Thermalization time scales for WIMP capture by the Sun in effective theories
PPrepared for submission to JCAP
Thermalization time scales forWIMP capture by the Sun ineffective theories
A. Widmark a,b a The Oskar Klein Centre for Cosmoparticle Physics, AlbaNova,SE-106 91 Stockholm, Sweden b Department of Physics, AlbaNova, Stockholm University,SE-106 91, Stockholm, SwedenE-mail: [email protected]
Abstract.
I study the process of dark matter capture by the Sun, under the assumption ofa Weakly Interacting Massive Particle (WIMP), in the framework of non-relativistic effectivefield theory. Hypothetically, WIMPs from the galactic halo can scatter against atomic nu-clei in the solar interior, settle to thermal equilibrium with the solar core and annihilate toproduce an observable flux of neutrinos. In particular, I examine the thermalization processusing Monte-Carlo integration of WIMP trajectories. I consider WIMPs in a mass range of10–1000 GeV and WIMP-nucleon interaction operators with different dependence on spin andtransferred momentum. I find that the density profiles of captured WIMPs are in accordancewith a thermal profile described by the Sun’s gravitational potential and core temperature.Depending on the operator that governs the interaction, the majority of the thermalizationtime is spent in either the solar interior or exterior. If normalizing the WIMP-nuclei interac-tion strength to a specific capture rate, I find that the thermalization time differs at most by3 orders of magnitude between operators. In most cases of interest, the thermalization timeis many orders of magnitude shorter than the age of the solar system. a r X i v : . [ h e p - ph ] M a r ontents Research in recent decades indicate that non-baryonic dark matter constitutes a majority ofthe Universe’s matter content. This is supported by observational evidence of a vast physicalrange, from the sub-galactic to the cosmological scale [1–3]. One of the most prominent andstudied dark matter particle candidates is the Weakly Interacting Massive Particle (WIMP),for which many different detection techniques are utilized [4–6].Indirect detection aims to infer the existence of dark matter particles by observing theirannihilation products. A variety of such experiments are conducted, observing various Stan-dard Model particles from various sources [7–11]. The focus of this paper is the hypothesizedprocess of dark matter capture by the Sun. WIMPs from the galactic dark matter halo canscatter against atomic nuclei in the Sun’s interior and become gravitationally bound. Withfurther collisions they lose energy, thermalize and settle in the Sun’s core. Annihilation ofcaptured WIMPs could potentially produce a detectable flux of neutrinos emanating fromthe Sun, differentiable from neutrinos produced by nuclear fusion due to their higher energyscale [12, 13]. Such a neutrino signal is currently sought after with neutrino telescopes, suchas IceCube, Super-Kamiokande, ANTARES and Baikal [14–17].In recent years, effort has gone into studying dark matter experiments in the frameworkof non-relativistic effective field theories [18–30]. This allows for a model independent analysisof observational data. The same approach has also been applied to the process of dark mattercapture by the Sun [31–36], for which capture rates of different WIMP-nucleon interactionoperators and WIMP self-interaction operators have been calculated.The aim of this article is to investigate the process of thermalization for WIMP captureby the Sun, and to do so in the framework of a non-relativistic effective field theory. Thereasons for studying this subject are the following. The time it takes for a WIMP boundin orbit to be down-scattered and thermalized to core temperature is most often neglectedand approximated as instantaneous. Furthermore, the resulting WIMP density distribution– 1 – O = χN ˆ O = i ˆS χ · (cid:16) ˆS N × ˆq m N (cid:17) ˆ O = i ˆS N · (cid:16) ˆq m N × ˆv ⊥ (cid:17) ˆ O = i ˆS N · ˆq m N ˆ O = ˆS χ · ˆS N ˆ O = i ˆS χ · ˆq m N ˆ O = i ˆS χ · (cid:16) ˆq m N × ˆv ⊥ (cid:17) ˆ O = ˆS χ · (cid:16) ˆS N × ˆv ⊥ (cid:17) ˆ O = (cid:16) ˆS χ · ˆq m N (cid:17) (cid:16) ˆS N · ˆq m N (cid:17) ˆ O = i (cid:16) ˆS χ · ˆv ⊥ (cid:17) (cid:16) ˆS N · ˆq m N (cid:17) ˆ O = ˆS N · ˆv ⊥ ˆ O = i (cid:16) ˆS χ · ˆq m N (cid:17) (cid:16) ˆS N · ˆv ⊥ (cid:17) ˆ O = ˆS χ · ˆv ⊥ ˆ O = − (cid:16) ˆS χ · ˆq m N (cid:17) (cid:104)(cid:16) ˆS N × ˆv ⊥ (cid:17) · ˆq m N (cid:105) Table 1 . All leading order non-relativistic interaction operators. is assumed to follow a thermal profile, given by the Sun’s gravitational potential and coretemperature. Any departure from these statements can significantly alter the distribution ofWIMPs inside the Sun, which in turn has a direct effect on the rate of annihilation and theresulting neutrino signal. There is also a possibility that the WIMP capture rate varies withtime, as an effect of the Sun traveling through substructures in the dark matter halo [37].In such a scenario, a long thermalization time would serve to smoothen fluctuations in theannihilation rate and neutrino signal.This paper is structured as follows. In section 2, I present a brief theoretical summary ofWIMP-nucleus scattering in effective field theory. In section 3, I review the theory of WIMPcapture by the Sun, including the method used for simulating the down-scattering process.In sections 4 and 5, I present the results and conclusions.
