# Impact of internal-delensing biases on searches for primordial $B$-modes of
CMB polarisation

Antón Baleato Lizancos  Anthony Challinor  and Julien Carron

###### Abstract

Searches for the imprint of primordial gravitational waves in degree-scale CMB
$B$-mode polarisation data must account for significant contamination from
gravitational lensing. Fortunately, the lensing effects can be partially
removed by combining high-resolution $E$-mode measurements with an estimate of
the projected matter distribution. In the near future, experimental
characteristics will be such that the latter can be reconstructed internally
with high fidelity from the observed CMB, with the $EB$ quadratic estimator
providing a large fraction of the signal-to-noise. It is a well-known
phenomenon in this context that any overlap in modes between the $B$-field to
be delensed and the $B$-field from which the reconstruction is derived leads
to a suppression of delensed power going beyond that which can be attributed
to a mitigation of the lensing effects. More importantly, the variance
associated with this spectrum is also reduced, posing the question of whether
the additional power suppression could help better constrain the tensor-to-
scalar ratio, $r$. In this paper, we show this is not the case, as suggested
but not quantified in previous work. We develop an analytic model for the
biased delensed $B$-mode angular power spectrum, which suggests a simple
renormalisation prescription to avoid bias on the inferred tensor-to-scalar
ratio. With this approach, we learn that the bias necessarily leads to a
_degradation_ of the signal-to-noise on a primordial component compared to
“unbiased delensing”. Next, we assess the impact of removing from the lensing
reconstruction any overlapping $B$-modes on our ability to constrain $r$,
showing that it is in general advantageous to do this rather than modelling or
renormalising the bias. Finally, we verify these results within a maximum-
likelihood inference framework applied to simulations.

##  1 Introduction

The theoretical framework best fitting cosmological observations includes a
period of accelerated expansion at very early times called inflation. It is a
general feature amongst inflationary models that a background of gravitational
waves (tensor perturbations) would have been generated during that era along
with fluctuations in the density (scalar perturbations). Although difficult to
be detected directly, primordial gravitational waves should have left an
imprint on the temperature and polarisation of the cosmic microwave background
(CMB) that may be detectable if inflation occurred at a sufficiently high
energy scale [1, 2, 3].

Although scalar perturbations are now known to overwhelm any contribution from
tensor perturbations to the temperature anisotropy ($T$) or the gradient-like
$E$-mode component of the polarisation, it is possible to form a curl-like
component ($B$-mode) that at recombination is sourced only by tensor
perturbations (in linear theory). A detection of a primordial $B$-mode is
therefore widely regarded as a conclusive test of inflation. Furthermore, a
measurement of the ratio of primordial tensor-to-scalar power, $r$, would
reveal the energy scale of inflation. Current experimental bounds place
$r<0.06$ with $95\,\%$ confidence [4].

As we search for a signal below this level, it is not sufficient just to
improve the sensitivity of experiments, for gravitational lensing of CMB
photons by the large-scale distribution of matter in the Universe contaminates
the signal by converting part of the primordial $E$-mode signal into $B$-mode
[5]. The $B$-modes induced by gravitational lensing were first detected by
[6]. These have a power spectrum resembling that of white noise with
$\Delta_{P}=5\mu\text{K\,arcmin}$ on the large angular scales where the
primordial signal is expected to be strongest. Consequently, the uncertainty
on measurements of the primordial tensor signal will henceforth be dominated
by the large variance associated with the lensing component unless the latter
can be removed.

Unfortunately, the removal of lensing $B$-modes cannot be achieved via multi-
frequency cleaning, since lensing merely induces a remapping of photons by
small deflection angles (of order a few arcminutes) while preserving the
blackbody spectrum of the unlensed CMB. Progress can be made, however, by
estimating the projected matter distribution responsible for the deflections,
and undoing them (at least partially). Several groups have already applied
this procedure, known as delensing, to real $B$-mode data [7, 8, 9, 10, 11,
12]. For the foreseeable future (as long as lensing residuals remain larger
than $O(1\%)$ of the original amplitude), remapping by the inverse deflections
can be approximated by the more analytically transparent and computationally
efficient subtraction of a template estimating the lensing $B$-modes to
leading order [13], that is, a weighted convolution of $E$-mode observations
with some estimate of the lensing deflections on the sky. This procedure,
which we refer to as _template delensing_ , has successfully been applied to
data [14, 8].

The extent to which the lensing signal can be removed depends on the fidelity
with which the lensing potential can be estimated. For sufficiently low
experimental noise levels, the lensing potential, $\phi$, can be reconstructed
internally from the CMB maps themselves by employing either the quadratic
estimators of ref. [15] or the more powerful, albeit analytically and
computationally complex, maximum-likelihood [16] or Bayesian methods [17, 18].
For experimental noise levels available already in the next generation of CMB
observatories, a quadratic combination involving $E$\- and $B$-fields will
provide a sizeable fraction of the signal-to-noise, dominating over other
estimators in the regime where $\Delta_{P}<5\,\mu\text{K\,arcmin}$.

However, ref. [19] showed that whenever there is an overlap in modes between
the field we wish to delens and the fields from which a lensing reconstruction
is derived, the delensed power is subject to a bias that leads to a
suppression in power going beyond that which can be attributed to a mitigation
of the lensing effects. More importantly, the variance associated with this
delensed spectrum is also reduced, posing the question of whether the bias
could help better constrain the tensor-to-scalar ratio, $r$.

In this paper, we set out to understand the angular power spectrum of
$B$-modes after internal delensing with an $EB$ quadratic estimator, as a
function of experimental characteristics and, importantly, primordial $B$-mode
power, assuming no foreground or survey non-idealities. In section 2, we
introduce the quadratic estimators for lensing reconstruction, focusing on the
$EB$ combination. In section 3, we show how an estimate of the lensing
$B$-modes at the map level can be made by combining observations of $E$ and
estimates of $\phi$. Then, in section 4, we introduce analytic models for the
angular power spectrum of delensed $B$-modes in both the cases with and
without overlapping modes, and compare them to simulations. (These models are
derived carefully in appendix A, and the simulations are described in appendix
B.) In section 5, we consider simple models for the covariance of the delensed
$B$-mode power spectrum. We combine the power spectrum models and covariances
in a maximum-likelihood framework in section 6, where we simulate inferences
of $r$ to compare quantitatively the different ways in which the bias can be
dealt with. Finally, in section 7, we briefly study the impact of the bias on
an alternative estimator which correlates observed and delensed $B$-modes.

The results in this paper will often refer to the experimental specifications
of the upcoming Simons Observatory (SO)[20], which will feature a large-
aperture telescope (LAT) responsible, among other scientific targets, for
lensing reconstruction; and separate small-aperture telescopes (SATs) for
observations of $B$-modes on large angular scales. However, the insights
developed here apply more widely, and will likely be relevant to any
application of internal delensing that uses information from the $B$-modes for
the purpose of lensing reconstruction.

##  2 Quadratic estimators for lensing reconstruction

In order to undo the deflections induced by lensing, an estimate of the
projected matter distribution on the sky – which determines the lensing
potential – is required. For the sensitivities and resolution of current CMB
experiments, the best possible estimate is obtained from tracers external to
the CMB such as the cosmic infrared background (CIB) or very deep galaxy
surveys [21, 22, 23, 24]. The CIB retains a high degree of correlation with
the smaller-scale lenses at high redshift that are important for converting
$E$-modes into $B$-modes, something that is not yet possible with internal
lensing reconstruction, which is very noisy on the relevant angular scales.
However, this situation will change with planned CMB polarisation surveys,
which can provide high signal-to-noise lensing reconstructions on nearly all
scales relevant for $B$-mode delensing [25]. Consequently, reconstructions of
$\phi$ derived from the CMB fields themselves will ultimately provide the best
delensing performance.

Internal reconstruction techniques use the fact that, if we could average over
realisations of the CMB while keeping the lensing potential fixed, lensing
would break statistical isotropy by inducing correlations between fluctuations
on different scales; consequently, the lensing potential can be reconstructed
by combining many off-diagonal correlations. In general, the optimal internal
reconstruction of $\phi$ will be obtained from studying the likelihood
function of the lensed CMB temperature and polarisation anisotropies [26, 16,
17, 18]. However, for the noise levels attainable in the near future, the
optimal lensing reconstruction has been shown (see, e.g., ref. [27]) to be
equivalent to that from the more tractable and computationally efficient
“quadratic estimators” of refs. [28, 15]. In fact, in the upcoming era of
high-resolution CMB experiments such as the Simons Observatory [20] and SPT3G
[29], which feature experimental noise levels
$1\,\mu\mathrm{K\,arcmin}<\Delta_{P}<10\,\mu\mathrm{K\,arcmin}$, the optimal
reconstruction will arise from a combination of external tracers and internal
reconstructions involving quadratic estimators [22].

A minimum-variance internal reconstruction of the lensing potential can be
obtained from combining different quadratic estimators, as seen in figure 1.
For low enough noise levels, the quadratic estimator involving $E$ and $B$
fields is expected to provide the highest signal-to-noise reconstruction – the
reason being that, in this case, the dominant source of reconstruction noise
involves the Gaussian contraction $\langle EE\rangle\langle BB\rangle$, and
the lensing and primordial contributions to $\langle BB\rangle$ are small
[27]. It is because of its relevance in upcoming efforts to delens the CMB
that we focus on the $EB$ estimator in this paper. It takes the form

$\hat{\phi}_{EB}(\bm{L})=A^{EB}_{L}\int\frac{d^{2}\bm{l}}{(2\pi)^{2}}W(\bm{L}-\bm{l},-\bm{l})\frac{\tilde{C}^{EE,\,\mathrm{fid}}_{l}}{C^{EE,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{l}C^{BB,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{|\bm{l}-\bm{L}|}}E^{\text{obs},\text{LAT}}(\bm{l})B^{\text{obs},\text{LAT}}(\bm{L}-\bm{l}),$
(2.1)

where $E^{\text{obs},\text{LAT}}$ and $B^{\text{obs},\text{LAT}}$ are beam-
deconvolved observed fields111In the case where an observatory consists of
separate telescopes with different characteristics and mapping strategies, the
one with the larger aperture – labeled here as LAT, in contrast with the SAT,
or small aperture telescope – will provide the superior lensing reconstruction
as it can more finely resolve the small scales where lensing dominates.,
$A^{EB}_{L}$ is a normalisation factor,
$C^{XX,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{l}$ is the fiducial, total
power observed by the LAT which is used to inverse-variance filter222For
simplicity, we take this filter to be diagonal. In practical applications with
masking and/or inhomogeneous noise, the optimal filtering function will no
longer be diagonal. the field $X$, and the geometric coupling

