Dark matter interferometry

The next generation of ultralight dark matter (DM) direct detection experiments, which could confirm sub-eV bosons as the dominant source of DM, will feature multiple detectors operating at various terrestrial locations. As a result of the wavelike nature of ultralight DM, spatially separated detectors will each measure a unique DM phase. When the separation between experiments is comparable to the DM coherence length, the spatially varying phase contains information beyond that which is accessible at a single detector. We introduce a formalism to extract this information, which performs interferometry directly on the DM wave. In particular, we develop a likelihood-based framework that combines data from multiple experiments to constrain directional information about the DM phase space distribution. We show that the signal in multiple detectors is subject to a daily modulation effect unique to wavelike DM. Leveraging daily modulation, we illustrate that within days of an initial discovery multiple detectors acting in unison could localize directional parameters of the DM velocity distribution such as the direction of the solar velocity to subdegree accuracy, or the direction of a putative cold DM stream to the subarcminute level. We outline how to optimize the locations of multiple detectors with either resonant cavity (such as ADMX or HAYSTAC) or quasistatic (such as ABRACADABRA or DM-Radio) readouts to have maximal sensitivity to the full three-dimensional DM velocity distribution.

Figure 1

The imprint of DM interferometry. A single wavelike DM experiment is sensitive to the DM speed distribution $f(v)$. Two detectors separated by a vector ${\mathbf{x}}_{12}$, however, are sensitive to the speed distribution modulated by the $\mathbf{k}·{\mathbf{x}}_{12}$ phase of the DM wave, replacing $f(v)$ with functions ${\mathcal{F}}_{12}^{c,s}(v)$ as defined in (3). As the figures demonstrate, the modified speed distributions exhibit daily modulation and carry additional information about the velocity distribution $f(\mathbf{v})$ that would be invisible to a single detector. For this example we take $m_{\mathrm{DM}}=25.2\mu \mathrm{eV}$ [49], near the window where the HAYSTAC collaboration is searching for axion DM. Taking the Standard Halo Model ansatz for $f(\mathbf{v})$ in (50), we place one detector at a latitude and longitude of (41° N, 73° W), and a second instrument $\sim 20\mathrm{m}$ to the north, corresponding to $d\sim 2{\lambda }_c$. A curve is shown for every ten minutes starting from midnight on January 1, 2020. Note that as ${\mathcal{F}}_{12}^{c,s}(v)$ are functions of $m_{\mathrm{DM}}d$, qualitatively similar effects exist for, e.g., $m_{\mathrm{DM}}\sim {10}^{-9}\mathrm{eV}$, in the mass range probed by ABRACADABRA and DM-Radio, for $d\sim 500\mathrm{km}$.

Figure 2

The modified speed distribution, ${\mathcal{F}}_{12}^c(v)$, that carries the imprint of DM interferometry. Here we show the particularly simple example of an isotropic SHM for $\mathcal{N}=2$ detectors, in which case the expression is given in (40). The result is shown for various choices of the two detector separation $d$ as compared to the axion coherence length ${\lambda }_c=(m_a{v}_0{)}^{-1}$, with $v_0=220\mathrm{km}/\mathrm{s}$. The limiting cases of ${\mathcal{F}}_{12}^c(v)\to f(v)$ for $d\ll {\lambda }_c$ and ${\mathcal{F}}_{12}^c(v)\to 0$ for $d\gg {\lambda }_c$ are apparent. For $d\sim {\lambda }_c$, however, the profile is modulated with the interference inherent in the cross spectrum. In this simple case, there is no additional information about the velocity distribution that may be extracted by having multiple detectors.

Figure 3

The expected uncertainty on the angle between the detector axis and solar velocity, ${\theta }_{\odot }=\arccos ({\widehat{\mathbf{v}}}_{\odot }·{\widehat{\mathbf{x}}}_{12})$, as a function of $d/{\lambda }_c=d\times m_a{v}_0$. In this example we have set the true orientation to ${\theta }_{\odot }^t=\pi /4$. With this configuration, we find that the maximum precision is obtained for $d\approx 2{\lambda }_c$.

Figure 4

Left: a Mollweide projection of the Asimov test statistic $\tilde{\mathrm{\Theta }}(\theta ,\varphi )$ for the SHM divided by the colocated detection significance ${\mathrm{TS}}_0$. The detectors are configured so that the displacement vector between them is parallel to the SHM boost velocity, and the Mollweide plot is rotated so that it is centered on the maximum test statistic. Center: as on the left, but for a detector configuration where the displacement vector is at a 45° angle to the north (${\theta }_{\odot }^t=\pi /4$) with respect to the SHM boost velocity. Right: as on the left, but for a detector configuration where the displacement vector is perpendicular to the SHM boost velocity (${\theta }_{\odot }^t=\pi /2$). In this configuration the location of the boost velocity can only be localized to a great circle on the celestial sphere.

