Constraint on Light Dipole Dark Matter from Helioseismology
aa r X i v : . [ a s t r o - ph . S R ] F e b Draft version September 24, 2018
Preprint typeset using L A TEX style emulateapj v. 12/16/11
CONSTRAINT ON LIGHT DIPOLE DARK MATTER FROM HELIOSEISMOLOGY
Il´ıdio Lopes , Kenji Kadota , and Joseph Silk (Dated: September 24, 2018) Draft version September 24, 2018
ABSTRACTWe investigate the effects of a magnetic dipole moment of asymmetric dark matter (DM) in theevolution of the Sun. The dipole interaction can lead to a sizable DM scattering cross section evenfor light DM, and asymmetric DM can lead to a large DM number density in the Sun. We find thatsolar model precision tests, using as diagnostic the sound speed profile obtained from helioseismologydata, exclude dipolar DM particles with a mass larger than 4 . . × − e cm. Subject headings: cosmology: miscellaneous – dark matter – elementary particles – primordial nucle-osynthesis – Sun: helioseismology INTRODUCTION
The universe is composed of baryons and unknownnonbaryonic particles, commonly called dark matter. Al-though the gravitational interaction between baryonsand DM is well established, other types of interactionswith the standard particles are less well known.Several experiments seek to detect the DM particleby observing its scattering off nuclei, by detecting someby-product resulting from its annihilation into high en-ergy particles, or by producing them in acceleratorsthrough the collision of standard particles. The goal isfor one or more of these experiments to obtain a sig-nal of the DM interaction, other than the well-knowngravitational interaction. The first positive hints ofdirect DM observations are intriguing, although con-troversial: DAMA/LIBRA (Bernabei et al. 2010), Co-GeNT (Aalseth et al. 2011), CRESST (Angloher et al.2012) and CDMS (Agnese et al. 2013b) collaborationsreport indications of positive signals which do not com-ply with standard explanations of weak interactingmassive particle interactions. Furthermore, other col-laborations, such as XENON (Aprile et al. 2012) andCDMS (Agnese et al. 2013a), can nearly exclude the pos-itive results found by the previous experiments.This experimental data has stimulated interest in lightDM .
10 GeV as a candidate. For the purpose of il-lustrating the potentially stringent constraints on lightDM, we consider an operator which can induce largeenough DM-nuclear interactions even for small momen-tum transfer due to a small DM mass. One of the sim-plest extensions of the standard model for this purpose Centro Multidisciplinar de Astrof´ısica, Instituto SuperiorT´ecnico, Universidade de Lisboa , Av. Rovisco Pais, 1049-001Lisboa, Portugal;[email protected],[email protected] Departamento de F´ısica,Escola de Ciencia e Tecnologia, Uni-versidade de ´Evora, Col´egio Luis Ant´onio Verney, 7002-554´Evora, Portugal Department of Physics, Nagoya University, Nagoya 464-8602, Japan;[email protected] Institut d’Astrophysique de Paris, UMR 7095 CNRS, Uni-versit´e Pierre et Marie Curie, 98 bis Boulevard Arago, F-75014Paris, France; [email protected] Department of Physics and Astronomy, The Johns HopkinsUniversity, Baltimore, MD21218, USA Department of Physics, University of Oxford, UK would be the dipole operator which is the only gauge in-variant operator up to dimension five, letting fermionicDM with an intrinsic dipole moment couple to thephotons (Bagnasco et al. 1994; Sigurdson et al. 2006;Masso et al. 2009; Heo 2010). Such magnetic dipoledark matter (MDDM) can successfully explain recentclaims of direct detections, including DAMA/LIBRA andCoGeNT (An et al. 2010; Barger et al. 2011). More-over, several constraints have been set on MDDMusing direct search data (Lin & Finkbeiner 2011;Del Nobile et al. 2012), astrophysical and cosmologicalobservations (Fukushima & Kumar 2013) and the LargeHadron Collider data (Fortin & Tait 2012; Barger et al.2012).Here, we use helioseismic data to set constraints onasymmetric MDDM. In particular, we focus on the solarsound speed radial profile, for which there is a reliableinversion obtained from seismic data (Basu et al. 2009;Turck-Chieze et al. 1997). PROPERTIES OF MAGNETIC DIPOLE DARK MATTER
