Explore the Axion Dark Matter through the Radio Signals from Magnetic White Dwarf Stars
EExplore the Axion Dark Matter through the RadioSignals from Magnetic White Dwarf Stars
Jin-Wei Wang a,b,c, ¶ , Xiao-Jun Bi d,e, † , Run-Min Yao d,e, ‡ , Peng-Fei Yin d, § a Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136Trieste, Italy b INFN, Sezione di Trieste, via Valerio 2, 34127 Trieste, Italy b Institute for Fundamental Physics of the Universe (IFPU), via Beirut 2, 34151 Trieste, Italy d Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, ChineseAcademy of Sciences, Beijing, China e School of Physical Sciences, University of Chinese Academy of Sciences, Beijing, China
Axion as one of the promising dark matter candidates can be detectedthrough narrow radio lines emitted from the magnetic white dwarfstars. Due to the existence of the strong magnetic field, the axionmay resonantly convert into the radio photon (Primakoff effect) whenit passes through the corona of the magnetic white dwarf, where thephoton effective mass is equal to the axion mass. We show that forthe magnetic white dwarf WD 2010+310, the future experiment SKAphase 1 with 100 hours of observation will set an upper limit on theaxion-photon coupling g aγ of ∼ − GeV − for the axion mass rangeof 0 . ∼ . µ eV. ¶ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] J a n ontents The existence of dark matter (DM) has been established by solid astrophysical and cosmologicalobservations [1, 2]. For quite a long time the weakly interacting massive particles (WIMPs) areregarded as the most promising DM candidates, because they can naturally explain the DMrelic density [3–5]. However, so far no convincing dark matter signal has been found in the directdetection, indirect detection, and collider detection experiments. Furthermore, the limitationson the couplings between the DM particles and standard model particles are becoming moreand more stringent [4, 6, 7]. In this case, the experimental searches for other DM candidateshave thus attracted increasingly attention in recent years [8, 9].Among many other alternatives, the QCD axion, a light neutral pseudoscalar particle as-sociated with the U(1) Peccei-Quinn symmetry [10], is one of the best options due to severalexcellent theoretical characteristics: (1) it can resolve the strong CP problem very well [11–13];(2) it can explain the observed DM abundance [14–16]. For more details we refer the reader tothe excellent reviews of axion physics [17–19].Based on the possible couplings between axion and the electromagnetic sector, a numberof experiments have been set up to search for axion DM signals. These interactions predicttwo different phenomena: (1) the conversion between an axion particle and a photon undermagnetic fields (so-called Primakoff effect [20]), e.g. axion helioscope [21, 22], ”light shiningthrough a wall” experiments [23, 24], and so on; (2) the photon birefringence under axionbackground [25–30]. In this paper we focus on the former phenomenon.The compact stars, e.g. magnetic white dwarf stars (MWDs) and neutron stars, are verypromising probes to search for the axion DM, since these stars host strong magnetic fields, inwhich the axion can be converted into detectable photon signals. For example, there are studiesin the literature using the X-ray observations of MWDs to detect the star-born axions [31]and detecting the radio signals from axion DM conversion in the magnetospheres of neutronstars [32–36]. 2n this work we focus on the signals of axion DM from MWDs whose magnetic fields areat order of 10 ∼ G. With such a strong magnetic field, the axion DM may be convertedinto photons within the coronae of these MWDs. With the number density of plasma at thebase of the MWD corona ∼ cm − , the effective photon mass m γ in the corona is ∼ µ eV,which corresponds to a frequency of ∼ GHz. This means that the axion DM with mass of m a ∼ µ eV could fulfill the resonant conversion condition m a ∼ m γ and thus the conversionprobability can be enhanced greatly. The frequency of the corresponding signal happens to bein the sensitive region of the terrestrial radio telescopes, such