Searching for Dark Photon Dark Matter in LIGO O1 Data
SSearching for Dark Photon Dark Matter in LIGO O1 Data
Huai-Ke Guo, Keith Riles, Feng-Wei Yang,
3, 4, ∗ and Yue Zhao Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109 Department of Physics and Laboratory for Space Research,The University of Hong Kong, PokFuLam, Hong Kong SAR, China Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112, USA
Dark matter exists in our Universe, but its nature remains mysterious. The remarkable sensitivityof the Laser Interferometer Gravitational-Wave Observatory (LIGO) may be able to solve thismystery. A good dark matter candidate is the ultralight dark photon. Because of its interaction withordinary matter, it induces displacements on LIGO mirrors that can lead to an observable signal.In a study that bridges gravitational wave science and particle physics, we perform a direct darkmatter search using data from LIGO’s first (O1) data run, as opposed to an indirect search for darkmatter via its production of gravitational waves. We demonstrate an achieved sensitivity on squaredcoupling as ∼ × − , in a U (1) B dark photon dark matter mass band around m A ∼ × − eV. Substantially improved search sensitivity is expected during the coming years of continued datataking by LIGO and other gravitational wave detectors in a growing global network. Introduction
Although there is little doubt that dark matter (DM)exists in our Universe, its nature remains mysterious, in-cluding its component mass(es). It may be an ultralightelementary particle, such as fuzzy DM with mass ∼ − eV [1–4], or it may arise from stellar-mass objects, suchas primordial black holes [5].One promising DM candidate in the ultralight massregime is the dark photon (DP), which is the gauge bo-son of a U (1) gauge group. The DP can acquire its massthrough the Higgs or Stueckelberg mechanism. As ul-tralight DM, the DP must be produced non-thermally, e.g. , production from the misalignment mechanism [6–8],parametric resonance production or tachyonic instabilityof a scalar field [9–12], or from the decay of a cosmicstring network[13].It was recently proposed in [14, 15] that a gravita-tional wave (GW) detector may be sensitive to dark pho-ton dark matter (DPDM). The Advanced Laser Inter-ferometer Gravitational-Wave Observatory (LIGO) con-sists of two 4-km dual-recycled Michelson Fabry-Perot in-terferometers in Livingston Louisiana (L1) and Hanford,Washington (H1). From the first two observing runs (co-incident with the Virgo detector for several weeks of theO2 run), detections of ten binary black hole mergers andone binary neutron star merger have been reported [16].These measurements require a differential strain mea-surement sensitivity better than 10 − for broadbandtransients with central frequencies of O (100 Hz), basedon detecting minute changes in distance between the mir-ror pairs forming the Fabry-Perot interferometer arms.Relevant to this search, the mirror separations canalso change in response to a gradient in a DPDM field ∗ Electronic address: [email protected] due to non-zero photon velocity. More explicitly, weconsider a DP with mass m A between 10 − ∼ − eV. The DPDM is an oscillating background field, forwhich the rest-frame oscillation frequency satisfies: f ≈ (cid:2) m A − ev (cid:3) (241 Hz). We assume the DP is the gauge bo-son of the gauged U (1) B group so that any object, in-cluding a LIGO mirror, that carries baryon number willfeel its oscillatory force, similar to that experienced by amacroscopic, electrically charged object in an oscillatingelectric field.Using LIGO to look for DPDM bridges GW scienceand particle physics. In this paper, we present a U (1) B DPDM search using data from Advanced LIGO’s first ob-serving run, O1. We confirm that the data from LIGO’sfirst observation run yields results already more sensitivethan limits from prior experiments in a narrow DPDMmass range. The sensitivity will be improved significantlywith more LIGO data, as well as with the growth of theglobal network of GW detectors. Meanwhile, the samesearch strategy can be directly applied to search for manyother ultralight DM scenarios, with excellent sensitivitiesachievable.
Results
Estimating DPDM induced effects.