This study focuses on the DM fermion χ which cou-ples to the photon by the magnetic dipole interactionLagrangian L MDDM = µ χ ¯ χσ µν F µν χ where F µν is theelectromagnetic field strength tensor, σ µν is the commu-tator of two Dirac matrices and µ χ is the magnetic dipolemoment. We notice this interaction vanishes for Majo-rana fermions, so the fermionic DM particle has to beDirac.The fermionic DM particle with a magnetic dipole mo-ment arises in many models of DM, including amongothers models related to technicolor (e.g., Foadi et al.2007). In particular, we are interested in the caseof the interaction of DM with the baryons thattakes place by means of a massless mediator to yieldlong-range interaction, a process quite distinct fromthe contact-like interaction (Del Nobile & Sannino 2012;Del Nobile et al. 2012). The photon is the obvious me-diator, among more exotic candidates such as the darkphoton (Fornengo et al. 2011; Chun et al. 2011).The DM differential scattering cross section with re-spect to the nuclei recoil energy E R off the nucleus witha spin I with a DM particle of spin S χ , via the electro-magnetic interaction between the nuclear magnetic mo- ( c m od − c ss m ) / c ss m ( % ) Fig. 1.—
Comparison of the sound speed radial profile betweenthe SSM (Lopes & Turck-Chieze 2013) and different solar modelsevolving within an environment rich in MDDM. The red-dotted-green curve corresponds to the difference between inverted soundspeed profile (Turck-Chieze et al. 1997; Basu et al. 2009) and ourSSM (Turck-Chieze & Lopes 1993; Lopes & Turck-Chieze 2013).The continuous curves correspond to DM particles that have amass m χ of 1 – 20 GeV (blue curve m χ ≤ ≤ m χ ≤
12 GeV and cyan curve m χ ≥
12 GeV) and a magneticdipole that takes values from 10 − e cm to 10 − e cm. In thecore of the Sun, the variation caused by the presence of MDDM ismuch larger that the current sound speed difference between theoryand observation. ment µ Z,A (of a nucleus of mass A and charge Z ) andthe magnetic moment µ χ of the DM particle, is describedby (Barger et al. 2011): dσ χ dE R = e µ χ πE R S χ + 13 S χ (cid:2) A E | G E | + B M | G M | (cid:3) (1)where A E and B M are the electrical and magnetic mo-ment terms, and G E ( E R ) and G M ( E R ) are the nuclearform factors (normalized to G E (0) = 1 , G M (0) = 1) totake into account the elastic scattering off a heavy nu-cleus. A E and B M read A E = Z (cid:18) − E R m A v r − E R m χ v r (cid:19) (2) B M = I + 13 I (cid:18) µ Z,A e/ (2 m p ) (cid:19) m A E R m p v r , (3)where e and m p are the elementary electric charge andthe mass of the proton (Barger et al. 2012). m A and m χ are the masses of the baryon nucleus and DM par-ticle, respectively. E R and v r are defined from thetransfer momentum q and the center-of-mass momen-tum p , such that q = 2 m A E R and | p | = m r v r where m r = m A m χ / ( m A + m χ ).At first glance, it is reasonable to expect that the con-tribution of heavy elements for the interaction inside theSun could be quite significant, due to the strong depen-dence in Z and I (see Equations (1)–(3)). However, be-cause metals inside the Sun contribute to less than 2%of the total mass of the Sun, their contribution can beconsidered to be negligible. The electromagnetic inter-action between MDDM and baryons will be importantfor hydrogen and helium which correspond to 98% of the
00 00 0 000 111111 22 33 444 55667000 0888 0 m χ (GeV) Log µ χ ( e c m ) Fig. 2.—
Exclusion plot for magnetic dipole DM parameterspace ( m χ – µ χ ) from present day low-Z SSM and helioseismol-ogy data. The possible candidates must lie in the light region,above the iso-contour with 2%. The different isocontour curvesrepresent the maximum difference, i.e., max (cid:2) ( c − c ) /c (cid:3) in the region below 0 . R ⊙ – the percentage of the maximum sounddifference between the SSM and the MDDM solar models. TheMDDM halo is assumed to be an isothermal sphere with localdensity ρ χ = 0 .
