# Cosmological constraints from the tomographic cross-correlation of DESI
Luminous Red Galaxies and Planck CMB lensing

Martin White  Rongpu Zhou  Joseph DeRose  Simone Ferraro  Shi-Fan Chen
Nickolas Kokron  Stephen Bailey  David Brooks  Juan García-Bellido  Julien Guy
Klaus Honscheid  Robert Kehoe  Anthony Kremin  Michael Levi  Nathalie
Palanque-Delabrouille  Claire Poppett  David Schlegel  Gregory Tarle

###### Abstract

We use luminous red galaxies selected from the imaging surveys that are being
used for targeting by the Dark Energy Spectroscopic Instrument (DESI) in
combination with CMB lensing maps from the Planck collaboration to probe the
amplitude of large-scale structure over $0.4\leq z\leq 1$. Our galaxy sample,
with an angular number density of approximately $500\,\mathrm{deg}^{-2}$ over
18,000 sq.deg., is divided into 4 tomographic bins by photometric redshift and
the redshift distributions are calibrated using spectroscopy from DESI. We fit
the galaxy autospectra and galaxy-convergence cross-spectra using models based
on cosmological perturbation theory, restricting to large scales that are
expected to be well described by such models. Within the context of
$\Lambda$CDM, combining all 4 samples and using priors on the background
cosmology from supernova and baryon acoustic oscillation measurements, we find
$S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}=0.73\pm 0.03$. This result is lower
than the prediction of the $\Lambda$CDM model conditioned on the Planck data.
Our data prefer a slower growth of structure at low redshift than the model
predictions, though at only modest significance.

##  1 Introduction

The growth of the large-scale structure of the Universe from small
perturbations laid down at early times, as probed by fluctuations in the
cosmic microwave background, to the galaxies, clusters and superclusters of
the modern era is a prediction of our standard cosmological model. Measuring
the growth rate and comparing to this prediction provides a strong test of
many facets of this theory, including the contents, expansion history and laws
of gravity in our Universe. One means of measuring the inhomogeneity in the
modern Universe is through gravitational lensing. In particular, gravitational
lensing of CMB photons provides us with a relativistic tracer that gives an
unbiased measure of the gravitational potentials associated with large-scale
structure (see refs. [1, 2] for reviews). With the statistical properties and
redshift of the source (primary CMB anisotropies) extremely well
characterized, CMB lensing is a robust probe and one that is rapidly
increasing in statistical power.

Like other lensing probes CMB lensing is a projected signal, sensitive to a
wide range of redshifts, and this makes isolating a particular epoch
difficult. By cross-correlating CMB lensing with low-redshift tracers (such as
galaxies) that can be confined to narrow slices in redshift we can extract the
information as a function of redshift (lensing tomography). Moreover, by
combining tracers with different dependences on galaxy bias, we can
efficiently break the degeneracy between bias and the amplitude of
fluctuations at low redshift, providing tight cosmological constraints on the
low-redshift Universe as a function of epoch.

While the $\Lambda$CDM model, conditioned on CMB data, makes highly precise
predictions for the amplitude of large-scale structure ($\sigma_{8}$) as a
function of redshift in the low redshift Universe [3] the observational
situation as regards testing these predictions is currently mixed (see also
§5.3). Some inferences from the abundance of clusters suggest lower values of
$\sigma_{8}$ than the CMB-based model predicts (see §6.4 of ref. [3] or ref.
[4] for discussion), while other inferences from the thermal Sunyaev-Zeldovich
(tSZ; [5]) power spectrum suggest values in good agreement with CMB [6]
depending on assumptions about the relation between the observable and halo
mass. Surveys measuring cosmic shear tend to find lower values than high
redshift inferences [7, 8, 9, 10, 11, 12, 13] while studies of velocities of
galaxies via redshift-space distortions find “low” [14, 15, 16, 17, 18, 19],
“intermediate” [20, 21] and “high” [22] results. Earlier analyses of galaxies,
clusters and quasars in combination with Planck lensing [23, 24, 25] found
either consistent [23] or lower [26, 27, 28, 29, 30, 31] clustering amplitudes
than expected from CMB, though with low statistical significance. By contrast
the combination of BOSS galaxies at $z\approx 0.6$ and the ACT lensing maps is
consist with the CMB-preferred normalization [32]. In a recent analysis of a
wide range of data sets, ref. [33] find evidence for a deficit in the growth
of large scale structure in the redshift regime $0.2<z<0.6$.

Given this situation, the purpose of this paper is to measure the amplitude of
large scale structure for $z\simeq 0.5-1$ by combining recently acquired
samples of luminous red galaxies covering 18,000 sq.deg. of the extragalactic
sky [34] with the CMB lensing map probed by Planck [25]. In a companion paper
[35] we present the samples of galaxies we use, drawn from the imaging surveys
[34] being used for targeting by the Dark Energy Spectroscopic Instrument
(DESI; [36]). These samples included those being targeted by DESI for follow-
up spectroscopy, split by photometric redshift into different redshift ranges
[35]. In this paper we present the cross-correlation between those maps and
the Planck 2018 CMB lensing maps [25] and provide an initial cosmological
investigation. A particular goal is to measure the amplitude of fluctuations,
$\sigma_{8}$, using these samples. In the future we will be able to combine
these lensing inferences (or those with higher signal-to-noise ratio lensing
maps) with constraints from galaxy-galaxy lensing, baryon acoustic
oscillations and redshift-space distortions measured from the same samples of
galaxies and thus provide a more comprehensive test of our cosmological model.

The outline of the paper is as follows. In §2 we review the data going into
our analysis including computation of the (pseudo-)spectra, window functions
and covariances. We describe the modeling in §3 and tests of our pipeline in
§4. Our results are presented in §5 while §6 contains our conclusions and
directions for future work. We use units where $c=1$ throughout unless
otherwise specified. All distances are comoving and in $h^{-1}$Mpc units.

##  2 Data

###  2.1 Luminous Red Galaxies

Sample | $\bar{z}$ | $\Delta z$ | $\bar{n}_{\theta}$ |  $10^{6}\,$SN | $s_{\mu}$ | $z_{\rm eff}$ | $\ell_{\rm max}$ | $b_{E}$ | $\ell_{SN}$
---|---|---|---|---|---|---|---|---|---
1 | 0.47 | 0.064 | 83 | 4.02 | 0.982 | 0.47 | 250 | 1.9 | 400
2 | 0.63 | 0.075 | 149 | 2.24 | 1.033 | 0.62 | 300 | 2.1 | 530
3 | 0.79 | 0.079 | 162 | 2.07 | 0.961 | 0.78 | 350 | 2.4 | 575
4 | 0.92 | 0.097 | 149 | 2.26 | 1.013 | 0.91 | 400 | 2.5 | 425
Table 1: Some properties of the samples we analyze. The sample number refers
to the pz_bin index in the catalogs [35]. The redshift location and scale of
each sample are provided, as well as the mean number of galaxies per square
degree ($\bar{n}_{\theta}$) and the Poisson shot-noise level, SN. The
(log)slope of the magnitude distribution, $s_{\mu}$, is used for the
magnification calculation (Eq. 3.5) while the effective redshift is defined in
Eq. (3.9). In the last three columns we give the maximum $\ell$ used in the
fits (§3.2), the inferred large-scale (Eulerian) bias ($b$; §5) and the $\ell$
at which Poisson shot noise equals the cosmological power in the best-fitting
auto-spectrum model.

Figure 1: The LRG overdensity for the full sample (top) and CMB-determined
$\kappa$ (bottom), in Mollweide projection and galactic coordinates. We follow
the astronomical convention that east is left. The grey region shows parts of
the sky that are masked (see text). The common, unmasked area is approximately
18,000 sq.deg. The maps use the Healpix pixelization, with $N_{\rm side}=256$.

Our galaxy samples are drawn from the imaging data [34] being used for target
selection by the DESI spectroscopic redshift survey [36]. The samples are
described in detail in ref. [35] and some of the key features are summarized
in Table 1. Briefly they consist of DESI luminous red galaxies (LRGs) [37],
selected from the imaging by applying a series of color cuts on extinction-
corrected magnitudes in the $g$, $r$, $z$, and $W1$ bands, that have been
divided into 4 redshift slices using photometric redshifts with 1.4–2.8
million galaxies per slice. We refer to these photometric redshift bins as
pz_bin 1 through 4. The combination of all 4 slices has 500 galaxies per
square degree and covers 44% of the sky ($\approx 18$K square degrees; Fig.
1). The photometric redshift cuts have been designed to give narrow redshift
distributions with well-controlled tails at low and high redshift and modest
overlap between samples as this simplifies the modeling of the cross-
correlations (see ref. [35] and later for further discussion).

In this paper we will focus on the ‘fiducial’ sample of ref. [35]. This is
drawn from the LRG sample for which DESI will obtain spectroscopic redshifts
over the course of the survey. While the legacy imaging surveys also provide
deeper samples, with higher number density in the same redshift slices, we
will be limited not by galaxy shot noise but rather by noise in the Planck
$\kappa$ map and the limitations of our models. For this reason we focus on
the fiducial sample, with the goal of eventually combining these constraints
with those from 3D clustering using DESI redshifts. Other cross-correlation
projects, including cross-correlations with higher fidelity CMB lensing maps,
may benefit more from deeper samples which have lower shot noise while
maintaining the narrow redshift distributions. Since the DESI survey obtained
deeper spectroscopy during its commissioning, the redshift distributions for
these samples can still be spectroscopically calibrated.

We use the masks provided by ref. [35], which are chosen to eliminate bad
data, avoid the galactic plane and regions of high extinction or high stellar
density and to mask out bright stars. While we make every effort to keep them
to a minimum during catalog construction, the mask contains a large number of
small holes and apodizing the mask (to smoothly connect the signal and masked
regions) would lead to a non-trivial loss of area and invalidate some of the
approximations we will use later to compute the covariance matrix (see e.g.
ref. [38] for an extended and clear discussion). Thus we do not apodize the
galaxy mask. Tests with Gaussian mock catalogs show that we can recover the
bandpowers on the scales relevant to our analysis with 0.1% precision whether
or not we apodize the mask. This is much smaller than our statistical errors.

Figure 2: The redshift distribution of our 4 photo-$z$-selected samples (Table
1), as determined by DESI spectroscopy. The bins are relatively narrow and
somewhat disjoint while maintaining a high angular number density, making them
good for tomographic cross-correlation analyses. The shaded band is
proportional to the CMB lensing kernel.

