Abstract
Ultralight bosons such as axionlike particles are viable candidates for dark matter. They can form stable, macroscopic field configurations in the form of topological defects that could concentrate the dark matter density into many distinct, compact spatial regions that are small compared with the Galaxy but much larger than the Earth. Here we report the results of the search for transient signals from the domain walls of axionlike particles by using the global network of optical magnetometers for exotic (GNOME) physics searches. We search the data, consisting of correlated measurements from optical atomic magnetometers located in laboratories all over the world, for patterns of signals propagating through the network consistent with domain walls. The analysis of these data from a continuous monthlong operation of GNOME finds no statistically significant signals, thus placing experimental constraints on such dark matter scenarios.
Main
The nature of dark matter—an invisible substance comprising over 80% of the mass of the Universe^{1,2}—is one of the most profound mysteries of modern physics. Although evidence for the existence of dark matter comes from its gravitational interactions, unravelling its nature likely requires observing nongravitational interactions between dark matter and ordinary matter^{3}. One of the leading hypotheses is that dark matter consists of ultralight bosons such as axions^{4} or axionlike particles (ALPs)^{5,6,7}. Axions and ALPs arise from spontaneous symmetry breaking at an unknown energy scale f_{SB}, which—along with their mass m_{a}—determines many of their physical properties.
ALPs can manifest as stable, macroscopic field configurations in the form of topological defects^{8,9,10} or composite objects bound together by selfinteractions such as boson stars^{11,12}. Such ALP field configurations could concentrate the dark matter density into many distinct, compact spatial regions that are small compared with the Galaxy but much larger than the Earth. In such scenarios, Earthbound detectors would only be able to measure signals associated with dark matter interactions on occasions when the Earth passes through such a dark matter object. It turns out that there is a wide range of parameter space—consistent with observations—for which such dark matter objects can have the required size and abundance such that the characteristic time between encounters could be of the order of one year or less^{9,10,12}. This opens up the possibility of searches with terrestrial detectors. Here we present the results of such a search for ALP domain walls, a class of topological defects that can form between regions of space with different vacua of an ALP field^{8,9}. We note that although some models suggest that axion domain walls cannot survive to the present epoch^{13,14,15}, there do exist a number of ALP models demonstrating the theoretical possibility that ALP domain walls or composite dark matter objects with similar characteristics^{12,16,17} can survive to modern times^{18,19,20} and have the characteristics of cold dark matter^{9,10,21}.
Since ALPs can interact with atomic spins^{3}, the passage of Earth through an ALP domain wall affects atomic spins similar to a transient magneticfield pulse^{9,12}. Considering a linear coupling between the ALP field gradient \({{{\bf{\nabla }}}} a ({{{\bf{r}}}},t)\) and atomic spin S, the interaction Hamiltonian can be written as
where ℏ is the reduced Planck’s constant, c is the speed of light, r is the position of spin, t is time, and f_{SB}/ξ ≡ f_{int} is the coupling constant in units of energy described with respect to the symmetrybreaking scale f_{SB} (ref. ^{22}); here ξ is unitless. In most theories, the coupling constants f_{int} describing the interaction between standard model fermions and the ALP field are proportional to f_{SB}; however, f_{int} can differ between electrons, neutrons and protons by modeldependent factors that can be substantial^{3,5}.
Analogous to equation (1), the Zeeman Hamiltonian describing the interaction of magnetic field B with atomic spin S can be written as
where γ is the gyromagnetic ratio. Since equations (1) and (2) have the same structure, the gradient of the ALP field—even though it couples to the particle spin rather than the magnetic moment—can be treated as a ‘pseudomagnetic field’ as it causes energy shifts of Zeeman sublevels. An important distinction between the ALPspin interaction (equation (1)) and the Zeeman interaction (equation (2)) is that although γ tends to scale inversely with the fermion mass, no such scaling of the ALPspin interaction is expected^{3}.
The amplitude, direction and duration of the pseudomagneticfield pulse associated with the transit of the Earth through an ALP domain wall depends on many unknown parameters such as the energy density stored in the ALP field, coupling constant f_{int}, thickness of the domain wall, and relative velocity v between Earth and the domain wall. The dynamical parameters, such as the velocities of dark matter objects, are expected to randomly vary from encounter to encounter. We assume that they are described by the standard halo model for virialized dark matter^{23}. Furthermore, the abundance of domain walls in the Galaxy is limited by physical constants, namely, m_{a} and f_{SB}, as these determine the energy contained in the wall, and the total energy of all the domain walls is constrained by estimates of the local dark matter density^{24}. The expected temporal form of the pseudomagneticfield pulse can depend on the theoretical model describing the ALP domain wall as well as particular details of the terrestrial encounter (such as the orientation of Earth). The relationships between these parameters and characteristics of the pseudomagneticfield pulses searched for in our analysis are discussed in Supplementary Section II and other studies^{9,12,22}.