$W(\bm{l},\bm{l}^{\prime})\equiv\bm{l}^{\prime}\cdot(\bm{l}-\bm{l}^{\prime})\sin
2\left(\psi_{\bm{l}}-\psi_{\bm{l}^{\prime}}\right).$ (2.2)

Here, $\psi_{\bm{l}}$ is the angle between $\bm{l}$ and the $x$-axis used to
define positive Stokes parameter $Q$, and similarly for
$\psi_{\bm{l}^{\prime}}$. Notice the use of the fiducial lensed
$\tilde{C}_{l}^{EE,\mathrm{fid}}$ instead of its unlensed counterpart, since
this has been shown to optimise the correlation of estimates of the lensing
potential with the underlying truth333For the experimental configurations
considered in this paper, the delensing performance obtained by using these
weights appears to be indistinguishable from the case where the unlesed
spectra are used instead. [30, 31]. If eq. (2.1) were to minimise exactly the
variance of $\hat{\phi}_{EB}$, it would include an additional term
proportional to the primordial $B$-mode spectrum444While we denote as
$\tilde{C}_{l}^{EE}$ the lensed $E$-mode spectra (including primordial
$E$-modes), we follow the standard convention that $\tilde{C}_{l}^{BB}$
comprises only the lensing contribution to the $B$-mode spectrum, with the
primordial part given separately by $C_{l}^{BB,t}$, $C_{l}^{BB,t}$, like the
one shown but replacing
$W(\bm{L}-\bm{l},-\bm{l})\tilde{C}_{l}^{EE,\mathrm{fid}}\rightarrow-W(-\bm{l},\bm{L}-\bm{l})C_{|\bm{L}-\bm{l}|}^{BB,\mathrm{t,fid}}$.
We ignore it here, however, as its effect is negligible for values of $r$
compatible with experimental bounds.

Figure 1: Lensing reconstruction noise levels associated with the five
different quadratic estimators, and their minimum-variance combination, for an
experiment with resolution $\theta_{\mathrm{FWHM}}=1.5\,\text{arcmin}$ (full
width at half maximum) and isotropic white noise with
$\Delta_{T}=6\,\mu\text{K\,arcmin}$ for temperature and
$\Delta_{P}=\sqrt{2}\Delta_{T}$ for polarisation. The maximum multipole used
in the reconstruction is $l_{\mathrm{max}}=3000$. These specifications are
similar to the goals for the SO LAT at 145 GHz. The dashed blue line is the
lensing potential power spectrum.

##  3 Template delensing of the $B$-mode

Once we have an estimate for the lensing potential, delensing can be performed
by using it to build a template approximating the lensing $B$-modes,
$B^{\text{temp}}$, which can then be subtracted from the observed
$B$-modes,555In multi-telescope observatories, small aperture telescopes
(SATs), which have proven stability to large-scale signals, are being adopted
to pursue primordial $B$-modes. $B^{\text{obs},\,\text{SAT}}$, to yield a
delensed field

$B^{\text{del}}=B^{\text{obs},\,\text{SAT}}-B^{\text{temp}}.$ (3.1)

To leading order in $\bm{\nabla}\phi$ (what is usually referred to as the
“gradient approximation”, and is known to be an excellent approximation to the
non-perturbative calculation of the lensed $B$-mode spectrum on large angular
scales [32]), the template can be constructed as

$B^{\text{temp}}(\bm{l})=\int\frac{d^{2}\bm{l}^{\prime}}{(2\pi)^{2}}f(\bm{l},\bm{l}^{\prime})W(\bm{l},\bm{l}^{\prime})E^{\text{obs},\text{LAT}}(\bm{l}^{\prime})\hat{\phi}(\bm{l}-\bm{l}^{\prime}),$
(3.2)

with $W(\bm{l},\bm{l}^{\prime})$ as defined in eq. (2.2). The function
$f(\bm{l},\bm{l}^{\prime})$ can be chosen to minimise the power spectrum of
the delensed field (3.1), which ref. [21] shows to be the case when the fields
are filtered with

$f(\bm{l},\bm{l}^{\prime})=\mathcal{W}^{E}_{l^{\prime}}\mathcal{W}^{\phi}_{|\bm{l}-\bm{l}^{\prime}|},$
(3.3)

where

$\mathcal{W}^{E}_{l}\equiv\frac{\tilde{C}_{l}^{EE,\mathrm{fid}}}{C_{l}^{EE,\text{\text{obs},}\mathrm{fid},\,\text{LAT}}}\quad\text{and}\quad\mathcal{W}^{\phi}_{L}\equiv\frac{{C}_{L}^{\phi\phi,\mathrm{fid}}}{{C}_{L}^{\phi\phi,\mathrm{fid}}+N_{L}^{\phi\phi}},$
(3.4)

are Wiener filters for the $E$-modes and the estimate of the lensing
potential. Here, $N_{L}^{\phi\phi}$ is the reconstruction noise on
$\hat{\phi}$. In practice, we use $N_{L}^{\phi\phi}\approx N^{(0),EB}_{L}$,
the disconnected (Gaussian) contribution to the power spectrum of
$\hat{\phi}$, and ignore higher-order terms such as $N^{(1),EB}_{L}$, as the
former dominates over the latter by several orders of magnitude on all
relevant scales [33]. Notice also that eq. (3.2) uses the observed (lensed and
noisy) $E^{\text{obs},\text{LAT}}$, rather than unlensed (or delensed)
$E$-modes. This is to be preferred _for template delensing_ since an
approximate cancellation suppresses higher-order contributions in
$C_{L}^{\phi\phi}$ to the delensed power spectrum, which would otherwise be as
big as 30 % on large scales [13].

##  4 Angular power spectrum of delensed $B$-modes

Most inflationary models predict a spectrum of primordial perturbations that
are very nearly Gaussian distributed. This character extends approximately to
the fluctuations of the primary CMB, as linear-theory transfer functions are
an excellent approximation in the early universe. For such Gaussian fields,
the data can be losslessly compressed into a power spectrum, making the latter
one of the most powerful tools for studying the CMB. Although the lensed CMB
is actually non-Gaussian (as a result of the Gaussian primary CMB fields being
displaced by the nearly-Gaussian lensing potential), delensing has been shown
to mitigate its non-Gaussian character to the point where the delensed
$B$-modes – if delensed in such a way that the $EB$ delensing bias is avoided
– deviate only slightly from Gaussianity [34] and the information lost from
working with the power spectrum is relatively small.

More importantly, in practical applications involving masked observations and
anisotropic noise, any computation of the exact likelihood of the data becomes
intractable and one must work instead with approximate forms, most typically
involving estimators of the power spectra and their covariance [35].

For these reasons, most efforts to detect primordial $B$-modes work with the
power spectrum of the delensed field of eq. (3.1):

$\displaystyle\langle
B^{\text{del}}(\bm{l}_{1})B^{\text{del}}(\bm{l}_{2})\rangle$
$\displaystyle=\langle
B^{\text{obs},\,\text{SAT}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle-2\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle$
$\displaystyle\qquad+\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{temp}}(\bm{l}_{2})\rangle$
$\displaystyle\equiv(2\pi)^{2}\delta^{(2)}(\bm{l}_{1}+\bm{l}_{2})C_{l_{1}}^{BB,\text{del}},$
(4.1)

where in the last line we have used the statistical isotropy of the CMB to
define $C_{l}^{BB,\text{del}}$, the angular power spectrum of the delensed
$B$-modes. In the remainder of this section we examine this expression in
detail. In section 4.1, we evaluate it in the case where the statistical
errors in the estimated lensing potential are independent of the lensed CMB
fields, exploring the reduction in power associated with removal of part of
the lensing signal. Then, in section 4.2, we study eq. (4) in the case where
$\hat{\phi}$ is derived from an $EB$ quadratic estimator; new couplings then
appear that further suppress the delensed power spectrum beyond a simple
removal of lensing power. We conclude the section by proposing an analytic
model to capture the behaviour of such “bias” terms.

###  4.1 The unbiased case: reconstruction errors statistically independent of
the lensed CMB

If the noise on $\hat{\phi}$ were independent of the $B$-mode we would like to
delens, as would be the case if an external tracer were used for $\hat{\phi}$,
the delensed power spectrum would take the form

$C_{l}^{BB,\text{del},\text{unbiased}}=N_{l}^{BB,\text{SAT}}+C_{l}^{BB,t}+C_{l}^{BB,\text{res}}.$
(4.2)

In addition to the instrumental noise, $N_{l}^{BB,\,\text{SAT}}$, and
primordial component, $C_{l}^{BB,\mathrm{t}}$, there is a contribution from
residual deflections – imperfect delensing – given to leading order in lensing
by [21]

$\displaystyle C_{l}^{BB,\text{res}}$
$\displaystyle\approx\int\frac{d^{2}\bm{l}^{\prime}}{(2\pi)^{2}}W^{2}(\bm{l},\bm{l}^{\prime})C_{l^{\prime}}^{EE}C_{|\bm{l}-\bm{l}^{\prime}|}^{\phi\phi}\big{[}1-\mathcal{W}^{E}_{l^{\prime}}\mathcal{W}^{\phi}_{|\bm{l}-\bm{l}^{\prime}|}\big{]}$
$\displaystyle=\tilde{C}_{l}^{BB}-C_{l}^{W},$ (4.3)

where we have defined

$C_{l}^{W}\equiv\int\frac{d^{2}\bm{l}^{\prime}}{(2\pi)^{2}}W^{2}(\bm{l},\bm{l}^{\prime})\big{[}\mathcal{W}^{E}_{l^{\prime}}C_{l^{\prime}}^{EE,\mathrm{fid}}\big{]}\big{[}\mathcal{W}^{\phi}_{|\bm{l}-\bm{l}^{\prime}|}C^{\phi\phi,\mathrm{fid}}_{|\bm{l}-\bm{l}^{\prime}|}\big{]}$
(4.4)

and assumed that our fiducial model for the lensing power spectrum,
$C^{\phi\phi,\mathrm{fid}}_{l}$, is correct. We note that $C_{l}^{W}$ is
simply the fiducial power spectrum of the $B$-mode template. In appendix A.1,
we explain how eq. (4.1) can be recovered in the analytic framework developed
in section 4.2 to characterise the delensing bias. Notice that, in the limit
of no observational noise and a perfect $\hat{\phi}$, we have
$\mathcal{W}^{E}_{l}\rightarrow 1$ and $\mathcal{W}^{\phi}_{l}\rightarrow 1$
and all of the (leading-order) lensing signal is removed. Higher-order terms
mean that template delensing can, in fact, be used to reduce the lensing
contribution to the power spectrum to approximately $1\,\%$ of its original
level. This is in the case where lensed $E$-modes are employed in the
construction of the $B$-mode template – the alternative of using
unlensed/delensed $E$-modes performs worse with a lensing residual of order
$30\,\%$ as noted above [13]. Were estimates of the lensing potential accurate
enough to reduce the delensed power to the $1\,\%$ level, template delensing
should be replaced with non-perturbative methods whereby the Wiener-filtered
$\hat{\phi}$ is used to remap the observed CMB directly (see, e.g., [23, 7,
10, 9] for demonstrations of this method).