Through viri-alization, DPDM particles in our galaxy have a typicalvelocity around v ≡ − of the speed of light, andthus they are highly non-relativistic. The total energyof a DM particle is then the sum of its mass energy andkinetic energy, i.e. m A (1 + v / i.e. c = (cid:125) = 1. Therefore theoscillation frequency of the DP field is approximately aconstant, ω (cid:39) m A , with O (10 − ) deviations.Therefore within a small period of time and spatialseparation, the DP field can be treated approximately asa planewave, i.e. , A µ (cid:39) A µ, cos[ m A t − k · x + θ ] . (1) a r X i v : . [ h e p - ph ] D ec FIG. 1: An example of dark photon dark matter signal powerspectrum and corresponding detector sensitivity. The darkphoton dark matter (DPDM) signal power spectrum is shownin terms of characteristic strains h c (red), with U (1) B couplingparameter (cid:15) = 10 − , DPDM oscillation frequency f = 500Hz and typical velocity of DPDM v = 10 − of the speedof light. The Advanced LIGO design sensitivity in a smallfrequency window is approximately flat, which is shown asthe black dashed line. Here A µ, is the amplitude of the DP field and θ is a ran-dom phase. The DP field strength can be simply writtenas F µν = ∂ µ A ν − ∂ ν A µ . We choose the Lorenz gauge, ∂ µ A µ = 0, in what follows. In the non-relativistic limit,the dark electric field is much stronger than the darkmagnetic field, and A t is negligible relative to A . Themagnitude of the DP field can be determined by the DMenergy density, i.e. , | A | (cid:39) √ ρ DM /m A .In Eq. 1, we neglect the kinetic energy contributionto the oscillation frequency. We also set the polariza-tion and propagation vectors, i.e. , A and k , to be con-stant vectors. This approximation is valid only whenthe observation is taken within the coherence region, i.e. t obs < t coh (cid:39) πm A v and l obs < l coh (cid:39) πm A v vir . For ex-ample, if the DP field oscillates at 100 Hz, the coherencetime is only 10 s, much shorter than the total obser-vation time. In order to model the DPDM field for atime much longer than the coherence time, we simulatethe DPDM field by linearly adding up many planewavespropagating in randomly sampled directions. More de-tails are given in the “Methods” section below.From the DPDM background field A ( t, x i ), one canderive the acceleration induced by the DPDM on eachtest object, labeled by index i . This can be written: a i ( t, x i ) (cid:39) (cid:15)e q D ,i M i ∂ t A ( t, x i ) . (2)Here we use the approximation E (cid:39) ∂ t A ( t, x i ) for thedark electric field. q D ,i /M i is the charge-mass-ratio of the test object in LIGO. Treating a DP as the gaugeboson of U (1) B , and given that the LIGO mirrors (testmasses) are primarily silica, q D ,i /M i = 1 / GeV. We notethat results from [17] impose very strong constraints onthe gauged U (1) B scenario, due to gauge anomaly. How-ever the results derived in these papers rely on an as-sumption of how to embed the model into a completetheory at high energy in order to cancel U (1) B anoma-lies. Extending the electroweak symmetry breaking sec-tor or other anomaly cancellation mechanism can avoidsuch severe constraints. If one takes the model inde-pendent constraint on an anomalous gauge symmetry,new particles need to be added at energy scale O ( πm A (cid:15)e ),which gives O (TeV) for the parameter space we are in-terested in. Since the required new particles carry onlyelectroweak charges, they are safe from various collidersearches. We label the DP-baryon coupling as (cid:15)e where e is the U (1) EM coupling constant. We emphasize thatwe choose DP to be a U (1) B gauge boson as a bench-mark model. The same analysis strategy presented inthis study can be directly applied to many other scenar-ios, such as a U (1) B − L gauge boson or scalar field whichcouples through Yukawa interactions. More details onvarious models, as well as subtleties when observationtime is longer than coherence time, will be described inthe future work. Signal-to-Noise Ratio Estimation.
