Search for dark-photon dark matter in the SuperMAG geomagnetic field dataset

In our recent companion paper [Fedderke et al. Phys. Rev. D 104, 075023 (2021)], we pointed out a novel signature of ultralight kinetically mixed dark-photon dark matter. This signature is a quasimonochromatic, time-oscillating terrestrial magnetic field that takes a particular pattern over the surface of Earth. In this work, we present a search for this signal in existing, unshielded magnetometer data recorded by geographically dispersed, geomagnetic stations. The dataset comes from the SuperMAG Collaboration and consists of measurements taken with one-minute cadence since 1970, with $\mathcal{O}(500)$ stations contributing in all. We aggregate the magnetic field measurements from all stations by projecting them onto a small set of global vector spherical harmonics (VSH) that capture the expected vectorial pattern of the signal at each station. Within each dark-photon coherence time, we use a data-driven technique to estimate the broadband background noise in the data, and search for excess narrow-band power in this set of VSH components; we stack the searches in distinct coherence times incoherently. Following a Bayesian analysis approach that allows us to account for the stochastic nature of the dark-photon dark-matter field, we set exclusion bounds on the kinetic-mixing parameter in the dark-photon dark-matter mass range $2\times {10}^{-18}\mathrm{eV}\lesssim m_{A^{\prime }}\lesssim 7\times {10}^{-17}\mathrm{eV}$ (corresponding to frequencies $6\times {10}^{-4}\mathrm{Hz}\lesssim f_{A^{\prime }}\lesssim 2\times {10}^{-2}\mathrm{Hz}$). These limits are complementary to various existing astrophysical constraints. Although our main analysis also identifies a number of candidate signals in the SuperMAG dataset, these appear to either fail or be in tension with various additional robustness checks we apply to those candidates. We report no robust and significant evidence for a dark-photon dark-matter signal in the SuperMAG dataset.

Figure 1

Locations of SuperMAG geomagnetic observatories whose data are included in the analysis in this paper [43, 44, 45]. World map (equirectangular projection) created using cartopy [62].

Figure 2

Number of stations in the SuperMAG dataset [43, 44, 45] reporting as a function of the date. We discuss the manifest annual trends in Sec. 3d.

Figure 3

Graphical representation of the scheme used to approximate $T_{\mathrm{coh}}(f)$ as outlined in Sec. 5e. The solid black line shows the values of $T_n$ employed in the analysis, as a function of the frequency. The solid red line shows the approximate coherence time $(f{v}_{\mathrm{DM}}^2{)}^{-1}$ for the DPDM signal, while the red shaded band gives a 3% tolerance around this approximate value; note importantly that for all $f$ such that $(f{v}_{\mathrm{DM}}^2{)}^{-1}<T_{\mathrm{tot}}$, $T_n$ lies within this tolerance of $(f{v}_{\mathrm{DM}}^2{)}^{-1}$. The dotted green line shows the total duration of the dataset: Note that once $(f{v}_{\mathrm{DM}}^2{)}^{-1}>T_{\mathrm{tot}}$, $T_n=T_{\mathrm{tot}}$ is assumed (i.e., the data are all analyzed in a single coherent block).

Figure 4

Exclusion bounds on the kinetic-mixing parameter $ϵ$ of the dark-photon dark matter as a function of the dark-matter mass $m_{A^{\prime }}$ (frequency $f_{A^{\prime }}$). The darker blue line (appearing as a band owing to frequency-to-frequency limit fluctuations) shows our 95% credible upper limit (local significance) [cf. Eqs. (64) and (70)] on the kinetic-mixing parameter $ϵ$ as a function of the dark-photon dark-matter mass (corresponding Compton frequency noted on upper axis), assuming that the dark photon constitutes all of the local dark-matter density, ${\rho }_{\mathrm{DM}}=0.3{\mathrm{GeV}/\mathrm{cm}}^3$, but taking into account the stochastic variations expected for classical-field dark matter (see, e.g., Refs. [66, 67, 69, 70, 72]). These limits include the effect of the 25% degradation factor discussed in Sec. 5f. To guide the eye and give a sense of the relative density of stronger vs weaker limits in narrow frequency bands, we also show as the lighter blue solid line the sliding average of the limit taken over the neighboring $\pm 25000$ frequencies. Various sharply rising narrow spikes in our limits provide a variety of potential candidate signals; we examine these in detail in Sec. 6, where we conclude that none constitute robust evidence for a real signal. The various other lines show a variety of existing astrophysical limits arising from dark-photon dark-matter heating of gas in a number of astrophysical environments: the ionized interstellar medium in the Milky Way (dotted orange) [46], the intergalactic medium around helium reionization (short-dashed red, labeled “${\mathrm{He}}^{++}$”) [48], and gas in the Leo T dwarf galaxy (dot-dashed purple) [49]. A DM-depletion limit from nonresonant dark-photon–photon conversion [48] is also shown (long-dashed green, labeled “$\mathrm{\Delta }{\rho }_{\mathrm{cdm}}$”). Our limits are complementary to these existing bounds as they arise from terrestrial experimental data (analogous to “direct detection”), and are thus subject to completely different sources of systematic uncertainty as compared to the other bounds shown (which are analogous to “indirect detection”).