Interactions between WIMPs and nuclei can be described in the framework of a non-relativisticeffective field theory of WIMP-nucleon interactions. The possible quantum operators thatdescribe such interactions are restricted by Galilean symmetry and can only be constructedas a combination of these five Hermitian operators: χN , ˆS χ , ˆS N , i ˆq , ˆv ⊥ , (2.1)where index χ ( N ) refers to a WIMP (nucleon), S denotes a spin vector, q is the transferredmomentum of the collision, and v ⊥ ≡ v + q / µ N is the transverse velocity as given bycollisional velocity v and the reduced mass of the WIMP-nucleon system µ N .Following the convention as established by [22, 23], in table 1 I have listed all linearlyindependent leading order operators that can be constructed from these building blocks,constrained by assuming a force mediating heavy particle of spin 1 or less. Because there aretwo types of nucleons, the parameter space of possible WIMP-nuclei interactions are doubledto a total number of 28.The WIMP-nucleon Hamiltonian density can be written in a basis of isospin, representedby an upper index τ , which is 0 for isoscalar and 1 for isovector coupling. In the former,WIMPs scatter off of all nucleons the same way; in the latter, the scattering off of protons and– 2 –eutrons have opposite signs. By labeling each nucleon with an index i , the total Hamiltoniandensity for a nucleus of mass number A can be written ˆ H ( r ) = A (cid:88) i =1 (cid:88) τ =0 , (cid:88) k =1 c τk ˆ O ( i ) k ( r ) t τ ( i ) , (2.2)where t τ ( i ) is a matrix that projects a nucleon state onto an isospin basis. In the basis ofproton and neutron couplings, the coupling coefficients are related like c pk = ( c k + c k ) / and c nk = ( c k − c k ) / .As has been shown in great detail in other sources [22, 23, 35], this leads to a differentialWIMP-nucleus cross section given by d σ ( E r , w )d E r = 12 J T + 1 2 m T w (cid:88) τ,τ (cid:48) (cid:40) (cid:88) k = M, Σ (cid:48)(cid:48) , Σ (cid:48) R ττ (cid:48) k (cid:18) v ⊥ T , q m N (cid:19) W ττ (cid:48) k ( q )++ q m N (cid:32) (cid:88) k =Φ (cid:48)(cid:48) , Φ (cid:48)(cid:48) M, ˜Φ (cid:48) , ∆ , ∆Σ (cid:48) R ττ (cid:48) k (cid:18) v ⊥ T , q m N (cid:19) W ττ (cid:48) k ( q ) (cid:33)(cid:41) , (2.3)where E r is the recoil energy in the rest frame of the target nucleus, w is the collisionalvelocity, J T is the nucleus’ spin, and m T ( m N ) is the nucleus (nucleon) mass. The quantitiesdenoted R ττ (cid:48) k are the WIMP response functions, as found in appendix A. The quantities W ττ (cid:48) k are the nuclear response functions, different for each nuclear isotope. Nuclear responsefunctions for the 16 most abundant isotopes in the Sun are taken from [35], where they havebeen calculated from ground state one-body density matrix elements. These 16 isotopes are,in order of mass abundance: H, He, O, C, Ne, N, Fe, Si, Mg, S, He, Ni, Ar, Ca, Al, Na.
