#  LiteBIRD Science Goals and Forecasts: Improving Sensitivity to Inflationary
Gravitational Waves with Multitracer Delensing

T. Namikawa  A. I. Lonappan  C. Baccigalupi  N. Bartolo  D. Beck  K. Benabed
A. Challinor  P. Diego-Palazuelos  J. Errard  S. Farrens  A. Gruppuso  N.
Krachmalnicoff  M. Migliaccio  E. Martínez-González  V. Pettorino  G.
Piccirilli  M. Ruiz-Granda  B. Sherwin  J. Starck  P. Vielva  R. Akizawa  A.
Anand  J. Aumont  R. Aurlien  S. Azzoni  M. Ballardini  A. J. Banday  R. B.
Barreiro  M. Bersanelli  D. Blinov  M. Bortolami  T. Brinckmann  E. Calabrese
P. Campeti  A. Carones  F. Carralot  F. J. Casas  K. Cheung  L. Clermont  F.
Columbro  G. Conenna  A. Coppolecchia  F. Cuttaia  G. D’Alessandro  P. de
Bernardis  T. de Haan  M. De Petris  S. Della Torre  E. Di Giorgi  H. K.
Eriksen  F. Finelli  C. Franceschet  U. Fuskeland  G. Galloni  M. Galloway  M.
Georges  M. Gerbino  M. Gervasi  T. Ghigna  S. Giardiello  C. Gimeno-Amo  E.
Gjerløw  M. Hazumi  S. Henrot-Versillé  L. T. Hergt  E. Hivon  K. Kohri  E.
Komatsu  L. Lamagna  M. Lattanzi  C. Leloup  M. Lembo  M. López-Caniego  G.
Luzzi  B. Maffei  S. Masi  M. Massa  S. Matarrese  T. Matsumura  S. Micheli
A. Moggi  M. Monelli  L. Montier  G. Morgante  B. Mot  L. Mousset  R. Nagata
P. Natoli  A. Novelli  I. Obata  A. Occhiuzzi  L. Pagano  A. Paiella  D.
Paoletti  G. Pascual-Cisneros  V. Pavlidou  F. Piacentini  M. Pinchera  G.
Pisano  G. Polenta  G. Puglisi  M. Remazeilles  A. Ritacco  A. Rizzieri  J.
Rubino-Martin  Y. Sakurai  D. Scott  M. Shiraishi  G. Signorelli  S. L. Stever
Y. Takase  H. Tanimura  A. Tartari  K. Tassis  L. Terenzi  M. Tristram  L.
Vacher  B. van Tent  I. K. Wehus  G. Weymann-Despres  M. Zannoni  and Y. Zhou
LiteBIRD Collaboration

###### Abstract

We estimate the efficiency of mitigating the lensing $B$-mode polarization,
the so-called delensing, for the LiteBIRD experiment with multiple external
data sets of lensing-mass tracers. The current best bound on the tensor-to-
scalar ratio, $r$, is limited by lensing rather than Galactic foregrounds.
Delensing will be a critical step to improve sensitivity to $r$ as
measurements of $r$ become more and more limited by lensing. In this paper, we
extend the analysis of the recent LiteBIRD forecast paper to include multiple
mass tracers, i.e., the CMB lensing maps from LiteBIRD and CMB-S4-like
experiment, cosmic infrared background, and galaxy number density from
Euclid\- and LSST-like survey. We find that multi-tracer delensing will
further improve the constraint on $r$ by about $20\%$. In LiteBIRD, the
residual Galactic foregrounds also significantly contribute to uncertainties
of the $B$-modes, and delensing becomes more important if the residual
foregrounds are further reduced by an improved component separation method.

##  1 Introduction

Measuring the polarization of the cosmic microwave background (CMB)
anisotropies will be at the forefront of observational cosmology in the next
decade. In particular, measurements of the curl component ($B$ modes) in the
CMB polarization will be of great importance, as these provide us with a
unique window to probe inflationary gravitational waves (IGWs) and gain new
insights into the early Universe [1, 2] (see also reviews [3, 4]). The
BICEP/Keck Array collaborations recently placed constraints on the IGW
background, parameterized by the tensor-to-scalar ratio $r$ (at a pivot scale
of $0.05\,$Mpc-1) as $r<0.036$ ($2\,\sigma$) for a fixed cosmology [5, 6],
while Ref. [7] combine the BICEP/Keck Array data with Planck PR4 and baryon
acoustic oscillations, finding $r<0.032$ ($2\,\sigma$) by simultaneously
constraining the cosmological parameters. Several ongoing and upcoming CMB
experiments, including the BICEP Array [8], Simons Array [9], Simons
Observatory (SO) [10], LiteBIRD [11], and CMB-S4 [12], are targeting detection
of IGW $B$-mode polarization over the next decade.

A high-precision measurement of the large-scale $B$-mode polarization can
tightly constrain $r$. However, the precision of the IGW $B$-mode measurement
is limited by other sources of $B$ modes. On large angular scales, the
Galactic foregrounds dominate the IGW $B$-mode polarization [13]. To overcome
this issue, many studies have developed some methodology to substantially
reduce the Galactic foreground contaminants (e.g., Refs. [14, 15, 16, 17, 18,
19, 20, 21, 22, 23, 24, 25], and Ref. [11] for the LiteBIRD case). In addition
to Galactic foregrounds, gravitational lensing leads to $B$ modes from the
conversion of part of the $E$ modes [26]; this lensing $B$-mode polarization
is as an additional noise component when constraining $r$. The lensing
$B$-mode polarization on large scales behaves as white noise, and its
amplitude is comparable to that of the IGW $B$-mode polarization with $r=0.01$
at the recombination bump ($\ell\sim 10$ – $100$) [27]. The constraint on $r$
is already limited by the variance from the lensing $B$-mode polarization more
than by Galactic foregrounds, at least, in low dust region [5]. Reducing
statistical uncertainties by subtracting off the lensing $B$-mode polarization
or equivalent methods – a process usually referred to as “delensing” – will
hence be critical for improving the constraints on $r$ (e.g., Refs. [28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]).

To estimate the lensing $B$-mode polarization in some survey region, known as
a lensing $B$-mode template, the simplest method is to combine the measured
$E$-mode polarization with a reconstructed lensing map derived from CMB data,
and multiple works have studied this technique (e.g., Refs. [29, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54]). In addition, other mass tracers,
such as the spatial distribution of optical and radio galaxies, as well as the
cosmic infrared background (CIB) are also possible to include measurement of
the lensing $B$-mode polarization (e.g., Refs. [55, 56, 57, 58, 59, 60, 61]).
Recently, Ref. [62] demonstrated for the first time $B$-mode delensing to
improve constraints on IGWs using the CIB as a mass tracer. In the LiteBIRD
case, the CIB and galaxies are useful for delensing. In addition, the lensing
potential reconstructed from CMB-S4 data will be available in the CMB-S4
observing patch.

This work is part of a series of papers that present the science achievable by
the LiteBIRD space mission, expanding on the overview published in Ref. [11]
(hereafter, LB23). In particular, this work focuses on the potential of the
LiteBIRD $B$-mode delensing to improve sensitivity to constrain IGWs by
combining multiple external mass tracers. In LB23, we consider a simple
forecast using CIB as a possible mass tracer for delensing. In this paper, we
extend the delensing study in several ways. We include the CMB lensing maps of
LiteBIRD and CMB-S4, the CIB, and galaxy number density from Euclid and the
Vera Rubin Observatory Legacy Survey of Space and Time (LSST). In addition, we
make our forecast more realistic by adding survey windows of each survey, a
realistic lensing map from LiteBIRD, component separation of the LiteBIRD
$E$-mode polarization used for the lensing $B$-mode template, and the large-
scale $B$-mode polarization after the component separation obtained in LB23.
We also note that LB23 provides the delensing forecast only for the case if
the true value of $r$ is $r_{0}=0$. If $r_{0}$ is $\mathcal{O}(10^{-3})$, the
cosmic variance from IGWs significantly dominates the statistical uncertainty
at the reionization bump but does not at the recombination bump. On the other
hand, if $r_{0}$ goes to zero, the information from the reionization bump also
becomes important to constrain $r$. Since the lensing cosmic variance is
important only at the recombination bump, delensing would become more
important for $r_{0}=\mathcal{O}(10^{-3})$ than the case with $r_{0}=0$. Also,
if $r_{0}$ is close to or less than the LiteBIRD target sensitivity,
$10^{-3}$, the analysis needs an accurate treatment of possible concerns, such
as instrumental systematic effects and foreground contaminations, whose
characterization is beyond the scope of this paper. Thus, this paper explores
the case for a nonzero $r_{0}$.

This paper is organized as follows. Section 2 presents the baseline strategy
for LiteBIRD delensing after briefly reviewing the lensing effect on the CMB.
Section 3 describes our simulation setup. Section 4 shows the improvement in
constraint on $r$ by the multi-tracer delensing, including realistic survey
effects. Section 5 is devoted to summary and discussion. This paper is a
companion paper of the LiteBIRD lensing study [63] (hereafter, LB-Lensing) and
uses some of the results presented in that paper.

