Pipeline diagram 

Figure 1: Flow diagram of the axionoscillation analysis pipeline. The three main components are indicated on the left, and more detail is presented in the rest of the figure. The mapmaking step is similar to that of a standard CMB analysis, though the reobservation must now include a component that represents polarization oscillations (Secs. III B, III D and III G). The coadds are estimates of the timeaveraged Stokes parameters \( \left\langle Q(\hat{n}) \right\rangle \) and \( \left\langle U(\hat{n}) \right\rangle \). The global rotation angle \(f(t) / 2\) is obtained through the correlation method of Sec. III E with some of the computational speedups of Sec. III G. The likelihood and Bayesian analysis are described in Sec. IV. The ensemble of simulations is used to construct a model distribution, which we can resample to form large numbers of Gaussian pseudosimulations (Sec. IV A 1). The model distribution also implies a likelihood function (Sec. IV B) that is used to set Bayesian upper limits (Sec. IV C). At the same time, we estimate test statistics to check consistency with the background model (Sec. IV D) and to test for spurious systematic signals (Sec. V). Based on the results of the systematics checks, which we call “jackknife tests”, we unblind the real nonjackknife data (Sec. VI B).

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Illustration of correlation method 

Figure 2: An illustration of the correlation method introduced in Sec. III E for a simulation of a single detector and a single observation. (Top) The pair difference \( \bar{D}_i ( \hat{\bf n}, \tau ) \) (blue) from a constantelevation scan is plotted against right ascension (RA). The pair difference is dominated by atmospheric loading and fluctuates strongly. In this simulation, a global rotation of 3° has been imposed. The oscillating component \( \bar{D}_i^\mathrm{(osc)} ( \hat{\bf n}, \tau) \) (red) is hidden underneath the atmospheric fluctuations. The rotated map \( \bar{r}_i (\hat{\bf n}, \tau ) \) (orange) is used as a template to pick out global polarization rotations. (Bottom) The correlation \( \rho(\tau) \) (Eq. 20 summed over detector \(i\) only) between the template and the full pair difference (blue) has a large variance but mean zero. The correlation with the oscillating component (red) is biased positive.

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Simulated time series 

Figure 3: A simulated time series \( \hat{f}(\tau) \) (blue) with an input rotation amplitude \(A/2 = 3\)°, chosen to be relatively large in order to illustrate the effect more clearly. The background \( \hat{f}^\mathrm{(bkg)} (\tau) \) (cyan, defined explicitly in Eq. 51) dominates over the oscillating component \( \hat{f}^\mathrm{(osc)} (\tau) \) (red, Eq. 50). The template maps \( \bar{Q} (\hat{\bf n}) \) and \( \bar{U} (\hat{\bf n}) \) are constructed from standard BICEP simulations of only the 2012 observing season of the Keck Array, so the underlying true oscillation \( f(t) \) (orange) is not recovered at full strength but is instead suppressed by ∼30% as described in Sec. III F.

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Test statistic for consistency with background model 

Figure 4: The test statistic \( \Delta q_m \) for consistency with the background model (Eq. 65) for real data from the 2012 observing season of the Keck Array. We plot \( \Delta q_m^{1/4} \) on the vertical axis in order to compress the distribution for visual purposes, and we plot frequency \( m / (2 \pi) \) in units of inverse days (\( \mathrm{d}^{1} \)) on the horizontal axis. The maximum and minimum values are indicated in the legend with their corresponding oscillation periods. The levels for global \(2.5 \sigma\) fluctuations in both directions are indicated by horizontal red lines, i.e., there is a 1.2% probability in the background model that at least one value of \( \Delta q_m \) will lie outside the region bounded by the red lines.

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Bayesian 95% confidence upper limits 

Figure 5: Bayesian 95%confidence upper limits on rotation amplitude \(A/2\) (Sec. IV C). We also provide the median expectation (black dashed) from backgroundonly simulations as well as \(1 \sigma\) (green) and \(2 \sigma\) (yellow) regions. These expectations represent local rather than global percentiles. With nearly \(10^4\) frequencies under consideration, we expect several values outside of the \(2 \sigma\) region. The question of background consistency is addressed by Fig. 4 and the test statistic \( \Delta \hat{q} \) (Eq. 66). The median limit for oscillation periods larger than 24 hr (frequency less than 1 \(\mathrm{d}^{1}\)) is 0.68°. For shorter periods (larger frequencies), the limits are degraded due to binning observations in ∼1hr scansets (Sec. III C). Additionally, we plot a smoothed approximation to our upper limits (Eq. 76) in cyan.

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Exclusion plot 

Figure 6: Excluded regions in the masscoupling parameter space for axionlike dark matter (cf. Fig. 3 in [12]). All constraints push the allowed regions to larger masses and smaller coupling constants, i.e., toward the bottom right of the figure. If the dark matter is assumed to consist entirely of axionlike particles, i.e., if \( \kappa = 1 \), then our constraints (blue) are immediately implied by Eq. 78 and the results of Fig. 5. A smoothed approximation is shown in cyan (Eq. 79). The orange dotdashed and dotted lines show the constraints that would be achieved if the rotation amplitude were constrained to 0.1° and 0.01°, respectively. The green solid line shows the constraint set by Fedderke et al. [12] by searching for the washout effect (Sec. I) in publicly available Planck power spectra. The dashed green line shows the cosmicvariance limit for the washout effect. The dashed grey horizontal line shows the limit from searching for a gammaray excess from SN1987A [23]. The solid grey horizontal line is the limit set by the CAST experiment [26]. The dotted grey vertical line is a constraint on the minimum axion mass from observations of smallscale structure in the Lymanα forest [27], though we note that several similar bounds have also been set by other considerations of smallscale structure [28, 29].

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