esc Θ( u − v esc ) . (A14)However, if ∆ E falls in the interval v esc w < ∆ EE < u w (A15)The target particle will be ejected and the incoming particle will still have a velocity thatis larger than the escape velocity and thus also escape. For this case, w Ω is found to be w Ω = σn ( r )( u − v esc )Θ( u − v esc ) . (A16) [1] L. Bergstrom, Rept. Prog. Phys. , 793 (2000), arXiv:hep-ph/0002126 [hep-ph].[2] G. Bertone, D. Hooper, and J. Silk, Phys. Lett. BRept. , 279 (2005), arXiv:hep-ph/0404175 [hep-ph].[3] G. Jungman, M. Kamionkowski, and K. Griest, Phys. Rept. , 195 (1996), arXiv:hep-ph/9506380.[4] W. H. Press and D. N. Spergel, Astrophys. J. , 679 (1985).[5] Q. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. , 011301 (2002), arXiv:nucl-ex/0204008.[6] M. Asplund, N. Grevesse, and J. Sauval, Nucl. Phys. A777 , 1 (2006), arXiv:astro-ph/0410214[astro-ph].[7] M. Asplund, N. Grevesse, A. J. Sauval, and P. Scott, Ann. Rev. Astron. Astrophys. , 481(2009), arXiv:0909.0948.[8] A. Gould and G. Raffelt, Astrophys. J. , 654 (1990).[9] J. N. Bahcall and A. Ulmer, Phys. Rev. D , 4202 (1996), arXiv:astro-ph/9602012.[10] D. Spergel and W. Press, Astrophys. J. , 663 (1985).
11] D. Dearborn, K. Griest, and G. Raffelt, Astrophys. J. , 626 (1991).[12] M. T. Frandsen and S. Sarkar, Phys. Rev. Lett. , 011301 (2010), arXiv:1003.4505.[13] M. Taoso, F. Iocco, G. Meynet, G. Bertone, and P. Eggenberger, Phys. Rev. D , 083509(2010), arXiv:1005.5711.[14] D. T. Cumberbatch, J. Guzik, J. Silk, L. S. Watson, and S. M. West, Phys. Rev. D , 103503(2010), arXiv:1005.5102.[15] I. Lopes and J. Silk, Astrophys. J. , 130 (2012), arXiv:1209.3631.[16] I. Lopes, K. Kadota, and J. Silk, Astrophys. J. Lett. , L15 (2014), arXiv:1310.0673.[17] C. Kouvaris, Phys. Rev. D , 023006 (2008), arXiv:0708.2362 [astro-ph].[18] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 063531 (2010), arXiv:1004.0586 [astro-ph.GA].[19] C. Kouvaris and P. Tinyakov, Phys. Rev. Lett. , 091301 (2011), arXiv:1104.0382 [astro-ph.CO].[20] S. D. McDermott, H.-B. Yu, and K. M. Zurek, Phys. Rev. D , 023519 (2012),arXiv:1103.5472 [hep-ph].[21] F. Iocco, M. Taoso, F. Leclercq, and G. Meynet, Phys. Rev. Lett. , 061301 (2012),arXiv:1201.5387.[22] M. Aartsen et al. (IceCube collaboration), Phys. Rev. Lett. , 131302 (2013),arXiv:1212.4097.[23] S. Adrian-Martinez et al. (ANTARES Collaboration), JCAP , 032 (2013),arXiv:1302.6516.[24] A. R. Zentner, Phys. Rev. D , 063501 (2009), arXiv:0907.3448.[25] M. Fairbairn, P. Scott, and J. Edsj¨o, Phys. Rev. D , 047301 (2008), arXiv:0710.3396.[26] K. M. Zurek, Phys. Rept. , 91 (2014), arXiv:1308.0338 [hep-ph].[27] K. Petraki and R. R. Volkas, Int. J. Mod. Phys. A28 , 1330028 (2013), arXiv:1305.4939 [hep-ph].[28] I. Lopes, P. Panci, and J. Silk, Astrophys . J. , 162 (2014), arXiv:1402.0682 [astro-ph.SR].[29] A. C. Vincent, P. Scott, and A. Serenelli, Phys. Rev. Lett. , 081302 (2015),arXiv:1411.6626 [hep-ph].[30] A. Gould, Astrophys. J. , 560 (1987).[31] G. Busoni, A. De Simone, and W.-C. Huang, JCAP , 010 (2013), arXiv:1305.1817 [hep- h].[32] K. Griest and D. Seckel, Nucl. Phys. B , 681 (1987).[33] J. Bovy and S. Tremaine, Astrophys. J. , 89 (2012), arXiv:1205.4033.[34] J. Read, J. Phys. G , 063101 (2014), arXiv:1404.1938.[35] D. Akerib et al. (LUX Collaboration), Phys. Rev. Lett. , 091303 (2014), arXiv:1310.8214.[36] E. Aprile et al. (XENON100 Collaboration), Phys. Rev. Lett. , 181301 (2012),arXiv:1207.5988.[37] E. Aprile et al. (XENON100 Collaboration), Phys. Rev. Lett. , 021301 (2013),arXiv:1301.6620.[38] R. Agnese et al. (SuperCDMS Collaboration), Phys. Rev. Lett. , 241302 (2014),arXiv:1402.7137.[39] R. Agnese et al. (SuperCDMS Collaboration), Phys. Rev. Lett. , 041302 (2014),arXiv:1309.3259 [physics.ins-det].[40] S. Archambault, F. Aubin, M. Auger, E. Behnke, B. Beltran, et al. , Phys. Lett. B , 185(2009), arXiv:0907.0307.[41] S. W. Randall, M. Markevitch, D. Clowe, A. H. Gonzalez, and M. Bradac, Astrophys. J. ,1173 (2008), arXiv:0704.0261.[42] C. Bennett et al. (WMAP Collaboration), Astrophys. J. Suppl. , 20 (2013),arXiv:1212.5225 [astro-ph.CO].[43] P. Ade et al. (Planck), (2015), arXiv:1502.01589 [astro-ph.CO].[44] A. Serenelli, S. Basu, J. W. Ferguson, and M. Asplund, Astrophys. J. , L123 (2009),arXiv:0909.2668.[45] A. Gould, Astrophys. J. , 571 (1987)., 571 (1987).