LiteBIRD science goals and forecasts: a full-sky measurement of gravitational lensing of the CMB

We explore the capability of measuring lensing signals in LiteBIRD full-sky polarization maps. With a 30 arcmin beam width and an impressively low polarization noise of 2.16 μK-arcmin, LiteBIRD will be able to measure the full-sky polarization of the cosmic microwave background (CMB) very precisely. This unique sensitivity also enables the reconstruction of a nearly full-sky lensing map using only polarization data, even considering its limited capability to capture small-scale CMB anisotropies. In this paper, we investigate the ability to construct a full-sky lensing measurement in the presence of Galactic foregrounds, finding that several possible biases from Galactic foregrounds should be negligible after component separation by harmonic-space internal linear combination. We find that the signal-to-noise ratio of the lensing is approximately 40 using only polarization data measured over 80% of the sky. This achievement is comparable to Planck's recent lensing measurement with both temperature and polarization and represents a four-fold improvement over Planck's polarization-only lensing measurement. The LiteBIRD lensing map will complement the Planck lensing map and provide several opportunities for cross-correlation science, especially in the northern hemisphere.

A Constraint on A lens 18 1 Introduction Cosmic microwave background (CMB) polarization has been measured by multiple CMB experiments to improve constraints on cosmology.A CMB linear polarization map contains two spatial patterns: parity-even E-modes and parity-odd B-modes [1,2].In linear theory, density perturbations are the dominant source of the temperature anisotropies and E-mode polarization.The density perturbations do not produce the B-mode polarization without non-linear effects [3].However, inflationary gravitational waves could generate the B-mode polarization [4,5].The main goal of the LiteBIRD experiment is the measurement of Bmode polarization produced by inflationary gravitational waves, which would be considered conclusive evidence for inflation in the early Universe [6].
The lensed CMB polarization data has off-diagonal correlations between angular multipoles, which can be utilized to reconstruct the gravitational lensing potential [53,54].Multiple CMB experiments have reconstructed the CMB lensing mass map.Its angular power spectrum has been measured, by the Atacama Cosmology Telescope [55][56][57][58][59][60], BICEP [61,62], Planck [40,[63][64][65], POLARBEAR [66,67], and the South Pole Telescope [68][69][70][71][72]. Upcoming and future ground-based CMB experiments, including the Simons Observatory [73], CMB-S4 [74] and Ali CMB Polarization Telescope (AliCPT) [75], are planning to make high-sensitivity measurements of CMB polarization.Their target sensitivities are enough to measure the small-scale B-mode signal caused by lensing and significantly improve the precision of lensing measurements.After Planck however, only LiteBIRD can reconstruct a full-sky lensing mass map including both the northern and southern hemispheres, which is impossible to achieve with a single ground-based experiment.Therefore, the LiteBIRD lensing mass map provides an exciting opportunity to cross-correlate the CMB lensing with galaxies on the sky.The LiteBIRD lensing map in the northern hemisphere would also be complementary with the AliCPT lensing map [75].
This work is part of a series of papers that present the science achievable by the Lite-BIRD space mission, expanding on the overview of the mission published in Ref. [6] (hereafter, LB23).In particular, this work focuses on the initial investigation of the capability of measuring lensing signals with LiteBIRD.Multiple studies have explored the practical issues and developed mitigation techniques for these issues, such as survey boundary and source masking [76][77][78][79][80], inhomogeneous noise [78,81], extragalactic foregrounds [79,[82][83][84][85][86][87][88][89], and instrumental systematics on lensing measurements [90,91].Among practical concerns on lensing analysis, the Galactic foregrounds are one of the most important issues for LiteBIRD.Reference [92] shows that the residual Galactic foregrounds should be negligible in CMB-S4-like experiments that utilize small-scale CMB anisotropies.However, we use large-scale CMB polarization data in the LiteBIRD lensing measurement, where the Galactic foregrounds could be impactful [92].This paper addresses this question by simulating the lensing reconstruction from a nearly full-sky LiteBIRD polarization map, including the component-separation procedure.We further investigate some applications of the LiteBIRD lensing map, including the primordial non-Gaussianity, the late-time ISW effect, and inflationary gravitational waves.This paper is organized as follows.In Section 2, we explain our method for lensing reconstruction for LiteBIRD.In Sec. 3, we describe our setup for simulation.In Sec. 4, we show our main results for the lensing reconstruction for LiteBIRD.In Sec. 5, we investigate potential applications of the LiteBIRD lensing map for cosmology.Finally, Sec. 6 is devoted to a summary and discussion.Throughout this paper, we use the cosmological parameters for the flat ΛCDM model adopted in LB23.In a companion paper [93] (hereafter LB-Delensing), we explore the feasibility of delensing for LiteBIRD.We choose the cosmological parameters obtained from [94].This section describes our methodology for the lensing reconstruction from LiteBIRD polarization data.After briefly reviewing the principal impacts of weak gravitational lensing on CMB anisotropies, we summarize the quadratic estimator for the lensing reconstruction [53,54], the internal-linear combination (ILC) in harmonic space (hereafter HILC) [95] for component separation, and filtering of the polarization map.The temperature map obtained from LiteBIRD will not offer significant improvement over the results already obtained by the Planck mission due to its restricted angular resolution.However, LiteBIRD will be able to precisely measure CMB polarization over the entire sky, providing a complementary fullsky lensing map that will have better accuracy at large angular scales (L < 10) since the EB quadratic estimator will be much less sensitive to several potential mean fields [79].Our study focuses on the polarization measurements of LiteBIRD, where the polarization quadratic estimators yield the highest signal-to-noise ratio (SNR).The rest of this article will thus concentrate on polarization analysis.