The global network of optical magnetometers for exotic (GNOME) physics searches is a worldwide network searching for correlated signals heralding beyondthestandardmodel physics that currently comprises more than a dozen optical atomic magnetometers, with stations (each with a magnetometer and supporting devices) in Europe, North America, Asia, the Middle East and Australia. A schematic of a domainwall encounter with GNOME is shown in Fig. 1. The measurements from the magnetometers are recorded with custom dataacquisition systems^{25}; synchronized to the global positioning system (GPS) time; and uploaded to servers located in Mainz, Germany, and Daejeon, South Korea. Descriptions of the operational principles and characteristics of GNOME magnetometers are presented in Methods, Extended Data Table 1, and ref. ^{26}.
The active field sensor at the heart of every GNOME magnetometer is an optically pumped and probed gas of alkali atoms. Magnetic fields are measured by variations in the Larmor spin precession of the optically polarized atoms. The vapour cells containing the alkali atoms are placed inside multilayer magneticshielding systems that reduce background magnetic noise by orders of magnitude^{27} despite retaining sensitivity to exotic spin couplings between ALP dark matter and atomic nuclei.
If the ALP field only couples to electron spins, interactions between the ALP field and magnetic shield will reduce the ALPinduced signal amplitudes in each magnetometer by roughly the magnetic shielding factors of 10^{6}–10^{7}, as discussed in ref. ^{28}. Therefore, in the present work, we only consider interactions between ALP fields and atomic nuclei. Since all the GNOME magnetometers presently use atoms whose nuclei have a valence proton, the signal amplitudes measured by GNOME due to an ALPspin interaction are proportional to the relative contribution of proton spin to nuclear spin (as discussed in Supplementary Section II and ref. ^{29}). This pattern of signal amplitudes (equation (1)) can be characterized by a pseudomagnetic field B_{j} measured with sensor j:
where
is the normalized pseudomagnetic field describing the effect of the ALP domain wall on proton spins and μ_{B} is the Bohr magneton. The ratio between the Landé gfactor and the effective proton spin (g_{F,j}/σ_{j}) accounts for the specific protonspin coupling in the respective sensor. This ratio depends on the atomic and nuclear structure in addition to details of the magnetometry scheme (Supplementary Section II). Since each GNOME magnetometer measures the projection of the field along a particular sensitive axis, the factor η_{j} is introduced to account for directional sensitivity. This factor, given by the cosine of the angle between \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) and the sensitive axes, takes on values between +1 and –1.
In spite of the unknown properties of a particular terrestrial encounter with an ALP domain wall, GNOME measures a recognizable global pattern of the associated amplitudes of the pseudomagneticfield pulse described by equation (3), as illustrated in Fig. 1b. The associated pseudomagneticfield pulses would point along a common axis, have the same duration and exhibit a characteristic timing pattern. The dataanalysis algorithm used in the present work to search for ALP domain walls is described in Methods and ref. ^{30}. The algorithm searches for a characteristic signal pattern across GNOME, having properties consistent with the passage of Earth through an ALP domain wall. Separate analyses to search for transient oscillatory signals associated with boson stars^{12} and bursts of exotic lowmass fields from cataclysmic astrophysical events^{31} are presently underway.
Here we report the results of a dark matter search with GNOME: a search for transient couplings of atomic spins to macroscopic dark matter objects, thereby demonstrating the ability of GNOME to explore the parameter space previously unconstrained by direct laboratory experiments. Searches for macroscopic dark matter objects based on similar ideas were carried out using atomic clock networks^{10,23,32,33}, and there are a number of experimental proposals utilizing other sensor networks^{34,35,36,37}. All these networks are sensitive to bosonic dark matter with a scalar coupling to standard model particles^{3}. GNOME is sensitive to a different class of dark matter: bosons with pseudoscalar couplings to standard model particles. Pseudoscalar bosonic dark matter generally produces no observable effects in clock networks^{3}, but it does couple to atomic spins via the interaction described by equation (1). Thus, GNOME is sensitive to a distinct—so far, mostly unconstrained—class of interactions compared with other sensor networks.
Search for ALP domainwall signatures
There have been four GNOME science runs between 2017 and 2020, as discussed in Methods. Here we analyse the data from Science Run 2, which had comparatively good overall noise characteristics and consistent network operation (as shown in Extended Data Fig. 1). Nine magnetometers took part in Science Run 2 that spanned from 29 November 2017 to 22 December 2017. The characteristics of the magnetometers are summarized in Extended Data Table 1.