We approximatethe DPDM field as a planewave within a coherence re-gion. For a DP field oscillating at frequency O (100) Hz,the coherence length is O (10 m), much larger than theseparation between the two LIGO GW detectors at Han-ford and Livingston. Thus these two GW detectors expe-rience a nearly identical DPDM field, inducing stronglycorrelated responses. Exploiting the correlation dramat-ically reduces the background in the analysis.The DPDM signal is exceedingly narrowband, mak-ing Fourier analysis natural. We first compute discreteFourier transforms (DFT) from the time-domain data.The total observation time is broken into smaller, con-tiguous segments, each of duration T DFT , with a totalobserving time T obs = N DFT T DFT . Denote the value ofthe complex DFT coefficient for two interferometers 1and 2, DFT i and frequency bin j to be z ,ij . Theone-sided power spectral densities (PSD) for two inter-ferometers are related to the raw powers as PSD ,ij =2 P ,ij /T DFT . P ,ij are taken to be the expectationvalues for | z ,ij | , as estimated from neighboring, non-signal frequency bins, assuming locally flat noise (usinga 50-bin running median estimate).To an excellent approximation, the noise in the twoLIGO interferometers is statistically independent, withthe exception of particular very narrow bands with elec-tronic line disturbances [18], which are excluded from theanalysis. Detailed descriptions of broadband LIGO noisecontributions may be found in [19], including discussionof potential environmental contaminations that could becorrelated between the two LIGO detectors, but none ofwhich would mimic a DPDM signal. The normalized sig-nal strength using cross correlation of all simultaneousDFTs in the observation time can be written as S j = 1 N DFT N DFT (cid:88) i =1 z ,ij z ∗ ,ij P ,ij P ,ij . (3)In the absence of a signal, the expectation value is zeroand the variance of the real and imaginary parts is σ j = 1 N DFT (cid:28) P ,j P ,j (cid:29) N DFT , (4)where (cid:104) (cid:105) N DFT denotes an average over the N DFT
DFTs,which may have slowly varying non-stationarity. Thesignal-to-noise ratio (SNR) can be defined bySNR j ≡ S j σ j . (5)Taking into the account the small separation between theinterferometers relative to the DP coherence length andtheir relative orientation (approximate 90-deg rotationof one interferometer’s arms projected onto the other in-terferometer’s plane), we expect the SNR j for a strongDPDM field to be primarily real and negative. Efficiency factor.
In order to use the observedreal(SNR) values to set limits on DPDM coupling asa function of frequency, one must correct for the signalpower lost from binning. The suggested nominal binningproposed in [15] is ∆ f /f = 10 − , based on a Maxwellvelocity distribution. The binsize in frequency space isset by T DFT , i.e. , ∆ f = 1 /T DFT , which is optimal at only f opt (cid:39) /T DFT . For a frequency higher than f opt , therelative frequency binning is finer, implying loss of signalpower in single-bin measurements. At frequencies lowerthan f opt , the relative frequency binning is coarser, im-plying full capture of the signal power, but at the cost ofunnecessarily increased noise. We note that it is possi-ble the DM velocity distribution deviates from Maxwelldistribution by an O (1) factor, e.g. [20, 21]. However,the impact is small, as the single-bin search used heredepends on the integrated power within a frequency binand not so much on its shape.In Fig. 1, we show the DPDM signal power spectrum asa function of frequency, where f = m A / π . We chooseto normalize the x-axis by the intrinsic signal width, de-termined by the typical kinetic energy of DM particles.In this calculation, we include the Earth rotation effect.Without it, the signal PSD is proportional to vf ( v ) where f ( v ) is the Maxwell distribution. The Earth’s rotationbroadens our signal by ∆ f ≈ f E . Different choices of f result in slightly different deformations after includingthe rotation, but the changes are negligible in the fre-quency regime of interest. An analytical understandingof the PSD will be presented in the future work. - - - - - - FIG. 2: Signal power single-bin detection efficiency as afunction of relative frequency resolution for a fixed coherencetime of 1800 s. The upper (red) curve is for an optimal binboundary choice ( a priori unknown) for a given signal. Thelower (blue) curve shows the worst-case efficiency for the leastoptimal boundary choice. The middle (green) curve shows anaverage over randomly chosen boundary choices.
The power spectrum from numerical simulation is usedto determine empirically the fractions of power fallinginto a single fixed ∆ f /f bin, where bin boundaries aresystematically varied over the allowed range. Fig. 2shows the resulting efficiencies (power fractions) for T DFT set to be 1800s. The red dotted curve shows the best case,for which the bin boundary is optimal. The blue dashedcurve shows the worst case, which necessarily approaches50% for coarse binning (low frequency), while the greensolid curve shows the average maximum efficiency overall bin boundary choices. A fit to the green solid curveis used for deriving upper limits on DPDM coupling.
