Optimal constraints on primordial gravitational waves from the lensed CMB

We demonstrate how to obtain optimal constraints on a primordial gravitational wave component in lensed cosmic microwave background (CMB) data under ideal conditions. We first derive an estimator of the tensor-to-scalar ratio, $r$, by using an error-controlled close approximation to the exact posterior, under the assumption of Gaussian primordial CMB and lensing deflection potential. This combines fast internal iterative lensing reconstruction with optimal recovery of the unlensed CMB. We evaluate its performance on simulated low-noise polarization data targeted at the recombination peak. We carefully demonstrate our $r$-posterior estimate is optimal and shows no significant bias, making it the most powerful estimator of primordial gravitational waves from the CMB. We compare these constraints to those obtained from $B$-mode band-power likelihood analyses on the same simulated data, before and after map-level quadratic estimator delensing, and iterative delensing. Internally, iteratively delensed band powers are only slightly less powerful on average (by less than 10%), promising close-to-optimal constraints from a stage IV CMB experiment.

Figure 1

Upper panel: The three curves show the Monte Carlo noise $\mathrm{\Delta }\widehat{c}(r)/c(r)$ (one simulation equivalent) root variance for three estimators $\widehat{c}(r)$ of the lensed data covariance $r$ derivative, Eq. (3.2). The estimator accelerated by the isotropic reference estimator using unlensed spectra and neglecting lensing (green) is already accurate to well below a percent with a single Monte Carlo estimate. Lower panel: The red curve shows the relative deviation of the analytic perturbative prediction to the isotropic, unlensed approximation $c_0^{\mathrm{unl}}(r)$ to $\widehat{c}(r)$ (multiplied by a factor 100), Eq. (3.4). Blue points show estimates of the exact $c(r)$ from independent Monte Carlo simulations at each $r$ point, where the iterative lensing solution ${\varphi }^{*}(r)$ was obtained using ten iterations starting from the quadratic estimator. It is apparent that the exact result is both very close to the isotropic estimation and correctly captured by the analytical perturbative expansion. The error bars, independent from point to point, are the empirical standard deviations across nine Monte Carlo simulations. The figure was built using a data realization with input $r^{\mathrm{in}}=0.05$ where the $r$ grid is densest.

Figure 2

Data dependence of the Hessian log-determinant. Comparison of the $r$-posterior reconstruction on two simulated data maps spanning $645{\mathrm{deg}}^2$ with $2.1\mu \mathrm{K}\text{−}\mathrm{arc}\min$ polarization sensitivity. The filled colors use the expensive, realization-dependent curvature determinant estimate. The width shows the $2\sigma$ Monte Carlo uncertainties from the determinant derivative calculation. For each realization, errors on opposite sides of the peak of the posterior are strongly anticorrelated. The black lines show the same reconstructions but swap the log-determinant between the two realizations (black lines), with virtually identical results. For the cases we investigated, the determinant needs only be calculated on a single data realization.

Figure 3

Corrections to posterior and alternative expansion. Tensor-to-scalar ratio $r$-posterior estimates for one data realization (with input value $r_{\mathrm{in}}=0.05$), without any cumulant correction (solid blue), including first (solid orange) and second (solid green) cumulant corrections. The dashed lines show the same curves, but use a different posterior expansion scheme, with a less accurate but simpler curvature matrix neglecting the lensing. The agreement between the two methods is very good and provides a good consistency check, but in the latter case, the inclusion of the cumulant corrections becomes mandatory. The dashed bands contain 68% of the probability.

Figure 4

Posterior constraints on the tensor-to-scalar ratio $r$ for different reconstruction methods on one data realization with an input $r_{\mathrm{in}}=0.05$, 0.01, and 0 from top to bottom. The black line shows our new estimator. The green, orange, and blue lines show the constraints from nominal, quadratic estimator delensed, and iteratively delensed $B$-mode band powers, respectively, with likelihood built as described in the text. For comparison, the red line shows the case of perfect knowledge of the input lensing map, with posterior given by Eq. (5.1). In the upper two panels, the dashed bands contain 68% of the probability.

Figure 5

Comparison of constraints on the tensor-to-scalar ratio $r$ obtained from band-power analysis and the $r$ posterior obtained in this work (black), for 21 simulated data maps. Shown are constraints from nominal $B$-mode band powers (blue), quadratic estimator delensed band powers (orange), and iteratively delensed band powers (green). The red curves show the constraints when the lensing deflection potential is perfectly known for comparison. The grey shaded area shows the $2\sigma$ uncertainty on the constraint due to the Monte Carlo evaluation of one of the deflection Hessian curvature determinants in the posterior. The three panels use maps with different input tensor amplitudes $r_{\mathrm{in}}=0.05$, 0.01, and 0 (top to bottom). We quote two standard deviations (multiplied by 100) in the first two panels, where the tensor modes are well constrained, and the 95% confidence limit (also multiplied by 100) in the lowest panel with vanishing input tensor amplitude. All reconstructions are performed on $645{\mathrm{deg}}^2$, with a white polarization noise level of $2.12\mu \mathrm{K}\text{−}\mathrm{arc}\min$.