Before the data are searched for evidence of domainwall signatures, they are preprocessed by applying a rolling average, highpass filters, and notch filters to the raw data. The averaging process enhances the signaltonoise ratio for certain pulse durations, avoids complications arising from different magnetometers having different bandwidths, and reduces the amount of data to be analysed. The highpass and notch filters reduce the effects of longterm drifts and noisy frequency bands. We refer to the filtered and rollingaveraged dataset as the ‘search data.’
The search data are examined for the evidence of collective signal patterns corresponding to planes with uniform, nonzero thickness, crossing Earth at constant velocities. The imprinted pattern of amplitudes depends on the domainwallcrossing velocity^{30}. We assume that the domainwallvelocity probability density function follows the standard halo model for virialized dark matter. The signature of a domain wall crossing the magnetometer network depends on the component of the relative velocity between the domain wall and the Earth that is perpendicular to the domainwall plane, v_{⊥}. A lattice of points in the velocity space is constructed such that the search algorithm covers 97.5% of the velocity probability density function. The algorithm scans over the velocity lattice and, for every velocity, the data from each magnetometer are appropriately timeshifted so that the signals in different magnetometers from a hypothetical domainwall crossing with the given velocity occur at the same time. For each velocity and at each measurement time, the amplitudes measured by each magnetometer are fit to the ALP domainwallcrossing model described in ref. ^{30}. As a result, estimations for signal magnitude and domainwall direction, along with associated uncertainties, are obtained for each measurement time and all the lattice velocities. The magnitudetouncertainty ratio of an event is given by the ratio between the signal magnitude and its associated uncertainty.
The search algorithm uses two different tests to evaluate if a given event is likely to have been produced by an ALP domainwall crossing: a domainwall model test and a directionalconsistency test^{30}. The domainwall model test evaluates whether the event amplitudes measured by the GNOME magnetometers match the signal amplitudes predicted by the ALP domainwallcrossing model, and is quantified by the Pvalue, as discussed in Methods and ref. ^{30}. The directionalconsistency test checks the agreement between the direction of the scanned velocity and the estimated domainwall direction, and is quantified by the angle between the two directions normalized by the angle between the adjacent lattice velocities. The thresholds on these tests are chosen to guarantee an overall detection efficiency ϵ ≥ 95% for the search algorithm, considering both incomplete velocity lattice coverage and detection probability (Extended Data Fig. 2).
The search data are analysed for domainwall encounters using the algorithm presented in ref. ^{30}. The cumulative distribution of candidate events as a function of their magnitudetouncertainty ratio is shown as the solid green line in Fig. 2. The candidate event in the search data with the largest magnitudetouncertainty ratio (namely, 12.6) had a significance of less than one sigma. Therefore, we find no evidence of an ALP domainwall crossing during Science Run 2. Rare domainwallcrossing events that produce signals below a magnitudetouncertainty ratio of 12.6 are indistinguishable from the background. Therefore, we base constraints on the ALP parameters on the absence of any detection above the ‘loudest event’ in a manner similar to that described, for example, in ref. ^{38}.
To evaluate the domainwall characteristics excluded by this result, the observable domainwallcrossing parameters above a magnitudetouncertainty ratio of 12.6 during Science Run 2 are determined. GNOME has nonuniform directional sensitivity^{30}; we conservatively estimate the network sensitivity assuming the domain wall comes from the leastsensitive direction. Figure 3 shows the active time T(\({{\Delta }}t,{{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}\)), that is, how long the network was sensitive to domain walls as a function of sensitivity of the pseudomagneticfield magnitude, \({{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}\), and pulse duration, Δt. A signal with pseudomagneticfield magnitude \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) produces a magnitudetouncertainty ratio of \(\zeta ={{{{\mathcal{B}}}}}_{{{\rm{p}}}}/{{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}\). The active time, T(\({{\Delta }}t,{{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}\)), can be used to constrain the ALP domainwall parameter space, as discussed in Supplementary Section II.
If one assumes a probability distribution for the number of domainwall encounters, an upper bound on the rate R_{C} of such encounters can be calculated with confidence level C. We assume a Poisson probability distribution for the domainwall crossings. Since the excess number of events in the search data compared with the background data was not statistically significant, the upper bound on the observable rate is given by the probability of measuring no events during the effective time^{38}. Note that since T depends on the parameters of the domainwall crossing, our constraint on the observed rate depends on the ALP properties. We choose the confidence level to be C = 90%.
