New Limits on Axionic Dark Matter from the Magnetar PSR J1745-2900

The Peccei-Quinn mechanism offers a solution to the strong charge-parity (CP) problem in quantum chromodynamics (QCD) with the introduction of the axion, a spin-zero chargeless massive particle (Peccei & Quinn 1977; Weinberg 1978; Wilczek 1978). Moreover, QCD axions are a promising cold dark matter candidate (Abbott & Sikivie 1983; Dine & Fischler 1983; Preskill et al. 1983), and may explain the matter−antimatter asymmetry in the early universe (Co & Harigaya 2020).

If axions (a) exist, they would have a two-photon coupling g_aγγ such that the electromagnetic interaction is

$$\begin{eqnarray}&&{{ \mathcal L }}_{a\gamma \gamma }=-(1/4){g}_{a\gamma \gamma }a\,{F}_{\mu \nu }{\tilde{F}}^{\mu \nu }={g}_{a\gamma \gamma }a\,{\boldsymbol{E}}\cdot {\boldsymbol{B}}.\end{eqnarray} \tag{ 1 }$$

Axion−photon conversion can thus occur in the presence of a magnetic field, but the axion−photon coupling is weak: g_aγγ ∼ 10⁻¹⁶ GeV⁻¹ for axion mass m_a = 1 μeV (Kim 1979; Shifman et al. 1980; Zhitnitsky 1980; Dine et al. 1981; Sikivie 1983). Theory predicts g_aγγ ∝ m_a, but the mass is not constrained. Axion searches must therefore span decades in m_a while reaching very small g_aγγ.

Axion experiments include CAST, which searched for Solar axions (Arik et al. 2014, 2015), and "haloscopes" that use narrow-band resonant cavities to detect dark matter axions, such as ADMX and HAYSTAC (Asztalos et al. 2001, 2010; Brubaker et al. 2017; Zhong et al. 2018). There are also natural settings where one may use telescopes to conduct sensitive and wide-band QCD axion searches toward pulsars or galaxies (Hook et al. 2018; Huang et al. 2018; Day & McDonald 2019; Battye et al. 2020; Edwards et al. 2020; Leroy et al. 2020; Mukherjee et al. 2020).

Of particular interest is the natural axion−photon conversion engine, the Galactic Center magnetar PSR J1745−2900. PSR J1745−2900 has a strong magnetic field (1.6 × 10¹⁴ G; Mori et al. 2013), and it should encounter the highest dark matter flux in the Galaxy (Hook et al. 2018). Axions will encounter a plasma frequency at some radius in the magnetosphere that equals its mass, allowing the axion to resonantly convert into a photon (Hook et al. 2018). The likely axion mass range, 1–100 μeV, equates to radio frequencies 240 MHz to 24 GHz.

In this Letter, we present archival observations of PSR J1745−2900 from the National Science Foundation's (NSF's) Karl G. Jansky Very Large Array (VLA¹ ) that expand upon our previous work (Darling 2020). We obtain 95% confidence limits on resonant axion−photon conversion emission line flux density from the magnetar spanning 62% of the 1–40 GHz band. Limits on the axion−photon coupling, g_aγγ, rely on a neutron star magnetosphere model and are bracketed by two limiting-case uncored Galactic dark matter profiles. We present model caveats, discuss observational limitations, and suggest observational and theoretical work to expand the g_aγγ versus m_a space probed by this technique.

Darling (2020) used archival VLA observations of Sgr A* and/or PSR J1745−2900 (both are present in every primary beam), that have the highest angular resolution (A-array) in order to separate the magnetar from Sgr A* and to resolve out extended spectral line-emitting Galactic Center gas and extended continuum emission. The work presented here adds observations that are sub-optimal (B- and C-array) but which still enable the angular separation of Sgr A* from PSR J1745−2900. Unlike for the observations presented in Darling (2020), the PSR J1745−2900 radio continuum cannot be separated from the surrounding emission in these lower angular resolution data. Based on our previous confirmation of the astrometry of PSR J1745−2900 conducted by Bower et al. (2015), we are confident in our ability to extract a spectrum of PSR J1745−2900 from interferometric image cubes.