##  2 Multitracer delensing

###  2.1 Lensing-induced B-modes and delensing

The distortion effect of lensing on the primary CMB polarization anisotropies
is expressed by remapping the primary polarization anisotropies from the last-
scattering surface, $Q\pm{\rm i}\hskip 0.50003ptU$. The lensed Stokes
parameters in sky direction $\hat{\bm{n}}$ can be approximately expressed as a
remapping [27]:

$\displaystyle[\widetilde{Q}\pm{\rm i}\hskip
0.50003pt\widetilde{U}](\hat{\bm{n}})$ $\displaystyle=[Q\pm{\rm i}\hskip
0.50003ptU](\hat{\bm{n}}+\bm{d}(\hat{\bm{n}}))\,,$ (2.1)

where tildes indicate lensed quantities and $\bm{d}$ is the deflection angle
used to describe the lensing displacements on the spherical sky (see Ref. [64]
for more details). The quantity $\bm{d}$ is related to the lensing
convergence, $\kappa$, as $\bm{\nabla}\cdot\bm{d}=-2\kappa$, where
$\bm{\nabla}$ is the covariant derivative on the unit sphere. The $E$\- and
$B$-mode polarization are then obtained by [65, 66]

$\displaystyle E_{\ell m}\pm{\rm i}\hskip 0.50003ptB_{\ell
m}=-\int\\!\\!\,{\rm d}^{2}\hat{\bm{n}}\,\,{}_{\pm 2}Y_{\ell
m}^{*}(\hat{\bm{n}})[Q\pm{\rm i}\hskip 0.50003ptU](\hat{\bm{n}})\,,$ (2.2)

where we denote the spin-$2$ spherical harmonics as ${}_{\pm 2}Y_{\ell
m}(\hat{\bm{n}})$. Similarly with the spin-$0$ (scalar) spherical harmonics,
$Y_{\ell m}(\hat{\bm{n}})$, the harmonic coefficients of the lensing
convergence is defined as

$\displaystyle\kappa_{LM}$ $\displaystyle=\int\\!\\!\,{\rm
d}^{2}\hat{\bm{n}}\,\,Y_{LM}^{*}(\hat{\bm{n}})\kappa(\hat{\bm{n}})\,.$ (2.3)

Expanding the right-hand side of Eq. (2.1) to first order in the lensing
potential and transforming the Stokes $Q/U$ parameters to $E/B$ modes using
Eq. (2.2), the $B$ modes of the lensed polarization field can be expressed as
[41]:

$\displaystyle B^{\mathrm{lens}}_{lm}$ $\displaystyle={\rm i}\hskip
0.50003pt\sum_{\ell^{\prime}m^{\prime}}\sum_{LM}\begin{pmatrix}\ell&\ell^{\prime}&L\\\
m&m^{\prime}&M\end{pmatrix}p^{-}F^{(2)}_{\ell
L\ell^{\prime}}E_{\ell^{\prime}m^{\prime}}^{*}\kappa_{LM}^{*}\,,$ (2.4)

where we ignore the primary $B$-mode polarization. The quantity in round
brackets is the Wigner-3$j$ symbol, $p^{-}$ is unity if $\ell+L+\ell^{\prime}$
is an odd integer and zero otherwise, and $F^{(2)}_{\ell L\ell^{\prime}}$
represents the mode coupling induced by the lensing [67]:

$\displaystyle F^{(s)}_{\ell L\ell^{\prime}}$
$\displaystyle=\frac{2}{L(L+1)}\sqrt{\frac{(2\ell+1)(2\ell^{\prime}+1)(2L+1)}{16\pi}}$
$\displaystyle\times[-\ell(\ell+1)+\ell^{\prime}(\ell^{\prime}+1)+L(L+1)]\begin{pmatrix}\ell&\ell^{\prime}&L\\\
-s&s&0\end{pmatrix}\,.$ (2.5)

From Eq. (2.4), the lensing $B$-mode polarization is simply expressed in terms
of a convolution between the unlensed $E$-mode polarization and the lensing
potential. Once we obtain an estimate for the $E$ mode and lensing maps, we
simply approximate the lensing $B$-mode polarization as a convolution of the
Wiener-filtered $E$-mode polarization and lensing convergence:

$\displaystyle\widehat{B}^{\mathrm{temp}}_{\ell m}={\rm i}\hskip
0.50003pt\sum_{\ell^{\prime}m^{\prime}}\sum_{LM}\begin{pmatrix}\ell&\ell^{\prime}&L\\\
m&m^{\prime}&M\end{pmatrix}p^{-}F^{(2)}_{\ell L\ell^{\prime}}(\widehat{E}^{\rm
WF}_{\ell^{\prime}m^{\prime}})^{*}(\widehat{\kappa}^{\rm WF}_{LM})^{*}\,,$
(2.6)

where $\widehat{E}^{\rm WF}_{\ell m}$ and $\widehat{\kappa}^{\rm WF}_{LM}$ are
the Wiener-filtered observed $E$-mode polarization and lensing convergence,
respectively. In idealistic cases, we compute the Wiener-filtered multipoles
as

$\displaystyle\widehat{E}^{\rm WF}_{\ell m}=\frac{C_{\ell}^{EE}}{C^{EE,{\rm
obs}}_{\ell}}\widehat{E}_{\ell m}\,,$ (2.7)
$\displaystyle\widehat{\kappa}^{\rm WF}_{LM}=\frac{C_{L}^{\kappa
m}}{C^{mm,{\rm obs}}_{L}}\widehat{m}_{LM}\,,$ (2.8)

where $m$ is a lensing mass tracer. The $\widehat{E}$ and $\widehat{m}$ denote
the observed $E$-mode polarization and $m$, respectively, $C^{EE}_{\ell}$ is
the $E$-mode power spectrum, $C_{L}^{\kappa m}$ is the cross-power spectrum
between $\kappa$ and $m$, and $C_{l}^{EE,{\rm obs}}$ and $C^{mm,{\rm
obs}}_{L}$ are the best estimates of the angular power spectra of
$\widehat{E}$ and $\widehat{m}$, respectively. The template provides a good
approximation for the large-scale $B$-mode polarization [68, 69]. Using this
template, we can remove part of the lensing $B$-mode polarization from the
observed map [29], or construct the multi-spectrum likelihood [62, 70] to
improve constraints on $r$.

###  2.2 Lensing mass tracers

To estimate the lensing mass map, we can use the internally reconstructed
lensing map from LiteBIRD observation. However, the LiteBIRD lensing mass map
alone cannot remove the lensing $B$-mode polarization efficiently [71]. A more
optimal way is to employ the lensing convergence field estimated from
observations of the large-scale structure tracers, such as the spatial
distribution of galaxies or the CIB [58, 57, 59, 72]. The delensing efficiency
of these tracers depends on the correlation coefficients between the CMB
lensing signal and the observed mass tracer. In this section, we provide our
formulas for computing the correlation coefficients and assumptions for the
lensing mass tracers.

####  2.2.1 Angular power spectrum

We compute the angular auto and cross-power spectra of the CMB lensing, CIB,
and galaxies as follows. In a spatially flat cosmological model, the angular
power spectra are given by (see, e.g., Refs. [73, 74])

$\displaystyle C^{XY}_{L}$ $\displaystyle=4\pi\int_{0}^{\infty}\\!\\!\,{\rm
d}\ln k\,\,\int_{0}^{\infty}\\!\\!\,{\rm
d}\chi\,\,j_{L}(k\chi)\int_{0}^{\infty}\\!\\!\,{\rm
d}\chi^{\prime}\,\,j_{L}(k\chi^{\prime})$ $\displaystyle\quad\times
W^{X}(k,\chi)W^{Y}(k,\chi^{\prime})\Delta_{\rm m}(k;\chi,\chi^{\prime})\,,$
(2.9)

with the quantity $\chi$ being the comoving radial distance. Here, $X$ and $Y$
denote the observables from either the CMB lensing ($\kappa$), galaxy number
density fluctuations ($g$), or CIB ($I$). The function $\Delta_{\rm
m}(k;\chi,\chi^{\prime})$ is the dimensionless power spectrum of the matter
density fluctuations, and $j_{\ell}$ is the spherical Bessel function. The
function $W^{\rm X}(k,\chi)$ is the weight function of an observable X, the
functional form of which will be specified for each mass tracer as follows.

####  2.2.2 CMB Lensing

Since the lensing comes from the last-scattering surface of the CMB photons
which is approximately described by a single-source plane, the weight function
of the lensing convergence is given by (e.g., Ref. [27])

$\displaystyle W^{\kappa}(k,\chi)=-\frac{L(L+1)}{2}\frac{3\Omega_{\rm
m,0}H_{0}^{2}(1+z)}{k^{2}}\frac{\chi_{*}-\chi}{\chi_{*}\chi}\qquad(0\leq\chi\leq\chi_{*})\,,$
(2.10)

and becomes zero otherwise. Here, the quantity $\chi_{*}$ denotes the comoving
radial distance to the last-scattering surface, $\Omega_{\rm m,0}$ is the
fractional energy density of matter at present, and $H_{0}$ is the current
expansion rate.

####  2.2.3 Galaxy number density

We follow the theoretical modeling of the galaxy number density fluctuations
described in, e.g., Refs. [75, 70]. The weight function of the galaxy number
density fluctuations in the $i$th redshift bin becomes

$\displaystyle W^{{\rm g},i}(\chi)=\dfrac{{\rm d}n^{i}_{\rm gal}}{{\rm
d}\chi}(z(\chi))\,b_{\rm gal}(z(\chi))\,,$ (2.11)

where ${\rm d}n^{i}_{\rm gal}/{\rm d}\chi$ and $b_{\rm gal}(z)$ are the
normalized redshift distribution of galaxies in the $i$th bin and the linear
galaxy bias, respectively. For the redshift distribution function, we adopt
the following form [76, 77]:

$\displaystyle\dfrac{{\rm d}n^{i}_{\rm gal}}{{\rm d}\chi}(z)=\frac{\beta
H(z)}{\Gamma[(\alpha+1)/\beta]}\frac{z^{\alpha}}{z_{0}^{\alpha+1}}\exp\left[-\left(\frac{z}{z_{0}}\right)^{\beta}\right]p^{i}_{\rm
gal}(z,\sigma_{z})\,,$ (2.12)

where $\alpha$, $\beta$, and $z_{0}$ are parameters determining the
distribution. The parameter $z_{0}$ is related to the mean redshift $z_{\rm
m}$ as

$\displaystyle z_{\rm
m}=\frac{\Gamma[(\alpha+2)/\beta]}{\Gamma[(\alpha+1)/\beta]}z_{0}\,.$ (2.13)

$H(z)$ is the expansion rate, simply from the conversion between $z$ and
$\chi$. The function, $p^{i}_{\rm gal}(z,\sigma_{z})$, specifies the redshift
distribution of galaxies in $i$th redshift bin with the modification of the
redshift distribution by the photometric redshift errors. We assume that
$p^{i}_{\rm gal}(z,\sigma_{z})$ has the following form:

$\displaystyle p^{i}_{\rm gal}(z,\sigma_{z})=\frac{1}{2}\left[{\rm
erfc}\left(\frac{z_{i-1}-z}{\sqrt{2}\sigma(z)}\right)-{\rm
erfc}\left(\frac{z_{i}-z}{\sqrt{2}\sigma(z)}\right)\right]\,,$ (2.14)

where $\sigma(z)=\sigma_{z}(1+z)$, $z_{i-1}$ and $z_{i}$ are the minimum and
maximum photometric redshift of the $i$th bin, and the function ${\rm
erfc}(x)$ is the complementary error function defined by

$\displaystyle{\rm
erfc}(x)\equiv\frac{2}{\sqrt{\pi}}\int_{x}^{\infty}\\!\\!\,{\rm d}z\,\,\hskip
1.00006pt{\rm e}^{-z^{2}}\,.$ (2.15)

In this paper, we consider Euclid and LSST as galaxy surveys. For Euclid, we
consider $\alpha=2$, $\beta=1.5$, $\sigma_{z}=0.05$, $z_{0}=0.9/\sqrt{2}$, and
$b_{\rm gal}(z)=\sqrt{1+z}$ [76] and the following five tomographic bins;
$[0,0.8,1.5,2.0,2.5,6.]$. For the LSST, we follow Refs. [77, 75] and choose
$\alpha=2$, $\beta=1$, $z_{0}=$, $\sigma_{z}=0.05$, $b_{\rm gal}(z)=1+0.84z$,
and the five redshift tomographic bins: $[0,0.5,1.0,2.0,3.0,6.]$.

