Statistics of Daily Modulation in Dark Matter Direct Detection Experiments

Abstract

The time-dependent modulation of the event rate in dark matter direct detection experiments, arising from the motion of the Earth with respect to the Galactic rest frame, is a distinctive signature whose observation is crucial for claiming a discovery of dark matter. While annual modulation has been well studied for decades, daily modulation due to the Earth's rotation has attracted increased attention recently due to the identification of anisotropic solid-state detector materials that yield a direction-dependent scattering rate without sacrificing the overall rate. We perform a statistical analysis of daily modulation in dark matter scattering experiments, with the goal of maximizing the statistical significance of a modulating signal in the presence of an unknown background rate, which may be either flat (non-modulating), or modulating over a 24-hour period with a known or unknown phase. In the background-dominated regime, we find that the discovery significance scales as $f_\text{RMS} \sqrt{T}$, where $T$ is the total exposure time and $f_\text{RMS}$ is the root-mean-square modulation amplitude; in particular, the significance continues to improve with exposure rather than saturating due to systematic uncertainties in the background rate. Using anisotropic trans-stilbene detectors for sub-GeV dark matter as a benchmark example, we provide prescriptions for optimizing the significance for a given total detector mass and location. In an example analysis using three detectors, optimizing the detector orientations can reduce the required exposure by a factor of $\sim 5$ for a desired discovery or exclusion significance, even after profiling over an unknown modulating background phase.

Figures (11)

• Figure 1: (Left) A comparison of signal and background templates used in our Monte Carlo data-generation procedure. In black, we depict the signal rate $s(t)$ for $A = 1$ while in dashed red, we depict the uniform background rate for $B_\mathrm{tot} = 100$. (Right) A comparison of the asymptotic sensitivity estimates with the results of the Monte Carlo procedure for generating limits under the null as a function of the number of bins used to resolve the total event rate jointly produced by signal and background processes. As a function of $N_\mathrm{bins}$, the black dashed line indicates the median expected $A_{95}$ in the asymptotic limit, while the interval in green represents the central 68% confidence interval for $A_{95}$ under the null; the interval in yellow represents the $84^\mathrm{th}$ to $97.5^\mathrm{th}$ percentile values of $A_{95}$ in its expected distribution under the null. The grey data points indicate the median of a Monte Carlo sampling of $A_{95}$ for data generated under the null, while the lower error bar (first upper error bar) [second upper error bar] indicates the 16$^\mathrm{th}$ (84$^\mathrm{th}$) [97.5$^\mathrm{th}$] percentile value in the sampled distribution. See text for the details of this construction. The relatively good agreement across $N_\mathrm{bins}$ indicates that our asymptotic sensitivity estimates already are trustworthy in these instances with only $\sim 100$ events in the dataset, with correspondingly better convergence expected for datasets with more events. We emphasize that the level of agreement is constant across $N_\mathrm{bins}$, indicating that our results hold in the narrow or even unbinned limit.
• Figure 2: The right ascension (RA) and declination (dec) of the Earth velocity vector in the galactic rest frame as a function of time of year, with the RA angle given in hours ($360^\circ = 24$ hours). The speed of the local standard of rest $v_0$ is not known precisely: following Baxter:2021pqo we adopt $v_0 = 238$ km/s as the standard value. The dashed and dotted lines show $v_\oplus(t)$ assuming $v_0 = 230$ km/s and 220 km/s, respectively. For reference, the red dots in the center show the constant Sun velocity $v_\odot$ for each value of $v_0$. The first day of each month is marked on the plot, starting with 2026 Jan. 1 (00:00:00 UTC). Note that the angle between the north pole and the Earth velocity is $(90^\circ - \text{dec})$.
• Figure 3: Two schemes for stacking daily modulation data are shown, for 366 sidereal days worth of exposure time. The lower panel tracks $R(t)$ according to the sidereal time, while the upper panel uses the shifted sidereal time $\tilde{t}$. The rates on day zero (Jan 1, 2026) and day 135 (May 16) are highlighted in black and purple, with the average value of $R(t)$ shown in red. On the other days of the year, $R(t)$ varies within the gray envelope.
• Figure 4: The crystal-centric coordinate system. On the left, we show the translationally invariant unit cell of trans-stilbene (outlined in purple). It is symmetric with respect to rotations of $180^\circ$ about the $\hat{z}$ axis, or inversions $\mathbf x \rightarrow -\mathbf x$ through the central point of the molecule. On the right, we show the Earth velocity in the same crystal-centric system, with the sample oriented to place the north pole along the $\hat{n} = (\theta_n, \phi_n)$ direction. As the Earth rotates, the Earth velocity and DM wind velocity sweep out a cone with opening angle $\theta_N$.
• Figure 5: Optimized orientations for a single trans-stilbene detector, assuming a constant background, with $m_\chi = 5 \ {\rm MeV}$, $M_T = 1 \ {\rm kg}$, and $\bar{\sigma}_e = 10^{-38} \ {\rm cm}^2$, for both limits of the mediator mass. In the left panel, we plot $\chi^2 = f_{\rm RMS}^2 N$ as a function of the detector orientation $(\theta_n, \phi_n)$. To the right, we plot $R(t)$ for the best and worst orientations, corresponding to $(\theta_n, \phi_n) = (46^\circ, 16^\circ)$ and $(90^\circ, 16^\circ)$ in black and red, respectively, for the heavy mediator. For the light mediator, the best and worst scores are found (respectively) at $(\theta_n, \phi_n) = (46^\circ, 170^\circ)$ and $\theta_n = 0^\circ$. The respective $f_{\rm RMS}$ scores of the best and worst $R(t)$ functions are $6.72\%$ and $1.18\%$ for the heavy mediator, and $10.26\%$ and $0.63\%$ for the light mediator.