Radio-frequency Dark Photon Dark Matter across the Sun

Dark photon as an ultralight dark matter candidate can interact with the Standard Model particles via kinetic mixing. We propose to search for the ultralight dark photon dark matter using radio telescopes with solar observations. The dark photon dark matter can efficiently convert into photons in the outermost region of the solar atmosphere, the solar corona, where the plasma mass of photons is close to the dark photon rest mass. Due to the strong resonant conversion and benefiting from the short distance between the Sun and the Earth, the radio telescopes can lead the dark photon search sensitivity in the mass range of $4 \times 10^{-8} - 4\times 10^{-6} \, \rm{eV}$, corresponding to the frequency $10 - 1000 \, {\rm MHz}$. As a promising example, the operating radio telescope LOFAR can reach the kinetic mixing $\epsilon \sim 10^{-13}$ ($10^{-14}$) within 1 (100) hour solar observations. The future experiment SKA phase 1 can reach $\epsilon \sim 10^{-16} - 10^{-14}$ with $1$ hour solar observations.

Introduction-The ultralight bosonic fields are attractive dark matter (DM) candidates. Within them, the QCD axions, axion-like particles, and dark photons are well-studied scenarios [1,2]. Kinetic mixing dark photon is one of the simplest extension of new physics beyond the Standard Model (SM) via a marginal operator, which is well-motivated at low energies. It can also constitute DM [3][4][5][6][7] and may reveal the theories beyond the SM [8][9][10][11][12]. There are many searches looking for dark photon or dark photon DM. For mass 10 −9 eV, the dark photon DM can be constrained by the observation of astronomical radio sources [13], CMB spectrum distortion, BBN, Lyman-α and heating of primordial plasma [6,[14][15][16][17][18][19][20][21]. In the optical mass range of 0.1 − 10 eV, dark photon DM can be detected by the optical haloscope [22]. For dark photon DM with a mass larger than about O(10) eV, it can be absorbed in the underground DM detectors and produce electronic recoil signals [23][24][25][26]. Dark photon lighter than the temperatures at the center of stars can also be produced inside stars and suffer stellar cooling constraints [27][28][29][30]. Dark photons produced inside the Sun can be detected by DM direct detection experiments [30][31][32].
In this letter, we focus on the radio mass window (10 −8 − 10 −6 eV) for dark photon and assume it constitutes all the DM. This mass window is of particular interest because it overlaps with the regions that dark photon DM is naturally produced by mechanisms including the inflationary fluctuations [7,33], parametric resonances [34][35][36][37], cosmic strings [38], the misalignment with non-minimal coupling to the gravity [6,39] (see the ghost instability discussion in [40]), and production by inflaton motion [41]. The relevant searches for dark photon DM are haloscope experiments [42][43][44][45][46][47], dish antenna experiments [48,49], plasma telescopes [50] and CMB spectrum distortion [6,19]. The searches include direct detection of local dark photon DM in laboratories and observation on its impact in the early universe. Differently, we proposal to look for resonant conversion of dark photon DM A → γ at the Sun through the radio telescopes for solar observations. This is an indirect detection of dark photon DM signal from the closest astronomical object, the Sun. It provides competitive sensitivities even with existing radio telescopes and opens vast new parameter space with future setups.
Below the electroweak scale, the minimal coupling between the dark photon and the Standard Model particles can be described by the following Lagrangian density where F µν is the photon field strength, A is the dark photon field, F µν is the dark photon field strength, and is the kinetic mixing. With this mixing term, the dark photons can oscillate resonantly into photons in thermal plasma once the plasma frequency ω p ≈ m A . The plasma frequency for non-relativistic plasma relies on the electron density n e , where α and m e are the fine structure constant and electron mass, respectively.
