Searching for anisotropic cosmic birefringence with polarization data from SPTpol

We present a search for anisotropic cosmic birefringence in $500{\mathrm{deg}}^2$ of southern sky observed at 150 GHz with the SPTpol camera on the South Pole Telescope. We reconstruct a map of cosmic polarization rotation anisotropies using higher-order correlations between the observed cosmic microwave background (CMB) $E$ and $B$ fields. We then measure the angular power spectrum of this map, which is found to be consistent with zero. The nondetection is translated into an upper limit on the amplitude of the scale-invariant cosmic rotation power spectrum, $L(L+1)C_L^{\alpha \alpha }/2\pi <0.10\times {10}^{-4}{\mathrm{rad}}^2$ ($0.033{\mathrm{deg}}^2$, 95% C.L.). This upper limit can be used to place constraints on the strength of primordial magnetic fields, $B_{1\mathrm{Mpc}}<17\mathrm{nG}$ (95% C.L.), and on the coupling constant of the Chern-Simons electromagnetic term $g_{a\gamma }<4.0\times {10}^{-2}/H_I$ (95% C.L.), where $H_I$ is the inflationary Hubble scale. For the first time, we also cross-correlate the CMB temperature fluctuations with the reconstructed rotation angle map, a signal expected to be nonvanishing in certain theoretical scenarios, and find no detectable signal. We perform a suite of systematics and consistency checks and find no evidence for contamination.

Figure 1

Noise biases for the cosmic rotation reconstruction. The theoretical scale-invariant cosmic rotation power spectrum with unit amplitude ($A_{\mathrm{CB}}=1$) is shown by the black solid line. The main source of noise, the Gaussian $N_L^{(0),\mathrm{RD}}$ bias, is shown by the yellow solid line and is estimated with the realization-dependent approach. The blue solid (dashed) line shows the positive (negative) values of the lensing bias $N_L^{\mathrm{lens}}$. The sum of $N_L^{(0),\mathrm{RD}}$ and $N_L^{\mathrm{lens}}$ is the total noise bias (cyan solid line) that we subtract from the measured raw power spectrum $C_L^{\widehat{\alpha }\widehat{\alpha }}$. For reference, the $N_L^{(1)}$ bias (calculated for $A_{\mathrm{CB}}=1$ and not subtracted from the observed spectrum) is shown by the red solid line. See the text for further details.

Figure 2

Distribution of the reconstructed amplitudes $A_{\mathrm{CB}}$ of the scale-invariant cosmic rotation power spectrum from unrotated (yellow histogram) and rotated (light green histogram) simulations. The corresponding $A_{\mathrm{CB}}$ value found from the observed data is shown by the red vertical line.

Figure 3

Consistency tests summary. The difference band power $\mathrm{\Delta }{\widehat{C}}_{L_b}^{\alpha \alpha }$ between the baseline/alternate analyses and their uncertainties are scaled by the $1\sigma$ cosmic rotation uncertainties in that specific bin. The grey shaded regions indicate the $1\sigma$ uncertainties on the baseline measurement of ${\widehat{C}}_L^{\alpha \alpha }$. As can be seen, the induced shifts are generally only a small fraction of the statistical band-power uncertainties.

Figure 4

Impact of non-Gaussian polarized Galactic dust. Mean difference cosmic rotation power spectrum between simulations that include the non-Gaussian Galactic dust realizations from [60] and the nominal realization with Gaussian foregrounds. The band-power difference $\mathrm{\Delta }{C}_L^{\alpha \alpha }$ is normalized by the $1\sigma$ statistical uncertainty at each multipole bin.

Figure 5

Top: a map of the reconstructed cosmic birefringence fluctuations $\widehat{\alpha }$ from the SPTpol $500{\mathrm{deg}}^2$ polarization data using the $EB$ quadratic estimator. The map has been smoothed by a 1 deg FWHM Gaussian beam. Bottom: simulated $\alpha$ maps plotted with the same color scale as the top panel and smoothed by a 1 deg FWHM Gaussian beam. The left panel shows the input $\alpha$ map generated from a scale-invariant CB power spectrum with $A_{\mathrm{CB}}=1$, the middle panel shows the reconstructed map estimated from the noisy simulation that has been rotated using the input map on the left, and the right panel shows the reconstructed $\alpha$ map obtained from the corresponding unrotated simulation. The pattern of the CB fluctuations reconstructed from the data appears similar to what is seen in the unrotated case, providing a visual indication that the amplitude of the CB signal in the data must be $A_{\mathrm{CB}}\ll 1$.

Figure 6

Anisotropic cosmic rotation power spectrum measured from SPTpol $500{\mathrm{deg}}^2$ polarization data (red circles) and from the ACTpol experiment (blue squares) [21]. The black solid line shows the fiducial scale-invariant cosmic rotation power spectrum assuming $A_{\mathrm{CB}}=1$ [see Eq. (2.7)]. The PTE under the no-rotation hypothesis is 76.5% and therefore cannot be rejected.

Figure 7

Cross-power spectrum between the SPTpol CMB temperature fluctuations and the anisotropic CB angle. The inset panel shows the distribution of ${\chi }_{\mathrm{null}}^2$ from simulations (blue histogram) and the value from data (red vertical line). The cross power is consistent with the hypothesis of no signal between the maps.