Galen Carter Senior Thesis 2024

Analysis of Experimental Data to Identify Axion-like Dark Matter Spectral Signatures

Galen Carter Senior Thesis | 2024

Analysis of experimental data to identify axionlike dark matter spectral signatures Abstract Dark matter has been theorized to exist and may account for the “missing mass” in cosmological models. Axions are theoretically motivated candidate particles which may comprise the majority of dark matter. If they exist, axions may interact with the physical world through a mechanism that acts like a tiny magnetic field.

The Sushkov Quantum Lab has produced ferromagnetic-based experiments with the intent of capturing these magnetic-field interactions. Such experiments are abbreviated Search for Halo Axions with Ferromagnetic Toroids (SHAFT). Because the shape of what these axion-like signatures is specified by theory, computational programs can greatly assist in the scanning of extremely large data sets to identify these specific signatures.

Newly developed code for MATLAB 2022a takes previouslycollected SHAFT experimental data as an input and processes it to generate a list of potential axion candidates. This pipeline includes a Fourier transform to convert the data into Power-Spectral Density (PSD), a series of Savitzky-Golay filtering operations before convolving, and finally outputting potential axion candidates with corresponding coupling powers. This paper discusses SHAFT electric (SHAFT-e), a successor to earlier SHAFT experiments which uses ferromagnetic rods instead of toroids and details both the design and analytical portions of the experiment.

Introduction What is dark matter?

There is sufficient theoretical and experimental evidence for the existence of dark matter that has accumulated over the past several decades, yet the nature of what exactly dark matter is remains unknown. However, there are several characteristics which we generally attribute to dark matter, and hence reduce the number of potential possibilities for dark matter candidates. Dark matter is generally agreed to be cold, meaning it is low-energy and moves slower than the speed of light; dark, meaning it does not emit light and does not interact with the electromagnetic force; and nonbaryonic, meaning it is not composed of particles belonging to the standard model (i.e. quarks, leptons, and bosons and carrier particles for the strong and weak nuclear forces). Furthermore, dark matter is not antimatter because gamma rays (high-energy photons) are not produced from interactions with matter unlike when antimatter annihilates matter. This definition comes from the Lambda cold dark matter (LCDM) model which has done a fantastic job in explaining the expansion of the universe. Note that there are many other proposed forms of dark matter (i.e. hot, warm, etc.) but these are much less theoretically motivated.

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As early as 1933, galactic rotational velocity observations have served as concrete evidence for the existence of dark matter. Fritz Zwicky showed that the rotational velocities of galaxies in the Coma cluster were much higher than expected based on the sum of the relevant galactic masses. There must then be some extra, unobservable mass to allow the galaxies to maintain such a high rotational velocity with increasing radius. Dark matter provides an explanation for this missing mass. Additionally, there is ample cosmological evidence to motivate dark matter. The cosmic 1

microwave background (CMB) which is essentially a historic record of the universe’s composition and therefore also mass.

Analysis of the CMB has shown that a small fraction of the energy budget of the universe is composed of baryonic particles, a larger fraction from dark matter, and a still larger fraction from the much less understood dark energy, the scope of which is beyond this paper. Furthermore, the formation of primordial (i.e. happening towards the beginning of the formation of the universe) galaxies could not have happened with purely baryonic matter. Essentially, if baryonic matter alone created primordial galaxies, there would simply not be as many galaxies as is observed today. Fluctuating gravitating dark matter provides a reasonable explanation to galactic formation because it would be unaffected by the radiation pressure of the early universe.

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Dark Matter Candidates Over the decades since the first observation, numerous candidates for dark matter have been hypothesized. A popular candidate is weakly-interacting massive particles (WIMPs). Definitions for what exactly constitutes a WIMP, as well as other candidates in general, are loose. However, WIMPS are generally larger (hence “massive”) but still particle-size and interact with gravity. Supersymmetry is one strong theoretical motivation for WIMPs. Supersymmetry proposes that each particle in the standard model has a dark, heavier “superpartner” with similar properties to the original particle. WIMPs, if discovered, could potentially be these superpartners. However, narrowing of the predicted mass range and the failure to detect WIMPs at previously-successful particle accelerators such as the Large Hadron Collider has cast doubt on their existence. Nonetheless, WIMPs remain a strong candidate for dark matter.