####  2.2.4 Cosmic Infrared Background

For the CIB, we assume a model to mimic the result of Ref. [75], where they
consider the CIB map constructed via the Generalized Needlet Internal Linear
Combination (GNILC) algorithm applied to the Planck PR2 data [78, 79]. For a
theoretical computation of the CIB angular power spectrum, they adopt the
single spectral energy distribution (SED) model suggested by the Planck
observations [80, 81, 82]. Specifically, the window function is defined as

$\displaystyle W^{I}(\chi)=b_{\rm
c}\frac{\chi^{2}}{(1+z)^{2}}f[\nu(1+z)]\exp\left[-\frac{(z-z_{\rm
c})^{2}}{2\sigma_{z}^{2}}\right]\,,$ (2.16)

where $z_{\rm c}=\sigma_{z}=2$, $b_{\rm c}$ is a normalization factor, $z$ is
the redshift as a function of $\chi$, and

$\displaystyle f(\nu)=\begin{cases}(e^{h\nu/k_{\rm
B}T}-1)^{-1}\nu^{\beta+3}&(\nu\leq\nu^{\prime})\,,\\\ (e^{h\nu^{\prime}/k_{\rm
B}T}-1)^{-1}\nu^{\prime\beta+3}\left(\frac{\nu}{\nu^{\prime}}\right)^{-\alpha}&(\nu>\nu^{\prime})\,,\end{cases}$
(2.17)

with $T=34$ K, $\alpha=\beta=2$, $\nu=353$ GHz and $\nu^{\prime}=4955$ GHz.

####  2.2.5 Kernel function

Figure 1:  Kernel function for CMB lensing, CIB, Euclid, and LSST galaxies
(see text in Sec. 2.2.5 for the definitions). The different colors represent
different redshift bins for galaxy surveys defined in Sec. 2.2.3. The function
is normalized so that the peak is unity.

Figure 1 shows the kernel functions for each mass tracer defined as follows.
For CMB lensing, we show $-(k\chi)^{2}W^{\kappa}/(L(L+1))/H(z)$. For CIB and
galaxies, we show $W^{I}/H(z)$ and $W^{g,i}/H(z)$, respectively. The galaxies
provide a lensing mass map at lower redshifts, while the CIB data have
information on relatively higher redshifts. Thus, combining these tracers will
provide a lensing mass map for a wide redshift range.

###  2.3 Correlation coefficients

Figure 2:  Left: Correlation coefficients between the ideal full-sky CMB
lensing and observed lensing mass-tracer maps obtained by combining several of
the LiteBIRD lensing maps with the CIB, galaxy number density from Euclid and
LSST, and CMB-S4 lensing maps. Right: Same as the left panel but for the
correlation coefficients for each external data set taken separately, i.e.,
the CIB, galaxies from Euclid and LSST, and CMB-S4.

From the theoretical angular power spectra, we can estimate how significantly
each mass tracer correlates with the true CMB lensing mass map. To assess the
significance of the correlation, we introduce the correlation coefficients of
an observed mass tracer map with the true CMB lensing mass map:

$\displaystyle\rho^{\kappa\kappa^{i}}_{L}\equiv\frac{C_{L}^{\kappa\kappa^{i}}}{\sqrt{C_{L}^{\kappa\kappa}C_{L}^{\kappa^{i}\kappa^{i},{\rm
obs}}}}\,.$ (2.18)

Here, $C_{L}^{\kappa\kappa^{i}}$ is the cross-power spectrum between CMB
lensing and $i$th observed lensing mass maps, $C_{L}^{\kappa\kappa}$ is the
CMB lensing power spectrum, and $C_{L}^{\kappa^{i}\kappa^{i},{\rm obs}}$ is
the auto-power spectrum of the $i$th observed lensing mass map, which contains
both signal and noise. We compute the signal angular power spectra defined in
Eq. (2.9) with CAMB [74] by providing the appropriate window functions given
above. To compute the correlation coefficients for the combined cases, we
combine the mass tracers following Ref. [58]. For the CIB, we employ the same
noise spectrum given by Ref. [75] and remove large-scale CIB data ($L<100$) to
avoid contamination from Galactic foreground residuals. For galaxy surveys, we
employ a shot noise determined by the number density of galaxies and assume
the total galaxy number density of $30$ arcmin-2 [76] and $40$ arcmin-2 [77]
for Euclid and LSST, respectively. Note that we also check cases in which we
increase the number of the tomographic bins, but the correlation coefficients
are almost unchanged.

Figure 2 shows the results of the correlation coefficients. For the large-
scale $B$-mode delensing, the most important scale for efficient delensing is
$L\simeq 300$ – $500$ [57, 58]. From the left panel, adding external mass
tracers significantly improves the correlation coefficient and the delensing
efficiency of the large-scale $B$-mode polarization. In the right panel, we
show that the correlation coefficients of CIB and Euclid galaxies are not
better than that of the LSST galaxies at $L\simeq 400$. However, the sky
coverage of LSST is limited compared to other mass tracers, and a net
contribution to the improvement on $r$ constraints with the CIB becomes not so
different from that obtained with galaxies.

##  3 Simulation setup

Figure 3:  Survey windows for each observation in Galactic coordinate used in
our analysis.

###  3.1 Survey window

We plot the survey windows in Fig. 3. For the LiteBIRD data, we use the same
Galactic mask used in LB23 for the LiteBIRD observation, which covers
approximately $50\%$ of the sky. We also apply the same mask to the CIB map to
mimic the sky fraction of the survey window used in Ref. [75]. For the CMB-S4
observation, we create a survey window that mimics the CMB-S4 LAT survey
window multiplied by the LiteBIRD Galactic mask to avoid the Galactic plane.
In practice, we would need further masks to avoid extragalactic sources, but
we ignore these masks in this paper. For Euclid and LSST, we follow Refs. [83]
and [77], respectively, to create the survey window. We further multiply the
LiteBIRD Galactic mask to those of the Euclid and LSST survey windows to
consider only the region overlapping with the galaxy number density map.

###  3.2 Wiener-filtered E-modes

To obtain the Wiener-filtered $E$-mode polarization from the component-
separated polarization map, we use the same method described in LB-Lensing for
the LiteBIRD-only case. We also consider the case of combining LiteBIRD and
CMB-S4 data for delensing. In this case, the Wiener-filtered $E$-mode
polarization maps are obtained by solving the following equation [84]:

$\displaystyle\left(1+\sum_{t}{\bm{\mathrm{C}}}^{1/2}{\bm{\mathrm{B}}}_{t}{\bm{\mathrm{Y}}}_{2}^{\dagger}{\bm{\mathrm{N}}}^{-1}_{t}{\bm{\mathrm{Y}}}_{2}{\bm{\mathrm{B}}}_{t}{\bm{\mathrm{C}}}^{1/2}\right)({\bm{\mathrm{C}}}^{-1/2}\bm{\widehat{p}}^{\rm
WF})=\sum_{t}{\bm{\mathrm{C}}}^{1/2}{\bm{\mathrm{B}}}_{t}{\bm{\mathrm{Y}}}^{\dagger}_{2}{\bm{\mathrm{N}}}^{-1}_{t}\bm{\widehat{p}}^{\rm
obs}_{t}\,.$ (3.1)

Here, $t$ corresponds to indices for the input maps specifying the experiment
(LiteBIRD or CMB-S4), the vector $\bm{\widehat{p}}^{\rm WF}$ has the harmonic
coefficients of the Wiener-filtered $E$\- and $B$-mode polarization,
${\bm{\mathrm{C}}}$ is the diagonal signal covariance of the lensed $E$\- and
$B$-mode polarization in spherical-harmonic space, and ${\bm{\mathrm{B}}}_{t}$
is a diagonal matrix to include the beam smearing. The matrix
${\bm{\mathrm{C}}}^{1/2}$ is defined so that its square is equal to
${\bm{\mathrm{C}}}$. The real-space vector $\bm{\widehat{p}}^{\rm obs}_{t}$
contains the Stokes $Q$ and $U$ maps observed by experiment $t$, and
${\bm{\mathrm{N}}}_{t}$ is the pixel-space covariance matrix of the
instrumental noise in these maps. The matrix ${\bm{\mathrm{Y}}}_{2}$ is
defined to transform the multipoles of the $E$\- and $B$-mode polarization
into real-space maps of the Stokes parameters $Q$ and $U$. Note that this
filtering optimally mitigates the $E$-to-$B$ leakage due to the masking. We
use the code implemented in cmblensplus [85].

To generate an observed CMB map from a CMB-S4-like experiment, we use the same
realizations of the input lensed CMB polarization given in LB-Lensing by
simply adding a random Gaussian noise component generated by the polarization
noise spectrum obtained from the CMB-S4
wiki.111https://cmb-s4.uchicago.edu/wiki/index.php/Survey_Performance_Expectations
The noise curve contains the effect of the foreground cleaning and atmospheric
$1/f$ noise. We assume the standard internal-linear-combination noise spectrum
for this paper. The polarization map is then multiplied by the CMB-S4 mask
described above and combined with the LiteBIRD polarization map to obtain the
optimal $E$-mode polarization. We assign infinite noise in the noise
covariance for unobserved pixels. In this paper, we do not include any
extragalactic foregrounds, but in practice, we should include masks for
extragalactic contaminants as unobserved pixels, which is left for future
work.

