# Detection of cross-correlation between CMB Lensing and low-density points

Fuyu Dong<EMAIL_ADDRESS>School of Physics, Korea Institute for
Advanced Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of
Korea  Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao
Tong University, Shanghai, 200240, China  Pengjie Zhang<EMAIL_ADDRESS>Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai, 200240, China  Division of Astronomy and Astrophysics,
Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai, 200240,
China  Key Laboratory for Particle Astrophysics and Cosmology (MOE)/Shanghai
Key Laboratory for Particle Physics and Cosmology, China  Le Zhang
<EMAIL_ADDRESS>School of Physics and Astronomy, Sun Yat-Sen
University, 2 Daxue Road, Tangjia, Zhuhai, 519082, P.R. China  Ji Yao
Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai, 200240, China  Zeyang Sun  Department of Astronomy,
School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai,
200240, China  Changbom Park  School of Physics, Korea Institute for Advanced
Study (KIAS), 85 Hoegiro, Dongdaemun-gu, Seoul, 02455, Republic of Korea
Xiaohu Yang  Department of Astronomy, School of Physics and Astronomy,
Shanghai Jiao Tong University, Shanghai, 200240, China  Division of Astronomy
and Astrophysics, Tsung-Dao Lee Institute, Shanghai Jiao Tong University,
Shanghai, 200240, China  Key Laboratory for Particle Astrophysics and
Cosmology (MOE)/Shanghai Key Laboratory for Particle Physics and Cosmology,
China

###### Abstract

Low Density Points (LDPs, Dong et al. (2019)), obtained by removing high-
density regions of observed galaxies, can trace the Large-Scale Structures
(LSSs) of the universe. In particular, it offers an intriguing opportunity to
detect weak gravitational lensing from low-density regions. In this work, we
investigate tomographic cross-correlation between Planck CMB lensing maps and
LDP-traced LSSs, where LDPs are constructed from the DR8 data release of the
DESI legacy imaging survey, with about $10^{6}$-$10^{7}$ galaxies. We find
that, due to the large sky coverage (20,000 deg2) and large redshift depth
($z\leq 1.2$), a significant detection ($10\sigma$–$30\sigma$) of the CMB
lensing-LDP cross-correlation in all six redshift bins can be achieved, with a
total significance of $\sim 53\sigma$ over $\ell\leq 1024$. Moreover, the
measurements are in good agreement with a theoretical template constructed
from our numerical simulation in the WMAP 9-year $\Lambda$CDM cosmology. A
scaling factor for the lensing amplitude $A_{\rm lens}$ is constrained to
$A_{\rm lens}=1\pm 0.12$ for $z<0.2$, $A_{\rm lens}=1.07\pm 0.07$ for
$0.2<z<0.4$ and $A_{\rm lens}=1.07\pm 0.05$ for $0.4<z<0.6$, with the r-band
absolute magnitude cut of $-21.5$ for LDP selection. A variety of tests have
been performed to check the detection reliability, against variations in LDP
samples and galaxy magnitude cuts, masks, CMB lensing maps, multipole $\ell$
cuts, sky regions, and photo-z bias. We also perform a cross-correlation
measurement between CMB lensing and galaxy number density, which is consistent
with the CMB lensing-LDP cross-correlation. This work therefore further
convincingly demonstrates that LDP is a competitive tracer of LSS.

Cosmology: cosmic background radiation – Cosmology: large-scale structure of
Universe – Cosmology: gravitational lensing

##  1 INTRODUCTION

Due to the gravitational lensing effect, the Large-Scale Structures (LSSs)
along the line of sight deflects the Cosmic Microwave Background (CMB) photons
trajectories, which can leave measurable imprints on the CMB. The CMB lensing,
as a late-time observable, therefore can place competitive constraints on
cosmological parameters and help to resolve the Hubble tension (Planck
Collaboration et al., 2020a) between the early- and the late-Universe.

Reconstructed lensing maps contain the integrated information of the overall
matter distribution in the universe, and hence such tomography approach allows
us to precisely measure the evolution of the growth of structure, as well as
increasing the significance of the detection of CMB lensing. Nowadays, many
large and deep redshift surveys, which have broad overlap with the CMB lensing
kernel, have made significant progress in finding high-significance detection
of cross-correlations.

Over the past few years, many studies have been carried out to investigate the
cross-correlation between CMB lensing and LSS tracers, such as galaxy-CMB
lensing correlation (Hirata et al., 2008; Das et al., 2011; Bleem et al.,
2012; van Engelen et al., 2012; Sherwin et al., 2012; Holder et al., 2013;
Geach et al., 2013; Omori & Holder, 2015; Allison et al., 2015; Bianchini et
al., 2015; Giannantonio et al., 2016; Peacock & Bilicki, 2018; Giusarma et
al., 2018; Omori et al., 2019; Kitanidis & White, 2020; Singh et al., 2020;
Krolewski et al., 2020; Hang et al., 2021b, a, a; Shekhar Saraf et al., 2021),
cosmic shear-CMB (polarization) lensing correlation (Vallinotto, 2012; Kuntz,
2015; Giusarma et al., 2018; Baxter et al., 2016; Namikawa et al., 2019),
thermal SZ-CMB lensing correlation (Hill & Spergel, 2014).

Furthermore, contrary to the high-density peaks of the matter distribution,
the vast low-density regions (known as cosmic voids) has recently achieved
great importance for cosmology. This is because the voids have many desirable
properties for probing the dark energy and can be accurately modeled by the
linear perturbation theory, even on small scales. The studies of the
correlation between CMB lensing maps with vast low-density regions have been
performed recently (Cai et al., 2016; Chantavat et al., 2017; Vielzeuf et al.,
2020; Raghunathan et al., 2020; Hang et al., 2021a), reporting non-zero
detection about 3–5$\sigma$.

Similar to the void method, Dong et al. (2019) have suggested measuring the
stacked shear signals around Low-Density-Points (LDPs), which enables to
differentiate several dark energy models. LDPs are obtained by removing high-
density regions of observed galaxies within a certain radius, yielding void-
like profiles with negative surface density. In addition, Dong et al. (2021)
demonstrate that, using a correlation with CMB temperature maps, the LDP
method is a competitive alternative to the existing methods for measuring the
integrated Sachs-Wolfe (ISW) effect. In this study, we use the reconstructed
convergence map from the Planck observation (Planck Collaboration et al.,
2020a) and LDPs from the DESI legacy imaging surveys DR8 to examine the
performance of the CMB lensing-LDP cross correlation.

The paper is organized as follows: In Sect. 2, we describe the data sets used
in our analysis, and explain the methodology adopted in this study, and then
we obtain theoretical predictions based on a numerical N-body simulationin in
§3. In Sect. 4, we present our results and consistency checks are performed in
Sect. 5. Our conclusions are given in Sect. 6.

##  2 Data analysis

In this study, we perform the analysis using the photo-$z$ galaxy catalog from
the DESI imaging surveys and the Planck CMB lensing maps, in the Galactic
coordinate system.

###  2.1 Galaxy Catalog

The photometric galaxy catalog of the DESI imaging surveys is selected upon
the three optical bands ($g$, $r$, $z$) and mid-infrared bands observed by the
Wide-field Infrared Survey Explorer satellite (Zou et al., 2019; Silva et al.,
2016; Flaugher, 2005; Blum et al., 2016; Abbott et al., 2018), for which the
DESI Legacy Survey DR8 data covers $\sim 20000~{}\rm deg^{2}$ in both northern
and southern Galactic caps, and source detection uses stacked images of the
$(g,r,z)$ bands.

We select galaxy samples by using the DESI DR8 galaxy catalog (Zou et al.,
2019), in which bright stars are excluded and only consider the galaxies with
$r<23.0$ with full five-band photometric measurements. The photo-$z$ of each
galaxy is estimated by a local linear regression algorithm (Beck et al., 2016;
Gao et al., 2018), based on the nearest neighbors with spectroscopic
redshifts, and the redshift error is given as the root-mean-square between the
spectroscopic redshifts of these neighbors and the best-fit one.

###  2.2 CMB Lensing Map

We use the latest PLANCK reconstructed lensing convergence maps and analysis
masks provided by Planck Collaboration et al. (2020a). In this work our
fiducial analysis is based on the CMB lensing maps estimated from the SMICA
foreground-cleaned CMB maps (Planck Collaboration et al., 2020b) with the
temperature-only estimator, where the impact of the thermal Sunyaev-Zeldovich
(tSZ) effect has been deprojected using a multi-frequency component separation
approach. Moreover, the lensing map determined from both the temperature and
polarization maps with the minimum-variance (MV) estimator, is also taken into
account for comparison, where the tSZ bias in lensing reconstruction is not
removed. Throughout the paper, these two lensing maps will be referred to as
“tSZ-deproj” and “MV”, respectively.