In this section I present a theoretical background for dark matter accumulation in the Sun. Insubsection 3.1, I review the theory contingent on the assumptions of a cold Sun and instantthermalization to a thermal density profile. In subsection 3.2, I relax the assumption of a coldSun and present the theory by which I explore the thermalization process. In subsection 3.3,I provide some remarks about capture and thermalization in the case of significant WIMPself-interaction.
WIMPs traveling through the Sun can collide with atomic nuclei in the solar interior. Insome of these interactions the WIMP loses enough kinetic energy to be bound in orbit. Theprobablity per unit time for a WIMP to scatter to less than local escape velocity v ( r ) is[35, 38] Ω − v ( w ) = (cid:88) T n T w Θ (cid:32) µ T µ ,T − u w (cid:33) (cid:90) E k µ T /µ ,T E k u /w d E r d σ ( E r , w )d E r , (3.1)where m χ ( m T ) is the dark matter (target nucleus) mass, n T is the target species numberdensity, u and w = (cid:112) u + v ( r ) are the dark matter particle velocities at point of scatter– 3 –nd at infinite radius, and d σ/ d E r is the differential cross section, as given by equation (2.3).The lower bound of the integral represents the minimal energy transfer necessary for capture,while the upper limit is the highest possible energy transfer in an elastic collision, given bythe WIMP’s kinetic energy E k = m χ w / , and dimensionless parameters µ T = m χ /m T and µ + ,T = ( µ T + 1) / . The Heaviside function, Θ , ensures that capture is kinematically possible.Because interactions are weak and the Sun is optically thin, the WIMP capture rate byatomic nuclei per volume is given by d C c d V = (cid:90) ∞ d u f ( u ) u w Ω − v ( w ) , (3.2)where f ( u ) is the WIMP halo velocity distribution. Integrating over the full volume of theSun gives the total capture rate C c = (cid:90) R (cid:12) πr d C c d V d r, (3.3)where R (cid:12) is the solar radius.The other process which governs the amount of captured WIMPs is annihilation, whichwill come into effect when the concentration of WIMPs inside the Sun has become sufficientlyhigh. A canonical value for the thermally averaged annihilation cross-section is (cid:104) σ A v (cid:105) (cid:39) · − cm s − . However, recent studies have made more precise evaluations of this value[39]. Following these results, the value used in this article is (cid:104) σ A v (cid:105) = 2 · − cm s − .The total number of WIMPs annihilated per unit time is C a N , where N is the totalnumber of trapped WIMPs and C a is an annihilation factor. The latter is given by C a = 4 π (cid:104) σ A v (cid:105) N (cid:90) R (cid:12) (cid:15) ( r ) r d r, (3.4)where (cid:15) ( r ) is the WIMP number density function. It is commonly assumed that the WIMPsthermalize to core temparature T c and follow a thermal profile, (cid:15) ( r ) ∝ exp (cid:18) − m χ φ ( r ) k B T c (cid:19) , (3.5)where φ ( r ) is the gravitational potential. Because the Sun’s core has a more or less constantdensity, the annihilation factor follows very closely the proportionality relation C a ∝ (cid:104) σ A v (cid:105) m / χ . (3.6)Assuming an instantaneous thermalization, the amount of WIMPs trapped within theSun, N , is described by the following differential equation, d N d t = C c − C a N . (3.7)It has solution N ( t ) = (cid:114) C c C a tanh (cid:16)(cid:112) C c C a t (cid:17) , (3.8)– 4 –hich approaches an equilibrium solution N eq = (cid:112) C c /C a as t → ∞ . The number of annihi-lation events per unit time follows the form Γ( t ) = 12 C a N ( t ) = 12 C c tanh (cid:16)(cid:112) C c C a t (cid:17) , (3.9)where the factor 1/2 comes from the fact that every annihilation event involves a WIMP pair. The aim of this article is to study the process of thermalization, to evaluate the thermalizationtime scale and eventual density profile, for different types of WIMP-nucleon interactions. Thisis done by Monte-Carlo integration, by following WIMP trajectories from the first scatteringevent that binds a WIMP to the Sun’s gravitational field, to down-scattering to orbits thatare in thermal equilibrium with the Sun’s core. In order to accurately randomize thesetrajectories, I evaluate the probability density functions and 3-dimensional kinematics thatgovern this process.A WIMP’s orbit in or around a spherically symmetric massive body is completely de-scribed by its total energy E and angular momentum J . The innermost and outermost radiiof a WIMP’s orbit fulfill the relation E = m χ φ ( r ) + J m χ r , (3.10)where φ ( r ) is the gravitational potential. The distance traveled through a shell of thickness d r , per orbital period, is d s = 2d r (cid:18) − ( Jm χ rw ) (cid:19) − / , (3.11)where the factor is due to the