Testing parity symmetry with the polarized cosmic microwave background

New physics in the early Universe could lead to parity-violation in the late Universe, sourcing statistics whose sign changes under point reflection. The best constraints on such phenomena have come from the Planck temperature fluctuations; however, this is already cosmic-variance limited down to relatively small scales. Thus only small improvements are expected in the future. Here, we search for signatures of parity violation in the polarized cosmic microwave background, using the Planck PR4 $T$- and $E$-mode data. We perform both a simulation-based blind test for any parity-violating signal at $\ell <518$, and a targeted search for primordial $U(1)$ gauge fields (and the amplitudes of a generic collapsed model) at $\ell <2000$. In all cases, we find no evidence for new physics, with the model-independent test finding consistency with the $\mathrm{FFP}10/\mathrm{NPIPE}$ simulation suite at $(-)0.4\sigma$, and the gauge field test constraining the fractional amplitude of gauge fields during inflation to be below $6\times {10}^{-19}$ at 95% confidence level for a fiducial model. The addition of polarization data can significantly improve the constraints, depending on the particular model of primordial physics, and the bounds will tighten significantly with the inclusion of smaller-scale information.

Figure 1

Comparison between the Planck data (left) and a single FFP10 simulation (right). The three plots show $T$, $Q$, and $U$ maps (in $\mu \mathrm{K}$ units), all of which are used in the analysis. Black regions show pixels that are inpainted in our analysis, whilst those in white are masked. We note that the FFP10 simulations capture the large-scale (systematic) non-Gaussian features in the polarized maps.

Figure 2

Comparison of empirical (left) and theoretical (right) correlation matrices for the trispectrum, $t^u(\mathbf{b},\beta )$. The empirical matrices are computed using 600 FFP10 simulations, whilst the theoretical prediction is given by the inverse trispectrum normalization matrix, itself obtained via Monte Carlo methods, using only the survey mask and weighting scheme. In each panel, we stack all 2386 $T$- and $E$-mode trispectrum measurements with $4\le \ell <518$ into one dimension (ordered from low- to high-$\ell$), separating out the six nontrivial combinations of fields, which are marked in green. For example, the submatrix in the second column and third row contains the $TTTE\times TTEE$ covariance. Note that different correlators contain different numbers of bins, due to labeling degeneracies (i.e. restricting to ${\ell }_1<{\ell }_2$ for fields $X_1\ne X_2$). We subtract the (unit) diagonal of the matrix for clarity, and truncate the scale to $\pm 5\%$. The empirical correlation matches the inverse normalization to high precision (up to noise), implying that the latter can be used to decorrelate the measured data, as shown in Fig. 4.

Figure 3

Rescaled parity-odd trispectrum measured from Planck PR4 data, alongside measurements from 600 FFP10 and 1000 GRF simulations for all combinations of $T$- and $E$-modes with $4\le \ell <518$. The top panel shows the raw measurements, whilst the bottom shows their simulated variances. The $\tau$-statistic is defined in (6), with weighting chosen to decorrelate the various trispectrum bins. As in Fig. 2, we stack the bins into one dimension, with different polarizations indicated by dotted lines. We find no obvious detection of the trispectrum across the 2386 bins considered herein. The variance of the simulations is larger than unity, implying that the estimators are slightly suboptimal, particularly for polarization. Whilst this does not bias our analysis, it suggests slightly stronger constraints could be wrought with a more advanced weighting scheme.

Figure 4

Correlation matrix of the rescaled trispectrum statistic $\tau$, defined by (6). This is obtained using both 600 FFP10 simulations (upper left) and 1000 GRF simulations (bottom right). Unlike the unscaled trispectrum correlations shown in Fig. 2, we find no evidence for off diagonal contamination in the $\tau$ covariances, with both matrices consistent with noise. As in Fig. 2, we stack all bins into one dimension and subtract the leading diagonal (which is unity only if the estimator is optimal).

