A Search for Axionic Dark Matter Using the Magnetar PSR J1745-2900
AA Search for Axionic Dark Matter Using the Magnetar PSR J1745 − Jeremy Darling
Center for Astrophysics and Space AstronomyDepartment of Astrophysical and Planetary SciencesUniversity of Colorado, 389 UCBBoulder, CO 80309-0389, USA ∗ (Dated: August 19, 2020)We report on a search for dark matter axion conversion photons from the magnetosphere ofthe Galactic Center magnetar PSR J1745 − a No significant spectral features are detected. Using a hybrid model for PSRJ1745 − g aγγ > × − GeV − with 95% confidence over the massranges 4.2–8.4, 18.6–26.9, 33.0–41.4, 53.7–62.1, and 126.0–159.3 µ eV. If there is a dark matter cusp,the limits reduce to g aγγ > × − GeV − , which overlap some axion models for the observedmass ranges > µ eV. These limits may be improved by modeling the stimulated emission that canboost the axion-photon conversion process. INTRODUCTION
The axion is a spin zero chargeless massive particle in-troduced to address the strong CP (charge-parity) prob-lem in quantum chromodynamics (QCD; [1–3]). If theyexist, QCD axions are likely to be produced in the earlyuniverse, are a promising cold dark matter candidate [4–6], and may explain the cosmic matter-antimatter asym-metry [7].Axions couple to QCD and electromagnetism. Theelectromagnetic coupling L aγγ = − (1 / g aγγ a F µν ˜ F µν = g aγγ a E · B suggests that axion-photon conversion canoccur in the presence of magnetic fields, but the axion-photon coupling is weak ( g aγγ ∼ − GeV − for axionmass m a = 1 µ eV) [8–12]. The relationship between g aγγ and m a is linear, but the mass is not constrained; axionsearches must span decades in m a .Recent axion searches include CAST, which searchedfor Solar axions [13, 14], and haloscopes such as ADMXand HAYSTAC that use narrow-band resonant cavitiesto detect dark matter axions [15–18]. There are also nat-ural settings where telescopes may conduct sensitive andwide-band QCD axion searches [19–27].The Galactic Center magnetar PSR J1745 − . × G) [28] and should seethe highest possible dark matter flux [20]. Axions willencounter a plasma frequency at some radius that equalsits mass, and the axion can resonantly convert into aphoton at that location [20]. The most promising axionmass range, 1–100 µ eV, corresponds to radio frequencies200 MHz to 20 GHz.In this Letter , we present broad-band radio telescope a The National Radio Astronomy Observatory is a facility of theNational Science Foundation operated under cooperative agree-ment by Associated Universities, Inc. observations of PSR J1745 − g aγγ versus m a based on a hybrid neutronstar magnetosphere model and two limiting-case Galacticdark matter profiles. Finally, we discuss model caveats,limitations of the observations, and future observationaland theoretical work to expand the g aγγ vs. m a spaceprobed by this technique. OBSERVATIONS
Interferometric observations of Sgr A* were obtainedfrom the National Science Foundation’s Karl G. Jan-sky Very Large Array (VLA [29]) data archive. We se-lected sessions: (1) to maximize on-target integrationtime, (2) with adequate angular resolution to separatePSR J1745 − − − a r X i v : . [ a s t r o - ph . C O ] A ug TABLE I. Very Large Array Summary of Observations.Band Frequency Program Channel Width Median Integration Median Beam PA MJD a rms b Obs. Smoothed Velocity(GHz) (MHz) (MHz) (km s − ) (s) (arcsec) ( ◦ ) (mJy)L 1.008–2.032 14A-231 1 10 2000 10591 2 . × . c d C 4.487–6.511 14A-231 2 14 763 10591 0 . × . − . × . − e Ku 12.988–15.012 14A-231 2 20 428 16156 0 . × . − . × . − . × . − . × . − . × . − a Modified Julian Date b The rms noise in spectra Gaussian-smoothed to channels of width ∆ f (Column 5 and Equation 2). c The quoted L-band beam is the continuum beam; the synthesized beam in the spectral cube is highly variable due to RFI and thelarge fractional bandwidth. d The rms noise includes residual unmitigated RFI. e The rms noise omits the central RFI feature and band edges. spectral windows were divided into 2 or 4 slightly over-lapping baseband groups. The L-band configuration dif-fered: it used spectral windows of 64 MHz bandwith and1 MHz channels. All sampling was 8-bit except for theKa-band session, which used 3-bit sampling. Correlatordump times were 1–5 s.