The maps are generated in the standard manner: the LRGs for each slice are
binned into Healpix [39] pixels at $N_{\rm side}=2048$ to form an “LRG map”.
Random points are uniformly distributed within the footprint, and then passed
through a series of masks that mirror the cuts applied to the data and include
BRIGHT, GALAXY and CLUSTER
masks111https://www.legacysurvey.org/dr9/bitmasks/#maskbits plus a custom
bright star mask [35]. We apply systematics weights to each random point to
account for the effects of the largest systematics on the density field [35].
The weights include corrections for trends with extinction, PSF size and depth
in each band as discussed in detail in ref. [35]. These weights are applied to
the random catalog rather than the LRGs themselves to reduce the impact of
shot-noise from the small number of galaxies per pixel at $N_{\rm side}=2048$
(we have approximately 0.1 galaxies and 60 randoms per pixel per slice). These
weights are also used in the computation of the Poisson shot-noise level,
which is listed in Table 1. The LRG samples that we use are already very
clean, and we find that these systematics weights have a very small effect on
the measured clustering. For all redshift bins including the weights removes
some large-scale power associated with the very different imaging depths in
the SGC compared to the NGC region (by $1-2\%$ at $\ell=50$; see ref. [35] for
more details and figures). The weighted random counts in each Healpix pixel
then form the “random map”. The overdensity field is defined as the “LRG map”
divided by the “random map”, normalized to mean density and mean subtracted.

Since the photometric systems between the northern and southern imaging
differ, there is a possibility for systematics to affect one sample
differently than the other. We have checked that our results are stable
(within errors) to selecting only galaxies from the northern vs. the southern
hemisphere. Our results are also stable to making a more stringent $E(B-V)$
cut, or to different treatments of the stellar mask. Upon cross-correlating
the galaxy maps with alternative extinction maps produced by the Planck
collaboration [40] we find only a $1-5\%$ correlation, suggesting this is not
a major contribution to our systematic error budget.

The redshift distribution is obtained from spectroscopic redshifts obtained by
DESI, as described in detail in ref. [35], and is shown in Fig. 2. The LRG
sample has a spectroscopic redshift completeness of 98%. The 2% LRGs without
secure redshifts are mostly a result of low spectroscopic S/N, and we correct
for this redshift incompleteness by up-weighting the fainter LRGs (that do
have secure redshifts) in the $dN/dz$ estimation. Note the definition of the
redshift bins through pz_bin leads to quite compact redshift distributions
with minimal tails, which greatly simplifies the modeling as we describe
later.

From the spectroscopic followup performed by DESI, we know that the stellar
contamination is very low ($\ll 1\%$) for these samples and can be safely
neglected. There is a small contribution from active galactic nuclei,
frequently an active nucleus within a luminous red galaxy, which have very
similar $dN/dz$ and bias to our main sample and thus we shall not try to
correct for this contamination.

Finally we require the slope of the galaxy number counts as a function of
magnitude, $s_{\mu}$, in order the model the magnification bias of our samples
(Table 1 and Eq. 3.5). The slope is determined independently for each redshift
bin by finite differences (shifting all magnitudes by $\delta m=0.01$ and re-
applying the selection) with two additional corrections. First, since the LRG
selection includes a fiber-flux cut we need to account for how the fiber-flux
of each galaxy changes when it is magnified or demagnified. This is estimated
as a function of the shape parameters for each morphology type, as described
in detail in ref. [35]. A second subtlety is that the redshift binning is done
using random-forest-based photo-$z$s that use magnitude and galaxy size as
features. To account for the change in photo-$z$s under magnification we
recompute the photo-$z$s with the shifted magnitudes and sizes and redo the
redshift selection. These two corrections change the slope from the naive
finite difference estimate by $10-30\%$ (depending upon sample [35]).

###  2.2 Planck lensing convergence

We use the latest CMB lensing maps from the Planck 2018 release [25] and their
associated masks, downloaded from the Planck Legacy Archive.222PLA:
https://pla.esac.esa.int/ These maps are provided as spherical harmonic
coefficients of the convergence, $\kappa_{\ell m}$, in HEALPix format [39] and
with $\ell_{\rm max}=4096$. In particular, for our fiducial analysis we use
the minimum-variance (MV) estimate obtained from both temperature and
polarization, based on the SMICA foreground-reduced CMB map.

We low-pass filter the $\kappa_{\ell m}$ to remove power above $\ell=2500$
before generating the convergence map in Healpix format [39] at $N_{\rm
side}=2048$ with the Healpix routine alm2map (Fig. 1). We apodize the mask
provided by the Planck team with a $30^{\prime}$ “C2” filter333This applies a
filtering $(1/2)[1-\cos\pi x]$ to all pixels near ($x<1$) a pixel in the mask
with zero value (i.e. a masked pixel). The variable
$x\equiv\sqrt{(1-\cos\theta)/(1-\cos\theta_{\star})}\approx\theta/\theta_{\star}$
with $\theta_{\star}$ the apodization scale. using the NaMaster [41] routine
mask_apodization. This makes the window functions we derive later more compact
in harmonic space.

Since the MV reconstruction is dominated by temperature, residual galactic and
extragalactic foregrounds may contaminate the signal. Extensive testing has
been performed by the Planck team, indicating no significant problems at the
current statistical level. However, as a test, we repeated the analysis with a
lensing reconstruction on SMICA foreground-reduced maps where the thermal tSZ
effect [5] has been explicitly deprojected [25]. For the highest two redshift
samples we found negligible impact from this change. For the second bin the
change was larger but not statistically significant. However for the lowest
redshift bin the cross-correlation with the tSZ-deprojected maps revealed
$2\,\sigma$ excess power at $\ell\simeq 50$. We believe this may be due to
non-tSZ foregrounds in the lensing map (e.g. dust or CIB) correlating with
systematics in the LRG map. It is known that SZ-deprojection increases the
impact of non-SZ foreground on the $\kappa$ map [42]. We also know that the
lowest redshift bin could be more affected by galactic dust or extinction due
to its use of the $g$-band to reject low $z$ galaxies. The combination could
plausibly explain the excess power at low $\ell$. Though we have no evidence
that the cross-correlation with the fiducial lensing maps is similarly
contaminated, this may suggest the cross-correlations with the LRGs in the
first redshift bin are slightly less reliable, at least on large scales. Since
most extinction maps are correlated with large-scale structure due to
contamination from the CIB [43, 44], one direction for future work could be a
study of the correlation with CIB-free extinction or reddening maps such as
those derived from galactic Hi emission [43]. We also note444We thank the
referee for emphasizing this point. that the provided CMB lensing mask removes
regions around high $S/N$ tSZ sources to avoid contamination. These are mostly
low-redshift sources that could potentially correlate with the lowest redshift
bin [45]. We shall leave further investigation of these issues to future work.

###  2.3 Angular power spectra and covariance

The estimation of angular power spectra from sky maps is now routine (see
refs. [38, 46, 47, 28, 32, 29, 31, 48, 49, 50, 33, 51, 52] for some examples
within just the last year). We use the pseudo-$C_{\ell}$ method [53] as
implemented within the NaMaster package [41] to estimate our angular power
spectra. As a reminder, this approach uses fast spherical harmonic transforms
to expand a field, $X(\hat{n})$ into spherical harmonics, $Y_{\ell m}$, as

$\tilde{a}_{\ell m}=\int_{4\pi}d\Omega\ X(\hat{n})\ W(\hat{n})\ Y^{*}_{\ell
m}(\hat{n})$ (2.1)

where $W(\hat{n})$ is the mask. The (pseudo) angular power spectrum,
$\tilde{C}_{\ell}$, is then an average over $m$-modes,

$\tilde{C}_{\ell}=\frac{1}{2\ell+1}\sum_{m=-\ell}^{\ell}|\tilde{a}_{\ell
m}|^{2}$ (2.2)

The $\tilde{C}_{\ell}$ are related to the true spectrum via a convolution with
the mask such that their ensemble averages are related by
$\langle\tilde{C}_{\ell}\rangle=\sum_{\ell^{\prime}}M_{\ell\ell^{\prime}}\langle
C_{\ell^{\prime}}\rangle$ where the mode-mode coupling matrix $M$ can be
determined purely from the mask [53]. This $\ell$-by-$\ell$ matrix is
generally singular in the case of large sky cuts. In order to perform matrix
inversion, a common method is to use a set of discrete bandpower bins $L$ and
assume the angular power spectrum is a step-wise function in each bin. While
not strictly necessary, the $L$-by-$L$ mode-mode coupling matrix is typically
inverted and applied to extract the binned angular power spectrum from the
binned pseudo angular power spectrum ($\langle
C_{L}\rangle=\sum_{L^{\prime}}M_{LL^{\prime}}^{-1}\langle\tilde{C}_{L^{\prime}}\rangle$).
The final result is a linear relationship between the binned power spectrum,
$C_{L}$, and the underlying $C_{\ell}$ that holds for expectation values. We
use the compute_full_master method in NaMaster [41] to calculate the binned
power spectra and the bandpower window functions relating them to the
underlying theory: $\langle C_{L}\rangle=\sum_{\ell}W_{L\ell}C_{\ell}$. We
choose a conservative binning scheme with linearly spaced bins of size
$\Delta\ell=50$ starting from $\ell_{\rm min}=25$. The bin width is larger
than expected correlations between modes induced by the survey masks, while
being narrow enough to preserve the structure in our angular spectra. To avoid
power leakage near the edge of the measured range we perform the computation
to $\ell=6000$, and simply discard the bins beyond some $\ell_{\rm max}$ [47].
The associated $W_{L\ell}$ for the auto- and cross-power spectra are shown in
Fig. 3 for the bandpowers up to $L=750$.

Figure 3: The bandpower window functions for the auto- (left) and cross-
(right) spectra defined such that $\langle C_{L}^{\rm
obs}\rangle=\sum_{\ell}W_{L\ell}C_{\ell}^{\rm thy}$. Since we bin $C_{L}^{\rm
obs}$ with $\Delta\ell=50$ we expect $W_{L\ell}$ to be approximately top-hats
of height $0.02$ and width $50$. The effect of masking modifies this
expectation somewhat, with the largest impact on the auto-correlation window
function.

While it is a small effect on the scales of interest here, we correct our
power spectra for the effects of the pixel window function (obtained from the
Healpix routine pixwin). The cross-spectrum between $\kappa$ and the galaxy
density is divided by one power of the pixel window function while the shot-
noise-subtracted auto-spectrum is corrected by two powers. The shot-noise
itself receives no correction. These choices were checked against Gaussian and
Poisson sky simulations.

Figure 4: The angular pseudo-spectra ($C_{\ell}^{gg}$ in blue and
$C_{\ell}^{g\kappa}$ in red) for each of our 4 samples in bins of
$\Delta\ell=50$. The points, with errors, show the measurements while the
solid lines show the fit used to compute the analytic covariance matrix. The
second and fourth rows show the ratio of the data to the fit, so that it is
easier to see the errors (which are very small, especially for the
autocorrelation).

Fig. 4 shows the measured bandpowers, $C_{\ell}^{gg}$ and $C_{\ell}^{\kappa
g}$, for each of our samples. The error bars are the diagonal of the
covariance matrix, but the individual bandpowers are largely decorrelated with
bins of width $\Delta\ell=50$ and our sky coverage. Fig. 5 shows the cross-
spectra between galaxies in different slices, which we use to compute the
covariance matrix of the joint samples. Since the photo-$z$ bins are
relatively compact and have little overlap the cross-power is a small fraction
of the auto-power in each slice.

