Listening for Dark Photon Radio from the Galactic Centre

Dark photon dark matter that has a kinetic mixing with the Standard Model photon can resonantly convert in environments where its mass $m_{A'}$ coincides with the plasma frequency. We show that such conversion in neutron stars or accreting white dwarfs in the galactic centre can lead to detectable radio signals. Depending on the dark matter spatial distribution, future radio telescopes could be sensitive to values of the kinetic mixing parameter that exceed current constraints by orders of magnitude for $m_{A'} \in \left(6\times 10^{-6},7\times 10^{-4}\right)$ eV.

GBT and ALMA, operating in GHz to THz frequencies, could surpass current constraints on the kinetic mixing for a wide range of DP masses.
The remainder of this paper is structured as follows. In Sec. II we provide a schematic overview of the resonant conversion process in plasma. In Sec. III and IV we describe the environments of neutron stars and white dwarfs and details of the conversion process there. In Sec. V we study the sensitivity of radio telescope to dark photon dark matter. The uncertainties on the dark matter profile are discussed in Sec. VI and the white dwarf environments are revisited in Sec. VII. Finally, in Section VIII we discuss our results and describe future refinements to our analysis. Technical material is provided in the Appendices: we derive the equations governing the conversion in generality and show how these reduce to the expressions in the main text. We also discuss the propagation of photons after production and analyse the impact of additional processes that can affect the conversion.

II. THEORETICAL FRAMEWORK
We consider a DP with Lagrangian density where F (F ′ ) is the SM photon (DP) field, and we assume that the dynamics that give rise to the DP mass m A ′ are decoupled. The kinetic mixing, with coupling κ, allows conversion between photons and DPs. In a stellar environment this process can be enhanced in the presence of an electron plasma, the properties of which are affected if there is a magnetic field. 1 The dynamics of the plasma are described by the permittivity tensor χ χ χ p , reviewed in Appendix A. DPs and photons of energy ω . For each telescope we plot two lines: the solid lines assume a gNFW dark matter profile and the dashed lines assume a density spike near the galactic centre (the BL limit is only visible with a density spike). The grey line depicts the cosmological constraint on dark photon dark matter (DPDM) from Arias et al [9]. Overlaid shaded regions are limits from haloscopes [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37]  propagating in the z direction evolve according to where A A A (′) ≡ (A x (′) , A y (′) , A z (′) ) and D 2 A A A ≡ ∇(∇ · A A A)/ω 2 . Conversion is efficient in locations where the photon and DP dispersion relations match, and in these regions the fields can be written as A A A (′) (x x x, t) = A A A (′) (x x x)e iωt−ikz . To calculate the photon field sourced by DPs we use the WKB approximation. This gives, schematically, where s is the direction in which the amplitude of the photon field increases, which may not coincide with z in an anisotropic plasma. The relevant photon polarisation, labeled j, along with the functions h j (set by the plasma frequency) and g j (set by the mixing of DPs and photons) depends on the particular environment. Eq. The integrand in Eq. (4) is highly oscillatory so the dominant contribution is from the position where the phase in the exponential is stationary s = s c , i.e.
which sets the condition for resonant conversion. In what follows we make the simplifying assumption that the photon and DP both travel on exactly radial trajectories in the conversion region, which for an approximately isotropic plasma implies ∂ s = ∂ r , where r is the distance from the star's centre. For an isotropic plasma the effects due to the true trajectories not being exactly radial are small. Moreover, we expect that the corrections due to the non-isotropic environment of an accreting WD are relatively small although future detailed modeling would be valuable. We assume these relations also hold in NSs, although there may be important effects in this case [55,56]. Additional details of the conversion process are provided in Appendices B and C, and the validity of our assumptions is examined in Appendix I.

III. CONVERSION IN NEUTRON STARS
We describe NS magnetospheres by the Goldreich-Julian (GJ) model [69], which is believed to be accurate in the vicinity of the star [70][71][72]. The charge density at position r r r above a NS's surface is n GJ ≃ |2Ω Ω Ω · B B B(r r r)/e|, where the angular velocity Ω Ω Ω is related to the spin period P by |Ω Ω Ω| = 2π/P (we drop a term of relative importance ∼ |Ω Ω Ω|r, which is negligibly small). The magnetic field B B B(r r r) has a dipolar distribution such that its projection on the rotation axis Ω Ω Ω · B B B(r r r)/|Ω Ω Ω| = 1 2 B 0 (r 0 /r) 3 β, where r 0 the radius of NS, and β ≡ 3 cos θ pm m m ·r r r − cos θ m . Here θ p is the angle between the position vector r r r and the rotation axis, θ m is the angle between the magnetisation axism m m and the rotation axis, andm m m ·r r r = cos θ m cos θ p + sin θ m sin θ p cos Ωt.
We assume the free electron number density n e = n GJ ; the resulting plasma frequency is For typical NSs B 0 ∼ 10 14 G and P ∼ 1 s, ω p ≲ 70 µeV i.e. ω p /(2π) ≲ 17 GHz. The typical strong magnetic field in a NS crucially affects dark photon conversion through its effects on the plasma. In the presence of such a field, only DP polarisations in the plane spanned by the DP propagation direction and the magnetic field can efficiently convert to photons, and therefore the resultant photon signals are polarised. As before, we take the DP to be propagating in the z-direction, and we fix the magnetic field to lie in the y-z plane at an angle θ tok k k. The induced photon field has a transverse polarisation A y and a longitudinal polarisation A z that are interwoven with A z = −m 2 A ′ cot θA y /ω 2 (see Appendix B for details). In the non-relativistic limit A A A is aligned with B B B and its amplitude increases in a direction s that is orthogonal to B B B. The photon field's dispersion relation implies a superluminal phase velocity and, given its polarisation, it therefore corresponds to the Langmuir-O (LO) mode [73], which evolves adiabatically into transverse waves as it propagates out the NS.
