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Precision measurement noise asymmetry and its annual modulation as a dark matter signature

Dark matter may be composed of ultralight quantum fields that form macroscopic objects. As the Earth moves through the galaxy, interactions with such objects may leave transient signatures in terrestrial experiments. These signatures may be sought by analyzing correlations between multiple devices in a distributed network. However, if the objects are small (<~10^3 km) it becomes unlikely that more than one device will be affected in a given event. Such models may, however, induce an observable asymmetry in the noise distributions of precision measurement devices, such as atomic clocks. Further, an annual modulation in this asymmetry is expected. Such an analysis may be performed very simply using existing data, and would be sensitive to models with a high event rate, even if individual events cannot be resolved. For certain models, our technique extends the discovery reach beyond that of existing experiments by many orders of magnitude.

Introduction.-Despite composing the majority of matter in the universe, the nature and composition of dark matter (DM) remain a mystery. Most of the particle physics experiments so far have focused on weaklyinteracting massive particles (WIMPs) with ∼ GeV -TeV masses. Despite the extensive effort, there is no solid evidence for WIMPs in such experiments [1,2]. Besides WIMPs, there are a multitude of other DM candidates with masses that span many orders of magnitude. Here, we consider ultralight boson fields with masses 1 eV. If the fields have sufficient self-interactions, they may form macroscopic objects, such as topological defects [3,4], Qballs [5], or axion stars [6]. Encounters with such objects may leave transient signals in measurement device data.
A general challenge with searching for transient signals is that they are difficult to distinguish from conventional noise. One possibility, as per Refs. [7][8][9], is to use a network of devices, and search for the correlated propagation of transients through the network. However, objects of spatial extent smaller than the node separation would not produce such a signature. Then one has to rely on unique signatures of the interactions with a single device that may differentiate them from the conventional noise. Gravitational wave searches, for example, make use of both correlated signal propagation across a network and a distinct signal frequency pattern [10].
Here, we present a method to search for transient DM signatures when the number of DM encounters in the observation time is high. If DM interacts with standard model particles, recurring encounters may cause perturbations in precision measurement devices. The magnitude of these perturbations will depend on the geometry of individual collisions. However, in some models, the sign will remain the same. If this effect were to lead only to a shift in the mean of the data it would be undetectable, as DM is always present. Such interactions may, however, cause an asymmetry in the data noise distribution, which is observable. Further, we show that there would be an appreciable annual modulation in this asymmetry, arising due to the Earth's solar orbit through the galactic DM halo. The period, phase, and amplitude of the modulation serve as unique DM signatures [11]. This approach has been used in WIMP searches, with a positive result claimed by the DAMA Collaboration [12] (see also [13,14]).
Following these ideas, one may perform DM searches that are many orders of magnitude more sensitive than the existing constraints for certain models. The range of parameter space that can be probed is complimentary to that of other ultralight DM searches [15][16][17][18][19]. The technique proves particularly useful for the case of small or low density objects, where the expected event rate is high. Moreover, such searches may be carried out using existing data, making this an inexpensive avenue for potential discovery. Finally, we note that while we focus on atomic clocks, the ideas apply also to other precision measurement devices, such as magnetometers [8,20], interferometers [21], and dipole moment searches [22][23][24].
Expected dark matter signal.-We consider interactions that lead to transient shifts in atomic transition frequencies. Generally, this can be expressed in the form where ω c is the unperturbed clock frequency, Γ is the effective coupling constant, φ is the DM field, and n = 1, 2... For macroscopic DM objects, the field amplitude φ → A inside the object and φ = 0 outside [7], so that the frequency excursion is realized only when the clock and DM object overlap. The exact form of φ is model dependent, as discussed below. Here, we take n = 2. The frequency excursion (1) leads to an additive term in the clock phase s (0) → s (0) + χ (0) , with where the data is recorded for discrete values of time t j . Clock phase noise is dominated by random-walk processes. To form a stationary data set, we apply the firstorder differencing procedure and define These data are proportional to clock frequency excursions, and we refer to them as pseudo-frequencies.
Before introducing specific models, we remark on generic properties of macroscopic DM objects. We denote the characteristic size (width) of the objects as d, and define T to be the mean time between consecutive events. Along with Γ, these form the free parameters in our model. By "event" we mean a collision between a point-like measurement device and a DM object that occurs within a distance (impact parameter) of d. The energy density inside the object, ρ inside , is linked to T as where R 0 is the mean event rate, v 0 ≈ 300 km/s is the mean relative speed of the DM objects, and ρ gal is their total galactic energy density. In the assumption that these objects dominate the DM density, we have ρ gal = ρ DM ≈ 0.4 GeV cm −3 [25]. The event rate scales inversely with the field amplitude. Indeed, if the field amplitude is made larger, the number density of the objects (and hence the event rate) must become correspondingly smaller to avoid over-saturating the local DM density. Transient variation of fundamental constants.-Following Ref. [7], we now consider the specific case of DM objects with quadratic scalar couplings to SM particles and fields. Such interactions lead to the effective redefinition of fundamental masses and couplings [26]. Those relevant to atomic clocks are the fine structure constant α, the electron-proton mass ratio m e /m p , and the ratio of the light quark mass to the QCD energy scale m q /Λ QCD .