Data selection and analysis.
The strain data usedin this analysis was downloaded from the GravitationalWave Open Science Center (GWOSC) web site [22] andtransformed to create 1786 1800-second coincident DFTsfrom the L1 and H1 interferometers. The GWOSC datasets exclude short periods during which overall data qual-ity is poor. The choice of coherence time in this firstDPDM search is somewhat arbitrary, but allowed conve-nient comparison of spectral line artifacts observed withthose reported from 1800-s DFTs in LIGO continuousgravitational wave searches, for which 1800-s is a com-mon DFT duration chosen. A shorter coherence timewould be more optimal at frequencies above ∼
500 Hz forthis single-bin detection analysis. In principle, a longertime would be more optimal for lower frequencies, butin practice, sporadic interruptions of interferometer op-erations during data taking lead to significant livetimeloss for DFTs requiring very long contiguous periods ofcoincident Hanford-Livingston operations. * * * ** * ** * * *************************************************************************************************************************** * ** ** ******************************************************************************************************** - - Distribution of the Real Part of SNR * * * * * * * * * * *** * * * * * *** ************** *** - - - - *** * ** *** **** ******************************************************************************************************************************* ** *** ******************************************************************************************************** - - Distribution of the Imaginary Part of SNR * * * * * * * * * ** * * * * ***************** ** * **** - - - - FIG. 3: Distributions of the real and imaginary parts of the signal-to-noise ratio. The signal-to-noise ratio (SNR) for the signalbins (“zero lag”) are labeled in magenta and the lagged (control) bins in black, along with the ideal Gaussian expectation ingreen.
The search for detections and the setting of upper lim-its in the absence of detection is based on “loud” valuesof the detection statistic (Eq. 5). Specifically, we lookfor large negative real values of the SNR. Since we searchover ∼ < − .
8, cor-responding to a ∼
1% false alarm probability, assumingGaussian noise. In practice, the noise in some frequencybands is not truly Gaussian, leading to excess counts atlarge SNR. To assess the severity of this effect, we alsodefine and examine control bands (“frequency lags”) inwhich a DFT frequency bin in one interferometer is com-pared to a set of offset bins from the other interferome-ter such that a true DPDM signal would not contributeto a non-zero cross correlation, but for which single-interferometer artifacts or broadband correlated artifactslead to non-zero correlation. This frequency lag methodis analogous to the time lag method used in transientgravitational wave analysis. Specifically, we choose 10lags of ( − −
40, ..., −
10, +10, ..., +50) frequency binoffsets to assess the non-Gaussian background from theseinstrumental artifacts. To avoid contamination of bothsignal and control bands from known artifacts, we ex-clude from the analysis any band within ∼ .
056 Hz ofa narrow disturbance listed in [18], where the extra vetomargin is to reduce susceptibility to spectral leakage fromstrong lines. We also exclude the band 331.3-331.9 Hz,for which extremely loud narrow calibration excitationsin the two interferometers lead to significant overlappingspectral leakage and hence non-random correlation.Fig. 3 shows the distributions of the real and imagi-nary parts of the SNR (Eq. 5) for both the signal bins(“zero lag”) in magenta and the lagged bins in black.The distributions follow quite closely the ideal Gaussian curve shown, except for a slight excess visible in the tailsbeyond | SNR | > ∼
10 times as manylagged bins as signal bins in the graphs). The only signalbins with | SNR | > . New constraints from LIGO O1 data.