38 GeVcm − , and thermal velocity (dispersion) v th = 270 kms − . total mass of the Sun, from which Helium abundance inthe Sun’s core changes during its evolution from an ini-tial abundance of 27% to 62% for the current age of theSun. Nevertheless, this increase of helium occurs only inthe very center of the Sun, the atomic number of helium( Z = 2) is relatively small (when compared with otherchemical elements), and the helium abundance is alwayssmaller than that of hydrogen. Furthermore, during theSun’s evolution, to a good approximation, it is reasonableto consider only the interaction of MDDM with hydrogento be relevant. The total elastic scattering cross sectionof DM with hydrogen σ χp , from Equations (1)–(3) with auniform form factor, can then be expressed as σ SI χp + σ SD χp where σ SD χp and σ SI χp are the conventional effective scat-tering cross sections (Barger et al. 2012): σ SI χp = µ χ e π (cid:18) − m r m p − m r m p m χ (cid:19) (4)and σ SD χp = µ χ e π (cid:18) µ p e/ (2 m p ) (cid:19) m r m p , (5)with µ p being the magnetic moment of the proton. Both σ SD χp and σ SI χp contribute equally for the total scatteringcross section, σ χp . However, in the case that the massof the DM particle is much larger than the mass of theproton,i.e., m χ ≫ m p , it follows that m r ≈ m p and the σ χp becomes independent of the mass of the DM particle,and σ χp is only proportional to the square of the DMmagnetic moment, σ χp ≈ . e / (2 π ) µ χ . DARK MATTER AND THE STANDARD SOLAR MODEL
Our Galaxy and the Sun are assumed to be immersedin an isothermal sphere composed of MDDM parti-cles. The observational determination of the DM density ρ χ in the neighborhood of the Sun is quite uncertain,varying from 0 . .
85 GeVcm − (Bovy & Tremaine2012; Garbari et al. 2012). Here we set ρ χ =0 .
38 GeVcm − (Catena & Ullio 2010).DM particles flow through any celestial object suchas the Sun. A few of these particles will occasionallyscatter with the atomic nuclei of the star, mostly withprotons, losing energy in the process, in some cases theenergy reduction is such that the velocity of the DMparticle becomes smaller than the Sun’s escape velocity,i.e., the DM particle becomes gravitationally bound tothe star. As time passes, the DM particle undergoes morecollisions, until the particle reaches thermal equilibriumwith local baryons.The accretion of DM by the Sun depends on the bal-ance between three processes: capture of DM via energyloss of the colliding particle with the baryon nuclei; anni-hilation of two DM particles into standard particles; and evaporation of DM particles, important for light DM par-ticles for which the collisions with nuclei result in escapefrom the Sun’s gravitational field. Therefore, the totalnumber of DM particles inside the Sun is determined bythe relative magnitude of these three processes.At each step of the Sun’s evolution, the total num-ber of particles N χ that accumulate inside the staris computed by solving the following equation numeri-cally (Lopes & Silk 2012a): dN χ ( t ) dt = C c − C a N χ ( t ) − C e N χ ( t ) , (6)where C c is the rate of capture of particles from theMDDM halo, C a is the annihilation rate of particles,and C e in the evaporation rate of DM particles fromthe star. The capture rate C c is computed numeri-cally at each step of the evolution from the expressionobtained by Gould (1987). The cross section used inthe computation of C c corresponds to the ones givenby Equations (4)–(5). In the following, we will con-sider the asymmetric DM scenario and hence C a can beneglected. A concrete realization could include asym-metric composite DM which can induce a sizable dipolefor the bound state of the particles (Foadi et al. 2007;Ryttov & Sannino 2008; Del Nobile et al. 2011).The evaporation of DM particles is not relevant for m χ ≥ C e , whichwe use in our calculations to define the lower bound ofless massive DM particles. This expression reproducesthe full numerical results with an accuracy better than15% (Kappl & Winkler 2011).The reference solar model used in our compu-tation corresponds to the standard solar model(SSM), which has been updated with the most re-cent physics (Turck-Chieze & Lopes 1993). In par-ticular, we choose to use the solar composition de-termined by Asplund et al. (2009), usually knownas the low-metallicity (or low-Z) SSM, as discussedby Haxton, et al. (2013). Figure 1 shows the soundspeed difference of reference that corresponds to the sound speed difference between the low-Z SSM and thesound speed obtained by inversion of the seismic data ofthe Global Oscillations at Low Frequencies (GOLF) andMichelson Doppler Image (MDI) instruments of the Solarand Heliospheric Observatory Satellite and BiSON obser-vational network (Turck-Chieze et al. 1997; Basu et al.2009). A detailed discussion about the physics of ourSSM can be found in Lopes & Turck-Chieze (2013). ThisSSM is identical to others low-Z SSM published in