Figure 5

Left panel: the (local) $p_0$-values (blue) for every frequency bin analyzed in our range of interest, computed per Eq. (73). The threshold value for declaring a naive candidate signal at 95% confidence $p_{\mathrm{crit}}\approx 1.6\times {10}^{-8}$ [see Eq. (74)] is shown by the horizontal dashed orange line; this threshold takes into account a trials factor (i.e., the significance is global). We identify 30 naive signal candidates in the range $6\times {10}^{-4}\mathrm{Hz}<f_{A^{\prime }}<[(1\min {)}^{-1}-6\times {10}^{-4}]\mathrm{Hz}$ (see text). We investigate these naive signal candidates in Sec. 6b. Right panel: histogram of all of the $p_0$-values that are shown in the left panel, showing the expected smoothly falling distribution with the identified signal candidates as clear outliers above the threshold $p_{\mathrm{crit}}$, which is shown by the vertical dashed orange line.

Figure 6

The results presented in red as exclusion bounds on the kinetic-mixing parameter $ϵ$ as a function of the dark-photon mass $m_{A^{\prime }}$ of our analysis pipeline as applied to mock-signal dataset consisting of an injected signal added to the SuperMAG data, as described in Sec. 6c. Overlaid in blue, and mostly obscuring the mock-signal exclusion bounds, are the exclusion bounds obtained using the unadulterated SuperMAG dataset; see Fig. 4. In both cases, these limits account for the degradation factor $\zeta =1.25$ discussed in Sec. 5f. The presence of strong peaks in the exclusion bounds from the mock-signal dataset at the injected signal frequency $f_{*}=7.5\times {10}^{-3}\mathrm{Hz}$, and at its reflection across the Nyquist frequency $(1\min {)}^{-1}-f_{*}=9.2\times {10}^{-3}\mathrm{Hz}$ (both indicated by the vertical dashed green lines) show that the analysis reconstructs the injected signal at the appropriate frequencies, but that the exclusion bounds are otherwise largely unaffected away from the vicinity of these peaks. The inset axes show an enlarged view of the peak at $f=f_{*}$, with the gray shaded region corresponding to $f_{*}\pm 2{\sigma }_f$; the width of the region of the weakened exclusion bounds near $f=f_{*}$ is clearly consistent with our choice of ${\sigma }_f={10}^{-6}{f}_{*}\sim \mathrm{\Delta }f$. Moreover, the exclusion bounds at the frequencies where these strong peaks appear are close to the injected signal size ${ϵ}_{*}={10}^{-3}$ (indicated by the horizontal dashed green line), as expected.

Figure 7

Shaded contour plots of the real and imaginary parts of all the nonzero $\widehat{\mathit{\theta }}$- and $\widehat{\mathit{\varphi }}$-components of the vector spherical harmonics ${\mathbf{\Phi }}_{11}$ and ${\mathbf{\Phi }}_{10}$; the cognate plots for ${\mathbf{\Phi }}_{1,-1}$ can be read from those of ${\mathbf{\Phi }}_{11}$ using Eq. (b4). Red (blue) indicates positive (negative) values, with the color range for each plot independently normalized to span the range of values plotted. Overlaid are the outlines of Earth’s continents (white), and the locations of the SuperMAG stations (green points); see also Fig. 1.

Figure 8

Noise stationarity validation; on-diagonal elements of $S_{mn}^a(f)$. Upper panels: the solid black line shows representative components of the full-year averaged [see Eq. (e1)] noise autopower spectra ${\overline{S}}_{mn}^a$ ($m=n$) for some selected representative years $a$, while the various shaded colored bands give the values of ${\overline{S}}_{mn}^a(f)\pm {\sigma }_{mn}^a(f)$ that are computed using data only from one of each of the four quarters within that year $a$ (see legend). The left-hand side shows a case where there is some mild tension between the quarter-by-quarter noise determinations; the right-hand side shows a case where the four quarter-by-quarter determinations agree excellently. Lower panels: for three selected frequencies (vertical gray dashed lines marked in the upper panels), we show histograms (colored bars; see legend) of the values of $S_{mn}^a(f_q)$ for $f_q$ falling within the averaging window used to determine the quarter-by-quarter values of ${\overline{S}}_{mn}^a(f)$ [see discussion around Eq. (e1)], along with the full-year average ${\overline{S}}_{mn}^a(f)$ (vertical black line). Each histogram panel is displayed immediately below the relevant vertical gray dashed line in the upper panel which marks the frequency to which it corresponds (i.e., in order from left to right, the histograms correspond to the same three frequencies marked, in order from left to right, by the vertical gray lines in the upper panel).