Gravitational Lensing of CMB
The trajectories of CMB photons passing through a mass distribution are deflected by the gravitational potential which is referred to as gravitational lensing.The gravitational potential of the large-scale structure in the late-time Universe causes the gravitational lensing effect on the CMB, and the lensing effect distorts the observed CMB anisotropies.The lensed polarization anisotropies described by the Stokes Q and U parameters that we measure today are approximated as (e.g., Refs.[96][97][98]) where n is the line-of-sight unit vector and d is the deflection vector.We define the lensing potential, ϕ, as d = ∇ϕ, where ∇ is the covariant derivative on the sphere.The lensing potential is related to the line-of-sight projection of the three-dimensional gravitational potential (the so-called Weyl potential [97]), Ψ, sourced by the matter distribution, as Here, χ is the conformal distance, η 0 is the conformal time today, and χ * is the conformal distance to the last-scattering surface. 1 The gravitational potential, Ψ(χ n, η 0 − χ), is evaluated along the unperturbed trajectory χ n, the so-called Born approximation [100][101][102], with conformal time η 0 − χ.We ignore the curl mode, which vanishes in the linear theory of perturbations having only scalar density perturbations (see, e.g., Ref. [103] and reference therein).Furthermore, the lensing converts part of the E-mode polarization into B-mode polarization [104].The lensing-induced B-mode polarization is roughly comparable to a 5 µKarcmin white noise up to half-degree angular scale and has been detected by ground-based CMB experiments (e.g., Ref. [46]).This lensing B-mode polarization impedes the detection of inflationary gravitational waves [51].LB-Delensing focuses on overcoming this issue with delensing techniques, which we will also mention in Sec. 5.

Reconstruction of Lensing Potential
The lensing potential can be reconstructed using the fact that averaging over CMB realizations, while keeping the lensing potential unchanged, violates the statistical isotropy of the CMB, thereby introducing correlations between different CMB polarization multipoles.The off-diagonal elements of the CMB polarization covariance generated by lensing up to linear order in ϕ are given as [53,105] 2 where X, Y ∈ {E, B} and the quantity in parentheses is the Wigner 3j symbol that represents the angular momentum coupling.The function f XY ℓℓ ′ L quantifies the response of the off-diagonal correlations to lensing whose explicit form is given by Table 1 in Ref. [53].Thus, the lensing potential can be estimated as a quadratic combination of the CMB anisotropies at different angular scales.
In the LiteBIRD case, the EB estimator dominates the SNR of lensing.The improvement of the SNR using the iterative estimator is negligible for the LiteBIRD case.Therefore, this paper only focuses on the EB quadratic estimator.In an idealistic case, the estimator is given as [53] where we define In an idealistic case, the filtered multipoles, Ēℓm and Bℓm , are obtained by multiplying their inverse variance for each multipole, Xℓm = X ℓm /C XX ℓ , with X = E or B. In this study, we employ Wiener filtering, including the pixel-space noise covariance, the details of which are described in Sec.2.4.The normalization in the idealistic full-sky case is given by In the following, we describe how to implement the estimator for LiteBIRD.