Figure 5

Left: as in Fig. 3, but for the Sagittarius (SGR) stream rather than for the SHM. As before, the maximum precision for the inferred value of ${\theta }_{\mathrm{str}}$ is achieved at $m_a{v}_{0}d\approx 2$, although the overall dependence is somewhat softened outside of the extremes at $m_a{v}_{0}d=0$ and $m_a{v}_{0}d=2\pi$. The values of ${\sigma }_{{\theta }_{\mathrm{str}}}\times \sqrt{{\mathrm{TS}}_0}$ are also considerably smaller than those found in the SHM example, indicating that the angle ${\theta }_{\mathrm{str}}$ can be reconstructed with much greater precision for the SGR stream as compared to the SHM. Right: the Asimov TS $\tilde{\mathrm{\Theta }}({\theta }_{\mathrm{str}})$ for the SGR stream rescaled by the colocated detection significance ${\mathrm{TS}}_0$ as a function of ${\theta }_{\mathrm{str}}$ for a detector configuration where the true stream direction is ${\theta }_{\mathrm{str}}^t=\pi /4$ (dashed vertical line). We have fixed $m_a{v}_{0}d=2$. The TS $\tilde{\mathrm{\Theta }}({\theta }_{\mathrm{str}})$ is maximized at the true value of ${\theta }_{\mathrm{str}}$, but there is considerable nontrivial global structure with a large number of local minima and maxima in $\tilde{\mathrm{\Theta }}$.

Figure 6

As in Fig. 4, we construct Mollweide projections of the Asimov test statistic $\tilde{\mathrm{\Theta }}(\theta ,\varphi )$ for the SHM rescaled by the colocated detection significance ${\mathrm{TS}}_0$. However, we now perform a joint likelihood over data collected over a 24-hour period so that the daily modulation of the detector displacement vector produces a time-varying signal, which helps break degeneracies in the reconstructed directional parameters. The Mollweide projection for a configuration in which the detectors are oriented along an east-west (north-south) orientation is shown on the left (right). While the results obtained in an east-west configuration do not depend on the latitude of the detectors, the north-south configuration results do, so for definiteness, we have taken the detectors to be located in New Haven, CT, the site of the HAYSTAC detector. In both configurations, the SHM boost velocity direction can be localized effectively, although there remains a nontrivial degeneracy in the east-west map between two points on the sphere.

Figure 7

As in Fig. 6, but for the Sagittarius Stream example. For a fixed axion mass, the physical detector separation $d=2{\lambda }_c$ is a factor of 20 larger than in Fig. 6 because of the larger coherence length of the stream. While there are many local maxima in both configurations, the north-south orientation produces only a single global maximum, at the true detector localization, while the east-west orientation leads to two degenerate global maxima (one at the true detector location and the other displaced). An animated version of these figures, showing how the localization improves throughout the day as more orientations of ${\mathbf{x}}_{12}$ are sampled, can be found at github.com/joshwfoster/DM_Interferometry.

Figure 8

The posterior distribution for a model with daily modulation where the signal strength is at the threshold of an expected $5\sigma$ detection for a 100 second observation with a single detector. Monte Carlo data are generated for 24 hours of data collection with two detectors separated along the north-south direction by a distance with $2\times (m_a{v}_0{)}^{-1}$. The true parameters are indicated in blue, with the $1\sigma$ confidence intervals on the parameter estimations are indicated by the dashed black lines in the single-parameter posteriors. The two parameter posteriors show the $1\sigma$ and $2\sigma$ contours. On the left, we display the posterior distributions for the overall signal strength, the boost speed of the SHM, and the velocity dispersion of the SHM, all of which are parameters accessible in a single detector configuration. On the right, we display the posterior distributions for the angles $\mathrm{\Delta }{\theta }_{\odot }={\theta }_{\odot }-{\theta }_{\odot }^t$ and $\mathrm{\Delta }{\varphi }_{\odot }={\varphi }_{\odot }-{\varphi }_{\odot }^t$, which specify the orientation of SHM boost velocity and are only accessible in a multiple-detector configuration. Both ${\theta }_{\odot }$ and ${\varphi }_{\odot }$ are determined at degree precision in this scenario.

Figure 9

Monte Carlo validation that the statistics of DM interferometry are as claimed in Appendix pp2. In the left figure we confirm that the variances of the real and imaginary signal-only datasets, collected for the $\mathcal{N}=2$ experiments, is as claimed in (16). This was proven directly in the text, but in the plot we show that the average of 4000 Monte Carlo simulations provides a consistent prediction for the variances as a function of frequency in the different cases. On the right figure, for the frequency where $⟨R^{(1)}{R}^{(1)}⟩$ achieves its maximum, we show the distribution of values across the simulations. In detail, we see that the real and imaginary components are normally distributed, and consistent with a mean-zero normal distribution, where the variance is given as on the left, here ${\sigma }^2\approx 25{\mathrm{Wb}}^2/\mathrm{Hz}$. We found that the distributions were consistent with the Gaussian expectation at the level of $p>.05$ using the D’Agostino and Pearson omnibus normality test [73, 74]. In both cases, each Monte Carlo simulation involves a direct construction of the axion field starting from (b15) with $N_a=100$,000, taking $m_a=2\pi \mathrm{Hz}$, and $A=1{\mathrm{Wb}}^2$. Further, we take the velocity distribution to follow a variant of the SHM in (50), but with $v_0=0.07$ and ${\mathbf{v}}_{\odot }=(0,0.08,0)$, both in natural units. The (unphysically) large velocity helps simplify the computation of the Fourier transform. The detector separation is ${\mathbf{x}}_{12}=d(0,1,0)$, with $d\approx 4.4{\lambda }_c$.