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More complex possibilities for dark matter include (as mentioned above) warm and hot components, meaning traveling at or faster than the speed of light. Among these candidates include primordial black holes – ancient black holes formed after the Big Bang that might explain the gravitational effects in galaxies and galaxy clusters; sterile neutrinos – particles that interact only via gravity (as opposed to regular neutrinos which also interact with the weak force); self-interacting dark matter – dark matter which interacts strongly as opposed to normal cold dark matter; and modified gravity – a theory which explains galactic gravity effects by modifying Newtonian gravitational dynamics. There are many possibilities but a common issue among them is that they are very hard to prove (either experimentally or observationally).

Axion-like particles (ALPs) are a competing, promising dark matter candidate. ALPs are generally defined to be less massive than WIMPs. The matter-antimatter asymmetry is a natural motivation for the existence of ALPs. Essentially, each Standard Model particle is required, by the laws of quantum mechanics and special relativity, to have a corresponding antimatter pair (note that this is not a super-symmetric pair). These antimatter particles have been experimentally proven to exist through particle collisions and cosmological observations have shown that there is more matter than antimatter in the universe. The charge-conjugation-parity (CP) violation provides an explanation for the matter-antimatter asymmetry. It shows that CP-symmetry can indeed be violated and thus might sensibly explain the predominance of matter over antimatter. But while CP violation has been shown with weak force decay, CP violation has not been shown to appear in strong force interaction (as would be expected in the Standard Model).

Another, perhaps simpler, way of thinking about this “strong CP problem” is the existence of a term (! bar, the effective periodic strong CP-violating term) which could theoretically take on any value between 0 and 2π ends up having a value equal to 0 (or very 4

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close). The quantum chromodynamics (QCD) axion provides an explanation for ! bar. This is called the Peccei-Quinn theory. It is worth noting that CP violation has been shown to happen indirectly via kaon decay however this process cannot nearly account for the major asymmetry and there must be some other reason (i.e. the QCD axion).

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The QCD axion is part of the broader category of ALPs. Note that ALPs, in general, do not solve the strong CP problem and therefore could be much more difficult to discover experimentally than the theorized axion, which has a discrete mass range. Other axions (i.e. not the QCD axion) are motivated by string theory. Another range, axion theorists are concerned with is the coupling values of axions to particles, however efforts to define this range are still undergoing.

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Axion mass has a very large theoretical range (but many restrictions). Therefore, for the most part, experiments for the detection of axions attempt to find the axion itself but also to, more importantly, eliminate portions of the range in which the axion might be. There are astrophysical bounds – e.g. supernovae observations, stellar evolution theory, white dwarf luminosity function, and galactic globular clusters – that greatly constrain the range of axions (see the blue and green ranges in Figure 1). For example, the axion electron coupling value is astrophysically limited to just above 10 . However, the remaining range is still large spanning some 10 eV to 10 eV and therefore many different detection methods need to be used.

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Axions, if they exist, are known to interact via gravity. Gravity is weak (about by a factor of roughly 10 ) so other interactions must occur for axions to be directly detectable. There are three such theorized interactions: electromagnetic fields, QCD field strength 40

tensors, and standard model fermions. The first interaction, with electromagnetic fields (i.e. photons), has a strength characterized by a coupling power. The second interaction couples axions to nuclei creating electron dipole moments (an intrinsic property of an electron related to potential energy and electric field strength).

It is this interaction that would resolve the strong CP problem. The third and final interaction creates torques (“axion wind”) on the spins of standard model fermions (quarks and leptons). It appears as a wind to an observer on Earth due to the motion of the Earth (and Solar System) through the local cold dark matter cloud surrounding the Milky Way. 10

Axion-detection Experiments There are a vast variety of past and ongoing axion-detection experiments as demonstrated in Figure 1. Several experiments focus on the assumptions that axions (or ALPs) are not the dominant component of dark matter and instead can be detected through solar bodies or through artificial laboratory construction.

Figure 1. Summary of axion-like particle photon-coupling space with current experimental and astrophysical constraints. The green and blue areas lining the middle left corner and right side are all various astrophysical limitations, signifying what axions could not possibly be. The red areas are various experiments and the ranges they have eliminated. The goal of these experiments is to reach the KSVZ and DFSZ II lines representing two theoretical axion models. If axion-like particles were to be found, it would most likely be in that range.

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These are indirect searches for dark matter. One such of these experiments is XENON1T, a xenon-based detector, has observed a strange excess of low-energy recoil events. There are several possible explanations for this such as the beta decay of tritium or other novel new-physics scenarios but the one the collaboration focuses on is that the events originate from the solar axions (i.e. axions originating from the sun) interacting with electrons. A 12

similar xenon-based detection experiment is LUX (Large Underground Xenon). The CERN Axion Search Telescope (CAST) at CERN attempts to detect axions produced in the charged core of the Sun via electromagnetic x-ray scattering. Such experiments called “helioscopes” are getting much closer to reaching theoretical astrophysical bounds thereby limiting potential axion ranges. Another experiment Any Light Particle Search (ALPS) at Deutsches Elektronen-Synchrotron uses magnets and lasers to induce an axion-photon-axion conversion. Additionally, Neutron star magnetospheres could also convert axions into photons which might be visible by modern telescopes.