as the Square Kilometer Array(SKA) that partially covers the 50 ∼ g aγ at the SKA. Conclusions and further discussions are givenin Sec. 5. The X-radiation search can be used to set stringent constraints on the parameters of the MWDs’coronae [43, 44], which is suggested by several theories [45–47]. For example, the Chandraobservation of the single cool MWD GD 356 sets limits on the plasma density of the hot coronaas n e < . × cm − with the temperature of corona T cor ∼ K [44], while in Ref. [43]the upper limit on the plasma density is n e ∼ cm − with T cor (cid:38) K for the MWDG99-47 (WD 0553+053). In the following sections, we show that the MWDs satisfying theseconstraints can be promising probes to detect the axion DM.In this work, for the properties of the MWDs’ coronae, we adopt the same assumptions asin Ref. [43]: (1) the corona is composed of fully ionized hydrogen plasma uniformly covering theentire surface of the white dwarf; (2) the field-aligned temperature of the electrons T cor ∼ Kis a constant throughout the corona. Under these conditions the distribution of the electrondensity at r is described by the barometric formula [43, 44] n e ( r ) = n e exp (cid:18) − r − R WD H cor (cid:19) , (1)3here n e is the density at the base of the corona, R WD is the radius of the MWDs, and H cor = 2 k B T cor m p g = 21 . (cid:18) T cor K (cid:19) (cid:18) M WD M (cid:12) (cid:19) (cid:18) R WD km (cid:19) − km (2)is the scale height of the isothermal corona, k B is the Boltzmann constant, m p is the protonmass, g is the free-fall acceleration at the surface of MWDs, M WD is the mass of the MWDs.Highly MWDs may give a very complex magnetic field structure [48]. In this work, forsimplicity we take the dipole configuration as in Ref. [31]: B = B R r (cid:16)
3( ˆ m · ˆ r ) ˆ r − ˆ m (cid:17) for r > R WD , (3)where B is the value of the magnetic field at the MWDs’ surface in the direction of the magneticpole, m = 2 πB R ˆ m is the magnetic dipole moment, r = r ˆ r is the spatial coordinate, and r = | r | represents the distance from the center of the MWDs. Furthermore, we simply set θ = π/ B = | B | = B R r for r > R WD . (4) In this section we present a more general formalism to calculate the axion-photon conversionprobability in the magnetic fields of MWDs. The general interactions between axion and photonare given by the following Lagrangian [31]: L = 12 ( ∂ µ a ) − m a a − F µν F µν + 12 m A A µ A µ (5) − g aγγ a F µν ˜ F µν + α m e (cid:104)(cid:0) F µν F µν (cid:1) + 74 (cid:0) F µν ˜ F µν (cid:1) (cid:105) , where a represents the axion field with the mass m a , F µν is the electromagnetic field tensor,˜ F µν ≡ (cid:15) µνρσ F ρσ is its dual, m A is the photon’s effective mass , g aγγ is the axion-photoncoupling, α em is the electromagnetic fine structure constant, and m e is the electron mass. Inthe nonrelativistic plasma with nonzero electron density n e , the effective mass of the photonis given by m A = (cid:112) πα em n e /m e . The last term in eq. (5) is the Euler-Heisenberg effectiveLagrangian arising from the vacuum polarizability [50]. It describes the photon’s self-interaction Here we have ignored the term A µ j µ , in which j µ is the electromagnetic current density, since its physicalconsequence has been replace by the photon’s effective mass m A when photon propagates in the plasma [49].
4n the limit where the photon’s frequency is small in comparison with the electron mass m e .Its contribution to the m A is estimated as [51, 34] Q EH A = 7 α em π ω B B (6)where B cri ≡ m e /e = 4 . × G is a critical field strength, ω is the photon’s frequency. Forthe MWDs considered in this work, the strength of the magnetic field is about 10 ∼ G(see Table.1), which indicates B /B (cid:28)
1. Therefore, we can safely ignore the effects of theEuler-Heisenberg term.As shown below, the resonant conversion between the axion and photon occurs when thephoton’s effective mass m A is equal to the axion mass m a as m γ ( r ) = 4 πα n e ( r ) m e = m a . (7)By solving the eq. (1) and eq. (7), the resonant conversion radius r c can be expressed as r c = R WD + 21 . × (cid:20) .