In the Sun's corona, n e ∼ 10 6 − 10 10 cm −3 is shown in Fig. 1. Hence the range of the plasma frequency ω p is from 4 × 10 −8 to 4 × 10 −6 eV. If m A falls in this range, A can resonantly convert into a monochromatic radio wave in the corona, with the peak frequency corresponding to m A , which is in the range of about 10 − 1000 MHz. This frequency range happens to be in the sensitive region of the terrestrial radio telescopes, such as the LOw-Frequency ARray (LOFAR) [51] and Square Kilometer Array (SKA) [52]. Therefore, we propose to use radio telescopes to search for dark photon DM in this mass range.
Resonant Conversion in the Sun's Corona-The average conversion probability of a dark photon particle flying across the Sun's corona is the time integral of the decay rate of A → γ, written as Here we take average of the initial state of A . During the structure formation, the momentum direction of A is randomly rotated in the gravitational potential. Therefore, each mode (either transverse or longitudinal) has the equal probability, 1/3. Since only the transverse modes of photon can survive outside the plasma and propagate to the Earth, we only sum over the transverse polarizations in the final state. In the second line, v r is the velocity projected on the radial direction of the Sun. Due to the spherical distribution of n e , ω p only changes in the radial direction.
In Eq. (3), it utilized the quantum field method to calculate the 1 → 1 conversion and the matrix element M is derived by directly using the kinetic mixing operator 1 2 F µν F µν . Due to the momentum conservation, it only applies for the resonant conversion ω p = m A . An equivalent way to calculate the conversion rate is to solve the linearized wave equations for the photon and dark photon [53], which works for both resonant and non-resonant conversion. After applying the saddle point approximation, the result is the same as in Eq. (3). It can be explicitly shown that the non-resonant contribution is negligible. The detailed calculations for the two methods are given in the Supplemental Material. Finally, the above result is in agreement with the probability for inverse conversion γ → A [14].
Given the conversion probability, the radiation power P per solid angle dΩ at the conversion radius r c is where we consider DM density ρ DM = 0.4 GeV cm −3 completely composed of dark photon. Its average velocity v 0 220 km/s and the resonant conversion happens at the solar radius r c . The parameter z is the impact parameter at infinity for the incoming A , while b is the largest value of the impact parameter such that A can reach the conversion shell at r = r c . Due to the gravitational focusing enhancement, b = r c v(r c )/v 0 will be larger than r c in general, by a factor of about 2-3 in numeric calculations. The velocity of A at radius r c is given by v(r c ) = v 2 0 + 2G N M /r c , with G N being the gravitational constant and M the solar mass. The radial direction velocity at the conversion point is v r (z) = 2G N M /r c + v 2 0 − v 2 0 z 2 /r 2 c . The factor 2 in Eq. (4) counts the DM coming in and going out of the resonant layer. The converted photon from DM coming in will be reflected, because when the photon frequency is smaller than the plasma mass, the total reflection will happen.
The spectral power flux density emitted per unit solid angle is given as where d = 1AU is the distance from the Earth to the Sun, B is the optimized bandwidth, which is set as the larger one of the signal bandwidth B sig and the telescope spectral resolution B res , namely, B = max(B sig , B res ). The signal bandwidth B sig is due to the dispersion of the dark photons, which is normally smaller than B res . And the telescope spectral resolution B res depends on the property of the telescope.
The Photon Propagation-After the conversion, the propagation of the radio waves in the thermal plasma follows the refraction law, n sin θ = const, where n is the refractive index and θ is the incident angle. In nonrelativistic plasma, n can be expressed as where n(ω) equals to the group velocity of the radio waves, i.e. the photon speed. In the resonant region, the dark photon DM has a velocity of about v ∼ 10 −3 −10 −2 .