The wave equation of A y is given by Eq. (2) with where ξ = sin 2 θ/(1 − ω 2 p ω 2 cos 2 θ). Hence, the resonance condition is m 2 A ′ = ξω 2 p and, approximating ω ≃ m A ′ , the resonant conversion radius The conversion probability where β ′ ≡ (cos θ − sin 3 θ) 2 / sin 2 θ and v c is the DP velocity at r c . We assume the DP velocity has a Maxwell-Boltzmann distribution in the galactic rest frame, Starting from an asymptotic velocity v i far away from NS of mass M NS , the infalling DP accelerates to near r c , so v c ≫ v i . By Liouville's theorem, the DP's phase space density is conserved during infalling, so its density near r c is enhanced to where the factor of 2 accounts for conversion when approaching and leaving the NS. We assume that the magnetisation axis aligns with the rotation axis, i.e. θ m = 0, so in the GJ model β = 3 cos 2 θ p − 1 and cos θ = 2 cos θ p / 3 cos 2 θ p + 1. The conversion probability diverges as θ tends to 0 due to the relation between A y and A z . This would be regulated by the inclusion of vacuum polarisation effects, the variation of the resonance condition within the conversion length, or the back conversion of photons to DPs. However, we simply impose p NS (θ) ≤ 1000p NS (π/2) or 1.8 • ≤ θ ≤ 178.2 • , which is expected to be conservative as it leads to p NS (θ) ≪ 1 in all the parameter space of interest. When considering the signal from a collection of stars, we average over the θ p angular dependence in Eq. (11). We also average over the asymptotic dark photon velocity: . The mean emission power per NS is then where we take r 0 = 10 km and M NS = M ⊙ . The produced photons travel out of the neutron star with negligible absorption or scattering.

IV. CONVERSION IN ACCRETING WHITE DWARFS
Mass accretion onto a WD from a companion main sequence star converts gravitational energy to heat and produces a hot and dense plasma [75]. We focus on nonmagnetic accreting WDs, in particular non-magnetic cataclysmic variables (CVs). In these, the accreting mass forms a disk, which, near the surface of the WD, is decelerated resulting in a boundary layer. If the accretion rateṀ ≲ 10 16 g/s, the boundary layer is thought to be an optically thin plasma that is heated to a temperature T s ≃ 10 8 K, explaining the observation of hard X-rays from such systems [76][77][78][79]. The boundary layer extends from the surface of the WD at r = r 0 up to r 0 +b. Throughout this region the gravitational potential is balanced by the radial pressure gradient, which implies [79] b ≃ 600 km T s b also sets the scale over which physical properties vary in the radial direction, i.e. ∂/∂r ∼ 1/b [79]. We assume the α-disk model [79,80]. In this, the disk's scale height at the outer edge of the boundary layer H = 2 × 10 3 km α −1/10 dṀ 3/20 16 r 0 + b 10 5 km and the matter density at the centre of the disk just outside the boundary layer n d = 2 × 10 16 cm −3 α −7/10 dṀ 11/20 16 whereṀ 16 =Ṁ /(10 16 g/s), α d parameterises the disk viscosity (we set this to 1), f r = (1 − r 0 /(r 0 + b)) 1/4 , and we fix M WD = M ⊙ . The matter density in the transverse direction drops as n d exp(−h 2 /H 2 ), where h is the distance perpendicular to the disk. Given that the boundary layer is fed by the accretion disk, we assume that the electron density inside the boundary layer has the same transverse profile, i.e. [78] and that the temperature is constant throughout. Because there is not a strong magnetic field, the longitudinal polarisation of the photon does not propagate in the boundary layer plasma and only conversion of transverse DPs is relevant for the signal. This is described by Eq. (3) with The resonance condition is m 2 A ′ = ω 2 p , which sets the conversion radius to be where n 0 (h) = n d exp(1 − h 2 /H 2 ). The resulting dark photon-photon conversion probability is n 0 does not enter p WD and instead simply sets the maximum DP mass for which conversion is possible. Resonant conversion occurs for m 2 A ′ m e /(4πα) ≤ n d , taking place on both sides of the disk.
The photons produced can be absorbed by inverse bremsstrahlung as they travel out of the WD, and we define p IB s to be the survival probability (an explicit expression is given in Appendix H). Additionally, because the boundary layer is not exactly isotropic, the photons will be deflected slightly in the direction of the density gradient, however we leave a detailed modelling for future work and continue to assume exactly radial trajectories. Given the boundary layer's finite transverse depth, photons are only emitted in some directions; the power per solid angle along these is analogously to Eq. (11), where we have fixed r c ≃ r 0 = 0.01R ⊙ and M WD = M ⊙ .

V. SIGNALS AND DETECTION SENSITIVITY
We consider the radio signals from compact stars in the GC, where the dark matter density is greatly enhanced relative to the Earth's local environment. To quantify the uncertainties from the dark matter density distribution, we compare the signals from two representative profiles: the generalised Navarro-Frenk-White (gNFW) profile [81,82] and a density spike near the central black hole [83][84][85]. However, we note that a cored profile is not ruled out [86], and, as we will discuss in Sec. VI, would lead to weaker limits. We assume that the DP makes up the entirety of the dark matter abundance.