Our main results are presented in Fig. 4. We show thederived 95% confidence level (CL) upper limits on theparameter (cid:15) for DP-baryon coupling, as a function of -47 -46 -45 -44 -43 -42 -41 -40 -39 O1 95% CL limits (1800s SFTs)Nominal SNR O1 95% CL limitsAve optimal O1 limits (893 hours)Ave optimal O4-O5 limits (2 years)Eot-Wash limits -13 -12 FIG. 4: Derived 95% confidence level upper limits on the coupling parameter (cid:15) for dark photon-baryon coupling. The broadred band shows the actual upper limits with 1 / f/f = 10 − for the same 893-hour observation time. Themagenta curve shows the “optimal” upper limit expected for a 2-year, 100%-livetime run at Advanced LIGO design sensitivity(“O4-O5”). The dashed curve shows upper limits derived from the E¨ot-Wash group [23, 24]. This is a fifth-force experiment,whose constraint does not rely on dark photon (DP) being dark matter (DM). The large spikes of red and blue curves, overlaidon top of each other, are induced by known sources of noise, such as vibrations of mirror suspension fibers. DPDM oscillating frequency. The broad red band showsthe range of upper limits obtained with 1 / f /f = 10 − for the same 893-hour obser-vation time, for the same efficiency correction and for anaveraged detector sensitivity equal to that in the analy-sis. The yellow and dark blue curves agree well with eachother at around 500 Hz, where 1 / / m A ∼ × − eV.Future searches in more sensitive data will probedeeper into an unexplored (cid:15) – m A parameter space. As-suming no discovery and a negligible true GW stochas-tic background, the magenta curve shows the “optimal”upper limit expected for a 2-year, 100%-livetime run atAdvanced LIGO design sensitivity (“O4-O5”). This limitlooks smoother, as it uses a design sensitivity curve thatshows only fundamental noise sources, while the bluecurve includes additional, non-fundamental noise arti-facts that have not yet been mitigated in LIGO detectorcommissioning, such as power mains contamination at60 Hz and harmonics and environmental vibrations. Thesimulations discussed below uncovered an error of a fac-tor of 4 in the (cid:15) – m A sensitivity plot in [15]. This errorhas been corrected in the current study. As GW detec-tors become more sensitive in the future, one expects astochastic GW background from compact binary coales-cence mergers to emerge eventually, with an integratedbroadband stochastic signal perhaps detectable as earlyas the O4-O5 run [26]. Nonetheless, the stochastic GWstrain power from mergers in a single DPDM search binwill remain negligible for years to come.The inclusion of a global network of detectors, such asVirgo, KAGRA and LIGO-India, improves the DPDMsearch sensitivity, in principle. The degree of improve-ment depends, however, upon the relative alignmentsamong these detectors as well as their sensitivities. TheVirgo detector is currently less sensitive than the twoLIGO detectors. In addition, its orientation is not wellaligned with those of the LIGO detectors. Future third-generation detectors, such as Einstein Telescope and Cos-mic Explorer, will have much lower noise, permitting stillmore sensitive DPDM searches. Discussion
In this paper, we present a direct DM search usinggravitational wave detector strain data. We choose the U (1) B DP as our benchmark scenario; our early resultsalready improve upon prior searches in a narrow DP massrange, and future searches will probe deeper in DPDMcoupling strength and wider in mass range. This firstanalysis uses a non-templated, single-Fourier-bin cross-correlation detection statistic. Refinements to be exam-ined for analysis of future data sets include multiple DFTcoherence times, tuned according to search band, tem-plated filtering over multiple Fourier bins and exploita-tion of extremely narrow features expected in the DPDMspectrum, resolvable by GW detectors for loud enoughSNR.With more data to be collected by LIGO and othergravitational wave detectors in the coming years, as wellas with improved search strategies, we expect DPDMsearches to probe steadily deeper in DPDM parameterspace. The same strategy can be implemented directlyin searches for many other ultralight DM scenarios. Thisnovel use of data from a gravitational wave detectordemonstrates the versatility of these remarkable instru-ments for directly probing exotic physics.
Methods
Simulating the DPDM background.
