The search for low-mass axion dark matter with ABRACADABRA-10cm
Chiara P. Salemi, Joshua W. Foster, Jonathan L. Ouellet, Andrew Gavin, Kaliroe M. W. Pappas, Sabrina Cheng, Kate A. Richardson, Reyco Henning, Yonatan Kahn, Rachel Nguyen, Nicholas L. Rodd, Benjamin R. Safdi, Lindley Winslow
TThe search for low-mass axion dark matter with ABRACADABRA-10 cm
Chiara P. Salemi, ∗ Joshua W. Foster,
2, 3, 4, † Jonathan L. Ouellet, ‡ Andrew Gavin, Kaliro¨e M. W. Pappas, Sabrina Cheng, Kate A. Richardson, Reyco Henning,
5, 6
Yonatan Kahn,
7, 8
Rachel Nguyen,
7, 8
Nicholas L. Rodd,
3, 4
Benjamin R. Safdi,
2, 3, 4 and Lindley Winslow § Laboratory of Nuclear Science, Massachusetts Institute of Technology, Cambridge, MA 02139 Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, MI 48109 Berkeley Center for Theoretical Physics, University of California, Berkeley, CA 94720 Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, CA 94720 Department of Physics and Astronomy, University of North Carolina, Chapel Hill, Chapel Hill, NC, 27599 Triangle Universities Nuclear Laboratory, Durham, NC 27710 Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801 Illinois Center for Advanced Studies of the Universe,University of Illinois at Urbana-Champaign, Urbana, IL 61801 (Dated: February 16, 2021)Two of the most pressing questions in physics are the microscopic nature of the dark matter thatcomprises 84% of the mass in the universe and the absence of a neutron electric dipole moment.These questions would be resolved by the existence of a hypothetical particle known as the quantumchromodynamics (QCD) axion. In this work, we probe the hypothesis that axions constitute darkmatter, using the ABRACADABRA-10 cm experiment in a broadband configuration, with world-leading sensitivity. We find no significant evidence for axions, and we present 95% upper limits onthe axion-photon coupling down to the world-leading level g aγγ < . × − GeV − , representingone of the most sensitive searches for axions in the 0 . − .
27 neV mass range. Our work pavesa direct path for future experiments capable of confirming or excluding the hypothesis that darkmatter is a QCD axion in the mass range motivated by String Theory and Grand Unified Theories.
The axion is a well-motivated candidate to explainthe particle nature of dark matter (DM) [1–3]. Thispseudoscalar particle is naturally realized as a pseudo-Goldstone boson of the Peccei-Quinn (PQ) symmetry,which is broken at a high scale f a ; the axion would beexactly massless but for its low-energy interactions withquantum chromodynamics (QCD) [4–7]. The axion massis tied to the scale f a by m a ≈ . GeV /f a ) neV [8].The range of scales f a ≈ − GeV is particularlycompelling because of connections to String Theory [9]and Grand Unification [10, 11], and in the correspondingmass range of m a ∼ −
10 neV the axion may naturallyexplain the observed DM abundance [11, 12]. In this Ar-ticle we provide the most sensitive probe of axion darkdatter (ADM) in this mass range to date.ADM that couples to photons modifies Amp`ere’s lawsuch that in current-free regions ∇ × B = ∂ E ∂t − g aγγ (cid:18) E × ∇ a − ∂a∂t B (cid:19) , (1)with E and B the electric and magnetic fields, re-spectively, a ( x , t ) the ADM field, and g aγγ the axion-electromagnetic coupling constant. In the presence of astatic external magnetic field ADM behaves like an ef-fective current density J eff = g aγγ ( ∂ t a ) B . If the axion ∗ These authors contributed equally; [email protected] † These authors contributed equally; [email protected] ‡ These authors contributed equally; [email protected] § [email protected] makes up all of the observed DM then, to leading or-der in the DM velocity, ∂ t a = √ ρ DM cos( m a t ), with ρ DM ≈ . / cm the local DM density [13]. It waspointed out in [14, 15] that the effective current inducesan oscillating secondary magnetic field which may be de-tectable in the laboratory without the aid of a resonantcavity for sufficiently small m a . The oscillation frequencyis given by f = m a / (2 π ), with bandwidth δf /f ≈ − arising from the finite axion velocity dispersion [16]. Inthis work we leverage this theoretical principle to searchfor axions in the laboratory.The most common detection strategy for ADM isthrough the electromagnetic coupling g aγγ , which for theQCD axion is directly proportional to the mass m a . Un-til recently, experiments have focused on searching foraxions in the mass range 1 (cid:46) m a (cid:46) µ eV, whichis well-suited to microwave cavity searches [17–21]. Inthe low-mass regime targeted in this work, the Comptonwavelength of the axion λ C ∼ km is much larger thanthe experimental apparatus, and so the sensitivity of theexperiment improves with volume as V / , roughly inde-pendent of m a until the size of the experiment approaches λ C [15]. This scaling is important because the ex-pected coupling g aγγ is smaller at lower masses, requiringever-more-sensitive experiments to achieve a detection.ABRACADABRA is an experimental program designedto detect axions at the Grand Unification scale using astrong toroidal magnetic field [15]. ABRACADABRA ispart of a suite of ADM experiments which together aim toprobe the full QCD axion parameter space [19–28]. Theexperiment we report on here, ABRACADABRA-10 cm,is a prototype for a larger ADM detector that would be a r X i v : . [ h e p - e x ] F e b sensitive to the QCD axion. This Article presents datacollected in 2020 that is up to an order of magnitudemore sensitive than our previous results [29] and placesstrong limits on ADM in the 0 . − .
27 neV range ofaxion masses.