634 + ln (cid:18) n e cm − (cid:19) + ln (cid:18) µ eV m a (cid:19)(cid:21) × (cid:18) T cor K (cid:19) (cid:18) M WD M (cid:12) (cid:19) (cid:18) R WD km (cid:19) − km . (8)The equation of motion for the axion and photon field can be derived by applying thevariational principle to eq. (5):¨ a − ∇ a + m a a = − g aγγ ˙ A · B , (9)¨ A − ∇ A + m A A = g aγγ ˙ a B − g aγγ ∇ a × ˙ A . (10)Here we have chosen the temporal gauge A = 0 and the Coulomb gauge ∇ · A = 0. Fol-lowing [33, 51] we adopt the radial plane wave solution a ( r, t ) = ie iωt − ikr ˜ a ( r ) and A (cid:107) ( r, t ) = e iωt − ikr ˜ A (cid:107) ( r ), where k = (cid:112) ω − m a represents the momentum of the axion. Note that we havetemporarily ignored the damping effects of the outgoing photon wave [33] and will include thiseffect in the finally answer.Plugging the above plane wave solutions into the eq. (9) ∼ (10) and using the WKB approx-imation | ˜ A (cid:48)(cid:48)(cid:107) ( r ) | (cid:28) k | ˜ A (cid:48) ( r ) | and | ˜ a (cid:48)(cid:48) ( r ) | (cid:28) k | a (cid:48) ( r ) | near r c , we can rewrite the above mixingequations into a more compact first-order ordinary differential equation [33, 51] (cid:34) − i ddr + 12 k (cid:32) m a − m A − ∆ B − ∆ B (cid:33)(cid:35) (cid:32) ˜ A (cid:107) ˜ a (cid:33) = 0 , (11)where ∆ B = Bg aγγ ω , ω = m a / (cid:112) − v c . Using the successive approximation method with theinitial conditions ˜ A (cid:107) ( R WD ) = 0 and ˜ a ( R WD ) = a , at the first order of ∆ B the eq. (11) can be5olved as [33, 51] p aγ ( r ) = (cid:12)(cid:12)(cid:12)(cid:12) i (cid:90) rR WD dr (cid:48) ωB ( r (cid:48) ) g aγγ k × e if ( r (cid:48) )2 k (cid:12)(cid:12)(cid:12)(cid:12) , (12)where f ( r (cid:48) ) = (cid:90) r (cid:48) R WD d ˜ r (cid:2) m a − m A (˜ r ) (cid:3) . (13)Since 1 / k (cid:28)
1, we can evaluate eq. (12) by the method of stationary phase. With thecondition df ( r (cid:48) ) /dr (cid:48) = 0, we get m a − m A ( r c ) = 0, which exactly is the resonant conversioncondition mentioned in eq. (7). By explicitly performing the integral in eq. (12) and includingthe damping effect of the photon wave, the probability of the axion-photon conversion at finite r is given by [33] p aγ ( r ) ≈ v c g aγγ B ( r c ) L × G (cid:18) r − r c L (cid:19) × (cid:40) , r < r cm a v c D ( r ) , r ≥ r c , (14)with L = (cid:112) πr c v c / (3 m a ) and D ( r ) = (cid:112) ω − m A . The function G ( x ) is defined by G ( x ) = (cid:0) + C ( x ) (cid:1) + (cid:0) + S ( x ) (cid:1) C and S integrals C ( x ) = (cid:90) x dt cos( πt / , S ( x ) = (cid:90) x dt sin( πt / . (16)The last term in eq. (14) represents the damping effect of the photon wave [33]. Consider thatall the astrophysical objects we are considering are very far away from us, i.e. ∼ p ∞ aγ = lim r →∞ p aγ ( r ) ≈ v c g aγγ B ( r c ) L . (17)Here we have used the fact that lim x →∞ G ( x ) = 1 and lim r →∞ m a v c /D ( r ) = v c . In this work we consider the following conversion picture: the axion DM particle starts outnon-relativistic ( ∼ − c ) far away from the MWDs and is accelerated as it moves toward the We have compared the numerical solutions by solving eq. (11) with this analytic approximation, and wefind that these two results matched very well. p ∞ aγ . Besides, because of the high density of the MWDs, all the incident photons that areconverted from axons will be totally reflected back out [33, 52].Once converted, the outgoing photons can be absorbed or scattered in the MWDs’ coronae,which is characterized by opacity. There are two important processes: the inverse bremsstrahlungprocess and Compton scattering. The