###  3.3 Combining mass tracers

Next, we discuss an estimate of lensing mass for LiteBIRD delensing analysis
in the presence of different sky coverage of mass tracers. Let us assume that
we have $n$ observed mass tracers, $\kappa^{{\rm obs},1},\kappa^{{\rm
obs},2},\dots,\kappa^{{\rm obs},n}$. We obtained the combined lensing-mass map
as

$\displaystyle\widehat{\kappa}^{\rm
WF}_{LM}=\sum_{i=1}^{n}C^{\kappa\kappa^{i}}_{L}\widehat{\kappa}^{\rm
inv,i}_{LM}\,,$ (3.2)

where we obtain the inverse-variance filtered lensing-mass tracers,
$\widehat{\kappa}^{\rm inv,i}_{LM}$, as follows:

$\displaystyle\left(1+{\bm{\mathrm{Y}}}^{\dagger}_{0}\overline{{\bm{\mathrm{N}}}}^{-1}_{\kappa}{\bm{\mathrm{Y}}}_{0}{\bm{\mathrm{C}}}\right)\bm{\widehat{\kappa}}^{\rm
inv}={\bm{\mathrm{Y}}}^{\dagger}_{0}\overline{{\bm{\mathrm{N}}}}^{-1}_{\kappa}\bm{\widehat{\kappa}}^{\rm
obs}\,.$ (3.3)

Here, $\bm{\widehat{\kappa}}^{\rm obs}$ is the data vector of the lensing mass
maps in pixel space. The lensing-mass signal covariance is block diagonal in
$(L,M)$, and each block matrix is symmetric:

$\displaystyle\\{{\bm{\mathrm{C}}}\\}_{LM,L^{\prime}M^{\prime}}\equiv\delta_{LL^{\prime}}\delta_{MM^{\prime}}\begin{pmatrix}C^{\kappa^{1}\kappa^{1}}_{L}&\cdots&C^{\kappa^{1}\kappa^{n}}_{L}\\\
\vdots&&\vdots\\\
C^{\kappa^{1}\kappa^{n}}_{L}&\cdots&C^{\kappa^{n}\kappa^{n}}_{L}\end{pmatrix}\,.$
(3.4)

The matrix ${\bm{\mathrm{Y}}}_{0}$ converts the multipoles of the lensing
convergence to the lensing mass map. The matrix
$\overline{{\bm{\mathrm{N}}}}_{\kappa}$ denotes the pixel-pixel noise
covariance.

We assume that the pixel-pixel noise covariance matrix for each lensing mass
map is given by

$\displaystyle\\{\overline{{\bm{\mathrm{N}}}}_{\kappa}\\}^{i}={\bm{\mathrm{M}}}_{i}{\bm{\mathrm{Y}}}_{0}{\bm{\mathrm{N}}}^{i}_{\kappa}{\bm{\mathrm{Y}}}^{\dagger}_{0}{\bm{\mathrm{M}}}_{i}\,,$
(3.5)

where ${\bm{\mathrm{M}}}_{i}$ is a diagonal matrix for the survey binary mask
of the $i$th mass tracer, and the noise covariance in harmonic space,
${\bm{\mathrm{N}}}^{i}_{\kappa}$, is defined as a diagonal matrix:

$\displaystyle\\{{\bm{\mathrm{N}}}^{i}_{\kappa}\\}_{LM,LM}=N^{i}_{L}\,.$ (3.6)

Here, $N^{i}_{L}$ is the noise power spectrum of the $i$th lensing mass map.
For LiteBIRD and CMB-S4, $N^{i}_{L}$ is the disconnected bias for the
reconstructed CMB lensing maps, which we approximate by the estimator
normalization. Although the noise of CMB lensing maps from LiteBIRD and CMB-S4
could be correlated, we ignore such correlations because the lensing signals
of LiteBIRD and CMB-S4 are mostly reconstructed from CMB anisotropies at large
($\ell\lesssim 500$) and small angular scales ($\ell\gtrsim 500$),
respectively, and the correlation of the two reconstruction noise terms would
be negligible. In the idealistic full-sky observation where
${\bm{\mathrm{M}}}_{i}=1$, the Wiener-filtered lensing mass map becomes that
obtained by Ref. [58]. We use the code implemented in cmblensplus [85].

We use the LiteBIRD reconstructed lensing mass map of LB-Lensing but exclude
$B$-mode polarization at multipoles $\ell\leq 190$ for the lensing
reconstruction to avoid the delensing bias, which primarily originated from
the fact that $B$-mode polarization used for the lensing reconstruction
correlates with that to be delensed (see, e.g., Refs. [47, 86] for more
detailed discussion). The details of the simulation production and
experimental setups for LiteBIRD for the lensing reconstruction are given in
LB-Lensing. We also generate the CIB and galaxy tracers in the full sky as a
constrained realization of multiple random Gaussian fields for a given
realization of the input CMB lensing map. We do not consider the nonlinear
evolution of the external tracers since its impact would be small [87].

###  3.4 Large-scale LiteBIRD B-modes

For the delensing of the large-scale $B$-modes, we prepare the $Q$ and $U$
polarization maps, which contain lensed CMB and Galactic foreground noise. We
optionally further include the IGW contribution to the polarization map. The
input full-sky lensed CMB of LB-Lensing are projected onto a map with $N_{\rm
side}=128$ [88] resolution, including the pixel-window convolution and
$80$-arcmin Gaussian beam convolution. The IGW $B$-mode polarization is also
generated as a random Gaussian field with $r=1$ and projected onto a map with
the same map projection. We use the noise realizations after the Galactic
foreground cleaning obtained from LB23. The maps are provided at $N_{\rm
side}=64$ resolution with $80$-arcmin Gaussian beam convolution, and we
transform the resolution to $N_{\rm side}=128$ to match the lensed CMB maps.

Note that the noise realizations used for the lensing reconstruction and
lensing $B$-mode template construction are independent of LB23. This means
that we ignore the correlations between the noise involved in the lensing
$B$-mode template and that in the large-scale $B$-modes to be delensed. The
correlation is negligible as long as we use different CMB multipoles for the
lensing $B$-mode template and $B$-mode polarization to be delensed, as
demonstrated by multiple existing works [29, 47, 52].

##  4 Results

###  4.1 Delensed B-mode power spectrum

Figure 4:  Partial maps of the full-sky $B$-mode polarization (Left) and
lensing $B$-mode template (Right). For the lensing $B$-mode template, we use
mass tracers from the CMB-S4 lensing map, the CIB, galaxies, and the internal
lensing map.

We first show the lensing $B$-mode template in Fig. 4. We use mass tracers
from the CMB-S4 lensing map, the CIB, galaxies, and the internal lensing map.
We compute the input $B$-mode map by transforming the input $B$-mode harmonic
coefficients using the spin-$0$ inverse spherical harmonics. We include
multipoles $10\leq\ell\leq 190$ for visualization. We can see a clear
correlation between the input $B$-mode map and lensing $B$-mode template in
Fig. 4.

Figure 5:  Correlation coefficients between the input full-sky $B$-mode
polarization and the lensing $B$-mode template. We show the cases using the
mass tracers of the LiteBIRD lensing map (“LiteBIRD internal”), adding CIB
(“$+$CIB”), adding galaxy number density fluctuations (“$+\delta_{\rm g}$”),
adding CIB and galaxy number density fluctuations (“$+$CIB$+\delta_{\rm g}$”),
and further adding CMB-S4 $E$-mode polarization and lensing map
(“$+$CIB$+\delta_{\rm g}+$CMB-S4”).

Figure 6:  Left: Same as Fig. 5, but for the ratio of the delensed $B$-mode
power spectrum to the non-delensed $B$-mode power spectrum. Right: Same as
Left, but for the $1\,\sigma$ error at each multipole bin.

We also show the efficiency of delensing by computing the delensed $B$-mode
power spectrum. For a given realization of the LiteBIRD large-scale $B$-mode
polarization, we compute the Wiener-filtered $B$-mode polarization,
$\hat{B}^{\rm WF}_{lm}$, as described in Eq. (3.1), to reconstruct the
$B$-modes optimally and to mitigate the $E$-to-$B$ leakage induced by the
survey window. We then compute the angular power spectrum of the Wiener-
filtered $B$-mode polarization as $\hat{C}_{l}=\sum_{m}|\hat{B}^{\rm
WF}_{lm}|^{2}/(2l+1)$ and obtain the binned power spectrum, $\hat{C}_{b}$. For
multipole binning, we divide the multipoles between $\ell=2$ and $190$ into
$15$ bins.

For the delensing case, we define the delensed $B$ modes as

$\displaystyle B^{\rm del}_{\ell m}=\hat{B}_{\ell m}-\alpha_{\ell}B^{\rm
temp}_{\ell m}\,,$ (4.1)

where $\alpha_{\ell}$ is determined so that the variance of $B^{\rm del}$ is
minimized, i.e., $\alpha_{\ell}=C^{BB,{\rm cross}}_{\ell}/C_{\ell}^{BB,{\rm
temp}}$. Here, $\hat{C}_{\ell}^{\rm cross}$ is the cross-power spectrum
between the Wiener-filtered $B$-mode polarization and lensing $B$-mode
template, and $\hat{C}_{\ell}^{\rm temp}$ is the lensing template power
spectrum. We thus compute the binned angular power spectrum of the delensed
$B$-mode power spectrum as

$\displaystyle\hat{C}^{\rm del}_{b}=\hat{C}_{b}-2\alpha_{b}\hat{C}^{\rm
cross}_{b}+\alpha_{b}^{2}\hat{C}^{\rm temp}_{b}\,,$ (4.2)

where $\alpha_{b}=\langle\hat{C}^{\rm cross}_{b}\rangle/\langle\hat{C}^{\rm
temp}_{b}\rangle$ is evaluated from simulations.

Figure 5 shows the correlation coefficients between the input full-sky
$B$-mode polarization and lensing $B$-mode template constructed from mass
tracers. We show the cases using the LiteBIRD lensing map, adding CIB, adding
galaxy surveys, adding both CIB and galaxy surveys, and further adding the
CMB-S4 $E$-mode polarization and lensing map. The correlation coefficient for
the LiteBIRD case is much lower than other cases. Combining all available mass
tracers will provide a lensing $B$-mode template that correlates with the
full-sky input $B$-mode map by roughly $50\%$ at $\ell<100$.