DATA REDUCTION
All data reduction tasks were performed using CASA[30]. After data flagging, we applied the flux, delay, at-mospheric transmission, complex bandpass, and complexgain calibration to the target field. We then performedin-beam phase self-calibration using the Sgr A* contin-uum, which ranged from 0.5 to 5.6 Jy (L- to Ka-band).We imaged the continuum field and identified the contin-uum emission of PSR J1745 − − α, ∆ δ ) = (1 . , − . − CO and 36.795 GHz CH CN. Sidelobes from these featuresare cleaned during cube deconvolution and do not signif-icantly contaminate the magnetar spectrum. The zero-mean magnetar spectra typically reach the theoreticalnoise.At the position of the magnetar, we extract a spec-trum by summing pixels over the (frequency-dependent)synthesized 2D Gaussian beam. We correct the spectrumfor the beam size to capture the total flux density of apoint source separately for every channel. In order toassess the significance of features in the magnetar spec-trum, which can be highly variable channel-to-channeldue to radio frequency interference (RFI) or band edges,we create a noise spectrum by measuring the off-sourcerms sky noise in each channel. This allows us to create asignificance spectrum for each band. We scale the noisespectrum to match the spectral noise measured towardPSR J1745 − f /f = ( v /c ) [20], where v is theaxion velocity dispersion, to ( v /c ) [21]. We adopt thespinning mirror model whereby the Doppler broadeningis dominated by the neutron star rotation [24], produc-ing, on average, ∆ f /f (cid:39) Ω r c ε /c , where Ω is the neu-tron star’s angular frequency, ε is the eccentricity of theelliptical critical surface (which we and [24] set to unity),and r c is the axion-photon conversion radius. For thelatter, we adopt the formulation of [20], assuming po-lar orientation angle θ = π/ θ m = 0 (see below, where we marginalize over these un-known angles; as [25] show, ray-tracing is the technicallycorrect approach, and tends to show less variation with FIG. 1. 95% confidence limits on emission line flux densityfor channel width ∆ f = 8 . × ( m a c / . µ eV) / . these angles than the analytic treatment): r c = 224 km (cid:16) r
10 km (cid:17) (cid:34) B G 12 π Ω1 Hz (cid:18) . µ eV m a c (cid:19) (cid:35) / (1)where r is the neutron star radius, B is the magneticfield, and 4.1 µ eV corresponds to an observed frequencyof 1 GHz. This leads to an expected line bandwidth:∆ f = 3 . (cid:18) Ω1 Hz (cid:19) / (cid:18) m a c . µ eV (cid:19) / (cid:18) B G (cid:19) / . (2)The spin period of PSR J1745 − . × G [28]. The expected linewidth is therefore ∆ f = 8 . × ( m a c / . µ eV) / .The VLA correlator spectral windows have a strongresponse dropoff on edge channels and on the edges of thebasebands. These channels lack the signal-to-noise forcalibration and are flagged, which typically creates single-channel gaps in the spectrum and decreased sensitivityon the spectrum edges. The flagged channels are muchnarrower than the expected line width. We smooth themagnetar and noise spectra to the expected line width(Equation 2) using a Gaussian kernel [33], and interpolateacross missing channels during smoothing (the resultingstatistics are not substantially altered by this loss of strictchannel-to-channel independence). In L-band there aregaps in the spectrum where RFI-flagged spectral regionsare wider than the smoothing kernel.Table I lists synthesized beam parameters, channelwidths, and spectral rms noise values. The Supplemen-tal Material [34] presents the magnetar flux, noise, and signal-to-noise spectra. RESULTS
We detect no significant single-channel emission fea-tures in the observed bands. The only two channelsabove 3 σ in emission were found in Ka-band (3.1 σ and3.5 σ ). These departures are consistent with Gaussiannoise statistics.After establishing the lack of significant detections inthe observed spectra, we form 95% confidence limits fromthe sky noise spectra. Figure 1 presents these single-channel confidence limits. These are model-independentlimits on the flux density of photons produced by ax-ion conversion in the magnetosphere of PSR J1745 − ANALYSIS