VLA observations of Sgr A* and PSR J1745−2900 were selected (1) to maximize on-target integration time, (2) with adequate angular resolution to separate PSR J1745−2900 from Sgr A* (1.7" in both coordinates), (3) to maximize total bandwidth, and (4) to adequately sample the expected emission line bandwidth. In addition to the VLA A-configuration programs 14A-231 and 14A-232 presented in Darling (2020), programs 12A-339, BP198, and 15A-418 meet these criteria and are analyzed in this Letter (Table 1).

Observing sessions used J1331 + 3030 (3C286) and J0137 + 331 (3C48) for flux calibration, 3C286, 3C48, and J1733−1304 for bandpass calibration, and J1744−3116, J1745−283, and J1751−2524 for complex gain calibration. Right- and left-circular polarizations were combined to form Stokes-I spectral cubes. Bandwidths of spectral windows were either 32 MHz or 128 MHz, subdivided into 0.5 or 2 MHz channels, respectively, and grouped into two to four overlapping basebands. All programs used 8-bit sampling except for 15A-418, which used 3-bit sampling. Correlator dump times were 1–5 s.

We used CASA (McMullin et al. 2007) for interferometric data reduction. Data were flagged and calibrated (flux, delay, atmospheric transmission, complex bandpass, and complex gain). After applying calibration to the target field, we did in-beam phase self-calibration on the Sgr A* continuum (0.8–1.7 Jy from S- to Q-band).

We imaged the continuum in the target field and fit a 2D Gaussian to Sgr A* to set the origin for relative astrometry. We locate PSR J1745−2900 using the bootstrap proper motion solution obtained by Bower et al. (2015). Offsets were consistent between epochs in each band and between bands and were consistent with the observed continuum position of the magnetar detected in programs 14A-231 and 14A-232 (Darling 2020).

After linear continuum subtraction in uv space, we formed spectral image cubes and cleaned these down to five times the theoretical noise. Sgr A* shows narrow-band spectral structure after the continuum subtraction, particularly in X-band where we see a comb of radio recombination lines (RRLs) in emission, but also due to spectral window edge effects. The RRLs are presumably mildly stimulated. We also see extended RRL emission in many spectral cubes due to the low angular resolution, and we see RRL emission toward the magnetar in some bands (see below). Sgr A* sidelobes are cleaned during cube deconvolution and do not significantly contaminate the magnetar spectrum, and the magnetar spectra typically reach the theoretical noise. Synthesized beams vary across each spectral cube due to the natural frequency-dependent angular resolution of the interferometer and due to data flagging and radio-frequency interference (RFI).

We extract the magnetar spectrum over the 2D Gaussian beam and correct for the beam size for each channel in order to capture the total point source flux density. The spectral noise varies channel-to-channel, which can create false peaks in the magnetar spectrum. To assess significance of spectral features, we form a noise spectrum using a measurement of the sky rms noise in each channel. The overall noise spectrum is scaled to the spectral noise of PSR J1745−2900, which typically differ by a few percent.

For single-channel detection, we need to smooth spectra to the expected axion–photon conversion line width, but there is theoretical disagreement about the expected bandwidth of the emission line. Hook et al. (2018) made a conservation of energy argument to derive a fractional bandwidth that depends quadratically on the axion velocity dispersion v₀: Δf/f = (v₀/c)² (contrary to the intuitive expectation for the line width to reflect the velocity dispersion as a Doppler shift; Huang et al. 2018). Battye et al. (2020), however, suggested that the line width is dominated by the neutron star's spinning magnetosphere. We adopt this spinning-mirror model that produces, on average, a bandwidth Δf/f ≃ Ω r_c /c, where Ω is the rotation angular frequency and r_c is the axion–photon conversion radius. Hook et al. (2018) showed that r_c depends on the neutron star's radius r₀, magnetic field B₀, angular frequency, polar orientation angle θ, and magnetic axis offset angle θ_m:

$$\begin{eqnarray}\begin{array}{rcl}{r}_{c} & = & 224\,\mathrm{km}\times {\left|3\cos \theta \hat{m}\cdot \hat{r}-\cos {\theta }_{m}\right|}^{1/3}\\ & & \times \,\left(\displaystyle \frac{{r}_{0}}{10\,\mathrm{km}}\right){\left[\displaystyle \frac{{B}_{0}}{{10}^{14}{\rm{G}}}\displaystyle \frac{1}{2\pi }\displaystyle \frac{{\rm{\Omega }}}{1\mathrm{Hz}}{\left(\displaystyle \frac{4.1\mu \mathrm{eV}}{{m}_{a}{c}^{2}}\right)}^{2}\right]}^{1/3}\end{array}\end{eqnarray} \tag{ 2 }$$

where $\hat{m}\cdot \hat{r}=\cos {\theta }_{m}\cos \theta +\sin {\theta }_{m}\sin \theta \cos {\rm{\Omega }}t$. For now, we assume that θ = π/2 and θ_m = 0 (we deal appropriately with these angles in Section 5) to obtain the expected line width:

$$\begin{eqnarray}&&{\rm{\Delta }}f=3.6\,\mathrm{MHz}{\left(\displaystyle \frac{{\rm{\Omega }}}{1\mathrm{Hz}}\right)}^{4/3}{\left(\displaystyle \frac{{m}_{a}{c}^{2}}{4.1\mu \mathrm{eV}}\right)}^{1/3}{\left(\displaystyle \frac{{B}_{0}}{{10}^{14}{\rm{G}}}\right)}^{1/3}\end{eqnarray} \tag{ 3 }$$

(4.1 μeV corresponds to 1 GHz as observed). PSR J1745−2900 has a 3.76 s rotation period (Kennea et al. 2013) and a magnetic field of 1.6 × 10¹⁴ G (Mori et al. 2013). The expected axion–photon conversion line width is thus ${\rm{\Delta }}f=8.3\,\mathrm{MHz}\times {({m}_{a}{c}^{2}/4.1\mu \mathrm{eV})}^{1/3}$. This corresponds to 2500 km s⁻¹ at 1 GHz and 215 km s⁻¹ at 40 GHz, which is generally broader than the expected dark matter dispersion, ∼300 km s⁻¹, except at the highest observed frequencies.

Spectra have flagged channels due to RFI and due to spectral window edges that lack the signal-to-noise for calibration. Flagged channels are much narrower than the expected line width, so we interpolate across these channels when smoothing using a Gaussian kernel (Price-Whelan et al. 2018). In some bands, such as L, S, and K, entire spectral windows can be flagged and all information is lost.

RRL emission lines are seen toward the magnetar in X-band (15A-418), Ku 12–13 GHz, Ku 17–18 GHz, and K 26 GHz. To remove RRLs, two 2 MHz channels are flagged per line, and subsequent smoothing interpolates across the flagged channels (the expected axion conversion line width is 18 MHz in X-band and 24 MHz at 26 GHz).

We combine spectra obtained from multiple observing sessions using an error-weighted average, and the sky (noise) spectra are combined in quadrature. When different observing programs overlap in frequency, we select the lowest noise observation. This is effectively the same as combining overlapping spectra in quadrature because the less sensitive spectra contribute negligibly to an error-weighted mean.

Table 1 lists synthesized beam parameters, channel widths, and spectral rms noise values. Appendix appendix presents the new magnetar flux, noise, and signal-to-noise spectra.

After smoothing to the expected axion–photon conversion line width, the new spectra show no significant (>3σ) single-channel emission features (Appendix appendix). The exception is a 3.2σ channel at 2.34 GHz (9.67 μeV) in an RFI-affected region of S-band (there also two channels at 3.1σ and 3.5σ in previous Ka-band spectra; Darling 2020).

We obtain single-channel 95% confidence limit flux density spectra from the sky noise spectra. Figure 1 shows the combined limits from this work and Darling (2020). These limits do depend on the assumed line width (Equation (3)), which depends on the magnetar model, but these limits may be scaled as needed for magnetar models not treated in this Letter.