ABRACADABRA-10 CM DETECTOR
The ABRACADABRA-10 cm detector is built arounda 12 cm diameter, 12 cm tall, 1 T toroidal magnet fab-ricated by Superconducting Systems Inc [30]. The ax-ion interactions with the toroidal magnetic field B drivethe effective current, J eff , which oscillates parallel to B and sources a real oscillating magnetic field through thetoroid’s center. The oscillating magnetic flux is readout with a two-stage DC-SQUID via a superconductingpickup in the central bore. Unlike other axion detectordesigns, this novel geometry situates the readout pickupin a nominally field-free region unless axions are present[15]. The detector can be calibrated by injecting fake ax-ion signals ( i.e. , AC currents) through a wire calibrationloop that runs through the body of the magnet. Thedetector, illustrated schematically in Fig. 1, is located onMIT’s campus in Cambridge, MA.In 2019, we performed several detector upgrades fromthe Run 1 configuration in order to improve our sensitiv-ity [29, 31]. In this Article we report the results of thesubsequent data campaign (Run 3), collected after thedetector upgrade. Run 3 data consists of ∼
430 hours ofdata collected from June 5 to June 29, 2020.Before the upgrades were complete, we took additional,uncalibrated data (Run 2), which is not presented here.A subset of that data was instead used to develop ourdata analysis procedure in order to run a blind analysison the Run 3 data, as described in detail below.The total expected axion power, A , coupled into ourreadout pickup is related to the axion-induced flux Φ a as A ≡ (cid:104)| Φ a | (cid:105) = g aγγ ρ DM G V B , (2)where G is a geometric coupling, V is the magnetic fieldvolume, B max is the maximum value of | B | , and the an-gle brackets denote the time average [15, 32]. Run 1 uti-lized a 4.02 cm diameter pickup loop made from a 1 mmdiameter wire, giving G ≈ . µ m-thick Nb sheet wrapped arounda polytetrafluoroethylene (PTFE) cylinder. This designyields a stronger geometric coupling to J eff of G ≈ . G using electromagnetic simulations in the COM-SOL Multiphysics package [31, 33].To amplify our signal, Φ a is coupled into the read-out SQUID through the pickup circuit (see Fig. 1) yield-ing a transformer gain M in /L T , where M in is the inputcoupling to the SQUID, and L T ≡ L p + L in + L wires isthe total inductance of the pickup circuit, with L p the C R L R R p L P L in M R B a B R J e↵ L p L wires L in a M in m a Frequency P o w e r FFT J e↵ L p L wires L in a M in G R f M f m a Frequency P o w e r FFT L in C Figure 1.
Top : Schematic of ABRACADABRA-10 cm show-ing the effective axion-induced current (blue), sourced by thetoroidal magnetic field, generating a magnetic flux (magenta)through the pickup cylinder (green) in the toroid bore.
Bot-tom : Simplified schematic of the ABRACADABRA-10 cmreadout (full circuit diagram in Supp. Fig. S1). The pickupcylinder L p is inductively coupled to the axion effective cur-rent J eff . The power spectrum of the induced current is readout through a DC SQUID inductively coupled to the circuitthrough L in . An axion signal would appear as excess powerabove the noise floor at a frequency corresponding to the ax-ion mass. pickup cylinder inductance, L in the input inductance ofthe SQUID package, and L wires the parasitic inductance,dominated by the twisted pair wiring. The SQUID, man-ufactured by Magnicon [34], is read out using Magnicon’sXXF-1 SQUID electronics operating in closed feedbackloop mode. The Run 1 sensitivity was limited by par-asitic inductance in the NbTi wiring of this circuit thatplaced a lower limit on L T (cid:38) . µ H. During the upgrade,we replaced this wiring, moving the SQUIDs closer to thedetector to reduce the wire length. Based on calibrationdata, we found that the total impedance in the circuit is ∼
800 nH. Finally, the SQUID was operated at a higherflux-to-voltage gain setting of 4.3 V/Φ in Run 3, com-pared to the previous Run 1 which we ran at 1.29 V/Φ due to higher levels of environmental noise. This changedoes not directly improve the signal gain, but does reducesystem noise. We also improved our noise floor by re-ducing the operating temperature of the SQUID packagefrom ∼
870 mK to ∼
450 mK. All together, the upgradecampaign increased the expected power coupled into ourreadout and reduced the total system noise.The improved sensitivity of the upgraded readout cir-cuit also amplified the low-frequency vibrational back-grounds seen in Run 1, which caused the SQUID am-plifier to rail when the magnet was on. In order to cor-rect this, we implemented an active feedback stabilization(AFS) system to reduce the low-frequency noise, which isdiscussed further in the Supplementary Information (SI).As in Run 1, the magnet and pickup were placed in-side a superconducting tin-copper shield and hung from apassive vibration isolation system, consisting of a stringpendulum and spring, within an Oxford InstrumentsTriton 400 dilution refrigerator [31]. The magnet andpickup were operated at (cid:46) ∼
400 mK, which kept the readout circuit superconduct-ing over the course of the run and kept thermal noise sub-dominant to SQUID flux noise. Following the procedureof Run 1, the output of the SQUID was run into an 8-bitAlazarTech AT9870 digitizer via a 70 kHz-5 MHz band-pass filter. The digitizer was locked to a Stanford Re-search Systems FS725 Rubidium frequency standard inorder to maintain clock accuracy over the coherence timeof the axion signal, ∼ Frequency [Hz]0510152025 F l u x t o V o l t ag e C o n v e r s i o n [ m V / Φ ] Run 3 Run 1 (prev. work) Calculated gain, Run 3
Figure 2. The gain shown here is defined as the change inamplifier output voltage over a corresponding change in in-put flux amplitude on the pickup cylinder ( ∂V out /∂ Φ a ). Bothtransfer functions roll off at high frequencies due to the ampli-fier bandwidth, which we estimate to have a cutoff frequencyof approximately 1 MHz. We believe the difference in calcu-lated and measured gain is due to inconsistency in the totalinductance of the pickup circuit. We performed in situ magnet-on and magnet-off cali-brations in the data-taking configuration by attaching aharmonic signal generator to the calibration circuit andscanning across frequencies and amplitudes. The cali-bration signal was attenuated and fed into the calibra-tion loop, mimicking the axion effective current signal J eff up to geometric factors. The geometry is modeledin COMSOL Multiphysics [33], from which we extractthe coupling between both the calibration loop and ax-ion effective current signal to the pickup cylinder. Bycombining the results of the calibration scans and ge-ometric modeling, we can determine the effective gain, ∂V out /∂ Φ a , of the SQUID amplifier output voltage as a function of flux on the pickup cylinder (see Fig. 2). Thisprocedure is analogous to that used in Run 1 [31].The gain measured by the calibrations for Run 3 dif-fers from the calculated gain by a factor of ∼ DATA COLLECTION
The axion search data was collected using an identi-cal procedure as in Run 1 [31]. The SQUID amplifieroutput voltage was sampled at a frequency of 10 MS/s,with a ±
40 mV voltage window. The data were stored asa series of power spectral densities (PSDs), which werecomputed on-the-fly: ¯ F with a Nyquist frequency of5 MHz and frequency resolution of ∆ f = 100 mHz, ¯ F with a Nyquist frequency of 500 kHz and frequency res-olution of ∆ f = 10 mHz, and a continuous data streamsampled at 100 kHz that can be analyzed offline. ¯ F ( ¯ F ) is averaged over 800 s (1600 s) before being writ-ten to disk. In this work, we used the ¯ F to searchthe frequency range from 500 kHz − F spectra to search from 50 −
500 kHz.