Limits on the axion-photon coupling g aγγ obtainedfrom observed flux density limits depend on the magne-tar magnetospheric and axion-photon conversion modeland on the behavior of the dark matter at the Galac-tic Center. Our choices — as well as alternatives — arediscussed below. The Magnetar Model
Hook et al. [20] predict the photon flux from axion con-version in a neutron star magnetosphere using a variantof the Goldreich and Julian [36] model. The observed fluxdensity depends on the local dark matter density ( ρ ∞ )and its velocity dispersion ( v ) and on the neutron starmass, radius, magnetic field, rotation period, distance,viewing angle with respect to the rotation axis, and mag-netic misalignment angle from the spin axis ( M NS , r , B , P , d , θ , and θ m , respectively). It also depends on m a and g aγγ . Equations 11–13 in [20], modified for theexpected bandwidth (Equation 2), yield an expressionfor the emission line flux density normalized to fiducialvalues for a nearby pulsar: S ν = 1 . × − mJy (cid:18)
100 pc d (cid:19) (cid:16) m a (cid:17) / (cid:18)
200 km s − v (cid:19) (cid:18) g aγγ − GeV − (cid:19) (cid:16) r
10 km (cid:17) × (cid:18) B G (cid:19) / (cid:18) Ω1 Hz (cid:19) − / (cid:18) ρ ∞ . − (cid:19) (cid:18) M NS (cid:12) (cid:19)
3( ˆ m · ˆ r ) + 1 | θ ˆ m · ˆ r − cos θ m | / v c c (3)where ˆ m · ˆ r = cos θ m cos θ + sin θ m sin θ cos Ω t . The axion velocity v c at the conversion point r c (see Supplement) isincluded to correct the conversion probability presentedin [20], as discussed in [25]. For a given line sensitivity(or observed flux density limit spectrum), one can ob-tain a track in g aγγ versus m a space above which g aγγ isexcluded at some confidence level.For PSR J1745 − r = 10 kmand mass M NS = 1 M (cid:12) . Using the Navarro-Frenk-White (NFW [37]) Galactic dark matter profile γ = 1 model of[38], which has Galactic Center distance d = 8 . r s = 18 . ρ (cid:12) = 0 .
38 GeV cm − (seebelow), one obtains an axion mass-dependent expressionfor g aγγ : g aγγ = 3 × − GeV − (cid:18) S ν µ Jy (cid:19) / (cid:16) m a (cid:17) − / (cid:18) v
200 km s − (cid:19) / × (cid:18) ρ ∞ . × GeV cm − (cid:19) − / (cid:18)
3( ˆ m · ˆ r ) + 1 | θ ˆ m · ˆ r − cos θ m | / v c c (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ( i ) − / . (4)The angular term ( i ) in Equation 4 relies on the unknownviewing and magnetic field misalignment angles, so wemarginalize over all possible orientations while keepingtrack of the fraction of the magnetar spin period (if any)that has r c < r (see Supplement). If the conversionradius r c is less than the neutron star radius, then noconversion occurs and the photon flux vanishes [20]. Theanalytic value of r c < r is a conservative estimate; ray-tracing suggests that the actual axion-photon conversioncan occur over a larger range of angles [25]. Contraryto [20], we find that conversion can still occur up to andbeyond 10 GHz. At the low end of the observed Ka-band range, 2.1% of all possible ( θ, θ m ) have r c < r at alltimes; at the high end of the band, this increases to 5.6%.At the high end of the observed Ku band, the fraction oforientations with no emission decreases to 0.16%. Axion Constraints from Dark Matter Models
Following [20], we use two dark matter models — ageneric NFW model and the NFW model plus a centraldark matter spike — that roughly bound the weakest andstrongest constraints on g aγγ given the magnetar modelunless the Galactic Center dark matter is cored (see be-low and Discussion). We use the NFW dark mattermodel from [38] with γ = 1, R = 8 . r s = 18 . v = 300 km s − , and ρ (cid:12) = 0 .