Naturally, the DM-induced transient variations in fundamental constants lead to transient shifts in atomic energy levels, and therefore transition frequencies: with X = α, m e /m p , m q /Λ QCD . Here, K X are sensitivity coefficients that quantify the response of the atomic transition to the variation in a given fundamental constant, and are known from atomic and nuclear calculations [27,28]. For consistency with previous literature, we also define the effective energy scales Λ X = 1/ |Γ X |. For topological defects, the internal energy density is is set by the Compton wavelength [7]. Combing with Eq. (4), and assuming the objects dominate the local DM density, leads to the expression for the field amplitude: Finally, we remark on the form of the field, φ, which we assume to have a Gaussian profile, where t 0 is the time of closest approach of the DM object to the measurement device, ρ is the impact parameter, and v is the relative velocity. All the above formulas are general and apply to any topological defects: monopoles, strings, and domain walls (for domain walls ρ = 0). DM induced asymmetry & skewness.
-We consider the case where there are many events during the observation time, as shown in Fig. 1. Not every event imparts the same signal magnitude, as the DM velocities and impact parameters differ. However, the sign of the perturbation remains the same, since it is set only by the sign of the coupling Γ in Eq. (1). This leads to an asymmetry in the observed data noise distribution. It may be possible to observe this asymmetry, even if individual events cannot be resolved or the perturbations are well below the noise.
The observed clock noise value at a given time will be s = n + χ if there was a DM interaction during the sampling interval, and s = n otherwise. Here, n is the conventional physics noise. If p χ is the distribution for DM signals (in the absence of noise), the observed probability distribution for clock excursions reads where p n is the intrinsic noise distribution, and τ 0 is the data sampling interval. For p n , we assume Gaussian noise with variance σ 2 . Formally, this is the assumption of white frequency noise, which is typically dominant for atomic clocks. For clocks, σ is related to the Allan deviation as σ ≈ τ 0 σ y (τ 0 ). While other noise processes exist in clocks, we assume that p n is symmetric. Even if p n is not symmetric, the annual modulation discussed below still presents an observable DM signature.
The skewness, defined as the third standard moment, is a measure of the asymmetry in the distribution for variable x. The uncertainty in the skewness is δκ 3 = 6/N , where N is the number of data points. For the DM-induced skewness, we have with means and variance σ 2 s calculated from p s . A kurtosis (fourth standard moment), κ 4 , is also induced.
To compute the expected DM-induced skewness, we must know the distribution of DM signals, p χ . The magnitude of each DM signal depends on v and ρ. We take the velocity distribution, f v , to be that of the standard halo model (see, e.g., Ref. [11]). The impact parameter distribution comes from geometric arguments. For monopole-like (spherically-symmetric) objects it is 2ρ/d 2 , which is normalized by our definition of an event as a collision that occurs with ρ < d.
-For objects small enough such that they traverse the clock within one sampling interval, i.e., d vτ 0 , the expected DM signal per event contributes to just a single data point. For Gaussianprofile monopoles, the magnitude is where χ 0 ≈ ΓA 2 d/v 0 is the most probable DM signal. Without loss of generality, we take χ 0 > 0 from here on. The resulting DM signal distribution is for χ > 0, and p χ = 0 otherwise, where y ≡ v 0 χ 0 √ π/χ, and e is the base of the natural logarithm. To find the resultant p s distribution, p χ is convolved with the p n distribution in Eq. (8). This can be done numerically, however, in order to extract analytic results we make an approximation, and confirm its adequacy numerically [29].
As an approximation, we form a "uniform ball" model, where instead of Eq. (7) we take φ 2 = A 2 for r < d and φ 2 = 0 otherwise (r is the distance from the center of the ball). For uniform balls with d < vτ 0 , the DM signal is χ = χ 0 v0 v 1 − ρ 2 /d 2 , for ρ < d, and χ = 0 otherwise; χ 0 is the same as in Eq. (11). Further, instead of f v , we simply assume v = v 0 . Then, the DM signal distribution is p χ (χ) = 2χ/χ 2 0 . This approximation underestimates the asymmetry, due to the missing low-v contributions. Therefore, results from this model should be considered conservative. The DM-induced skewness can be found analytically (to leading order in R 0 τ 0 ): Requiring that κ 3 > δκ 3 , and noting that N = T obs /τ 0 , where T obs is the total observation time, implies the smallest detectable signal satisfies  [29]. The skewness is calculated for each week of data (purple squares, with 6/N error bars). The extracted modulation amplitude is κ (19). The solid blue curve is the best-fit cosine, and the dotted lines are the uncertainties. The dashed red curve shows the mean κ3.