Figure 5: The angular pseudo-cross-spectra ($C_{\ell}$) for each of our 4
galaxy samples cross-correlated with each of the others. In each panel the
dashed line shows the fit to the auto-spectrum of the lower redshift sample
while the dotted line shows the fit to the higher redshift sample for
comparison. The points show the measured cross-correlation while the light
cyan line shows a polynomial fit used in computing the covariance matrix.

Our signal comes predominantly from large scales (Figs. 6, 7) where the
approximation of a Gaussian (disconnected) covariance matrix holds. We compute
this matrix with the spectra of the best-fitting theoretical model from an
earlier iteration using the NaMaster function gaussian_covariance, including
both the auto-correlations of $C_{\ell}^{gg}$ and $C_{\ell}^{\kappa g}$ and
their covariance. The signal and noise of the $\kappa$ autospectrum are taken
from the Planck data release. We checked that the simpler approximation in §3
of ref. [53], which involves the $2^{\rm nd}$ and $4^{\rm th}$ moments of the
window function and $f_{\rm sky}$, also works well, with the full expression
giving slightly stronger bin-bin correlations but quite similar $\chi^{2}$
values in our fits (i.e. differences much less that $\sqrt{2\,N_{\rm dof}}$).
We also checked our covariance matrix against one produced by Monte-Carlo
sampling 2048 Gaussian realizations, with the same masking, binning, etc. The
agreement was also good. Denoting $C_{\rm sim}$ as the Monte-Carlo derived
covariance and $C_{\rm fid}$ as our fiducial (analytically derived) one ${\rm
tr}\left[C_{\rm fid}\ \Delta\Omega\right]$, with $\Delta\Omega=C_{\rm
fid}^{-1}-C_{\rm sim}^{-1}$, gives the ‘typical’ change in $\chi^{2}$ induced
by using the different covariance matrices. For our data in the most
constraining slice (pz_bin 4) ${\rm tr}\left[C_{\rm fid}\
\Delta\Omega\right]\approx 0.2$ for the 8 data points in the galaxy auto-
spectrum below $\ell_{\rm max}=400$ most affected by masking, much smaller
than $\sqrt{2\,N_{\rm dof}}$. We note that these approximations become less
accurate as the number of holes in the mask increases, as this leads to a
broader window and violation of some of the approximations made in NaMaster
[38]. It is for this reason that we have kept the number of holes as small as
practically possible [35]. The agreement between the Monte-Carlo and NaMaster-
produced covariance was not as good for $C_{\ell}^{\kappa g}$, with the Monte-
Carlo predicting slightly smaller variance at low $\ell$. However, the
variance at higher $\ell$ and the structure of the correlation matrix was very
similar (up to noise) so we have chosen to use the NaMaster-produced
covariance as the conservative choice. As a cross check if we use the Monte-
Carlo covariance instead of our default covariance when fitting $\Lambda$CDM
models to each pz_bin sample the best-fit parameters change by $<0.2\,\sigma$.
For all of our approximations the covariance matrices are strongly diagonally
dominant, with the first off-diagonal terms typically at the percent level and
second off-diagonals even lower. This is because our bin width,
$\Delta\ell=50$, and our sky coverage are both simultaneously large.

Figure 6: The (cumulative) signal to noise ratio for $C_{\ell}^{g\kappa}$ for
our 4 samples. The signal-to-noise ratio of $C_{\ell}^{gg}$ is much higher
than $C_{\ell}^{\kappa g}$, thus the latter dominates the error when
determining cosmological parameters. While the samples are not independent, so
the SNRs cannot be simply added, this figure serves to indicate the range of
scales and which samples dominate our constraints.

When we fit multiple samples at once we do not include the galaxy cross-
spectra (which are quite small; Fig. 5) in the data vector but we do include
their contributions to the covariance (based on each of the
$C_{\ell}^{g_{i}g_{j}}$ and $C_{\ell}^{g_{i}\kappa}$). The inclusion of the
galaxy cross-spectra as data would be helpful in calibrating $dN/dz$ [54], but
since we have a spectroscopically calibrated $dN/dz$ they contain little
additional information.

##  3 Modeling

Figure 7: The fractional contributions to $C_{\ell}^{gg}$ (dashed) and
$C_{\ell}^{g\kappa}$ (dotted) for three different multipoles ($\ell=200$, 300
and 400; blue, orange and green) for the lowest (left; $z\approx 0.5$) and
highest (right; $z\approx 0.9$) redshift samples. We use the best-fitting Anzu
model for the signal model. Within the Limber approximation the contributions
are given by Eq. (3.1), with a given multipole probing scales
$k\approx\ell/\chi_{0}$ if $\chi_{0}$ is the characteristic distance to the
sample. The black, downward-pointing arrows show the non-linear scale, $k_{\rm
nl}$, defined as the inverse of the mean Zeldovich displacement (i.e. the same
convention as ref. [55]). The scales where our S/N peaks are in the linear or
quasi-linear regime, with negligible contributions from baryonic effects,
simplifying the analysis.

###  3.1 Spectra

The galaxy overdensity, $\delta_{g}$, and CMB lensing convergence, $\kappa$,
are both projections of 3D density fields. Within the Limber approximation
[56] the angular cross-spectrum between two such fields $X$ and $Y$ is given
by555The largest correction to the Limber approximation, arising at low
$\ell$, comes from redshift-space distortions (RSD) in the galaxy auto-
spectrum [57, 58]. These effects are smaller than our errors for the
bandpowers we consider except for the first bin where for some samples it is
comparable. Rather than include RSD in our predictions, we add a “theory
error” to the first bin conservatively larger than the difference between RSD
and no-RSD predictions of our fiducial model. an integral over comoving
distance $\chi$

$C_{\ell}^{XY}=\int d\chi\
\frac{W^{X}(\chi)W^{Y}(\chi)}{\chi^{2}}P_{XY}{\Big{(}}k=(\ell+1/2)/\chi;z{\Big{)}}$
(3.1)

where the projection kernels for galaxy overdensity and CMB lensing
convergence are, respectively,

$\displaystyle\begin{aligned}
W^{g}(\chi)&=H(z(\chi))\phi(z(\chi))+W^{\mu}(\chi)\\\
W^{\kappa}(\chi)&=\frac{3}{2}\Omega_{m0}H_{0}^{2}(1+z)\frac{\chi(\chi_{\rm*}-\chi)}{\chi_{\rm*}}\end{aligned}$
(3.2)

with $\chi_{\rm*}=\chi(z_{\rm*}{\approx}1100)\approx 9400$ $h^{-1}$Mpc the
distance to the surface of last scattering and $\phi(z)$ the normalized
redshift distribution of the galaxy sample. The $W^{\mu}$ term in $W^{g}$
accounts for magnification bias and, to first order, is given by

$\displaystyle W^{\mu}(\chi)$ $\displaystyle=(5s_{\mu}-2)\
\frac{3}{2}\Omega_{m0}H_{0}^{2}(1+z)\int_{z}^{z_{\star}}dz^{\prime}\
g(z^{\prime})$ (3.3) $\displaystyle g(z^{\prime})$
$\displaystyle=\frac{\chi(z)(\chi(z^{\prime})-\chi(z))}{\chi(z^{\prime})}\phi(z^{\prime})$
(3.4)

with $z$ implicitly a function of $\chi$ and $s_{\mu}$ the slope of the
cumulative magnitude function, i.e. the response of the number density of the
sample to a multiplicative change in brightness at the limiting magnitude of
the survey:

$\displaystyle s_{\mu}\equiv\left.\frac{d\log_{10}n(m<m_{\rm
lim})}{dm}\right|_{m=m_{\rm lim}}\quad.$ (3.5)

Due to the relatively narrow $dN/dz$ of our samples the magnification
correction is small for the auto-spectrum. It is larger for the cross-
spectrum, but even there amounts to only 5% of the signal even for the worst
case (the highest redshift bin), so we are quite insensitive to uncertainties
in this term. Since the magnification term probes smaller scales than the
other contributions to the auto- and cross-spectra but high accuracy is not a
requirement we use the HaloFit fitting function for $P_{mm}$ ([59, 60] as
implemented in CAMB [61, 62]) for the magnification contributions. We have
checked that using perturbation theory with the lowest order counterterm
instead leads to indistinguishable posterior distributions.

To build intuition we note that for a thin shell of galaxies at distance
$\chi_{0}$ of width $\Delta\chi$, if the other parts of the integrands are
changing slowly we have

$C_{\ell}^{gg}\approx\mathcal{V}^{-1}P(k=\ell/\chi_{0})\quad,\quad
C_{\ell}^{g\kappa}\approx W^{\kappa}(\chi_{0})\chi_{0}^{-2}P(k=\ell/\chi_{0})$
(3.6)

where $\mathcal{V}=\chi_{0}^{2}\,\Delta\chi$ is the volume per steradian and
we have neglected the magnification terms for simplicity. Note the galaxy
auto-spectrum increases as we decrease the width of the shell (and there is
less “washing out” of the clustering signal) while the cross-spectrum
amplitude is independent of $\Delta\chi$. This implies the cross-correlation
coefficient between $\kappa$ and the galaxy overdensity decreases for thinner
shells, which probe a smaller fraction of the total $\kappa$ signal.

Since the Planck lensing map has high signal-to-noise predominantly at low
$\ell$ (Fig. 6), our measurement is dominated by large scales. The relatively
compact nature of $dN/dz$ (Fig. 2) means each $\ell$ mode probes only a narrow
range of physical scales, $k$, and these modes will in general be quasi-linear
at the redshifts of relevance (Fig. 7). This, plus the spectroscopically
determined $dN/dz$ and the well-known redshift of the CMB (which is always
well behind and well separated from our lenses) makes our measurement robust
to many common modeling uncertainties. Figures 6 and 7 also nicely illustrate
the gains from moving to higher redshift galaxy samples. The increase in the
CMB lensing kernel from $z\sim 0$ to $z\sim 2$, the shift to probing more
modes at higher $\ell$ for a fixed $k$ and the larger $k$ range for which the
models are accurate at higher redshift all contribute to making the higher
redshift samples more powerful probes.

###  3.2 Power spectrum models

Key to interpreting our measurements are models for the galaxy-galaxy and
galaxy-matter power spectra accurate on linear and quasi-linear scales. Our
cosmological results are based on a perturbative model (§3.2.1), though we use
an N-body based emulator (§3.2.2) to validate some of our results. We next
describe each model, before discussing our parameter priors and some technical
details of the implementation.