Constraints on ALP domain walls
Analysis of the GNOME data did not find any statistically significant excess of events above the background during Science Run 2 that could point to the existence of ALP domain walls, as shown in Fig. 2. The expected rate of domainwall encounters (r) depends on the ALP mass (m_{a}), domainwall energy density in the Milky Way (ρ_{DW}), typical relative domainwall speed \((\bar{v})\) and symmetrybreaking scale (f_{SB}). The region of parameter space to which GNOME is sensitive is defined by the ALP parameters expected to produce signals above the magnitudetouncertainty ratio of 12.6 with rates r ≥ R_{90%} during Science Run 2 (Fig. 3). Based on the null result of our search, the sensitive region is interpreted as the excluded ALP parameter space.
The ALP parameters and the phenomenological parameters describing the ALP domain walls in our Galaxy, namely, thickness Δx, surface tension or energy per unit area σ_{DW}, and the average separation L, can be related through the ALP domainwall model described elsewhere^{9,22}. A full derivation of how the observable parameters are related to the ALP parameters is given in Supplementary Section II.
The coloured region in Fig. 4a describes the symmetrybreaking scales up to which GNOME was sensitive with 90% confidence. The parameter space is spanned by ALP mass, maximum symmetrybreaking scale, and ratio between the symmetrybreaking scale and coupling constant. The shape of the sensitive area shown in Fig. 4a is determined by the event with the largest magnitudetouncertainty ratio and the characteristics of preprocessing applied to the raw data.
Figure 4b shows the various cross sections for different ratios between the symmetrybreaking scale and the coupling constant, as indicated by the dashed lines in Fig. 4a. The upper bound of f_{SB} that can be observed by the network is shown in Fig. 4b for different values of ξ ≡ f_{SB}/f_{int}. Because \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\propto {m}_{\mathrm{a}}\) (Supplementary equation (10) in Supplementary Section II), there is a sharp cutoff for low ALP mass where the corresponding field magnitude falls below the network sensitivity. Even though \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) increases for large m_{a}, the mean rate of domainwall encounters decreases with increasing mass (equations (11) and (12)). Correspondingly, the upper limit for the symmetrybreaking scale f_{SB} is \(\propto 1/\sqrt{{m}_{\mathrm{a}}}\). Given that no events were found, the sensitive region of the ALP domainwall parameter space during Science Run 2 can be excluded.
Our experiment explores the ALP parameter space up to f_{int} ≈ 4 × 10^{5} GeV (Fig. 4). This goes beyond that excluded by previous direct laboratory experiments searching for ALPmediated exotic pseudoscalar interactions between protons that have shown that f_{int} ≳ 300 GeV over the ALP mass range probed by GNOME^{39}. Although astrophysical observations suggest that f_{int} ≳ 2 × 10^{8} GeV, there are a variety of scenarios in which such astrophysical constraints can be evaded^{40,41}. The parameter space for f_{int} and m_{a} explored in this search is well outside the typical predictions for axions in quantum chromodynamics^{42,43}. However, for ALPs, a vast array of possibilities for the generation of ALP masses and couplings are opened by a variety of beyondthestandardmodel theories, meaning that the values of f_{int} and m_{a} explored in our search are theoretically possible^{44,45}.
Future work of the GNOME collaboration will focus on both upgrades to our experimental apparatus and new dataanalysis strategies. One of our key goals is to improve the overall reliability and duration of continuous operation of GNOME magnetometers. The intermittent operation of some magnetometers due to technical difficulties during Science Runs 1–3 made it difficult to search for signals persisting for ≳1 h. Additionally, magnetometers varied in their bandwidths and reliability, as well as stability of their calibration. These challenges were addressed in Science Run 4 through a variety of magnetometer upgrades and instituting daily worldwide test and calibration pulse sequences. However, GNOME suffered disruptions due to the COVID19 pandemic. We plan to carry out Science Run 5 in 2021 to take full advantage of the improvements. Furthermore, by upgrading to noblegasbased comagnetometers^{46,47} for future science runs (advanced GNOME), we expect to considerably improve the sensitivity to ALP domain walls. Additionally, GNOME data can be searched for other signatures of physics beyond the standard model, such as boson stars^{12}, relaxion halos^{48} and bursts of exotic lowmass fields from blackhole mergers^{31}.
In terms of the dataanalysis algorithm used to search for ALP domain walls, recent studies^{49} have considered a possible backaction that the Earth may have on a domain wall when certain interactions are important, namely, uptoquadratic coupling terms between a scalar field and fermions. In contrast to another study^{49}, the present work analyses a completely different interaction, namely, a linear coupling between a pseudoscalar field and fermion spins, which produces no major backaction effect. Regardless, it would be worthwhile to consider interactions generating similar backaction effects of the Earth on domain walls and the ALP field in later analysis. Further, in future work, we aim to improve the efficiency of the scan over the velocity lattice. The number of points in the velocity lattice to reliably cover a fixed fraction (for example, 97.5%) of the ALPvelocity probability distribution grows as (Δt)^{–3} (where Δt is given by equation (9)). This makes the algorithm computationally intensive. We are investigating a variety of analysis approaches, such as machinelearningbased algorithms, to address these issues.