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There are also many direct dark matter searches. These experiments mostly utilize the interaction of axions (and ALPs) with photons, utilizing large magnetic fields and volumes to maximize the conversion rate of axions to photons. In other words, the emergence of electromagnetic radiation out of a vacuum when magnetic fields are present might signify the presence of axions. Large magnetic fields are therefore needed to have a chance at even a slight axion signal. An experiment that pioneered this phenomenon, ADMX (Axion Dark-Matter eXperiment) at the University of Washington, uses resonant conversion to convert axions into monochromatic (fixed energy) microwave photons via a cavity within a large superconducting magnet. The experiment has sensitivity to search from axion dark matter between 2.6 - 4.2 μeV, thereby excluding associated QCD couplings with that mass range. Another similar experiment is QUAX (QUest for AXions).

QUAX uses a ferrimagnetic yttrium iron garnet sphere to detect potential rare-earth-dark-matter interactions. CASPEr experiments on the other hand use NMR (nuclear magnetic resonance) to look for effects of axion-gluon coupling (CASPErelectric) and “axion-wind” coupling (CASPEr-wind).

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The predecessor to our experiment, Search for Halo Axions with Ferromagnetic Toroid (SHAFT), utilized, as the name suggests, four permeable toroids made of an iron-nickel alloy. The toroids had the effect of increasing the magnitude of the static magnetic field and thereby increasing the sensitivity to ALPs (which would interact with this field). SHAFT was split into two independent detection channels (two toroids each) monitored by a SQUID (Superconducting Quantum Interference Device) which translated potential axion dark matter detection signatures into an analyzable output signal whose amplitude corresponded to toroid magnetization. SQUIDs additionally reduced noise, a central experimental consideration of axion detection – experiments need to make sure that any potential candidate is not merely a byproduct of any extraneous interference in the magnetic field. Additionally, if an axion signal was detected, having a second channel allows verification of the signal. If the signal only appeared in one of the channels, one can be fairly confident that it was not real, whereas if it appeared in both, then it might signify that there is actually something there. Also present was a lead superconducting magnetic shield which had a purpose of suppressing both electromagnetic interference and ambient magnetic field noise.

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The main issue that axion-detection (and in general, all dark matter detection) experiments deal with is maximization of sensitivity. In SHAFT, the use of permeable toroids specifically had material magnetization in addition to a field created by the free current (i.e. the unbounded motion of electrons as opposed to motion restricted by alignment of electron dipole moments) in the toroidal magnetization coil. The largest overall limitations to the experimental sensitivity were the noise penetration caused by vibrations of the coil for low frequencies and SQUID feedback electronics bandwidth roll-off (a frequency past which the devices begin to degrade in performance) for high frequencies. Ultimately

the experiment was capable of detecting axions within the 10 eV to 10 eV range.

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This paper will detail the SHAFT electric (SHAFT-e) experiment, the successor to SHAFT, and in particular the data analysis process with the end goal of producing results which confirm the existence of axion dark matter.

Methods The SHAFT-e experiment attempts to search for axions through the third interaction mentioned above, that is the effects created by axion-wind coupling to electron spins. There are some complex equations and physics topics that provide a full theoretical explanation. However, an adequate simplification is that SHAFT-e looks for the magnetization oscillations induced by axion-wind.

Experimental Apparatus SHAFT-e follows the same experimental design as SHAFT but differs in several ways. First, instead of using ferromagnetic toroids, three mutually orthogonal rods (positioned like the x-y-z axes of a graph) are used. The reason for this is so that the experiment is sensitive to all three components of the effective magnetic field created by the axion-wind interaction. Each rod is coupled to a SQUID. As a result, there are three SQUID channels instead of two. Second, there are two phases of the experimental design. The first one mounts the apparatus in a 4 K liquid helium bath. This is simple and serves the purpose of performing a preliminary test, making sure equipment is working as intended.

For the second stage, the apparatus was mounted in a dilution refrigerator, which cools the set-up to 4 millikelvin (mK). This huge difference in temperature increases the performance of the

SQUIDs (by reducing noise) by a factor of 10. Also present are a set of superconducting magnetic shields (inner and outer) which attempt to further reduce external noise. Lastly, there is a G-10 (a fiberglass laminate) structure which supports the rod configuration.