####  3.2.1 Lagrangian perturbation theory

Our fiducial model is based on convolution Lagrangian effective field theory
(CLEFT; [63] and the references contained therein), as implemented in the
velocileptors666Code available at https://github.com/sfschen/velocileptors
code [64, 65]. This is the same formulation and code as used in ref. [31].
Under this formalism, the galaxy-galaxy and galaxy-matter power spectra are:

$\displaystyle P_{\rm gg}$
$\displaystyle=\left(1-\frac{\alpha_{a}k^{2}}{2}\right)P_{\rm Z}+P_{\rm
1-loop}+b_{1}P_{\rm b_{1}}+b_{2}P_{\rm b_{2}}+b_{1}b_{2}P_{\rm
b_{1}b_{2}}+b_{1}^{2}P_{\rm b_{1}^{2}}+b_{2}^{2}P_{\rm b_{2}^{2}}$ (3.7)
$\displaystyle P_{\rm gm}$
$\displaystyle=\left(1-\frac{\alpha_{\times}k^{2}}{2}\right)P_{\rm Z}+P_{\rm
1-loop}+\frac{b_{1}}{2}P_{\rm b_{1}}+\frac{b_{2}}{2}P_{\rm b_{2}}$ (3.8)

where we have dropped the terms corresponding to shear bias as we find they
mainly affect small scales and are somewhat degenerate with the $\alpha_{i}$
[66]. Here, $P_{\rm Z}$ and $P_{\rm 1-loop}$ are the Zeldovich and 1-loop dark
matter contributions, $b_{1}$ and $b_{2}$ are the Lagrangian bias parameters
[67] for the galaxy sample, and $\alpha_{\times}$ and $\alpha_{a}$ are free
parameters encapsulating the small-scale physics not modeled by perturbation
theory (see e.g. ref. [68] for further discussion). Each term can be written
as an integral of one or two powers of the linear theory power spectrum, times
an analytically known kernel function. Explicit formulae for the power
spectrum contributions can be found in the above-mentioned references and are
collected in Appendix A.

To reduce model expense and avoid needing to model redshift-dependent bias, we
evaluate the power spectrum terms at a single777Since the magnification terms
depend upon a wider redshift range, we include the expected linear growth of
$P_{mm}(k)$ in the kernel. This is only an approximation on small scales, but
the magnification is a small correction and we probe primarily large scales so
the approximation is quite good. effective redshift,

$z_{\rm eff}^{XY}=\frac{\int d\chi\ z\ W^{X}(\chi)W^{Y}(\chi)/\chi^{2}}{\int
d\chi\ W^{X}(\chi)W^{Y}(\chi)/\chi^{2}}\quad.$ (3.9)

These redshifts are listed in Table 1 for each sample for the fiducial Planck
cosmology, though in our inferences we recompute $z_{\rm eff}$ and $P(k,z_{\rm
eff})$ for each cosmology being tested. Given the narrow redshift distribution
and almost passive bias evolution of our galaxy sample, the $z_{\rm eff}$
approximation does not affect the $C_{\ell}$’s significantly [66] and is very
weakly cosmology dependent. We validate this approximation using simulations
later.

It is by now well established that halos and galaxies can be well approximated
as biased tracers of the dark matter and baryon fluids only [69, 70, 71, 72,
73, 74, 75]. For this reason we use $P_{cb}$, the dark matter plus baryon
spectrum as computed by the Boltzmann code CAMB [61, 62], as input to
velocileptors for $P_{gg}$ and $P_{gm}$. We use the full matter power spectrum
for $P_{mm}$. This substitution assumes the Green’s function associated with
higher order perturbative solutions is very well approximated by the Einstein-
de-Sitter one, implying the structure of the loops remains unchanged. This is
an excellent approximation for the small fiducial value of neutrino masses
considered in this work [76]. In principle one needs to consider the scale-
dependence of the growth rate, which we shall neglect for the same reason
mentioned above.

Fig. 8 shows $C_{\ell}^{gg}$ and $C_{\ell}^{\kappa g}$ for pz_bin=4 as we
successively add $b_{2}$ and counter terms to the model with just $b_{1}$ and
shot noise. The models for the other redshifts look qualitatively similar. For
illustrative purposes we vary the parameters for this sample at fixed
cosmology ($\Omega_{m}=0.31$, $h=0.68$, $\sigma_{8}=0.77$). Note that the
scales we will probe are dominated by the linear contributions, however we
find that using a “minimal model” with scale-independent bias and no counter
terms can lead to shifts in parameters of $\mathcal{O}(1\,\sigma)$ depending
upon the redshift.

Figure 8: Model predictions for $C_{\ell}^{gg}$ (solid) and $C_{\ell}^{\kappa
g}$ (dotted) for the pz_bin=4 sample as we successively add terms to the
model. The left panels show the Lagrangian PT results with the blue line being
a base model with just $b_{1}$ ($=1.4$) and shot noise ($=5490$), the orange
line adding $b_{2}\simeq 0.5$ and the green line including the counter terms
($\alpha_{a}=22$, $\alpha_{\times}=8$). The right panels (plotted to higher
$\ell_{\rm max}$) show the results for Anzu, again with the blue line
including just $b_{1}$ ($=1.4$) and shot noise ($=5300$), the orange line
adding $b_{2}\simeq 1$ and the green line adding $b_{s}\simeq-0.1$ and
$b_{\nabla}\simeq-2.6$. In each case we vary the bias parameters at fixed
cosmology ($\Omega_{m}=0.31$, $h=0.68$, $\sigma_{8}=0.77$). Note that on the
scales plotted the lowest order results dominate over the effects of scale-
dependent bias (and counterterms) for both the auto- and cross-spectra.
However, the observational errors are sufficiently small that the corrections
are not insignificant.

####  3.2.2 Lagrangian bias expansion: Anzu

The second model we employ is Anzu888Code available at
https://github.com/kokron/anzu. [77], an emulator based on the Aemulus
simulations [78] that implements the Lagrangian bias prescription with
component spectra measured from N-body simulations [79]. Anzu provides
$P_{gg}$ and $P_{gm}$ as a function of four bias parameters (the same999The
definition of $b_{2}$ employed in Anzu differs from that in velocileptors by a
factor of two. We have standardized to the velocileptors definition. $b_{1}$,
$b_{2}$ as above plus a shear bias, $b_{s}$, and a derivative bias,
$b_{\nabla}$) for a range of $\Lambda$CDM cosmologies by interpolating from a
grid. Anzu does not (yet) include massive neutrinos so for this case we
consider pure $\Lambda$CDM cosmologies. In order to better match the shape of
the linear power spectrum including massive neutrinos we reduce $n_{s}$ by
0.005 for Anzu. This is a small change, and within the current uncertainty on
$n_{s}$, but achieves a better match to the broad-band power between the
models.

Fig. 8 shows $C_{\ell}^{gg}$ and $C_{\ell}^{\kappa g}$ for pz_bin=4 as we
successively add the higher order biases to the model with just $b_{1}$ and
shot noise. For illustrative purposes we choose best-fit parameters for this
sample at fixed cosmology ($\Omega_{m}=0.31$, $h=0.68$, $\sigma_{8}=0.77$).
Again, the large scales we probe are dominated by the linear contributions.

Unfortunately the Planck lensing maps are relatively noisy, leading to broad
posteriors on many parameters. The grid of models used to train Anzu is not
wide enough to encompass the full posterior, so we cannot use Anzu in its
present form for our cosmological inferences. However, for models within the
training volume we expect Anzu to be more accurate than CLEFT and so we have
employed Anzu throughout the analysis as a systematic check.

###  3.3 Parameter priors, sampling and emulation

For our parameter inference we employ a now-standard Markov Chain Monte Carlo
technique, using the Cobaya code [80, 81] with the CosmoMC sampler [82, 83,
84] and GetDist [85] visualization software. For each point in parameter space
we generate $C_{\ell}^{XY}$ using the Limber approximation from the underlying
real-space power spectra and use a Gaussian likelihood.

The evaluation of the theoretical models involves computation of the linear
theory power spectrum (for which we use CAMB [61, 62]) and higher order
corrections as described above. The codes for doing this are quite efficient,
but still take several seconds per call. When many chains need to be run for
many samples with many different settings per chain this can add up to a large
computational cost. In order to make the chains more efficient we do not call
the theory model directly, but rather use an additional
emulation/interpolation stage based on neural networks. Neural networks can be
used as “universal” function interpolators, whose accuracy is limited
primarily by the amount of data available to train them. In this regard the
combination of neural networks and perturbative models is particularly
attractive, because it is possible to generate millions of models to train the
network at modest computational cost. In a sense we pay the cost of evaluating
models in a given family up front, and such models can be used within many
chains. This technique is gaining currency in cosmology, and similar
approaches have been presented in refs. [86, 87, 88, 89, 90, 91, 92, 93, 94].

We present the details of the neural network architecture in ref. [95].
Briefly we use an architecture and activation function similar to that in ref.
[96]. The network has $4$ fully connected layers of $128$ neurons, with an
activation function [96]

$a(\mathbf{x})=\left[\mathbf{\gamma}+\frac{1-\mathbf{\gamma}}{1+\exp(-\mathbf{\beta}\odot\mathbf{x})}\right]\odot\mathbf{x}$
(3.10)

where $\odot$ indicates element-wise multiplication, $\beta$ and $\gamma$ are
parameters and $\mathbf{x}$ are the inputs. A final layer performs a linear
offset and scaling. A grid of $10^{6}$ models covering a broad range of
cosmological and bias parameters and redshifts is generated. The spectra are
normalized, the mean subtracted and the arcsinh taken to reduce the dynamic
range, and then compressed into $64$ principal components. The coefficients of
these principal components are then predicted by our neural network. We find
the neural network is able to reproduce the spectra in our training and test
sets with a median error of $0.1\%$ with the prediction step now several
orders of magnitude faster than the theory codes. We find the posterior
contours using the full model and the emulator are indistinguishable in all
tests we have performed.

Cosmological | Nuisance
---|---
$\theta_{\star}$ | 0.0104109 | $b_{1}$ | $\mathcal{U}(0,3)$
$n_{s}$ | 0.97 | $b_{2}$ | $\mathcal{U}(-5,5)$
$\tau$ | 0.07 | $\alpha_{a}$ | $\mathcal{N}(0,50)$
$\omega_{b}$ | 0.022 | $\alpha_{\times}$ | $\mathcal{N}(0,50)$
$\omega_{c}$ | $\mathcal{U}(0.08,0.16)$ | SN | $\mathcal{N}({\rm Poiss},30\%)$
$\ln(10^{10}A_{s})$ | $\mathcal{U}(2,4)$ | $s_{\mu}$ | $\mathcal{N}(-,0.1)$
Table 2: Priors used in our fits. The first 2 columns contain the priors on
cosmological parameters, with the symbols having their usual meanings (e.g.
ref. [97]). The last 2 columns show the nuisance parameters (biases and
counterterms) for the PT model described in §3.2. The bias terms $b_{1}$,
$b_{2}$ and the slope $s_{\mu}$ are dimensionless, the counterterms
($\alpha_{i}$) are in $h^{-2}\mathrm{Mpc}^{2}$ and the shotnoise is in
$h^{-3}\mathrm{Mpc}^{3}$. A numerical value indicates a fixed parameter,
$\mathcal{U}$ is the uniform distribution while $\mathcal{N}$ is a normal
distribution. For the shotnoise (SN) we use the shorthand “Poiss” to refer to
the Poisson value and for $s_{\mu}$ we use $-$ to refer to the mean value,
which are given in Table 1 for each sample.