absorption and scattering rate are given as [52]Γ inv ≈ πn e n N α ω m e (cid:18) πm e T cor (cid:19) / log (cid:18) T m A (cid:19) (cid:16) − e − ω/T cor (cid:17) , (18)Γ Com = 8 πα m e n e , (19)where n N is the number density of the charged ions . Then the probability for the convertedphotons to be scattered or absorbed during the propagation can be expressed as P s/a (cid:39) exp (cid:20) − (cid:90) ∞ R WD dr (Γ inv + Γ Com ) (cid:21) . (20)For the MWDs candidates in Table.1 we calculate that the P s/a is ∼ . P in a solid angle d Ω at r c can be estimated as [33] d P d Ω ≈ × p ∞ aγ ρ r c DM v c r c , (21)where ρ r c DM is the DM mass density at r = r c . The factor of two is from the fact that the DMmay convert into photons either on its way in to or out of the resonant layer. In the limit v / ( GM WD /r c ) (cid:28)
1, the ρ r c DM can be expressed as ρ r c DM = ρ ∞ DM √ π v (cid:115) GM WD r c + · · · , (22)where ρ ∞ DM is the DM density at infinity ∼ . / cm and v ∼
200 km / s is the DM virialvelocity. The raido flux density at the Earth can be estimated as S aγ = d P d Ω 1 B d , (23)where d represents the distance from the MWD to us, B = max { B sig , B res } is the optimizedbandwidth, B sig ∼ v m a / (2 π ) is the signal bandwidth, which is determined by the velocitydispersion in the asymptotic DM distribution, and B res is the telescope spectral resolution. Itis worth noting that in general B sig is smaller than B res (see Table. 2). Since the plasma is electric neutral, so we have n e = n N . M WD [ M (cid:12) ] R WD [ R (cid:12) ] T eff [K] B [MG] d WD [pc] S aγ [ µ Jy]RE J0317-853 1 .
32 0 . .
54 1 . . .
77 33 . .
937 0 . .
09 2 . .
02 0 . .
33 19 . .
13 0 . .
93 0 . µ Jy. Some typical parameters are taken as m a = 10 − eV, g aγγ = 10 − GeV − , n e = 10 cm − , T cor = 10 K. With these parameters we can derivethat B = 1 . ∼ (23) we can calculate the radio flux density for specific MWDs. InTable.1 we list the parameters of five MWD candidates as well as their radio flux densities S aγ . The following typical parameters are selected for the calculation: m a = 10 − eV, g aγγ =10 − GeV − , n e = 10 cm − , T cor = 10 K. With these parameters we can derive that B = 1 . S min = SEFD η s (cid:112) n pol B t obs , (24)where SEFD = 2 k B T sys A eff (25)is the system-equivalent flux density, n pol = 2 is the number of polarization, t obs is the observa-tion time, η s is the system efficiency, k B is the Boltzmann constant, T sys is the antenna systemtemperature, and A eff is the antenna effective area of the array. In this work, we take η s = 0 . B res for SKA are listed in Table.2.Next we propose to use the SKA phase 1 (SKA1) as a benchmark to search for the radiosignals converted from axion DM at MWDs. As shown in Table. 2, it consists of a low-frequency aperture array (SKA1-Low) and a middle frequency aperture array (SKA1-Mid) [37].The SKA1-Low covers the (50 , , , , , , , m A ∼ m a as well as the plasma density n e (cid:46) cm − , there will be an8arameters and sensitivity of the SKAName f [MHz] B res [kHz] (cid:104) T sys (cid:105) [K] (cid:104) A eff (cid:105) [m ] SEFD [Jy] S min [ µ Jy]SKA1-Low (50, 350) 1.0 680 2 . × . × B res , averaged system temperature T sys , averaged effective area A eff , SEFD, and minimum detectable flux density in the differentfrequency bands for SKA1.upper bound on the frequency of the photon f A (cid:46)
903 MHz. Therefore, we only need to usethe first frequency band SKA1-Mid B1 in our analysis. The specific parameters of the differentfrequency bands of the SKA are listed in Table. 2. - - - - - - - - - - - - - - - m a [ eV ] g a γ [ G e V - ] f = m a /( π ) [ MHz ] A D M X A D M XP r o j e c t i on CAST Q C D A x i o n SKA1