DATA ANALYSIS AND RESULTS
An axion signal is expected to manifest as a narrowpeak in the PSD data, as illustrated in Fig. 1. The widthand overall shape of the signal are set by the local DMvelocity distribution, which we take to be the StandardHalo Model with a velocity dispersion of v = 220 km/sand a boost from the halo to the solar rest frame of v (cid:12) = 232 km/s [35]. With the speed distribution andlocal DM density fixed, the two free signal parametersare the axion mass, m a , which determines the minimumfrequency of the signal, and the coupling g aγγ , which de-termines its amplitude through Eq. (2). Our analysisprocedure closely follows the approach used in the Run 1search [29, 31] based on [32], which constrains the allow-able values of g aγγ at each possible value of m a .The search is performed with a frequentist log-likelihood ratio test statistic (TS); exact expressions areprovided in the SI (see also [31]). Our broadband searchprocedure probes ∼ . . − .
27 neV (100 kHz − χ -distribution [32], which was indeed thecase in Run 1 [29, 31]. However, the increased sensitiv-ity from the detector upgrades introduced non-Gaussiannoise sources that required us to modify our Run 1 anal-ysis procedure. We developed and validated our new pro-cedure on a randomly-selected sample of 10% of Run 2’s ∼ . m a , the search was performed in a narrow windowaround that mass with the background level allowed tovary independently in each window. For the Run 2 andRun 3 analyses we allow the mean background level ofthe noise to vary linearly with frequency uniquely in eachsliding window. We use sliding windows of relative width δf /f ≈ . × − , starting at f = (1 − − ) × m a / (2 π ).As in Run 1, we use the magnet-off data to veto fre-quency ranges that also display statistically significantTS values when | B | = 0 and thus the axion power shouldvanish. However, we observed narrow single-bin ‘spikes’that appear to drift in frequency on the timescale of ourdata collection (see Supp. Fig. S6 for an example). If in-terpreted in isolation, these spikes sometimes correspondto statistically-significant excesses. Nevertheless, theyare inconsistent with axion signals and are most likelydue to unknown environmental noise sources near the de-tector, persisting throughout Runs 2 and 3; indeed, manyof the peaks are distributed at multiples of 50 Hz. To re-move these artifacts, we leverage the fact that the PSDsare saved periodically to disk yielding a time evolution ofthe environmental backgrounds; we veto single-bin spikesthat move in frequency. We place a 1.0 Hz veto windowaround these single-bin spikes. These cuts remove 3.8%of the axion mass points from our search in the Run 3data. The magnet-off vetoing procedure removes an ad-ditional 0 .
07% of mass points.After implementing the vetoes, we found the distribu-tion of TS values in the 10% Run 2 validation sampledeviated from the expected χ distribution; for example,there were 27 mass points with TS >
25 whereas from the χ distribution we would have expected less than one. Toaccount for the deviation in the TS distribution from the χ distribution in a data-driven fashion, we follow theprescription developed and implemented in [36–38] forsearches for DM-induced lines in astrophysical gamma-ray data sets. At each mass point, we introduce andprofile over a systematic nuisance parameter that wouldbe degenerate with the signal parameter but for a priorthat is determined by forcing the TS distribution to fol-low the χ distribution. Specifically, we force the TSdistribution to match the null hypothesis distribution at4 σ local significance. This is described further in the SI.After the nuisance parameter and vetoing procedures,we construct a likelihood as a function of g aγγ at eachmass point. The final distribution of TS values computedfrom the likelihoods is shown in Fig. 3; no TS values werefound in excess of the 5 σ look-elsewhere effect-correcteddiscovery threshold. In the calibration of our analysisprocedure, we found one signal candidate in the Run 2data at over 5 σ global statistical significance (see Supp. TS10 − − − Su r v i v a l F r a c t i o n RawData CleaningOff VetoesFinal (Nuisance Tuned)Expected5 σ Detection Threshold
Figure 3. The survival function of TS values from the likeli-hood analysis of the Run 3 results. The y -axis indicates thefraction of mass points tested with a discovery TS at or abovethe value on the x -axis. Under the null hypothesis, the dis-tribution should follow the survival function of the one-sided χ distribution with one degree of freedom (“Expected,” dot-ted gray). This is indeed the case after data cleaning for e.g. single-channel excesses in time slices, magnet-off vetoes,and the inclusion of a systematic nuisance parameter, whichis tuned in a sliding window at 4 σ local significance to givethe correct number of excesses at or above that significance,masking the signal of interest. No excesses are found beyondour indicated 5 σ LEE-corrected discovery threshold.
Fig. S6, where a corresponding feature can be seen in themagnet-off data), but that mass point is not significantin the Run 3 analysis.In the absence of an excess consistent with an ADMorigin, we can determine 95% one-sided upper limits on g aγγ as a function of the mass, m a . The average 95%upper limits from the Run 3 analysis along with their1 σ and 2 σ expectations under the null hypothesis are in-dicated in Fig. 4. In that figure we compare our upperlimits to those found from the ADM experiment SHAFT[27] along with results from the solar axion experimentCAST [39] and astrophysical X -ray searches (SSC) [40],both of which do not require the axion to comprise theDM. The fraction of vetoed mass points is illustrated ina sliding window in Supp. Fig. S4, which also shows thedistribution of data fractions included in the analyses. InSupp. Fig. S5 we illustrate the magnitude of the system-atic nuisance parameter g nuis .aγγ , while in Supp. Fig. S7we show what the limit would be without the nuisanceparameter tuning. Supp. Fig. S8 shows that the 95%upper limit and discovery TS behave as expected whensynthetic axion signals are injected into the real data. m a [neV]10 − − g % a γγ [ G e V − ] S HA F T S S C CAST
95% U.L. [This Work] 1 / σ Containment Run 1 [Ouellet et al., PRL 2019] − Frequency [MHz]
Figure 4. The one-sided 95% upper limit (U.L.) on g aγγ from this work excludes previously unexplored regions of ADMparameter space. The 1 σ and 2 σ containment regions are constructed by taking the appropriate percentiles of the distributionsof the limits over narrow mass ranges; note that this means that ∼
16% of the upper limits lie at the bottom of the greenband. While ∼ g aγγ (cid:46) . × − GeV − at m a ∼ .