fact that a WIMP travels through a shell twice per orbitalperiod. The time it takes to pass through this shell is d t = w − d s , which gives the orbitaltime by integration from minimal to maximal radius.The WIMP-nucleus interactions are weak and the Sun is optically thin. For the case ofa cold Sun, neglecting any thermal motion of the target nuclei, the probability of scatter ina thin shell of thickness d r during one orbital period is d P sc = d s n T (cid:90) E k ( w ) µ T /µ ,T d E r d σ ( E r , w )d E r . (3.12)By including the thermal motion of the target nuclei, the probability of scatter per orbitalperiod becomes d P sc = d s n T w (cid:90) ∞−∞ d ˜ w (cid:90) E k ( ˜ w ) µ T /µ ,T d E r d σ ( E r , ˜ w )d E r f T ( ˜ w ) ˜ w, (3.13)where ˜ w is the collisional velocity between the WIMP and target nucleus, and f T ( ˜ w ) isthe velocity distribution of a nucleus in the WIMP rest frame, assumed to be a thermalMaxwell-Boltzmann distribution boosted by velocity w and normalized to 1. The factor d s/w corresponds to the time spent in the shell, while the factor f T ( ˜ w ) ˜ w in the integrand accountsfor the number of encounters of a certain collisional velocity. In the limit w (cid:29) k B T /m T ,– 5 – T ( ˜ w ) becomes a narrow function peaked around ˜ w = w , such that equation (3.13) becomesequivalent to (3.12). Integrating d P sc over all radii gives the total scattering probability perorbital period and, given the time of such a period, the average time spent on this orbit.Expressing d P sc as a probability density function over r allows for a randomization of thescattering radius.Given a scattering event where the collisional velocity ˜ w and the WIMP velocity w areknown, the thermal velocity of the nuclei, v th , is not unique. Its component along the WIMPtrajectory axis (in the solar rest frame), v thz , can be anything in range w − ˜ w ≤ v thz ≤ w + ˜ w .The perpendicular component of the thermal velocity fulfills that v th ⊥ + ( v thz − w ) = ˜ w .Expressing this condition as a delta function over ˜ w and integrating over all other variablesgives the probability density function for v thz : ˜ f T ( v thz ) d v thz == (cid:90) d θ v th ⊥ d v th ⊥ δ (cid:18) ˜ w − (cid:113) ( v thz − w ) + v th ⊥ (cid:19) (cid:18) m T πk B T (cid:19) / exp (cid:18) − m T k B T v th (cid:19) d v thz == 2 π ˜ w (cid:18) m T πk B T (cid:19) / exp (cid:18) − m T k B T ( ˜ w − w + 2 wv thz ) (cid:19) d v thz , (3.14)where there is one factor v th ⊥ from the cylindrical coordinate system Jacobian and one factor ˜ w/v th ⊥ from the inner derivative of the delta function. Integrating this function over therange of possible v thz gives the boosted Maxwell-Boltzmann distribution. After finding thecollisional velocity, thermal velocity and recoil energy of a collision, there are two remainingdegrees of freedom. They are found in the angular orientation of v th ⊥ and in one scatteringangle, both of which are trivially randomized in the rest frame of the target nucleus whereall angles are equiprobable.Thus I have all the probability density functions necessary in order to randomize thescattering target, scattering radius, collisional velocity, target thermal velocity, recoil energy,scattering angles, and subsequent WIMP energy and angular momentum of the new orbit.The very first scattering event, when a WIMP from the halo scatters and becomes boundin orbit, is randomized in almost the same manner. In this case, the radius of scattering isgiven by the integrand in equation (3.3), and the WIMP velocity by the integrand in equation(3.2). The angular momentum after first scattering is trivially randomized, given by the factthat all incoming solid angles are equiprobable, due to the homogeneity of the WIMP halodistribution and spherical symmetry of the system.I do not concern myself with the influence of planets and how they affect WIMP tra-jectories, as this is beyond the scope of this article. This issue is non-trivial and has beendiscussed for decades, with many twist and turns with regards to the evaluated capture rates[40–47]. Most recently, is has been argued that completely neglecting the planets and regard-ing the Sun as being in free space is a fair approximation [47], both in terms of the totalcapture rate and the thermalization process itself. It has been pointed out that WIMP self-interaction can increase the capture rate and amplifythe resulting neutrino signal [48], as an already amassed concentration of trapped WIMPswill itself constitute a scattering target for WIMPs from the galactic halo. It has been– 6 –emonstrated that, within current limits to the WIMP self-interaction cross section, sucheffects can amplify the neutrino signal by several orders of magnitude [36].In