Especially for the PT model, for which we will have limited dynamic range,
some of the higher order bias parameters are degenerate with the stochastic
term (i.e. the shot-noise). For this reason we apply loose priors to these
parameters to improve the efficiency of the chains (Table 2). We can gain some
intuition about the expected size of our bias parameters by considering Fig. 8
of ref. [98], generalized to mock galaxies in refs. [99, 100]. For the shot-
noise term we choose a Gaussian prior on SN, centered on the Poisson value
(Table 1) with a width of $30\%$ [101]. For the slope of the number counts
($s_{\mu}$) we use a Gaussian prior centered on the measured value (Table 1)
with a width of $0.1$. This is approximately an order of magnitude larger than
the statistical uncertainty in the determination of $s_{\mu}$ [35]. For the
cosmological parameters and several of the nuisance parameters the priors are
broad enough to be irrelevant, however in the case of the counterterms the
posterior depends upon the imposed shot-noise prior and our constraint on
$s_{\mu}$ is determined entirely by the prior. With only one exception (see
§5) we find these nuisance parameters do not influence the cosmological ones,
so our cosmological constraints are very weakly dependent on our prior choices
for these parameters. We hold the primordial power spectrum slope, $n_{s}$,
and the optical depth, $\tau$, fixed. We are quite insensitive to the specific
choices: changing these by $2\,\sigma$ does not appreciably alter our
constraints.

The fact that our data are projected, and thus do not strongly resolve the
baryon acoustic oscillations that can be used as a standard ruler, and of
limited dynamic range means that we are sensitive primarily to a combination
of $\Omega_{m}$ and $\sigma_{8}$ rather than each individually. When doing
fits to $\Lambda$CDM parameters we hold the angular size of the sound horizon,
$\theta_{\star}$, fixed. This parameter is known to exquisite precision and
depends very weakly on model assumptions as it comes from the observed
locations of the CMB acoustic peaks [97] (changes in the precise value of
$\theta_{\star}$ hardly impact our constraints, nor does holding $H_{0}$ fixed
instead of $\theta_{\star}$). We additionally jointly fit to the Pantheon SNe
data compilation [102] and BAO data [103, 104, 105]. The combination of the
SNe data and $\theta_{\star}$ provides relatively strong constraints on
$\Omega_{m}$, constraining it to be $\Omega_{m}\approx 0.30\pm 0.02$ in
agreement with a host of other observations [106]. The combination with BAO
approximately halves the error as the BAO data provide a consistent, but
tighter constraint on $\Omega_{m}$, while being primarily a geometric
measurement. This better constrains the parameter combination that our auto-
and cross-correlations do not constrain well themselves and improves the
convergence of the chains by downweighting low-$\Omega_{m}$ solutions that are
allowed by just our data. When fitting the $\Lambda$CDM model to the
combination of the 4 redshift slices we have 52 data points and 26 parameters,
9 of which are at least partially prior dominated.

##  4 Simulations and pipeline test

We use mock catalogs [107] derived from N-body simulations in order to
validate our models and determine our scale cuts. The theoretical models have
been extensively compared to simulations and data elsewhere [66, 64, 65, 31,
108, 77] but we include these tests to validate the full pipeline for samples
similar to our data. Since none of our data, covariance or models are
calibrated from these simulations our requirements are relatively relaxed; we
desire a sufficiently complex input model with approximately the same redshift
distribution and clustering power.

###  4.1 Mock catalogs

The mock samples are generated from the Buzzard N-body simulations [109, 110,
111] using the Addgals algorithm, which assigns galaxy positions, velocities,
rest-frame absolute magnitudes and spectral energy distributions to each
galaxy. This allows us to generate photometry in the DECAM bandpasses, so that
we can make galaxy sample selections similar to those used in our measurements
on the data. In particular, we make small adjustments to the color cuts used
to select the DESI LRG sample in order to match the angular number density of
the LRG sample. After tuning to the total angular number density, our
selection in the simulations agrees well with the comoving number density and
projected clustering, $w_{p}$, for the LRGs measured during survey validation
[107]. In order to place our simulated sample into redshift bins, we assign
photometric redshift estimates to each galaxy. We do this by first binning the
DESI LRG data into narrow ($\Delta z=0.075$) redshift bins, and estimating the
mean photometric redshift uncertainty in these narrow bins, providing a model
for $\sigma_{z}(z)$. For each simulated galaxy, we then assign a photometric
redshift by perturbing the true redshift by a Gaussian error with standard
deviation given by $\sigma_{z}(z)$. We use the Born approximation and the dark
matter distribution in the simulation to produce mock CMB-lensing $\kappa$
maps. The angular number densities of the mock samples are 85, 155, 164, and
$145\,{\rm deg}^{-2}$, very close to those of the data (Table 1).

###  4.2 Mock spectra

Figure 9: A comparison of the angular power spectra of the data and the mocks
that we use to validate our pipeline. For each of the 4 samples we show
$C_{\ell}^{gg}$ and $C_{\ell}^{g\kappa}$ for both the mocks and data, and we
see that the agreement is very good. We have omitted the error bars to avoid
crowding, but they are the same as in Fig. 4. Though we do not show it here,
the number density and $dN/dz$ of the mocks and data are also in good
agreement.

To improve their value as a cross-check we ensured that the pipeline to create
the mock measurements was completely independent of the fitting pipeline,
though we computed the auto- and cross-spectra using NaMaster (with the same
binning as for the data) in both cases. A comparison of $C_{\ell}$ between
data and mock is shown in Fig. 9. For all simulated analyses we use the the
mean of the measurements made from 7 quarter-sky Buzzard simulations.

We then fit the mock data with our models. For this test we used the window
functions and data from the mocks and the covariance matrix of the data. Since
the Buzzard simulations are pure dark matter simulations we set $m_{\nu}=0$
and similarly match $n_{s}$, $\Omega_{b}\,h^{2}$ and the central value of the
shot-noise to the values used in the simulations. To mimic the constraints
from the SNe we generate a mock dataset with the same magnitude errors and
redshifts as the Pantheon data but distances calculated using the mock
cosmology. Our pipeline returns unbiased estimates of the cosmological
parameters for all 4 of our mock samples, as shown in Fig. 10. Since the
volume of the mock catalogs is approximately $3.5\times$ the volume of the
data we would expect biases in the parameters at $\approx\frac{1}{2}\sigma$,
and so regard agreement at this level as a successful cross-check of the
model. The pipeline is also unbiased for the combination of the 4 samples (not
shown in the figure to avoid crowding) with $\Omega_{\mathrm{m}}=0.285\pm
0.014$ and $\sigma_{8}=0.800\pm 0.037$ to be compared to $0.286$ and $0.825$.
The best-fit model has $\chi^{2}\simeq 6.6$ for the clustering data (for 52
data points and 26 parameters, 9 of which are at least partially prior
dominated) on a volume $3.5\times$ larger than our survey. Note that our data
do not strongly constrain $\Omega_{m}$ and $\sigma_{8}$ individually, but are
primarily sensitive to a combination as mentioned above. Movement along the
$\Omega_{m}-\sigma_{8}$ degeneracy direction incurs only a small $\chi^{2}$
penalty.

Figure 10: Fits of our $\Lambda$CDM models, with fiducial scale cuts as in
Table 1, to the mocks. The fits are nearly unbiased in all of the parameters,
with the differences being consistent with residual noise in the mock data
arising from finite volume (see text).

###  4.3 Blinding

Our analysis was performed blind to the cosmological parameters. We did not
blind the input catalogs, weights or maps nor did we blind the angular power
spectra or covariances (though the pipeline is straightforward and there are
no user-defined “knobs” aside from binning). The choice of samples, maps and
masks were fixed before any cosmology fits were done. Similarly, our choice of
priors and range of scales to be fit for each model was determined in advance
of fitting the data using measurements from mock catalogs (described above).
On the few occassions when tests of the full pipeline were performed on the
data (primarily to test the code for bugs in I/O), we “blinded” the $dN/dz$
used in the inference by applying an unknown affine transformation to $z$
before we did any fits and removed tick labels from any plots. Once the data
were unblinded we made no changes to these inputs or choices.

We flagged the pz_bin 1 sample as being sensitive to our choice of foreground
subtraction in the CMB $\kappa$ map before unblinding, but did not find any
reasons to remove it at that time. After unblinding it was found to lead to
noticeably lower $\sigma_{8}$ than the other samples, though it is not highly
discrepant statistically given its large error bar (see below). We have chosen
to include this sample in our fiducial results though future work should
investigate this sample more closely.

##  5 Results

We now describe the fits to the galaxy autospectra and the galaxy-convergence
cross-spectra for the 4 samples individually and combined. We begin by showing
fits to the amplitude at fixed power spectrum shape, before discussing fits to
the $\Lambda$CDM model.

###  5.1 Fixed shape

Within the $\Lambda$CDM paradigm, CMB anisotropies largely fix the epoch of
matter radiation equality, the slope of the primordial power spectrum and the
physical densities of baryons, cold dark matter and neutrinos [97]. These are
the parameters that determine the shape of the power spectrum [3]. Between
$z\approx 10^{3}$ and today the linear theory power spectrum shape changes
only modestly (due to neutrinos going non-relativistic) while the amplitude
grows by orders of magnitude. Testing whether the growth predicted by the
$\Lambda$CDM model matches current observations is one of the goals of this
paper.

Figure 11: Posteriors for the power spectrum amplitude ($\sigma_{8}$) in the
“fixed shape” fits. We present the results for two assumptions about the
matter density, $\Omega_{m}=0.31$ (left) and $\Omega_{m}=0.27$ (right), as
described in the text.

Our data, like other lensing surveys, are consistent with the power spectrum
shape determined from the CMB but unfortunately do not provide competitive
(compared to Planck [97]) constraints on this shape. Our spectra best
constrain a combination of the matter density and the clustering amplitude.
One consistency test of the model is therefore to hold the background
cosmology and power spectrum shape fixed while allowing the low $z$ amplitude
determined by lensing to vary and comparing the results to that predicted by
$\Lambda$CDM fit to the CMB data. We do this for two values of the matter
density, differing by about 10%.

For the chain with $\Omega_{m}\simeq 0.31$ we find $\sigma_{8}=0.640\pm
0.066$, $0.723\pm 0.057$, $0.739\pm 0.047$ and $0.755\pm 0.042$ for bins 1-4.
The combined constraint (including the inter-sample covariance) is $0.73\pm
0.03$. Note there is a tendency for the higher redshift bins to prefer higher
$\sigma_{8}$, i.e. the observed level of clustering is not growing as rapidly
as the $\Lambda$CDM model predicts. For the $\Omega_{m}\simeq 0.27$ chain the
corresponding values are $0.675\pm 0.072$, $0.756\pm 0.059$, $0.767\pm 0.048$
and $0.779\pm 0.044$ for bins 1-4 and $0.75\pm 0.03$ for the full sample. The
clustering amplitude is higher due to the lower matter density, and the trend
of increasing best-fit $\sigma_{8}$ with redshift persists. The marginalized
posterior distributions for the 4 bins and the combined sample are shown in
Figure 11 for each of the two values of $\Omega_{m}$. We find that our data
constrain a combination of $\Omega_{m}$ and $\sigma_{8}$ that is intermediate
between the $\Sigma_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.25}$ constrained by CMB
lensing autocorrelations and the $S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$
constrained by recent cosmic shear surveys. To avoid introducing yet another
summary statistic, we shall quote $\Sigma_{8}$ and $S_{8}$ constraints below
when comparing to CMB and cosmic shear results.