99 neV. Our limits surpass those from a number of indicatedastrophysical and laboratory searches in this mass range (see text for details).
DISCUSSION
In this work we present the results fromABRACADABRA-10 cm’s second physics campaign,searching for ADM in the mass range 0.41-8.27 neV.We find no evidence for ADM and constrain theaxion-photon coupling down to the world-leading level g aγγ (cid:46) . × − GeV − at 95% confidence. Our workmotivates key elements of the design of future larger-scaleexperiments. These include the mitigation of stray fieldsfrom the magnet and vibrations induced by a modernpulse-tube-based cryogenic system, which limits our cur-rent low-frequency reach. The ABRACADABRA-10 cmresults presented in this Article demonstrate the powerof mature simulations for optimizing the design of thedetector and for modeling the calibration response.An advanced and novel analysis framework was usedto identify noise sources and account for systematicuncertainties in a data-driven fashion. Our work identifies three areas that can be addressed inthe next physics campaign: (i) moderate improvements(up to a factor ∼ g aγγ ) could be achieved by fur-ther reducing the wire and SQUID inductances, (ii) bet-ter shielding from environmental noise could increase thesensitivity to g aγγ by an order of magnitude at low fre-quencies, so long as (iii) the fringe fields are reduced orbetter vibrationally isolated (see Supp. Fig. S2). Tosignificantly increase the sensitivity of the experiment,larger magnets with higher fields are needed since thesensitivity to g aγγ scales with the detector volume V and field B as g − aγγ ∼ B V / [15]. The addition ofa resonant readout circuit could enhance the reach in g aγγ by an additional ∼ [1] John Preskill, Mark B. Wise, and Frank Wilczek, “Cos-mology of the Invisible Axion,” Phys. Lett. B120 , 127–132 (1983).[2] L. F. Abbott and P. Sikivie, “A Cosmological Bound onthe Invisible Axion,” Phys. Lett.
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IF1 (2020).
METHODS
The ABRACADABRA-10 cm detector is mounted inan Oxford Instruments Triton 400 dilution refrigerator(DR). The detector is suspended from a vibration iso-lation system consisting of a ∼ f ≈ . F . The data stream is simultaneouslydown-sampled to a sampling frequency of 1 MS/s, Fouriertransformed in 100 s windows and accumulated into arunning PSD called ¯ F . ¯ F ( ¯ F ) is written to diskevery 800 s (1600 s) and then reset, yielding a detailedtime evolution of each spectrum. Two collections, whichwe refer to as Run 2 and Run 3, were performed. Run 2contained 1,168,000 s (324 h) of magnet on data in 1,460¯ F spectra and 700 ¯ F spectra and 320,000 s (89 h)of magnet off data in 388 ¯ F spectra and 196 ¯ F spectra. Run 3 contained 1,091,200 s (303 h) of magneton data in 1,364 ¯ F spectra and 682 ¯ F spectra and448,000 s (124 h) of magnet off data in 560 ¯ F spectraand 280 ¯ F spectra. Due to differences in the readoutconfiguration, Run 2 was not as sensitive as Run 3 andthe results are not presented here, though the Run 2 dataare used for tuning the analysis procedure.The data were analyzed in search of the expected ax-ion line-shape using a profiled Gaussian likelihood with alinear background model, estimating the data variance asa floating nuisance parameter at 11.1 million axion masspoints between 100 kHz and 2 MHz. Data from Run 3was analyzed independently based on an analysis vali-dated on 10% of Run 2 data. The data were cleaned byremoving spectral excesses confined to a single frequencybin. The analysis was initially applied to independentlydivided time-continuous subintervals of the data (each ofsize approximately 5% of the full data volume), allowingfor a filtering of transient excesses applied independentlyat each mass. Data which passed the filtering were thenstacked and analyzed for each run, then joined to pro-duce a 95 th percentile upper limit and detection signifi-cance. The detection significance was then corrected bya Gaussian penalty term with a hyperparameter to ac-commodate systematic effects that may produce spuriousdetections. DATA AVAILABILITY
Source data for this paper are made publicly available; all other data may be made available upon reasonablerequest. CODE AVAILABILITY
The code that supports the results presented in thispaper may be made available by the corresponding au-thors upon reasonable request.