As a result, the refractive index at the resonant region is in the range of n res ∼ 10 −3 −10 −2 , which is much smaller than one. From Fig. 1, the electron density n e decreases quickly with the increase of r. Consequently, once the photon leaves the resonant region, the refractive index will quickly go back to 1, n out ∼ 1. Thus, according to the refractive law, the incident angle outside the resonant region can be written as Therefore, the direction of the converted photon is approximately along the gradient of the electron density −∇n e . Considering the conversion happened when the dark photon flies into the Sun and the converted photon moving into the denser region, we expect that electromagnetic waves is always total reflected away from the region where ω < ω p . Hence the above discussion of the final photon direction applies after the total reflection. If the electron distribution in the Sun's corona is spherical, the converted radio waves will all propagate along with the radial direction of the Sun. In this case, all the converted radio waves observed on the Earth's surface are from the center of the solar plate. However, there are turbulences and flares in the Sun's corona, which makes n e non-spherical and even evolve with time. It will affect the gradient direction of n e , thus modify the out-going direction of the photon. However, such modification should not have preferred directions, unless there are underline substructures. Therefore, we ignore those modifications and assume that in average, the out-going converted photons are isotropic. Once converted, the radio waves can be absorbed or scattered in the plasma, which is characterized by opacity. It turns out that the dominant absorption process is the inverse bremsstrahlung process. In the corona sphere, the temperature is as high as 10 6 K, which is much larger than the ionization energy of the hydrogen atom. As a result, the Born approximation can be used to calculate the absorption rate. Since we are interested in the radio wave frequency, it satisfies ω T m e . The absorption rate of the inverse bremsstrahlung process can be calculated as where the singularity at ω = 0 clearly shows the effect of the infrared enhancement. n N is the number density of charged ions. The logarithmic factor is from the longrange effect of the Coulomb interaction, which is cut-off by the Debye screening effect. The factor (1 − e −ω/T ) is due to the stimulated radiation. The above calculation is in good agreement with Ref. [55], except for a minor difference in the argument of the logarithmic factor. Besides the inverse bremsstrahlung process, there is also a contribution from the Compton scattering with the rate given as The Compton scattering can shift the photon energy by a few percent due to the velocity of the electrons. This change is normally larger than the optimized bandwidth. As a conservative consideration, we add up the two contributions and have the attenuation rate Γ att = Γ inv + Γ Com for the converted photon. Numerically, the inverse bremsstrahlung dominates. The survival probability P s for the converted photons to escape the Sun is to add the two rates, which represents the chance of the photons being not scattered or absorbed during the propagation. We terminate the integration at r max = 10 6 km + r ps due to the available electron density data [54], where r ps = 695, 510 km is the photosphere radius. Further extending the range will not change the result significantly, because the electron density is too low such that the interaction rate is negligible. Dark photon DM with mass > 4 × 10 −6 eV can also convert resonantly to photons in the Sun's chromosphere. However, the temperature of the chromosphere is only about 10 3 K, which is about three orders of magnitude smaller than the temperature of the corona. This makes the inverse bremsstrahlung absorption much stronger in the chromosphere than in the corona. Furthermore, the electron number density, as shown in Fig. 1, is also orders of magnitude larger, and so does the density of charged ions. Therefore, the radio waves produced in the chromosphere cannot propagate out.
In summary, the dark photon DM's resonant conversion happening in the Sun's corona can propagate to the Earth's surface. In terms of distance, the region 2300 km above the photosphere (higher than the solar transition region) is our signal region. This corresponds to the unshaded region in Fig. 1. The relevant observed photon frequency is 1000 MHz and dark photon mass is m A 4 × 10 −6 eV. In the above discussions, we only use the well-accepted electron density and temperature profiles as shown in Fig. 1. They are good approximations and have acceptable uncertainties for the signal calculation. More discussions on the solar models and the corresponding uncertainties are given in the Supplemental Material [56][57][58][59][60][61][62].