The dark matter distribution inferred from the circular velocity profile of luminous stars can be well described by a generalised Navarro-Frenk-White (gNFW) profile [81,82] where R is the distance to the galactic centre, the dark matter density local to the Earth ρ 0 = 0.47 GeV/cm 3 , and the Earth's distance to the galactic centre R 0 = 8 kpc. The scale radius R s = 20 kpc and the profile index γ = 1.03 from fits to data (the choice of the parameters is motivated by the fit using 'CjX' baryonic morphology in [81], which is also consistent with the more recent analysis [82]). This yields a dark photon dark matter density of 1.03 × 10 5 GeV/cm 3 at a distance 0.1 pc from the galactic centre. However, the dark matter density near the galactic centre supermassive black hole, Sgr A * , is highly uncertain. If the central supermassive black hole grows adiabatically, the dark matter density within a pc of the galactic centre can be enhanced by orders of magnitude, forming a dark matter spike [83][84][85] (although such a spike is not guaranteed to form [87] and might not survive to the present day [88,89]). For non-annihilating dark matter the spike density is characterised by a power low at distances R < R sp where we take the spike extension R sp = 80 pc, and γ sp = 7/3, which yields a dark photon density of 6.2 × 10 8 GeV/cm 3 at 0.1 pc. The difference between the gNFW and spike profiles gives a quantitative estimate of the uncertainties on the dark matter density in the galactic centre. The distribution of WDs and NSs in the GC is detailed in Appendix F. Because only a small fraction of WDs are accreting, we consider the signal from an individual star. The analysis of Chandra in [90] suggests that there are about 11 hard X-ray point sources within 8 ′′ (0.3 pc) of the GC, which are likely to be a mixture of magnetic and non-magnetic CVs [90,91]. It is reasonable to assume at least one non-magnetic CV will be aligned such that the radio signal is observable given that the X-ray emissions are expected to be similarly directional, and we conservatively consider the signal from a non-magnetic CV at 0.3 pc. We assume a boundary layer temperature T s = 4.4 × 10 8 K in a WD with mass M WD = 0.83 M ⊙ and the expected radius r 0 = 0.01 R ⊙ [92,93], inspired by the study in [91]. We also takeṀ 16 = 1, which yields b = 3188 km, H = 107 km, n d = 4.6 × 10 17 g/cm 3 . Note that the boundary layer plasma could be partly relativistic at such high temperature, where the relativistic effect will modify the dielectric tensor of the plasma, which in turn affects the dispersion relation and hence the propagation of photons in the plasma [94]. We leave a more dedicated study of such effects in future work. We estimate the importance of absorption of converted photons by inverse bremsstrahlung as they travel out through the boundary layer by assuming a travelling distance of 500 km (absorption in the cold accretion disk is expected to be less efficient). The resulting attenuation is significant for m A ′ ≥ 10 −4 eV, but we stress the true effect depends on the production location and a complete model and a full simulation would be required for a fully reliable analysis. The signal power S sig is the energy flux at Earth divided by the bandwidth B, which we take to be the maximum of signal line-width B sig and the detection bandwidth of a particular telescope B det . For a single WD, energy conservation For NSs we consider the collective signal from all stars that are a distance R between R min and R max from the GC. This leads to where d = 8 kpc is the distance of the Earth from the GC. We take the distributions of the magnetic field f B and spin period f P of NSs to be log-normal centred at log 10 B 0 /G = 12.65, log 10 P/ms = 2.7, with standard deviations σ log 10 B0/G = 0.55 and σ log 10 P/ms = 0.34, respectively [95][96][97][98]. The lower limit on the integral over to facilitate resonant conversion. The population of compact stars in the GC has been studied with Monte Carlo simulations [99]. For NSs the population distribution n NS can be fit with a power law that is accurate for R > R min = 0.025 pc. We assume the NSs have a radius of 10 km and an average mass of 1.4 M ⊙ . The signal from a collection of NSs is broader because the frequencies from the individual sources are Doppler shifted differently due to the motion of the stars, leading to a total width B sig ≃ m A ′ σ s ∼ 10 −3 m A ′ , where σ s is the stars' velocity dispersion [54]. We conservatively use the velocity dispersion at R = 0.1 pc, outside which most of the stars reside.
To set a limit on, or find evidence for, a DP we require the signal power to be larger than the minimum detectable flux density of a radio telescope. This is defined as the fluctuation of the telescope receiver output in a frequency band cumulated over the observation time.
Given the narrow bandwidth of the signal, the minimum detectable flux could be orders of magnitude below the continuous background [100]. In Fig. 1 we plot the sensitivity reach of SKA [101], GBT [102] and ALMA [103] with 100 hours of observation of the GC, which together cover a broad frequency range from 50 MHz to 950 GHz, as described in Appendix E. For the signal from NSs we set R max = 3 pc, motivated by the field view of GBT (at high frequencies ≳ 10 GHz this may require beams to be combined or a prolonged observation time). Additionally, the signal from NSs allows us to set constraints on dark photons in the mass window of 15 to 35 µeV using the flux density limit data (with background correction) based on observations of the GC with GBT [60] in the Breakthrough Listen (BL) project, which covers a range of 2.9 pc from the GC.
We note that non-accreting WDs can also lead to resonant DP conversion. One possibility is conversion in a WD's atmosphere, which consists of a dense plasma. Due to the relatively low temperatures, the signals from this environment are weaker than those from an accreting WD, but future observations might still surpass the cosmological constraint depending on the assumed dark matter density profile. Additionally, some isolated WDs might be surrounded by a hot corona, which would lead to strong signals if present in a sizeable fraction of stars. However, as yet there is no compelling evidence of such corona and instead only upper limits on the would-be plasma densities for particular WDs. Details of the signals from these environments and the resulting detection prospects are presented in Appendix H.

VI. EFFECTS OF THE DARK MATTER DENSITY PROFILE
In Sec. V we consider two cuspy dark matter density profiles, the gNFW profile and a density spike in the galactic center. Observation shows galaxies with high stellar density are more likely to be cuspy than cored [104], and recent studies varying baryon models indicate that the NFW profile generally fits the rotation curve data better than the cored Burkert profile [105] in the Milky Way. Meanwhile, dedicated simulations including baryon feedback suggest that the dark matter profile might be even further contracted close to the galactic center than in an NFW profile [106] (which, although we do not investigate this possibility in detail, would strengthen our projected sensitivity). However, we note that a cored profile in the Milky Way is not ruled out [86]. To explore the impact of this scenario, we assume the dark matter in the Milky Way follows Burkert where we take ρ s = 1.79 GeV/cm 3 and the core radius R c = 7.8 kpc, corresponding to the 'B4D4C1' baryon model in [105] which produces the minimum χ 2 in the fit for Burkert halo. The resulting dark photon sensitivity is displayed in Fig. 2 with dotted lines. We also note that, even assuming a gNFW profile, there is a residual uncertainty on the fit from the choice of baryon model (or morphology). To illustrate the impact of this, in Fig. 2 we plot the dark photon sensitivity with the alternative gNFW parameters with γ = 0.8 ('E2 HG' morphology in [108]) and γ = 1.39 ('G2 CM' morphology in [108]), corresponding to the least and most cuspy dark matter profiles for the baryon models analyzed in [108]. Fig. 2 show that the least cuspy gNFW profile leads to slightly weaker sensitivity to κ, while the most cuspy one will enhance the sensitivity of the gNFW profile in Fig. 5 by an order of magnitude. The 'Breakthough' constraint from neutron stars is already visible even without assuming a density spike in the galactic center in this case. A cored profile, on the other hand, will reduce the sensitivity by about two orders of magnitude compared with the gNFW profile in Sec. V.
