Search for transient ultralight dark matter signatures with networks of precision measurement devices using a Bayesian statistics method

We analyze the prospects of employing a distributed global network of precision measurement devices as a dark matter and exotic physics observatory. In particular, we consider the atomic clocks of the global positioning system (GPS), consisting of a constellation of 32 medium-Earth orbit satellites equipped with either Cs or Rb microwave clocks and a number of Earth-based receiver stations, some of which employ highly-stable H-maser atomic clocks. High-accuracy timing data is available for almost two decades. By analyzing the satellite and terrestrial atomic clock data, it is possible to search for transient signatures of exotic physics, such as “clumpy” dark matter and dark energy, effectively transforming the GPS constellation into a 50 000 km aperture sensor array. Here we characterize the noise of the GPS satellite atomic clocks, describe the search method based on Bayesian statistics, and test the method using simulated clock data. We present the projected discovery reach using our method, and demonstrate that it can surpass the existing constrains by several order of magnitude for certain models. Our method is not limited in scope to GPS or atomic clock networks, and can also be applied to other networks of precision measurement devices.

Figure 1

Geometry of a domain wall encounter with the Earth. Here, $d$ is the domain wall width, $\mathit{v}$ is the relative velocity of the encounter with component ${\mathit{v}}_{\perp }$ perpendicular to the wall surface, and $\eta$ is the angle between $\mathit{v}$ and ${\mathit{v}}_{\perp }$. The incident direction of the wall, $\widehat{\mathit{n}}$, is defined as pointing away from the Earth center, so that $\widehat{\mathit{n}}=-\mathit{v}/|\mathit{v}|$ and ${\widehat{\mathit{n}}}_{\perp }\equiv -{\mathit{v}}_{\perp }/|{\mathit{v}}_{\perp }|$.

Figure 2

Probability densities for the DM scalar velocity (left), incident angles (middle), and transit time for the GPS constellation (right). The forward-facing angle $\psi =\pi$ points in the direction of the galactic motion of the Solar system, towards the Cygnus constellation. The blue curves show the standard halo model distributions (relevant for monopole-like DM objects), and the red curves show the distributions for velocities perpendicular to the wall (relevant for domain walls and strings).

Figure 3

Example geometry for a monopole crossing.

Figure 4

Annual variation in the direction of the Earth’s galactic motion (ECI frame, $\theta \in [0,\pi ]$ is the polar angle), which is the most probable incident DM direction. The central point, ${\widehat{\mathit{n}}}_g$, is the average direction, corresponding to the direction of the Sun’s velocity through the galaxy.

Figure 5

Comparison of (polynomial-reduced) real GPS clock data for a few satellites from 21 June 2015 UTC (left) with simulated data for the corresponding SVNs (right). Each time-series is shifted by a constant offset for clarity. The curve labels encode the clock type, GPS block, and SVN.

Figure 6

Comparison of the autocorrelation functions (left) and Allan variances (right) for the real data to those for the simulated data. Clocks are the same as in Fig. 5. The solid lines are from the real data, and the dotted lines are from the simulated data; they are practically indistinguishable.

Figure 7

Rate of statistical false positives as a function of the odds ratio threshold, $O_{\text{thresh}}$, for thin domain walls. The rate of false positives from simulated GPS networks typical for the given years: 2000 (1 Rb-II, 7 Rb-IIA, 3 Rb-IIR, 5 Cs-II, 11 Cs-IIA), 2005 (1 Rb-II, 8 Rb-IIA, 12 Rb-IIR, 1 Cs-II, 8 Cs-IIA), 2010 (5 Rb-IIA, 19 Rb-IIR, 2 Rb-IIF, 5 Cs-IIA, 2 Earth-based H-masers), 2017 (19 Rb-IIR, 10 Rb-IIF, 5 Earth-based H-masers), and a possible future network (30 Rb-IIF–style satellites, 20 Earth-based H-masers). Each curve corresponds to 4 years of 30 s sampled simulated data.

Figure 8

Bayesian detection of an injected thick domain wall ($d={10}^4\mathrm{km}$) signal. The wall sweeps the GPS network of 32 satellite clocks (with $\sigma =0.01\mathrm{ns}$) at time $t_0=0$. For this simulation, $h=0.02\mathrm{ns}$. Bottom panel: simulated clock biases shown for the first 8 clocks (including the injected thin-wall signal). Each time-series is shifted by a constant offset for clarity. Top panel: the corresponding odds ratio using the same time scale.