this case, the differential equation (3.7) that describes how the number of capturedWIMPs change over time is modified to read d N d t = C c − C a N + C s N, (3.15)where the last term corresponds to the capture rate by WIMP self-interaction. The quantity C s has unit of inverse time, such that C − s corresponds to the average time it takes before athermalized WIMP interacts with a WIMP from the galactic halo. (There are some minorand negligable corrections to this statement, as in some cases a collision can result in oneof the two WIMPs being ejected, resulting in no net gain or loss of captured WIMPs; inextremely rare cases both WIMPs can be ejected, resulting in evaporation.) The total energythat is distributed between the two WIMPs in the collisions is of order m χ v esc ( r = 0) / . Ascollisions between WIMPs are resonant, very long orbits are unlikely and need not be takeninto account. Rather, the two WIMPs involved in such a capture by self-interaction willspend most of their thermalization time at small radii. If that thermalization time is longeror of about equal value with C − s , the assumption of instant thermalization is broken. Thekinetic energy of the WIMPs would not dissipate quickly enough to counteract the energyinput from the halo, resulting in a heating up of the density profile of thermalized WIMPs.Given time, an equilibrium between capture and annihilation would still be reached, althoughwith a lower annihilation coefficient C a and a higher total concentration of trapped WIMPs.In the calculations of neutrino signal amplification due to WIMP self-interaction [36], theself-interaction cross-section is limited by N-body simulations [49] to an approximate uppervalue of σ χχ < . m χ g cm = 1 . × − m χ GeV cm . (3.16)This corresponds to a value of C − s = 6 . × years. If the thermalization time, starting froma mid-range energy of order ∼ m χ v esc ( r = 0) / , is longer than this time scale, then WIMPswill not be able to re-thermalize before being subject to another halo WIMP interaction. The thermalization process has been simulated by Monte-Carlo integration; by sampling alarge number of WIMP trajectories I have calculated thermalization time scales and sub-sequent thermal density profiles. I have considered a WIMP in the mass range of 10–1000GeV and spin 1/2. The galactic WIMP halo is assumed to follow a Maxwell-Boltzmanndistribution with velocity dispersion of 270 km/s, a Local Standard of Rest velocity of 220km/s, and a local dark matter density of 0.4 GeV/cm . Solar densities and temperatures aretaken from [50], as used also in [51]. The atomic nuclei of the Sun are assumed to follow aMaxwell-Boltzmann velocity distribution, given by the local solar temperature. It is commonly assumed that thermalized WIMPs follow a thermal profile, as described byequation (3.5). This assumption has been tested by following a single WIMP’s trajectory,starting from a state of thermal equilibrium, as it collides with atomic nuclei within the– 7 – igure 1 . Thermalization profiles for a WIMP of mass 100 GeV, with WIMP-nucleon interactionoperators ˆ O (left) and ˆ O (right), with isoscalar (solid red) and isovector (dashed red) coupling. Thethermal profile (solid blue) is the commonly assumed distribution, given by the Sun’s core temperatureand gravitational potential. The upper panels show the time averaged number density as a function ofradius, normalized to unity. The lower panels show the probability density for a WIMP to be locatedat radius R at a given point in time. solar medium. I have considered all WIMP-nucleon interaction operators of table 1, in bothisoscalar and isovector couplings, with WIMP masses of 10, 100 and 1000 GeV.Thermalization profiles for operators ˆ O and ˆ O are shown in figure 1. Although thesetwo operators are of different nature, the density profiles are practically the same. The upperpanels of figure 1 display the number density as a function of radius; the innermost binsare somewhat noisy, explained by the fact that these bins represent very small volumes. Analternative representation of the distribution is visible in the lower panels, which display theprobability that a WIMP is located at a certain radius. The time averaged total energy ofthe thermalized WIMPs takes the approximate value (cid:104) E (cid:105) (cid:39) k B T , which fits well with thefact that a particle in the potential well of the Sun has 6 degrees of freedom. Note that theaverage energy is higher than the median due to the high-energy tail of the distribution. Fora given point in time, most of the thermalized WIMPs have energies below ∼ . k B T .I have found that all operators tend to the same profile, very accurately described by thestandard assumption of a