Methods
GNOME consists of over a dozen optical atomic magnetometers, each enclosed within a multilayer magnetic shield, distributed around the world^{27}. GNOME magnetometers are based on a variety of different atomic species, optical transitions and measurement techniques: some are frequency or amplitudemodulated nonlinear magnetooptical rotation magnetometers^{50,51}, some are radiofrequencydriven optical magnetometers^{26}, whereas others are spinexchangerelaxationfree magnetometers^{52}. A detailed description and characterization of six GNOME magnetometers are given in ref. ^{26}. A summary of the properties of the GNOME magnetometers active during Science Run 2 is presented in Extended Data Table 1.
Each GNOME station is equipped with auxiliary sensors, including accelerometers, gyroscopes and unshielded magnetometers, to measure local perturbations that could mimic a dark matter signal. Suspicious data are flagged^{26} and discarded during the analysis.
The number of active GNOME magnetometers during the four science runs and the combined network noise, as defined in ref. ^{30}, are shown as a function of time in Extended Data Fig. 1. Although Science Run 4 was carried out over a longer period of time than Science Run 2, it featured poorer noise characteristics and consistency of operation compared with Science Run 2. Since many GNOME stations underwent upgrades in 2018 and 2019, further characterization of the data from Science Run 4 is needed, and the results will be presented in future work. The number of active magnetometers during Science Runs 1 and 3 was often less than four, which in insufficient to characterize a domainwall crossing. We thus present the analysis efforts on the data from Science Run 2.
Here we provide more details on the analysis procedure. The identification of events likely to be produced by ALP domainwall crossings comprise three stages: preprocessing, velocity scanning and postselection^{30}. First, in the preprocessing stage, a rolling average and filters are applied to the raw data from the GNOME magnetometer, which are originally recorded by the GPSsynchronized dataacquisition system at the rate of 512 samples per second (ref. ^{25}). The rolling average is characterized by a 20 s time constant. Noisy frequency bands are suppressed using a firstorder Butterworth highpass filter at 1.67 mHz together with notch filters corresponding to powerline frequencies of 50 or 60 Hz with a quality factor of 60. These filters are applied forward and backward to remove any phase effects. This limits the observable pulse properties to a frequency region to which all the magnetometers are sensitive. Additionally, it guarantees that the duration of the signal is the same for all the sensors. We note that these filter settings may be changed in future analyses.
The local standard deviation around each point in the magnetometer’s data is determined using an iterative process. Outliers are discarded until the standard deviation of the data in the segment converges. The local standard deviation is calculated taking 100 downsampled points around each data point.
Additionally, auxiliary measurements have shown that the calibration factors used by each magnetometer to convert raw data into magneticfield units experience change over time due to, for example, changes in the environmental conditions. Upper limits on the errors in the calibration factor due to such drifts over the course of Science Run 2 have been evaluated, as listed in Extended Data Table 1. Calibration errors result in magneticfield measurement errors proportional to magnetic field B_{j}. The uncertainty resulting from the calibration error is later used to determine the agreement with the domainwall model, but not in the magnitudetouncertainty ratio estimate resulting from the model, since the calibration error affects the signal and noise in the same way.
Second, at the velocityscanning stage, data from the individual magnetometers are timeshifted according to different relative velocities between Earth and the ALP domain walls. To sample 97.5% of the velocity probability distribution, a scan of the speeds from 53.7 to 770 km s^{–1} with directions covering the full 4π solid angle is chosen; therefore, the domain walls can take any orientation with respect to the movement of Earth. Note that this distribution considers just the observable perpendicular component of the relative domainwall velocity and neglects the orbital motion of the Earth around the Sun. For low relative velocities, both time between signals at different magnetometers and signal duration diverge. Therefore, the velocity range is determined by the chosen 97.5% coverage and the maximum relative speed of the domain walls travelling at the Galactic escape speed.
The corresponding timeshifted data along with their local standard deviation estimate are fetched from each magnetometer’s rollingaverage fullrate data at the rate of 0.1 samples per second. This reduces the amount of data to process, even though keeping the full timing resolution.
The step size used in the speed scan is chosen so that a single step in speed corresponds to timeshift differences of less than the downsampled sampling period. For each speed, a lattice of directions covering the full 4π solid angle is constructed. The angular difference between adjacent directions is informed by the sampling rate and speed^{30} such that, as for the speed scan, a single step in direction results in timeshift differences of less than the downsampled sampling period. With the settings used, the velocityscanning lattice consists of 1,661 points. This number scales with the cube of the downsampled sampling rate.