Figure 2 gives an overview of the experiment.

Candidates are checked in the same process as SHAFT, crossverifying signals using amplitudes and phases expected of an axion-induced oscillation between all three channels. Having three channels instead of two allows for a higher accuracy in identifying true from false axion signals. The same detection limitations are present for SHAFT-e as for SHAFT. On the lower side, vibrations and magnetic noise prevent accuracy, and on the higher side, SQUID feedback electronic bandwidth roll-off limits sensitivity.

Before any data collection is done, it is worth noting that the apparatus needs to be calibrated to ensure the set-up is working as expected. This is done via calibration coils on each of the cylinder rods

Figure 2. SHAFT-e apparatus. (a) depicts the effective magnetic field (B ) created by axion interactions. That magnetic field then creates a magnetization (M) on the ferromagnetic rod which is then relayed via the blue pickup coil to a SQUID. (b) Circuit model of the experiment. Φa is the axion-flux (i.e. the oscillating magnetization created by the axion); SQ is the SQUID; DAQ is the data acquisition system; L is the inductance of the pickup coil; L is the inductance of the twisted pair leads; L is the inductance of the input coil. (c) Experimental apparatus. Note the three permeable rods. Credit: SHAFT-e proposal. * a

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Data Collection Outline MATLAB 2022a was used to perform all analyses of the experiment.

Data points were recorded over a set period of time as mentioned in the time domain. That is, frequency of oscillation vs. time. Then a discrete fast fourier transformation (FFT) was applied to the data, bringing it from the time domain to the frequency domain. FFTs work by fitting sinusoids to a curve, and using the properties of

those sine and cosine functions to create graphs that display frequency vs. signal amplitude (how strong the signal is). From the frequency domain, the data was converted into the power spectral domain (PSD), the space where the spectral signatures of the axions are able to be seen. A Savitsky-Golay filtering operation was then applied to eliminate peaks beyond a certain threshold that were either too broad (broad SG filter) or too narrow (narrow SG filter). Next, a convolution was applied on the cleaned up data.

This operation overlays the expected spectral signature, given by a user-inputted function that is extremely dependent on the specifics of the experiment, on the data and graphs the overlap. Essentially, the highest convolved points are those where there is expected to be an axion.

Finally, a Gaussian distribution was applied to the convolved points. Points above a 95% threshold are the actual candidates and a graph matching each of the candidates to their associated coupling powers are generated.

Candidates, as mentioned in the apparatus section, were confirmed through the two-channel detection system. There are four tests: the signal above the threshold ( = present) in Channel A, the signal above the threshold in Channel B, is the signal below the threshold in symmetric combination, and is the signal above the threshold in the antisymmetric combination. Antisymmetric and symmetric data are obtained from an average of the two channels; symmetric is the sum of A and B divided by 2 while antisymmetric is the difference of A and B divided by 2. For an axion to be even considered as a possible detection it needs to pass all four tests.

The Operations FFT / DFT – A fast fourier transform; utilizes the Nyquist frequency – a frequency whose cycle length / period is half the sampling length. An FFT at its essence transforms data into a different set of basis functions, in this case complex exponentials or sinusoids. FFT units are amplitude vs. frequency (called the frequency domain) and this is the domain in which any possible axion spectral signatures can be deciphered; FFT are functions used to calculate DFT (discrete fourier transform) because they have efficiency O(n log n) vs. O(n ). This means the program will take significantly shorter when running huge chunks of data, like these experiments require.

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Figure 3 depicts a great example of a fourier transformation. The square wave (time domain, i.e. frequency vs. time) is an infinite sum of sinusoids gradually decreasing in amplitude. These amplitudes are then graphed against frequency in the frequency domain and the transformation is complete. Note that the FFT greatly simplified the square wave and this is a signal processing trick that the SHAFT program uses to smooth the analysis.

Figure 3. Fourier transformation of a square wave.

PSD – Power spectral domain; simply, takes data produced by a DFT or FFT and alters it into a form (called the PSD) that is much easier to interpret visually (accentuates peaks and diminishes lows). Calculated from the mean squared amplitude of each frequency component averaged over the samples, the PSD can also be thought about the frequency distribution of the signal variance.

SG Filter – Savitzky-Golay filtering operations. To eliminate unnecessary noise, the SG filter uses a lower-order polynomial fit to smooth out the data. The script employs two of such filters: a broad SG filter which eliminates peaks that have a width a certain multiple larger than would be expected and a narrow SG filter which does the same for narrower peaks.