Figure 9

Efficacy of the method for detecting injected thin-wall DM signals, for the same simulated networks as in Fig. 7. Each point represents 128 trials, each curve has $\sim 60$ points. Top panel shows the fraction of injected thin-wall events that were correctly identified, as a function of the event magnitude $h$. The odds threshold was set to allow fewer than 10 false positives per year (see Fig. 7). Bottom panel shows the average log-odds ratio as a function of $h$ on the same scale.

Figure 10

Monte-Carlo simulations for thin domain walls, using a network of pure white-noise (in $d^{(1)}$) devices. The green, red, and blue curves are for a network of 20, 30, and 50 identical devices, respectively. Top panel shows the fraction of events that were correctly identified, as a function of the injected event magnitude $h$ (scaled by $\sigma$, the standard deviation of the data noise). This is done requiring an odds threshold such that there are fewer than 10 false positives per year (solid lines), or 1 false positive per day (dotted lines). Bottom left panel shows the average log-odds ratio as a function of $h/\sigma$ on the same scale. Bottom right panel shows the yearly rate of false positives as a function of the odds threshold, $O_{\text{thresh}}$.

Figure 11

Example normalized histograms for the difference between the injected event parameters and the best-fit values extracted from the Bayesian analysis. Results for 2048 randomized simulations of a 25 satellite clock homogeneous network. Each trial has a single $\sim 1\sigma$ thin wall event injected with $v\simeq 300\mathrm{km}{\mathrm{s}}^{-1}$. Left: for the incident arrival time, $t_0$, middle: for the scalar speed, $v$, and right: for the incident polar angle, $\theta$. We have resolution of better than $\sim \pm 0.1\pi$ radians for the incident angle, and $\sim \pm 10\mathrm{s}$ for the incident time (note that this is with 30 s sampled data).

Figure 12

Normalized histograms for the distribution of the best-fit values extracted from the false positives of the Bayesian analysis, left: for the scalar speed $v$, middle: for the polar angle $\theta$, and right: the azimuthal angle $\varphi$. (The hump in the $\theta$ histogram is due to the solid angle volume element $\sin \theta$.) Here, a low threshold ($O_{\text{thresh}}=10$) was chosen to increase the statistics; when increasing $O_{\text{thresh}}$, the shape of the histograms remains constant (it is prohibitively computationally intensive to run enough simulations to form false positive histograms for large $O_{\text{thresh}}$, see Fig. 7). For these simulations, the priors were excluded (i.e., flat priors were assumed).

Figure 13

Projected discovery reach for topological defect dark matter, along with existing constraints for comparison. The red shaded region are the limits (on domain walls) from the initial GPS.DM search using the Rb GPS network [6], the shaded orange regions are limits set by optical Sr clock [63] and from astrophysics observations [50]; these apply for walls, strings, and monopoles. The curves represent the projected sensitivities for our method, with the red, green, and blue colors for domain walls, strings, and monopolelike dark matter, respectively. For monopoles and strings, we require that at least 3 clocks are affected in the DM crossing, which causes the sharp cutoff for low $d$. The solid lines are the projections for the global network of GPS microwave clocks, and the dashed lines are the reach for the case when a single optical clock can be incorporated into the analysis. The sensitivity is slightly lower for large $\mathcal{T}$, since we rely on the older GPS clocks.

Figure 14

Averaged autocorrelation functions for first-order (top) and second-order (bottom) differenced clock data.

Figure 15

Allan variance (2) for each GPS satellite block, averaged over all available SVNs. See also the Supplemental Material [31].

Figure 16

The averaged power spectral densities (for $d^{(1)}$) for the various clock/satellite block combinations. Note that this includes noise from the H-maser reference clock. See also the Supplemental Material [31].

Figure 17

Geometry of a monopole object crossing the GPS constellation. The monopole (labeled $\chi$) enters along unit vector $\widehat{\mathit{n}}$ (which points into the page), at perpendicular distance $R$ from ECI0 (the Earth center), and makes an angle $\alpha$ with respect to the ${\widehat{z}}^{\prime }$-axis in the plane perpendicular to $\widehat{\mathit{n}}$; ${\rho }^a$ is the impact parameter for satellite $a$.