Limitations of CMB $B$-mode template delensing

Efforts to detect a primordial $B$-mode of cosmic microwave background polarization generated by inflationary gravitational waves ought to mitigate the large variance associated with the $B$-modes produced by gravitational lensing, a process known as delensing. A popular approach to delensing entails building a lensing $B$-mode template by mimicking the lensing operation, either at gradient order or nonperturbatively, using high-resolution $E$-mode observations and some proxy of the lensing potential. By explicitly calculating all contributions to two-loop order in lensing to the power spectrum of $B$-modes delensed with such a template in the noise-free limit, we are able to show that: (i) corrections to the leading-order calculation of the lensing $B$-mode power spectrum only enter at the $O(1)\%$ level because of extensive cancellations between large terms at next-to-leading order; (ii) these cancellations would disappear if a gradient-order template were to be built from unlensed or delensed $E$-modes, giving rise to a residual delensing floor of $O(10)\%$ of the original power; (iii) new cancellations arise when the lensed $E$-modes are used in the gradient-order template, allowing for the delensing floor to be as low as $O(1)\%$ of the original power in practical applications of this method; and (iv) these new cancellations would disappear for a nonperturbative template constructed from the lensed $E$-modes, reintroducing a residual delensing floor of $O(10)\%$. We further show that the gradient-order template outperforms the nonperturbative one in realistic scenarios with noisy estimates of the $E$-mode polarization and lensing potential. We, therefore, recommend that in practical applications of $B$-mode template delensing, where the template is constructed directly from the (filtered) observed $E$-modes, the gradient-order approach should be used rather than a nonperturbative remapping.

Figure 1

Power spectrum of ${\tilde{B}}^{(1)}(\mathbf{l})$ estimated from a single flat-sky simulation (black points) compared to the nonperturbative spherical calculation of the lens-induced $B$-mode power from camb [36, 38] (black solid). The leading-order difference between these spectra involves the sum of $⟨{\tilde{B}}^{(2)}(\mathbf{l}){\tilde{B}}^{(2)}({\mathbf{l}}^{\prime }){⟩}^{\prime }$ (blue) and $2⟨{\tilde{B}}^{(1)}(\mathbf{l}){\tilde{B}}^{(3)}({\mathbf{l}}^{\prime }){⟩}^{\prime }$ (with minus this shown in dashed orange). Separately, these two spectra are large in magnitude, of $O(10)\%$ of the lensed $B$-mode spectrum on large scales, but they cancel rather precisely to leave only sub-percent-level corrections to the lensed $B$-mode power spectrum (red solid for positive values, red dashed for negative). For reference, the thin grey line shows the nonperturbative lensing power spectrum, scaled to 1% of its original amplitude. Analytic evaluations are shown as solid lines and estimates derived from a single flat-sky simulation as points.

Figure 2

Real-space scalar fields derived from ${\tilde{B}}^{(2)}(\mathbf{l})$ (left) and the contribution to the gradient-order $B$-mode template from the linear lensing correction to $E$-modes, $\mathrm{\Delta }{\tilde{B}}^{\mathrm{temp}}(\mathbf{l})$ (right). The plotted intensity ranges from $-0.5\mu \mathrm{K}$ (dark blue) to $0.5\mu \mathrm{K}$ (yellow). The similarity of these fields and their non-Gaussian nature are clearly apparent.

Figure 3

Analytic (solid lines) versus simulated (dots) lensing $B$-mode residual spectra after delensing by: (i) building a gradient-order template from unlensed (blue) or lensed (grey) $E$-modes; and (ii) forming a nonperturbative template involving lensed $E$-modes (orange). We also show the simulated residual $B$-mode power after antilensing polarization fields containing lensed $E$- and $B$-modes (red dots). Noiseless polarization fields and perfect $\varphi$ are used in all cases. Also shown is the lensing $B$-mode power spectrum (black solid line) and a primordial $B$-mode spectrum for $r=0.001$ (dashed) generated using camb [38].

Figure 4

Residual $B$-mode maps after delensing in the limit of noiseless CMB fields and access to the true $\varphi$ and using either a gradient-order template built from unlensed $E$-modes (top left), a gradient-order template built from lensed $E$-modes (top right), a nonperturbative template built from lensed $E$-modes (bottom left), or by antilensing polarization maps containing both $E$- and $B$-modes (bottom right). The plotted intensity ranges from $-0.5\mu \mathrm{K}$ (dark blue) to $0.5\mu \mathrm{K}$ (yellow).

Figure 5

Left: Fraction of lensing power remaining after delensing noiseless $B$-modes with a nonperturbative template built from (Wiener-filtered) noisy lensed $E$-modes and noisy $\varphi$. Right: Ratio of residual power after delensing with a gradient-order template to that after delensing with a nonperturbative template, in both cases, using filtered, noisy, lensed $E$-modes and $\varphi$ to delens noiseless $B$-modes. When computing the ratios, we first average spectra over the multipole range $l\in [25,325]$ and over 20 simulations. In all cases, we add white noise to the simulated lensed $E$-modes used to construct the templates with variance ${\mathrm{\Delta }}_P^2$ (and ignore the effects of finite instrumental angular resolution). For the lensing proxy, we add white noise to the true lensing convergence ($\kappa =-{\nabla }^2\varphi /2$) with variance ${\mathrm{\Delta }}_{\kappa }^2$ per $\mathrm{arc}{\min }^2\mathrm{pixel}$. Note the $O(10)\%$ floor in delensed power when using the nonperturbative template in the ideal limit (top-left corner) and how the gradient-order template consistently outperforms the nonperturbative one, even before the latter has hit its delensing floor. The improvement will start to be significant for noniterative lensing reconstructions in the era of CMB-S4 [44] (marked with stars).