thermal profile. The resulting value for the annihilation coefficient C a , given by equation (3.4), differs at most by a few percent. Thermalization time scales are calculated by sampling a large number of WIMP trajectories,where each trajectory begins with a first scattering event that binds a WIMP from the galactichalo to the Sun’s gravitational field, followed by down-scattering to lower energies. The– 8 –rajectory ends when the WIMP can be considered thermalized, chosen as the first time thatits energy goes below the time-averaged energy of the thermal distribution (cid:104) E (cid:105) (cid:39) k B T . Ipresent my results as a time median of these trajectories (the mean value is not very illustrativedue to orbit outliers of very long radii).These results are presented using three different normalizations. In figure 2, the WIMP-nucleon cross section is normalized to a specific value. While this serves a purpose of recordkeeping, it does not illustrate very well the greater picture when it comes to the capture rateand expected neutrino signal. Furthermore, for a lot of operators direct detection experimentshave excluded such large cross sections. This is shown in figure 3, where the coupling constantsare set to the limits provided by the Large Underground Xenon (LUX) direct detectionexperiment. In figure 4, the interaction strength is normalized such that all operators giverise to the same total capture rate. In this manner, it is possible to relate the thermalizationtime scales to not only to the capture rate, but also to the rate of annihilation and resultingflux of neutrinos.The thermalization time medians for all operators ˆ O i are visible in figure 2, where thecoupling constants are normalized to values such that the WIMP-nucleon cross section atcollisional velocity 1000 km/s is σ χN = 10 − cm . The cross section and thermalizationtime have an inverse proportionality. The longest thermalization times in this figure are fromoperators that scatter almost exclusively on hydrogen, while other operators also interact withheavier nuclei. A lot of operators with isovector couplings have significantly longer time scalesthan their isoscalar counterparts, which is due to destructive interference between proton andneutron scattering. Although the general feature is that the thermalization time increaseswith WIMP mass, some operators have their shortest thermalization time for a mid-rangeWIMP. This is due to resonant effects; for example, operator ˆ O (isoscalar and isovector)scatters predominantly off of Fe and has its shortest thermalization time for a correspondingWIMP mass. Depending on the governing operator, the 90th percentile to the thermalizationtime is a factor 1.5–10 larger than the median, where this factor is strongly correlated withthe fraction of time spent on long orbits. Such behavior, of spending a majority of time onlong orbits, is exhibited by operators ˆ O , ˆ O and the isovector component of ˆ O , and in thehigher mass range also ˆ O , ˆ O and the isoscalar component of ˆ O . In the remainder ofparameter space, however, the majority of the thermalization time is actually spent in thesolar interior. This behavior is especially pronounced for operators with a strong dependenceon transferred momentum, such as ˆ O and ˆ O . For these operators, the WIMPs down-scatterto orbits within the solar interior very quickly, but on the other hand the cross section dropsdramatically with lower collisional velocities, resulting in a very slow energy loss in the veryend of the thermalization process.In figure 3, the coupling constants are set to the limit provided by the LUX directdetection experiment. These limits were calculated in [34], using the first publication of resultsfrom LUX [52]. Since then, stronger limits have been provided, most recently with [53]. Theexperiment’s sensitivity with respect to the WIMP-nucleon cross section has increased byabout a factor 5. They have also seen a downward fluctuation in their background signal,which have put an even stricter limit to the cross section in the higher WIMP mass range(an improvement of about a factor 8 in the 90 % C.L. between the old and new source). Forthe sake of simplicity, I use the coupling constant limits provided by [34], but increase thetime medians by a factor 5 to account for the sensitivity difference between the old and newLUX limits. There is a very large spread also in this figure, due to the varying quality of thecoupling constant limits for the different interaction operators. The limits are contingent on– 9 – igure 2 . Thermalization time scales for operators ˆ O i , with isoscalar (upper panel) and isovector(lower panel) couplings. The respective coupling constants are set such that