Galactic Foreground Cleaning
We perform lensing reconstruction on the foreground-cleaned polarization map.Our simulation includes both Galactic dust and synchrotron emission, with spatially varying spectral indices.Point sources and free-free emission are excluded, since their impact on large angular CMB multipoles is insignificant [108,109].The harmonic coefficients of our frequency maps at each observed frequency ν are modeled as where S CMB ℓm is the CMB component, S FG,ν ℓm is the total foreground contribution at the observing frequency ν, and N ν ℓm is the noise at the frequency ν.The assumption of a known frequency scaling permits the utilization of ILC in harmonic space for the cleaning.An additional critical assumption underlying the ILC method is the Gaussianity of the CMB.In HILC, we define the foreground-cleaned map as [95] where w ℓ contains the weights for each frequency map, and the bold letters indicate the vectors containing all observed frequency maps.We can derive the HILC weights by minimizing the variance from the foregrounds and noise contributions under the constraint for an unbiased estimate, w T ℓ • a = 1, with a = (1, 1, . . ., 1) T .The weights are then given by where the covariance of frequency maps in harmonic space is given by (2.10) The HILC method requires the total power spectrum, which includes contributions from the CMB, Galactic foregrounds, and noise, to compute the covariance of the frequency maps.
In practice, this total power spectrum is estimated from observed data.On the other hand, the method does not include the non-Gaussianity of the Galactic foregrounds.Unlike other component-separation methods, the HILC method focuses on minimizing the variance in the foreground cleaned map while preserving the signal power, rather than globally suppressing the statistical noise.HILC assumes isotropy to derive the weights, in contrast to, for example, Needlet ILC [110].This isotropic assumption limits the ability of HILC to optimally account for spatial variations in foreground morphology and spectral energy distribution (SED), potentially introducing bias in the lensing measurements due to the spatial variation of Galactic foregrounds.This paper tests whether the HILC method works even if the Galactic foregrounds have some spatial variation.

Filtering of CMB anisotropies
We compute the Wiener-filtered E-and B-mode polarization starting from the above componentseparated polarization map as inputs.Specifically, we solve the following equation [111,112]: Here, we solve for the vector p WF , which has the harmonic coefficients of the Wiener-filtered E-and B-mode polarization, C s is the diagonal signal covariance of the lensed E-and Bmode polarization in spherical-harmonic space, and B is a diagonal matrix to include the beam smearing.The matrix C 1/2 s is defined so that its square is equal to C s .The real-space vector p obs contains the observed Stokes Q and U maps after adopting HILC.The matrix Y 2 is defined to transform the multipoles of the E-and B-modes into real-space maps of the Stokes parameters Q and U .Finally, N pix is the pixel-space covariance matrix of the instrumental noise in these maps.Foreground residuals and noise presented in our component-separated polarization maps.The solid black line represents the signal.The left and right panels display the E-and B-mode residuals, respectively.We illustrate the residuals for two foreground models, s0d0 (red) and s1d1 (green).Note that the green curves deviate significantly from the assumed model during component separation, substantially increasing residuals.Both spectra are computed from 500 fullsky simulations.
To compute the noise covariance, we assign infinite noise for pixels not used in the analysis.We assume isotropic noise in observed pixels, following a model of E-and B-mode noise power spectra.Explicitly, we assume that the inverse noise covariance is given by where W is a matrix that takes the value 1 for pixels used in the analysis and zeroes otherwise, and N is the noise covariance of E and B modes.We assume that the noise covariance of E-and B-modes is diagonal in harmonic space, with elements given by a model of the Eand B-mode noise power spectra, N EE ℓ and N BB ℓ .Our simulation pipeline uses the noise spectra obtained from the HILC weights during component separation with the reference noise spectra of 15 frequency channels.We do not include any extra masks besides the Galactic mask.