Figure 6 shows the $B$-mode power spectrum ratio with delensing compared to
that without delensing. The LiteBIRD lensing map does not help much in
removing the lensing $B$-mode power spectrum. The lensing $B$-mode power is
reduced by $15\%$ at $50\lesssim\ell\lesssim 150$ if we use both CIB and
galaxies and is further reduced by $20\%$ in the same multipole range if we
combine with CMB-S4 data. Figure 6 also compares the $1\,\sigma$ error of the
$B$-mode power spectrum with delensing to the one without delensing. The ratio
has a similar trend to the one shown in the Left panel of the same figure.

###  4.2 Improvement on constraining tensor-to-scalar ratio

Figure 7:  Posterior distribution of $r$ normalized to unity where the fiducial value is $r=5\times 10^{-3}$. We show the cases without (No-delensing) and with delensing using LiteBIRD data alone (LiteBIRD internal), adding the CIB (+CIB), galaxy surveys (+$\delta_{\rm g}$), the CIB and galaxy surveys (+CIB+$\delta_{\rm g}$), and combining LiteBIRD, CMB-S4, the CIB and galaxies (+CIB+$\delta_{\rm g}$+CMB-S4).  | $\sigma(r)\times 10^{3}$
---|---
No-delensing | 1.44
LiteBIRD internal | 1.41
\+ CIB | 1.30
\+ $\delta_{\rm g}$ | 1.31
\+ CIB + $\delta_{\rm g}$ | 1.25
\+ CIB + $\delta_{\rm g}$ \+ CMB-S4 | 1.21
Table 1: Expected constraints on $r$ without (No-delensing) and with delensing
using LiteBIRD data alone (LiteBIRD internal), adding the CIB (+CIB), galaxy
surveys (+$\delta_{\rm g}$), the CIB and galaxy surveys (+CIB+$\delta_{\rm
g}$), and combining LiteBIRD, CMB-S4, the CIB and galaxies (+CIB+$\delta_{\rm
g}$+CMB-S4). We show the $1\,\sigma$ constraint on $r$ if the true value of
$r$ is $5\times 10^{-3}$. In the last case, we use $E$-mode data obtained by
combining LiteBIRD and CMB-S4.

To assess the importance of delensing on constraining $r$, we here compute the
expected constraint on $r$. We consider the following simple forecast based on
the direct likelihood method which has been used in, e.g., Refs. [89, 90, 91],
where the posterior distribution is evaluated from the simulation without
assuming a specific likelihood function.

We first construct an estimator for $r$. At a single multipole, the tensor-to-
scalar ratio, which maximizes the full-sky Gaussian likelihood function of
$B$-mode multipoles, is obtained by, e.g., differentiating Eq. (3) of Ref.
[21] in terms of $r$:

$\displaystyle\hat{r}_{\ell}=\frac{\hat{C}_{\ell}-C^{r=0,\rm
obs}_{\ell}}{C^{\rm IGW}_{\ell}}\,.$ (4.3)

Here, $\hat{C}_{\ell}$ is the observed $B$-mode power spectrum, $C^{r=0,\rm
obs}_{\ell}$ is a model for the observed power spectrum without the IGW
signal, and $C^{\rm IGW}_{\ell}$ is the IGW-induced $B$-mode power spectrum
with $r=1$. In our simulation, we adopt the above equation for the binned
spectrum and obtain an estimate of $\hat{r}_{b}$ at each multipole bin. We
then combine $\hat{r}_{b}$ to estimate the single number for $r$ using a
weighted mean over multipole bins:

$\displaystyle\hat{r}=\frac{\sum_{b}a_{b}\hat{r}_{b}}{\sum_{b}a_{b}}\,.$ (4.4)

The weights, $a_{b}$, are taken from the covariance of $\hat{r}_{b}$ obtained
from simulation as

$\displaystyle
a_{b}=\sum_{b^{\prime}}{\bm{\mathrm{Cov}}}_{bb^{\prime}}^{-1}\,.$ (4.5)

The binned angular power spectra, $C^{r=0,\rm obs}_{b}$ and $C^{\rm IGW}_{b}$,
are both computed from simulations by averaging over realizations. In the
direct likelihood method [89], for a given $r$, we compute the distribution of
$\hat{r}$ and obtain the probability $P(\hat{r}|r)$ and then obtain the
posterior distribution by assuming $P(r|\hat{r})\propto P(\hat{r}|r)$. The
estimator, $\hat{r}$, is unbiased, and we set $\hat{r}=r_{0}$ where $r_{0}$ is
the true value assumed in this paper. We use the posterior distribution,
$P(r|\hat{r}=r_{0})$, to evaluate the expected $1\,\sigma$ error on $r$. In
our forecast, we choose a nonzero true value, $r_{0}=5\times 10^{-3}$, to
highlight delensing as we described in the introduction. We find that the
posterior distribution is close to a Gaussian distribution and compute the
$1\,\sigma$ statistical error on $r$, $\sigma(r)$, by fitting the posterior
distribution with a Gaussian function.

Figure 7 shows the posterior distribution obtained from the direct likelihood
approach. We compare the results with and without delensing. For all the
delensing cases, we use the lensing map internally reconstructed from LiteBIRD
data. Table 1 shows the expected $1\,\sigma$ statistical uncertainty on $r$ if
$r_{0}=5\times 10^{-3}$. The $r$ constraint does not improve much by using the
internally reconstructed lensing map. Adding either the CIB or galaxies shows
a similar improvement on the $r$ constraint. By combining available mass
tracers of the CIB, galaxies, and the internally reconstructed lensing map,
the delensing improves the constraint by approximately $15\%$ by combining
LiteBIRD and external tracers. If we further combine with CMB-S4 data, the
constraint on $r$ improves by $\sim 20\%$. The results are almost insensitive
to the number of multipole bins if the number chosen is around $15$.

##  5 Summary and Discussion

As a further investigation of delensing for LiteBIRD from LB23, we have
explored the potential of the LiteBIRD delensing using multiple tracers,
including the CIB from the Planck GNILC method, galaxy spatial distributions
from Euclid and LSST tomographic data, and high-resolution CMB data from
CMB-S4. We included some realism, such as the survey window for each data set,
a realistic LiteBIRD lensing map, and LiteBIRD $E$-mode polarization after
component separation. We developed algorithms to combine multiple tracers
optimally and to combine $E$-mode polarization from LiteBIRD and CMB-S4, and
we constructed our lensing $B$-mode template from the optimally combined
lensing mass map and the $E$-mode maps. We also used realistic component-
separated $B$-mode polarization obtained by LB23 to estimate the expected
constraint on $r$. We found that delensing can improve the constraint on $r$
by around $15\%$ by combining external data from the CIB and galaxies. The
improvement becomes $20\%$ when further adding CMB-S4. If the residual
foreground noise is reduced, the importance of delensing is further increased.

In our forecast, we have ignored the uncertainties in the mass tracers.
Multiple studies have already investigated the impact of these uncertainties
on the estimate of $r$ [58, 51, 70, 92]. A multiplicative bias in mass tracers
propagates into the lensing $B$-mode template and modifies its shape. However,
the spectral shape is close to a white noise spectrum, similar to the lensing
$B$-mode power spectrum. Thus, the bias in $r$ from the uncertainty of the
mass tracer would be absorbed into a nuisance parameter such as the amplitude
parameter of lensing, $A_{\rm lens}$. Additive biases to the mass tracer could
also lead to a bias in the lensing $B$-mode template. For example, the
internally reconstructed CMB lensing map from temperature could contain
extragalactic sources [92]. Such contributions would be mitigated by future
polarization-based reconstruction with several methods [93, 94, 95, 96, 97].
In CIB delensing, the residual foregrounds in the CIB could lead to a bias in
the delensed $B$-mode power spectrum through higher-order statistics, e.g.,
$\langle BEI\rangle_{\rm c}$ and $\langle EIEI\rangle_{\rm c}$ where $c$
denotes the connected part of the correlation. Ref. [51] showed that, for SO,
the bias is negligible for the constraint on $r$. For LiteBIRD, the map has a
region where the dust contributions are significant compared to the SO patch
assumed in Ref. [51]. However, Galactic foreground cleaning of polarization
can substantially reduce the bias. Also, multi-frequency data would improve
the accuracy of the component separation of both $E$-mode polarization and the
CIB map. We expect this bias to be insignificant, but a quantitative analysis
is important; however, we leave it for our future work.

As an initial investigation of multitracer delensing for LiteBIRD we have
focused on improving constraints on $r$ by adding external tracers and have
ignored inhomogeneous noise due to the scan pattern of LiteBIRD, as well as
any instrumental systematics in our analysis. We also did not include the
inhomogeneity of the delensing efficiency in the map. Inhomogeneous noise will
be accounted for by simply implementing the inhomogeneous noise covariance
into the filtering of the observed CMB anisotropies. For the instrumental
systematics, Ref. [98] recently performed a preliminary analysis of the impact
of instrumental systematics on delensing using a map-based simulation. The
results show that the instrumental systematics in the lensing $B$-mode
template do not significantly bias the delensed $B$-mode power spectrum as
long as the systematics are well suppressed in the measured $B$-mode power
spectrum. The inhomogeneity of the delensing efficiency would also be
included, similar to the inhomogeneous noise, by modifying the pixel
covariance matrix, which would further improve sensitivity to $r$. A more
detailed analysis of these effects will be addressed in future work.