ACKNOWLEDGMENTS
We would like to thank Kent Irwin and our DMRadiocolleagues for useful discussions and look forward to thenext-generation experiment. We would like to thank thosethat took part in Run 1 of ABRACADABRA-10cm in-cluding Zachary Bogorad, Janet Conrad, Joseph Formag-gio, Joe Minervini, Alexey Radovinsky, Jesse Thaler, andDaniel Winklehner. We thank Christopher Dessert foruseful analysis discussion. This research was supportedby the National Science Foundation under grant num-bers NSF-PHY-1658693, NSF-PHY-1806440. J.F. andB.R.S. were supported in part by the DOE Early CareerGrant DESC0019225, through computational resourcesand services provided by Advanced Research Computingat the University of Michigan, Ann Arbor, and by compu-tational resources at the Lawrencium computational clus-ter provided by the IT Division at the Lawrence BerkeleyNational Laboratory, supported by the Director, Office ofScience, and Office of Basic Energy Sciences, of the U.S.Department of Energy under Contract No. DE-AC02-05CH11231. Y.K. is supported in part by US Depart-ment of Energy grant DE-SC0015655. R.N. is supportedby the National Science Foundation Graduate Fellowshipunder Grant No. DGE–1746047. N.L.R. is supportedby the Miller Institute for Basic Research in Science atthe University of California, Berkeley. C.P.S. is sup-ported in part by the National Science Foundation Gradu-ate Research Fellowship under Grant No. 1122374. R.H.and K.R. are supported by the U.S. Department of En-ergy, Office of Science, Office of Nuclear Physics underAwards No. DEFG02-97ER41041 and No. DEFG02-97ER41033. We would like to thank the University ofNorth Carolina at Chapel Hill and the Research Com-puting group for providing computational resources andsupport that have contributed to these research results. https://github.com/joshwfoster/ABRA_Results_2020 Supplementary Material for the search for low-mass axion dark matter withABRACADABRA-10 cm
Chiara P. Salemi, Joshua W. Foster, Jonathan L. Ouellet, Andrew Gavin, Kaliro¨e M. W. Pappas, Sabrina Cheng,Kate A. Richardson, Reyco Henning, Yonatan Kahn, Rachel Nguyen, Nicholas L. Rodd, Benjamin R. Safdi, andLindley Winslow
I. DETECTOR UPGRADE ANDELECTROMAGNETIC SIMULATIONS
The sensitivity of ABRACADABRA-10 cm to ADMis set by the coupling strength between the axion in-duced current J eff and the readout SQUIDs. This cou-pling can be conceptually split into two parts: the cou-pling between J eff and the pickup, and the coupling be-tween the pickup and the SQUIDs. Before Run 2, theABRACADABRA-10 cm detector was upgraded in twoways to increase each of these two coupling strengths.The first step of the upgrade was the installation of thesuperconducting pickup cylinder. The pickup cylindergeometry more effectively cancels the flux induced by J eff and thus couples more strongly to it. The cylinder wasconstructed out of a 150 µ m thick sheet of Nb wrappedaround a PTFE tube, secured with Kapton tape. The re-sulting cylindrical pickup was 10 cm tall with a 5.1 cm di-ameter and centered vertically in the magnet bore. Thisis close to the maximum diameter that could practicallyfit. A 1 mm gap was left in the wrapping of the Nb sheetto prevent electrical contact and the formation of a com-plete loop. The PTFE tube was glued and clamped ontothe magnet support structure inside the superconductingshielding can. From experience in Run 1, a strong mount-ing was critical to reducing relative motion between thepickup and magnet.The second step of the upgrade was a replacement ofthe wiring between the pickup cylinder and the SQUIDreadouts. The new wiring – along with the new pickupcylinder – reduced the total inductance of the readoutcircuit, resulting in more current in the SQUIDs. Thenew wiring consists of 75 µ m superconducting solid NbTitwisted-pair wires that are spot welded to two corners ofthe Nb sheet. Spot welding ensures a superconductingconnection between the Nb sheet and the NbTi wires.We used a series of four spot welds on each corner for re-dundancy, in case of breakage during handling or due todifferential thermal contraction. These wires were thentaped to the PTFE cylinder with Kapton in order to min-imize stress on the connections. The wires run ∼ ∼ II. ACTIVE FEEDBACK SYSTEM
A major challenge encountered in Run 1 was low fre-quency vibrations converting stray fields from the mag-net into low frequency noise. Since this noise is generallybelow the frequency ranges of interest, we were able tosimply filter it and ignore it. In Runs 2 and 3, the highergain of the upgraded detector amplified this vibrationalnoise enough to rail the SQUID amplifier. Because mostof this noise was relatively slow – below ∼ III. DETECTOR CALIBRATION
At a basic level, the ABRACADABRA-10 cm readoutconverts the flux through the pickup cylinder Φ p to anoutput voltage from the SQUID amplifier, V SQUID . Thedetector calibration provides an end-to-end measurementof the detector response ∂V SQUID /∂ Φ p to an axion-likesignal across the full range of frequencies being searched.A schematic of the detector configuration for the Run 3calibration can be seen in Fig S1. We generate a fixedfrequency signal of known amplitude using an StanfordResearch Systems SG380 signal generator. This signal isattenuated by 93 dB before passing into the calibrationloop in the detector. The current in the calibration loopgenerates a flux through the pickup cylinder, inducing acurrent and response in the readout circuit in the sameway that an axion signal would, up to geometric factors.The response of the system to a calibration signal canbe written as ∂V ADC ∂V Sig = ∂V ADC ∂V SQUID ∂V SQUID ∂ Φ p ∂ Φ p ∂I C ∂I C ∂V Sig (S1)where V ADC is the RMS voltage measured by the digi-tizer, I C is the RMS current entering the calibration loop, Signal Generator 1 kHz LPFFeedback L C L P R P L in SQUID M CP M in < SignalGenerator LPFFeedback -50 dB L C L P R P L in SQUID M CP M in < SignalGenerator -40dB LPFFeedback -10 dB -40 dB L P L wires L in SQUID M CP M in < Figure S1. ABRACADABRA-10 cm Run 3 calibration cir-cuit diagram. A fake axion signal generated in the signalgenerator is attenuated by 93 dB (including 3dB loss in thecombiner) before being coupled into the pickup cylinder anal-ogously to an axion signal. The resulting power excess isreadout on the SQUID and measured in the ADC digitizer.In Run 3, calibration is performed with the magnet turned onand the active feedback circuit running. During data taking,the signal generator is replaced with a 50 Ω terminator. Theflux-lock feedback loop (FLL) feedback resistor and inductorare omitted for clarity. and V Sig is the peak-to-peak voltage output by the signalgenerator. The first and last terms in this conversion aredetermined by the warm electronics and cold attenua-tors, and can be measured directly, while the third termis the mutual inductance between the calibration loopand pickup cylinder, which is modeled in COMSOL. Bydividing the measured end-to-end calibration by thesethree terms, we are left with the resulting flux to volt-age conversion of the ABRACADABRA-10 cm readoutcircuit ∂V SQUID /∂ Φ p .During Run 3, the calibration was performed in anidentical configuration to data taking, namely with themagnet on and the AFS active. The resulting calibrationcan be seen in Fig. 2, and agreed very well with our calcu-lated signal gain. The rolloff above ∼ g aγγ underthe null hypothesis scales like the square root of the fluxnoise, which is shown in Fig. S2. In that figure we il-lustrate three different noise levels through the SQUID:(i) the measured magnet on flux, which is the relevantflux for the axion signal analysis; (ii) the magnet off flux;(iii) the flux measured in a similar SQUID that is notconnected to the pickup loop circuit (labeled “open in-put”). The increased noise level in the magnet off datarelative to the open input SQUID is likely the result ofimperfect shielding, with environmental noise magnified Frequency [Hz]10 − − S Q U I D fl u x [ µ Φ / H z ] Magnet onMagnet offSimilar SQUID, open input
Figure S2. The SQUID flux for Run 3 over the 70 kHz to2 MHz frequency range at which we collect data. The magneton noise level (magenta) is elevated compared to data takenwith the magnet off (gold) primarily due to vibrating fringemagnetic fields. For comparison, the noise level from a similarSQUID without anything plugged into its input is plotted inteal. by the pickup loop. On the other hand, when the mag-net is on increased noise is apparent at low frequencies,which is the result of the magnetic fringe fields givingfrequency-dependent flux noise because of vibrations. In-creasing the quality of the shielding and decreasing eitherthe magnitude of the fringe fields or their vibrational cou-pling to the pickup loop would improve the sensitivity.