Additionally, although we demonstrate the potential of radio telescopes to discover dark photon dark matter by considering the Milky Way, signals from nearby galaxies are also interesting. It is likely that some nearby galaxies host cuspy dark matter profiles or even density spikes, and these could potentially lead to strong constraints, although we leave an analysis to future work.

VII. EFFECTS OF THE WHITE DWARF ENVIRONMENT
Here we describe our assumptions about the white dwarf environment in more detail and analyse the resulting uncertainties on the projected sensitivity to dark photon dark matter conversion. We focus on non-magnetic cataclysmic variables.
Our assumption that there is at least one accreting white dwarf within 0.3 pc of the galactic centre is supported by observational evidence. In particular, [90] shows that a significant fraction of the detected hard Xray point sources in the galactic center is attributable to the non-magnetic cataclysmic variables (CVs) that we consider, in addition to magnetic CVs (it is also thought that magnetic CVs only make up about 10% of all CVs [75]). Furthermore, the observed cumulative hard X-ray spectrum can be well fit by thermal bremsstrahlung [91], suggesting that most of the detected X-rays come from the thermal plasmas formed in accretion, with non-magnetic CVs contributing significantly.
Our analysis also involves assumptions about the shape of the boundary layer and the electron density distribu-tion. Since dark photons travel approximately in the radial direction, Eq. (19) indicates that the dark photon conversion probability is only related to the derivative of the radial electron density profile, not the electron density in the vertical direction. The scale height H of the boundary layer, inferred from the α-disk model, will slightly change the anisotropy of the plasma as well as the emission region, but this has little effect on the resulting signal power. For similar reasons, n d which describes the matter density only sets the maximum plasma frequency or the maximum dark photon mass, but does not significantly affect the conversion probability. As discussed in Appendix G, the radial extension of the boundary layer b is derived from hydrostatic equilibrium where the pressure gradient of the gas balances the gravitational potential, so that b is solely determined by the plasma temperature and the mass of white dwarf. This relation has been used to infer the electron density profile of white dwarf corona [109][110][111] as well as the properties of the boundary layer [79]. We stress that the radial electron density profile in the boundary layer is presently uncertain and an important topic for future dedicated study. To estimate the effects of the uncertainty on the density profile, in Fig. 3 we plot the sensitivity varying b independently of the plasma temperature. The effects are two-fold. On the one hand, increasing b enhances the conversion probability through the density gradient ∂ r ω p . On the other hand, a larger b also affects absorption and leads to signal loss at high dark photon mass.
The temperature of the boundary layer T s is determined from the flux ratio of the Fe XXVI to Fe XXV emission lines (I 7.0 /I 6.7 ) of the hard X-rays observed in the galactic center [91]. We use the low luminosity samples (GCXE-L) which are likely to come from nonmagnetic cataclysmic variables instead of the magnetic ones [90,91]. The resulting inferred temperatures range from 30 to 50 keV, with a mean of 38 keV (4.4 × 10 8 K). To illustrate the effects of a different T s we allow the temperature to vary by a factor of 2 in Fig. 4 (we also restore the relation between T s and b to highlight the effect of T s only). In general, a higher plasma temperature both facilitates conversion and suppresses absorption.

VIII. DISCUSSION AND OUTLOOK
Future telescopes searching for signals from an accreting WD could cover a substantial region of viable parameter space with m A ′ ≳ 10 −5 eV. This is the case even with conservative assumptions about the dark matter distribution, and the sensitivity is greatly enhanced if the dark matter profile has a spike. The projected reach from signals from an accreting WD surpasses that from NSs due to the dependence of the emission power on the radius of the resonant conversion region as well as the relatively high temperatures and plasma densities in the boundary layer. However, it will be important to study the WD's properties in more detail in the future, The dependence of the sensitivity of radio telescopes to dark photon dark matter on the assumed dark matter spatial distribution. As in Fig. 5, we assume 100 hours of observation of the cumulative signal from neutron stars within 3 pc of the galactic centre (NSs, left) and the signal from an individual accreting white dwarf 0.3 pc from the galactic centre (WD, right). Coloured lines give the projected sensitivities of ALMA, GBT, SKA1, and SKA2. The red shaded region is our constraint derived from the Breakthrough Listen (BL) project [60]. For each telescope we plot three lines: the dotted lines assume the Burkert dark matter profile, the dashed lines assume the gNFW profile with the 'G2 CM' baryonic morphology in [108] (the BL limit is only visible with this profile), and the dash-dotted lines assume the same profile with the 'E2 HG' baryonic morphology in [108]. Unlike Fig. 5, we do not plot the results assuming a density spike in the galactic center. The grey line depicts the cosmological constraint on dark photon dark matter (DPDM) from Arias [9], and the black line shows the constraint from CMB distortion from [44] (CMB).  Fig. 5), 6376 km and 1594 km, respectively. The plasma temperature Ts = 4.4 × 10 8 K is assumed in the boundary layer. Left: Results obtained assuming the dark matter follows the gNFW profile. Right: Same as the left, but a density spike is assumed in the galactic center. The grey line depicts the cosmological constraint on dark photon dark matter (DPDM) from [9], and the black line shows the constraint from CMB distortion from [44] (CMB). e.g. modeling the boundary layer and the accretion disk in detail. A component of the observed X-rays from CVs might be generated by magnetic reconnection [112] instead of accretion, and the resulting environment may also lead to interesting signals. In addition, since the boundary layer may not be exactly isotropic, signal photons could be refracted when they propagate out of the WD, potentially reducing the signal power. This is to be scrutinized in the future with a more realistic boundary layer profile. The temperature profile of the boundary Left: Results obtained assuming the dark matter follows the gNFW profile. Right: Same as the left, but a density spike is assumed in the galactic center. The grey line depicts the cosmological constraint on dark photon dark matter (DPDM) from [9], and the black line shows the constraint from CMB distortion from [44] (CMB).