After the time shift, the pulses produced by a domainwall crossing simultaneously appear as if all the magnetometers were placed at the Earth’s centre. This process results in a timeshifted dataset for each lattice velocity on which χ^{2} minimization is performed for each time point to estimate the domainwall parameters. An ALP domainwallcrossing direction and magnitude \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) with the corresponding Pvalue quantifying the agreement is obtained. The Pvalue is evaluated as the probability of obtaining the given χ^{2} value or higher from χ^{2} minimization. The Pvalue is calculated using the quadrature sum of the standard deviation of the data and the uncertainty due to drifts in the calibration factors. All the data points in every timeshifted dataset are considered to be potential events, characterized by time, Pvalue, and direction and magnitude \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) with their associated uncertainties. The magnitudetouncertainty ratio of an event ζ is the ratio between \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) and its associated uncertainty.
Third, in the postselection stage, two tests are carried out to check if a potential event is consistent with an ALP domainwall crossing. The domainwall model test evaluates if the observed signal amplitudes are consistent with the expected pattern of a domainwall crossing from any possible direction. It is quantified by the aforementioned Pvalue. The directionalconsistency test is based on the angular difference between the estimated domainwallcrossing direction and the direction of velocity corresponding to the particular timeshifted dataset being analysed. In a real domainwallcrossing event, these two directions should be aligned.
To evaluate the consistency of a potential event with a domainwall crossing, we impose thresholds on the Pvalue and the angular difference normalized with respect to the angular spacing of the lattice of velocity points for that speed. The thresholds are chosen to guarantee a detection probability of 97.5% with the minimum possible falsepositive probability. The falsepositive analysis is performed on the background data. The truepositive analysis is performed on the test data consisting of background data with randomly inserted domainwall signals as described below.
A single signal pattern may appear as multiple potential events in the analysis, whereas we are seeking to characterize a single underlying domainwallcrossing event. For example, a signal consistent with a domainwall crossing lasting for multiple sampling periods would appear as multiple potential events in a single timeshifted dataset. Furthermore, even if such a signal lasts only for a single sampling period, the corresponding potential events appear in different timeshifted datasets. Since it is assumed that domainwall crossings rarely occur, such clusters of potential events are classified as a single ‘event’. To reduce the double counting of these events, conditions are imposed. If potential events passing the thresholds occur at the same time in different timeshifted datasets or are contiguous in time, the potential event with the greatest magnitudetouncertainty ratio is classified as the corresponding single event.
To evaluate the detection probability of the search algorithm, a wellcharacterized dataset that includes domainwallcrossing signals with known properties is required. For this purpose, we generate a background dataset by randomly time shuffling the search data so that the relative timing of measurements from different GNOME stations is shifted by amounts so large that no truepositive events could occur. By repeating the process of time shuffling, the length of background data can be made to far exceed the search data. This method is used to generate background data with noise characteristics closely reproducing those of the search data^{53}. A set of pseudomagneticfield pulses matching the expected amplitude and timing pattern produced by the passages of Earth through the ALP domain walls are inserted into the background data to create the test data.
The truepositive analysis studies the detection probability as a function of the thresholds. Multiple test datasets are created featuring domainwallsignal patterns with random parameters by inserting Lorentzianshaped pulses into the background data of the different GNOME magnetometers. The domaincrossing events have magnitudes of \({{{{\mathcal{B}}}}}_{{{\rm{p}}}}\) randomly selected between 0.1 and 10^{4} pT and durations randomly selected between 0.01 and 10^{3} s. The distributions of the these randomized parameters are chosen to be flat on a logarithmic scale. Additionally, the signals are inserted at random times with random directions. To simulate the effects of calibration error, the pulse amplitudes inserted in each magnetometer are weighted by a random factor whose range is given in Extended Data Table 1. The crossing velocity is also randomized within the range covered by the velocity lattice. For each inserted domainwallcrossing event, the Pvalue, normalized angular difference and magnitudetouncertainty ratio are computed.
Extended Data Fig. 2a shows the detection probability as a function of the threshold on the lower limit of the Pvalue and the threshold on the upper limit of the normalized angular difference. We restrict the analysis in Extended Data Fig. 2a to events inserted with a magnitudetouncertainty ratio between 5 and 10. This enables a reliable determination of the truepositive detection probability without major contamination by falsepositive events, since the background event probability above ζ = 5 is below 0.01% in a 10 s sampling interval. Since the detection probability increases with the signal magnitude, we focus on the events below ζ = 10. The detection probability is then the number of detected events divided by the number of inserted events. The black line marks the numerically evaluated boundary of the area, guaranteeing at least 97.5% detection. All points along this black line yield the desired detection probability; therefore, this particular choice is made to minimize the number of candidate events when applying the search algorithm to the background data. Here the values determined for the Pvalue threshold and directionalconsistency threshold are 0.001 and 3.5, respectively (represented as the white dot in Extended Data Fig. 2a). Extended Data Fig. 2b shows that the detection probability is greater than 97.5% for events featuring a magnitudetouncertainty ratio above 5 and guarantees ϵ ≥ 95%. This results in an overall detection efficiency of ϵ ≥ 95% for the search algorithm, considering both incomplete velocity lattice coverage and detection probability.