IV. LIKELIHOOD ANALYSIS
In this section, we describe the implementation of theanalysis framework used to produce upper limits on g aγγ and determine detection significances for potential ex-cesses. We first define the profiled Gaussian likelihoodused herein, followed by our procedure for cleaning thedata to enable the removal of spurious excesses and con-founding backgrounds. We then detail our treatment ofa nuisance hyperparameter used to address potential sys-tematics in the data, describe our results in terms of sur-vival functions and upper limits on g aγγ , and demon-strate the efficacy of our analysis pipeline with injectedsignal tests. Though uncalibrated and not presentedhere, the Run 2 data was used while constructing thelikelihood analysis framework. As such, we include it inthe discussion below. A. Likelihood for Axion Signal Detection
The likelihood analysis utilized in this work is per-formed using the signal modeling formalism developedin [32], which was also used in studying the Run 1 re-sults [29, 31]. Our starting point is a series of N samplesof the flux in the pickup loop { Φ n } , made over a collec-tion time T and at a sampling frequency f = 1 / ∆ t (suchthat N ∆ t = T ). In the presence of an axion, this fluxwill receive a contribution from both the DM signal andany background. The mean expectation for the PSD ata frequency f k = k/T isΦ k = A ( g aγγ ) s k ( m a ) + µ k , (S2)with µ k is the mean expected background at this fre-quency. The signal strength parameter A is given in (2)and is controlled by the unknown g aγγ , while s k is thesignal template for a specific axion mass: s k ( m a ) = (cid:26) πf ( v ω ) m a v ω f k > m a / π , f k ≤ m a / π . (S3)Here, v ω = (cid:112) πf k /m a − f ( v ) is the local axionspeed distribution, which we take to be the StandardHalo Model with boost velocity v (cid:12) = 232 km/s andvelocity dispersion v = 220 km/s. To the extent thebackground is Gaussian in the time domain, the PSDformed from this data will be exponentially distributed,and the sum of multiple PSDs formed during data stack-ing will be Erlang-distributed. Nevertheless, in the limitof a large number of stackings, the Erlang-distributionbecomes normally-distributed. For this reason we arejustified in analyzing the data using a Gaussian likeli-hood.In detail, the likelihood used is given by L ( d | m a , A ; a , σ )= (cid:89) k √ πσ exp (cid:20) − ( d k − As k − µ k ( a )) σ (cid:21) , (S4)where d k is the average stacked data, A and s k determinethe axion signal as described above, µ k is the backgroundmodel (specified by parameters a ), and σ is the standarddeviation which we will treat as a nuisance parameter,and therefore estimate directly from the data. For agiven axion mass m a , the signal only has support overa narrow frequency range, and therefore we truncate thelikelihood to k values between m a (1 − ( v (cid:12) + v ) / / π and m a (1 + 2( v (cid:12) + v ) ) / π . Over this narrow range, wefind the background is adequately described by a first or-der polynomial, defined by the two-component vector a (c.f. Run 1 where the background in each signal windowwas described by a flat white-noise spectrum). In sum-mary, our likelihood is a function of five parameters: m a and A , which define the location and normalization of thesignal, and nuisance parameters a and σ , which describethe mean size, slope, and fluctuations of the background.Our goal is to use the likelihood in (S4) to search fordeviations from the background only distribution indica-tive of the presence of an axion. To do so we define thefollowing test statistic (TS), which is a log-likelihood ra-tio of the signal and null models, t ( m a , A ) = 2 ln (cid:20) L ( d | m a , A ; ˆ a , ˆ σ ) L ( d | m a , A = 0; ˆ a , ˆ σ ) (cid:21) . (S5) Hatted background quantities are fixed to the value atwhich the likelihood attains its maximum value, giventhe specified signal values (i.e. for A (cid:54) = 0, ˆ a and ˆ σ willin general take different values in the numerator and de-nominator). In other words, in defining this TS, we pro-file over the background nuisance parameters. The abovetest statistic is defined for any m a and A . For a given m a , we then define the discovery TS asTS( m a ) = max A t ( m a , A ) . (S6)The maximization of A is initially performed over a rangeincluding positive and negative values, which is criticalfor the valid interpretation of TS as a χ -distributedquantity under Wilks’ theorem; intuitively, backgroundfluctuations below the mean are just as likely as thoseabove. However, as the presence of an actual axion sig-nal will only result in positive spectral excesses, we takeTS( m a ) = 0 when the test statistic is maximized with A <
0. Accordingly, the discovery test statistic is ex-pected to have the following asymptotic distribution p (TS) = 12 (cid:2) δ (TS) + χ k =1 (TS) (cid:3) , (S7)which is expressed in terms of χ k =1 , the probability den-sity function for the χ -distribution with one degree offreedom, and a Dirac δ function.Using the test statistic in (S5), we search for evidenceof ADM with masses m a such that the signal wouldappear within the frequency range f min = 100 kHz to f max = 2 MHz. The local significance of any excess canbe quantified by inverting the distribution in (S7). In or-der to cover our entire frequency between f min and f max ,this search is performed in 11 . s k , nearby windows are cor-related. We account for this self-consistently using theformalism developed in [32], from which we compute the N σ detection threshold, accounting for the LEE byTS thresh ( N ) = (cid:20) Φ − (cid:18) − v Φ( N )3 ln( f max /f min ) (cid:19)(cid:21) , (S8)expressed in terms of Φ, the cumulative density functionof the zero mean and unit standard deviation normaldistribution, with Φ − its inverse. Using this formalism,we find that the 5 σ detection threshold accounting forthe LEE is TS thresh ≈ B. Data Cleaning Procedure
Many of the excesses present in the uncleaned dataare characterized by a narrow spectral feature, oftenpresent in a single frequency bin. The features often driftthroughout our collection time, and appear at regularfrequency intervals. Although such features are incon-sistent with the axion signal expectation, which shouldbe distributed over several frequency bins, such narrowfeatures are far more consistent with our signal modelthan our linear background model, and therefore resultin high-significance TS values.A notable example is the background resulting fromAM radio broadcasts: these manifest as large excesses atuniform 10 kHz intervals from 560 kHz to 1.60 MHz. Weidentify these AM radio signals in our data and removethem with a mask of width 15 Hz centered on the radiosignal peak, beyond which the radio signal falls belowour noise floor. In other cases the origin of the featuresis unclear, although the fact that many appear at 50 Hzintervals suggests a universal environmental origin. Re-gardless, we remain agnostic to their origin and insteadremove them using a data-driven procedure we now out-line.For each frequency bin, we identify a single-bin excessas follows. We determine the mean and standard devia-tion of the data on either side of the bin of interest; inparticular, we use the 10 bins on both sides, ignoring theimmediately adjacent frequencies. We then use these re-sults to calculate the significance of the data observed inthe bin of interest. We repeat this procedure for each fre-quency bin in each independently collected dataset, i.e.the data before it is stacked. If the bin of interest at-tains a significance δχ >
100 in any one dataset, thenit is flagged for masking. If the bin is not flagged by this procedure, we then stack the dataset and repeat thisprocedure once more. If after stacking, the bin now has δχ >
35, then it is again flagged. For all flagged bins,we mask the 21 frequencies centered on the bin of inter-est. The motivation for considering the individual andstacked datasets is to identify both excesses that driftwith time and also those that are only significant in thestacked data where we perform our fiducial analysis.After the single-bin spikes have been identified and re-moved, we perform an initial analysis of the data. Weanalyze the ¯ F ( ¯ F ) data providing a frequency res-olution of 0 . .
