Modeling the Differential Rate for Signal Interactions in Coincidence with Noise Fluctuations or Large Rate Backgrounds

TL;DR

This work addresses how light-mass dark matter signals, which deposit energies near detector noise levels, alter the measured differential rate when coincident with noise. It derives the net differential response $\Delta f(E'|E_s) = f(E'|E_s) - f(E'|0)$ and shows that the observed spectrum obeys $\frac{dR}{dE'}(E'|S) = \frac{dR}{dE'}(E'|0) + \Delta t^{-1} \sum_{n_s\ge1} P(n_s|R_s\Delta t) \big(f(E'|n_s E_s) - f(E'|0)\big)$, highlighting two competing effects: signal-noise coincidences and reduced live-time for noise-only periods. The paper develops a discretized integrating-detector model, analyzes test cases (Gaussian and non-Gaussian noise, pileup regimes), and extends to continuous energy spectra, showing how to estimate $\Delta f$ (or measure $f(E'|0)$) and set conservative upper limits, notably via salting the data stream for analog detectors. It demonstrates that, in the no-pileup regime, conservative bounds can be obtained, while in the pileup regime limits may be non-conservative, motivating strategies to ensure conservativeness (restricting the energy-domain or using noise-based bounds). The framework is further applied to waveform detectors, where an effective coincidence timescale $\Delta t_{\rm OF}$ governs the nonlinearity of triggers, and salting remains a practical tool for accounting for nonlinearities. Overall, the results provide a principled, implementable approach to quantify dark matter sensitivities near threshold and to avoid over- or under-estimating experimental reach in the presence of noise fluctuations and background pileup.

Abstract

The characteristic energy of a relic dark matter interaction with a detector scales strongly with the putative dark matter mass. Consequently, experimental search sensitivity at the lightest masses will always come from interactions whose size is similar to noise fluctuations and low energy backgrounds in the detector. In this paper, we correctly calculate the net change in measured differential rate due to signal interactions that overlap in time with noise and backgrounds, accounting for both periods of time when the signal is coincident with noise/backgrounds and for the decreased amount of time in which only noise/backgrounds occur. Previous experimental searches have not accounted for this second fundamental effect, and thus either vastly overestimate their experimental search sensitivity (very bad) or use ad hoc conservative cuts which can underestimate experimental sensitivity (not ideal). We find that the detector response to dark matter can be trivially and conservatively understood as long as the true probability of dark matter pileup is small. We also show that introducing random events in the continuous raw data stream (a form of ``salting") provides a correct and practical implementation that correctly accounts for the decreased live time available for noise fluctuations and background events out of coincidence with a true dark matter signal.

Figures (7)

• Figure 1: Probability distribution functions for when $f(E', 0)$ (yellow) is Gaussian distributed noise.The net change $\Delta f(E'|E_s)$ (purple) is $f(E'|E_{s})$ (dashed green) subtracted by $f(E', 0)$. The DM signal energy $E_{s}=5\sigma$, where $\sigma$ is the Gaussian RMS. The signal rate $R_{s}=0.1/\Delta t$.
• Figure 2: Same probability distributions as Fig. \ref{['fig:f_gaussian']} but assuming a non-Gaussian noise. The signal model here represents a case with low DM energy, where $E_s=\sigma$ and $R_s=0.1/\Delta t$. Gray highlights the non-Gaussian noise and background region where $\Delta f(E'|E_s)$ is suppressed compared to $f(E'|E_s)$. The flat background differential rate is $0.02/\Delta t/\sigma$.
• Figure 3: Left: probability distributions for the case where there is DM contamination of the background data (see text). Here, the unknown true dark matter rate $\lambda_s = 1.5$, and the net response is negative at the single $E_s$ peak. Right: limit set by eq. \ref{['eq:R_lim_largelambda_largeEs']}. The gray line indicates the true DM background rate. The limit is conservative only when $1.5E_s<E'<2.5E_s$.
• Figure 4: A realistic detector background spectrum without dark matter, yellow, $f(E'|0)$, and with the presence of unknown DM signals, red, $f(E'|S)$. The background spectrum is shown in gray. The unknown signal, blue, is from 50MeV nuclear recoil DM with $\sigma_{n}=5.6e-33\square cm$ scattering in a 10g silicon detector. The resulting expected signal per integration time, $\lambda_s$, is 0.02 (2) for the fast (slow) integration time. The detected DM spectrum $dR/dE'$, black, is estimated using different methods. The solid black line uses the correct net differential rate change proposed in this work, Eq. \ref{["eq:dRdE'int"]}, the dotted black line uses the smearing without signal live-time correction, Eq. \ref{["eq:dRbaddE'"]}, and the dashed line is smeared assuming Gaussian baseline fluctuation and truncated at $3\sigma$, Eq. \ref{["eq:dRdE'_3sigma"]}.
• Figure 5: Limit of dark matter rate as a function of the measured energy $E'$ with respect to the true DM background rate, assuming infinite exposure. Line styles are the same as in fig. \ref{['fig:DMsim_dRdE']}. Limits in the left figure are conservative. In the right figure, long frame time results in $\lambda_s>1$, and the limits are underestimated at certain $E'$.