Since the noise has a nonzero probability of mimicking the signal pattern expected from an ALP domainwall crossing well enough to pass the Pvalue and directionalconsistency tests, we perform a falsepositive study on background data of length T_{b}. The analysis algorithm is applied to T_{b} = 10.7 years of timeshuffled data to establish the rate of events solely expected from the background. Because of the larger amount of background data analysed, lower rates and larger magnitudetouncertainty ratios are accessible compared with the search data. Based on the falsepositive study, the probability of finding one or more events in the search data above ζ is^{54}
where T = 23 days is the duration of Science Run 2 and n_{b}(ζ) is the number of candidate events found in the background data above ζ. The significance is then defined as \(S=\sqrt{2}{{{\mathrm{erf}}}\,}^{1}\left[12(1P)\right]\), where erf^{–1} is the inverse error function. The significance is given in units of the Gaussian standard deviation that corresponds to a onesided probability of P.
After characterizing the background for Science Run 2, the search data are analysed. The results are represented as a solid green line in Fig. 2. For ζ > 6, only a few events were found. The event with the largest magnitudetouncertainty ratio, ζ_{max}, was measured at 12.6 followed by additional events at 6.2 and 5.6. From equation (5), the significance associated with finding one or more events produced by the background featuring at least ζ_{max} is lower than one sigma. This null result defines the sensitivity of the search and is used to set constraints on the parameter space describing the ALP domain walls.
The observable rate of domainwall crossings depends on how long GNOME was sensitive to different signal durations and magnitudes. For the evaluation of this effective time, the raw data of each magnetometer are divided into continuous segments between one and two hours depending on the availability of data. The preprocessing steps are applied to each segment. Then, the data are binned by taking the average in 20 s intervals. To estimate the noise in each magnetometer, the standard deviation in each binned segment is calculated to define the covariance matrix Σ_{s}. The domainwall magnitude, crossing with the worsecase direction m, needed to produce ζ = 1 is calculated, as in ref. ^{30}, for each bin.
The matrix D_{Δt} contains the sensitivity axes of the magnetometers, factor σ_{p}/g, and effects of preprocessing as a function of signal duration (as described in ref. ^{30}). Such prepocessing effects rely on a Lorentzianshaped signal and give rise to the characteristic shape shown in Fig. 3. The effective time T is defined as the amount of time for which the network can measure a domain wall with duration Δt and magnitude \({{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}\), producing ζ ≥ 1. Monte Carlo simulations analysing segments with inserted domainwall encounters on the raw data show good agreement with the sensitivity estimation in equation (6).
Assuming that the domainwall encounters follow Poisson statistics, a bound on the observable rate of events above ζ_{max} with 90% confidence is set as^{38}
The domainwall thickness is determined by the ALP mass, and is of the order of the ALPreduced Compton wavelength ƛ_{a} (ref. ^{22}):
The constant prefactor of \(2\sqrt{2}\) is obtained by approximating the spatial profile of the fieldgradient magnitude as a Lorentzian and defining the thickness as the fullwidth at halfmaximum (FWHM). For a given relativevelocity component perpendicular to the domain wall v_{⊥}, the signal duration is
We assume that domain walls comprise the dominant component of dark matter. Thus, with the energy density ρ_{DW} ≈ 0.4 GeV cm^{–3} in the Milky Way^{24}, the energy per unit area (surface tension) in a domain wall, σ_{DW}, determines the average separation between the domain walls, L. The surface tension σ_{DW} is related to the symmetrybreaking scale^{9} as
The average domainwall separation is then approximated by
which results in the average domainwall encounter rate of
We assume the typical relative domainwall speed to be equal to the Galactic rotation speed of Earth.
The ALP parameter space is constrained by imposing r ≥ R_{90%}. The experimental constraint on the coupling constant is written as follows (Supplementary equation (13) in Supplementary Section II).