38 GeV cm − , which pre-dicts a dark matter density of 6 . × GeV cm − atthe 0.1 pc radial distance of the magnetar. The assumedphysical distance of PSR J1745 − γ sp = 7 / R sp = 0 . . × GeV cm − at the magnetar position. All elseequal, this dark matter spike model can be considered tobe a best-case constraint on g aγγ . The two models spana factor of 10 in dark matter density, which is a factor of100 in the g aγγ constraint. Future studies of the dynam-ics in the central parsec should improve constraints on thedark matter density encountered by PSR J1745 − | g aγγ | obtained from these end-case dark matter mod-els for each observed band. Figure 2 shows the limitsfor the two models versus frequency and axion mass.The limits obtained for the standard NFW profile reach | g aγγ | (cid:39) × − GeV − , which is a factor of ∼
40 abovethe strongest-coupling theoretical model [41]. The limitsobtained for the dark matter spike model impinge on thefamily of theoretical models [41] for m a in 33.0–41.4 µ eV,53.7–62.1 µ eV, and 126.0–159.3 µ eV, but do not excludethe canonical KSVZ or DFSZ models [8–11]. If the Galac-tic dark matter has a flat profile inward of 0.5 kpc (i.e.it is “cored”), then the predicted dark matter density is12 GeV cm − at the magnetar, the limits on g aγγ are afactor of 100 above the NFW profile limits, and the limitsare less constraining than the CAST limits. DISCUSSION
The limits on g aγγ presented here are less stringentthan those predicted by [20], even after scaling the tele-scope sensitivity and integration time. This is due to theexpected bandwidth of the conversion emission line; we FIG. 2. 95% confidence limits on g aγγ for a generic NFW γ = 1 model with ρ (cid:12) = 0 .
38 GeV cm − (purple, upper) and thesame NFW model plus a central 100 pc dark matter spike (blue, lower). The green bar shows the HAYSTAC limit [18], whichhas been scaled from a local axion density of 0.45 GeV cm − to 0.38 GeV cm − , and the yellow bar shows the CAST 95%confidence limit [42]. Orange loci indicate a range of possible QCD axion models [41], including the canonical KSVZ and DFSZmodels [8–11].TABLE II. Limits on g aγγ for two dark matter profile models.Axion Mass Median | g aγγ |
95% Confidence Limits( µ eV) NFW Profile DM Spike(GeV − ) (GeV − )4.2–8.4 a . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − There are gaps in the coverage of this mass range (see Figure 2). used a line width roughly 10 –10 times larger, followingthe rotating mirror conversion region treatment of [24],which leads to a factor of ∼
300 in g aγγ . This bandwidthis likely the most conservative choice, and a more phys-ical model for the axion-photon conversion bandwidthwill require detailed ray-tracing and a more sophisticatedmagnetosphere model [25].The magnetospheres of magnetars must be more com-plicated than current models assume. Moreover, the res-onant conversion of axions into photons will likely beboosted by stimulated emission caused by the ambientphoton bath [43]. The expected conversion emission linewould be enhanced by the photon occupation number f γ = π ρ γ /E γ , where ρ γ is the specific energy densityof stimulating photons (which must match the conver- sion photons in energy and momentum). Including thecontribution from stimulated emission, which is expectedto be particularly important in the Galactic Center envi-ronment [43], will significantly improve the constraints on g aγγ without the need for additional observations. Nu-merical simulation may be required to correctly charac-terize the role of stimulated emission in resonant axion-photon conversion in neutron star magnetospheres.Finally, the constraints on g aγγ depend critically on thedark matter energy density in the inner parsec, which re-mains poorly constrained compared to the other param-eters in Equations 3 and 4 (which have uncertainties oforder 