The spherical harmonic coefficients of the reconstructed lensing convergence
maps are provided in the format of HEALPix (Górski et al., 2005), with
$\ell_{\rm max}=4096$, and the associated masks are given as HELAPix maps with
the resolution of $N_{\rm side}$=2048. In order to match the resolution of the
LDP maps, we downgrade the lensing maps to a lower resolution with $N_{\rm
side}=512$ in an appropriate way, to avoid aliasing of power from small to
large scales (see details in Appendix A).

Furthermore, noise Planck Full Focal Plane
(FFP10111http://pla.esac.esa.int/pla/#cosmology) simulations (Planck
Collaboration et al., 2020a) (300 realizations in total) are used to determine
the mean-field bias and the associated statistical error, and specifically, 60
out of the 300 simulations for the mean-field removal and the rest for the
covariance matrix estimation.

###  2.3 Survey Masks

The mask is generated from the available random catalogs provided by the DR8
website222https://portal.nersc.gov/cfs/cosmo/data/legacysurvey/dr8/randoms/.
The random points contain the number of observations in the $g,r,z$ bands,
according to sky coordinates drawn independently from the observed
distribution.

With $N_{\rm side}=4096$ in the HEALPix scheme, we choose random points which
have observations in all $grz$ bands and MASKBITS$=0$ to produce the survey
mask, which can populate the same sky coverage and geometry with the galaxy
catalog. The objective of the MASKBITS is to provide a general purpose
artifact flagging capability. As defined, MASKBITS $\equiv\sum_{i}2^{{\rm
bit}_{i}}$, and the bit values, ${\rm{bit}}_{i}$, which are integers from 0 to
13, indicate the different causes of contamination and are defined
specifically for each point of the random catalogues released by DESI. In
general, for a given point, the MASKBITS $>0$ (i.e., ${\rm bit}_{i}\geq 0$)
corresponds to a “bad” one which may potentially suffer contamination that
requires flagging. Instead, in our case, since we only consider sources that
are not in corrupted pixels, nor near bright stars or globular clusters or
large galaxies, i.e., excluding objects with MASKBITS above zero set in the
Legacy Surveys catalogs, so that we simply use random points with MASKBITS
$=0$ to create a binary mask map that removes all the contaminated sources. In
the analysis, all sources lying in the masked regions are excluded.

###  2.4 LDP Identification and Over-Density Map

Figure 1: Map of LDP overdensity, $f_{ldp}$, using HEALPix scheme with resolution $N_{\rm side}$ = 512. A combined mask ($f_{sky}\sim 0.44$) from both DESI DR8 catalog and the Planck survey is applied to the full-sky LDP over-density (i.e., the fraction of low-density points in each pixel). LDPs are generated with a cut radius of $R_{s}=3^{\prime}$ based on the galaxy sample selected with absolute magnitude cut of $M_{c}=-21$ and photo-$z$ cut of $0.2<z<0.4$. Table 1: Galaxies that are brighter than $M_{c}$ in the DR8 photo-$z$ catalogue. |  |  | $n_{\rm gal}\,(\times 10^{6})$ |  |  |  | $\overline{f}_{\rm LDP}$ |  |
---|---|---|---|---|---|---|---|---|---
$z_{m}\setminus M_{c}$ | -20.5 | -21 | -21.5 | -22 |  | -20.5 | -21 | -21.5 | -22
0.1 | 1.73 | 0.81 | 0.28 | - |  | 0.55 | 0.75 | 0.9 | -
0.3 | - | 4.53 | 1.31 | 0.26 |  | - | 0.22 | 0.62 | 0.9
0.5 | - | 6.52 | 1.66 | 0.24 |  | - | 0.16 | 0.55 | 0.91
0.7 | - | 6.98 | 1.66 | 0.21 |  | - | 0.14 | 0.56 | 0.91
0.9 | - | 5.81 | 1.11 | 0.14 |  | - | 0.16 | 0.64 | 0.94
1.1 | - | 3.16 | 0.54 | - |  | - | 0.31 | 0.8 | -
$z_{m}$: the redshift slice centered at $z_{m}$ with bin-width of 0.1; |  |
$n_{\rm gal}$: the number of galaxies in each galaxy sample; |  |
$\overline{f}_{\rm LDP}$: the average value of $f_{\rm LDP}$, identified with $R_{s}=3^{\prime}$. |  |

The LDPs are certainly dependent on galaxy samples. Using the procedure
developed in Dong et al. (2021), we first use the $r$-band absolute
magnitudes333The K-correction is not applied to the absolute magnitudes since
the absence of K-correction is not an issue for the LDP generation. This is
because galaxies within the same photo-$z$ bin have similar K-corrections and
the LDPs are only sensitive to the relative brightness of these galaxies. to
select galaxies that are brighter than a certain absolute magnitude, $M_{c}$,
and within a redshift slice of $z_{m}\pm\Delta z$ with $\Delta z=0.1$ which is
chosen due to the photo-$z$ error dispersion $\lesssim 0.1$ on average.

The r-band absolute magnitude can be expressed in terms of

$M_{r}-5\lg h=m_{r}-5\lg\left(\frac{d_{L}}{h^{-1}\rm Mpc}\right)-25\,,$ (1)

where $d_{L}$ is the luminosity distance determined by photometric redshift,
$h$ is the dimensionless Hubble parameter. In the following, we use a
parameter $M_{c}$ for magnitude cut, which represents the absolute magnitude
limit on the faint end. In this galaxy sample, we surround each galaxy with an
angular radius of $R_{s}$ and remove all pixels lying within this radius from
the sky. The remaining regions are defined as “LDP” candidates.

We sample LDPs on equal-area high-resolution HEALPix grids at $N_{\rm
side}=4096$, corresponding to an angular resolution of 0.859’. According to
Dong et al. (2021), LDPs physically correspond to underdense regions, and the
generation of LDPs in principle depends on the choice of the cut radius
$R_{s}$ to each galaxy. Statistically, a larger $R_{s}$ would lead to a more
negative density contrast $\delta_{m}$. After generating LDPs, we apply the
same procedure to produce the pixelized LDP over-density field as in Dong et
al. (2021), to which we refer readers for more details, so that,

$\delta_{l}=\frac{f_{\rm LDP}-\overline{f}_{\rm LDP}}{\overline{f}_{\rm
LDP}},\,$ (2)

where $f_{\rm LDP}$ is the proportion of area occupied by LDPs in a coarse
cell. For a given LDP sample, the larger $f_{ldp}$ is, the more negative
$\delta_{l}$ becomes in each cell.

We make a choice on the coarse cells being used in order to reduce the impact
of mask and edge effect on our calculations on $\delta_{l}$. We require that
the random points satisfy the selection criteria in §2.3 should fill more than
$70\%$ of the area of each selected cell.

Moreover, based on the fact that, fewer LDPs will be identified by selecting
more galaxies, and the number of galaxy samples varies with redshift for a
given projected number density, we have to control the number of galaxy
samples such that the LDP distribution become neither too populated nor too
sparse, making the statistics of $\delta_{\rm LDP}$ accurate. In practice, for
photo-$z$ bin of $0.01\leq z\leq 0.2$, we choose $M_{c}$ to be -20.5, -21, and
-21.5, respectively. For $0.3\leq z\leq 0.9$, $M_{c}$ is set to be -21, -21.5,
and -22, and we then set $M_{c}$ to be -21.5 and -22 for $1.0\leq z\leq 1.2$

In Tab. 2, we summarize the galaxy samples and the corresponding average
density of LDP, by varying galaxy selection criterion. In Fig. 1, we show one
example of the $f_{\rm LDP}$ distribution, generated with the following
parameters, $z_{m}=0.3,M_{c}=-21$, and $R_{s}=3^{\prime}$. Both the galaxy and
CMB masks have been applied to the full-sky LDP map, corresponding to a sky
coverage of $f_{sky}\simeq 0.44$.

##  3 Theoretical prediction from Numerical Simulation

From a theoretical perspective, due to the complicated observational effects,
and the non-trival relationship between LDP and galaxy samples, one requires
simulations to validate and interpret the final cross-correlation signal,
although implementing LDP analysis directly to observed data is
straightforward. We will use an N-body simulation to generate mock data
including galaxy catalogs and LDPs as well as lensing convergence maps, under
a given observational condition, which are then used to predict the CMB
lensing-LDP cross-correlation signal.

###  3.1 Simulation Data

Figure 2: CMB lensing convergence map (upper panel) and corresponding power
spectrum (lower panel), generated from our N-body simulation by stacking
density field along LOS in mock light-cone for the redshift bins $z<0.972$.
The theoretical prediction (blue) computed in Limber approximation within the
same redshift range and using the Halofit nonlinear power spectrum is shown as
blue line, showing a good consistency between the simulated and the
theoretical results.

To accurately model the realistic LSS matter distribution and the CMB lensing
field, we use a high-resolution N-body simulation from the Cosmic-Growth
simulation series (Jing, 2019), which uses a flat $\Lambda$CDM cosmology with
parameters:
$\Omega_{m}=0.268,~{}\Omega_{\Lambda}=0.732,~{}\sigma_{8}=0.85,~{}h=0.71,~{}n_{s}=0.968$,
with $3072^{3}$ dark matter particles in a box size of 1200 ${\rm Mpc}/h$.
Halos in simulation are identified with FoF group finder (Turner & Gott, 1976;
Huchra & Geller, 1982; Davis et al., 1985), and subhalos with Hierachical
Bound-Tracing (HBT) method (Han et al., 2012).