01) Hz for axions which would producea signal in the 500 kHz - 2 MHz (50 - 500 kHz) fre-quency range. We stack the Run 2 ¯ F ( ¯ F ) data,which consists of 1460 (700) spectra, into 20 subinter-vals, each of which are initially analyzed independently.For each axion mass, each subinterval is analyzed inde-pendently, with the 50% of subintervals which realize thesmallest values of the TS for discovery and any addi-tional subintervals which have TS < F ( ¯ F ) data, consisting of1364 (682) spectra, are divided into 22 subintervals. ThisTS filtering procedure was implemented in order to mit-igate the impact of transient excesses that imitate anaxion signal in some of the subintervals and might pro-duce a spurious excess if included in the stacked data.With this exclusion criteria, under the null, each spec-trum is expected to be excluded with probability 0 . F and ¯ F datawhich are collected with the magnet off. Since no ax-ion detection can be made with the magnet off, any masspoints which are excesses at TS >
16 in both the magnet-on and magnet-off data are vetoed.
C. Nuisance Parameter Correction
After applying both our individual bin flagging andvetoing procedures, the remaining dataset is designatedclean. Nevertheless, the distribution of TS values re-mains inconsistent with that expected for the asymptoticone-sided χ distribution given in (S7), indicative of fur-ther background mismodeling. To resolve this, we imple-ment an additional nuisance parameter correction to ourlikelihood.In detail, we modify the likelihood and TS with addi- TS10 − − − Su r v i v a l F r a c t i o n RawData CleaningOff VetoesFinal (Nuisance Tuned)Expected5 σ Detection Threshold
Figure S3. As in Fig. 3, but evaluated on the 10% of unblinded Run 2 data against which we calibrated our analysis procedure. . . . . . . Data Acceptance Fraction − − − − F r a c t i o n o f M a ss e s m a [neV] − − F r a c t i o n M a ss e s R e m o v e d Figure S4. (
Left ) The histogrammed data acceptance fraction under the data filtering over all masses analyzed in Run 3 data.(
Right ) The fraction of masses removed by magnet off vetoes as a function of frequency in Run 3 data. The acceptance fractionis determined within 100 log-spaced bins between the minimum and maximum axion masses within our analysis range. Notethat while we display the Run 2 results, those were used only to develop our analysis protocols and not in the physics analysis. tional nuisance parameters A m and σ A m as follows, L ( d | m a , A ; a , σ, A m , σ A m ) = N ( A m | , σ A m ) (cid:89) k √ πσ exp (cid:20) − ( d k − ( A + A m ) s k − µ k ( a )) σ (cid:21) (S9)TS( m a | σ A n ) = 2 ln (cid:34) max A L ( d | m a , A ; ˆ a , ˆ σ, ˆ A m , σ A m ) L ( d | m a , A = 0; ˆ a , ˆ σ, ˆ A m , σ A m ) (cid:35) . (S10)The index m indicates that the nuisance parameters de-pend on the signal window under consideration.By construction, the additional nuisance parameter is– up to a penalty factor – fully degenerate with the sig-nal. This allows the background model the flexibilityto fit signal-like excesses, but at the cost of a Gaus-sian penalty factor given by N ( A m | , σ A m ), which is azero mean normal distribution of width σ A m evaluated at A m . The magnitude of this penalty is controlled bythe hyperparameter σ A m , which can be chosen to ensurethe above TS has the expected asymptotic distribution.To be specific, we determine σ A m for each mass (indexedby k ) by tuning the observed distribution TS( m a | σ A m )against the expected distribution in the vicinity of themass point of interest. We consider the ensemble of thediscovery test statistics belong to the nearest 94,723 masspoints, not including: the mass point of interest; the fivenearest mass points above and below the mass point ofinterest; or any mass points that are vetoed by compar-ison with the magnet off data. We then tune the valueof σ A m to its minimum value such that there are onlythree discovery test statistics in excess of 16 within theensemble, which would be expected if the discovery teststatistics were half-chi-square distributed. The nuisancehyperparameter σ A m translated into an effective nuisanceparameter g nuis aγγ is presented in Fig. S5, and can be un-derstood as an effective floor for our limit-setting powerthat competes with the statistical noise floor set by thebackground strength.We note that many of the procedures required to fixthe hyperparameter can be performed analytically. Asthe log-likelihood is approximately quadratic around itsmaximum, ˆ A , near the maximum we have t ( m a , A ) = TS( m a ) − (cid:32) A − ˆ A ˆ A (cid:33) , (S11)where at this stage we do not yet zero out test statis-tics associated with negative best-fit signal amplitudes.When including the correcting nuisance parameter, thedistribution becomes t ( m a , A, A m | σ A m )=TS( m a ) − (cid:32) A + A m − ˆ A ˆ A (cid:33) − (cid:18) A m σ A m (cid:19) , (S12)with the final term arising from the Gaussian penalty. Wecan now define a new test statistic for discovery includingthe background signal nuisance parameter asTS( m a | σ A m ) =max A,A m t ( m a , A, A n | σ A m ) − max A m t ( m a , A = 0 , A n | σ A m ) . (S13)Using this, for a given σ A m , the new test statistic fordiscovery with the nuisance background signal can bedirectly constructed from the test statistic without thenuisance background signal. In particular, since (S13)involves only maximizations of a quadratic function, theresult is given byTS( m a | σ A n ) = TS( m a ) ˆ A ˆ A + TS( m a ) σ A n , (S14)which has the effect of decreasing the computed TS. Asbefore, we then zero out TS( m a | σ A n ) when the best fitsignal strength parameter ˆ A is negative. D. Survival Functions, Unvetoed Excesses, andLimits
The analysis procedure was tuned on 10% of the Run 2data. Once validated, to the full Run 3 dataset, which had remained blinded. The survival function evaluated atvarious stages of our analysis procedure, realized on the10% of Run 2 data used for tuning, is shown in Fig. S3.Approximately 10% of masses in Run 2 and 5% of massesin Run 3 are removed by the peak exclusion and vetoingprocedure, with the fraction of masses removed as a func-tion of frequency shown in Fig. S4.Even after the nuisance parameter tuning, there re-main some discrepancies between the observed and ex-pected survival functions at moderate values (TS > σ LEEthreshold in Run 2 data. All of these excesses occur innearby frequencies, associated with a transient, and rel-atively broad, spectral feature which is shown in Fig. S6.Further, the mass points which are high significance ex-cesses in the Run 2 data are not significant in the Run 3data. Accordingly, we do not consider these excesses torepresent credible detections.The independent Run 3 limits are shown with andwithout the tuned nuisance parameter in Fig. S7.