## Acknowledgments

This work is supported in Japan by ISAS/JAXA for Pre-Phase A2 studies, by the
acceleration program of JAXA research and development directorate, by the
World Premier International Research Center Initiative (WPI) of MEXT, by the
JSPS Core-to-Core Program of A. Advanced Research Networks, and by JSPS
KAKENHI Grant Numbers JP15H05891, JP17H01115, and JP17H01125. The Canadian
contribution is supported by the Canadian Space Agency. The French LiteBIRD
phase A contribution is supported by the Centre National d’Etudes Spatiale
(CNES), by the Centre National de la Recherche Scientifique (CNRS), and by the
Commissariat à l’Energie Atomique (CEA). The German participation in LiteBIRD
is supported in part by the Excellence Cluster ORIGINS, which is funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under
Germany’s Excellence Strategy (Grant No. EXC-2094 - 390783311). The Italian
LiteBIRD phase A contribution is supported by the Italian Space Agency (ASI
Grants No. 2020-9-HH.0 and 2016-24-H.1-2018), the National Institute for
Nuclear Physics (INFN) and the National Institute for Astrophysics (INAF).
Norwegian participation in LiteBIRD is supported by the Research Council of
Norway (Grant No. 263011) and has received funding from the European Research
Council (ERC) under the Horizon 2020 Research and Innovation Programme (Grant
agreement No. 772253 and 819478). The Spanish LiteBIRD phase A contribution is
supported by the Spanish Agencia Estatal de Investigación (AEI), project refs.
PID2019-110610RB-C21, PID2020-120514GB-I00, ProID2020010108 and ICTP20210008.
Funds that support contributions from Sweden come from the Swedish National
Space Agency (SNSA/Rymdstyrelsen) and the Swedish Research Council (Reg. no.
2019-03959). The US contribution is supported by NASA grant no. 80NSSC18K0132.
We also acknowledge support from JSPS KAKENHI Grant No. JP20H05859 and No.
JP22K03682, and H2020-MSCA-RISE- 2020 European grant (Marie Sklodowska-Curie
Research and Innovation Staff Exchange). This research used resources of the
National Energy Research Scientific Computing Center (NERSC), a U.S.
Department of Energy Office of Science User Facility located at Lawrence
Berkeley National Laboratory.

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$\bigl{(}\not{D}_{X}^{\mathrm{skew}}\bigr{)}^{*}=-\not{D}_{X}^{\mathrm{skew}}.$
If $n\equiv 2,$ then $\not{D}_{X}^{\mathrm{skew}}$ is $\mathbb{C}$-linear and
$\bigl{(}\not{D}_{X}^{\mathrm{skew}}\bigr{)}^{*}=-\overline{\not{D}_{X}^{\mathrm{skew}}}.$
When we speak of ‘the’ _real Dirac operator_ $D_{X}$ in dimension $n,$ we mean
the positive Dirac operator $\not{D}_{X}^{+}$ if $n\equiv 0,4,$ the skew-
adjoint Dirac operator $\not{D}_{X}^{\mathrm{skew}}$ if $n\equiv 1,2,$ and the
full Dirac operator $\not{D}_{X}$ otherwise.

######  Example B.5.

(a) Let $X$ be a $1$-dimensional oriented Riemannian manifold. As
$\mathrm{Spin}(1)=\\{\pm 1\\}\subset C\ell_{1}=\mathbb{C},$ a spin structure
is a double covering $P_{\mathrm{Spin}}(X)\to X.$ Also $\Delta_{1}=\mathbb{C}$
with $\rho_{\Delta_{1}}(e_{1})=i$ and $\Sigma_{1}=\mathbb{C}$ is the sign
representation. Let $t$ be a local coordinate on $X$ where
$P_{\mathrm{Spin}}(X)$ is trivial. Locally, a spinor is then a complex-valued
function on which the full Dirac operator acts by $\not{D}_{X}=i\frac{{\rm
d}}{{\rm d}t}.$ Since $\Sigma_{1}^{+}=\mathbb{R},$
$\Sigma_{1}^{-}=i\mathbb{R}$ we can locally identify positive spinors with
real-valued functions and negative spinors with purely imaginary functions.
The positive Dirac operator is $\not{D}_{X}^{+}=i\frac{{\rm d}}{{\rm d}t}$ and
the skew-adjoint Dirac operator is $\not{D}_{X}^{\mathrm{skew}}=\frac{{\rm
d}}{{\rm d}t}.$

(b) Let $X$ be a Riemann surface. We have $\Delta_{2}=\mathbb{H}$ where
$\rho_{\Delta_{2}}(e_{1})(q)\allowbreak=qi,$ $\rho_{\Delta_{2}}(e_{2})(q)=qj,$
and the quaternionic structure on $\Delta_{2}$ is $I(q)=iq$ and $J(q)=jq.$ Fix
a complex coordinate $z=x+iy$ on $X$ where $P_{\mathrm{Spin}}(X)$ is trivial.
Locally, a spinor is then an $\mathbb{H}$-valued function $\psi(z)$ on which
the full Dirac operator acts by
$\not{D}_{X}(\psi)=\frac{\partial\psi}{\partial
x}i+\frac{\partial\psi}{\partial y}j.$ The spinor representation $\Sigma_{2}$
splits as $\Sigma_{2}^{\pm}=\mathbb{C}(1\pm j)\subset\mathbb{H},$ so half-
spinors $\psi^{\pm}(z)$ are locally complex-valued functions $f^{\pm}(z)$ and
the skew-adjoint Dirac operator is the Cauchy–Riemann operator
$\not{D}_{X}^{\mathrm{skew}}(f)=2\overline{\partial f/\partial\bar{z}}.$

###  B.3. Dirac operators in adjacent dimensions

Finally, we can explain the relationship between real Dirac operators in
adjacent dimensions. The result is summarized in Table B.2.

######  Proposition B.6.

(a)If $n\equiv 0,$ there is an isomorphism
$\Phi_{n+1}\colon\Sigma_{n}\otimes\mathbb{C}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$
of complex representations of $\mathrm{Spin}(n)$ under which the real
structure
$\sigma_{n+1}=\operatorname{id}_{\Sigma_{n+1}^{+}}\oplus-\operatorname{id}_{\Sigma_{n+1}^{-}}$
on $\Sigma_{n+1}=\Sigma_{n+1}^{+}\oplus\Sigma_{n+1}^{-}$ corresponds to the
complex conjugation on $\Sigma_{n}\otimes\mathbb{C}$ and the Clifford
multiplication to

$\displaystyle\rho_{\Delta_{n+1}}(e_{k})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ(\rho_{\Delta_{n}}(\omega_{n}e_{k})\otimes i),$
$\displaystyle(1\leqslant k\leqslant n)$ (B.4)
$\displaystyle\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ(\rho_{\Delta_{n}}(\omega_{n})\otimes i).$

  1. (b)

If $n\equiv 1,$ there is a $\mathbb{C}$-linear isomorphism
$\Phi_{n+1}\colon\Sigma_{n}\oplus\Sigma_{n}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$
under which the Clifford multiplication and quaternionic structure are

$\displaystyle\rho_{\Delta_{n+1}}(e_{k})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\begin{pmatrix}\rho_{\Delta_{n}}(e_{k})&0\\\
0&-\rho_{\Delta_{n}}(e_{k})\end{pmatrix},$ $\displaystyle\mathrlap{(1\leqslant
k\leqslant n)}$ (B.5)
$\displaystyle\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\begin{pmatrix}0&\operatorname{id}_{\Delta_{n}}\\\
-\operatorname{id}_{\Delta_{n}}&0\end{pmatrix},$ $\displaystyle
J\circ\Phi_{n+1}$ $\displaystyle=\Phi_{n+1}\circ\begin{pmatrix}0&\sigma_{n}\\\
-\sigma_{n}&0\end{pmatrix},$

where $\sigma_{n}$ is the real structure on $\Sigma_{n}$ from (a).

  2. (c)

If $n\equiv 2,$ there is an $\mathbb{H}$-linear isomorphism
$\Phi_{n+1}\colon\Sigma_{n}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$ with

$\displaystyle\rho_{\Delta_{n+1}}^{+}(e_{k})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\rho_{\Delta_{n}}(e_{k}),$
$\displaystyle{(1\leqslant k\leqslant n)}$ (B.6)
$\displaystyle\rho_{\Delta_{n+1}}^{+}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=-\Phi_{n+1}\circ\rho_{\Delta_{n}}(\omega_{n}).$

  3. (d)

If $n\equiv 3$ let $\mathbb{K}=\mathbb{H},$ and if $n\equiv 7$ let
$\mathbb{K}=\mathbb{R}.$ There is a pair of $\mathbb{K}$-linear isomorphisms
$\Phi_{n+1}^{\pm}\colon\Sigma_{n}\to\Sigma_{n+1}^{\pm}|_{\mathrm{Spin}(n)}$
under which

$\displaystyle\rho_{\Delta_{n+1}}(e_{k})\circ\Phi_{n+1}^{\pm}$
$\displaystyle=\Phi_{n+1}^{\mp}\circ\rho_{\Delta_{n}^{+}}(e_{k}),$
$\displaystyle(1\leqslant k\leqslant n)$ (B.7)
$\displaystyle\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}^{\pm}$
$\displaystyle=\pm\Phi_{n+1}^{\mp}.$

  4. (e)

If $n\equiv 4,$ there is an $\mathbb{H}$-linear isomorphism
$\Phi_{n+1}\colon\Sigma_{n}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$ with

$\displaystyle\rho_{\Delta_{n+1}}(e_{k})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\rho_{\Delta_{n}}(e_{k}),$
$\displaystyle(1\leqslant k\leqslant n)$ (B.8)
$\displaystyle\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=I\circ\Phi_{n+1}\circ\rho_{\Delta_{n}}(\omega_{n}).$

  5. (f)

If $n\equiv 5,$ then $\Sigma_{n}$ has a quaternionic structure $I,$ $J$ and
there is an isomorphism of representations
$\Phi_{n+1}\colon\Sigma_{n}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$ with

$\displaystyle\rho_{\Delta_{n+1}}(e_{k}e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\rho_{\Delta_{n}}(e_{k}),$
$\displaystyle(1\leqslant k\leqslant n)$ (B.9)
$\displaystyle\rho_{\Delta_{n+1}}(\omega_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ I,$
$\displaystyle\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ J.$

  6. (g)

If $n\equiv 6,$ there is an $\mathbb{R}$-linear isomorphism
$\Phi_{n+1}\colon\Sigma_{n}\to\Sigma_{n+1}|_{\mathrm{Spin}(n)}$ with

$\displaystyle\rho_{\Delta_{n+1}^{+}}(e_{k})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\rho_{\Delta_{n}}(e_{k}),$
$\displaystyle(1\leqslant k\leqslant n)$ (B.10)
$\displaystyle\rho_{\Delta_{n+1}^{+}}(e_{n+1})\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ I.$

The isomorphisms (a)–(g) are orthogonal and unique up to scalar factors.

###### Proof.

We discuss (a) in detail and give briefer arguments for (b)–(g).