###  3.2 Generation of LDPs with mock galaxies

Similar to the procedure in (Dong et al., 2021), the mock photo-$z$ catalog is
generated based on the HBT subhalo catalog. First, we draw correspondence
between galaxies and halos/subhalos through matching the galaxy-subhalo
abundance (SHAM). And then, subhalos from different snapshots are used to fill
the corresponding radial distance slices. Finally, the mask of DR8 is applied
in the simulation to ensure the same angular selection of galaxies as in the
observation. In addition, we also generate scatters in the redshifts and
luminosities of mock galaxies so as to better mimic the real observation.

In observation, galaxies are provided with photo-z errors, according to which
we disturb the redshift of mock galaxies. By assuming a Gaussian distributed
photo-$z$ error with zero mean and a scatter of $\delta_{z}$, we randomly
shift the galaxy redshifts and adjust the absolute magnitudes of mock galaxies
accordingly based on $z$-$z_{\rm photo}$. Moreover, to mimic the galaxy-
halo/subhalo relation, a constant scatter $\sigma_{\rm Mag}=0.375$ dex to each
galaxy is taken into account (Yang et al., 2008). After taking these
uncertainties into account, we re-performe the SHAM to guarantee that the
luminous function of the simulated galaxies is the same as the observed one.
By doing so, we generate a simulated photo-$z$ catalog in the simulation with
$z<0.8$.

Note that in the above procedures, subhaloes in the redshift bin of $0.8\leq
z<1$ are also used to generate realistic volume-limited mock galaxies in lower
redshift bins, since there is a photo-z induced leakage mixing galaxies
between higher and lower redshift bins.

###  3.3 Construction of full-sky lensing convergence map

Under the Born approximation (Cooray & Hu, 2002), the convergence $\kappa$ for
CMB photons at the last-scattering surface with $z_{s}\simeq 1100$) and
direction $\widehat{n}$ is the projected density fluctuation
$\delta_{m}(\hat{n},z)$ weighted by the lensing kernel along LOS, which reads

$\kappa(\hat{n})=\int_{0}^{\chi_{s}}\delta_{m}(\hat{n},z)W^{\kappa}(z,z_{s})d\chi\,,$
(3)

where $\chi$ is the comoving distance to redshift $z$, and $W^{\kappa}$ is the
lensing kernel expressed by

$W^{\kappa}(z,z_{s})=\frac{3}{2}\left(\frac{H_{0}}{c}\right)^{2}\Omega_{m}(1+z)\chi(z)\left[1-\frac{\chi(z)}{\chi(z_{s})}\right]\,.$
(4)

We could construct the map of lensing convergence by stacking projected
surface mass densities along the LOS.

Wei et al. (2018) has confirmed a good agreement among the predicted angular
power spectrum of convergence from the Born approximation by stacking density
field along LOS in mock light-cone and that from the full-sky ray-tracing
simulation at $\ell\lesssim 4000$, with relative deviation $<10\%$. In this
study, we focus on the multipoles $\ell<1024$ so that the Born-approximation
is accurate enough for this purpose.

To a reasonable approximation, Eq. 3 can be written in terms of:

$\kappa(\hat{n})\simeq\sum_{i}\delta^{\Sigma}_{i}(\hat{n})W_{i}\Delta\chi_{i}\,,$
(5)

where $\delta^{\Sigma}_{i}$ is the surface overdensity of the $i$-th redshift
bin ($z_{i}-\Delta z_{i}<z<z_{i}+\Delta z_{i}$), and $W_{i}\simeq
W(\chi_{i},\chi_{s})\Delta\chi_{i}$ and $\Delta\chi_{i}=\chi(z_{i}+\Delta
z_{i}/2)-\chi(z_{i}-\Delta z_{i}/2)$. Such approximation is reasonably good as
long as $\Delta z_{i}<0.2$ (Yu et al., 2016).

We construct the full-sky past light-cone of the observer in the redshift
range of $0\leq z\leq 0.97$. The light-cone is built based on the nine output
snapshots at redshifts of 0, 0.058, 0.151, 0.253, 0.364, 0.485, 0.616, 0.76
and 0.916, respectively, corresponding to spherical shells of comoving
thickness $\Delta\chi\simeq 300{\rm Mpc}/h$. This essentially guarantees that
the lensing convergence maps generated by mock samples within the redshift
range of $z<0.8$. The particles located inside each of these snapshot are
projected onto spherical shells of surface over-density $\delta^{\Sigma}_{i}$
of the $i$-th redshift shell in $[(z_{i-1}+z_{i})/2,(z_{i}+z_{i+1})/2]$,
sampled on an HEALPix grid at $N_{\rm SIDE}=1024$ (angular size 3.44’). During
this process, we assume periodic boundary conditions in order to avoid
discontinuities of matter distribution.

In Fig. 2, the upper panel shows the simulated CMB convergence map based on
our mock samples and, to verify the simulation results in terms of the
theoretical predictions, the lower one shows the measured angular power
spectrum of such convergence map is in good agreement with the theoretical
prediction. Theoretically, the CMB lensing auto-correlation in the Limber
approximation(Limber, 1953) is

$C_{l}^{\kappa\kappa}=\int_{0}^{\chi_{s}}d\chi\frac{W^{\kappa}(\chi)^{2}}{\chi^{2}}P_{mm}\left(k=\frac{l+1/2}{\chi},z\right)\,,$
(6)

where $W^{\kappa}(\chi)$ is the CMB lensing kernal, $P_{mm}(z)$ is the matter
power spectrum evaluated at wavenumber $k$ at redshift $z$, and one can
compute it using the Halofit dark matter nonlinear power spectrum (Smith et
al., 2003). For the theoretical prediction curve shown in Fig. 2, we adopt the
same redshift range as the simulation, $z<0.972$.

The angular cross-spectrum between LDP overdensity field and CMB lensing
convergence can be similarly written as

$C_{\ell}^{l\kappa}=\int_{0}^{\chi_{s}}d\chi\frac{W^{l}(\chi)W^{\kappa}(\chi)}{\chi^{2}}P_{lm}\left(k=\frac{l+1/2}{\chi},z\right)\,,$
(7)

where $W^{l}(\chi)$ is the projection kernel for LDPs, and $P_{lm}$ represents
the cross-power spectrum of matter and LDPs. However, instead of modelling
$W^{l}(\chi)$ and $P_{lm}$ analytically, the theoretical prediction for
$C_{\ell}^{l\kappa}$ is estimated by using our simulations.

##  4 Results

Using the data analysis pipeline and the simulated data described above, we
now begin to perform tomographic cross-correlation measurements between the
Planck CMB lensing maps and the LDPs, constructed from the DESI legacy imaging
surveys DR8, and present their theoretical interpretations.

###  4.1 Lensing of the CMB detected by LDPs in Observation

Figure 3:  The cross-correlation between CMB lensing and LDPs measured for six
redshift bins: $z_{m}=$0.1, 0.3, 0.5, 0.7, 0.9 and 1.1 with
$R_{s}=3^{\prime}$. All the signals are multiplied by minus one. The results
for LDPs generated with different parameters are distinguished in different
colored lines. There are two different lines shown for each color. The solid
line shows signal measured with tSZ-deproj map in observation, the dashed line
shows signal measured with MV map in observation, and the dotted line shows
signal measured from the simulation.

Using the photo-$z$ samples of the DESI DR8 galaxy catalog, in this study we
split the galaxy sample into 6 redshift bins: [0.01, 0.2], [0.2, 0.4], [0.4,
0.6], [0.6, 0.8], [0.8,1] and [1,1.2] , as shown in Tab. 2. In the following
analysis, we choose a binning scheme with 13 logarithmically-spaced bins from
$\ell_{\rm min}=12$ up to $\ell_{\rm max}=1024$, and the averaged amplitude of
power spectrum in each $\ell$-bin with $N_{\ell}$ being the number of
multipoles is estimated in terms of

$\overline{C}_{\ell}=\frac{1}{N_{\ell}}\sum_{\ell^{\prime}=l-N_{\ell}/2}^{\ell+N_{\ell}/2}C_{\ell^{\prime}}\,,$
(8)

where $\ell$ is the mean multipole value for each $\ell$-bin, and
$C_{\ell^{\prime}}$ measured with the healpy software package (Zonca et al.,
2019; Górski et al., 2005). Also note that, the band powers are essentially
uncorrelated if we set the bin widths to be sufficiently large. In our case
for $f_{sky}\sim 0.44$, we have the widths $\Delta\ell\gg f^{-1/2}_{sky}$, so
that the band powers and associated statistical errors between adjacent bins
will be essentially uncorrelated since the mode coupling matrix will be fairly
diagonal.