Henceforth, we will refer to eq. (4.2) as the _unbiased_ delensed power
spectrum in order to differentiate it from the case where the errors in
$\hat{\phi}$ are statistically dependent on the lensed CMB, as is the case
when internal delensing with overlapping modes. We now consider the new terms
that arise in the delensed power for this latter case.

###  4.2 The biased case: $\hat{\phi}$ obtained from an $EB$ quadratic
estimator

We discussed in section 2 that, in the upcoming era of low-noise CMB
experiments, the highest signal-to-noise reconstruction using quadratic
estimators will arise from combining $E$ and $B$ fields. Crucially, ref. [19]
first noticed that, if such a reconstruction is used to delens observations of
$B$-modes, the delensed power spectrum will be biased: while eq. (4.1)
continues to quantify the power spectrum due to residual deflections, the
total delensed spectrum will show an additional suppression of power beyond
that which can be attributed to delensing. In this section, we add to the work
of [19, 36, 37] by providing a detailed calculation of the biased delensed
$B$-mode power spectrum, including terms neglected by eq. (4.1).

In order to isolate the effects of the bias, we assume henceforth that the
lensing reconstruction is obtained exclusively from an $EB$ quadratic
estimator. Although this is set to be the dominant source of information on
lensing for upcoming experiments, in real applications the optimal
reconstruction will, in fact, arise as a minimum-variance (MV) combination of
several quadratic estimators. In that scenario, the bias arising from $EB$
reconstruction needs to be propagated through the co-adding procedure, and
will ultimately be reduced relative to the case where only the $EB$ estimator
is used because the other estimators it is combined with do not source a bias.
Hence, to obtain the amplitude of the bias on the delensed $B$-mode spectrum
and on $r$ in the MV case, the results quoted in this paper would need to be
scaled appropriately by the MV weight pertaining to the $EB$ estimator.

In this _biased_ case, the first term in the delensed $B$-mode spectrum of eq.
(4) is still

$\langle
B^{\text{obs},\,\text{SAT}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle=(2\pi)^{2}\delta^{(2)}(\bm{l}_{1}+\bm{l}_{2})C_{l_{1}}^{BB,\text{obs,\,\text{SAT}}},$
(4.5)

but the other two correlators receive important new contributions.
Substituting in eqs. (2.1) and (3.2), the second term becomes

$\displaystyle\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle=$
$\displaystyle\int\frac{d^{2}\bm{l}^{\prime}_{1}d^{2}\bm{l}^{\prime\prime}_{1}}{(2\pi)^{4}}\mathcal{W}^{E}_{l^{\prime}_{1}}\mathcal{W}^{\phi}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}|}$
$\displaystyle\times\frac{A^{EB}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}|}\tilde{C}^{EE,\,\mathrm{fid}}_{l^{\prime\prime}_{1}}}{C^{EE,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{l^{\prime\prime}_{1}}C^{BB,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1}|}}W(\bm{l}_{1},\bm{l}^{\prime}_{1})W(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1},-\bm{l}^{\prime\prime}_{1})$
$\displaystyle\times\langle
E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{1})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{LAT}}(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle.$
(4.6)

The last term, which correlates two templates, takes the form

$\displaystyle\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{temp}}(\bm{l}_{2})\rangle=$
$\displaystyle\int\frac{d^{2}\bm{l}^{\prime}_{1}d^{2}\bm{l}^{\prime}_{2}d^{2}\bm{l}^{\prime\prime}_{1}d^{2}\bm{l}^{\prime\prime}_{2}}{(2\pi)^{8}}\mathcal{W}^{E}_{l^{\prime}_{1}}\mathcal{W}^{E}_{l^{\prime}_{2}}\mathcal{W}^{\phi}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}|}\mathcal{W}^{\phi}_{|\bm{l}_{2}-\bm{l}^{\prime}_{2}|}$
$\displaystyle\times\frac{A^{EB}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}|}A^{EB}_{|\bm{l}_{2}-\bm{l}^{\prime}_{2}|}}{C^{BB,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{|\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1}|}C^{BB,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{|\bm{l}_{2}-\bm{l}^{\prime}_{2}-\bm{l}^{\prime\prime}_{2}|}}\frac{\tilde{C}^{EE,\,\mathrm{fid}}_{l^{\prime\prime}_{1}}}{C^{EE,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{l^{\prime\prime}_{1}}}\frac{\tilde{C}^{EE,\,\mathrm{fid}}_{l^{\prime\prime}_{2}}}{C^{EE,\,\text{obs},\,\mathrm{fid},\,\text{LAT}}_{l^{\prime\prime}_{2}}}$
$\displaystyle\times
W(\bm{l}_{1},\bm{l}^{\prime}_{1})W(\bm{l}_{2},\bm{l}^{\prime}_{2})W(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1},-\bm{l}^{\prime\prime}_{1})W(\bm{l}_{2}-\bm{l}^{\prime}_{2}-\bm{l}^{\prime\prime}_{2},-\bm{l}^{\prime\prime}_{2})$
$\displaystyle\times\langle
E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{1})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{LAT}}(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1})$
$\displaystyle\qquad\times
E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{2})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{2})B^{\text{obs},\,\text{LAT}}(\bm{l}_{2}-\bm{l}^{\prime}_{2}-\bm{l}^{\prime\prime}_{2})\rangle.$
(4.7)

The evaluation of the four- and six-point functions that appear in eqs (4.5)
and (4.2) is discussed in detail in appendix A. These are expanded in terms of
connected $n$-point functions with $n=2$ and $4$ (the connected six-point
function is higher order in $C_{l}^{\phi\phi}$). A subset of these terms
combine to give the standard unbiased result (4.1), as shown in appendix A.1.
The remaining terms introduce corrections, with the most important of these
identified and evaluated in appendix A.2. Combining these results, we show
that the biased delensed $B$-mode power spectrum can be approximated as

$C_{l}^{BB,\text{del}}=(C_{l}^{BB,\text{res}}+C_{l}^{BB,t})(D_{l}-1)^{2}+D_{l}^{2}C_{l}^{W}+N_{l}^{BB,\text{SAT}}+N_{l}^{BB,\text{LAT}}D_{l}^{2}-2D_{l}N_{l}^{X},$
(4.8)

where the correlation of experimental noise in the SAT and the LAT is denoted
as

$N_{l}^{X}=\begin{cases}0&\text{if SAT and LAT are separate,}\\\
N_{l}^{BB,\text{LAT}}=N_{l}^{BB,\text{SAT}}&\text{for a single
telescope.}\end{cases}$ (4.9)

We have also defined

$D_{l}\equiv\frac{1}{C_{l}^{BB,\text{obs},\mathrm{fid},\text{LAT}}}\int\frac{d^{2}\bm{l}^{\prime}}{(2\pi)^{2}}W^{2}(\bm{l},\bm{l}^{\prime})\big{[}\mathcal{W}^{E}_{l^{\prime}}\tilde{C}_{l^{\prime}}^{EE}\big{]}\big{[}\mathcal{W}^{\phi}_{|\bm{l}-\bm{l}^{\prime}|}A_{|\bm{l}-\bm{l}^{\prime}|}^{EB}\big{]},$
(4.10)

whose origin and properties are discussed further below.

Equation (4.8) captures the general case where a large-aperture telescope
(LAT) focuses on lensing reconstruction while a separate small-aperture
telescope (SAT) makes observations of the $B$-modes on large angular scales.
In such a configuration, the experimental noise is uncorrelated between
instruments and $N_{l}^{X}=0$. On the other hand, the case where a single
telescope is used for both purposes can easily be recovered by letting
$N_{l}^{X}=N_{l}^{BB,\text{LAT}}=N_{l}^{BB,\text{SAT}}$, in which case the
biased delensed spectrum reduces to

$C_{l}^{BB,\text{del}}=(C_{l}^{BB,\text{res}}+C_{l}^{BB,t}+N_{l}^{BB})(D_{l}-1)^{2}+D_{l}^{2}C_{l}^{W}.$
(4.11)

All of the correction terms in eq. (4.8) are proportional to one or two powers
of $D_{l}$; for $D_{l}=0$ it reduces to the unbiased result (4.1). The
function $D_{l}$ arises when one contracts the $B$-mode template over the pair
of observed $E$-modes that enter explicitly. In detail,

$\langle
B^{\text{temp}}(\bm{l})\rangle_{E^{\text{obs},\text{LAT}}}=B^{\text{obs},\text{LAT}}(\bm{l})D_{l}\,.$
(4.12)

All of the correction terms retained in eq. (4.8) arise from such
contractions. For example, since $E$\- and $B$-modes are uncorrelated, the
disconnected contribution to $\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle$
gives a term

$\displaystyle\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle$
$\displaystyle\supset D_{l_{1}}\langle
B^{\text{obs},\text{LAT}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle$
$\displaystyle=(2\pi)^{2}\delta^{(2)}(\bm{l}_{1}+\bm{l}_{2})D_{l_{1}}\left(\tilde{C}^{BB}_{l_{1}}+{C}^{BB,\mathrm{t}}_{l_{1}}+{N}^{X}_{l_{1}}\right)\,.$
(4.13)

Note how this involves the cross-correlation of the experimental noise between
the SAT and the LAT, $N_{l}^{X}$, and the tensor $B$-mode power. It is the
main correction term and, given that it enters the delensed power through
$-2\langle
B^{\text{temp}}(\bm{l}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle$,
_suppresses_ the power further. (The other correction terms we retain increase
the delensed power.)