The signal duration can be written in terms of the mass of the hypothetical ALP particle and the specific domainwallcrossing speed, \({{\Delta }}t=\frac{2\sqrt{2}\hslash }{{v}{m}_{\rm{a}}c}\). When calculating the constraints on f_{int}, we fix the domainwallcrossing speed to the typical relative speed from the standard halo model, \(\bar{v}\) = 300 km s^{−1} (ref. ^{23}). In contrast to the signal duration, the pseudomagneticfield signal depends on all the parameters of the ALPs, mass, and ratio between the coupling and symmetrybreaking constants, namely, \({{{{\mathcal{B}}}}}_{{{{\rm{p}}}}}^{\prime}=\frac{4{m}_{\rm{a}}{c}^{2}\xi }{{\mu }_{\rm{B}}\zeta }\). The data shown in Fig. 4 are obtained using equation (13) by taking ζ = 12.6. The shape of the constrained space is given by the fact that T varies depending on the target m_{a} and ξ.
Data availability
Source data are provided with this paper. The datasets and analysis code used in the current study are available from the corresponding authors upon reasonable request. Also, see the collaboration website https://budker.unimainz.de/gnome/ where all the available data are displayed.
Code availability
The code used in the current study is available from the corresponding authors upon reasonable request.
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Acknowledgements
We are grateful to C. Pankow, J. R. Smith, J. Read, M. Givon, R. Folman, W. Gawlik, K. Grimm, G. Łukasiewicz, P. Fierlinger, V. Schultze, T. SanderThömmes and H. Müller for insightful discussions. This work was supported by the U.S. National Science Foundation under grants PHY1707875, PHY1707803, PHY1912465 and PHY1806672; the Swiss National Science Foundation under grant no. 200021 172686; the German Research Foundation (DFG) under grant no. 439720477; the German Federal Ministry of Education and Research (BMBF) within the Quantumtechnologien program (FKZ 13N15064); the European Research Council under the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 695405; the Cluster of Excellence PRISMA+; DFG Reinhart Koselleck Project; Simons Foundation; a Fundamental Physics Innovation Award from the Gordon and Betty Moore Foundation; HeisingSimons Foundation; the National Science Centre of Poland within the OPUS program (project no. 2015/19/B/ST2/02129); USTC startup funding; the National Natural Science Foundation of China (grant nos. 62071012 and 61225003); the National HiTech Research and Development (863) Program of China and IBSR017D12021a00 of the Republic of Korea. We acknowledge funding provided by the Institute of Physics Belgrade through a grant by the Ministry of Education, Science and Technological Development of the Republic of Serbia.
Funding
Open access funding provided by GSI Helmholtzzentrum für Schwerionenforschung GmbH.
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Contributions
All the authors have contributed to the publication, being responsible for the construction and operation of the different magnetometers, building the software infrastructure, assuring the quality of the data being taken, and establishing phenomenological motivation. The dataanalysis procedure presented here was led by J.A.S. and H.M.R., with collaboration from the other authors. D.F.J.K. coordinated collaboration between the different teams within GNOME. The manuscript was drafted by H.M.R., J.A.S., A. Wickenbrock and D.F.J.K. It was subject to an internal collaborationwide review process. All the authors reviewed and approved the final version of the manuscript.
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Extended data
Extended Data Fig. 1 Summary of the GNOME performance during the four Science Runs from 2017 to 2020.
The raw magnetometer data are averaged for 20 s and their standard deviation is calculated over a minimum of one and a maximum of two hours segments depending on the availability of continuous data segments. For each binned point, the combined network noise considering the worst case domainwall crossing direction is evaluated as defined in Ref. ^{30}. (a) Oneday rolling average of the number of active sensors. (b) Multicolored solid line represents the oneday rolling average of the combined network noise and the multicolored dashes show the noise of the individual sampled segments. The data are preprocessed with the same filters used for the analysis. The number of magnetometers active is indicated by the color of the line and dashes.
Extended Data Fig. 2 Summary of the truepositive analysis results.
(a) shows the probability of detecting a domainwallcrossing event with randomized parameters (as discussed in the text) as a function of pvalue and directionalconsistency thresholds. The inserted events have a magnitudetouncertainty ratio between 5 and 10. The black line indicates the combination of parameters corresponding to a 97.5% detection probability. The white dot indicates the particular thresholds chosen for the analysis. (b) Shows the mean detection probability reached for different magnitudetouncertainty ratios for the chosen thresholds.
Supplementary information
Supplementary Information
Supplementary text.
Source data
Source Data Fig. 2
Positive event data (Fig. 2).
Source Data Fig. 3
Time for which GNOME was sensitive to Lorentzian signals of various FWHM values and magnitudes (Fig. 3).
Source Data Fig. 4
Bounds on ALP parameter space (Fig. 4).
Source Data Extended Data Fig. 1
Noise data from each magnetometer (Extended Data Fig. 1).
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Afach, S., Buchler, B.C., Budker, D. et al. Search for topological defect dark matter with a global network of optical magnetometers. Nat. Phys. 17, 1396–1401 (2021). https://doi.org/10.1038/s4156702101393y
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DOI: https://doi.org/10.1038/s4156702101393y
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