First CMB constraints on direction-dependent cosmological birefringence from WMAP-7

A Chern-Simons coupling of a new scalar field to electromagnetism may give rise to cosmological birefringence, a rotation of the linear polarization of electromagnetic waves as they propagate over cosmological distances. Prior work has sought this rotation, assuming the rotation angle to be uniform across the sky, by looking for the parity-violating $TB$ and $EB$ correlations that a uniform rotation produces in the cosmic microwave background temperature/polarization. However, if the scalar field that gives rise to cosmological birefringence has spatial fluctuations, then the rotation angle may vary across the sky. Here we search for direction-dependent cosmological birefringence in the WMAP-7 data. We report the first cosmic microwave background constraint on the rotation-angle power spectrum $C_L^{\alpha \alpha }$ for multipoles between $L=0$ and $L=512$. We also obtain a 68% confidence-level upper limit of $\sqrt{C_2^{\alpha \alpha }/(4\pi )}\lesssim 1°$ on the quadrupole of a scale-invariant rotation-angle power spectrum.

Figure 1

The recovery of the CB signal with the analysis method presented in Sec. 3 is demonstrated using a suite of simulations that include realization of a rotated sky. The dashed blue line is the input signal power spectrum of $C_L^{\alpha \alpha }$, the thick red line represents the power spectrum of $\frac{1}{2}\sin [2\alpha (\widehat{\mathbf{n}})]$, and the thin black line is the mean recovered power from the simulations; the gray region is a $1\sigma$ confidence-level interval calculated from the same suite of simulations.

Figure 2

Top panel: Measurement of the rotation-angle power spectrum from $V$ band, shown before debiasing, along with the components of the noise bias: Monte Carlo measurement of the null-hypothesis mean $⟨{\widehat{C}}_L^{\alpha \alpha }⟩$, isotropic noise bias, and the mean isotropic bias. Middle panel: The same power spectrum after debiasing, with $1\sigma$ and $3\sigma$ confidence interval. Bottom panel: Binned version of the middle-panel power spectrum. The results are consistent with zero within $3\sigma$.

Figure 3

Same as Fig. 2, for $[f_1{f}_2][f_3{f}_4]=[\mathrm{WW}][\mathrm{VV}]$.

Figure 4

Same as Figure 2, for $[f_1{f}_2][f_3{f}_4]=[\mathrm{WV}][\mathrm{WV}]$.

Figure 5

Probability distributions of the best-fit amplitude $\widehat{A}$ of the scale-invariant rotation power $C_L^{\alpha \alpha }$ recovered from a suite of null-hypothesis realizations are shown for some of the cross-band correlations. The gray areas denote a 68% confidence interval around a mean value; the red vertical line represents the measurement of the best-fit $\widehat{A}$ for a given cross-band correlation. We find consistency with zero within $3\sigma$ for all five measurements.

Figure 6

Measurement of ${\widehat{C}}_L^{\alpha \alpha }$ from [WW][VV]. Results shown in black are obtained by using the analysis mask that covers all the point sources brighter than $\sim 1\mathrm{Jy}$, while the results in red (empty circles) are obtained after unmasking all the point sources. There is no apparent bias and the difference in every bin is less than the statistical uncertainty, despite the extreme variation in the source contamination. Scaling arguments in Sec. 5c2 imply that the unresolved point sources have a negligible contribution to the estimated measurement uncertainty for the most constraining cross-band correlation.

Figure 7

Measurement of ${\widehat{C}}_L^{\alpha \alpha }$ from [WW][VV] cross-band correlation, with the corresponding statistical uncertainty obtained from a suite of null-hypothesis foreground-free simulations. Black filled circles are measurements obtained from the foreground-reduced maps after applying the fiducial analysis mask, and they represent our fiducial results of Fig. 3. The colored data points (and the associated error bars) correspond to the two test analyses: the green $x$’s are obtained from the maps prior to foreground subtraction, but using the fiducial mask, while the red empty circles are measurements obtained from foreground-reduced maps after applying an extended mask (with an additional $\sim 35\%$ of the sky covered around the Galactic plane). The results from the two tests show no apparent bias. For the case of non-foreground-reduced analysis, the difference from the fiducial measurements is negligible compared to the statistical uncertainty; for the extended-mask case, the scatter between the two results is consistent with the difference in sky coverage. This implies that foregrounds and foreground residuals should not have a drastic impact on the estimated measurement uncertainty.

Figure 8

$TE$ correlation measured from WMAP-7 data (black) is compared to the primordial power spectrum, generated using CAMB for the best-fit cosmology with no rotation (red, solid); both power spectra are obtained from Ref. 23. The uncertainty on this measurement (gray) leaves room for a maximal rotation-angle RMS of roughly 9.5°.

Figure 9

The Wigner-3j geometric factors in the summands of Eqs. (8, 16) for three different values of $L$. The geometric factor is nonzero only in the region of the $l{l}^{\prime }$ space where the triangle inequalities are satisfied. The dominant contribution comes from the triangles in which $L\sim l$, or $L\sim l^{\prime }$, i.e., where either the temperature or the polarization mode has a length scale comparable to the length scale of the rotation-angle mode.

Figure 10

The power-spectrum kernel of the summands in Eqs. (8, 16). The types of triangles that contribute the most to the isotropic bias of $C_L^{\widehat{\alpha }\widehat{\alpha }}$ are set by the geometric properties of spin-2 Wigner-3j symbols illustrated by the kernel shown in Fig. 9, which is modulated by this kernel to produce summands in the expression for the bias.

Figure 11

The $L$ dependence of the $f_{\mathrm{sky}}$ factor used for the reconstruction of the scale-invariant rotation-angle power spectrum in Secs. 4c, 5b. The horizontal (green) line at $f_{\mathrm{sky}}=0.68$ represents the fraction of pixels admitted by the mask.

Figure 12

Fiducial temperature-analysis mask (with $\sim 78\%$ of the sky admitted), fiducial polarization-analysis mask ($\sim 73\%$), extended mask ($\sim 33\%$), and the temperature-analysis mask with point sources uncovered ($\sim 82\%$). Fraction of the sky admitted of the combined fiducial masks for polarization and temperature is $\sim 68\%$, which is twice the sky admitted as compared to the extended mask.