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Translating spectral flux density limits into limits on the axion–photon coupling g_aγγ depends on the axion–photon conversion in the magnetar magnetosphere and on the density of dark matter in the Galactic Center. Both of these rely on as-yet incompletely constrained models.

5.1. The Magnetar Model

We adopt the Hook et al. (2018) axion–photon conversion model for the magnetar magnetosphere, which is based on a variant of the Goldreich & Julian (1969) model (but note that there is substantial disagreement about the signal bandwidth and radiated power in the literature; e.g., Hook et al. 2018; Huang et al. 2018; Leroy et al. 2020; Battye et al. 2020). We modify this model for the bandwidth adopted above (Equation (3)) to obtain an expression for the observed flux density in the axion–photon conversion emission line that depends on the magnetar properties, distance, viewing angle θ, and local dark matter density (see Darling 2020, Equation (3)).

Darling (2020) shows that given a flux density limit spectrum, one can produce a limit on g_aγγ as a function of m_a that depends on the dark matter velocity dispersion v₀, the dark matter density ${\rho }_{\infty }$, and a time-dependent angular term involving θ, θ_m, and the axion velocity at the conversion point, ${v}_{c}^{2}\simeq 2{{GM}}_{{NS}}/{r}_{c}$ (Battye et al. 2020). For PSR J1745−2900 specifically, assuming a magnetar radius of 10 km, mass of 1 M${}_{\odot }$, and distance of 8.2 kpc, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{g}_{a\gamma \gamma } & = & 3\times {10}^{-11}\ {\mathrm{GeV}}^{-1}\ {\left(\displaystyle \frac{{S}_{\nu }}{10\mu \mathrm{Jy}}\right)}^{1/2}\\ & & \times \,{\left(\displaystyle \frac{{m}_{a}}{1\mathrm{GHz}}\right)}^{-2/3}{\left(\displaystyle \frac{{v}_{0}}{200\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{1/2}\\ & & \times \,{\left(\displaystyle \frac{{\rho }_{\infty }}{6.5\times {10}^{4}\mathrm{GeV}{\mathrm{cm}}^{-3}}\right)}^{-1/2}\\ & & \times \,{\left(\displaystyle \frac{3{\left(\hat{m}\cdot \hat{r}\right)}^{2}+1}{| 3\cos \theta \hat{m}\cdot \hat{r}-\cos {\theta }_{m}{| }^{4/3}}\displaystyle \frac{{v}_{c}}{c}\right)}^{-1/2}.\end{array}\end{eqnarray} \tag{ 4 }$$

The angular term relies on the unknown viewing and magnetic field misalignment angles and is time-dependent. Axion–photon conversion also relies on the conversion radius being outside the magnetar surface (r_c > r₀; see Equation (2)), which is axion mass-dependent. For a given (θ, θ_m) pair, there may be parts of the magnetar rotation period that do not radiate. Because the modulation time is much less than the integration time of the observations, we average the expected signal over the period of the magnetar separately for each frequency channel for each (θ, θ_m) pair, to form time-integrated flux density spectra. We then marginalize over all (θ, θ_m) to obtain a limit spectrum on g_aγγ given a dark matter density and velocity dispersion (see below).

The ray-tracing performed by Leroy et al. (2020) suggests that this analytic treatment is conservative and that axion–photon conversion can occur over a larger parameter space. Nonetheless, the signal losses caused by angles where r_c is always less than r₀ in this analytic treatment are a small fraction of the parameter space: at 10 GHz, 0.08% of all possible (θ, θ_m) always have r_c < r₀. This grows with frequency, and at 40 GHz the fraction of orientations with no emission rises to 6.6%.