layer will also affect the absorption of the signals. Finally, we stress that the dark matter distribution in the GC has a strong impact on the projected sensitivity, see Sec. V and VI, and it will be crucial to improve on this uncertainty in the future.
The signals we have studied complement future haloscope searches, e.g. [113], which have projected sensitivity to smaller κ but can only scan frequencies slowly. The discovery of a radio signal would provide experiments with a DP mass to target while direct detection searches could test the origin of a radio line unaffected by astrophysical uncertainties. Being independent of dynamics in the early Universe, searches for radio signals are also a useful addition to cosmological constraints, which are also subject to uncertainties and systematics. The more recent analysis of [44] gives limits a factor ≃ 2 weaker, can be seen from the difference between the 'CMB' and 'DPDM' lines in Fig. 2. In addition, the dark photon signals studied are insensitive to new physics in the early Universe, including the radiation dominated era, whereas the cosmological constraint depends on the dark photon dynamics at redshifts z ≳ 10 5 .
There are numerous possible extensions to our work. Having set up the formalism in generality, our analysis could be improved by solving the full 3-dimensional equations and utilising ray-tracing. The detection sensitivity might be improved by considering globular clusters (which have large concentration of compact stars and low velocity dispersion) [59] or nearby galaxies that might have very cuspy dark matter profile. It may also be possible to exploit the fact that the GC NS signal is composed of a forest of ultra-thin lines, or that the signal from a particular NS is polarised [114]. Other possibilities to consider include the signals from collisions of DP substructure such as DP stars [115] with astrophysical objects, the changes in theories in which the dynamics that give rise to the DP mass are not decoupled, and whether observable effects occur in theories with different interactions between the DP and the SM.
Finally, there are likely to be interesting signals from axion or DP conversion in other accretion environments. For example, accretion columns form around the magnetic poles in magnetic CVs and accreting NSs. The densities and temperatures in the resulting plasmas are expected to be similar to those in the boundary layer of non-magnetic CVs, which might allow for interesting signals for axion masses as large as an meV. Here we derive the equations of motion of photons and dark photons propagating in an anisotropic plasma, such as the magnetosphere of a neutron star, with a (possibly strong) external magnetic field B. We address the effects of the plasma and the non-linear interactions induced by a strong magnetic field in turn. Following the conventions of Ref. [68] we write the relevant parts of the photon and dark photon Lagrangian as It is convenient to redefine the photon field A µ → A µ + κA ′ µ to remove the mixing term, which yields with L a = −Ā µ J µ andĀ µ = A µ + κA ′ µ . The dark photon coupling to electrons is suppressed by κ and therefore weak in our parameter space of interest.Ā µ is the active state that interacts with the electromagnetic current J µ . In an anisotropic plasma, the Lagrangian of the current is modified to [116] whereF µν = F µν + κF ′ µν , J µ f is the free current density, andP µν is the polarisation tensor induced by the active statē Because the plasma is not expected to be ferromagnetic, we assume the magnetisationM M M = (M x ,M y ,M z ) = 0, although the collective motion of electrons could potentially induce magnetisation, see e.g. [117]. As in the main text, we assume that both the dark photon and photon propagate in the z direction, and we write their fields in the wave The polarisation induced by the active field is related to the electric fields by the electric permittivity tensor χ p χ p χ pP where j is summed over. In turn, the permittivity tensor is determined by the dielectric tensor ϵ ϵ ϵ [52, 55, 118] where we fix the external magnetic field to lie in the y − z plane at an angle θ from the propagation z direction, and R yz θ is the rotation matrix in the y − z plane. The entries in the dielectric tensor read In Eq. (A7), the plasma frequency ω p = 4παn e /m e (where n e is the free electron number density) and the electron cyclotron frequency ω c = √ αB/m e . The equations of motion of A µ and A ′ µ that follow from Eqs. (A2) and (A3) are Because no free current is expected in the plasma J J J f = 0, and E E E (′) = −∂ t A A A (′) , so Eq. (A8) can be rewritten as As expected, in the absence of a dark photon the time derivative of Eq. (A10) leads to the usual Maxwell equation of the electric field. Meanwhile, the propagation equation of dark photon, Eq. (A9), can be written as Applying ∂ ν to Eq. (A11) shows that ∂ µ A ′µ = 0 becauseP µν is anti-symmetric. Up to linear order in κ, Eq. (A11) reduces to Approximating ∇ 2 by ∂ 2 z , i.e. neglecting the second derivatives that do not involve the propagation direction, and combining Eq. (A10) and Eq. (A12) correspond to the photon and dark photon propagation equation in the main text.