The sensitivity of Radio Telescopes-The minimum detectable flux density of a radio telescope is [63] where n pol = 2 is the number of polarization, t obs is the observation time, and η s is the system efficiency. In our analysis, we take η s = 0.9 for SKA [63], and η s = 1 for LOFAR [64]. The values of the telescope spectral resolution B res for LOFAR and SKA are listed in Table I, which are much larger than the signal bandwidth B sig given in Eq. (6). Therefore, in our calculation, we always have B B res . In Eq. (12), SEFD is the system equivalent flux density, defined as where k B is the Boltzmann constant, T sys is the antenna system temperature, A eff is the antenna effective area of the array, and T nos is the antenna noise temperature increase when pointing to the Sun. We propose to use the radio telescope arrays SKA and LOFAR to search for the radio waves converted from dark photon DM at the Sun's corona. We consider SKA phase 1 (SKA1) as the benchmark of a future telescope to study the reach of dark photon DM. It has a low-frequency aperture array (SKA1-Low) and a middle frequency aperture array (SKA1-Mid) [63]. SKA1-Low covers the (50, 350) MHz frequency band. SKA1-Mid covers six frequency bands with frequency ranges (350, 1050) MHz, (950, 1760) MHz, (1650, 3050) MHz, (2800, 5180) MHz, (4600, 8500) MHz, and (8300, 15300) MHz. In this analysis, to partially cover the frequency range of the converted radio wave, we use the SKA-Low and the first two frequency bands of SKA-Mid, denoted as Mid B1 and Mid B2, respectively. LOFAR, as an existing radio telescope, can be used for dark photon hunting as well. Indeed, one of the key science projects for LO-FAR is to study solar physics. In its radio spectrometer mode, the intensity of the solar radio radiation over time is recorded. LOFAR covers the frequency ranges of (10, 80) MHz and (120, 240) MHz.
To calculate the minimum detectable flux S min given in Eq. (12), we need to determine the corresponding detector parameters, such as the telescope spectral resolution B res , the system temperature T sys , the solar noise temperature T nos and the effective area A eff . Table I lists the average values of these parameters for each telescope, and the details to achieve these parameters are given as follows: • spectral resolution B res : due to 2.5 × 10 5 fine frequency channels in SKA1-Low, its channel bandwidth can reach B res = 1 kHz, while the bandwidth for SKA1-Mid B1 and SKA1-Mid B2 are set to B res = 3.9 kHz [63]. For LOFAR, The spectral resolution B res is taken as 195 kHz [51,65].
• system temperature T sys and effective area A eff : the system temperature for SKA1-Low can be approximated as T Low sys ≈ T rec + T sky , where the sky noise T sky ≈ 1.23 × 10 8 K (MHz/f ) 2.55 and the receiver noise T rec = 40 K + 0.1T sky [63]. For SKA1-Mid, the average system temperatures for bands 1-5 are 28 K, 20 K, 20 K, 22 K, and 25 K, respectively [63]. The effective area A eff is derived using the system sensitivity A eff /T sys in [51]. The parameters of LOFAR like A eff can be directly found in [51], while T sys can be inferred from SEFD. Note that in the numeric calculation, the parameters A eff and T sys depend on the frequency.
• solar noise temperature T nos : T nos can be calculated under the blackbody assumption for a quiet Sun [66,67]. The ratio of T nos and the brightness temperature of the quiet Sun T b , T nos /T b , has been given for different half-power beamwidth (HPBW or -3dB beam width) and beam pointing offset. It is easy to understand that the ratio should always be smaller than 1, because the noise temperature cannot be higher than the source itself. The result shows that for the antenna with HPBW smaller than the angular diameter of the Sun disk, this ratio is close to 1 when beam is on the solar disk. The HPBW for SKA1-Low current design is about 4 arcminutes at the baseline frequency 110 MHz [68], while the angular diameter of the Sun is as large as 31.8 arcminutes. Therefore, SKA1-Low can be considered as a high-gain antenna with a very narrow beam. The SKA1-Mid has even smaller HPBW than SKA1-Low, thus throughout the calculation, we  Figure 2. The sensitivity reach of dark photon dark matter for LOFAR (blue) and SKA1 (red) telescopes with 1 or 100 hours solar observations. The constraints are obtained from the existing haloscope axion searches [6,[42][43][44][45][46], recent WIS-PDMX dark photon searches [47] and the CMB distortion [6,19]. For both signal and existing constraints, ρDM = ρ A is assumed.