The very low $\sigma_{8}$ preferred by the lowest redshift slice prompts
further investigation. This is also the point with the lowest S/N and smallest
dynamic range in scale. To assess goodness-of-fit and check for “volume
effects” we can also look at the profile likelihood. As this lowest redshift
slice has a very small contribution from magnification and is sample variance
dominated, our constraints on the shot-noise and magnitude slope are prior
dominated. Holding those parameters fixed at the centers of the prior and
holding the cosmology fixed we can find the minimum $\chi^{2}$, varying the
bias and counterterms. There are 10 data points and 4 free parameters, for 6
degrees of freedom (or between 5 and 6 if we account for the inflation of the
error bar in the autospectrum at $\ell=50$ due to beyond-Limber
contributions). By varying the values of $\sigma_{8}$ in the ‘fixed’ cosmology
for each run we can compute $\chi^{2}_{\rm min}(\sigma_{8})$ and hence
$\mathcal{L}_{\rm profile}(\sigma_{8})$. Taking $\Omega_{m}\simeq 0.31$ we
find for the Planck-preferred $\sigma_{8}\simeq 0.81$ that $\chi^{2}_{\rm
min}=9.3$ (PTE: 16%). This becomes $7.1$ (31%) for $\sigma_{8}=0.75$, $6.1$
(42%) for $\sigma_{8}=0.70$ and $5.8$ (45%) for $\sigma_{8}=0.65$. While the
$\Delta\chi^{2}$ is significant, all of these values are statistically
acceptable, indicating that for the scales analyzed this bin doesn’t provide
much constraining power. Indeed the lack of dynamic range in scale is partly
responsible for the low value of $\sigma_{8}$ we obtain for this bin, as we
find that $\sigma_{8}$ is quite degenerate with $\alpha_{\times}$ with low
$\sigma_{8}$ corresponding to large $\alpha_{\times}$. With $\alpha_{\times}$
largely unconstrained by the low $\ell$ data, the marginal posterior turns out
to be weighted to low $\sigma_{8}$ by the integration over $\alpha_{\times}$.
Our prior is broad enough that even large corrections from $\alpha_{\times}$
are permitted statistically, indicative of a breakdown of perturbation theory
for the lowest $\sigma_{8}$ values. Unfortunately the data are not
constraining enough to keep $\alpha_{\times}$ in the consistent region without
a strong prior; fortunately the low constraining power of this redshift bin
means our final results are not much affected by this issue.

###  5.2 CDM

Figure 12: Constraints on key $\Lambda$CDM parameters from each of our 4
pz_bin samples, and from the combination. Our constraint is better described
as a measurement of $\Sigma_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.25}$, as in CMB
lensing, than as a constraint on the $S_{8}$ more often constrained by low
redshift cosmic shear experiments.

Figure 13: Constraints on $\sigma_{8}(z)$ from the DESI data and from Planck.
In each case the uncertainty in the $\Lambda$CDM model parameters is
propagated into an uncertainty in the prediction for $\sigma_{8}(z)$ as a
function of redshift. For Planck the dark grey line indicates the mean
prediction while the light grey band covers the $\pm 1\,\sigma$ uncertainty
from the combination Planck primary CMB anisotropies and CMB lensing. For the
DESI data the constraints are from each pz_bin, plotted at the value of
$z_{\rm eff}$ for each sample. The open point at $z\simeq 0.47$ corresponds to
the constraint with $\alpha_{\times}=0$, which eliminates the projection
effect described in the text. The DESI data fall short of the Planck
predictions at all redshifts, with a weak preference for slower growth than
$\Lambda$CDM predicts. The dotted lines show the $\Lambda$CDM prediction for
$\Omega_{m}=0.3$ and $\sigma_{8}=0.72$, 0.74, 0.76 and 0.78 to guide the eye.

In order to asses the extent to which uncertainties in the matter density and
shape of the power spectrum affect our amplitude constraints, we also fit our
data directly to the predictions of the $\Lambda$CDM model. For these fits we
vary the matter density as well as the power spectrum amplitude, using the
known angular scale of the CMB ($\theta_{\star}$) to fix $h$ as a function of
$\Omega_{m}$. Although we cover a substantial range in redshift, so we have
some contour rotation to break degeneracies, our data still do not strongly
constrain movement along the $\Omega_{m}-\sigma_{8}$ degeneracy (see appendix
B). Fortunately, several other data sets do provide tight constraints on this
dimension within the $\Lambda$CDM framework. We combine our data with the
Pantheon SNe data sample and a combination of BAO data to better constrain
$\Omega_{m}$. Figure 12 shows the constraints on the $\Lambda$CDM parameters
from each of the 4 samples individually, as well as the combined constraint
(including the intersample covariance).

Our data can be most conveniently summarized as a constraint on
$\Sigma_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.25}$, with bins 1-4 giving $0.647\pm
0.068$, $0.729\pm 0.057$, $0.747\pm 0.048$ and $0.761\pm 0.042$ or as a
constraint on $S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$ with bins 1-4 giving
$0.649\pm 0.068$, $0.731\pm 0.057$, $0.747\pm 0.048$ and $0.762\pm 0.043$
respectively. Combining all 4 samples we find $\Sigma_{8}=0.727\pm 0.030$ or
$S_{8}=0.725\pm 0.030$. The best-fit model has $\chi^{2}\simeq 23.5$ for the
clustering data (for 52 data points and 26 parameters, 9 of which are at least
partially prior dominated). While not strongly inconsistent with the other
three, the first bin is clearly lower than the other three which are quite
compatible with each other. As described above this is partly a “volume
effect”. The lowest bin, having the smallest dynamic range in scale, cannot
strongly constrain $\alpha_{\times}$. This parameter is in turn correlated
with $\sigma_{8}$. Setting $\alpha_{\times}=0$ gives a reasonable fit to the
data with $\Sigma_{8}=0.76\pm 0.04$ indicating that the low value for the full
model is partly a result of parameter volume effects in the posterior. If we
omit this first redshift bin, our constraint becomes $\Sigma_{8}=0.742\pm
0.032$.

We noted above the tendency of the best-fit $\sigma_{8}$ to increase with
increasing sample redshift. This is responsible for the shift to lower
$\Omega_{m}$ of the combined samples constraint compared to each of the
individual samples (Fig. 12). Another view of this is shown in Figure 13 which
plots constraints on $\sigma_{8}(z)$, marginalized over the other parameters,
as a function of redshift. The grey lines show the predictions of the
$\Lambda$CDM model conditioned on the Planck primary CMB plus lensing data
(with the dark grey line being the mean and the grey band encompassing the
$\pm 1\,\sigma$ uncertainty). All of the points lie below the line, with the
lowest redshift points deviating the most strongly. We also show (dotted
lines) the $\Lambda$CDM prediction for $\Omega_{m}=0.3$ and $\sigma_{8}=0.72$,
0.74, 0.76 and 0.78 for reference. While of relatively low statistical
significance, the data suggest structure that is not growing as rapidly as
$\Lambda$CDM predicts with the largest shortfall appearing below $z\approx
0.6$.

Finally, while we are unable to use Anzu for the lowest redshift slices (where
it would be the most valuable) due to the likelihood having non-trivial
support outside of the range of validity of the emulator, we can compare the
results from Anzu to our LPT model for the higher redshift points. Fitting to
the same $\ell_{\rm max}$ the agreement on parameters is excellent. For
example, for pz_bin 4 the constraints differ by less than one percent between
the two models, significantly smaller than our statistical errors. Given the
kernels in Fig. 7 this is reassuring. If we increase the fitting range from
$\ell_{\rm max}=400$ to $\ell_{\rm max}=600$ the constraint on $\sigma_{8}$
also changes by less than 1%, and the error shrinks negligibly. This is not
unexpected, the Planck lensing map is signal dominated only at very low $\ell$
and the galaxies become shot noise limited for $\ell>425$. We would expect
more gain from Anzu in the lower redshift bins, where the non-linear scale is
shifted to much smaller $\ell$ (Fig. 7). Unfortunately, for these bins some of
the support for the likelihood lies outside of the grid of simulations upon
which the Anzu emulator was trained, so the emulator error is not well
controlled. We aim to increase the coverage of Anzu with more simulations in
the future.

###  5.3 Comparison

Figure 14: Posteriors for two key $\Lambda$CDM parameters constrained by our
data compared to a variety of recent experiments (as shown in the legend and
described in the text). Though the models fitted, priors assumed and
methodologies differ the comparison paints a picture of the level of agreement
of different low redshift probes of the growth of large-scale structure within
the context of $\Lambda$CDM and their comparison to primary CMB anisotropies
(in this case, Planck). See text for further discussion.

Figure 14 shows that our result gives a “low” clustering amplitude, compared
to Planck. In terms of $\Sigma_{8}\equiv\sigma_{8}(\Omega_{m}/0.3)^{0.25}$ we
find $\Sigma_{8}=0.727\pm 0.030$ for our full sample. Including CMB lensing,
Planck finds $\Sigma_{8}=0.821\pm 0.008$ [97]. This corresponds to a
$3\,\sigma$ “tension” with Planck when adding the statistical uncertainties in
quadrature. We have not included the Planck lensing auto-correlation in our
fits. Once BAO and weak priors are included that auto-correlation (P-lens in
figure 14) gives $\Sigma_{8}=0.815\pm 0.016$ [25]. Our result is $2.5\,\sigma$
lower than this, with approximately twice the uncertainty. If we had included
the $\kappa$ autospectrum (more closely following the “$3\times 2$pt” strategy
often employed by cosmic shear surveys) our result would have been drawn to
higher $\sigma_{8}$ by the statistically more powerful $\kappa\kappa$
constraint101010The $\kappa$ auto-spectrum has comparable signal-to-noise
ratio to our cross-correlations but obtains better $\sigma_{8}$ constraints
since it does not need to marginalize over bias parameters, e.g. ref. [55]..

Figure 15: A comparison of some recent constraints on the clustering amplitude
at low redshift. We show the posterior distribution of
$S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$, normalized to peak at unity for
readability, for a selection of surveys described in the main text. For the
DESI data we show the result using the full sample as a solid line, and
omitting the lowest redshift bin as the dashed line.