Next we consider the non-linear effects that are induced by a strong magnetic field, which modify the polarisation tensor discussed above. As we will show below, these are negligible for neutron stars, but they could be important in systems with even stronger magnetic fields, low plasma frequency or without a plasma. In the presence of a strong external field, the propagating fields experience non-linear QED effects due to the electron box diagram that couples them to the B field, known as vaccum polarisation [67,68,119,120]. This adds a non-linear contribution L nl to the Lagrangian in Eq. (A2), which is a function of I ≡F µνF µν (see [68,120] for an explicit expression). With the inclusion of L nl , the dielectric tensor and the magnetic permeability tensor can be calculated from [120]. The result is that the vacuum contribution modifies the dielectric tensor in Eq. (A6) to ϵ ϵ ϵ = 1 + χ χ χ p + χ χ χ vac where [118]. Additionally, the vacuum contribution induces a magnetisationM M M in Eq. (A4) that is related to the magnetic field of photon and dark photon through the permeability tensor, i.e.
which are accurate in both the weak field and b ≫ 1 limits [118]. Near the dark photon-photon conversion region both photons and dark photons are non-relativistic with k ≪ ω. Because the magnetic field involves the spatial derivative of A A A and A A A ′ , the in-medium magnetisation is suppressed by a factor k/ω compared with the polarisation densityP P P . Consequently we set µ µ µ −1 = 1 and leave a full exploration to future work. The resulting equations of motion are Eq. (A10) and Eq. (A12) with the replacement χ χ χ p → χ χ χ p + χ χ χ vac where χ χ χ p is as defined in Eq. (A6). We follow the prescription in [56] and neglect second order derivatives that do not involve z because the plasmas we consider are slowly varying. The wave equations of the photon are where ε ′ = ϵ − 1, η ′ = η − 1, with ϵ and η given in Eq. (A7), and we have defined ξ y ≡ ε ′ cos 2 θ + η ′ sin 2 θ + a + q sin 2 θ , (A23) ξ z ≡ ε ′ sin 2 θ + η ′ cos 2 θ + a + q cos 2 θ , (A24) ξ yz = (η ′ − ε ′ + q) cos θ sin θ . (A25) The corresponding wave equations for the dark photon are Eqs. (A19) to (A21) and (A26) to (A28) are analogous to Eq. (3.9) in [68], but are valid for k ≪ ω instead of the weak dispersion limit ω ≃ k. Moreover, one can straightforwardly obtain results for any k/ω by including the magnetisation from the vacuum polarisation. This is analogous to the axion-photon conversion described in [56], except that the source terms are now proportional to κ instead of g aγγ B. As illustrated in Fig. 5, in the non-relativistic limit the converted photons acquire both a transverse component A y and a longitudinal component A z . In combination, the photon polarisation lines up with the direction of the external magnetic field, and evolves in the s direction that is perpendicular to the magnetic field. As mentioned in the main text, the photon mode is identified as the Langmuir-O (LO) mode [73]. The dispersion relation of the LO mode (k 2 = ω 2 (ω 2 − ω 2 p )/ ω 2 − ω 2 p cos 2 θ ) transforms into the free-space dispersion relation as the photon travels outside the magnetosphere and the photon becomes purely transverse. The solution of Eq. (B5) is where the phase in the exponent The first exponential in Eq. (B7) is a pure phase that does not contribute to the conversion probability. The integrand in Eq. (B7) is highly oscillating and tends to cancel unless the phase is stationary, i.e. ∂ s f = 0. This gives the resonant conversion condition or The conversion peaks near s c and for practical purposes can be taken to be vanishing everywhere else. We can therefore expand Eq. (B8) as a Taylor series up to the second order, which yields where the conversion length is defined as We have also neglected the derivative of the dark photon momentum, which only varies slowly due to the gravitational potential of a star. Under this approximation, Eq. (B7) evaluates to [56] s should be defined at the location beyond which the momenta of photons and dark photons do not match anymore, or where the WKB approximation fails. For s ≳ 2L, the error function approximately evaluates to 1. For s ≲ 2L, √ 2L and the error function in Eq. (B13) is to be replaced by ∆z, the displacement in z direction where the WKB approximation applies. We refer readers to Refs. [56,122] for more extensive discussions and leave the exploration of this scenario in future work. In the former case, the photon field after conversioñ Similarly to axions, there are various effects that could modify Eq. (B14), e.g. due to bending of the photon path within the conversion length caused by refraction, gravitational curvature or the variation of the magnetic field. Moreover, if ∂ s ω p is close to 0 there is a divergence that will be cut-off by some additional dynamics. We also refer readers to Refs [55, 56] for detailed discussions. From Eq. (B3), ignoring the partial derivative and the dark photon mixing terms, we obtainÃ where we have used the exact resonant conversion condition in Eq. (B10). The conversion probability where we have assumed m A ′ ≫ k andÃ ′ x ≃Ã ′ y ≃Ã ′ z . The plasma frequency ω p at s c is given by Eq. (B10). We note that the factor of cot θ that relates A z to A y in Eq. (B15) when the full resonant condition Eq. (B10) is imposed leads to a divergence in p NS in Eq. (B16) as θ → 0. This is not dependent on the definition of L because A y is finite at θ = 0, but instead arises from the properties of the plasma which causes the mixing of A y and A z in Eq. (B3). This divergence would have been artificially removed if we we had approximated ω = ω p = m a (as done in e.g. Ref. [56]), which would lead to the relation However, there are several caveats to this analysis: 1) We have neglected the vacuum contribution in Eq. (B3), the inclusion of which will modify the sin θ in the denominator of Eq. (B15) and (B16). 2) Eq. (B10) varies slightly within the conversion length so the divergence only appears at s c and does not hold through the whole conversion region. As θ → 0 the conversion length L → 0 as well, and Eq. (B10) holds at this point, so it is not automatic that this regulates the divergence.