5.2. Dark Matter Models

The remaining unknowns in Equation (4) are the dark matter density ${\rho }_{\infty }$ and velocity dispersion v₀ at the location of the magnetar. The dark matter contribution in the Galactic Center has not been measured, so one must employ model-based interpolation. Following Hook et al. (2018), we adopt two models that roughly bracket the possible dark matter density (unless the dark matter distribution is cored; a multi-kpc central core is disfavored by observations (e.g., Hooper 2017), and there are reasons to believe that baryon contraction has occurred (Cautun et al. 2020), but the existence of a cusp or core in the inner kpc remains observationally untested): a Navarro–Frenk–White (NFW; Navarro et al. 1996) dark matter profile and the same model plus a maximal dark matter cusp. For both models, we assume ${v}_{0}=300$ km s⁻¹ and a 0.1 pc separation between PSR J1745−2900 and Sgr A* (we identify Sgr A* with the center of the dark matter distribution). The 0.1 pc separation between PSR J1745−2900 and Sgr A* is projected, but absent an acceleration measurement one cannot know their true physical separation (Bower et al. 2015).

For the NFW dark matter profile, we adopt the McMillan (2017) fit with scale index γ = 1, scale radius r_s = 18.6 kpc, local dark matter energy density ρ_⊙ = 0.38 GeV cm⁻³, and Galactic Center distance R₀ = 8.2 kpc, which agrees with the measurement by Abuter et al. (2019) of the orbit of the star S2 about Sgr A*. This model predicts a dark matter energy density of 6.5 × 10⁴ GeV cm⁻³ at 0.1 pc.

The dark matter cusp model adds a spike to the NFW model with scale R_sp = 100 pc and scale index γ_sp = 7/3. This is the maximal dark matter spike corresponding to a 99.7% upper limit derived from a lack of deviations of the S2 star from a black hole-only orbit about Sgr A* (Lacroix 2018). The maximal dark matter energy density encountered by the magnetar is thus 6.4 × 10⁸ GeV cm⁻³, a factor of 10⁴ larger than the NFW-only model. This enhanced dark matter density corresponds to a 100-fold smaller constraint on g_aγγ.

We present band-median 95% confidence limits on $| {g}_{a\gamma \gamma }|$ for each dark matter model in Table 2. Figure 2 shows the limit spectra spanning 62% of the 1–40 GHz (4.2–165.6 μeV) range, previous limits from CAST and HAYSTAC (Anastassopoulos et al. 2017; Zhong et al. 2018), and the family of theoretical axion models (di Luzio et al. 2017). Limits obtained from the NFW model exclude $| {g}_{a\gamma \gamma }| \gtrsim 6\mbox{--}34\times {10}^{-12}$ GeV⁻¹, which is 1.5–3.5 dex above the strongest-coupling theoretical prediction. The maximal dark matter spike model limits do, however, exclude portions of theoretical parameter space for m_a = 33.0–165.6 μeV. The canonical KSVZ or DFSZ models are not excluded (Kim 1979; Shifman et al. 1980; Zhitnitsky 1980; Dine et al. 1981).

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Table 2.  Limits on the Axion–Photon Coupling g_aγγ

Axion Mass	Median $| {g}_{a\gamma \gamma }|$ 95% Confidence Limits
	NFW Profile	DM Spike
(μeV)	(GeV−1)	(GeV−1)
4.2–8.4a	3.4 × 10−11	3.4 × 10−13
8.9–10.0	2.9 × 10−11	2.9 × 10−13
12.3–16.4	1.7 × 10−11	1.7 × 10−13
18.6–26.9	1.3 × 10−11	1.3 × 10−13
33.0–41.3	6.9 × 10−12	7.0 × 10−14
41.3–49.6	1.1 × 10−11	1.1 × 10−13
49.8–53.8	1.0 × 10−11	1.0 × 10−13
53.8–62.1	6.5 × 10−12	6.5 × 10−14
70.1–74.3	1.0 × 10−11	1.0 × 10−13
78.1–80.7	1.1 × 10−11	1.2 × 10−13
105.5–109.6	1.0 × 10−11	1.0 × 10−13
111.6–115.2	1.5 × 10−11	1.5 × 10−13
126.0–155.1	1.4 × 10−11	1.4 × 10−13
155.1–159.3	1.3 × 10−11	1.3 × 10−13
162.5–165.6	2.2 × 10−11	2.2 × 10−13

Note.

^aThere are gaps in the coverage of this mass range (see Figure 2).