Figure 9

Noise stationarity validation; off-diagonal elements of $S_{mn}^a(f)$. Because the off-diagonal components of the Hermitian matrices $S_{mn}^a(f)$ are complex, we cannot show the quarter-by-quarter agreement as a function of the frequency as simply for the off-diagonal elements as we did in Fig. 8 for the real, on-diagonal components. In this figure, at three selected representative frequencies (the same ones indicated by the vertical gray dashed lines in Fig. 8), we select some representative off-diagonal components $(m,n)$ for some representative years $a$, and show scatter plots in the complex plane of the values of $S_{mn}^a(f_q)$ for the $f_q$ that lie within the corresponding averaging windows used to determine the quarterly values of ${\overline{S}}_{mn}^a(f)$ [see discussion around Eq. (e1)] (shaded colored circular points; see legend). Note that the density of points, and not the depth of shading, indicates the clustering of values within each quarter (with some loss of resolution in the denser regions); we have fixed the depth of shading for each quarter to be independent of the density of points in order to make the differences in the clustering of points from quarter to quarter clearer. Also shown are the 68% coverage ellipses for two-dimensional Gaussian fits to the scattered points for each quarter (like-colored solid ellipses; see legend), along with the full-year average value of ${\overline{S}}_{mn}^a(f)$ (black cross). The left-hand column shows a case where there is some mild tension (within factors of $\sim 2–3$) between the intrayear statistical stationary of the noise assumed in our analysis, and the realized noise (i.e., the fitted 68% coverage ellipses for different quarters vary somewhat), while the right-hand column shows a case where intrayear noise stationarity is realized well.

Figure 10

Dependence of noise estimates on the parameter ${\tau }_{\min }$. Left column: frequency-space averaged noise spectra $|{\overline{S}}_{mn}^a(f)|$ for ${\tau }_{\min }=2^j\min$ for choices $j=11,\dots ,15$ shown for multiple representative choices of year $a$ and components $(m,n)$. The colors as annotated in the legend on the lower left distinguish the various cases. Note that the sliding window used to compute the frequency-space average is varied ($w=2^{j-5}$ for the integers $j$ defined above), so that the sliding window width used to compute the frequency-space averages is maintained at $5.2\times {10}^{-4}\mathrm{Hz}$ in all cases. Note also that, in this plot, where relevant and in contrast to Fig. 9, we show the absolute value of the complex off-diagonal ${\overline{S}}_{mn}^a(f)$. Right column: dependence of $|{\overline{S}}_{mn}^a(f)|$ on ${\tau }_{\min }$ for three representative fixed frequencies: $f=1.04\times {10}^{-3}\mathrm{Hz}$, $f=2.95\times {10}^{-3}\mathrm{Hz}$, and $f=8.33\times {10}^{-3}\mathrm{Hz}$ (the locations of these frequencies are also indicated as dashed vertical gray lines in the left panels). The gray bands represent the spread in the values in the sliding windows, ${\sigma }_{mn}^a(f)$; see Eq. (e2). It is clear that, in most cases, for ${\tau }_{\min }\ge 16384\min$, the dependence of $|{\overline{S}}_{mn}^a(f)|$ on ${\tau }_{\min }$ is much smaller than the spread ${\sigma }_{mn}^a(f)$. Note that the top row shows an anomalous off-diagonal component from an early year ($a=1976$), whose behavior is likely strongly influenced by the small number of active stations. Additionally, the panels in the third row show a result for an off-diagonal component [$(m,n)=(3,1)$], which is somewhat smaller in normalization than the other on-diagonal components from the same year; residual variation of these results with ${\tau }_{\min }$ will thus have little effect on our results.

Figure 11

Validation of Gaussianity. The (lower) blue line shows $⟨|z_{ik}{|}^2⟩$ and the (upper) orange line shows the ratio $⟨|z_{ik}{|}^4⟩/(2⟨|z_{ik}{|}^2{⟩}^2)$, both as a function of the frequency. Both of these quantities are expected to be equal to 1 (horizontal gray line) per our analysis assumptions; cf. Eq. (60). This is realized (note that the vertical axis covers only a small range of values) for $⟨|z_{ik}{|}^2⟩$ to within $\sim 8\%$ and for $⟨|z_{ik}{|}^4⟩/(2⟨|z_{ik}{|}^2{⟩}^2)$ to within $\sim 2\%$ over the entire frequency range of interest.