E. Injected Signal Tests
To further validate the robustness of our analysisframework, we can inject a synthetic signal into the dataand confirm that: (1) we are able to recover the signalstrength, when expected; and (2) our limits will not ex-clude an injected signal. To perform this test, we selectfive representative mass points and inspect the real datain the vicinity of the expected location of an injected sig-nal. We generate independently drawn axion signals at arange of axion couplings strengths which we add on topof each of the spectra collected in Run 3. We then applyour analysis framework, adopting the tuned value of thenuisance parameter that was previously determined fromthe real data in the vicinity of the injected signal loca-tion, and evaluate the best-fit axion coupling, the 95 th percentile upper limit on that coupling, and the detec-tion significance as a function of the true axion couplingof the injected signal. As a further test of the perfor-mance of our analysis framework, for each of the fivemass points, we fit the Run 3 data under the null model.We then generate Monte Carlo data under null modelfits and repeat our procedure of injecting and analyzing,allowing us to compare the analysis of signal injectionson real data with expected performance of the analysisframework under the null model. With the exceptionof the tuned nuisance parameter, which we continue tokeep fixed at its value determined from the real data, thisrepresents an entirely self-contained test of the analysisprocedure.The results of these tests are shown in Fig. S8. Criti-cally, our analysis procedure is able to place a robust 95 th percentile upper limit which does not exclude the truecoupling strength at which the signal is injected moreoften than would be expected and accurately recovers m a [neV]0 . . . . − × g a γγ [ G e V − ] Hessian Statistical ErrorSystemic Nuisance Param.10 − Frequency [MHz]
Figure S5. The hyperparameter, σ A , converted to the units of g aγγ , for the systematic nuisance parameter, g nuis aγγ , as a functionof axion mass (labeled Systematic Nuisance Param.). We compare the systematic nuisance hyperparameter to the statisticaluncertainties (labeled Hessian Statistical Error), which are computed from the Hessian for the log-likelihood without systematicuncertainties about the best-fit axion coupling, ˆ g aγγ . the correct axion parameters at a detection significancewithin the simulated expectations. We also briefly com-ment on the somewhat jagged nature of the detectionsignificance in the real data as a function of the injectedsignal strength. These features are a product of the fil-tering included in our analysis procedure which removesat most 50% of the spectra in ∼
5% subintervals if those subintervals have a detection significance in excess of 3 σ .This has the effect of somewhat weakening the detectionsignificance in discretized steps and also slightly biasesthe 95 th percentile limit to slightly lower values. Thisbias is removed using a TS-dependent correction of atmost 8% that is incorporated in our limits. -
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12 02 - Time Stamp . . . F r e qu e n c y [ k H z ] N o r m a li ze d P o w e r Magnet On Magnet Off
Figure S6. The time evolution of the broad excess that is associated with the putative signal candidate in the Run 2 datathat survived all analysis cuts. The excess persists after the magnet is turned off and evolves in frequency, indicative of abackground source. The magnet off veto did not anticipate this level of time evolution and so did not remove these excesses.Since this feature was found after unblinding, we report it here but do not consider it to be a credible axion detection. m a [neV]10 − g % a γγ [ G e V − ] S H A F T CAST
S S C Limit w/ NuisanceLimit w/o Nuisance − Frequency [MHz]
Figure S7. A comparison of our fiducial limits that include a nuisance hyperparameter correction (black) and those without anycorrection (blue). Limits set with the nuisance hyperparameter are slightly weaker, but the features and limit-setting powerare broadly similar. The figure is smoothed for clarity. − − R ec o n s tr u c t e d g a γγ [ G e V − ] m a =0.41 neV
95% Upper LimitBest FitInjected Value1 / σ Containment m a =1.24 neV m a =2.48 neV m a =7.44 neV10 − T S ( m a ) Injected5 σ Threshold1 / σ Containment − − − Injected g aγγ [GeV − ] Figure S8. (
Top row ) The best fit and 95% upper limit on the recovered signal strength as a function of the injected signalstrength at five mass points evaluated on the real Run 3 data. The results are compared to the 1 σ and 2 σ expectations forthe 95 th percentile upper limit under the hypothesis of no axion signal as determined by 2560 Monte Carlo (MC) realizationsof the null model fits to the real data at each injected signal strength. ( Bottom row ) In black, the recovered detection teststatistic for the signal injected in the real data as a function of injected signal strength. The dashed red line indicates thethreshold for a 5 σ detection significance account for the look-elsewhere effect while the green and yellow bands indicate the1 σ and 2 σσ