Figure 2: Evaluation of $D_{l}$, defined in eq. (4.10), for several values of
the fiducial $r_{\mathrm{fid}}$ in the filter
$1/C_{l}^{BB,\text{obs},\mathrm{fid},\text{LAT}}$ applied to the $B$-modes in
lensing reconstruction with the $EB$ quadratic estimator. The experimental
set-up is as in figure 1.

We can gain further insight into $D_{l}$ by noting that in the limit of
noiseless $E$-mode observations, and setting the normalisation of the
quadratic estimator to $A_{L}^{EB}=N^{(0),EB}_{L}$ (which is correct in the
usual limit that the fiducial spectra used to inverse-variance filter the CMB
fields in the lensing reconstruction are close to the true total power),
$D_{l}\rightarrow
C_{l}^{BB,\mathrm{res}}/C_{l}^{BB,\text{obs},\mathrm{fid},\text{LAT}}$ [cf.
eq. (4.1)]. Consequently, in this limit $D_{l}$ can be interpreted as the
ratio of residual lensing power to fiducial power used to inverse-variance
filter the $B$-modes in the lensing reconstruction. More generally, the Wiener
filter $\mathcal{W}^{E}_{l}$ will reduce $D_{l}$ below this value. In all
cases $0<D_{l}<1$. In figure 2, we see that $D_{l}$ depends strongly on large
scales on the fiducial primordial $B$-mode power contained in the inverse-
variance filters and parametrised by $r_{\mathrm{fid}}$. (The figure includes
large values of $r_{\mathrm{fid}}$ inconsistent with observations but included
for illustration.) To understand this behaviour recall that, for $r\sim 0.01$,
$C_{l}^{BB,\mathrm{t}}$ becomes comparable to the lensing power on large
angular scales between $10<l<100$. By raising $r_{\mathrm{fid}}$ above this
value, we are effectively down-weighting the large-scale lensing $B$-modes in
the fields from which the reconstruction is derived. Although this will not
significantly affect the fidelity of the lensing reconstruction – which is
derived from information coming chiefly from smaller angular scales – it will
give a sizeable reduction in $D_{l}$ and the bias in the total delensed power
on large scales.

The result in eq. (4.8) is a generalisation of that in ref. [36] (eq. A16
there), which assumes $r=0$ and separate LAT and SAT observations
($N_{l}^{X}=0$). An important insight from our result is that the bias term
responsible for the additional suppression of power, eq. (4.13), is
proportional to the _total_ observed $B$-mode power, and not simply its
lensing component666The model of eq. (4.11) (with $N_{l}^{X}=0$) can be
obtained from eq. (A.16) of ref. [36] by adding $-2D_{l}C_{l}^{BB,\mathrm{t}}$
and including a primordial component in all the $B$-mode auto spectra..
Crucially, our result thus predicts a suppression of the primordial signal
whenever it is present, as first noted in ref. [19]. This effect is also
hinted at in ref. [37], though the fact that their simulations have $r=0$
prevents them from making a quantitative statement.

In order to assess the validity of the model in eq. (4.8), we compare it to
the output of applying the reconstruction and delensing pipelines to
simulations (detailed in appendix B) of an experiment with characteristics
similar to those of SO [20]. Specifically, the LAT survey has angular
resolution $\theta_{\text{FWHM}}=1.5\,\text{armin}$ and temperature noise
level $\Delta_{T}=6\,\mu\text{K\,arcmin}$ (with the polarisation level
$\Delta_{P}=\sqrt{2}\Delta_{T}$), as in figure 1. It is the simulated data
from this telescope which we use for the purpose of lensing reconstruction,
with $l_{\mathrm{max}}=3000$. The SAT survey has
$\theta_{\text{FWMH}}=17\,\text{arcmin}$, $\Delta_{T}=2\,\mu\text{K\,arcmin}$
and $\Delta_{P}=\sqrt{2}\Delta_{T}$. Where needed for calculations of the
power spectrum variance, we assume the SAT survey covers approximately
$2.8\,\%$ of the sky (and is fully contained within the wider LAT survey). The
simulations have tensor-to-scalar ratio $r=0.01$. The results are shown in
figure 3. The biased delensed power spectrum (yellow curve in the top panel)
is seen to be in excellent agreement with eq. (4.8), shown as the dashed black
curve. The differences are smaller than $0.5\,\%$ on all scales. For lower
noise levels, the level of agreement is expected to worsen as bias terms
neglected by our model play an increasingly significant role. Since the most
relevant of those terms (the contribution to the six-point function from the
first term on the right of eq. A.10) makes a positive contribution to the
power, eq. (4.8) is expected to underestimate the amplitude of the biased
delensed power spectrum. Indeed, we simulate a single-telescope experiment
with $\theta_{\mathrm{FWHM}}=6\,\mathrm{arcmin}$ and
$\Delta_{T}=3\,\mu\mathrm{K\,arcmin}$ and find that the modelled biased
spectrum is around $2$–$3\,\%$ lower than its simulated counterpart.

Figure 3: Upper panel: noise-subtracted simulated power spectra of the
observed $B$-mode (green), biased residual after delensing with
$\hat{\phi}^{EB}$ (yellow) and residual after delensing with a reconstruction
cutoff at $l_{\mathrm{cut}}=300$ (cyan). The shaded regions represent the
variance of each of the delensed spectra, obtained from simulations. Notice
that the biased spectrum (yellow) has less variance than its unbiased
counterpart (cyan). Also plotted are noise-subtracted theoretical curves
modelling delensed spectra with (red, dot-dashed) and without (black, dashed)
an $l_{\mathrm{cut}}$. Middle Panel: ratio of model-to-simulated delensed
power spectrum amplitude for the biased (black, dashed) and unbiased (red,
dot-dashed) cases. Lower panel: ratio of delensed power spectrum variance in
the case where no $l_{\mathrm{cut}}$ is imposed to that where
$l_{\mathrm{cut}}=300$, obtained from simulations. All curves are for an
experiment with characteristics similar to the Simons Observatory, as
described in the text.

The bias in the delensed spectrum is around 80 % of the signal power,
$C_{l}^{BB,t}+C_{l}^{BB,\text{res}}$, in the unbiased delensed spectrum for
the configuration in figure 3 (see discussion below). This bias would need to
be modelled or otherwise mitigated in a likelihood analysis if inferences on
the tensor-to-scalar ratio are not to be biased. One simple way to remove the
bias in the delensed spectrum on the scales relevant for searches for
primordial $B$-modes was noted by ref. [19]: exclude from the reconstruction
any $B$-modes that overlap in scales with the $B$-modes we wish to delens.
This follows from the local character of the bias, whose origin is given in
eq. (4.12), meaning that it will be avoided at a given multipole as long as
that multipole is removed from the $B$-field used for the lensing
reconstruction. For the purpose of primordial $B$-mode searches, we shall
remove reconstruction $B$-modes on the largest angular scales $l\leq
l_{\mathrm{cut}}$. In this case, eq. (4.12) becomes

$\langle
B^{\text{temp}}(\bm{l})\rangle_{E^{\text{obs},\text{LAT}}}=B^{\text{obs},\text{LAT}}(\bm{l})D_{l}\Theta(l-l_{\text{cut}})\,,$
(4.14)

where $\Theta(l)$ is the Heaviside function, and so the bias terms in the
delensed power spectrum vanish for $l\leq l_{\text{cut}}$. Note that the
normalisation and noise of the $EB$ quadratic estimator are changed, albeit by
a small amount, on all scales by excising the input $B$-modes at certain
scales, and $D_{l}$ (for $l>l_{\text{cut}}$) needs to be calculated with the
modified normalisation and Wiener filter; see eq. (4.10). Equivalently, we can
think of effectively setting to infinity the noise on the $B$-modes used _in
the reconstruction_ on the scales that we wish to exclude. In this case,
$D_{l}$ goes to zero for $l\leq l_{\mathrm{cut}}$, and both eq. (4.8) and
(4.11) reduce to the unbiased spectrum of eq. (4.2). Averting the bias in this
way comes at the cost of a lower signal-to-noise lensing reconstruction as
information is being discarded, although the degradation is expected to be
minor since most of the lensing information is obtained from smaller scales of
the polarisation fields. In order to take this effect into account when
computing the delensed power spectrum below the cutoff using eq. (4.1), we
must make sure to calculate $C_{l}^{W}$ using a Wiener filter that includes
the higher reconstruction noise. When this modification is included, the
simple unbiased model agrees very well with the simulated spectrum, as shown
in figure 3 for the case of $l_{\text{cut}}=300$, with differences between the
two less than $0.2\,\%$ for an SO-like experiment. Imposing a cutoff at
$l_{\mathrm{cut}}=300$ leads to a lower signal-to-noise lensing reconstruction
and a small decrease in $C_{l}^{W}$ which, on large angular scales, results in
an increase of the residual lensing power spectrum and a degradation of
delensing efficiency by approximately $2\,\%$. Several techniques have been
suggested to reduce this small degradation, including realisation-dependent
methods [37] and an upgrade of the cutoff approach which involves splitting
the whole multipole space into “notches” within which the bias can be avoided
by excluding modes common to the notch and the reconstruction [19, 38].

Figure 4: Remaining fraction of lensing $B$-mode power after delensing with an
$EB$ quadratic estimator for different choices of the cutoff, for an
experiment with LAT specifications as in figure 1. Note that the curves with
$l_{\mathrm{cut}}=50$ and no $l_{\mathrm{cut}}$ overlap visually.

Although the bias from internal delensing can be easily removed on large
scales with the techniques just discussed, an alternative is to model
explicitly the biased delensed spectrum as part of any subsequent likelihood
analysis. Figure 3 suggests it is worth exploring this approach further since
the bias not only suppresses the delensed $B$-mode spectrum beyond what can be
attributed to the (partial) removal of lensing effects, but also reduces its
variance. Indeed, in figure 3 the variance of the biased delensed spectrum is
roughly $65\,\%$ of that associated with the unbiased case (the latter with
$l_{\text{cut}}=300$). If the power spectrum bias were independent of the
primordial $B$-mode power, this reduction in variance would translate directly
into improved constraints on the tensor-to-scalar ratio (assuming there is no
significant difference in the non-Gaussian correlations between the power
spectra at different multipoles; see section 5). However, part of the bias
does depend on the primordial spectrum, suppressing the contribution of
$C_{l}^{BB,t}$ in the biased delensed spectrum by a factor $(D_{l}-1)^{2}$
(see eq. 4.8).