(a) If $n\equiv 0,$ then (B.1) implies that the $\mathbb{C}$-linear
endomorphisms of $\Delta_{n}\otimes_{\mathbb{R}}\mathbb{C}$ defined by
$E_{k}=\rho_{\Delta_{n}}(\omega_{n}e_{k})\otimes i,$ $1\leqslant k\leqslant
n,$ and $E_{n+1}=\rho_{\Delta_{n}}(\omega_{n})\otimes i$ satisfy the Clifford
relations for $C\ell_{n+1}.$ Since $\Delta_{n+1}$ is the unique irreducible
complex representation of $C\ell_{n+1}$ and
$\dim_{\mathbb{C}}\Delta_{n}\otimes_{\mathbb{R}}\mathbb{C}=\dim_{\mathbb{C}}\Delta_{n+1},$
there exists a $\mathbb{C}$-linear isomorphism
$\Phi_{n+1}\colon\Delta_{n}\otimes_{\mathbb{R}}\mathbb{C}\to\Delta_{n+1},$
orthogonal and unique up to a scalar.

According to Theorem B.2 there is a real structure on
$\Delta_{n+1}|_{C\ell_{n+1}^{0}}$ (the irreducible representations of the
algebra $C\ell_{n}\cong C\ell_{n+1}^{0}$ are real), unique up to a sign, which
is fixed by the choice of splitting into $\Sigma_{n+1}^{\pm}.$ On
$\Delta_{n}\otimes_{\mathbb{R}}\mathbb{C}$ the complex conjugation defines a
real structure. By Schur’s Lemma, the real structures are automatically
preserved up to sign, so by passing to $i\Phi_{n+1}$ if necessary we can
arrange for them to be preserved exactly. Since
$\Sigma_{n}=\Delta_{n}|_{\mathrm{Spin}(n)}$ and
$\Sigma_{n+1}|_{\mathrm{Spin}(n)}=\Delta_{n+1}|_{\mathrm{Spin}(n)},$ the map
$\Phi_{n+1}$ defines the claimed isomorphism of representations satisfying
(B.4).

(b) If $n\equiv 1,$ the complex representation
$\Delta_{n}\oplus\alpha^{*}(\Delta_{n})$ of $C\ell_{n}$ is extended by

$E_{n+1}=\begin{pmatrix}0&\operatorname{id}_{\Delta_{n}}\\\
-\operatorname{id}_{\Delta_{n}}&0\end{pmatrix},\quad
J=\begin{pmatrix}0&\sigma_{n}\\\ -\sigma_{n}&0\end{pmatrix}$

to a quaternionic representation of $C\ell_{n+1},$ which is thus isomorphic to
$\Delta_{n+1}.$

(c) If $n\equiv 2,$ the $\mathbb{H}$-linear endomorphisms of $\Delta_{n}$
defined by

$E_{k}=\rho_{\Delta_{n}}(e_{k}),1\leqslant k\leqslant n,\quad
E_{n+1}=-\rho_{\Delta_{n}}(\omega_{n})$

satisfy the Clifford relations for $C\ell_{n+1}.$ Since $E_{1}\cdots
E_{n}E_{n+1}=1,$ there is an $\mathbb{H}$-linear isomorphism
$\Delta_{n}\to\Delta_{n+1}^{+}$ as in (B.6).

(d) If $n\equiv 3,7,$ the $\mathbb{K}$-linear endomorphisms of
$\Delta_{n}^{+}\oplus\Delta_{n}^{+}$ defined by

$E_{k}=\begin{pmatrix}\rho_{\Delta_{n}^{+}}(e_{k})&0\\\
0&-\rho_{\Delta_{n}^{+}}(e_{k})\end{pmatrix},\quad(1\leqslant k\leqslant
n)\quad E_{n+1}=\begin{pmatrix}0&\operatorname{id}_{\Delta_{n}^{+}}\\\
-\operatorname{id}_{\Delta_{n}^{+}}&0\end{pmatrix}.$

satisfy the Clifford relations for $C\ell_{n+1}.$ It follows that there is a
$\mathbb{K}$-linear isomorphism
$\Psi_{n+1}\colon\Delta_{n}^{+}\oplus\Delta_{n}^{+}\to\Delta_{n+1}$ taking
$E_{k}$ to $\rho_{\Delta_{n+1}}(e_{k})$ for $1\leqslant k\leqslant n+1.$ Now

$E_{1}\cdots E_{n}E_{n+1}=\begin{pmatrix}0&\omega_{n}\\\
\omega_{n}&0\end{pmatrix}=\begin{pmatrix}0&\operatorname{id}_{\Delta_{n}^{+}}\\\
\operatorname{id}_{\Delta_{n}^{+}}&0\end{pmatrix}\quad\text{as
$\omega_{n}|_{\Delta_{n}^{+}}=\operatorname{id}_{\Delta_{n}^{+}},$}$

so the splitting of $\Sigma_{n+1}=\Delta_{n+1}|_{\mathrm{Spin}(n+1)}$ into the
$(\pm 1)$-eigenspaces $\Sigma_{n+1}^{\pm}$ of the operator
$\rho_{\Delta_{n+1}}(\omega_{n+1})$ corresponds to the splitting of
$\Delta_{n}^{+}\oplus\Delta_{n}^{+}$ into the diagonal and anti-diagonal
spaces. Now define $\Phi_{n+1}^{\pm}(\phi)=\Psi_{n+1}(\phi,\pm\phi).$

(e) If $n\equiv 4,$ the $\mathbb{H}$-linear endomorphisms on $\Delta_{n}$
defined by $E_{k}=\rho_{\Delta_{n}}(e_{k})$ for $1\leqslant k\leqslant n$ and
$E_{n+1}=I\rho_{\Delta_{n}}(\omega_{n})$ satisfy the Clifford relations for
$C\ell_{n+1},$ so there is an $\mathbb{H}$-linear isomorphism
$\Phi_{n+1}\colon\Delta_{n}\to\Delta_{n+1}.$

(f) If $n\equiv 5,$ consider the real representation $j_{n}^{*}(\Delta_{n+1})$
of $C\ell_{n},$ where $e_{k}$ acts by $\rho_{\Delta_{n+1}}(e_{k}e_{n+1})$ for
$1\leqslant k\leqslant n.$ Observe that
$I^{\prime}=\rho_{\Delta_{n+1}}(\omega_{n+1})$ and
$J^{\prime}=\rho_{\Delta_{n+1}}(e_{n+1})$ anti-commute with the Clifford
multiplication by vectors, hence define a quaternionic structure on the
representation $j_{n}^{*}(\Delta_{n+1})|_{C\ell_{n}^{0}}.$ The unique
irreducible quaternionic representation $\Delta_{n}|_{C\ell_{n}^{0}}$ has the
same dimension, hence there exists an $\mathbb{H}$-linear isomorphism
$\Phi_{n+1}\colon\Delta_{n}|_{C\ell_{n}^{0}}\to
j_{n}^{*}(\Delta_{n+1})|_{C\ell_{n}^{0}}.$

(g) If $n\equiv 6,$ the $\mathbb{R}$-linear endomorphisms of $\Delta_{n}$
defined by $E_{k}=\rho_{\Delta_{n}}(e_{k}),$ $1\leqslant k\leqslant n,$ and
$E_{n+1}=-\rho_{\Delta_{n}}(\omega_{n})$ satisfy the Clifford relations for
$C\ell_{n+1}.$ Moreover, $E_{1}\cdots E_{n}E_{n+1}=1,$ so there is an
$\mathbb{R}$-linear isomorphism
$\Phi_{n+1}\colon\Delta_{n}\to\Delta_{n+1}^{+}.$ As $\omega_{n}^{2}=-1$ and
$\omega_{n}$ commutes with $C\ell_{n}^{0},$ we can arrange for
$I=\rho_{\Delta_{n}}(\omega_{n}).$ ∎

We now compute the Dirac operator over $Y=X\times\mathbb{R}$ in a local
orthonormal frame $e_{1},\ldots,e_{n}$ of $X$ and where
$e_{n+1}=\partial/\partial t.$ We have

$P_{\mathrm{Spin}}(TY)=(P_{\mathrm{Spin}}(TX)\times\mathrm{Spin}(n+1))/\mathrm{Spin}(n)\times\mathbb{R}$

and hence the isomorphisms of representations from Proposition B.6 induce
isomorphisms $\Phi_{n+1}^{(\pm)}$ between the spinor bundle of $Y$ and the
pullback of the spinor bundle of $X$ to $X\times\mathbb{R}.$

$\boldsymbol{n}\equiv$ Spinor bundles Dirac on $X^{n}$ On cylinder
$X\times\mathbb{R}$ $0$ $\Sigma_{X}^{\pm}$ real $\not{D}_{X}^{+}$
$\mathbb{R}$-linear
$\not{D}_{Y}^{\mathrm{skew}}=\begin{pmatrix}\partial/\partial
t&\not{D}_{X}^{-}\\\ -\not{D}_{X}^{+}&-\partial/\partial t\end{pmatrix}$ $1$
$\Sigma_{X}^{+}\cong\Sigma_{X}^{-}$
real $\not{D}_{X}^{\mathrm{skew}}$ $\mathbb{R}$-linear
skew-adjoint $\not{D}_{Y}^{\mathrm{skew}}=\overline{\frac{\partial}{\partial
t}+i(\not{D}_{X}^{\mathrm{skew}})_{\mathbb{C}}}$ $2$
$\Sigma_{X}^{+}\cong\overline{\Sigma_{X}^{-}}$
complex $\not{D}_{X}^{\mathrm{skew}}$ $\mathbb{C}$-linear
skew-adjoint $\not{D}_{Y}=\begin{pmatrix}-i\partial/\partial
t&-\overline{\not{D}_{X}^{\mathrm{skew}}}\\\
\not{D}_{X}^{\mathrm{skew}}&i\partial/\partial t\end{pmatrix}$ $3$
$\Sigma_{X}$ quat. $\not{D}_{X}$ $\mathbb{H}$-linear
self-adjoint $\not{D}_{Y}^{+}=\frac{\partial}{\partial t}+\not{D}_{X}$ $4$
$\Sigma_{X}^{\pm}$ quat. $\not{D}_{X}^{+}$ $\mathbb{H}$-linear
$\not{D}_{Y}=\begin{pmatrix}i\partial/\partial t&\not{D}_{X}^{-}\\\
\not{D}_{X}^{+}&-i\partial/\partial t\end{pmatrix}$ $5$ $\Sigma_{X}$ quat.
$\not{D}_{X}$ $\mathbb{C}$-linear
self-adjoint (anti-
commutes with $J$) $\not{D}_{Y}=J\bigl{(}\frac{\partial}{\partial
t}+\not{D}_{X}\bigr{)}$ $6$ $\Sigma_{X}$ complex $\not{D}_{X}$
$\mathbb{C}$-anti-linear
self-adjoint $\not{D}_{Y}=I\frac{\partial}{\partial t}+\not{D}_{X}$ $7$
$\Sigma_{X}$ real $\not{D}_{X}$ $\mathbb{R}$-linear
self-adjoint $\not{D}_{Y}^{+}=\frac{\partial}{\partial t}+\not{D}_{X}$

Table B.2. Dirac operators on a spin $n$-manifold $X$ and over a collar
neighborhood of a spin $(n+1)$-manifold $Y$ which can be identified with a
cylinder $X\times[0,\varepsilon)\subset X\times\mathbb{R},$ up to deformation.