3) The conversion probability obviously cannot exceed 1, and when p NS ∼ 1 photons will convert back to dark photons. A dedicated study is required to determine the impact of each of these and to determine the dominant effect that cuts off the divergence. Instead, as mentioned in the main text, we simply require the conversion probability at an arbitrary θ not to be larger than 1000 times the conversion probability at θ = π/2 where only A y is produced and A z vanishes. This factor is rather artificial, but, given that we work in the parameter space where p NS ≪ 1 and a, q ≪ η ′ , it is reasonable to expect that it is conservative. In the non-relativistic limit, our approximation amounts to imposing cot 2 θ ≤ 1000, which yields 1.8 • ≤ θ ≤ 178.2 • . We leave the exploration of smaller angles to future study. Away from the divergence we can safely use the approximation ω ≃ m A ′ to simplify Eq. (B16) and obtain Dedicated simulations of their trajectories of dark photons and photons would be required to properly evaluate the conversion probability in Eq. (B18) [55,56]. The approximations we make in the main text of assuming that both dark photons and photons travel on radial trajectories and that ∂ s = ∂ r , amount to effectively neglecting the derivative ∂ y in Eq. (B6). We note that important corrections are likely to arise for specific angles θ in a proper derivation, as analysed in [56].
obtained by removing dark photon terms in Eq. (A10) and setting B = 0. We write A A A(r, t) = e iωt−ik(r)rÃ A A(r) and assume the photon field varies slowly so that |d 2Ã A A/dr 2 | ≪ k|dÃ A A/dr|. Using the WKB approximation Eq. (D1) simplifies to Eq. (D2) yields the relationÃ A A(r) ∝ 1/ k(r). In addition, the amplitude of the photon field will drop as a function of radius, analogously to the case of black body radiation. In combination, we find at a radius r > r c , where as before v c is the velocity of dark photon at the resonant conversion radius. Eq. (D3) simply shows the total photon flux, proportional to |Ã A A| 2 r 2 k, is conserved during the photon propagation, as in [56,74]. The photon propagation in an anisotropic plasma with a strong magnetic field (e.g. neutron stars) is more complicated, and in this case the propagation equation should be solved explicitly. Likewise the propagation in a nonmagnetic cataclysmic variable will be complicated by the anisotropy of the plasma, which can lead to photons refracting despite the lack of a strong magnetic field. We leave a dedicated study for future work, and instead we assume Eq. (D3) to hold also in these environments (which is reasonable given the interpretation as conservation of the total photon flux).
Assuming Eq. (D3), the photon signal power per unit solid angle outside the plasma is where the Poynting flux S S S = E E E ×H H H. The converted photon fieldÃ Ã A(r c ) can be inferred from the conversion probability and the energy density of dark photon where p is the conversion probability given in Eq. (B16) and (C8) for neutron stars and white dwarfs, respectively. We have used the approximation ω ≃ m A ′ and included a factor of 2 to account for dark photon conversion when entering and leaving the resonant conversion region. GBT. We take the SEFD of GBT from Ref. [102] (with typical galactic background). We also assume the same detection bandwidth in SKA phase 2 as in phase 1 in similar frequency bands. The configuration sensitivity of SKA2 is roughly 15 times better than SKA1. We take the detector efficiency η = 0.9 for SKA1 [101] and η = 1 for SKA2 and GBT. The configurations of different radio telescopes are listed in Table I.
SKA. The configurations of SKA1 and SKA2 are obtained from [101] and their sensitivities are computed in Ref. [123].
ALMA. For ALMA we follow the prescription in Ref. [103]. The flux sensitivity for the 12-meter and 7-meter arrays is given by where w r = 1.1 is the weighting factor, η q = 0.96 is the quantization efficiency, η c = 0.88 is the correlator efficiency, and N = 50 (12) for the number of antennas in the 12 (7)-meter arrays. We take the shadowing fraction f s = 0. The effective area A eff = A an η ap . We focus on the 12-meter arrays with physical area A an = 113.1 m 2 and the aperture efficiency η ap ranges from 71% to 31% from Band 3 to Band 10. ALMA uses two types of correlators to identify fine spectral lines. We adopt the general analysis channel width of 15.3 kHz as suggested in the handbook, which is typically below the signal bandwidth of dark photons with masses ≳ 10 −4 eV (both for emission from a single white dwarf and also from a collection of neutron stars). We use the T sys at the first octile of PWV (best weather conditions) in [103], and, for plotting in Fig. 6, we show SEFD ≡ w r T sys /(η q η c (1 − f s )N A eff ) for the 12-meter arrays. ALMA's sensitivity is lost at particular frequencies that correspond to gaps between different bands or values where atmospheric absorption is significant. Finally, we note that the analysis bandwidth of the Breakthrough Listening (BL) project is 91.6 kHz, narrower than the width of the signal from neutron stars. Consequently, the bound on S min that we impose to obtain the limit from neutron stars in the main text is stronger than the experimental published value by a factor of m A ′ σ s /91.6 kHz due to Eq. (E1). Finally, we consider the distribution of cataclysmic variables. Recently the X-ray point sources in the galactic centre have been revisited in [90] based on Chandra observations [90]. Due to the gravitational potential of the central black hole, frequent encounters of massive stars facilitate the formation of binaries and cataclysmic variables, which are believed to be the origin of such X-rays. The surface number density (number density per area) of observed 2 − 8 keV X-ray point sources in the galactic centre can be well described by [90] log 10 Σ arcsec 2 = −0.0596y 3 + 0.00262y 2 − 0.188y − 1.52 , where y = log 10 (R/arcsec) and R ≤ 100 ′′ . Given this number density, our assumption in the main text of at least one appropriately aligned cataclysmic variable within 0.3 pc is reasonable. where we take the mean molecular weight µ = 0.5 for fully ionized hydrogen plasma. Based on spectroscopic studies, we assume a maximum electron number density n 0 = 10 17 cm −3 [127,128] (the same argument also applies to the boundary layer and is consistent with the description in [109][110][111]). Unlike the boundary layer around accreting white dwarfs, the atmosphere of a general white dwarf is expected to extend to a much larger radius than r 0 + l a . The magnetic field in white dwarfs is highly uncertain, but it is believed that only about 10% of white dwarfs have a magnetic field stronger than 0.1 MG [59, 129-131] so we set B = 0. The radio signals from the white dwarf atmosphere are therefore emitted isotropically. The signal power where we have fixed M WD = M ⊙ and r 0 = 0.01R ⊙ . This is markedly lower than the signal from accreting white dwarfs, but the signal can be enhanced by considering a collection of stars. Similarly to neutron stars, the signal flux density from the galactic centre where we take R min = 0.01 pc from simulations and R min = 3 pc from the view of radio telescopes. We assume the average white dwarf mass of 0.6 M ⊙ and the corresponding radius of 0.012 R ⊙ [92,93]. The resulting sensitivity is shown in the left panel of Fig. 7. Additionally, it is possible that some white dwarfs might be surrounded by a hot envelope in the outer part of their atmosphere, a so-called "corona" analogous to the solar corona [132,133]. Such corona was originally proposed to account for observations of X-ray emission by telescopes in Earth orbits, including by Einstein, EXOSAT, ROAST and Chandra [134], which could be the result of plasma emission in such a region. However, these observations were later revisited and found to be either consistent with emissions from the photosphere or with a non-detection [134][135][136]. Upper limits were set on the electron density in the corona, which range from 4.4 × 10 11 to 5 × 10 12 cm −3 [135,137].