The suppression of the primordial signal in the biased delensed spectrum acts
as a multiplicative bias. We can remove this by renormalising by
$(D_{l}-1)^{-2}$, in which case the contribution from $C_{l}^{BB,t}$ is the
same as for unbiased delensing:

$\frac{C_{l}^{BB,\text{del}}}{(D_{l}-1)^{2}}=C_{l}^{BB,\text{del},\text{unbiased}}+\left(\frac{D_{l}}{D_{l}-1}\right)^{2}\left[C_{l}^{W}+N_{l}^{BB,\text{LAT}}+N_{l}^{BB,\text{SAT}}\left(\frac{2}{D_{l}}-1\right)-\frac{2}{D_{l}}N_{l}^{X}\right],$
(4.15)

with $C_{l}^{BB,\text{del},\text{unbiased}}$ as defined in equation (4.2). The
last three noise terms on the right-hand side vanish for the case of
$B^{\text{obs},\text{LAT}}=B^{\text{obs},\text{SAT}}$, i.e., when the same
maps are used in all parts of the analysis, and are positive when
$B^{\text{obs},\text{LAT}}$ and $B^{\text{obs},\text{SAT}}$ have independent
instrument noise ($N_{l}^{X}=0$) since $0<D_{l}<1$. On the other hand, the
term proportional to $C_{l}^{W}$ is always positive. Crucially, this means
that, in presence of the bias, the renormalised delensed power is generally
larger than from the unbiased approach.777Mitigating the bias by excluding
large-scale $B$-modes from the reconstruction will increase
$C_{l}^{BB,\text{del},\text{unbiased}}$ a little since the reconstruction is
noisier (figure 4). For our instrument configuration, this increase is only
around 2 % of $\tilde{C}_{l}^{BB}$ or 8 % of $C_{l}^{W}$ on large scales. By
way of comparison, for the $N_{l}^{X}=0$ case, the instrument noise terms in
square brackets in equation (4.15) are about $25C_{l}^{W}$ (taking
$\tilde{C}_{l}^{BB}$ equivalent to $5\,\mu\text{K\,arcmin}$ white noise and
$C_{l}^{W}=0.25\tilde{C}_{l}^{BB}$), and on multiplying by
$D_{l}^{2}/(D_{l}-1)^{2}\approx 0.045$, the additional power over
$C_{l}^{BB,\text{del},\text{unbiased}}$ is about $1.1C_{l}^{W}$, which is
worse than doing no delensing at all. In other words, the primordial power is
always suppressed by a larger fraction than the sources of (lensing and
experimental) noise. This is illustrated in figure 5 where we see that, for
the experimental configuration adopted in figure 3, the signal-to-noise noise
on a primordial component is lower when the bias is allowed to play a part and
modelled than in the alternative approaches where the bias is avoided – worse,
even, than in the case of no delensing. In section 6, we shall see that this
indeed translates to errors on estimates of $r$ from a maximum-likelihood
inference pipeline that inevitably increase (in a statistical sense) whenever
the bias is modelled instead of avoided. For noise levels lower than those
considered here, we expect the approximate expression for the biased delensed
spectrum, equation (4.8), to underestimate the true power, as noted above,
with the dominant correction coming from the first term on the right of eq.
(A.10). In this term, $B$-modes only enter through a connected four-point
function of the form $\langle BEEB\rangle_{c}$, which receives no contribution
from tensor modes (ignoring lensing for these). It follows that the additional
term adds power but does not couple to $C_{l}^{BB,t}$, and so further reduces
the signal-to-noise on the primordial signal.

Figure 5: Effective noise power (i.e., the difference between the total
delensed power and the primordial contribution) for different estimators from
which a primordial $B$-mode signal (black dashed) is to be extracted. In the
biased case, the spectrum is first renormalised so that the amplitude of the
primordial power is the same as in the unbiased cases. The experimental set-up
is the same as in figure 3.

The renormalised spectrum of eq. (4.15) is lower, all other things being
equal, if the same maps are used for all parts of the analysis. The reason is
that in this case the experimental noise in the reconstruction and in the
$B$-modes to be delensed are correlated, allowing the disconnected couplings
(A.15) and (A.16) to suppress the experimental noise contribution to the
delensed power spectrum by the same fraction as for the primordial signal. If,
on the contrary, the noise is uncorrelated between the reconstruction and the
$B$-modes to delens, it will not contribute to the disconnected term (A.16)
and will consequently be suppressed by a smaller amount than the signal – in
fact, it will be amplified relative to the unbiased case by coupling (A.15).

The degradation in signal-to-noise on the primordial $B$-mode power in the
biased case can be understood by noticing that, in the case
$N_{l}^{BB,\text{LAT}}=N_{l}^{BB,\text{SAT}}=N_{l}^{X}$, the disconnected
couplings (A.15) and (A.16) combine with the terms retained in the unbiased
calculation to yield a suppression of the primordial power and experimental
noise by an equal fraction, while for the lensing contribution there is an
additional (positive) term, eq. (A.2), which reinstates some lensing power
while leaving the primordial signal unaffected. This last contribution arises
from the connected coupling of the $B$-mode in one of the templates and the
$EEB$ fields in the other template. This is not the only term that restores
lensing power, since the partially-connected six-point function (A.6) coupling
the lensing part of $\hat{{\phi}}$ across templates has a similar effect.
However, the latter is already included in the unbiased calculation.

##  5 Covariances of delensed power spectra

The precision with which we can isolate a primordial component from the
observed (or delensed) $B$-mode power spectrum is ultimately determined by the
covariance of the latter. The non-Gaussian covariance of lensed CMB spectra is
well understood by now [39, 40, 41, 42] and models exist that allow for its
numerical evaluation. In this work, we use the publicly-available code LensCov
[42] to compute the theoretical bandpower covariances for observations
corresponding to our choice of cosmology and experimental parameters. We find
good agreement between this theoretical computation and results from
simulations after applying the appropriate binning and sky fraction
corrections, as illustrated by figures 6 and 7. For visualisation purposes, we
plot the cross-correlation coefficient of bandpowers, defined as

$R_{l_{1}l_{2}}(\tilde{C}^{BB}_{l})\equiv\frac{\mathrm{Cov}(\tilde{C}^{BB}_{l_{1}},\tilde{C}^{BB}_{l_{2}})}{\sqrt{\mathrm{Cov}(\tilde{C}^{BB}_{l_{1}},\tilde{C}^{BB}_{l_{1}})\mathrm{Cov}(\tilde{C}^{BB}_{l_{2}},\tilde{C}^{BB}_{l_{2}})}},$
(5.1)

where $\mathrm{Cov}(\tilde{C}^{BB}_{l_{1}},\tilde{C}^{BB}_{l_{2}})$ is the
covariance of the binned angular power spectrum

$\mathrm{Cov}(\tilde{C}^{BB}_{l_{1}},\tilde{C}^{BB}_{l_{2}})=\langle\tilde{C}^{BB}_{l_{1}}\tilde{C}^{BB}_{l_{2}}\rangle-\langle\tilde{C}^{BB}_{l_{1}}\rangle\langle\tilde{C}^{BB}_{l_{2}}\rangle,$
(5.2)

and the angle brackets denote averaging over realisations of either the lensed
or delensed CMB and noise.

The delensed bandpower covariance is more complicated, but it can begin to be
understood by studying the case where the lensing reconstruction is
independent of the CMB. Reference [34] presents the following extension of the
lensed $B$-mode power covariance to this delensed case under the assumptions
that the $E$-modes used in the template are cosmic-variance limited on all
relevant scales (i.e., for multipoles $l\lesssim 2000$) and that $\phi$ is
Gaussian:

$\displaystyle\mathrm{Cov}(\tilde{C}^{BB,\mathrm{del}}_{l_{1}},\tilde{C}^{BB,\mathrm{del}}_{l_{2}})$
$\displaystyle\approx\frac{2}{2l_{1}+1}\big{(}\tilde{C}_{l_{1}}^{BB,\mathrm{del}}\big{)}^{2}\delta_{l_{1}l_{2}}+\sum_{l}\bigg{(}\frac{\partial\tilde{C}_{l_{1}}^{BB,\mathrm{res}}}{\partial
C_{l}^{EE}}\frac{2}{2l+1}(C_{l}^{EE})^{2}\frac{\partial\tilde{C}_{l_{2}}^{BB,\mathrm{res}}}{\partial
C_{l}^{EE}}\bigg{)}$
$\displaystyle+\sum_{l}\bigg{(}\frac{\partial\tilde{C}_{l_{1}}^{BB,\mathrm{res}}}{\partial
C_{l}^{\phi\phi}}\frac{2}{2l+1}(C_{l}^{\phi\phi})^{2}\frac{\partial\tilde{C}_{l_{2}}^{BB,\mathrm{res}}}{\partial
C_{l}^{\phi\phi}}\bigg{)}.$ (5.3)

The assumption that $E$-modes are limited by cosmic variance on the relevant
scales is not strictly true for the specifications adopted here. Although it
is not difficult to generalise eq. (5) to account for noise in the $E$-modes,
ref. [34] finds that the covariance without $E$-mode noise still matches well
that obtained from simulations of experiments with the noise levels of current
and upcoming experiments.

We evaluate equation (5) by modifying the LensCov code for the calculation of
lensed $B$-mode covariances. Along with a modified Gaussian variance (the
first term on the right of eq. 5), we must also replace the derivative terms
with those appearing above. We calculate these analytically from the leading-
order expression of equation (4.1). The results are shown in figure 6. From
this figure, and more acutely from figure 7, it is clear that delensing
reduces the non-Gaussian correlation between scales introduced by lensing
[39], effectively increasing the number of independent pieces of information
available. Figures 6 and 7 also suggest that the simple model of ref. [34] is
consistent – up to Monte-Carlo errors from the finite number (25 000) of
simulations888It can be shown (assuming that the errors on estimates of
individual bandpowers are distributed as Gaussian random variables with
appropriate correlations) that the fractional error on an element of the
correlation matrix, estimated from $N$ simulations, is
$\frac{\sigma(R_{{ij}})}{|R_{ij}|}=\sqrt{\frac{1}{N}\left(1+\frac{1}{R^{2}_{ij}}\right)},$
where $R_{ij}$ are the true correlation coefficients between bandpowers and
the mean bandpowers are known in advance. In the more usual case where the
variances are also estimated from the data, the relation is replaced with
$\frac{\sigma(R_{{ij}})}{|R_{ij}|}=\frac{1}{\sqrt{N}}\left|\frac{1}{R_{ij}}-R_{ij}\right|.$
– with the simulated covariance of delensed $B$-mode bandpowers below the
cutoff, where $\hat{\phi}$ is effectively independent of $B$.