Simulations
This section overviews the simulation sets used in this analysis and in LB-Delensing to produce the results.We use the experimental specifications described in table 13 of LB23 to generate multiple frequency maps with frequency coverage ranging from 40 GHz to 402 GHz in 15 frequency channels.Note that while table 13 of LB23 provides 22 frequency channels, we exclusively consider the combined values for each frequency.However, our investigation finds that the changes in the noise properties remain negligible in the foreground-cleaned map, even when utilizing the full 22-frequency configuration.
We generate realizations of the frequency maps and the post-component-separation map through the following steps.
1. We create a lensed CMB polarization map of the full sky.We first generate unlensed CMB polarization maps and a lensing potential map on the full sky as random Gaussian fields drawn from the input fiducial angular power spectra.We then remap the unlensed CMB maps with the lensing potential map using lenspyx, 3 where the algorithm utilizes bicubic interpolation in an oversampled equidistant-cylindrical-projection grid.
2. We produce 15 frequency maps of synchrotron and dust emission using pysm3. 4For our baseline simulation set we use s1 and d1 models for synchrotron and dust, respectively.In the s1 model, a power law scaling is applied to the synchrotron emission templates [113,114] with a spatially varying spectral index [115].The thermal dust, d1, is modeled as a single-component modified blackbody.The Planck dust template is scaled to different frequencies with a modified-blackbody spectrum using spatially varying temperature and spectral index [116].In our analysis, we also consider the d0 model for dust and the s0 model for synchrotron radiation, both sourced from PySM.Here, the d0 model corresponds to a simplified version of the d1 model, characterized by a fixed spectral index of 1.54 and a blackbody temperature of 20 K. On the other hand, the s0 has a constant spectral index of −3.
3. Each frequency map containing the CMB signal and the foreground components is convolved with the associated Gaussian beam, and then the corresponding white noise is added to each frequency map.
4. We deconvolve the aforementioned beam-convolved frequency maps and perform component separation using the HILC algorithm with fgbuster5 to obtain the full-sky component-separated map.HILC uses an evenly binned covariance, with bin size, ∆ℓ = 10.
We repeat this process for each realization and generate 500 realizations (hereafter SET1).These maps are defined on a Healpix [117] grid with N side = 512.In addition, we generate a simulation set to specifically compute the so-called N (1) bias in the lensing measurement [118], where the input lensing map is kept fixed for all realizations.This fixed-lensing simulation is generated for 100 realizations (hereafter SET2).
Figure 1 shows the E-and B-mode angular power spectra of the component-separated CMB maps on the full sky.At large angular scales (ℓ ≲ 100), the residual foregrounds become non-negligible, especially for the B-mode power spectrum, which is dominated by residual foregrounds at ℓ ≲ 100.At smaller angular scales (ℓ ≳ 600), the instrumental noise becomes dominant.

Lensing Reconstruction
In this section, we present the results of the reconstruction of lensing potential from our simulation set.For the forecast, we consider the s1d1 foreground model as our baseline simulation.We also highlight the biases in our estimates of the lensing power spectrum.

Reconstructed Lensing Map
We reconstruct the lensing potential from the component-separated maps using the public Planck Galactic mask6 that keeps 80% of the sky.Before this reconstruction, we apply the filtering as described in Eq. (2.11) of Sec.2.4 to the component-separated maps.The HILCweighted noise spectra averaged over the SET1 simulation are used for the noise covariance, N. We employed the EB quadratic estimator with cmblensplus7 [119] for reconstructing the lensing deflection field.For the analysis of this paper, CMB multipoles 2 < ℓ < 600 were used.The maximum multipole is determined so that the SNR of the lensing signals is saturated. 8Figure 2 shows one realization of the reconstructed lensing-convergence map.
We can see a clear correlation between the input and reconstructed lensing maps.