A tomographic cross-correlation measurement between LDPs at the 6 redshift
bins and CMB lensing convergence maps from Planck Collaboration et al. (2020a)
is shown in Fig. 3 (note the minus sign added in $C_{\ell}^{\kappa l}$), where
the LDPs are produced with galaxies with $R_{s}=3^{\prime}$. As shown, (i) the
negative correlations indicate that the underlying matter densities
$\delta_{m}$ at LDPs are statistically negative and hence the LDP field indeed
probes underdense regions as expected; (ii) the cross-correlation signals by
using the “MV” CMB convergence map and the “tSZ-deproj” map are well
consisitent with each other, clearly lying within 1-$\sigma$ of one another
with a mean difference of about $0.99\pm 0.03$ in the amplitude averaged over
$\ell\geq 17$ and $1\pm 0.07$ over $\ell\geq 91$; (iii) the error bars on the
cross-spectra are measured with the Planck noise simulations, and the errors
from “MV” map are slightly smaller than those from “tSZ-deproj” as the “MV”
map constructed by all temperature and polarization modes rather than the
temperature-only “tSZ-deproj” map.

###  4.2 Covariance Matrix and Detection Significance in Observation

Figure 4: Example of the normalized covariance matrix for the LDP-CMB lensing
cross-correlation with parameters of $M_{c}=-21.5$ and $z_{m}=0.3$, estimated
from 240 simulated realizations of Planck convergence maps. Table 2: S/N of
$C_{\ell}^{\kappa l}$ derived from the observational measurements over the
multipole range of $12\leq\ell\leq 1024$, with $R_{s}=3^{\prime}$.

$\kappa$ map |  $z_{m}$ $M_{c}$ ${\rm{S/N}_{null(model)}}$a |  | -20.5 | -21 | -21.5 | -22 |  Max(${\rm{S/N}^{tot}_{null}}$)b
---|---|---|---|---|---|---|---
”tSZ-deproj” | 0.1 |  | 10.7 (10.1) | 10.6 (10.) | 9.2 (8.8) | - | 53.3
0.3 |  | - | 16.6 (16.) | 16.8 (16.3) | 15.7 (15.1)
0.5 |  | - | 21.2 (20.7) | 21.4 (21.1) | 16.5 (16.1)
0.7 |  | - | 21.4 (21.) | 22.3 (22.) | 13.5 (13.1)
0.9 |  | - | 25.5 ( - ) | 23.8 ( - ) | 10.1 ( - )
1.1 |  | - | 28.9 ( - ) | 17.7 ( - ) | -
”MV” | 0.1 |  | 12.2 | 12.3 | 11.3 | - | 60.6
0.3 |  | - | 19.4 | 19.7 | 17.3
0.5 |  | - | 23.8 | 23.3 | 18.4
0.7 |  | - | 24.5 | 24.7 | 15.5
0.9 |  | - | 31. | 27.1 | 12.
1.1 |  | - | 31.6 | 21.2 | -

  * a

${\rm{S/N}_{null}}=(\chi^{2}_{\rm null})^{1/2}$ and
${\rm(S/N)_{model}}=(\chi^{2}_{\rm null}-\chi^{2}_{\rm min})^{1/2}$, defined
in Eqs. 9, 11.

  * b

The maximum ${\rm(S/N)^{tot}_{null}}$ achieved by combining the respective
maximum ${\rm(S/N)_{null}}$ of each bin (i.e., choosing appropriate $M_{c}$
listed here).

We cross-correlate the LDP maps with the publicly available Planck FFP10
simulations to empirically estimate the covariance matrix for the cross-power
spectrum through, $C_{ij}=\langle C_{\ell_{i}}C_{\ell_{j}}\rangle-\langle
C_{\ell_{i}}\rangle\langle C_{\ell_{j}}\rangle$ with $i$($j$)=$1,\dots,n$,
where $\ell_{i}$ denotes the $i$-th bandpower bin, $n$ the number of total
bandpower bin, and the ensemble average is taken over 240 realizations of CMB
lensing and LDP maps.

One example of the normalized covariance matrix
($C_{ij}/\sqrt{C_{ii}C_{jj}}$), with parameters of $z_{m}=0.3$ and
$M_{c}=-21.5$, is shown in Fig. 4, clearly indicating that the off-diagonal
components of the covariance matrix are almost negligible, $<20\%$ of the
diagonal ones. The 1-$\sigma$ error for each $\ell$-bin shown in Fig. 3 can
thus be well approximated by $C_{ii}^{1/2}$. For the purpose of accuracy, we,
however, still use the full covariance matrices for all calculations in our
results

To evalute the significance of the cross-correlation detection, we use the
$\chi^{2}$ statistic. For LDPs at a given photo-z bin $z_{i}$, the cumulative
$\chi^{2}$ for the null hypothesis, $\chi^{2}_{\rm null}$, is calculated by
running over all $\ell$-bins (denoted by the indices $i,j$) by following

$\chi^{2}_{\rm null}(z_{i})=\boldsymbol{d^{T}C^{-1}d}=\sum_{i,j}C^{\kappa
l}_{\ell_{i},\rm data}C_{ij}^{-1}C^{\kappa l}_{\ell_{j},\rm data}\,,$ (9)

where ${\boldsymbol{d}}$ denotes the data vector and $C^{-1}$ is the inverse
of the covariance matrix described above. Due to the fact that LDP sampes of
different photo-z bins are almost uncorrelated, the total $\chi^{2}_{\rm
null}$ is calculated by summing over the entire 6 photo-$z$ bins
($z_{i}=1,2,\dots,6$). The resulting $\rm(S/N)_{null}$ then read,

${\rm(S/N)^{tot}_{null}}=(\sum_{z_{i}}{\rm(S/N)}^{2}_{\rm
null}(z_{i}))^{1/2}=\left(\sum_{z_{i}}\chi^{2}_{\rm
null}(z_{i})\right)^{1/2}\,.$ (10)

Furthermore, the significance of the cross-correlation detection for the
preference of the best-fit model over the null hypothesis can be simply
defined as,

$\rm({S/N})_{model}=\sqrt{\chi^{2}_{\rm null}-\chi^{2}_{\rm min}}\,.$ (11)

Here $\chi^{2}_{\rm min}$ corresponds to the minimum $\chi^{2}$ for the best-
fit theoretical model for the cross signal, which is calculated by

$\displaystyle\chi^{2}$ $\displaystyle=$ $\displaystyle\sum_{i,j}\delta
C_{\ell_{i}}^{\kappa l}C_{ij}^{-1}\delta C_{\ell_{j}}^{\kappa l}\,,$ (12)
$\displaystyle{\rm with\quad}\delta C_{\ell_{i}}^{\kappa l}$
$\displaystyle\equiv$ $\displaystyle C_{\ell_{i},\rm data}^{\kappa l}-A_{\rm
lens}C_{\ell_{i,\rm sim}}^{\kappa l}\,\,,$

where the theoretical model has been chosen as $A_{\rm lens}C_{\ell_{i},\rm
sim}^{\kappa l}$, a template $C_{\ell_{i},\rm sim}^{\kappa l}$ measured from
our simulations multiplied by a free scaling factor $A_{\rm lens}$ for the
lensing amplitude. The best-fit $A_{\rm lens}$ can be determined by minimizing
the quantity $\chi^{2}$.

The S/N values for the various maps and LDP catalogs are summarized in Tab. 2.
We find that the $\rm(S/N)_{null}$ in each photo-$z$ bin becomes different
according to different choices of magnitude cut in sample selection and CMB
lensing maps (“MV” or “tSZ-deproj”). For example, when using the “tSZ-deproj”
map, the ${\rm(S/N)_{null}}$ for $z$-bin $0.4\leq z\leq 0.6$ will vary from
22.1 to 16.6 for $M_{c}=$ -21 and -22, respectively. Interestingly, a higher
redshift bin usually yields a higher detection significance, which is due to
the fact that CMB lensing efficiency will reach the maximum at $\sim 2$.
Finally, the total S/N from all the 6 photo-$z$ bins can be achieved,
${\rm(S/N)^{tot}_{null}}=53.3,60.6$, for the “tSZ-deproj” and “MV” cases,
respectively. These high signal-to-noise measurements strongly validate the
LDP as an effective tracer of LSS.

###  4.3 Comparison with the N-body Simulation

Figure 5: Left panel: dependence of the estimate of $A_{\rm lens}$ derived
from the observed cross-spectrum $C_{\ell}^{\kappa l}$ on the magnitude cut
$M_{c}$ and the photo-$z$ bin. Right panel: the corresponding reduced
$\chi^{2}$, with $dof=11$.

In the above we show tomography CMB lensing measurements spanning the redshift
range of $0.01<z<1.2$. But in view of the quality of redshift measurement of
galaxies and reliable construction of mock samples, we define sources with
$z\leq$ 0.6 as our baseline samples.

Using our simulated both the CMB convergence and LDP maps, described in §3, we
calculate cross-power spectra of the CMB-LDP cross-correlation as a
theoretical prediction. These results are shown as dotted lines in Fig. 3. For
a given photo-$z$ bin, the best-fit amplitude, $A_{\rm lens}$, can be
determined by minimizing the $\chi^{2}$, as defined in Eq. 12, where we
consider the multipole range of $22\leq\ell\leq 724$ for fitting.