10–50%). Cuspy versus cored dark matter profilesdepend on the dominant baryon physics. Baryons cancontract and steepen the central dark matter density, andthere is evidence that this has happened in the Galaxy[44], but baryon feedback (e.g. star formation and super-novae) can flatten the central dark matter profile. Whilea multi-kpc core is disfavored [45], current studies cannotyet probe the inner kpc, so the dark matter density at themagnetar remains an extrapolation. It seems likely, how-ever, that observations of stars and gas combined withdynamical models will soon provide meaningful measure-ments of the role of dark matter in the Galactic Center.The sensitivity of future telescopic observations will alsoimprove with the advent of large collecting-area facilitiessuch as the Square Kilometer Array, although added sen-sitivity above a few GHz would require a next-generationVLA. CONCLUSIONS
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Jeremy DarlingThis Supplementary Material presents the radio spectra, noise spectra, and significance spectra of the individualbands used to derive limits on the axion-photon coupling g aγγ versus axion mass m a presented in the main Letter.We also present a discussion of the impact of the magnetar’s rotation and magnetic field axes on the translation offlux density limits into limits on g aγγ . SPECTRA OF THE MAGNETAR PSR J1745 − Figures 1–5 show the continuum-subtracted flux density spectra, noise spectra, and significance spectra used forflux density and g aγγ limits. FIG. 1. L-band flux density, noise, and signal-to-noise spectra. The upper spectrum provides limits on g aγγ , while the lowerspectrum indicates the significance of spectral features. FIG. 2. C-band flux density, noise, and signal-to-noise spectra.FIG. 3. X-band flux density, noise, and signal-to-noise spectra.
FIG. 4. Ku-band flux density, noise, and signal-to-noise spectra.FIG. 5. Ka-band flux density, noise, and signal-to-noise spectra. The four overlapping basebands listed in Table I of the mainLetter were combined into a single spectrum.
FIG. 6. Left: Time-averaged angular term at 10 GHz plotted for the unknown viewing ( θ ) and magnetic dipole orientation( θ m ) angles of the magnetar. Right: Angle-marginalized and time-averaged angular term ( i ) versus frequency. IMPACT OF THE MAGNETAR SPIN AND MAGNETIC FIELD AXES ON AXION LIMITS
The angular term ( i ) in Equation 4 of the main Letter modulates the signal, and the modulation time is less thanthe integration time of the observations, so we average the expected signal over the period of the magnetar for eachfrequency channel. Moreover, the unknown magnetar spin axis orientation angle θ and magnetic dipole offset angle θ m impact both the emitted flux density (main Letter, Equation 3) and the axion-photon conversion radius r c [52]: r c = 224 km (cid:16) r
10 km (cid:17) (cid:34) B G 12 π Ω1 Hz (cid:18) . µ eV m a c (cid:19) (cid:35) / | θ ˆ m · ˆ r − cos θ m | / . (1)When r c < r no axion conversion occurs, so the rotation period-averaged flux density will be “censored” in afrequency-dependent manner. The velocity at the conversion point is v c (cid:39) GM NS /r c [24].For each observed frequency channel and possible ( θ, θ m ) pair, we create an emission profile over the period ofthe magnetar rotation that includes no emission when r c < r . We calculate a time-integrated flux density, andwe marginalize these flux densities at each channel over all ( θ, θ m ). Figure 6 shows an example at 10 GHz of theinfluence of θ and θ m on the angular term ( i ) that includes time-averaging including times when r c < r . We also plotthe net result of the marginalization over these unknown angles versus frequency and equivalent axion mass. Thesetime-averaged and angle-marginalized angular terms are used in the calculation to place limits on g aγγaγγ