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The limits on g_aγγ presented here for the NFW profile are conservative compared to the Hook et al. (2018) predictions. This is due to the choice of a spinning-mirror bandwidth that seems more physically plausible (Battye et al. 2020). This bandwidth is ${ \mathcal O }({v}_{0}/c)$, roughly 10³ times larger than the Hook et al. (2018) ${ \mathcal O }{({v}_{0}/c)}^{2}$ bandwidth. This is a factor of ∼300 in g_aγγ. A better treatment of this issue will require axion–photon conversion ray-tracing as proposed by Leroy et al. (2020).

Our limits may also be conservative because resonant axion–photon conversion may be stimulated by the local photon occupation number, which would boost any signal thereby improving constraints on g_aγγ (Caputo et al. 2019). It seems likely that stimulated emission would be particularly important in the Galactic Center photon bath, but it may also arise naturally from the magnetar itself. This effect and numerical ray-tracing may significantly improve the constraints on g_aγγ based on the current observations alone.

As Figure 2 shows, the highly uncertain dark matter energy density in the inner parsec allows a large range of possible constraints on axion parameter space. Moreover, if the central dark matter is cored with a fixed NFW density of 12 GeV cm⁻³ inward of 500 pc , then the limits on g_aγγ are degraded by 2 dex and lie above the CAST limits. We look forward to observational measurements of or constraints on the Galactic Center dark matter encountered by PSR J1745−2900 based on stellar and gas dynamics.

We have expanded the axion mass range searched for the axion–photon conversion signal originating from the magnetosphere of the Galactic Center magnetar PSR J1745−2900. New limits span 62% of the 4.2–165.6 μeV (1–40 GHz) axion mass range, excluding at 95% confidence g_aγγ > 6–34 ×10⁻¹² GeV⁻¹ if the dark matter energy density follows a generic NFW profile at the Galactic Center. For a maximal dark matter spike, the limit reduces to g_aγγ > 6–34 ×10⁻¹⁴ GeV⁻¹, which excludes some possible axion models for m_a > 33 μeV.

This work gets close to exhausting the appropriate data in the VLA archive. Lower-resolution interferometric observations cannot separate the magnetar from the Sgr A* continuum, and we demonstrate how the extended Galactic Center continuum and line emission impairs the identification of the magnetar continuum and impacts the low angular resolution spectrum (particularly the RRL emission). Future observations designed to fill in the axion mass coverage or to increase sensitivity should use sub-arcsec resolution arrays, but high-resolution observations will exclude any axion–photon conversion signal that may arise from an extended population of Galactic Center neutron stars (Safdi et al. 2019).

It is unclear whether an axion–photon conversion signal will be pulsed. Analytic models suggest that it should be (e.g., Hook et al. 2018), but detailed ray-tracing and magnetosphere simulation are needed (Leroy et al. 2020). If emission is pulsed, future observations could in principle increase signal-to-noise by observing spectra in a gated pulsar mode.

We thank the operations, observing, archive, and computing staff at the NRAO who made this work possible. We also thank Konrad Lehnert, Marco Chianese, Andrea Caputo, and Richard Battye for helpful discussions. This research made use of CASA (McMullin et al. 2007), NumPy (van der Walt et al. 2011), Matplotlib (Hunter 2007), and Astropy,² a community-developed core Python package for Astronomy (Astropy Collaboration et al. 2013; Price-Whelan et al. 2018).

Facility: VLA - Very Large Array

Software: CASA (McMullin et al. 2007), astropy (The Astropy Collaboration 2013, 2018), NumPy (van der Walt et al. 2011), Matplotlib (Hunter 2007)

Here we present the new radio spectra of the individual bands used to derive limits on the axion–photon coupling g_aγγ versus axion mass m_a presented in the main Letter. Figures A1–A9 show the flux density spectra, noise spectra, and significance spectra used for flux density and g_aγγ limits (Figures 1 and 2). Spectra obtained from VLA programs 14A-231 and 14A-232 (L-, C-, X-, Ku-, and Ka-bands listed in Table 1) are presented in Darling (2020).

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