In the “tSZ-deproj” case, the best-fit amplitudes and the corresponding
reduced chi-square $\chi^{2}_{\rm min}/dof$ are shown in Fig. 5, where the LDP
maps are generated from the DESI DR8 catalog. The best-fit values of $A_{\rm
lens}$ are mostly consistent with the theoretical prediction, within
2-$\sigma$, thus providing a good fit to the data. One may notice that the
amplitude from the high redshift bin ($z_{m}=0.7$) with magnitude cuts of
$M_{c}=22$ will be underestimated, significantly deviating from the simulation
result with about 4-$\sigma$, which might be due to potentially large
photo-$z$ errors in the photometric measurements. More detailed discussions
about photo-$z$ errors as well as various sysmetiactic effects are present in
Sect. 5.

Table 3: Best-fit $A_{\rm lens}$ and associated $1$-$\sigma$ uncertainty as
well as the reduced $\chi^{2}_{\rm min}$ with $dof=32$, estimated from
different combinations of LDP baseline samples from the three $z$-bins with
$z_{m}=0.1,0.3,0.5$ and using Planck ”tSZ-deproj” lensing map.

Sample |  Max($M_{c}$)a | -21.5 |  Min($M_{c}$)b
---|---|---|---
$A_{\rm lens}$ |  1.14$\pm$0.04 |  1.06$\pm$0.04 |  1.05$\pm$0.05
$\chi^{2}_{\rm min}/dof$ | 1.27 | 0.94 | 1.01

  * a

Max($M_{c}$) refers to selecting fainter but more galaxies, by setting
$M_{c}=-20.5,-21,-21$ for the three $z$-bins, respectively.

  * b

Min($M_{c}$) refers to selecting brighter but fewer galaxies, by setting
$M_{c}=-21.5,-22,-22$ for the $z$-bins, respectively.

To fully exploit the the measured the cross-power spectra at various
redshifts, we combine the spectra at each redshift bin to make a total
constraint on $A_{\rm lens}$. Since the estimate of $A_{\rm lens}$ from a
given redshift also depends on the choice of LDP samples as well, we summarize
the total constraints on $A_{\rm lens}$ based on different magnitude cuts,
presented in Tab. 3. As seen, if using LDPs generated from a relatively larger
number but fainter galaxies (as referred to ”Max($M_{c}$)”) by setting
$M_{c}=$ -20.5, -21 and -21 for $z_{m}=$ 0.1, 0.3, 0.5, respevtively, we
obtain the overall constraint of $A_{\rm lens}=1.14\pm 0.04$ by combining all
signals in the redshift range of $0\leq z\leq 0.6$. Alternatively, if
considering LDPs from relatively brighter but smaller number of galaxy samples
(referred to ”Min($M_{c}$)”) where $M_{c}=$ -21.5, -22, -22, respectively, for
these three bins, the derived $A_{\rm lens}$ is $1.05\pm 0.05$, about 8% lower
than the ”Max($M_{c}$)” case. Furthermore, if fixing the magnitude cut to
$M_{c}=-21.5$ for all redshift bins, the similar constraint to the case
”Min($M_{c}$)” is obtained, $A_{\rm lens}=1.06\pm 0.04$. In all these three
cases, the corresponding minimum reduced $\chi^{2}$, are $\chi^{2}_{\rm
min}/dof=$ 1.27, 0.94 and 1.01, with $dof=32$ for the cases of ”Max($M_{c}$)”,
”$M_{c}$ fixed” and ”Min($M_{c}$)”, respectively, so that the the best-fit
model is consistent with data. To make this statement more quantitatively, we
calculate the probability-to-exceed (PTE) of the best-fit model. Using
$\chi^{2}_{\rm min}=$ 40.6, 30.1 and 32.3 for $dof=32$, we find PTE $=0.1415$
for ”Max($M_{c}$)”, 0.56 for ”$M_{c}$ fixed”, and 0.45 for ”Min($M_{c}$)”.
Therefore, we find no strong evidence of potential systematics and the model
thus provides a good fit to the data.

From Fig. 3, we see that, although the measured $C_{\ell}^{\kappa l}$ from
different LDP samples can vary significantly in amplitude, even by an order of
magnitude, the good agreement on $A_{\rm lens}$ derived from the observation
and from the simulation clearly indicates a robust detection on the lensing-
LDP signal. Thus, the LDP method provides a robust test on the $\Lambda$CDM
model.

In addition, Hang et al. (2021b) has reported the overall scaling in
amplitude, $A_{\rm lens}$, to be $0.901\pm 0.026$, by using a cross-
correlation between Planck lensing convergence maps and DESI LRG samples,
which appears to be lower than our results. This is probably because they
utilized a different sample of galaxies to us. To cross-check, we also conduct
a cross-correlation measurement by using galaxies instead of LDPs, and find
the derived $A_{\rm lens}=1\pm 0.04$ for our baseline samples and
$M_{c}=-21.5$, as discussed in Appendix B.

##  5 Consistency Checks

In this section, we will present several consistency checks carried out to
ensure that our measurements are accurate and robust to the data selection and
observational effects.

###  5.1 Hemispherical Asymmetry

Table 4: Best-fit $A_{\rm lens}$ and associated $1$-$\sigma$ uncertainties measured in each individual $z$-bin with $M_{c}=-21.5$, for hemispherical asymmetry test. sky $z_{m}$ $A_{\rm lens}$ | 0.1 | 0.3 | 0.5 | 0.7
---|---|---|---|---
North+South |  1$\pm$ 0.15 |  1.07$\pm$ 0.07 |  1.07$\pm$ 0.05 |  0.92$\pm$0.04
North |  0.93$\pm$ 0.19 |  1.06$\pm$ 0.12 |  1.06$\pm$ 0.09 |  0.93$\pm$0.07
South |  1.15$\pm$ 0.3 |  1.1$\pm$ 0.19 |  1.1$\pm$ 0.14 |  0.91$\pm$0.11

Eriksen et al. (2004) first report the anomalous CMB North-South asymmetry
that leads to statistically significant anomalies of the CMB temperature power
spectrum in the low-$\ell$ regime ($\ell\lesssim 50$). Hansen et al. (2009)
find this asymmetry extends to much smaller scales at $\ell\lesssim 600$.
Recently, such hemispherical asymmetry has been confirmed by the Planck data
(Planck Collaboration et al., 2014). In this context, the Planck lensing
convergence constructed from the temperature maps may also potentially have a
North-South asymmetry. We now explore the hemispherical asymmetry in the CMB
lensing-LDP cross-correlation signals. By selecting LDPs from the northern and
the southern ecliptic hemispheres, with the same magnitude cut of
$M_{c}=-21.5$ as an example (without loss of generality), we find that there
are essentially no systematic deviations in the scaling factor $A_{\rm lens}$
in the 4 redshift bins, $z_{m}=0.1,0.3,0.5,0.7$, as summarized in Tab. 4.

###  5.2 High-redshift $A_{\rm lens}$

Table 5: Similar to Tab. 3, but adding the data of the redshift bin of $[0.6,0.8]$ to constrain $A_{\rm lens}$, where $dof=43$, increased by 10 (as 10 $\ell$-bins). Sample: |  Max($M_{c}$) | -21.5 |  Min($M_{c}$)
---|---|---|---
$A_{\rm lens}$ |  1.1$\pm$0.03 |  1$\pm$0.03 |  0.94$\pm$0.04
$\chi^{2}_{\rm min}/dof$ | 1.17 | 0.9 | 0.94

As seen in Fig. 5, the measured $A_{\rm lens}$ from the individual high-
redshift bin of $0.6\leq z\leq 0.8$ is lower than those from other low-
redshift $z$-bins, regardless of the value of $M_{c}$. Thus we now compare the
scaling factor $A_{\rm lens}$ derived with and without the high redshift bin
as a check. As shown in Tab. 5, adding such high-redshift data will give a
slightly lower estimate on $A_{\rm lens}$ than those from the low-redshift
bins (see Tab. 3), differing by only a few percent to 10% level. In general,
there is no obvious discrepancy between the highest $z$-bin and the lower
ones.

In fact, due to relatively lower quality of optical data and fewer
spectroscospic training sets when $z\gtrsim 0.6$, the photo-$z$s of galaxies
may be somewhat biasedly estimated. In the case of $M_{c}=-22$ which leads to
more brighter galaxies are taken, the best-fit value of $A_{\rm lens}$ from
the single bin of $z_{m}=0.7$ is $0.75\pm 0.06$ (see Fig. 5), and such
underestimate is probably due to the following reasons: 1) the $r$-band
magnitude may no longer be a good indicator in mass and the luminosity-halo
mass relation may have a large dispersion that we have not considered; 2) the
photo-$z$s and their errors of the galaxies at such high redshift might be
mis-estimated and the probability distribution $p(z_{phot}|z_{obs})$ in the
real situation is likely to be non-Gaussian. As illustrated in (Sun et al. in
preparation), a more reliable cross-correlation signal between the CMB lensing
and DESI clusters are measured if one select the clusters with more subhalos,
as the redshifts of clusters could be determined more accurately.