Our results reinforce the tendency for (some) lower redshift measurements to
have lower matter density or less clustering than inferred by Planck within
the context of $\Lambda$CDM. This is typically summarized in terms of
$S_{8}=\sigma_{8}(\Omega_{m}/0.3)^{0.5}$. In terms of this statistic we find
$S_{8}=0.73\pm 0.03$ for the combined sample, lower than Planck’s
$S_{8}=0.832\pm 0.012$. If we omit the first bin we find $S_{8}=0.74\pm 0.03$.
Earlier work using similar samples drawn from previous releases of the same
imaging data [29, 30, 31] also found lower $S_{8}$. Substituting unWISE-
selected galaxies for the optical samples also found lower $S_{8}$ [108]. We
are in good agreement with the unWISE analysis ($S_{8}=0.784\pm 0.015$) that
used the same CMB lensing maps but a different galaxy sample, extending to
higher redshift but with a more uncertain $dN/dz$ and a different theoretical
model. In general we also find good agreement with recent galaxy lensing
measurements, which tend to lie lower than the Planck value (Fig. 15). In
particular, we find agreement with the DES Y3 “$3\times 2$” analysis
($S_{8}=0.776\pm 0.017$ [10]), which includes both the auto-correlation of
shear111111Adding the Planck lensing auto-correlation to be more consistent
with the DES methodology would considerably tighten our contours and move our
result upwards, however it would also introduce sensitivity to the higher-
redshift matter fluctuations. We have decided to omit
$C_{\ell}^{\kappa\kappa}$ in order isolate the lower redshifts. and tracer
galaxies, as well as their cross-correlation. The shear-only analysis [11, 12]
finds $S_{8}=0.772\pm 0.017$, again in good agreement. We are also consistent
with the lensing and galaxy clustering results from the KiDS+BOSS+2dFlens
surveys, with $S_{8}=0.766^{+0.020}_{-0.014}$ [8], and the reanalysis of KiDS
by ref. [13] who find $S_{8}=0.754\pm 0.028$. Similarly “low” values of
$S_{8}$ have also been found by the first year cosmic shear analysis of HSC
[9], and by CFHTLenS [7].

Hints of a “low” value of $\Omega_{m}$ and $S_{8}$ in the low-redshift
Universe do not rely entirely on lensing. For example a recent analysis of the
“full-shape” redshift-space BOSS galaxy power spectrum, BAO and bispectrum
[112] reports $\Omega_{m}=0.338^{+0.016}_{-0.017}$ and
$\sigma_{8}=0.692^{+0.035}_{-0.041}$, roughly $3.5\sigma$ away from Planck,
though this analysis frees the spectral tilt, which is constrained to be
$n_{s}=0.870^{+0.067}_{-0.064}$; when $n_{s}$ is fixed to that measured by
Planck, the corresponding analysis finds $\Omega_{m}=0.320^{+0.010}_{-0.010}$
and $\sigma_{8}=0.722^{+0.032}_{-0.036}$, reducing the tension to less than
$3\sigma$. The analysis of ref. [112] utilized window-free estimators of
galaxy spectra, avoiding a normalization issue that affected some earlier
analyses of the BOSS data resulting in artificially low $\sigma_{8}$
constraints. Another recent full-shape analysis of BOSS by ref. [18] finds
results lower than but consistent with both the DES Y3 and Planck results
(with $\Omega_{m}=0.303\pm 0.008$ and $\sigma_{8}=0.733\pm 0.047$), using the
updated BOSS window functions of ref. [113]. Their fits imply
$\Sigma_{8}=0.74\pm 0.05$, which is in good agreement with our result. This
constraint is shown in Fig. 14, where we can see that it has a different
degeneracy direction than the lensing-based measurements. Since the model of
ref. [18] is the same as the one employed in our analysis, in future we should
be able to replace this BOSS constraint with one from the spectroscopic sample
of LRGs produced by DESI for which a joint analysis with our lensing
measurement can be performed. This will allow us to reduce our dependence upon
external datasets while retaining tight constraints. In addition, recent
constraints [20] from the BOSS correlation function, plus a range of external
data, give $\Omega_{m}=0.306\pm 0.011$ and $\sigma_{8}=0.766\pm 0.055$; again
in good agreement with our results (and also in agreement with Planck).
Finally an analysis of BOSS using an N-body based emulator [21] also returns
constraints very close to our result. By contrast recent analyses of RSD by
the eBOSS collaboration find $\sigma_{8}=0.84\pm 0.06$ [22] and
$\sigma_{8}=0.814\pm 0.044$ [114], considerably higher than our result though
still with large enough error bars that the tension with other results is not
high.

In terms of other inferences drawn from galaxies at comparable redshifts,
earlier analyses of BOSS galaxies at $z\simeq 0.57$ in combination with Planck
lensing [26, 27] have also found lower clustering amplitude than expected from
Planck (i.e. $C_{\ell}^{\kappa g}$ is lower on large scales or small $\ell$),
though also with large errors so the statistical significance is relatively
weak. Similarly, in an analysis of Planck CMB lensing with BOSS galaxies, ref.
[28] found consistency with Planck, albeit with 10% lower amplitude. By
contrast the earlier Planck analysis [23] and the combination of BOSS galaxies
at $z\simeq 0.57$ and the ACT lensing maps, over a much smaller area, finds a
consistency with the Planck-preferred normalization [32]. In a recent analysis
of a wide range of data sets, ref. [33] find evidence for a deficit in the
growth of large scale structure in the redshift regime $0.2<z<0.6$, which
encompasses our lowest redshift bin. The amplitude of the deficit is smaller
than, though consistent with, the shortfall we see. However much of that
shortfall arises from cosmic shear data rather than CMB lensing. When
analyzing their full redshift range and using CMB lensing but not cosmic shear
they find agreement with $\Lambda$CDM growth with $S_{8}=0.825\pm 0.023$,
which is higher than our result. The disagreement is puzzling since the
lensing map is the same and much of their constraint comes from a similar
redshift range though there are differences in the imaging data used, sample
construction and galaxy type, stellar contamination level, $dN/dz$
determination, masks, modeling and nuisance parameters. It would be
interesting to investigate these issues further.

##  6 Conclusions

We have presented constraints on cosmological models from a cross-correlation
of Planck CMB lensing [25] and LRGs from the DESI imaging surveys [34]. We use
photometric redshifts to divide the LRGs into 4 samples with mean redshifts of
0.47, 0.63, 0.79 and 0.92. Our LRG data cover 44% of the sky ($\approx$18,000
sq.deg.; Fig. 1) and the galaxy auto-spectrum is sample variance limited for
all samples on all of the scales that we model. We report a strong cross-
correlation signal in multiple redshift bins (Fig. 4), allowing us to
constrain a degenerate combination of the matter density and the amplitude of
the matter power spectrum.

Our cross-correlation signal comes primarily from large scales (Fig. 7) and we
model it using a combination of perturbation theory (§3.2.1) and a Lagrangian
bias expansion based on N-body simulations (§3.2.2). We show that these models
produce unbiased constraints on mock catalogs for the analysis choices we
adopt (§4; Fig. 10).

Our data alone do not strongly constrain all of the $\Lambda$CDM parameters,
so we have held some of them fixed and incorporated other data sets in order
to break some degeneracies. In particular we hold the spectral index
($n_{s}$), optical depth ($\tau$), baryon density ($\omega_{b}$) and angular
size of the sound horizon ($\theta_{\star}$) fixed as shown in Table 2. Our
results are robust to small changes in these values. We vary the amplitude of
the spectrum ($\ln[10^{10}A_{s}]$) and the cold dark matter density, plus the
bias parameters, counter terms and stochastic terms in the model. Our data are
consistent with a wide range of $\Omega_{m}$, so to improve convergence and
focus attention on the physically interesting region we combine with the
Pantheon supernova data and a collection of BAO constraints (but see appendix
B). Our results are summarized in Figs 12 and 13 and compared to a selection
of other measurements in Fig. 14. We find results that are lower than the
predictions of the $\Lambda$CDM model constrained by the primary CMB
anisotropy data as measured by Planck. Our results also indicate that
structure grows with time more slowly than the model predictions, though this
is driven primarily by our lowest redshift bin ($z\approx 0.47$) and is of
modest significance.

It is of concern that our most discrepant bin is also the one with the lowest
signal-to-noise ratio, the smallest dynamic range in scale and the one most
sensitive to changes in the Planck lensing map used. However, despite these
factors we have been unable to find a systematic that would account for the
lower $\sigma_{8}$ that we measure. At present we believe this is most likely
to be a statistical fluctuation. Indeed the best-fitting model with
$\Omega_{m}\simeq 0.31$ and $\sigma_{8}\simeq 0.81$ (consistent with Planck)
provides a statistically acceptable fit to the lowest redshift slice, though
the best fit is for lower $\sigma_{8}$. Part of the downward shift in
$\sigma_{8}$ for that bin is due to parameter volume effects (mostly
$\alpha_{\times}$) in the marginal posterior due to the limited dynamic range
of the data in scale, with the sense that lower $\sigma_{8}$ correspond to
regions where the perturbative model is least reliable (i.e. large values of
the counterterm). Further investigation awaits improved modeling.

Our results should have very different systematic uncertainties than other
probes. The galaxy sample and its dependence on observational conditions is
well known, and will become increasingly well characterized as the DESI survey
proceeds. The redshift of our source screen (the CMB) is very well known [3]
and the redshift distribution of our lenses (which are all well separated from
the source) has been spectroscopically determined [35]. As further redshifts
are gathered, we will be able to further investigate the dependence of $dN/dz$
on sample selection, imaging properties and photometry. Spectroscopic
redshifts will also allow us to infer the 3D clustering of the imaging sample
and, in combination with imaging-spectro cross-correlations, enable us to
bound any non-cosmological contribution to $C_{\ell}^{gg}$ (which could bias
our $\sigma_{8}$ results low). The availability of accurate photometric
redshifts means we can select galaxies to lie in narrow redshift bins,
obviating the need to model evolving bias. We are not affected by intrinsic
alignments, blending or other systematics that need to be carefully handled in
cosmic shear surveys. Our signal is dominated by large scales where linear or
quasi-linear theory holds and where the impact of baryonic effects is minimal.
As such we provide a valuable check of previous measurements.

Our results are sufficiently tight to begin to constrain the low-redshift
amplitude of structure, but fundamentally limited by the CMB lensing maps that
we use and by our modeling. Luckily, we anticipate rapid progress in both
directions in the very near future. Combining more sensitive, ground-based
measurements of the CMB with the existing Planck data in maps optimized for
cross-correlation should improve the signal to noise on $C_{\ell}^{\kappa g}$
for all samples. In combination with models capable of working to higher
$\ell$ at low $z$ this will improve the constraints on the power spectrum
amplitude, $\sigma_{8}$. The well-characterized, wide-area and deep LRG sample
that we have used in this analysis, combined with the Lagrangian bias
framework, should be ideal for such future work. Looking forward, DESI will
provide spectroscopic redshifts for most of the galaxies used in this work,
improving control of systematic errors and allowing us to combine 3D
clustering (including RSD and BAO) with lensing for the same sample. The
distance information inherent in the BAO signal will allow us to measure
$\Omega_{m}$ and $h$ internally, while the RSD signal will give us a probe of
the amplitude of large-scale structure from the Newtonian potential which can
be compared with the Weyl potential measurement from CMB or galaxy-galaxy
lensing.