Biases in the Lensing Power Spectrum Estimate
The estimation of the angular power spectrum of the lensing potential requires the subtraction of several biases because the estimates of the lensing potential are quadratic in CMB polarization anisotropies, and the power spectrum of the lensing estimator is the four-point The mean and standard deviation are obtained from 400 realizations of the SET1 simulation.The red and green solid lines are the MCN0 and MCN1 biases, respectively.The mean field is shown with a blue line.We use the s1d1 model for this analysis.We show the power spectrum up to L = 600 since the power spectrum at L > 600 is significantly noise-dominant.
correlation of the CMB anisotropies.The CMB four-point correlation consists of contributions from disconnected and connected parts, and the latter contains the lensing potential power spectrum.The remaining terms emerge as a bias in estimating the lensing power spectrum [118,[120][121][122].The power spectrum of the quadratic estimator, C φ φ L , is described by The first term is the lensing potential power spectrum, C ϕϕ L , multiplied by a factor, R L , due to a mismatch between the analytic and true normalizations.The second term, N (0) L , arises from the disconnected part of the four-point correlation, the so-called N 0 bias, which receives contributions from the CMB, foreground, and noise.The third term, N L , is the so-called N 1 bias that arises from the secondary contraction of the connected parts by lensing at first order in C ϕϕ L [118], and the last term, ∆C ϕϕ L , contains the other remaining biases of C ϕϕ L due to, e.g., possible issues of the estimation of normalization, mixing of power between different multipoles, and biases at O((C ϕϕ L ) 2 ).C φφ . Top: Comparison plot of the N0 (upper, colored solid) and mean-field biases (lower, colored dash-dotted), with and without the Galactic foregrounds.For the N0 bias, we show two foreground cases, s0d0 and s1d1, in addition to the no foreground case ('No FG').We also show the Monte Carlo error in estimating the mean-field bias, C MC L , (dotted), and the lensing power spectrum signal (gray solid).Bottom: The ratio of the measured lensing power spectrum to the input lensing power spectrum with (red) and without (green) the Galactic foregrounds.Note that the power spectrum shown in red is shifted to the right as L → L + 1 for clarity.
We estimate the mean-field bias, ⟨ φLM ⟩, which arises primarily due to the mask and foreground residuals, using 100 realizations of the SET1 simulation.The mean-field bias is subtracted at the map level as shown in Eq. (2.4), and these realizations are not included in computing other bias terms.To check the level of the mean-field bias, we also compute the power spectrum of the mean-field bias, C MF L .The N 0 bias approximately corresponds to the noise power spectrum of the reconstructed lensing map.We compute the N 0 bias with a Monte Carlo simulation (hereafter MCN0).Specifically, we compute the lensing estimator using the SET1 simulation in the following way: where i and j are simulation indices, and we define The ensemble average is over 500 realizations from SET1 with i and j = i + 1 changing cyclically.Since the MCN0 estimate is not optimal [123,124], we also compute the realizationdependent N 0 (hereafter RDN0) bias [64,79] for each realization in SET2, which is given by The average is over 100 realizations with j = i + 1, which is independent of that used for evaluating N (0),MC L .The N 1 bias is estimated with the SET2 simulation as (e.g., Ref. [69], hereafter MCN1) Here, we use 100 realizations of the SET2 simulations to evaluate the ensemble average.Figure 3 shows the significance of each bias term compared with the input theoretical lensing power spectrum.We also show the debiased angular power spectrum of lensing potential: where R MC L is the normalization correction obtained from Here, ϕ in is the full sky input lensing map.We use 400 realizations of the SET1 simulations to obtain the mean and standard deviation of Ĉϕϕ L .After correcting for the N0 bias, N1 bias, and normalization, the power spectrum is in good agreement with the input power spectrum.This indicates that ∆C ϕϕ L is negligible.The result also shows that the foreground-induced trispectrum and mean-field bias are negligible.
In Figure 4, we compare the N 0 and mean field biases of the lensing reconstruction in the presence and absence of Galactic foregrounds.Without Galactic foregrounds, the only non-idealistic effect is the Galactic mask, which primarily causes the mean-field bias.With the foreground cleaning, the noise level of the cleaned CMB polarization map increases, leading to an increase in the N 0 bias.The power spectrum of the mean-field bias is more than an order of magnitude lower than the signal power spectrum.The mean field does not increase even in the presence of the residual Galactic foregrounds.

The signal-to-noise ratio
We here show a simulation-based estimate of the SNR of the lensing signal.We first measure the amplitude of the lensing spectrum from a weighted mean over multipole bins: .SNR of the lensing power spectrum.The yellow bar shows the case with no foregrounds, while the light green and blue bars represent the cases with the s0d0 and s1d1 foreground models, respectively.The red bar shows the case with the s1d1 foreground model but with a wider sky coverage, f sky = 0.9.The pink bar is the SNR from the Planck polarization-based lensing analysis [65].
The quantity Âb is the relative amplitude of the power spectrum compared with a fiducial power spectrum for the Planck ΛCDM cosmology, C ϕϕ b , i.e., Âb ≡ Ĉϕϕ b /C ϕϕ b .The weights, a b , are taken from the band-power covariance as The fiducial band-power values and their covariances, including off-diagonal correlations between different multipole bins, are evaluated from the simulations.The SNR is then given by where σ A is the 1 σ constraint on A lens computed from SET1 simulations.Figure 5 shows a summary of the SNR for three cases: the baseline, s0d0, and noforeground cases.We find that the SNR of the LiteBIRD lensing reconstruction is comparable to that of the latest full-sky lensing measurement from Planck [40].In the presence of the s0d0 foregrounds, the SNR decreases by 10% compared to the no-foreground case.If we consider the foregrounds with the spatially varying spectral index, the SNR decreases by 3% compared to the s0d0 case.It is worth noting that LiteBIRD uses polarization alone, and the reconstructed map from LiteBIRD is complementary to the Planck lensing map, which is mostly based on temperature anisotropies.If we increase the sky fraction to f sky = 0.9, the SNR increases by 6.8%, a factor of 4 improvement compared to the Planck polarization-only reconstruction (referred to as Pol hereafter).
Figure 6 shows a comparison between N (0),MC L of the Planck polarization-only reconstruction and of LiteBIRD.The lensing maps from Planck full-data and LiteBIRD are reconstructed mostly from temperature and from polarization, respectively, and are almost statistically independent.Thus, the combined lensing map from Planck and LiteBIRD will between Planck and LiteBIRD.The gray solid line represents the signal, while the blue and red solid lines correspond to the noise bias in the Planck and LiteBIRD polarization-based estimates, respectively.have an SNR of approximately 60.We will show a more accurate estimate of the SNR for the combined lensing map in our future work.