We also try to get support for this conjecture by matching these bright
sources to a different photo-z catalog (Zhou et al., 2021) for comparison, but
found no systematical difference. But considering that these two catalogs have
used almost the same spectroscospic data sets (e.g. BOSS catalog) for
reference during the redshift measurements, similar systematic bias could be
introduced.

In short, we need to improve our understanding of photo-$z$ measurement
uncertainties and correct for photo-$z$ systematic errors in order to
significantly reduce the bias. Thanks to the future
release444https://desi.lbl.gov/trac/wiki/ClusteringWG/LSScat/SV3 of DESI
spectroscopic data, we will be able to make more accurate cross-correlation
analyses in the future.

###  5.3 Varying $\ell_{\rm min}$ and $\ell_{\rm max}$

Figure 6: Consistency check for $A_{\rm lens}$ by varying the multipole range.
All the results are based on a fixed magnitude cut of $M_{c}=-21.5$. The solid
lines in the upper left panel are the results when discarding the data points
with multipoles $\ell_{min}<\ell\leq 1024$. The dashed lines in the lower left
panel are the results when discarding the data points with multipoles larger
than $\ell_{max}$ while fixing the first data point to $\ell=16$. The dotted
line has similar meaning to the dashed line, but with the first data point
started from $\ell=65$. In the right two panels, we show the associated
minimum reduced chi-square. Figure 7: Consistency check for $A_{\rm lens}$
measured within three non-overlapping multipole bins: $\ell_{group1}=$[12,
46], $\ell_{group2}=[46,256]$ and $\ell_{group3}=[256,1024]$. These two panels
have similar meaning to Fig.6. All the results are based on a fixed magnitude
cut of $M_{c}=-21.5$. The solid lines are the results by comparing
observational signals to simulation. The dashed line has similar meaning to
the solid line, but using signals detected with noisy $\kappa$ map in
simulation instead of observational signals.

By varying the multipole range ($\ell_{\rm min}\leq\ell\leq\ell_{\rm max}$) in
the analysis of the cross-power spectrum, we can check consistency of the
scaling factor $A_{\rm lens}$ as more or fewer angular modes are included. We
run two cases: i) gradually increasing $\ell_{\rm min}$ from 8 to 90, but with
$\ell_{\rm max}=1024$ fixed; ii) gradually decreasing $\ell_{\rm max}$ from
1024 to 362, but fixing $\ell_{\rm min}=$ 16 or 64. As observed in Fig. 6, the
changes in $A_{\rm lens}$ are relatively small with respect to the associated
statistical errors when varying the multipole range. We also note that, for
the case of $z_{m}=0.1$, the values of $A_{\rm lens}$ when varying $l_{\rm
min}$ and $\ell_{max}$ are changed across 1, from 0.9 – 1.05, and the
corresponding statistical errors are somewhat larger than other $z$-bins. This
may be because of a large Poisson noise from the limited number of the
galaxies in such bin ($\sim 0.28\times 10^{6}$ galaxies for $z_{m}=0.1$, see
Tab. 1). In addition, the reduced $\chi^{2}$ fluctuates over all the scales
when varying the multipole range, which may indicate a different scale-
dependence of $C_{\ell}^{l\kappa}$ between the observed one and the template
model.

Furthermore, we divide data into three non-overlapping multipole bins to check
$A_{lens}$: [12, 46], [46, 256] and [256, 1024]. The results are shown as the
solid lines in Fig.7, from which we find $A_{lens}$ is consistent. Meanwhile,
we see fluctuations in both $A_{lens}$ and reduced chi-square, which might be
caused by the noise in the observational lensing map. With this concern, we
perform a further test with N-body simulation by adding one of the Planck
noise simulation map to the ideal $\kappa$ map to mimic this effect. And
similar results have been got, which are shown as the dashed lines in Fig.7,
indicating that the noise in the lensing map is possible to disturb the cross
powers at all scales. Based on these tests, we conclude that no obvious
systematic bias is found for $A_{lens}$ when varying the multipole range of
the observational signals.

###  5.4 Adding different Galactic Masks

Based on the two facts that, the reconstructed Planck CMB lensing maps may
suffer from foreground contamination especially in regions close to the
Galactic plane, and the galaxy data may also be contaminated by Galactic
foreground stars near the Galactic plane, we therefore add an additional mask
to the maps, while combing it with the fiducial one. We find that, when adding
the mask of $|{\rm Dec}|>30^{\circ}$ that vanishes the Galactic plane regions,
the change in the factor $A_{\rm lens}$ is relatively small ($A_{\rm
lens}=1.1\pm 0.08$), compared to the case of the fiducial mask together with
the parameters of $M_{c}=21$ and $z_{m}=0.3$. Also, when adding a relatively
larger mask of $|{\rm Dec}|>35^{\circ}$, the factor $A_{\rm lens}$ slightly
increases to $1.15\pm 0.1$. In the both cases, the derived $A_{\rm lens}$ are
consistent with the result from the fiducial mask (${A_{\rm lens}=1.12\pm
0.08}$, shown in Fig. 5), within 1-$\sigma$ level. We also find, using the
samples selected by $M_{c}=-21.5$ with the mask of $|{\rm Dec}|>30^{\circ}$,
$\rm{A_{lens}}$ are found to be $1.13\pm 0.14,1.08\pm 0.09,1.11\pm 0.06$ and
$0.96\pm 0.05$ for $z_{m}=0.1,0.3,0.5$ and 0.7, respectively, consistent with
the results shown in Fig. 5. We can thus conclude that, altering the region
near the Galactic plane used for calculating the cross-power spectrum does not
significantly change the estimate of $A_{\rm lens}$.

###  5.5 Introducing photo-$z$ bias

Figure 8: Impacts of Photo-$z$ bias ($dz$) on cross-power spectrum
$C_{\ell}^{\kappa l}$ and $A_{\rm lens}$ using LDPs selected in redshift bin
of [0.2,0.4] by $M_{c}=-21$. Left panel: the color gradient lines show the
resulting spectras by varying $dz$ from -0.09 to 0.05 by 0.01 ( i.e.,
$[-0.09,-0.08,\cdots,0.04,0.05]$ in simulation, and the red solid line with
error bars shows observational signal. Right panel: the reduced $\chi^{2}$ in
the two-parameter space of $dz$ and $A_{\rm lens}$, which reaches the minimum
by $dz_{\rm fit}=-0.06$ and $A_{\rm lens}=1.03$ (denoted by the blue point).

All the above analysis is based on the assumption of a Gaussian distribution
for $z_{\rm p}$ centered at the true redshift $z_{\rm true}$, without any
photo-$z$ bias. However, systematic photo-$z$ bias would propagate into errors
in cross-correlation measurements and hence lead to a misestimate on $A_{\rm
lens}$.

For checking this possibility, we focus on using LDP samples selected with
$M_{c}=-21$ in the $z$-bin of [0.2,0.4] as an illustration. Based on the
simulation data, by shifting each mock galaxy by a redshift of $dz$ that leads
to a systematic redshift bias generated at $z_{\rm true}$. We then add
photo-$z$ uncertainties to those shifted galaxies and recalculate those
absolute magnitudes according to $dz$, together with re-running the abundance
matching process. To clearly illustrate such effects, $dz$ is varied from
-0.09 to 0.05 by 0.01 in the analysis. With such new sets of mock galaxies,
the measured cross-correlations are shown in the left panel of Fig. 8. One can
find that, the above photo-$z$ bias does introduce visible errors, (leading to
changes of 10% – 20% level relative to the case without photo-$z$ errors) in
the cross-power spectrum $C_{\ell}^{l\kappa}$ and, consequently, the best-fit
parameters are obtained, $dz_{\rm fit}=-0.06$ and $A_{\rm lens}=1.03$ for
$\ell\geq 33$, and $dz_{\rm fit}=-0.06$ and $A_{\rm lens}=1.06$ for $\ell\geq
12$, by minimizing $\chi^{2}$. In other words, by introducing a negative $dz$,
one may reduce some overestimate on $A_{\rm lens}$. An incorrect photo-$z$
model thus would cause a shift in measured value of $A_{\rm lens}$.

Similarly, we obtain ($dz_{\rm fit}$, $A_{\rm lens})=(-0.05,0.99)$,
$(-0.04,0.89)$ and $(0.04,0.78)$ for the LDP samples selected by
$M_{c}=-21,-21.5$ and -22 in the redshift bin of $[0.6,0.8]$, respectively.
However, it is almost impossible for us to obtain an effective constraint when
$M_{c}=-22$, since the number of such galaxy sample is too small ($n_{\rm
gal}\sim 2\times 10^{5}$). In addition, for $z\in[0.4,0.6]$, the best-fit
$dz$, $dz_{\rm fit}$, is found to be almost zero and thus no obvious redshift
bias is found. Furthermore, notice that, we can not predict the photo-$z$
biased power spectrum from mock data at the lowest $z$-bin [0.01, 0.2], since
the redshifts of a large number of galaxies will become negative when shifting
the galaxies by offsets $dz$.