Figure 1: The projected sensitivity to g aγ as a function of the axion mass m a for SKA1 telescopeswith 100 hours observations of the WD 2010+310 is shown in the blue region. The lower masscutoff is set by the lowest available frequency of SKA, while the upper cutoff is set by requiringthe conversion radius to be larger than the MWDs’ radius. The QCD axion is predicted tolie within the yellow band. The limits set by CAST and ADMX (current and projected) areindicated by the gray and red regions, respectively.In Fig. 1 we show the sensitivity to g aγ for the WD 2010+310, which is the strongestsource in Talbe. 1. The blue regions show the physics potential of SKA1 with 100 hours ofobservation. Note that the lower mass cutoff is set by the lowest available frequency of SKA ∼
90 MHz, while the upper cutoff is set by requiring the conversion radius to be larger than theMWDs’ radius. We find that the axion DM in the mass range of 0 . ∼ . µ eV can be probedeffectively, and the limitation on g aγ is about (cid:46) − GeV − . For comparison, the limitationsgiven by the other two experiments, e.g. ADMX and CAST, are also shown in red and grayregions, respectively. In this work we propose to use MWDs as probes to detect the axion DM through the radiosignals. It is known that the MWDs can host very strong magnetic field (e.g. 10 ∼ G). Ifwe adopt the corona parameters that fulfill the X-ray constraints, such as n e ∼ cm − and T cor ∼ K, the effective photon mass is given by m γ ∼ µ eV (corresponding to the frequency ∼ GHz). We find that the resonant conversion may happen when axions pass through themagnetosphere that is a narrow region around the radius at which the photon effective mass isequal to the axion mass. Besides, we show that the effects of the inverse bremsstrahlung processand Compton scattering for the outgoing photons are negligible ( P a/s ∼ . ∼ µ eV, which happens to be inthe sensitive region of the terrestrial radio telescopes, such as SKA. In Sec. 4 we use the MWDWD 2010+310 as a target and show the sensitivity to g aγ from the future experiment SKA phase1 with 100 hours of observation. We find that the planned SKA1 will give an upper limit on theaxion-photon coupling g aγ of ∼ − GeV − for axion mass range of 0 . ∼ . µ eV, and thisresult may increase by more than one order of magnitude in the SKA phase 2 (SKA2) [37, 34].Note that all of the MWDs considered are isolated. In fact, one can consider another classof MWDs that occupy regions of high DM density and/or low velocity dispersion, such as thegalactic center, dwarf galaxies, and so on. In these regions, the DM density may be enhancedby a large factor. In addition, in dwarf galaxies the velocity dispersion of DM can be low as v ∼
10 km/s. These cases would significantly improve our results and are left for the futurework.
Acknowledgements
The work of JWW is supported by the research grant ”the Dark Universe: A Synergic Multi-messenger Approach” number 2017X7X85K under the program PRIN 2017 funded by theMinistero dell’Istruzione, Universit` a e della Ricerca (MIUR). The work of XJB, RMY, andPFY is supported by the National Key R&D Program of China (No. 2016YFA0400200), theNational Natural Science Foundation of China (Nos. U1738209 and 11851303).10 eferences [1] D. Clowe, M. Bradac, A. H. Gonzalez,M. Markevitch, S. W. Randall, C. Jones, andD. Zaritsky, “A direct empirical proof of theexistence of dark matter,” Astrophys. J. Lett. (2006) L109–L113, arXiv:astro-ph/0608407 .[2]
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