Impact of Foregrounds in Lensing estimate
In this section, we quantify the impact of the foregrounds on the lensing estimate, specifically, by computing the bias in the estimation of A lens .We assume that the s1d1 simulation represents real data.We measure the lensing power spectrum from s1d1 and estimate A lens from Eq. (4.8).We also repeat the same analysis but with a simulation generated from an incorrect model, i.e., the no-foregrounds and s0d0 cases.The incorrect simulation is used to evaluate the bias terms in the lensing power spectrum estimate.We then compute the covariance from the incorrect simulation and obtain A lens from the measured power spectrum.
Figure 7 presents the posterior distributions of A lens assuming that s1d1 is real data.The blue posterior corresponds to the estimation of A lens when assuming no foregrounds, resulting in a bias of 30%.This bias reduces to 10% when we utilize the s0d0 model, but is larger than the 1σ statistical error.This is because the two models, s0d0 and s1d1, have significant discrepancies already in the EE and BB power spectra, as we show in Fig. 1.This discrepancy leads to a significant mismatch between the true and simulation disconnected bias.In practice, however, we would not use no-foreground or s0d0 for our model since it does not fit the data of EE and BB power spectra.Thus, the bias shown here is more enhanced than what would be obtained in an analysis of real data.We finally mention when we correctly use the simulation from the s1d1 model.The bias in A lens is negligible compared to σ(A lens ), indicating that potential biases other than those considered in the power spectrum estimation are negligible.The blue curve represents the bias in A lens when employing the no foreground simulation set to fit the selected spectra.Similarly, the orange and green posterior distributions represent cases where the s0d0 and s1d1 models were used, respectively.The estimated A lens for each case are quoted at the respective peak of posterior.To investigate the agreement of the estimated lensing potential angular power spectra between our simulation sets and detect potential biases in our analysis, we apply the Kolmogorov-Smirnov (KS) test to compute p-values within each multipole bin of the angular power spectra.The p-values are calculated under the null hypothesis that the power spectra for both no-foreground and foreground cases follow the same distribution after bias mitigation (see Eq. 4.6).The p-values for both foreground cases, s0d0 and s1d1 are shown in Fig 8 .For both foreground cases, the p-values for angular scales below L = 400 exceeded the 0.05 threshold, indicating a consistent distribution with the no-foreground case.However, for angular scales beyond L = 400, the p-values fell below 0.05, suggesting a significant deviation.This deviation can be attributed to the non-negligible disconnected bias present in the foreground cases.Despite the discrepancies at high L, our analysis demonstrates the effective recovery of the lensing power spectrum, particularly within the angular scales of L ≤ 400, where the lensing signal dominates.

Applications of LiteBIRD Lensing Map
In this Section, we discuss some of the potential applications of the LiteBIRD lensing map.

Cross-correlations
One potential application of the LiteBIRD CMB lensing map is to cross-correlate it with other tracers of the large-scale structures in the Universe.The full-sky LiteBIRD lensing map can be used to calibrate tracers of the large-scale structure through cross-correlation and hence to constrain cosmology.Here, to estimate the SNR of the cross-correlation signal, we use the following equation: where X and Y represent the observables as specified below, N XX L and N Y Y L are the associated noise power spectra, and f XY sky is the sky fraction of the overlapping patch of the two observables.In this work, we use the baseline noise model, obtained with the s1d1 simulations (as described in Section 3).We do not consider potential biases arising from the bispectrum of the lensing potential, which only leads to a sub-percent level of bias using the EB estimator, even for high-resolution experiments, as shown in Ref. [125].