Nevertheless, it remains possible that corona could exist in some white dwarfs, so we briefly consider the radio signals that this would lead to. The suggested temperatures of corona T c range from 10 6 K to ≳ 10 7 K [135,137]. We therefore assume a log-normal distribution with centre log 10 (T c /K)=6 and σ log 10 (Tc/K) = 1. The form of the electron density profile would be similar to that in white dwarf atmospheres, and the signal power is given by Eq. (G4), albeit with a different temperature distribution. The signal flux density can be again computed from Eq. (G4). We remain agnostic about the maximum electron density in the corona n 0 , but we include a factor f that quantifies the fraction of white dwarfs with a corona that is dense enough that n 0 ≥ where r c is the radius of the conversion region and T m is the temperature of the magnetosphere, which ranges from ≃ 10 6 to ≃ 5 × 10 6 K [139]. The Compton scattering is also very inefficient in the magnetosphere and it seems likely this will remain the case after a full analysis. We show the survival probability of signal photons in Fig. 8. For neutron stars we assume T m = 2 × 10 6 K. As shown in the main text, the conversion radius r c is determined by the neutron star magnetic field, spin period and the emission angle. For higher magnetic field the absorption is more significant because r c is larger. We find that the effect of inverse bremsstrahlung is unimportant unless m A ′ ∼ 10 −4 eV and B 0 ∼ 10 15 G. On the other hand, the absorption in the white dwarf plasma could be important depending on the plasma temperature.
FIG. 8. The survival probability of the converted signal photons due to inverse bremmsstrahlung in the plasma. Left: The survival probability in the neutron star magnetosphere. Different colors correspond to the surface magnetic field of 10 13 G, 10 14 G and 10 15 G respectively. We assume the spin period P = 0.5 s, the plasma temperature Tm = 2 × 10 6 K and the angular dependence β = 1. Right: The solid lines depict the survival probability of photons in the atmosphere of isolated white dwarfs, with the plasma temperature of 10 4 K, 10 6 K and 10 8 K respectively. The dashed line delineates the absorption in non-magnetic cataclysmic variable with the boundary layer temperature Ts = 4.4 × 10 8 K.
FIG. 9. Left: Schematic illustration of photon and dark photon propagation in the conversion region. The z-axis is directed to the radial direction. Resonant conversion takes place at z = 0(r = rc). Right: The dephasing factor ϕ as a function of dark photon mass in white dwarf environments with different properties. Different line styles correspond to plasma temperatures of 10 4 K, 10 6 K, and 10 8 K, respectively. We fix an incident angle sin θi = 0.083 at the conversion region.
where s is interpreted as the trajectory of dark photon. Ignoring the gravitational curvature of the dark photon trajectory, which is small in the conversion region, the dark photon displacement ds = dr/ cos θ i . We can again expand Eq. (I2) to the second order, which gives the conversion length For the typical halo dark matter velocity in the galactic centre v i ≲ 400 km/s and the escape velocity at a white dwarf surface v c ∼ 4800 km/s, sin θ i ≲ 0.083. Comparing Eq. (I3) with Eq. (C7), this introduces at most 0.17% enhancement of the conversion length, which can be safely neglected. Additionally, Eq. (I2) continues to assume straight-line trajectories for both the photon and dark photon. However, if either trajectory deviates from this assumption in the conversion region the conversion probability is further modified. In particular this effect, known as dephasing [55], reduces the phase overlap. It typically occurs when photons are refracted while propagating in the plasma. Although a more sophisticated modification of the field equations, along with ray-tracing, would be required to properly account for it, we estimate the impact of dephasing by investigating the integral in the conversion region, where sin θ r ≡ k γ,y /k γ . This quantifies the relative phase induced due to the difference in the paths of the photon and dark photon. We stress that this is robust against the choice of the (somewhat arbitrary) definition of the location of the conversion region relative to the resonant conversion surface. If the dark photon and photon both follow the same trajectory, this integral vanishes. The dephasing effect would be important if |ϕ| ∼ 1. Because ϕ appears in the exponential, its effect is similar to that of the non-radial trajectory, and we expect that the effect of dephasing can be parameterised by a modified conversion length L ′ = L √ cos ϕ, with L given in Eq. (I3). Without loss of generality we set z = 0 to be at r = r c . Snell's law holds in the plasma, n(r) sin θ r = n i sin θ i , where n is the refraction index and we define n i ≡ n(z = 0). Indeed, in a plasma without a strong magnetic field the dispersion relation is ω 2 = k 2 γ + ω 2 p , n ≡ k γ /ω = (1 − ω p /ω) 1/2 , and we have the relation n sin θ = k γ,y /ω = const as dk γ,y /dt = −∂ y ω = −ω p (∂ y ω p /ω) = 0 for a plasma frequency that changes only in the z direction. Alternatively, θ r can be obtained from sin θ r = k γ,y / ω 2 − ω 2 p , with k γ,y = k A ′ sin θ i . With this setup, we obtain ϕ = r L rc nωdr (1 − w 2 sin 2 θ i ) −1/2 − cos −1 θ i ,