Oscillating massive scalar fields: Data analysis

PW P. Wcisło PA P. Ablewski KB K. Beloy SB S. Bilicki MB M. Bober RB R. Brown RF R. Fasano RC R. Ciuryło HH H. Hachisu TI T. Ido JL J. Lodewyck AL A. Ludlow WM W. McGrew PM P. Morzyński DN D. Nicolodi MS M. Schioppo MS M. Sekido RT R. Le Targat PW P. Wolf XZ X. Zhang BZ B. Zjawin MZ M. Zawada

This protocol is extracted from research article:

New bounds on dark matter coupling from a global network of optical atomic clocks

**
Sci Adv**,
Dec 7, 2018;
DOI:
10.1126/sciadv.aau4869

New bounds on dark matter coupling from a global network of optical atomic clocks

Procedure

For each frequency ω, i.e., for each mass *m*_{ϕ}, we performed the linear least-squares analysis of the normalized signal *R*(*t*) = *B* + *A*(ω)cos(ω*t* + 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 *A*^{2}(ω), can be treated as a new random variable. For the case of white noise [*N* samples of normalized signal *A*^{2}(ω)〉 = 4*σ*^{2}/*N*, and the cumulative probability distribution function (CDF) can be expressed by a simple analytical formula (*37*)*A*^{2} corresponding to *X* CL can be expressed as

The statistical interpretation of Eq. 10 is that the probability that *X*. For instance, for *X* = 95%, the corresponding *A*^{2} level is *N*/2 frequency channels available in the spectrum). That is, 1 per 20 frequency channels on average should exceed the *N*_{f} frequency channels, *13*) and (*37*): The random variable, *A*^{2}, exceeds the

As an *N*_{f}, we took the total considered frequency range (see Fig. 4) divided by Δω = 2π/*T*_{tot}, where *T*_{tot} 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 〈*A*^{2}〉 with the noise model function 〈*A*^{2}〉(ω). For instance, in the case of pink noise, 〈*A*^{2}〉(ω) ∝ 1/ω, the *A*^{2} corresponding to 5 and 99% CLs calculated as

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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