The DPDMbackground is a superposition of many DP wave func-tions, similar to the axion DM background as studied in[27]. In the galaxy frame, each DP has a random po-larization direction isotropically distributed. The mag-nitude of A is taken to be a fixed number for each DPparticle with normalization discussed below. As for thepolarization vector, the velocity direction also follows anisotropic distribution. The magnitude of the velocity isobtained from the Maxwell distribution f ( v ) ∼ v e − v /v , (6)where v is taken to be 0 . × − c [28]. In the non-relativistic limit, the polarization vector and the velocityvector are independent of each other.For the i -th DP particle, the wave function can bewritten as A i ( t, x ) ≡ A i, sin( ω i t − k i · x + φ i ) , (7) where ω i = (cid:112) k i + m ≡ πf i and k i = m A v i . TheDPDM background can be generated by superposingmany, N , of these wave functions A total ( t, x ) = N (cid:88) i =1 A i, sin( ω i t − k i · x + φ i ) . (8)Here the phase of the wave function for each particle, φ i ,is randomly chosen from a uniform distribution from 0to 2 π .To simulate DPDM background, we consider N = 10 DPDM particles. We note that having N = 10 sufficesto reveal essential features of the DPDM background,such as coherence length and coherence time. Further,this simulation provides a signal PSD which agrees wellwith analytic results ( N → ∞ limit). Thus we believethat the result from this simulation is reliable.Finally, the normalization of A i, is determined by thelocal DM energy density. In the non-relativistic limit,the energy density of DM can be calculated as1 V T (cid:90) V d x (cid:90) T dt m A = ρ DM (cid:39) . / cm . (9)In order to average out the fluctuations in numericalsimulation, the temporal and spatial integrals are takento be much longer than the coherence time and length, i.e. , T (cid:29) T coh and V (cid:29) l . Since the DPDM is ob-tained from a superposition of N DP particles in anuncorrelated manner, the total amplitude increases as √ N . For a fixed DM energy density ρ DM , one expects | A i, | (cid:39) √ ρ DM / ( m A √ N ), consistent with our numeri-cal results. Interface to LIGO simulations.
We use the LIGOScientific Collaboration Algorithm Library Suite (LAL-Suite) [29] for mimicking of GW detector response ofDPDM and for superposing random Gaussian noise. Thissuite of programs has been developed over two decadesfor simulating GW signals, detector response and for car-rying out GW analysis, including source parameter esti-mation.Below, we give a brief overview of the relevant LAL-Suite GW response model and explain what is modifiedto simulate DPDM induced effects. When the GW wave-length is much longer than the detector’s characteristicsize, i.e. , λ (cid:29) L , one can use the equation of the geodesicdeviation in the proper detector frame to calculate theGW induced effect, ¨ ξ i = 12 ¨ h TT ij ξ j , (10)where ξ is the coordinate of a test object in the properdetector frame. At leading order, the relative change ofthe arm length is R ≡ ∆ L x − ∆ L y L = h + F + + h × F × , (11)where F + and F × are antenna pattern functions, whichcan be written as F + = (cid:88) i,j D ij ( e + ) ij = 12 [( e + ) xx − ( e + ) yy ] ,F × = (cid:88) i,j D ij ( e × ) ij = 12 [( e × ) xx − ( e × ) yy ] , (12)with polarization tensors( e + ) ij = ( X ⊗ X − Y ⊗ Y ) ij , ( e × ) ij = ( X ⊗ Y + Y ⊗ X ) ij , (13)and detector tensors D ij = 12 ( n x ⊗ n x − n y ⊗ n y ) ij , (14)where vectors X and Y are the axes of the wave frame,and n x and n y are unit vectors along the x and y armsof LIGO respectively.In order to concretely estimate LIGO’s sensitivity to aDPDM signal, we calculate the DPDM induced relativechange of the arm length as a function of time, i.e. , R ( t ).Then we inject this as the signal into LALSuite. Thebackground is further added as a Gaussian white noise.As a benchmark, the DP oscillation frequency is set tobe 100 / √ (cid:15) to be 5 × − . For the simu-lation, we take T DFT = 1800 s, T obs = 200 hours and √ PSD = 10 − / √ Hz. The signal appears as a negativereal number, i.e. , SNR (cid:39) −
8. The sensitivity to (cid:15) scalesas (cid:15) (cid:48) (cid:15) = SNR (cid:48) SNR T coh T (cid:48) coh (cid:115) N DFT N (cid:48) DFT . (15)With this scaling, our simulations are consistent withupper limits shown in Fig. 4 based on the search of O1data. Data availability
Code availability
The source code for the analysis is available from thecorresponding author upon request.
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Acknowledgements
Author contributions
H.K.G., K.R., F.W.Y. and Y.Z. have contributed equallyto the intellectual content of the paper and the design ofthe analysis.