take T nos = T b . The spectral brightness temperature T b (f ) is calculated using the quiet Sun flux density from [67,69]. Regarding the LOFAR beamwidth, the HPBW of LOFAR ranges from (1.3, 19) degrees [51]. Therefore, it is much larger than the angular diameter of the Sun. Following the procedure of [67], we use the antenna diameters of LOFAR to calculate the ratio T nos /T b for the Sun as a function of frequency. This ratio is far smaller than one because much of the photon flux goes outside the HPBW. It is important to remark that this ratio should also work for the signals because both background and signal emissions are originated from the Sun. We find that for the frequency smaller than 55 MHz, the system temperature T sys dominates over the solar contribution T nos .
Results and Discussions-Requiring S sig ×P s = S min , one can obtain the sensitivities on the kinetic mixing from radio telescopes. The sensitivity reaches of dark photon DM for SKA and LOFAR are given in Fig. 2, where both the signal and constraints are plotted under the assumption ρ DM = ρ A . The blue regions show the physics potential of LOFAR with 1-hour and 100-hours observation time, which is 1-2 orders of magnitude better than the existing limits from haloscope limits [6,[42][43][44][45][46], recent WISPDMX constraint [47] and the CMB distortion [6,19]. SKA1 has smaller T sys , larger A eff , and better spectral resolution B res . Its sensitivities with 1hour and 100-hours observation time are shown in the red shaded region. With the same operation time, it can improve the reach of by another one or two orders of magnitude compared with LOFAR.
In conclusion, we propose to search for the radiofrequency dark photon DM from 10 − 1000 MHz, with radio telescopes. In this frequency regime, we show that the dark photon DM can convert resonantly into monochromatic radio waves in the solar corona. In this mass window, the existing LOFAR telescope can achieve a sensitivity of ∼ 10 −13 − 10 −14 on the kinetic mixing , and the planned SKA1 can achieve a sensitivity of ∼ 10 −14 − 10 −16 . Despite SKA and LOFAR, other radio telescopes that may be used in the dark photon DM search are MWA [70], Arecibo [71], JVLA [72] and FAST [73]. In future, the SKA phase 2 [52] can further improve the SEFD sensitivity to sub µJy and explore more parameter space of the dark photon DM. In this supplemental material, we show the derivation of the conversion probability of a dark photon particle P A →γ for the resonant conversion using quantum field method and linearized wave method. We also discuss the solar model we used in the study and compare it with the experimental observations. Lastly, we discuss the uncertainties in the calculation.
The first methodwe use the quantum field method to calculate the 1 → 1 conversion rate Γ A →γ . We further integrate this rate with the time it takes to fly across the solar corona and obtain the conversion probability P A →γ .
= dt 2ω We take average of the initial dark photon state. The factor 1/3 is the initial spin average for A . For the final state, we only sum over the transverse modes. After the structure formation, the A dark matter has fallen into the gravitational well of galaxies and clusters. The gravitational forces changes the momentum of A together with its direction. Therefore, we assume in the solar system the A polarization has equal probability for two transverse modes and one longitudinal mode. The amplitude M is given as For 1 → 1 process, the energy-momentum conservation implies p A = p γ ≡ p. For photon in the final states, we only count two transverse modes, because longitudinal photon cannot propagate to the Earth. Therefore, we have for the amplitude square. For Eq. (15), after integrating d 3 p, there is one δ function left for energy conservation. Together with the integration of dt, it has where we have used E γ = p 2 + ω 2 p and ω p is the plasma frequency which is location dependent. In our assumption, the electron density distribution is spherical symmetric, thus ω p only depends on radius r. We can further apply ∂t = v −1 r ∂r, because only radial movement changes ω p . Putting all the elements together, we arrive at the final result Since this is 1 → 1 process, the momentum conservation requires ω p (r c ) = m A , that the process happens at resonant region r c .