Note that, the effects of photo-$z$ bias may also degenerate with changes in
other cosmological parameters 555The lensing signal detected with LSS tracers
can be regarded as the matter-lensing cross power $C_{l}^{mk}$ multiplied by
$A_{lens}b_{g}$, where $b_{g}$ is the tracer bias. Unless we have numerical
simulations for different cosmologies, we are unable to quantitatively tell
the effect of cosmology on $A_{lens}$ measurements as it is coupled with
$b_{g}$. However, we could make a qualitative analysis of $A_{lens}$ here by
simply using $c_{l}^{mk}$. We find that Planck cosmology leads to a slightly
higher $C_{l}^{mk}$ than WMAP for $\ell>100$, and for $\ell<40$ it is on the
opposite. And the best-fit value of $A_{lens}b_{g}$ is decreased by $4\%-9\%$
on the whole within our fitting range. If assuming the same $b_{g}$ in both
cosmologies, we can imagine that $A_{lens}$ detected under Planck cosmology
would become slightly lower. Meanwhile, we find that the chi-square between
observation and the theoretical curve (with the best-fit $A_{lens}b_{g}$)
become slightly larger in most cases under Planck cosmology. such as
$\Omega_{m},\,\sigma_{8}$, and the tracer bias $b_{g}$ (as $C_{\ell}^{\kappa
m}=A_{\rm lens}b_{g}C^{m}_{\ell}$), so that the ongoing DESI spectroscopic
surveys are obviously necessary for precise constraints on cosmology.

###  5.6 Absolute Magnitude within a certain range

In the fiducial case, the LDP samples are generated with galaxies that are
selected by their absolute magnitudes, satisfying $M_{r}-5\lg h\leq M_{c}$.
For a given redshift bin but with different $M_{c}$, obviously,
$M_{c}$-selected galaxy samples are not completely independent and some of
them will be overlapped. With this concern, we have performed a further test
by producing LDPs for intermediate surface brightness galaxies ($-21.5\leq
M_{r}\leq-21$) at $z_{m}=$ 0.1, 0.3, 0.5, and 0.7, respectively. The $A_{\rm
lens}$ for $\ell\in[22,724]$ are constrained to be $1.09\pm 0.13,1.15\pm
0.08,1.15\pm 0.06,1.04\pm 0.05$, in full agreement with the fiducial case.

###  5.7 Cross-Correlation between $C_{\ell}^{\kappa l}$ and $C_{\ell}^{\kappa
g}$

Figure 9: Normalized covariance matrices of the vector ($C_{\ell_{1}}^{\kappa
l},\dots,C_{\ell_{n}}^{\kappa l},C_{\ell_{1}}^{\kappa
g},\dots,C_{\ell_{n}}^{\kappa g}$), calculated with varying $R_{s}$ from
$1.5^{\prime}$ to $7^{\prime}$ (from the left panel to the right) in the range
of $11<\ell\leq 1024$.

The cross-correlation between the lensing-LDP spectrum $C_{\ell}^{\kappa l}$
and the lensing-galaxy one $C_{\ell}^{\kappa g}$ provides a consistency check
on the validity of the methodology for the CMB lensing-LDP measurements. As
expected, the signal $C_{\ell}^{\kappa l}$ would anti-correlate with
$C_{\ell}^{\kappa g}$ since the LDP field and galaxies as desired trace
underdense and overdense regions of $\delta_{m}$, respectively, implying
negative correlations between them.

To do so, we calculate the auto- and cross-correlations between
$C_{\ell}^{\kappa l}$ and $C_{\ell}^{\kappa g}$ for different $\ell$-bins and
the corresponding normalized covariance matrices, shown in Fig. 9. In this
analysis, we consider different cases by varying $R_{s}$ from 1.5’ to 7’ but
keeping the magnitude cut $M_{c}=-21.5$ and the redshift range [0.2.0.4] fixed
for the LDP sample selection.

One can find strong anti-correlations between $C_{\ell}^{\kappa l}$ and
$C_{\ell}^{\kappa g}$ whereas the absolute values of the diagonal components,
$\left<C_{\ell i}^{\kappa l}C_{\ell i}^{\kappa g}\right>-\left<C_{\ell
i}^{\kappa l}\right>\left<C_{\ell i}^{\kappa g}\right>$ with $i=1,\dots,n$,
decrease with increasing $R_{s}$, as well as with increasing $\ell$. Such
decreasing trend can be interpreted as the following: 1) the correlation
between the overdense and underdense regions would become weaker or even
disappear when significantly increasing $R_{s}$; 2) the stronger correlations
at large scales than at small scales is due to that they may statistically
have a similar spatial distribution at large scales but behave very
differently at small scales.

Also, there is a transition scale ($\ell_{*}$) at which the $C_{\ell}^{\kappa
l}$-$C_{\ell}^{\kappa g}$ correlation drops to below $\sim$ 0.9 and the scale
depends on the value of $R_{s}$. We find an non-trivial but monotonic
dependence of the transition scale on $R_{s}$, approximately $R_{s}$ =1.5’ for
the transition scale of $\theta_{*}\sim 15^{\prime}$ (by $\theta_{*}\approx
180^{\circ}/\ell_{*}$), $R_{s}=3^{\prime}$ for $\sim 42^{\prime}$,
$5^{\prime}$ and $7^{\prime}$ for $\sim 2^{\circ}$ and $\sim 16^{\circ}$,
respectively.

In addition, the S/N for the null hypothesis over the range of $22\leq\ell\leq
1024$, are $({\rm S/N})^{l\kappa}_{\rm null}=$17, 16.9, 15.7 and 14.5 for
$R_{s}=1.5^{\prime}$, 3’, 5’ and 7’, respectively. If calculating the S/N from
the $\kappa$-galaxy correlation alone, we find $({\rm S/N})^{g\kappa,{\rm
tot}}_{\rm null}=17.3$, slightly higher than the LDP case. These results thus
imply LDPs can be served as a promising tracer of LSS. Moreover, one may
concern whether the joint analysis of $\kappa$-galaxy and $\kappa$-LDP could
enhance the S/N. To do so, we then consider the the full covariance by
inclusion of auto-correlations,
$\boldsymbol{C}^{l\kappa}$-$\boldsymbol{C}^{l\kappa}$ and
$\boldsymbol{C}^{g\kappa}$-$\boldsymbol{C}^{g\kappa}$, a cross-correlation,
$\boldsymbol{C}^{l\kappa}$-$\boldsymbol{C}^{g\kappa}$, as shown in Fig. 9,
leading to a combined significance of $\rm(S/N)^{joint}_{null}=$ 18, 18, 18.1
and 18.2 for $R_{s}=1.5^{\prime}$, 3’, 5’ and 7’, respectively. If using the
samples at the higher $z$-bin of $z_{m}=0.5$ and with $M_{c}=-21.5$, we find
$({\rm S/N})^{l\kappa}_{\rm null}=$21.5, 21.4, 19.6 and 16.1 for
$R_{s}=1.5^{\prime}$, 3’, 5’ and 7’, and $({\rm S/N})^{g\kappa}_{\rm
null}=21.2$, resulting a combined significance of $\rm(S/N)^{joint}_{null}=$
22, 22.2, 22.5 and 22.1 for $R_{s}=1.5^{\prime}$, 3’, 5’ and 7’, respectively.
Thus, the use of LDP fields certainly provides an additional information on
lensing detection, despite the fact that this improvement is not significant
(enhanced by $\sim 4\%-6\%$) due to the galaxy-LDP strong correlation. As a
further check, the samples chosen in the redshift bin of $z_{m}=0.7$ will lead
to a similar result as above. For redshifts $z_{m}=0.1,0.9$ and 1.1, the
$(S/N)^{\rm joint}_{\rm null}$ are found to be improved by $\sim 20\%$,
4-8$\%$ and $10\%$, respectively, compared to the lensing-galaxy case.

##  6 Conclusion

In this study, we have investigated the potential of utilizing LDPs (low-
density points, initially proposed in Dong et al. (2019)) to detect the CMB
lensing signal, with the aim to further verify the LDPs as a competitive
tracer of LSS. By cross-correlating the Planck CMB lensing convergence map
with the LDPs of different photometric redshift bins produced with galaxies
selected from the DESI DR8 photo-z catalog, we have achieved an unprecedented
detection significance of CMB lensing-LDP cross-spectrum of about $53\sigma$,
by combining the measurements over all the multipoles $12\leq\ell\leq 1024$
and photo-$z$ bins ($0<z\leq 1.2$).