Figure 6: Correlation coefficient of $B$-mode bandpowers ($\Delta l=30$) for
an experiment with characteristics similar to the Simons Observatory, as
described in section 4.2, and a SAT sky fraction of $2.8\,\%$ (with periodic
boundary conditions assumed). The fiducial cosmology includes a primordial
$B$-mode component with $r_{\mathrm{input}}=0.01$. The top row are obtained
from 25 000 simulations and the bottom row are analytic approximations
evaluated by modifying LensCov [42], as described in the text. Right column:
observed (lensed and noisy) bandpowers. Central column: delensed bandpowers in
the case where the bias is avoided by either introducing a cutoff at
$l_{\mathrm{cut}}=300$ (top) – thus avoiding the bias for $l<l_{\mathrm{cut}}$
– or by working with a $\hat{\phi}$ that is independent of $B$ yet just as
correlated with the actual $\phi$ as the reconstructed $\hat{\phi}^{EB}$
employed in the simulated analysis (bottom). Left column: biased delensed
bandpowers from simulations. Figure 7: Rows of the correlation matrices of
bandpowers of the observed (top) and delensed (bottom) $B$-modes shown in
figure 6. Dashed curves correspond to simulated covariances, with the shaded
region representing their approximate $1\,\sigma$ simulation error due to the
finite number (25 000) of simulations. In the delensed case, the simulations
involve a reconstruction $l_{\mathrm{cut}}=300$ so that the delensed power is
unbiased below $l<300$. Theory curves are shown as solid lines, taken from the
bottom panels of figure 6. Gaps are shown where the correlation coefficient is
one.

Finally, we study the bandpower covariance in the case where no
$l_{\mathrm{cut}}$ is imposed and thus the delensed $B$-mode spectrum is
biased. An analytic exploration of this case is beyond the scope of this
paper, since the framework of appendix A.2 suggests that the covariance
receives contributions from 12-point correlators of lensed polarisation fields
although some simplifications are likely possible. It can, however, be
obtained from simulations, as shown in figure 6. Interestingly, at the noise
levels considered here, the lower variance induced by the bias does not appear
to be accompanied by the larger off-diagonal correlations between bandpowers
seen by ref. [43]. The most likely explanation is that the Gaussian
experimental noise dominates the variance on the small angular scales where we
would expect the bias to induce strong covariance between delensed bandpowers
(cf. figure 11 of [43]). This suppresses the correlation and increases the
fractional Monte-Carlo error to the point where the signal gets buried in the
noise.

In summary, we have seen that, for an experiment with characteristics similar
to SO, the $EB$ lensing reconstruction bias brings about a reduction in the
variance of the power spectrum of delensed $B$-modes while the extent of the
correlations between bandpowers remains comparable to the unbiased case. If
the primordial signal were not suppressed by the bias, analysis of the biased
delensed bandpowers (with appropriate modelling of the bias) would lead to
improved constraints on the tensor-to-scalar ratio. However, we have already
seen in section 4.2, that the bias acts multiplicatively on the primordial
power, suppressing the signal more than the Gaussian variance. We thus expect
constraints on $r$ to worsen in this case, which we now demonstrate through a
simulated maximum-likelihood analysis.

##  6 Maximum-likelihood inference of $r$

Although the likelihood of CMB temperature and polarisation data can in
principle be written down exactly, even accounting for the effects of lensing
[26, 17, 18], in practical applications with realistic survey complexities,
easily-computable summary statistics (e.g., angular power spectra) and their
approximate sampling distributions are typically preferred (see [44] for a
review). For power spectra of approximately Gaussian fields, several such
likelihood approximations involving the estimated power spectra and their
covariance appear in the literature. We choose to work with the approximation
developed in ref. [35], and which was used by the BICEP/Keck collaboration to
analyse their most recent data [4]. For our application here, it takes the
form

$-2\ln\mathcal{L}(r|\\{\hat{C}^{BB}_{i}\\})=\sum_{ij}[g(\hat{C}^{BB}_{i}/C^{BB}_{i})C^{BB,\mathrm{fid}}_{i}]\mathrm{Cov}^{-1}(C^{BB,\mathrm{fid}}_{i},C^{BB,\mathrm{fid}}_{j})[C^{BB,\mathrm{fid}}_{j}g(\hat{C}^{BB}_{j}/C^{BB}_{j})],$
(6.1)

where $g(x)\equiv\text{sign}(x-1)\sqrt{2(x-\ln x-1)}$ and $\hat{C}^{BB}_{i}$
($C^{BB}_{i}$) denotes the $i$th empirical (model) bandpower of the delensed
$B$-mode spectrum. Only multipoles below $l_{\mathrm{max}}=280$ are used in
the inferences that follow. An advantage of using this approximate likelihood
is that the non-Gaussian character introduced by lensing can be accounted for
through the fiducial delensed bandpower covariance,
$\mathrm{Cov}(C^{BB,\mathrm{fid}}_{i},C^{BB,\mathrm{fid}}_{j})$, which only
needs to be computed (either analytically or via simulations) and inverted
once. As long as it matches the fiducial $C_{l}^{BB,\mathrm{fid}}$, the exact
form of this fiducial covariance has a negligible impact on the resulting
inferences. Indeed, we verify that there is no appreciable change in the
inferences when we vary the fiducial level of primordial $B$-mode signal.
Furthermore, ref. [35] shows that, even if the fiducial model deviates from
the truth, the likelihood is still exact in the full-sky, isotropic limit with
Gaussian fields. Hence, all analyses presented henceforth use fiducial
bandpower amplitudes and covariances with the same $r_{\text{input}}$.

The question remains whether the bandpower covariances ought to be modelled or
whether they could simply be obtained from simulations. In section 5 we saw
that, while it is possible to write a simple analytic model for the covariance
of unbiased delensed bandpowers, doing the same for the biased case is more
difficult. To address this question partially, we compare the distribution
across simulations of the maximum-likelihood estimates for the tensor-to-
scalar ratio $\hat{r}_{\mathrm{ML}}$ and their associated uncertainties
$\sigma(\hat{r}_{\mathrm{ML}})$ – derived from the second derivative of the
log-likelihood function at the maximum-likelihood point – in the cases where
the bandpower covariance is obtained analytically (using LensCov) or from
simulations. We do this for the cases of no delensing and unbiased delensing,
the latter excluding $B$-modes below $l_{\mathrm{cut}}=300$ from the simulated
fields used for reconstructing $\hat{\phi}^{EB}$. The simulations used in
these comparisons, and for all results presented in this section, assume the
experimental set-up described in section 4.2. The likelihood curves for these
two cases are shown in figure 8 for a typical realisation, and the
distribution of $\hat{r}_{\text{ML}}$ and $\sigma(\hat{r}_{\text{ML}})$ across
25 000 simulations are given in figure 9. We find only a slight degradation in
the errors on $r$ when simulated bandpower covariances are employed,
justifying our ensuing use of simulated matrices.

Figure 8: Typical likelihood functions for the tensor-to-scalar ratio $r$ for
analyses of simulated data with and without (unbiased) delensing. In both
cases, the use of simulated (solid lines) or analytic (dashed lines) power
spectrum covariances is compared and found to be in good agreement. All curves
are normalised to have the same value at the maximum-likelihood point. The
input value $r_{\text{input}}=0.01$ is indicated with the dotted line. The
simulations are for the experimental set-up described in section 4.2. Figure
9: Distributions of $\hat{r}_{\mathrm{ML}}$ (left) and
$\sigma(\hat{r}_{\mathrm{ML}})$ (right) across simulations comparing the use
of analytic (dashed lines) or simulated (solid lines) power spectrum
covariances in the cases of no delensing and unbiased delensing. The
distributions have been estimated from 25 000 simulated maximum-likelihood
inferences, with a typical likelihood curve as shown in figure 8, using a
Gaussian kernel density method with a bandwidth of $5\times 10^{-4}$ for the
left panel and $7\times 10^{-5}$ for the right.

We illustrate the need to account for (or mitigate) the bias in the delensed
$B$-mode power spectrum when delensing with $\hat{\phi}^{EB}$ with the
following naive analysis. For the empirical spectrum $\hat{C}_{l}^{BB}$ used
in the likelihood of eq. (6.1), we take the simulated biased delensed
spectrum, but all other fiducial and model spectra and covariances in the
likelihood are unbiased – that is, calculated assuming that $\hat{\phi}$ is
independent of the CMB but with the same correlation to $\phi$ as
$\hat{\phi}^{EB}$. In figure 10, we verify that ignoring the suppression in
the biased empirical spectrum leads to inferences on $r$ that are biased low
relative to the input $r_{\mathrm{input}}=0.01$, with a significant number of
likelihoods peaking (in the range $r\geq 0$) at $r=0$, and artificially small
typical errors on $r$. This is also true in the null case where
$r_{\mathrm{input}}=0$, as shown in figure 11.

We have already discussed, in section 4.2, several ways in which the bias can
be taken into account: modelling the biased spectrum; renormalizing the biased
spectrum to restore unit response to primordial power; or imposing a low-$l$
cutoff on the $B$ fields used for estimating $\hat{\phi}$. The first two
approaches should yield equivalent results on $r$, but are expected to be very
non-optimal since the bias reduces the primordial power by a larger fraction
than the rest of the power (see figure 5). Mitigating the bias with a cut-off
is preferred, although there is a small penalty due to the lower signal-to-
noise lensing reconstruction in this case. In figure 10, we quantitatively
compare these methods. For our experimental set-up and input value
$r_{\text{input}}=0.01$, we see that modelling the bias inflates the errors on
$r$ by typically $15\,\%$,999The exact figure depends on the experimental
characteristics and $r_{\text{input}}$. We remind the reader that our
likelihood results assume that lensing reconstruction is done exclusively by
an $EB$ quadratic estimator and so underestimate the actual internal-delensing
performance. compared to when a cutoff at $l_{\mathrm{cut}}=300$ is imposed,
translating to a slightly wider distribution of $\hat{r}_{\text{ML}}$ and a
lower number of detections of non-zero $r$. The figure also shows that, for
our set-up, modelling the bias degrades the sensitivity on $r$ to a level
comparable to (or even slightly worse than) the case of no delensing, as
suggested by our earlier results for the normalised power in this case (figure
5).

