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Oscillating massive scalar fields: Data analysis
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New bounds on dark matter coupling from a global network of optical atomic clocks

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For each frequency ω, i.e., for each mass mϕ, we performed the linear least-squares analysis of the normalized signal composed of the contributions from all the four laboratories. We fit to the function R(t) = B + A(ω)cos(ωt + δ), where B is a constant offset (the contributions from different laboratories are weighted with inverse variance). In the left panel in Fig. 4, the square of the fitted amplitude is shown as a function of ω. To interpret the obtained result, one has to verify whether the fitted amplitude considerably exceeds the noise level. First, let us consider the simplest case when the noise of is white. The square amplitude of the harmonic component, A2(ω), can be treated as a new random variable. For the case of white noise [N samples of normalized signal with SD σ], its expected value does not depend on ω, 〈A2(ω)〉 = 4σ2/N, and the cumulative probability distribution function (CDF) can be expressed by a simple analytical formula (37)(9)Then, the A2 corresponding to X CL can be expressed as(10)

The statistical interpretation of Eq. 10 is that the probability that is X. For instance, for X = 95%, the corresponding A2 level is . Equation 10 concerns a single frequency (one of the N/2 frequency channels available in the spectrum). That is, 1 per 20 frequency channels on average should exceed the level. Beyond the 95% CL criterion, we also defined a detection threshold for Nf frequency channels, , in a similar manner as performed in (13) and (37): The random variable, A2, exceeds the level for any frequency channel, on average, only 0.05 times per measurement. That is, if the measurement were repeated 100 times, then the noise would be interpreted as a positive detection, on average, only five times. To make this definition unique, we set an additional condition that the probability of exceeding the detection threshold is the same for all the frequencies. For the case of white noise, the detection threshold defined here can be expressed as(11)

As an Nf, we took the total considered frequency range (see Fig. 4) divided by Δω = 2π/Ttot, where Ttot is a total time of our measurements (i.e., the difference between the time of the last and first samples in the data combined from all the laboratories). Equations 10 and 11 are strictly valid for white noise. We tested with the Monte Carlo simulations, however, that Eqs. 10 and 11 well approximate the CL and detection threshold for other types of noise; for this, one should replace the white noise parameter 〈A2〉 with the noise model function 〈A2〉(ω). For instance, in the case of pink noise, 〈A2〉(ω) ∝ 1/ω, the A2 corresponding to 5 and 99% CLs calculated as is indistinguishable from accurate Monte Carlo simulations at the scale of Fig. 4. A slight difference occurs for the detection threshold, but it can be easily eliminated with a numerically determined correction factor(12)

We used the above expressions to determine the CLs and detection threshold. The periodogram in the left panel in Fig. 4 shows that in our measurements, the power distribution in the collected data is consistent with pink noise (37).

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