For comparison, we have tested our measurements using simulated data sets,
constructed specifically to precisely mimic various effects (such as sample
selection, observational effects, photo-$z$ errors) of our observables: LDP
catalog for DESI photometric survey and CMB convergence maps for the Planck
observation. When fitting a scaling factor in the amplitude of CMB lensing-LDP
cross-power spectrum, $A_{\rm lens}$, we find that, the cross-correlation
amplitudes for different redshift bins and magnitude cuts are well consistent
with the expected levels of cross-correlations predicted from our N-body
simulations with the $\Lambda$CDM model, i.e., $A_{\rm lens}=1\pm 0.12$
(1-$\sigma$ uncertainty) at $z<0.2$, $A_{\rm lens}=1.07\pm 0.07$ at
$0.2<z<0.4$ and $A_{\rm lens}=1.07\pm 0.05$ at $0.4<z<0.6$, with the magnitude
cut of -21.5 for the galaxy selection.

We also have carefully tested the various effects that could bias our result,
with consistency checks on i) the hemispherical asymmetry by selecting LDPs
from the northern and the southern ecliptic hemispheres, ii) selecting LDPs
from the highest redshift bin, iii) scale cut by varying the the multipole
range [$\ell_{min},\ell_{\rm max}$] used in the analysis, iv) masking the
regions near Galactic plane, (v) photo-$z$ bias in measurements, vi) absolute
magnitude within a certain range, vii) cross-correlation between lensing-LDP
and lensing-galaxy power spectra. Results are found to be consistent with the
measured result from the baseline samples. No changes in our data analysis
procedure have significant impacts on measured cross-correlation amplitudes,
within the statistical errors. According to our study, the CMB lensing-LDP
tomographic cross-correlation will be particularly important as a cosmological
tool and is expected to serve as an ingredient in the framework for the
analysis of cosmological parameters.

Still, many other possibilities need to be further explored. For example, the
CMB lensing kernel is broad, peaked at redshift around 2. Thanks to the Wide-
field Infrared Survey Explorer that allows to identify a large number of
galaxies up to $z\sim 1.5$ (see Tab. 1 in Schlafly et al. (2019)), therefore,
we will consider a combination of optical and (near) infrared measurements to
push high S/N CMB Lensing-LDP cross-spectrum measurements to a high-redshift
regime. Moreover, the high-redshift cross-correlations may help to effectively
break down the degeneracy among cosmological parameters and systematic
nuisance parameters such as galaxy bias in LSS measurements, thus providing an
accurate probe of the late-time cosmic acceleration.

## Acknowledgments

We thank Yipeng Jing for providing us the N-body simulation. We also thank
valuable suggestions from the referee. The author thanks the HEALPix/healpy
software package (Zonca et al., 2019; Górski et al., 2005). The author
acknowledges the Korea Institute for Advanced Study for providing computing
resources (KIAS Center for Advanced Computation) for this work. F.Y.D. is
supported by a KIAS Individual Grant PG079001 at Korea Institute for Advanced
Study. C.B.P. is supported by a KIAS Individual Grant PG016904 at Korea
Institute for Advanced Study. This work is supported by the National Key
R$\&$D Program of China (2018YFA0404504, 2018YFA0404601, 2020YFC2201600),
National Science Foundation of China (11621303, 11653003, 11773021, 11890691),
the 111 project, and the CAS Interdisciplinary Innovation Team (JCTD-2019-05).

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##  Appendix A Downgrading the Planck CMB lensing maps

Figure 10:  Original and downgraded convergence maps using our downgrading
procedure, and the corresponding power spectra. Top-left panel: the ”MV”
convergence map at a HEALPix resolution of $N_{\rm side}=2048$, generated from
original spherical harmonic coefficients $a_{\ell m}$ provided by the Planck.
Top-right panel: the masked ”MV” convergence map with $N_{\rm side}=512$,
produced from the low-pass filtered spherical harmonic coefficients with the
cutoff at $\ell=3\times 512$. Bottom panel: the angular power spectrum of the
downgraded map of $N_{\rm side}=512$ by using our downgrading procedure
(orange line), are well consistent with that of the original high-resolution
map of $N_{\rm side}=2048$ (blue), without any aliasing problem. If directly
downgrading the original map without using any low-pass filters, the resulting
power spectrum (green) is overestimated by several orders of magnitude, which
is due to the aliasing effect that leads to mapping of the noise-dominated
power (a nearly divergent spike at high $\ell$) to the signal.

In the analysis presented in this paper, we adopt the downgrading procedure as
follows, in order to prevent the so-called aliasing effect that causes a
mixture of signals at different scales. In principle, applying a low-pass
filter to the input signal before sampling can avoid aliasing.

In general, when downgrading a map from its original high resolution to a
lower one, 1) we first decompose the full-sky map into spherical harmonics
($a_{\ell m}$) of the input HEALPix resolution; 2) these coefficients are then
filtered to the low one by multiplying an appropriate low-pass filter
($w_{\ell m}$) that sets all high-$\ell$ coefficients to zero; 3) then the
modified filtered coefficients ($a^{\prime}_{\ell m}=a_{\ell m}w_{\ell m}$)
are used to directly synthesize a low-resolution map, without aliasing.

Specifically, in order to obtain a map at the resolution of $N_{\rm
nside}=512$, we can directly jump to step 2 and set $w_{\ell m}=0$ for
$\ell>3\times 512$ and $w_{\ell m}=1$ otherwise as the low-pass filter, since
the Planck only provides $a_{\ell m}$ up to $\ell_{\rm max}=4096$ rather than
maps. Additionally, when downgrading the binary mask ($N_{\rm side}$=2048)
into a lower resolution ($N_{\rm side}$=512) in this paper, the downgraded
mask is thresholded by setting pixels where the value between 0 and 1 are set
to zero and the rest are set to unity so as to again generate a binary mask.
From our tests, we find such procedure can prevent the aliasing of high-$\ell$
noise power spectrum (see the spike feature at right panel of Fig. 10).

##  Appendix B CMB LENSING SIGNALS MEASURED WITH GALAXIES

Figure 11: Similar to Fig.5, but the measured $A_{\rm lens}$ are derived from
galaxies rather than LDPs.

The overlap between DESI footprint and CMB surveys provides opportunities to
better understand the late-time universe by using cross-correlations between
galaxies and CMB lensing. Two detections have been reported so far: 1) a
cross-correlation over scales $30\leq\ell\leq 1000$ between Planck lensing
maps and DESI-like luminous red galaxies (LRGs) selected from the DECaLS
imaging is detected at a significance of 27.2$\sigma$ by Kitanidis & White
(2020); 2) Hang et al. (2021b) place a constraint on a scaling factor for the
lensing amplitude, $A_{\rm lens}$, determined by using Planck lensing
convergence and the $78.6\%$ of the selected DESI objects for $z\leq 0.8$, and
find $A_{\rm lens}=0.901\pm 0.026$, for $\Omega_{m}=0.275,\sigma_{8}=0.814$.

As a cross-check, we perform a cross-correlation measurement between Planck
lensing and DESI galaxies. Unlike the specifically defined LRG samples used in
Kitanidis & White (2020) or the sample selected by almost all the detectable
sources in Hang et al. (2021b), we select the volume-limited galaxy samples at
each photo-$z$ bin by using different absolute magnitude cuts, same as
described in the main text. The constraints on $A_{\rm lens}$ for the 4
$z$-bins are shown in Fig. 11, which are basically compatible to LDP results
(see Fig. 5). Furthermore, the galaxy samples at $z$-bin $0.6\leq z\leq 0.8$
yield an systematic underestimate on $A_{\rm lens}$ at beyond the 2-$\sigma$
level, relative to the theoretical model from our simulation. The obvious
underestimate for the brightest galaxy sample of $M_{c}=-22$ is consistent
with that of LDPs. This may be because of low-quality measurements of
photo-$z$ and the lack of galaxies at this high redshift bin. By combining the
different redshift slices with or without the highest bin, the measured
$A_{\rm lens}$ from different $M_{c}$ are summarized in Tab. 6. The inclusion
of galaxy samples from the highest $z$-bin indeed clearly lower the estimated
value of $A_{\rm lens}$.

Table 6: Best-fit $A_{\rm lens}$ with $1$-$\sigma$ uncertainties from different magnitude cuts $M_{c}$, with and without inclusion of the $z$-bin of [0.6,0.8], and the corresponding $\chi^{2}_{\rm min}/dof$ for each case. |  | $A_{\rm lens}$ | $\rm{\chi^{2}_{min}/dof}$ |
---|---|---|---|---
choice of $z_{m}$ |  | $Max(M_{c})$ | -21.5 | $Min(M_{c})$ |  |  | $Max(M_{c})$ | -21.5 | $Min(M_{c})$ |
w/o $z_{m}=0.7$ |  |  1.03$\pm$0.03 |  1$\pm$0.04 |  1.02$\pm$0.05 |  |  | 0.95 | 0.94 | 1.06 |
w $z_{m}=0.7$ |  |  0.96$\pm$0.02 |  0.92$\pm$0.03 |  0.9$\pm$0.04 |  |  | 0.88 | 0.9 | 0.97 |
Similar to Tab. 3, but the constraints are from galaxies alone. |