Figure 10: Distributions of $\hat{r}_{\text{ML}}$ (left) and
$\sigma(\hat{r}_{\text{ML}})$ (right) across simulations for different
delensing analyses. Blue: no delensing. Orange: biased delensing, based on a
reconstructed $\hat{\phi}^{EB}$ that does not impose a low-$l$ $B$-mode
cutoff, with no attempt to model the resulting power spectrum bias in the
likelihood. This results in $\hat{r}_{\text{ML}}$ being biased low (and a
significant number of simulated likelihoods that peak at $r=0$) and with
artificially small errors. Green: biased delensing, but with the power
spectrum bias modelled as described in section 4.2. This mitigates the bias in
$r$ but, for our experimental set-up, has comparable performance to no
delensing. Red: a cutoff at $l_{\mathrm{cut}}=300$ is imposed on the $B$-modes
used in the reconstruction, avoiding the bias to the delensed $B$-mode power
spectrum below $l_{\mathrm{cut}}$. There is no bias in $r$ in this case, and
delensing reduces $\sigma(\hat{r}_{\text{ML}})$ as intended. All analyses
employ simulated bandpower covariances. Figure 11: Same as figure 10, but in
the null scenario of $r_{\mathrm{input}}=0$.

##  7  The $B^{\text{obs}}(B^{\text{obs}}-B^{\text{temp}})$ estimator

In section 4.2, we saw that the bias from internal delensing with an $EB$
estimator _reduces_ the signal-to-noise on a primordial contribution to the
power spectrum of delensed $B$-modes. We attributed this to the coupling in
eq. (A.2), which arises in the cross-correlation of the $B$-mode lensing
template with itself, and that restores some of the lensing power while
leaving the primordial signal unaffected. This motivates the idea of
considering an alternative power spectrum estimator of the form
$B^{\text{obs}}(B^{\text{obs}}-B^{\text{temp}})$, i.e., the cross-correlation
of the observed $B$-modes with the delensed $B$-modes,
$B^{\text{del}}=B^{\text{obs}}-B^{\text{temp}}$. This estimator does not
involve the cross-correlation of the template with itself, and so avoids
couplings such as that in eq. (A.2). In the case of no $l_{\mathrm{cut}}$ in
the lensing reconstruction (such that the $B$-modes used for reconstruction
overlap with those to be delensed on all scales) and two separate surveys, LAT
and SAT, the expected value of the cross-correlation is

$\langle
B^{\text{obs}}(\bm{l})\left[B^{\text{obs}}(\bm{l}^{\prime})-B^{\text{temp}}(\bm{l}^{\prime})\right]\rangle_{\text{biased}}=(2\pi)^{2}\delta^{(2)}(\bm{l}+\bm{l}^{\prime})\bigg{[}(1-D_{l})\left(C_{l}^{BB,\text{res}}+C_{l}^{BB,\mathrm{t}}+N_{l}^{BB,\mathrm{SAT}}\right)\\\
+D_{l}\left(N_{l}^{BB,\mathrm{SAT}}-N_{l}^{X}\right)-D_{l}C_{l}^{W}\bigg{]}.$
(7.1)

In the case of a single survey, this reduces to

$\langle
B^{\text{obs}}(\bm{l})\left[B^{\text{obs}}(\bm{l}^{\prime})-B^{\text{temp}}(\bm{l}^{\prime})\right]\rangle_{\text{biased}}=(2\pi)^{2}\delta^{(2)}(\bm{l}+\bm{l}^{\prime})\bigg{[}(1-D_{l})\left(C_{l}^{BB,\text{res}}+C_{l}^{BB,\mathrm{t}}+N_{l}^{BB}\right)\\\
-D_{l}C_{l}^{W}\bigg{]}.$ (7.2)

The last term on the right-hand side is always negative. Renormalising the
cross-correlation by $(1-D_{l})$ to have unit response to the primordial
power, we see that its renormalised value is reduced below the power of the
unbiased, delensed spectrum (equation 4.2) by this negative term. While the
non-primordial power in both of these spectra decreases monotonically with the
experimental noise, the difference between the two is greatest when
$C_{l}^{W}D_{l}/(1-D_{l})$ peaks. This is the case when the experimental noise
level in polarisation is approximately $3\sqrt{2}\,\mu\text{K\,arcmin}$,
corresponding to a delensing efficiency of approximately $40\,\%$. For such an
experimental configuration, the term $-D_{l}C_{l}^{W}$ reduces the cross-
correlation in eq. (7.2) by an amount equivalent on large scales to white
noise with $\Delta_{P}\approx 2\,\mu\text{K\,arcmin}$ after normalising to
have unit response to a primordial signal.

It is instructive to compare the
$B^{\text{obs}}(B^{\text{obs}}-B^{\text{temp}})$ estimator above to another
defined along similar lines, but which features no overlap between
reconstruction $B$-modes and $B$-modes to be delensed (for example, by
introducing an $l_{\mathrm{cut}}$ in the reconstruction, such that $\langle
B^{\text{obs}}(B^{\text{obs}}-B^{\text{temp}})\rangle$ is unbiased on scales
$l<l_{\mathrm{cut}}$). This alternative estimator has expectation value

$\langle
B^{\text{obs}}(\bm{l})(B^{\text{obs}}(\bm{l}^{\prime})-B^{\text{temp}}(\bm{l}^{\prime}))\rangle_{\text{unbiased}}=(2\pi)^{2}\delta^{(2)}(\bm{l}+\bm{l}^{\prime})\left(C_{l}^{BB,\text{res}}+C_{l}^{BB,\mathrm{t}}+N_{l}^{BB}\right),$
(7.3)

which is the same as the unbiased, delensed power spectrum of eq. (4.2).
Comparing this to the biased estimator of eq. (7.2), we see that, after
appropriate renormalisation, the expectation value of the latter is lower than
the estimator in eq. (7.3), or even the standard power spectrum approach of
eq. (4.2). Our analytic models for all these different estimators are found to
be in excellent agreement with their values in simulations.

In assessing the performance of the cross-correlation estimators for
constraining the primordial signal, we remind the reader that, for a for a
cross-correlation, the variance is not determined by the expected value (even
in the limit of Gaussian fields). For a single survey, treating both
$B^{\text{obs}}$ and $B^{\text{del}}$ as Gaussian fields, the variances of the
estimators at hand are

$\displaystyle\frac{1}{(1-D_{l})^{2}}\text{Var}\left(B^{\text{obs}}B^{\text{del}}\right)_{\text{biased}}$
$\displaystyle=\frac{1}{2l+1}\left[2\left(C_{l}^{BB,\text{del},\text{unbiased}}\right)^{2}+\left(\frac{1-2D_{l}}{1-D_{l}}\right)^{2}C_{l}^{W}C_{l}^{BB,\text{del},\text{unbiased}}\right.$
$\displaystyle{}\left.+2\left(\frac{D_{l}}{1-D_{l}}\right)^{2}\left(C_{l}^{W}\right)^{2}\right]\,,$
(7.4)
$\displaystyle\text{Var}\left(B^{\text{obs}}B^{\text{del}}\right)_{\text{unbiased}}$
$\displaystyle=\frac{1}{2l+1}\left[2\left(C_{l}^{BB,\text{del},\text{unbiased}}\right)^{2}+C_{l}^{W}C_{l}^{BB,\text{del},\text{unbiased}}\right]\,,$
(7.5)
$\displaystyle\text{Var}\left(B^{\text{del}}B^{\text{del}}\right)_{\text{unbiased}}$
$\displaystyle=\frac{1}{2l+1}\left[2\left(C_{l}^{BB,\text{del},\text{unbiased}}\right)^{2}\right]\,,$
(7.6)

where, recall,
$C_{l}^{BB,\text{del},\text{unbiased}}=C_{l}^{BB,\text{res}}+C_{l}^{BB,t}+N_{l}^{BB}$.
Clearly, the Gaussian variance is smallest for the usual
$\left(B^{\text{del}}B^{\text{del}}\right)_{\text{unbiased}}$ estimator.
Moreover, we will now see that the (co)-variance of both the cross-estimators
is dominated by non-Gaussian effects, which further degrade their constraining
power.

In figure 12, we use simulations of an experiment with
$\Delta_{P}=3\sqrt{2}\,\mu\mathrm{K\,arcmin}$ and
$\theta_{\mathrm{FWHM}}=6\,\mathrm{arcmin}$ to quantify the variance
associated with the different estimators. Although the Gaussian variance of
eq. (7.6) appears to capture correctly the simulated behaviour of the unbiased
$\left(B^{\text{del}}B^{\text{del}}\right)$ estimator, the unbiased and
renormalised $\left(B^{\text{obs}}B^{\text{del}}\right)$ estimators show
variances well in excess of the Gaussian result. This can be attributed to the
predominance of non-Gaussian contributions, which also manifest themselves in
the strong correlations between bandpowers shown in figure 13. Intuitively,
the reason why the non-Gaussian character is less acute in
$\left(B^{\text{del}}B^{\text{del}}\right)$ than in
$\left(B^{\text{obs}}B^{\text{del}}\right)$ is that the former involves two
delensed fields and delensing is known to mitigate the correlations between
scales introduced by lensing.

Figure 12: Variances of different spectral estimators from which primordial
power is to be extracted, obtained from 25 000 simulations of an experiment
consisting of a single survey with
$\Delta_{P}=3\sqrt{2}\,\mu\text{K\,arcmin}$,
$\theta_{\mathrm{FWHM}}=6\,\mathrm{arcmin}$ and covering 2.8 % of the sky.
Yellow: the biased $(B^{\text{obs}}B^{\text{del}})_{\text{biased}}$ estimator
of eq. (7.2). Green: the same for the case where
$(B^{\text{obs}}B^{\text{del}})_{\text{biased}}$ is normalised to have unit
response to primordial $B$-mode power. Red: the
$(B^{\text{obs}}B^{\text{del}})_{\text{unbiased}}$ estimator of eq. (7.2)
(only below $l_{\mathrm{cut}}=300$). Black: the auto-spectrum of
$B^{\text{del}}$ in the case where the bias is averted below
$l_{\mathrm{cut}}=300$. Dotted lines show the Gaussian variance expected for
each estimator, calculated analytically.

Given these considerations, we believe that the estimator of eq. (7.2) should
lead to inferior constraints on $r$ compared to an analysis involving the
usual power spectrum estimator of eq. (4.2). A more careful demonstration of
this would require working within a maximum-likelihood inference framework, as
in section 6. Unfortunately, this is not straightforward since standard
likelihood approximations are not applicable for a single cross-spectrum.
These likelihood approximations are very convenient in the presence of real-
world effects such as masking, and for including non-Gaussian power spectrum
covariances. Extending these to a single cross-spectrum would require further
development.