Figure 5

Probability of parity violation in the large-scale ($4\le \ell <518$) Planck PR4 dataset. We compare the ${\chi }^2$ statistic of (7) measured in Planck to empirical distributions from FFP10 and GRF simulations (red and green respectively). We find excellent agreement between Planck results and the simulation histograms, with frequentist ranks and ${\chi }^2$ probabilities to exceed shown in the title. Both empirical distributions exhibit slightly heavier tails than the theoretical distribution, implying that classical ${\chi }^2$-analyses are insufficient for modeling the trispectrum noise properties. The frequentist detection probability for Planck corresponds to $-0.4\sigma$.

Figure 6

Theoretical models for the CMB $T$- and $E$-mode trispectra, alongside the empirical trispectrum error extracted from 100 FFP10 simulations. We show the $n=0$ and $n=1$ templates given in (8), the second of which well approximates the gauge field model. In contrast to previous figures, results are shown for $5\le \ell <2000$, with internal $2\le L<10$, where the model is large. The fiducial amplitudes of the model are set to $d_n^{\text{odd}}={10}^4$ for visualization (corresponding to $A_{\text{gauge}}={10}^4/3$), and we take the absolute value of the model correlators. Comparing theory and error, we expect that the constraints will be dominated by $TTTT$ and $TTTE$ trispectra on small scales.

Figure 7

Constraints on the amplitude of the gauge-field model [12] for the parity-violating trispectrum. We show the posterior for both the Planck PR4 data (up to $\ell =2000$) and the mean of 100 FFP10 simulations. The left panel shows the fiducial analysis with internal $L\ge 2$, whilst the central and right panels respectively show the results of removing $E$-modes and restricting to $L\ge 5$. The fiducial results find $A_{\text{gauge}}=(-0.9\pm 3.3)\times {10}^3$ (Planck) and $A_{\text{gauge}}=(-0.5\pm 3.3)\times {10}^3$ (FFP10), with no detection of parity violation. Constraints degrade by a factor $\approx 3$ with the excision of $E$ modes, or by a factor $\approx 35$ with the removal of low-$L$ configurations.

Figure 8

Constraints on the general parity-odd trispectrum given in (8) from the mean of 100 FFP10 simulations (blue) and Planck data (red) up to ${\ell }_{\max }=2000$, as in Fig. 7. We show two sets of contours: $T$-only (the wider and lighter contours), and $T+E$ (the darker and narrower contours). The best-fit Planck constraints are $d_0^{\mathrm{odd}}=(-0.9\pm 1.1)\times {10}^9$, $d_1^{\mathrm{odd}}=(-3.1\pm 9.8)\times {10}^3$ from temperature and polarization, or $d_0^{\mathrm{odd}}=(-7.9\pm 3.1)\times {10}^9$, $d_1^{\mathrm{odd}}=(4.5\pm 3.3)\times {10}^4$ from temperature alone.

Figure 9

Dependence of the gauge-model constraints shown in Fig. 7 on scale cuts, via the maximum multipole ${\ell }_{\max }$. We show the $1\sigma$ error bar both for the mean of 100 FFP10 simulations (blue) and Planck PR4 data (red). We include both $T$- and $E$-modes in all analyses and use internal $2\le L<10$. We find strong dependence on ${\ell }_{\max }$, though the scaling tempers slightly beyond ${\ell }_{\max }\gtrsim 1000$, likely due to projection effects.

Figure 10

As Fig. 5, but assessing the consistency of our analyses with various modifications to our fiducial assumptions. In the first three cases (described in Appendix pp2), we show the ${\chi }^2$ distributions from the analysis alongside the fiducial results in dashed lines; in the fourth case, we show only the new results, since the dimensionality differs. The first two tests use 100 FFP10 and GRF simulations, whilst the second two use the full fiducial sample. In all cases, the blind tests show no evidence for parity violation.