The second method-After the quantum field calculation, we use the linearized wave method to calculate the conversion probability. After eliminating the kinetic mixing term by redefinition, one can arrive at the coupled wave equations, We consider solutions with fixed frequency, ω. We define k = (ω 2 − m 2 A ) 1/2 . Then the solution of Eq. (20) can be written as A(r, t) = e i(ωt−rk)Ã (r) and A (r, t) = e i(ωt−rk)Ã (r). The plasma frequency is slowly varying compared with the k. As a result, we have |∂ rÃ (r)| k|Ã(r)| and |∂ 2 rÃ (r)| k|∂ rÃ (r)|, and the same is true for A field. Then, we can use the WKB approximation to rewrite Eq. (20) as a first-order differential equation, where Since H I is much smaller than H 0 , the first-order solution for the conversion probability is The result can be further simplified using the saddle point approximation, where f (r 0 ) = 0 and f (r) ≈ f (r 0 ) + 1 2 (r − r 0 ) 2 f (r 0 ). Recognizing f (r) = i r 0 dr , the probability P A →γ in Eq. (23) can be simplified to Eq. (19). One can explicitly expand f (r) to the next order and show that the correction is about f (r 0 )/(f (r 0 )) 3/2 ≈ v(r c )/(k∆r c ) 1/2 , where v(r c ) is the dark photon velocity at the resonant region, k −1 can be seen as the de Broglie wave length, and ∆r c is the resonant length. The dark photon velocity is about 10 −3 times the speed of light. The de Broglie wavelength of the dark photon is about 0.1 − 10 km. The size of the resonant region is at the scale of about 10 3 km. Therefore, the next-leading order effect in our case is suppressed by a factor of 10 −5 .
Clearly, the wave method is in good agreement with the quantum field method, which calculates only the resonant contribution. Another way to understand this is that outside the resonant region, the phase e −i dr··· in Eq. (23) oscillates quite fast, which cancels themselves in the probability amplitude.
Besides this linearized equation technique, one may also solve it similarly as neutrino oscillations with the mass matrix given in Eq. (20), see Ref. [14]. The result is in agreement with the above two methods .
The solar model-The A → γ conversion happens in the solar corona. Like the atmosphere on the Earth, it is a complex and vibrant environment. The corona can be divided into three regions. The active region holds most of the activities but makes up only a small fraction of the total surface area, like the cities on the Earth. The coronal hole region are the northern and southern polar zones of the Sun. The quiet Sun region is the rest of the surface area, which is not static but has minor dynamic processes with small scale phenomena comparing to the active regions.
We focus on the quiet Sun region for our study, because it has less active events like solar flares. Although it is not fully quiet, with some minor dynamic processes, we model it as a spherical symmetric and hydrostatic, in which the gas pressure is balanced by the gravitational force and is static in time. Indeed, the quiet Sun region does show perfect hydrostatic equilibrium, see Refs. [56,57].
The relevant quantities in our calculations are the electron number density n e and temperature T profiles. We take the profiles from Ref. [54], where they have calculated the temperature T and hydrogen density n H profiles for the quiet sun regime based on photospheric model from Ref. [56] and coronal model from Ref. [58,59]. With spherical symmetry assumption, hydrostatic equilibrium and radiative transfer assumption, they calculated the electron number density profile n e . We have not used the Pakal code developed in Ref. [54], but only n e and T profiles which are the input for Pakal code. Those profiles have also been calculated by different groups [60] using chromosphere model from Ref. [61] and again coronal model from Ref. [58,59]. Their results are in agreement with each other.