Convolution – the program uses a convolution to ultimately determine the potential presence of an axion (called a “candidate”).

Convolutions work by gradually overlaying a shape (in this case, the spectral signature) over another (in this case, the post-SGfiltered PSD data). As the convolution moves down the PSD, it

produces values that are the largest where the spectral signature has the most overlap.

Spectral signature - this is the shape of the axion. As mentioned, it is experimentally dependent but takes the general shape of a Maxwell-Boltzmann distribution. In the case of SHAFT and SHAFT-e, the standard halo model is assumed to convey some information from which the lineshape equation can then be derived. The standard halo model assumes a fully virialized halo is spherical and isothermal and therefore dark matter velocities in the galactic reference frame are Maxwellian. Thus, the expected dark matter signature for SHAFT looks like the following:

where v is the Compton frequency (a quantum mechanical property of an object) of the axion-like field, r is about root of two thirds, and beta-squared is a parameter that sets the spectral linewidth of the signal (i.e. how wide it is). a

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For SHAFT-e, the axion spectral signature utilizes the base line from SHAFT but makes the important difference of combining two components – perpendicular and parallel.

The final axion line (“finalAxline” in the program sample above) comes from a weighted, linear combination of the two components. Alpha, the only new variable, is the angle in radians between the velocity of the lab with respect to the galaxy and the vector normal to the pickup coil of the experiment. It varies with time, longitude, etc.

Figure 4 shows an example of the plot sequence for an axion analysis search using a fake axion insertion testing function. The

first plot shows an axion spectral signature embedded in the PSD. There are several intermediate plots, unshown in Figure 4: a raw PSD histogram, a corrected raw PSD histogram (removes the outliers), a broad corrected plot (post SG filter), and a fully-filtered SG histogram. Then, the second plot of Figure 4 appears which portrays the post-filtered, post-convolved PSD with the sharp peak where the axion was detected. The third plot is a histogram of that PSD and the fourth is a testing plot confirming the inputted fake axion coupling powers are found by the program.

Figure 4. Sample analysis plots. In this case, a fake axion was inserted using a testing function. (a) shows the experimental PSD with a synthetic axion-like signal inserted at 10.04 x 10 Hz (blue points). The red line is the axion lineshape from the SHAFT equation. This is the signal for which the program is scanning. (b) The convolved (using the standard halo model lineshape) and postfiltered data. Note that the same point at 10.04 x 10 Hz has a sharp peak because this is where the convolution has the highest overlap – by far. (c) Histogram of the post-filtered, convolved PSD with the dashed line representing the 3.355σ threshold (= 95% confidence) for candidacy. Note that the injected axion signal has a significance of 20 (those are the points way past the vertical line). (d) Verification that input signal coupling power matches output coupling power. From the SHAFT supplement.

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Results & Discussion Results are unavailable for as of March 2024.

Beyond the running time of the analysis (around 24 minutes for a period of 1 MHz to 4 MHz for 1 GB of data) which scales nearly linearly for larger amounts of data, there are several other problems which remain unsolved in the code to ensure it is functioning as intended. We recently upgraded MATLAB versions from 2019a to 2022a and, in changing versions, several features which seemed to be working before now have to be fixed. The axion coupling outputs (reference plot (d) of Figure 4) which were working before are now off by a factor of around 2. Interestingly, tuning the values of the SG filter alters the output values so there are certainly some scaling issues present. Additionally, sometimes some of the candidates would be found for certain input parameters of the SG filter but not others (ideally the candidates found would be consistent across all parameters). One last issue is that the expected candidate number which the code prints (this comes from the 95% threshold of the normal distribution histogram of PSD) is around 30-40% off the actual numbers.

We do not expect to detect a signal as we are simply not in a range where axions are theorized to be. The purpose of SHAFT-e is to demonstrate a novel technique and refine it so that future, higher sensitivity experiments may reach the coupling ranges where axions are theorized to exist. Likewise, the data analysis serves to improve techniques to reject noise and false positives and minimize time complexity, necessities for future data in which, just perhaps, an actual axion is found. Just as the team works to understand the behavior of the apparatus, the team continues to understand the software: in addition to the above changes, more recently we recognized odd patterns in which candidates were

being identified and we are considering issues with grouping of candidates to prevent any possibility of double-counting. SHAFT is an on-going process but already data such as the SHAFTWReSL (Weekend Relaxion-Search Laboratory) collaboration is moving through analysis to provide real-world tests of axiondetection.

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Supplementary Material The code is available online on Github (CASPEr-Collaboration).