the WIMP-nucleon crosssection at collisional velocity 1000 km/s has value σ χN = 10 − cm . the nuclear structure and evaluated nuclear response function of xenon (the LUX detectormedium), as well as the current local WIMP halo density. All thermalization time scales aresignificantly shorter than the age of the Sun, t (cid:12) (cid:39) . × years, so there is still a largemargin before direct detection experiments have excluded negligible thermalization times.In most scenarios where there is hope of detecting a high-energy neutrino signal comingfrom the Sun, the capture rate must be high enough for annihilation to have come intosignificant effect. Because the annihilation rate and the neutrino signal is proportional to theWIMP density squared, as is expressed in equation (3.7), significant annihilation presupposesthat the number of trapped WIMPs is close to its equilibrium amount. In figure 4, thethermalization time medians are presented for operator coupling constants that are normalizedsuch that the total capture rate is C c = 1 / ( C a t (cid:12) ) , where t (cid:12) is the age of the Sun. By usingthis value for C c , the number of annihilation events per unit time, as given by equation (3.9),– 10 – igure 3 . Thermalization time scales for operators ˆ O i , with isoscalar (upper panel) and isovector(lower panel) couplings. The respective coupling constants are set to the limit given by the LUXdirect detection experiment. This sets a lower limit to the thermalization time. is tanh (1) (cid:39) of its equilibrium value. The values for the capture rates that have beencalculated in this project are in accordance with the results of [35]. With this normalizationa new picture emerges. The thermalization time for a specific WIMP mass differs at most ∼ orders of magnitude between operators. The longest thermalization time scales are notfor operators that scatter against hydrogen into very long orbits (primarily ˆ O and ˆ O ), butrather for operators with a strong dependence on transferred momentum ( ˆ O and ˆ O ).For a few operators, a comparison between the last two figures shows that a close toequilibrium amount of trapped WIMPs is excluded by LUX, as the lower limit to the ther-malization time in figure 3 is higher than the value presented in figure 4. This is the case forisoscalar component of operators ˆ O , ˆ O and ˆ O , and the isovector component of operators ˆ O , ˆ O , ˆ O , ˆ O , ˆ O and ˆ O . Most of them are excluded by a small margin and only in thelower mass range, but for example the values of ˆ O with isovector coupling differ by almost– 11 – igure 4 . Thermalization time scales for operators ˆ O i , with isoscalar (upper panel) and isovector(lower panel) couplings. The respective coupling constants are normalized to a specific capture rate, C c = 1 / ( C a t (cid:12) ) , for which the amount of captured WIMPs is close to equilibrium and significantannihilation has come into effect. I have studied the thermalization process of WIMP capture by the Sun. I have considered aWIMP in the mass range of 10–1000 GeV, spin 1/2, and an interaction with atomic nucleidescribed by non-relativistic effective field theory with 28 degrees of freedom.I have found that the density profiles of thermalized WIMPs agree very well with thestandard assumption of a thermal profile. The thermalization time, on the other hand,varies dramatically depending on what operator that dominates the interaction. Using limitsprovided by the LUX direct detection experiment, most operators have a very large marginbefore the assumption of instantaneous thermalization breaks down.– 12 –n order to detect a neutrino signal coming from WIMP annihilation in the Sun, therate of annihilation must be sufficiently high. In most scenarios for which there is hope ofdetecting such a signal, the number of captured WIMPs must at the very least be close toits equilibrium solution. By normalizing the coupling strength such that C c = 1 / ( C a t (cid:12) ) , forwhich the neutrino flux is 58 % of its value at equilibrium, I find thermalization time mediansin the approximate range of – years. Compared to the 4.5 billion year life time of thesolar system, these time scales are short. In other words, if the capture rate is sufficientlylarge to give rise to an equilibrium (or almost equilibrium) number of trapped WIMPs atpresent day, then the assumption of instantaneous thermalization is valid. However, makingthe cross section one order of magnitude weaker could already be problematic in some cases,especially for WIMPs in the higher mass range.In terms of the thermalization process, the effective field theory operators of table 1 canbe categorized into two groups. In the first group, WIMPs spend most of their