## Acknowledgments

We would like to thank David Alonso for help with NaMaster. We also thank
Chris Blake and Alex Krolewski for helpful discussions during the preparation
of this manuscript. M.W. is supported by the DOE and the NSF. R.Z. is
supported by the DOE under Contract No. DE-AC02-05CH11231. J.D. is supported
by the Lawrence Berkeley National Laboratory Chamberlain Fellowship. S.F. is
supported by the Physics Division of Lawrence Berkeley National Laboratory. We
acknowledge the use of NaMaster [41], Cobaya [81, 80], GetDist [85], CAMB [61]
and velocileptors [64] and thank their authors for making these products
public. This research has made use of NASA’s Astrophysics Data System and the
arXiv preprint server. This research is supported by the Director, Office of
Science, Office of High Energy Physics of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231, and by the National Energy Research Scientific
Computing Center, a DOE Office of Science User Facility under the same
contract; additional support for DESI is provided by the U.S. National Science
Foundation, Division of Astronomical Sciences under Contract No. AST-0950945
to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the
Science and Technologies Facilities Council of the United Kingdom; the Gordon
and Betty Moore Foundation; the Heising-Simons Foundation; the French
Alternative Energies and Atomic Energy Commission (CEA); the National Council
of Science and Technology of Mexico; the Ministry of Economy of Spain, and by
the DESI Member Institutions.

The authors are honored to be permitted to conduct scientific research on
Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the
Tohono O’odham Nation.

The data used in this publication are available at
https://zenodo.org/record/5834378#.YeBonljMJUQ

##  Appendix A Lagrangian perturbation theory formulae

For the main results of the paper we have used a variant of Lagrangian
perturbation theory, as implemented in the velocileptors code. In the
Lagrangian basis galaxy clustering is described by an initial protohalo
density, $F(\textbf{q})$, dynamically advected from their initial positions,
q, to their observed positions, $\textbf{x}=\textbf{q}+\Psi(\textbf{q},t)$,
where $\Psi(\textbf{q},t)$ is the displacement, or trajectory, of a galaxy. By
number conservation this implies

$1+\delta_{g}(\textbf{x},t)=\int
d^{3}\textbf{q}\;F(\textbf{q})\,\delta_{D}(\textbf{x}-\textbf{q}-\Psi(\textbf{q},t)).$

The initial density $F(\textbf{q})$ can be written as a bias expansion in the
initial conditions; in this paper we adopt the restricted form

$F(\textbf{q})=1+b_{1}\delta(\textbf{q})+\frac{b_{2}}{2}(\delta(\textbf{q})-\langle\delta^{2}\rangle).$

The matter overdensity can be expressed in this formalism as an unbiased
tracer, i.e. $F=1$. We refer the reader to ref. [64] for details on higher-
order bias contributions since they will not be used in this paper.

The full expression for the galaxy power spectrum in real space is given e.g.
in §4.2.1 of ref. [64]. For completeness, however, we include in this appendix
a table of the relevant contributions to the galaxy autospectrum and galaxy-
matter cross spectra in Equation 3.8. These contributions, for
$(i,j)=\\{1,b_{1},b_{2}\\}$, can each be written as Hankel transforms

$P_{ij}(k)=4\pi\sum_{n=0}^{\infty}\int q^{2}dq\ e^{-k^{2}(X^{\rm lin}+Y^{\rm
lin})/2}\left(\frac{kY}{q}\right)^{n}f^{(n)}_{ij}(q)\ j_{n}(kq),$ (A.1)

where $j_{n}$ are spherical Bessel functions and $X$ and $Y$ are the
components of the large-scale displacement two-point function (defined below).
The functions $f^{(n)}_{ij}(q)$ are given in Table 3 and the angular kernels
$K_{n}$ in these expressions are given by

$K^{(n)}_{1}=\frac{q\Theta_{n>0}}{kY^{\rm lin}}\quad,\quad
K^{(n)}_{2}=1-\frac{2n}{k^{2}Y^{\rm lin}}\quad,\quad
K^{(3)}_{3}=\left[1-\frac{2(n-1)}{k^{2}Y^{\rm
lin}}\right]\frac{q\Theta_{n>0}}{kY^{\rm lin}}.$ (A.2)

where $\Theta_{n>0}$ is zero for $n=0$ and one for $n>0$. The $X$, $Y$,
$U,\ldots$ are correlators of Lagrangian operators. The linear contributions
are given by

$\displaystyle X^{\rm lin}=\int\frac{dk}{2\pi^{2}}\ P_{\rm
lin}(k)\Big{(}\frac{2}{3}-\frac{2}{3}[j_{0}(kq)+j_{2}(kq)]\Big{)}$
$\displaystyle Y^{\rm lin}=\int\frac{dk}{2\pi^{2}}\ P_{\rm
lin}(k)\Big{(}2j_{2}(kq)\Big{)}$ $\displaystyle U^{\rm
lin}=\int\frac{dk}{2\pi^{2}}\ k\Big{(}-P_{\rm lin}j_{1}(kq)\Big{)}$
$\displaystyle\xi^{\rm lin}=\int\frac{dk}{2\pi^{2}}\ k^{2}P_{\rm
lin}(k)j_{0}(kq).$

The one-loop contributions to the correlators are given by:

$\displaystyle X^{\rm 1-loop}=2X^{(13)}+X^{(22)},\quad Y^{\rm
1-loop}=2Y^{(13)}+Y^{(22)}$ $\displaystyle X^{(22)}=\int\frac{dk}{2\pi^{2}}\
\frac{9}{98}Q_{1}(k)\Big{(}\frac{2}{3}-\frac{2}{3}(j_{0}(kq)+j_{2}(kq))\Big{)}$
$\displaystyle Y^{(22)}=\int\frac{dk}{2\pi^{2}}\ \frac{9}{98}Q_{1}(k)\
2j_{2}(kq)$ $\displaystyle X^{(13)}=\int\frac{dk}{2\pi^{2}}\
\frac{5}{21}R_{1}(k)\Big{(}\frac{2}{3}-\frac{2}{3}(j_{0}(kq)+j_{2}(kq))\Big{)}$
$\displaystyle Y^{(13)}=\int\frac{dk}{2\pi^{2}}\ \frac{5}{21}Q_{1}(k)\
2j_{2}(kq)$ $\displaystyle V=3(2V_{1}+V_{3})$ $\displaystyle
V_{1}=\int\frac{dk}{2\pi^{2}}\ k^{-1}\
\Big{(}-\frac{3}{7}R_{1}(k)\Big{)}j_{1}(kq)+S(q)$ $\displaystyle
V_{3}=\int\frac{dk}{2\pi^{2}}\ k^{-1}\
\Big{(}-\frac{3}{7}Q_{1}(k)\Big{)}j_{1}(kq)+S(q)$ $\displaystyle
S=\int\frac{dk}{2\pi^{2}}\ k^{-1}\
\frac{3}{7}\Big{(}2R_{1}+4R_{2}+Q_{1}+2Q_{2}\Big{)}\frac{j_{2}(kq)}{kq}$
$\displaystyle T=\int\frac{dk}{2\pi^{2}}\ k^{-1}\
\left(-\frac{9}{7}\right)\Big{(}2R_{1}+4R_{2}+Q_{1}+2Q_{2}\Big{)}j_{3}(kq)$
$\displaystyle U^{(3)}=\int\frac{dk}{2\pi^{2}}\
k\Big{(}-\frac{5}{21}R_{1}j_{1}(kq)\Big{)}$ $\displaystyle
X^{10}=\int\frac{dk}{2\pi^{2}}\
\frac{1}{7}\Big{(}2(R_{1}-R_{2})-(4R_{2}+2Q_{5})j_{0}(kq)-(3R_{1}+4R_{2}+2Q_{5})j_{2}(kq)\Big{)}$
$\displaystyle Y^{10}=\int\frac{dk}{2\pi^{2}}\
\frac{3}{7}\Big{(}(3R_{1}+4R_{2}+2Q_{5})j_{2}(kq)\Big{)}$ $\displaystyle
U^{11}=\int\frac{dk}{2\pi^{2}}\
k\Big{(}-\frac{6}{7}(R_{1}+R_{2})j_{1}(kq)\Big{)}$ $\displaystyle
U^{20}=\int\frac{dk}{2\pi^{2}}\ k\Big{(}-\frac{3}{7}Q_{8}j_{1}(kq)\Big{)},$

where the functions $R_{n}$ and $Q_{n}$ [67] are quadratic in (integrals of)
$P_{\rm lin}$ whose definitions can be found in Eqs. (D.3-D.5) in Appendix D
of ref. [64].

$(i,j)$ | $f^{(n)}_{ij}(q)$
---|---
$(1,1)$ (Zeldovich) | $1$
$(1,1)$ (1-loop) | $-(k^{2}/2)(X^{\rm 1-loop}+Y^{\rm 1-loop}K^{(n)}_{2})+(k^{3}/6)(VK^{(n)}_{1}+TK^{(n)}_{3})$
$(1,b_{1})$ | $-2k(U^{\rm lin}+U^{3})K^{(n)}_{1}-k^{2}(X^{10}+Y^{10}K^{(n)}_{2})$
$(b_{1},b_{1})$ | $\xi^{\rm lin}-k^{2}U_{\rm lin}^{2}K^{(n)}_{2}-kU^{11}K^{(n)}_{1}$
$(1,b_{2})$ | $-k^{2}U_{\rm lin}^{2}K^{(n)}_{2}-kU^{20}K^{(n)}_{1}$
$(b_{1},b_{2})$ | $-2k\xi^{\rm lin}U^{\rm lin}K^{(n)}_{1}$
$(b_{2},b_{2})$ | $\frac{1}{2}\xi_{\rm lin}^{2}$
Table 3: Contributions to galaxy-galaxy and galaxy-matter power spectra in
LPT.

##  Appendix B Constraints without BAO and SNe data

In the main text (§5) we included constraints on the distance scale from BAO
and SNe data (§3.3), which together with the angular scale of the CMB serve to
tightly constrain $\Omega_{m}\approx 0.3$. It is also useful to look at the
constraints that come from the lensing and clustering data alone, which we
present here. In order to improve convergence of the chains we tighten the
prior on $\alpha_{\times}$ for the lowest redshift bin sample, which does not
have enough dynamic range to constrain $\alpha_{\times}$ on its own. Without
this tighter prior the chains explore regions where perturbation theory is
invalid (i.e. the contribution from $\alpha_{\times}k^{2}P(k)$ is a
significant contributor to the total power). Tightening the prior to
$\mathcal{N}(0,5)$ for this one parameter keeps the chains in the physical
region without leading to bad fits for that sample (modest changes in this
prior do not materially change our final constraints since this sample has
relatively low statistical weight). The marginalized posterior distribution
for $\Omega_{m}$ and $\sigma_{8}$ is shown in Fig. 16 both with and without
the additional data. The addition of the BAO and SNe data disfavor the
low-$\Omega_{m}$ region that is only weakly constrained by the DESI plus
Planck data, but does not appreciably narrow the constraint perpendicular to
the $\Omega_{m}-\sigma_{8}$ degeneracy.

Figure 16: Marginalized posterior distributions for the matter density,
$\Omega_{m}$, and power spectrum amplitude, $\sigma_{8}$, for the chains with
and without the BAO and SNe data that constrain the distance-redshift
relation. The dotted black line shows $S_{8}=0.73$.

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