theliterature (e.g., Serenelli et al. 2013).The solar models evolving in different MDDM halosare obtained by a similar procedure to the SSM. Like-wise these models are required to have the observedsolar radius and luminosity at the present age. Inour description of the impact of DM on the evolutionof the Sun, we closely follow recent developments inthis field (Cumberbatch et al. 2010; Lopes et al. 2011;Lopes & Silk 2012b,a; Casanellas & Lopes 2013). A de-tailed description of how this process is implemented inour code is discussed in Lopes et al. (2011).The accumulation of MDDM particles inside the Sunreduces the temperature in the Sun’s core and as a con-sequence, the sound speed drops, but is compensated forby an increase of sound speed in the radiative region andthe convection zone (see Figure 1). This results fromthe fact that these solar models are required to have aradius and luminosity consistent with observations. Thecalibration follows an iterative procedure identical to theone used to compute the SSM. In principle, we coulduse the sound speed and density profiles obtained frominversion of helioseismology data as a diagnostic tool,however, we prefer to use the sound speed because onlyfrequencies of acoustic modes are observed, consequentlythe sound speed inversion is the more reliable diagnosticmethod. In the future, if frequencies of gravity modesare measured with success, the density profile can be-come an independent method to probe the Suns core.Figure 1 shows that the sound speed differences of thesolar models computed for different values of m χ and µ χ are quite distinct from the sound speed difference of ref-erence. This effect is more important for DM particles ofrelatively low mass and high magnetic moment. In thecase of particles with a very low m χ the impact on thesound speed difference profile becomes insignificant dueto the occurrence of DM evaporation. Although the DMaffects the whole internal structure of the star equally,we focus our analysis on the Sun’s core where the directimpact of DM is detected. It is reasonable to considerthat for solar models for which the sound speed differenceis larger than the sound speed difference of the referencemodel, or equivalently if this difference is larger than 2%,then these solar models can be excluded on the basis thatthey cannot be accommodated with our current under-standing of the physics of the solar interior. It is truethat in the Sun’s deep core, the sound speed differenceof the reference solar model still contains a few uncer-tainties coming either from an insufficient description ofthe physics of the SSM, or poor inversion of the soundspeed profile due to lack of low degree seismic data. It isbelieved that some of the current problems in the SSMis related with abundances and opacities below the baseof the convection zone, but these localized uncertaintiesdo not affect the core of the Sun where this diagnosticis done. Moreover, their effect on the Sun’s structurewill be smaller than the observational sound speed dif-ference. Nevertheless, this uncertainty is at most of theorder of 1.5%. Alternatively, if we choose to use as ref-erence a high-Z SSM, as the sound speed difference withobservations is of the order of 0.3%, the constraint onthe MDDM parameters could be stronger. Nevertheless,due to the problem related with the chemical composi-tion in the solar interior ( Serenelli et al. 2011), we takethe conservative approach to use as reference the low-ZSSM which has the largest observational uncertainty.Figure 2 shows the MDDM exclusion plot computedfor different values of m χ and µ χ . We choose as diag-nostic the value corresponding to the maximum differ-ence between the square of the sound speed of the SSMand the sound speed of the DM solar models. There isa region of the parameter space for the relatively lightDM 4 . ≤ m χ ≤ . µ χ ≥ − ecm for which the sound speed differenceis larger than 2%. Accordingly, these models can be re-jected. We found the quantitatively same exclusion lim-its on the MDDM parameters even if we use the densityprofile, rather than the sound speed profile, as a diag-nostic method for which the observational density uncer-tainty is considered to be of the order of 4%. SUMMARY AND CONCLUSION
In this Letter, we use helioseismology to constrain themass and magnetic moment of MDDM. We find thatthere is an important set of parameters for which the im-pact of MDDM is much larger than the current differencebetween theory and observation. We have found that so-lar model precision tests using the sound speed profile ob-tained from helioseismology data as a diagnostic excludeMDDM with m χ ≥ . µ χ ≥ . × − e cm.DM particles with the above parameters produce changesin the Sun which are much larger than the current soundspeed difference between theory and observation.Furthermore, this new constraint does not affect the results found by Del Nobile et al. (2012) for which a DMparticle with m χ ∼