thermalizationtime on their first few orbits; operators that scatter almost exclusively off of hydrogen exhibitthis behavior. In the second group, WIMPs spend most of their thermalization time onshort orbits in the solar interior, a behavior that is especially pronounced for operators withquadratic or cubic dependence on transferred momentum, which is due to the decreasingWIMP velocity and rate of interaction. It must be noted that in my analysis I have assumedthe interactions to be dominated by only one operator, while in reality a combination ofoperators is expected. This is especially relevant for operators with a strong dependenceon collisional velocity. For example, operator ˆ O might dominate the first few scatteringevents and thus the probability for a WIMP to be captured, but as the collisional velocitydecreases operator ˆ O could start to dominate. Such a behavior would serve to hasten thethermalization process, as a higher interaction rate unequivocally makes the WIMP lose itsenergy quicker.As shown in [36], WIMP self-interaction could potentially amplify the capture rateand resulting neutrino signal, especially so if the capture by nuclei is insufficient in termsof creating an equilibrium at present time. In such a scenario, WIMP self-interaction couldincrease the capture rate to the extent that equilibrium is reached anyway. In fine-tuned casesof WIMP-nuclei interactions with quadratic or cubic dependence on transferred momentum,the thermalization time in the solar interior could be longer than the average time betweenself-interactions with halo WIMPs. This would result in a heating up of the density profile,an effect that suppresses the annihilation rate and the resulting neutrino signal.As is mentioned in section 1, the thermalization time is especially relevant if the solarsystem travels through substructures in the galactic WIMP halo, which would give rise toa time-varying capture rate. In [37] they consider the effect of passing through halo sub-structures with local over-densities of 2 and even 3 orders of magnitude, crossing-times inrange of – years, and mean time between encounters of years and upwards. Whilethe capture rate is proportional to the local WIMP density, the response in annihilation andneutrino flux is dependent on the thermalization process. If the thermalization time is longerthan the time scales of the density fluctuations, the response of the neutrino signal will beshifted and stretched. Given the results presented in this article, this can clearly be the case.In summary, the thermalization time and general behavior differs greatly between dif-ferent types of WIMP-nuclei interactions. For some operators most of the thermalizationprocess is spent on very long orbits; for other operators the very opposite is the case. Eitherway, the standard assumption of instant thermalization is valid in most cases of interest inthe effective field theory framework, although not necessarily by a large margin.– 13 – Dark matter response functions
Below are the dark matter response functions, as found in equation (2.3). R ττ (cid:48) M = c τ c τ (cid:48) + j χ ( j χ + 1)3 (cid:18) q m N v ⊥ T c τ c τ (cid:48) + v ⊥ T c τ c τ (cid:48) + q m N c τ c τ (cid:48) (cid:19) R ττ (cid:48) Φ (cid:48)(cid:48) = q m N c τ c τ (cid:48) + j χ ( j χ + 1)12 (cid:18) c τ − q m N c τ (cid:19) (cid:18) c τ (cid:48) − q m N c τ (cid:48) (cid:19) R ττ (cid:48) Φ (cid:48)(cid:48) M = c τ c τ (cid:48) + j χ ( j χ + 1)3 (cid:18) c τ − q m N c τ (cid:19) c τ (cid:48) R ττ (cid:48) ˜Φ (cid:48) = j χ ( j χ + 1)12 (cid:18) c τ c τ (cid:48) + q m N c τ c τ (cid:48) (cid:19) R ττ (cid:48) Σ (cid:48)(cid:48) = q m N c τ c τ (cid:48) + j χ ( j χ + 1)12 (cid:104) c τ c τ (cid:48) + q m N ( c τ c τ (cid:48) + c τ c τ (cid:48) ) ++ q m N c τ c τ (cid:48) + v ⊥ T c τ c τ (cid:48) + q m N v ⊥ T c τ c τ (cid:48) (cid:105) R ττ (cid:48) Σ (cid:48) = 18 (cid:18) q m N v ⊥ T c τ c τ (cid:48) + v ⊥ T c τ c τ (cid:48) (cid:19) + j χ ( j χ + 1)12 (cid:104) c τ c τ (cid:48) + q m N c τ c τ (cid:48) ++ v ⊥ T (cid:18) c τ − q m N c τ (cid:19) (cid:18) c τ (cid:48) − q m N c τ (cid:48) + q m N v ⊥ T c τ c τ (cid:48) (cid:19) (cid:105) R ττ (cid:48) ∆ = j χ ( j χ + 1)3 (cid:18) q m N c τ c τ (cid:48) + c τ c τ (cid:48) (cid:19) R ττ (cid:48) ∆Σ (cid:48) = j χ ( j χ + 1)3 (cid:16) c τ c τ (cid:48) − c τ c τ (cid:48) (cid:17) Acknowledgments
I would like to thank Joakim Edsjö and Sofia Sivertsson, who have provided valuable guidance,insightful discussion and cross-checking of results throughout the development of this project.I would also like to give thanks to Riccardo Catena, who first introduced me to the subjectof dark matter capture by the Sun.
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