More importantly, their predictions on profiles have been verified by various atomic lines observations at soft X-ray range [57] and extreme ultra-violet range [56]. For example, the n e profile for quiet Sun is in good agreement with the various observations [57] and the T profile gives the temperature in the right range (1-2 million Kelvin) [57] comparing with the extreme-ultraviolet line observations [62]. Therefore, the profiles used in the paper are simple and reliable.
The spherical and hydrostatic profile or model we used for solar atmosphere is not the most recent one, but is simple and consistent with the atomic line observations. The more recent development of the solar atmospheric model includes changing from hydrostatic equilibrium to hydrodynamic, by adding the continuity equation due to particle number conservation. The magnetic field is also very important for plasma movement. The combination of these effects is called magneto-hydrodynamics (MHD) model. For example, the particles not energetic enough will flow along the magnetic flux line. So, the plasma is treated as a fluid governed by gravitational, electromagnetic interactions. However, since we are only focusing on the quiet Sun region, and the relevant quantities are the density and temperature profiles, we believe that the spherical and hydrostatic model already provides a good description of the quiet Sun region.
The uncertainties in the calculation-Regarding the uncertainties from the solar model, the relevant errors come from n e and T profiles. The n e profile for quiet Sun is good within a factor of a few from the various observations [57]. Its square root determines the plasma frequency, which only shifts the location of resonant region. Its derivative on radius determines the conversion probability and the slope does fit nicely with the observational data [57].
On the other hand, the T profile determines the absorption of photon from inverse bremsstrahlung process, where the absorption rate is proportional to T −3/2 . The column emission measure, which is proportional to the line-of-sight integral of n 2 e , can be extracted from the broad range of extremeultraviolet and soft X-ray line observations. Its differential distribution over temperature for the model prediction [57] and the extreme-ultraviolet observation data [62] are peaked around log 10 (T[Kelvin]) ∼ 6.3 and 6.1 − 6.2 respectively, for low corona of quiet Sun. Therefore, the temperature profile is in pretty good agreement with data.
The other uncertainty in solar model is related to the spherical symmetric and hydrostatic assumption. In reality, the Sun has a vibrant environment, that the turbulences and flares in the corona can make n e non-spherical and even evolve with time. This will distort the spherical distribution of n e and leads to non-radial photon propagation direction outside the Sun. There are several reasons to alleviate the above concerns. Firstly, we have already chosen the quiet Sun region, which has the least dynamic activities comparing to the active regions. Second, the hydrostatic assumption together with spherical symmetry has been explicitly tested by Refs. [56,57] using soft X-ray and extreme-ultraviolet line observations. Therefore, the static and spherical symmetric picture is a good approximation in the sense of time and spatial average for the quiet Sun region. Thirdly, the activities can modify the above quantities but should not have preferred directions, unless there are underline substructures. Therefore, in the sense of the spatial average, out-going direction of the converted photon is isotropic.
As a result, we summarize for the solar corona model that we have used a simple model for the quiet sun. It is hydrostatic and spherical symmetric, but it is sufficient for our purpose of DM search. It only needs 1D profile which significantly simplify the signal calculation and the uncertainty for this model should be within a factor of a few.
Next, we move to the uncertainties from dark matter model. The first uncertainty is the local DM density. We have used the value ρ DM = 0.4 GeVcm −3 , which is an average number from the N-body simulation from the DM study. It is possible that the solar system sits in the DM substructure, that the density is boosted than other region. It is also possible that the density is much smaller than the average value due to fluctuation from the structure formation. As a result, the density provides the uncertainty as large as a factor of few.
The second possible uncertainty is from the local DM velocity. In our calculation, the inverse of velocity v −1 from conversion probability is canceled by the velocity in the DM flux, when calculating the radiation power. Therefore, the signal is less affected by the DM velocity comparing with the density.
Therefore, we conclude that the uncertainties from DM model that provides uncertainties from its density, which is a factor of a few. Together with the uncertainties from solar model, the predicted signal has an uncertainty of a few and should be within one order.