Therefore, in terms of the effective energy scale for the coupling leading to variation in the constant X, this technique should have sensitivity to This result depends only weakly on the event rate R 0 = 1/T . This is due to the assumption that the objects dominate the local DM density, see the discussion following Eq. (4). There is no sensitivity for T > T obs . Annual modulation.-As the Earth orbits the Sun, there is an annual change in the addition of their respective velocities in the galactic frame. This causes an annual modulation in the mean DM event rate through the modulation in the relative velocity of the DM objects. We may express the event rate as where ω = 2π/yr, ϕ is the phase with ωt + ϕ = 0 on June 2nd (when the Earth and Sun velocities add maximally), and ∆v/v 0 ≈ 0.05 [11]. Then, the skewness is expressed as a function of time: Note that the modulation is out of phase with that expected in WIMP searches [11]. We demonstrate this using simulated data in Fig. 2. The DM signal magnitude scales inversely with velocity, simply because slower DM objects interact with the device for a longer period. The DM-induced skewness scales linearly with the rate, and as the cube of the mean signal magnitude. Therefore, the modulation amplitude for the skewness is  [30], the Rb GPS atomic clock data (applies for domain walls only) [18], and astrophysics constraints [26]. The red curve shows the potential sensitivity for 1 year of data from an optical Sr clock, where the solid line is for the skewness (13), and the dashed line is for the annual modulation (18). The dotted black curve shows the potential reach for the Rb GPS network using the statistical asymmetry (13).
There is also a ∼ 15% modulation in κ 4 .
If the data is divided into M time bins, each consisting of N M = N/M points, with the skewness calculated for each bin, the modulation amplitude can be extracted as where κ 3 is the Fourier transform of κ 3 (t). The uncertainty, δκ 3 . This implies that we require signals with combination R 0 χ 3 0 that are larger by a factor v/∆v ≈ 20 compared to the result for the mean skewness (14). Or, for a fixed value of R 0 , signals that are ∼ 3 times larger. Nevertheless, it is important that there are signatures unique to DM (namely, the modulation phase, period, and amplitude) that can be sought in such experiments. If a skewness is present in the data, one may exclude DM origins if the modulation is absent.
Results & discussion.-In Figs. 3, 4, and 5 we show the estimated sensitivity of our technique to topological defect DM with couplings leading to variation in the fine structure constant α, the quark mass ratio m q /Λ QCD , and the electron to proton mass ratio m e /m p , respectively. These respective couplings are quantified by the effective energy scales, Λ α , Λ q , and Λ ep . The existing constraints come from an optical Sr clock experiment [30], the analysis of the Rb GPS atomic clock data (which only applies to domain walls) [18], and astrophysics observations [26,31,32].  [18]. The red curve shows the sensitivity for a Rb/Cs hyperfine frequency comparison, as in Ref. [17]. The dotted black curve shows the sensitivity for the Rb GPS network.
For the Λ α coupling, the sensitivity is estimated for 1 year of data from an optical Sr clock, assuming a stability of σ ∼ 10 −16 s with τ 0 = 1 s. The sensitivity coefficient for an optical transition in Sr compared against an optical cavity is K α = 1.06 [33]. The constraints from Refs. [30] and [18] scale as √ T up the the observation time (∼ 12 hrs and 16 yrs, respectively). The sensitivity of the method proposed in this work scales as T 1/3 . For the variation in the quark mass (Λ q ), we present the estimated sensitivity for 3 years of data from a comparison of Rb and Cs hyperfine frequencies, as in Ref. [17] (K q = −0.021 [28]). We take τ 0 = 864 s and σ ∼ 5 × 10 −15 s [17]. For the Λ ep coupling, we assume 4 days of data from the comparison of a Yb + optical to Cs hyperfine transition, with τ 0 ∼ 1 s, and σ ∼ 4 × 10 −16 s, as in Ref. [34] (K ep = 1).
We also present the sensitivity for the network of Rb GPS satellite clocks, which use a ground-based H-maser as reference (τ 0 = 30 s, σ ∼ 10 −11 s). For Λ α , they do not exceed the existing bounds (K α ≈ 0.34 [27]). For the coupling to the quark mass however, the existing constraints are less stringent, and an analysis of the GPS data would probe unexplored parameter space (K q = −0.08 [28]).
Conclusion.-We show that certain macroscopic dark matter models may induce an observable asymmetry in the data of precision measurement devices. Further, we demonstrate that a sizeable annual modulation in this asymmetry is expected. A search based on our proposal can be performed using existing measurement data, and would extend the discovery reach by many orders of magnitude beyond the existing constraints.
We thank M. Pospelov for discussions. This work was supported by the U.S. National Science Foundation.  [18]. Those limits are shown hatched because, due to differences in experimental design, they depend on a slightly different combination of parameters. a The red curve shows the sensitivity from a Yb + /Cs optical/hyperfine frequency comparison, as in Ref. [34].