Figure 13: Cross-correlation coefficients of different spectral estimators for
the experimental set-up in figure 12, estimated from 25 000 simulations.
Right: delensed $B$-mode bandpowers in the case where the bias is avoided
below $l<l_{\mathrm{cut}}$ by introducing a cutoff at $l_{\mathrm{cut}}=300$.
Centre: $(B^{\text{obs}}B^{\text{del}})_{\text{unbiased}}$ estimator of eq.
(7.3), unbiased below $l_{\mathrm{cut}}=300$. Left:
$(B^{\text{obs}}B^{\text{del}})_{\text{biased}}$ estimator of eq. (7.2),
renormalised to have unit response to any primordial power. All estimators are
binned with $\Delta l=30$.

##  8 Discussion

Searches for primordial $B$-modes in CMB data already necessarily require the
removal of the contamination from gravitational lensing. In the near future,
experimental characteristics will be such that the $EB$ quadratic estimator is
expected to provide a large fraction of the signal-to-noise on reconstructions
of the lensing potential, which is used to perform the delensing procedure. It
is well known that any overlap in modes between the $B$-field to be delensed
and the $B$-field from which the reconstruction is derived biases the
amplitude and variance of the delensed power spectrum [19, 37].

In this paper, we have refined the modelling of this bias, based on a cumulant
expansion and identification of the dominant terms, and used simulations to
verify its efficacy. Our model is consistent with others in the literature
[36, 37] that are valid in the absence of a primordial (gravitational wave)
signal, but extends our understanding to include the primordial component. The
power spectrum bias acts multiplicatively on the primordial power, suppressing
its amplitude. The bias also suppresses lensing and instrument noise power
but, crucially, generally less so than for the primordial signal. This leads
to a loss of constraining power for primordial $B$-modes, and this becomes
more acute whenever separate surveys (i.e., with independent instrument noise)
are used for lensing reconstruction and the study of $B$-modes on large
angular scales.

Fortunately, the bias is completely local in multipole space in the sense that
bias to the delensed $B$-mode power at multipole $l$ originates from $B$-modes
at that same multipole being used in the lensing reconstruction. The bias can
therefore be avoided by eliminating overlapping modes in the $B$-field
employed in the reconstruction, and within the context of searches for
primordial gravitational waves this can be done with little loss of
performance. We have shown that, for any reasonable $l_{\mathrm{cut}}$, such
elimination of overlapping modes is actually preferable over an approach where
the bias is modelled, as it yields improved constraining power. These findings
are verified by simulating the reconstruction and delensing procedures and
performing maximum-likelihood inferences of the primordial power on the
resulting delensed spectra.

Our model for the biased, delensed $B$-mode power spectrum is in excellent
agreement with simulations for experimental noise levels comparable or
inferior to those of the Simons Observatory, but we note that the agreement
worsens for lower noise levels as new terms that we ignore become relevant. In
this limit of low noise, we expect our model to over-estimate the signal-to-
noise so that our claim that retaining, but modelling, the bias hinders
efforts to constrain the primordial signal continues to hold. We therefore
recommend that any attempt to delens large-scale $B$-modes internally building
on lensing information gleaned from the $B$-modes themselves – be it via
quadratic estimators or likelihood-based techniques – feature the excision of
large-scale $B$-modes from the lensing inference.

Finally, we have considered alternative cross-correlation estimators for the
primordial power involving the observed and delensed $B$-modes, with a view to
removing some couplings in the biased spectrum that degrade the signal-to-
noise on $r$. We provided analytic models for the expectation values of these
cross-correlations and verified these against simulations. However, we showed
that ultimately these estimators have lower signal-to-noise to primordial
power than the auto-power spectrum of the (unbiased) delensed fields, due to
their cross-correlation character and their increased non-Gaussian covariance
between multipoles.

## Acknowledgements

ABL acknowledges support from an Isaac Newton Studentship at the University of
Cambridge and from the Science and Technology Facilities Council (STFC). AC
acknowledges support from the STFC (grant numbers ST/N000927/1 and
ST/S000623/1). JC acknowledges support from a SNSF Eccellenza Professorial
Fellowship (No. 186879) and from the European Research Council under the
European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant
Agreement No. [616170]. We thank Toshiya Namikawa and Blake Sherwin for useful
discussions.

##  Appendix A Internally-delensed $B$-mode power spectrum

In this appendix, we give details of the calculation of the $B$-mode power
spectrum after internal delensing, as given in eq. (4.8) in the main text.
Specifically, we evaluate the four- and six-point functions of the observed
CMB that appear in the cross-correlation of the lensing $B$-mode template with
the observed $B$-modes, eq. (4.2), and the power spectrum of the template, eq.
(4.2), respectively.

It is convenient to expand these four- and six-point functions in terms of
connected $n$-point functions. In the absence of lensing, the CMB would be
Gaussian and all connected $n$-point functions with $n>2$ would vanish. As the
CMB fields are zero mean, the evaluation of $n$-point functions would reduce
to products of two-point functions. However, in section 5 we saw that lensing
distorts the Gaussian primordial statistics, introducing significant non-
Gaussianities in the form of non-vanishing, higher-order connected $n$-point
functions, which we shall also refer to as “connected correlators”. In
particular, the connected four-point function, or trispectrum, induced by
lensing lies at the heart of any effort to infer $C_{L}^{\phi\phi}$ from the
lensed CMB [45, 33, 46, 30]. Note that Gaussian instrumental noise does not
contribute to the connected correlators of the observed CMB fields. Moreover,
if we ignore the impact of lensing on the (otherwise Gaussian) gravitational
wave contributions to the CMB (which should be a good approximation given
their power falls rapidly on intermediate and small scales), these
contributions are independent of the lensed, scalar contribution and so do not
contribute to the connected correlators either.

We organise the expansion of the $n$-point functions with the aid of a
graphical representation where fields are represented by nodes, drawing those
that are observed by the LAT in red, and those observed by the SAT in green.
Lines connecting $n$ nodes denote the connected $n$-point function of the
associated fields, which can be evaluated perturbatively to the desired order
in lensing. In order to preserve generality, we ensure that the case where a
single telescope is used for both construction of the delensing template and
the observation of the large-scale $B$-modes can be recovered by letting
$N_{l}^{X}=N_{l}^{BB,\text{LAT}}=N_{l}^{BB,\text{SAT}}$, with $N_{l}^{X}=0$
otherwise.

In this way, the four-point function appearing in eq. (4.2) can be represented
as

$\langle
E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{1})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{LAT}}(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{SAT}}(\bm{l}_{2})\rangle=\leavevmode\hbox
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{{{{{}}{}{}{}{{}{}}{{}{}{{}{}}{{}{}}{}{{}}}}}}{}{{{{{}}{}{}{}{{}{}}{{}{}{{}{}}{{}{}}{}{{}}}}}}{{}}{}{}{}{}\pgfsys@moveto{19.80508pt}{2.95647pt}\pgfsys@lineto{3.59526pt}{19.16628pt}\pgfsys@stroke\pgfsys@invoke{
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26.58034pt\lower-26.58034pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
\par{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.5,1,0.5}\pgfsys@color@rgb@fill{0.5}{1}{0.5}\pgfsys@invoke{
}{{{}{}}{{}{}}{{}{}}{{}{}}{}{}{}{}{}{}{}{{}}{}{{}{}}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{{}{}}{}{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgffillcolor}{rgb}{0.5,1,0.5}\pgfsys@color@rgb@fill{0.5}{1}{0.5}\pgfsys@invoke{
}\pgfsys@moveto{22.76186pt}{5.83327pt}\pgfsys@lineto{20.47607pt}{3.14616pt}\pgfsys@lineto{17.21405pt}{1.8026pt}\pgfsys@lineto{19.06332pt}{-1.20174pt}\pgfsys@lineto{19.33318pt}{-4.71922pt}\pgfsys@lineto{22.76186pt}{-3.88884pt}\pgfsys@lineto{26.19055pt}{-4.71922pt}\pgfsys@lineto{26.4604pt}{-1.20174pt}\pgfsys@lineto{28.30968pt}{1.8026pt}\pgfsys@lineto{25.04765pt}{3.14616pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.4}{0.0}{0.0}{0.4}{21.34521pt}{-1.36664pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{B}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \par{{}}{}{{}}
{{{{{}}{}{}{}{{}{}}{{}{}{{}{}}{{}{}}{}{{}}}}}}{}{{{{{}}{}{}{}{{}{}}{{}{}{{}{}}{{}{}}{}{{}}}}}}{{}}{}{}{}{}\pgfsys@moveto{3.59526pt}{19.16624pt}\pgfsys@lineto{19.805pt}{2.95647pt}\pgfsys@stroke\pgfsys@invoke{
} {{}}{}{{}}
{{{{{}}{}{}{}{}{{}}}}}{}{{{{{}}{}{}{}{}{{}}}}}{{}}{}{}{}{}\pgfsys@moveto{-20.00488pt}{-2.75659pt}\pgfsys@lineto{-2.75659pt}{-20.00488pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,,$ (A.1)

where terms of the form $\langle EB\rangle$ have vanished due to parity
invariance. The $B$\- and $E$-fields involved in the lens reconstruction are
at the top and bottom of the diagrams, respectively, and the $E$-modes used
further in the template are at the midpoint on the left. The two diagrams on
the right of eq. (A.1) correspond to a trispectrum $\langle
E^{\text{obs}}E^{\text{obs}}B^{\text{obs}}B^{\text{obs}}\rangle_{c}$ and a
product of two-point functions $\langle
E^{\text{obs}}E^{\text{obs}}\rangle\langle
B^{\text{obs}}B^{\text{obs}}\rangle$. Similarly, the six-point function
appearing in eq. (4.2) can be decomposed into

$\displaystyle\langle
E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{1})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{1})B^{\text{obs},\,\text{LAT}}(\bm{l}_{1}-\bm{l}^{\prime}_{1}-\bm{l}^{\prime\prime}_{1})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime}_{2})E^{\text{obs},\,\text{LAT}}(\bm{l}^{\prime\prime}_{2})B^{\text{obs},\,\text{LAT}}(\bm{l}_{2}-\bm{l}^{\prime}_{2}-\bm{l}^{\prime\prime}_{2})\rangle$
$\displaystyle\quad=\leavevmode\hbox to53.16pt{\vbox
to49.08pt{\pgfpicture\makeatletter\hbox{\hskip
26.58034pt\lower-23.53087pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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