####  B.3.1. Case $n\equiv 0$

The map $\Phi_{n+1}$ from Proposition B.6(a) restricts to an isomorphism
$\Phi_{n+1}^{+}\colon\Sigma_{n}\to\Sigma_{n+1}^{+}$ of the real points. Recall
that $\Sigma_{n}$ decomposes into the $(\pm 1)$-eigenspaces $\Sigma_{n}^{\pm}$
of $\omega_{n}.$ With respect to this decomposition, we have

$\displaystyle\not{D}_{X\times\mathbb{R}}^{\mathrm{skew}}$
$\displaystyle\circ\Phi_{n+1}^{+}=-I\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}^{+}$ by (B.2)
$\displaystyle=\Phi_{n+1}^{+}\circ\left(\sum_{k=1}^{n}\begin{pmatrix}0&\rho_{\Delta_{n}}(e_{k})\\\
-\rho_{\Delta_{n}}(e_{k})&0\end{pmatrix}\nabla_{k}+\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\frac{\partial}{\partial t}\right)$ by (B.4)
$\displaystyle=\Phi_{n+1}^{+}\circ\begin{pmatrix}\frac{\partial}{\partial
t}&\not{D}_{X}^{-}\\\ -\not{D}_{X}^{+}&-\frac{\partial}{\partial
t}\end{pmatrix}.$

####  B.3.2. Case $n\equiv 1$

The splitting of $\Sigma_{n+1}$ into the $(\pm i)$-eigenspaces
$\Sigma_{n+1}^{\pm}$ of $\omega_{n+1}$ corresponds under the isomorphism
$\Phi_{n+1}$ from Proposition B.6(b) to the splitting of
$\Sigma_{n}\oplus\Sigma_{n}$ into the diagonal and anti-diagonal subspaces
because

$\rho_{\Delta_{n+1}}(\omega_{n+1})\circ\Phi_{n+1}=\Phi_{n+1}\circ\begin{pmatrix}0&I\\\
I&0\end{pmatrix}$

by (B.5). In particular, $\Phi_{n+1}$ induces isomorphisms
$\Phi_{n+1}^{\pm}\colon\Sigma_{n}\to\Sigma_{n+1}^{\pm},$
$\phi\mapsto\Phi_{n+1}(\phi,\pm\phi).$ Using (B.3) and (B.5) we now compute

$\displaystyle\not{D}_{X\times\mathbb{R}}^{\mathrm{skew}}\circ$
$\displaystyle\Phi_{n+1}^{+}(\phi)=-J\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}\begin{pmatrix}\phi\\\ \phi\end{pmatrix}$
$\displaystyle=-J\circ\Phi_{n+1}\left(\sum_{k=1}^{n}\begin{pmatrix}e_{k}&0\\\
0&-e_{k}\end{pmatrix}\begin{pmatrix}\nabla_{k}\phi\\\
\nabla_{k}\phi\end{pmatrix}+\begin{pmatrix}0&1\\\
-1&0\end{pmatrix}\begin{pmatrix}\partial\phi/\partial t\\\
\partial\phi/\partial t\end{pmatrix}\right)$
$\displaystyle=\Phi_{n+1}\begin{pmatrix}0&-\sigma_{n}\\\
\sigma_{n}&0\end{pmatrix}\begin{pmatrix}\sum_{k=1}^{n}e_{k}\nabla_{k}\phi+\partial\phi/\partial
t\\\ \sum_{k=1}^{n}-e_{k}\nabla_{k}\phi-\partial\phi/\partial t\end{pmatrix}$
$\displaystyle=\Phi_{n+1}^{+}\circ\sigma_{n}(\not{D}_{X}+\partial/\partial
t)\phi$

Splitting $\Sigma_{n}$ further into the eigenspaces $\Sigma_{n}^{\pm}$ of
$\sigma_{n},$ we can write this as

$\sigma_{n}(\not{D}_{X}+\partial/\partial t)=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}\partial/\partial t&\not{D}_{X}^{-}\\\
\not{D}_{X}^{+}&\partial/\partial
t\end{pmatrix}=\overline{\frac{\partial}{\partial
t}+i(\not{D}_{X}^{\mathrm{skew}})_{\mathbb{C}}}.$

####  B.3.3. Case $n\equiv 2$

We have

$\displaystyle\not{D}_{X\times\mathbb{R}}\circ\Phi_{n+1}=\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}^{+}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}^{+}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\left(\sum_{k=1}^{n}\rho_{\Delta_{n}}(e_{k})\nabla_{k}-\rho_{\Delta_{n}}(\omega_{n})\frac{\partial}{\partial
t}\right)=\Phi_{n+1}\circ\begin{pmatrix}-i\partial/\partial
t&\not{D}_{X}^{-}\\\ \not{D}_{X}^{+}&i\partial/\partial t\end{pmatrix},$

where at the last step we split $\Sigma_{n}$ into the $(\pm i)$-eigenspaces
$\Sigma_{n}^{\pm}$ of $\omega_{n}.$ The composition of
$\Phi_{n+1}\colon\Sigma_{n}\to\Sigma_{n+1}$ with
$\operatorname{id}_{\Sigma_{n}^{+}}\oplus
J\colon\Sigma_{n}^{+}\oplus\overline{\Sigma_{n}^{+}}\to\Sigma_{n}^{+}\oplus\Sigma_{n}^{-}$
is an isomorphism
$\Psi_{n+1}\colon\Sigma_{n}^{+}\oplus\overline{\Sigma_{n}^{+}}\to\Sigma_{n+1}.$
The previous calculation and (B.2) imply

$\not{D}_{X\times\mathbb{R}}\circ\Psi_{n+1}=\Psi_{n+1}\circ\begin{pmatrix}-i\partial/\partial
t&-\overline{\not{D}_{X}^{\mathrm{skew}}}\\\
\not{D}_{X}^{\mathrm{skew}}&i\partial/\partial t\end{pmatrix}$

(the ‘$i$’ in lower right corner stands for the scalar multiplication in
$\overline{\Sigma_{n}^{+}},$ hence is equal to the action of $-i$ on
$\Sigma_{n}^{+}$).

####  B.3.4. Case $n\equiv 3,7$

Using (B.7) we find

$\displaystyle\not{D}_{X\times\mathbb{R}}^{+}\circ\Phi_{n+1}^{+}$
$\displaystyle=\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}^{+}$
$\displaystyle=\Phi_{n+1}^{-}\circ\left(\sum_{k=1}^{n}\rho_{\Delta_{n}}(e_{k})\nabla_{k}+\frac{\partial}{\partial
t}\right)=\Phi_{n+1}^{-}\circ\left(\not{D}_{X}+\frac{\partial}{\partial
t}\right).$

####  B.3.5. Case $n\equiv 4$

We have

$\displaystyle\not{D}_{X\times\mathbb{R}}\circ\Phi_{n+1}$
$\displaystyle=\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\left(\rho_{\Delta_{n}}(e_{k})\nabla_{k}+\rho_{\Delta_{n}}(\omega_{n})\frac{\partial}{\partial
t}\right)$ by (B.8)
$\displaystyle=\Phi_{n+1}\circ\begin{pmatrix}i\partial/\partial
t&\not{D}_{X}^{-}\\\ \not{D}_{X}^{+}&-i\partial/\partial t\end{pmatrix},$

where at the last step we split $\Sigma_{n}$ into the $(\pm i)$-eigenspaces
$\Sigma_{n}^{\pm}$ of $\omega_{n}.$

####  B.3.6. Case $n\equiv 5$

We have

$\displaystyle\not{D}_{X\times\mathbb{R}}\circ\Phi_{n+1}=\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}}(e_{n+1})\frac{\partial}{\partial
t}\right)\mathrlap{\circ\Phi_{n+1}}$
$\displaystyle=\rho_{\Delta_{n+1}}(e_{n+1})\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k}e_{n+1})\nabla_{k}+\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}$
$\displaystyle=\rho_{\Delta_{n+1}}(e_{n+1})\circ\Phi_{n+1}\circ\left(\sum_{k=1}^{n}\rho_{\Delta_{n}}(e_{k})\nabla_{k}+\frac{\partial}{\partial
t}\right)$ by (B.9) $\displaystyle=\Phi_{n+1}\circ
J\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}}(e_{k})\nabla_{k}+\frac{\partial}{\partial
t}\right)$ by (B.9) $\displaystyle=\Phi_{n+1}\circ
J(\not{D}_{X}+\partial/\partial t).$

####  B.3.7. Case $n\equiv 6$

Using (B.10) we find

$\displaystyle\not{D}_{X\times\mathbb{R}}\circ\Phi_{n+1}=\left(\sum_{k=1}^{n}\rho_{\Delta_{n+1}^{+}}(e_{k})\nabla_{k}+\rho_{\Delta_{n+1}^{+}}(e_{n+1})\frac{\partial}{\partial
t}\right)\circ\Phi_{n+1}$
$\displaystyle=\Phi_{n+1}\circ\left(\sum_{k=1}^{n}\rho_{\Delta_{n}}(e_{k})\nabla_{k}+I\frac{\partial}{\partial
t}\right)=\Phi_{n+1}\circ\left(\not{D}_{X}+I\frac{\partial}{\partial
t}\right).$

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