Galaxy-lensing cross-correlation as a probe of primordial non-Gaussianity
The fluctuations of the galaxy number density trace the underlying matter distribution of the large-scale structure and efficiently correlate with the CMB lensing maps.We estimate the SNR of the galaxy-CMB lensing cross-correlation signal (C ϕg L ) for two different galaxy surveys: the one provided by Euclid [126] and the one from the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST, [127]).To compute the total SNR, we consider a range of multipoles L = [2,1000].For the Euclid survey, we consider ten equipopulated tomographic galaxy bins and f ϕg sky = 0.36, obtaining SNR = 73.For the LSST survey, we consider ten equispaced redshift bins and f ϕg sky = 0.35, obtaining SNR = 56.Through the cross-correlation analysis between CMB lensing and galaxy distributions, we can constrain the local primordial non-Gaussianity f NL , which induces a scale-dependent bias due to the coupling between long and short wavelength modes (e.g., Refs.[128][129][130][131]).To determine the uncertainty of f NL using the cross-spectrum C ϕg L alone, we perform a χ 2 analysis in which we let only f NL free to vary.When considering the cross-correlation between Euclid galaxies and LiteBIRD CMB lensing, the resulting constraints on f NL are σ(f NL ) = 44.Note that in this analysis, we vary only f NL , and therefore, we do not explore potential degeneracies with other cosmological parameters.

CIB-lensing cross-correlation
The cosmic infrared background (CIB) is the integrated emission from unresolved dusty star-forming galaxies.Produced by the stellar-heated dust within galaxies, the CIB carries a wealth of information about the star formation process.The CIB traces the matter distribution at a relatively high redshift compared to galaxies in typical optical redshift surveys and is strongly correlated with CMB lensing.This cross-correlation can be used to constrain CIB models [132] and cosmology [133].Planck has measured the CIB-lensing cross-correlations with an SNR of 40 (statistical uncertainties only) [45].We estimate the SNR of the CIBlensing cross-correlation, assuming the CIB anisotropies measured by Planck which utilizes the model of CIB described in Appendix D of Ref. [64].We choose L min = 100 due to a contaminant from residual foregrounds in the CIB map.We find that the SNR of the crosscorrelation is 79 with f ϕI sky = 0.60, which is roughly a factor of two improvement compared with obtained by Planck.

ISW-lensing cross-correlation
The ISW effect provides information on large-scale structure through large-scale temperature fluctuations [134].As the same structure generates both the CMB lensing potential and the ISW effect, a substantial cross-correlation between the two observables is expected [135].The cross-correlation between ISW and CMB lensing is only significant at low multipoles (see, e.g., Ref. [97]).Thus, to measure the cross-correlation with the ISW effect, we need a nearly full-sky observation of the lensing map from space, such as Planck and LiteBIRD.We compute the SNR of the ISW-lensing cross-correlation signal, C ϕT ℓ , assuming cosmic-variance limited temperature fluctuations and f ϕT sky = 0.80.The estimated SNR for our baseline case is approximately 4.

Constraints on tensor-to-scalar ratio by delensing
Finally, we can use the internally reconstructed lensing potential for delensing.However, the lensing map of LiteBIRD does not significantly improve the constraint on this tensor-to-scalar ratio as shown in LB-Delensing.The improvement on this constraint is only at the level of a few percent if we use the LiteBIRD lensing map (see LB-Delensing for the details).

Summary and Discussion
We have conducted a lensing reconstruction study assuming the LiteBIRD experimental configuration, focusing on the impact of Galactic foregrounds from synchrotron and dust emission on the lensing analysis.We performed component separation on the frequency maps, applied filtering to the post-component-separated maps, and estimated the lensing potential map.We showed that the foreground-induced mean field and trispectrum have negligible biases on the measurement of the lensing power spectrum.Furthermore, we showed that the SNR of the LiteBIRD lensing map is approximately 40 which is comparable to that obtained from the latest Planck measurement.The LiteBIRD lensing map additionally holds potential for several cross-correlation analyses.
We focused on the lensing reconstruction from LiteBIRD data alone, but we can add Planck data to provide a more precise lensing map over the full sky, which will be investigated in future work.We assumed homogeneous white noise, but the LiteBIRD noise is inhomogeneous due to the scan strategy.This effect could introduce a larger mean-field bias [40,81].However, the EB estimator has no significant mean-field bias due to the parity symmetry [79,136].The inhomogeneous noise makes the reconstruction sub-optimal without including its effect in the analysis.However, we can optimally perform componentseparation and lensing analysis by modifying the covariance matrix, including the inhomogeneity of the noise [40].For the optimal lensing analysis, we can also incorporate the spatial variation in the estimator normalization [137].We have also ignored the instrumental systematic effect of the LiteBIRD experiment.Beam systematics could be one of the main sources of biases in the lensing measurement since the lensing analysis uses smaller scales available for a given dataset.These practical issues, including an optimal analysis for inhomogeneous noise and residual foregrounds, and instrumental systematics, will be investigated in our future works.The software used for this study is publicly available at https://github.com/litebird/LiteBIRD-lensing.

A Constraint on A lens
This appendix shows some details on estimating A lens .We first note that Eq.Differentiating the above likelihood in terms of A lens leads to Eq. (4.8).Instead of using Eq.(4.8), we can also estimate A lens by maximizing the above likelihood.For example, Fig. 9 shows the results with an Affine-Invariant Markov Chain Monte Carlo (MCMC) Ensemble sampler implemented in Python package emcee 9 for three different foreground cases.Note that the estimated A lens values are in agreement with A lens = 1 within the statistical errors.This highlights the agreement between our reconstructed lensing power spectrum in Eq. (4.6) and the theoretical input power spectrum.
To create the covariance matrix Cov, one can use MCN0 into Eq.(4.6) as the estimate of the disconnected bias.However, this choice introduces correlations within the covariance matrix, which, in turn, influences the precision of the SNR estimation (e.g., Refs.[61,123,124]).Instead, RDN0 has been used to reduce these correlations.For the LiteBIRD case, this reduction becomes visually evident when examining the correlation matrix shown in Fig. 10.Notably, the utilization of RDN0 leads to a reduction in correlations within the angular power spectra.
Figure 1.Foreground residuals and noise presented in our component-separated polarization maps.The solid black line represents the signal.The left and right panels display the E-and B-mode residuals, respectively.We illustrate the residuals for two foreground models, s0d0 (red) and s1d1 (green).Note that the green curves deviate significantly from the assumed model during component separation, substantially increasing residuals.Both spectra are computed from 500 fullsky simulations.

Figure 2 .
Figure 2. Lensing convergence map, which is defined in harmonic space as κ LM ≡ L(L + 1)ϕ LM /2 in harmonic space, obtained from one realization of the SET1 s1d1 simulation.The map contains multipole of 2 < L < 100.Left: The input map used to remap the primary CMB.Right: The Wiener-filtered reconstructed map.The map resolution is 5 arcmin-pixel −1 .

Figure 3 .
Figure 3. Binned angular power spectrum of the lensing potential shown in the black with error bars.The mean and standard deviation are obtained from 400 realizations of the SET1 simulation.The red and green solid lines are the MCN0 and MCN1 biases, respectively.The mean field is shown with a blue line.We use the s1d1 model for this analysis.We show the power spectrum up to L = 600 since the power spectrum at L > 600 is significantly noise-dominant.

Figure 5
Figure 5. SNR of the lensing power spectrum.The yellow bar shows the case with no foregrounds, while the light green and blue bars represent the cases with the s0d0 and s1d1 foreground models, respectively.The red bar shows the case with the s1d1 foreground model but with a wider sky coverage, f sky = 0.9.The pink bar is the SNR from the Planck polarization-based lensing analysis[65].

Figure 7 .
Figure 7. Posterior distribution of A lens when estimating it for realizations of the s1d1 simulation.The blue curve represents the bias in A lens when employing the no foreground simulation set to fit the selected spectra.Similarly, the orange and green posterior distributions represent cases where the s0d0 and s1d1 models were used, respectively.The estimated A lens for each case are quoted at the respective peak of posterior.

Figure 8 .
Figure 8.The Kolmogorov-Smirnov test p-values obtained for each bin.The red and blue lines represent the p-values when comparing s0d0 and s1d1 with the no-foreground case.

Figure 9 .
Figure 9. Box plots showing the distribution of samples, with blue, orange, and green denoting the no-foreground, s0d0, and s1d1 cases, respectively.A red dashed line serves as the reference value, indicating A lens = 1.Within each box plot, a box spans from the first quartile to the third quartile, while a horizontal black line extends through the box at the median.Lower and upper whiskers correspond to 4 σ, and individual points denote outliers (≥ 4 σ).

Figure 10 .
Figure 10.Correlation matrix of the binned angular power spectra for the s1d1 case.The left panel represents results obtained using MCN0, while the right panel illustrates outcomes with RDN0.Notably, the utilization of RDN0 leads to a reduction in correlations within the angular power spectra.