11institutetext:  Université Paris-Saclay, CNRS, Institut d’astrophysique
spatiale, 91405, Orsay, France.
11email<EMAIL_ADDRESS>22institutetext: Centro de Estudios de
Física del Cosmos de Aragón (CEFCA), Unidad Asociada al CSIC, Plaza San Juan,
1, planta 2, E-44001, Teruel, Spain 33institutetext: Donostia International
Physics Centre (DIPC), Paseo Manuel de Lardizabal 4, E-20018 Donostia-San
Sebastian, Spain 44institutetext: IKERBASQUE, Basque Foundation for Science,
E-48013, Bilbao, Spain

# High resolution tomography for galaxy spectroscopic surveys with Angular
Redshift Fluctuations

L. Legrand High resolution tomography for galaxy spectroscopic surveys with
Angular Redshift FluctuationsHigh resolution tomography for galaxy
spectroscopic surveys with Angular Redshift Fluctuations C. Hernández-
Monteagudo High resolution tomography for galaxy spectroscopic surveys with
Angular Redshift FluctuationsHigh resolution tomography for galaxy
spectroscopic surveys with Angular Redshift Fluctuations M. Douspis High
resolution tomography for galaxy spectroscopic surveys with Angular Redshift
FluctuationsHigh resolution tomography for galaxy spectroscopic surveys with
Angular Redshift Fluctuations N. Aghanim High resolution tomography for galaxy
spectroscopic surveys with Angular Redshift FluctuationsHigh resolution
tomography for galaxy spectroscopic surveys with Angular Redshift Fluctuations
Raúl E. Angulo High resolution tomography for galaxy spectroscopic surveys
with Angular Redshift FluctuationsHigh resolution tomography for galaxy
spectroscopic surveys with Angular Redshift FluctuationsHigh resolution
tomography for galaxy spectroscopic surveys with Angular Redshift
FluctuationsHigh resolution tomography for galaxy spectroscopic surveys with
Angular Redshift Fluctuations

In the context of next generation spectroscopic galaxy surveys, new
observables of the distribution of matter are currently being developed. Among
these we investigate the angular redshift fluctuations (ARF), which probe the
information contained in the projected redshift distribution of galaxies.
Relying on the Fisher formalism, we show how ARF will provide complementary
cosmological information compared to traditional angular galaxy clustering. We
test both the standard $\mathrm{\Lambda CDM}$ model and the $\mathrm{wCDM}$
extension. We find that the cosmological and galaxy bias parameters express
different degeneracies when inferred from ARF or from angular galaxy
clustering. As such, combining both observables breaks these degeneracies and
greatly decreases the marginalised uncertainties, by a factor of at least two
on most parameters for the $\mathrm{\Lambda CDM}$ and $\mathrm{wCDM}$ model.
We find that the ARF combined with angular galaxy clustering are a great probe
of dark energy by increasing the figure of merit of the $w_{0}$-$w_{\rm a}$
parameter set by a factor of more than 10 compared to angular galaxy
clustering alone. Finally we compare ARF to the CMB lensing constraints on the
galaxy bias parameters. We show that a joint analysis of ARF and angular
galaxy clustering improves constraints by $\sim 40\%$ on galaxy bias compared
to a joint analysis of angular galaxy clustering and CMB lensing.

###### Key Words.:

Cosmology: large-scale structure of Universe - observations - cosmological
parameters - dark energy

##  1 Introduction

In the coming years, large scale optical and infra-red (IR) surveys will map
with unprecedented accuracy our Universe from the present epoch up to when it
was roughly one tenth of its current age. A significant part of these surveys
will be spectroscopic, e.g., DESI (DESI Collaboration et al., 2016), 4MOST (de
Jong & 4MOST Consortium, 2015), WEAVE (Bonifacio et al., 2016), or NISP aboard
Euclid (Laureijs et al., 2011) and will provide us with spectra for large
samples of sources. Such spectra will not only enable deep insight on the
physics of those objects, but will also yield accurate estimates of their
redshift and thus of their distance to the observer. From the cosmological
point of view, this will enable a precise (statistical) characterization of
the (apparent) spatial distribution of those luminous tracers (via two- or
three-point statistics), and this itself should shed precious light on open
topics such as, e.g., the nature of dark energy, the possible interplay of
dark energy and dark matter, the mass hierarchy of neutrinos, or possible
deviations of gravity from General Relativity, to quote a few.

At the same time, a different family of surveys will scan the sky at greater
depths with of optical filters, and with exquisite image quality. These
photometric experiments build high quality, very large catalogues of sources,
with however relatively rough redshift estimations given their moderate number
of filters/colours. While mining the faint Universe, this type of surveys will
be particularly sensitive, from a cosmological perspective, to the angular
clustering of luminous matter, the cosmological aspects of gravitational
lensing throughout cosmic epochs, the satellite population in halos, and the
formation and evolution of the population of galaxy clusters. In this context,
the Dark Energy Survey (DES, Abbott et al., 2018) is currently providing
state-of-the-art cosmological constraints in the late universe, and these
should be further complemented by the Vera Rubin Observatory (LSST, Ivezić et
al., 2019), which, at the same time, will also explore the variability of the
night sky in a regime of depth and time domain that so far remains practically
unexplored.

An intermediate third class of experiments also exists. These are the spectro-
photometric surveys that conduct standard photometry in a relatively large set
(from $\sim 10$ up to $\sim 60$) of narrow-band optical filters. This strategy
combines the indiscriminate character of the photometric surveys with high
precision redshift estimates ($\Delta z/(1+z)\sim 10^{-3}$–$10^{-2}$) for a
large fraction ($>20$–$30\leavevmode\nobreak\ \%$) of the detected sources.
Given its multi-color character, these surveys are able to provide pseudo-
spectrum / photo-spectrum in each pixel of the surveyed area. The pioneer
example of COMBO-17 has been or is being followed by other efforts such as
COSMOS (Scoville et al., 2007), ALHAMBRA (Moles et al., 2008), SHARDS (Pérez-
González et al., 2013), PAU (Martí et al., 2014), J-PAS (Benitez et al.,
2014), SPHEREx (Doré et al., 2014), and J-PLUS (Cenarro et al., 2019).

In this work, we forecast the power of cosmological analyses in spectroscopic
and spectro-photometric surveys. In this type of surveys, it is customary to
convert redshift estimates into radial distances under the assumption of a
given fiducial cosmological model. Angular and redshift coordinates are thus
converted into the three-dimensional space, where standard 3D clustering
analysis techniques are applied.

In our case however, we choose to follow a different strategy. We focus on a
new cosmological observable, namely the angular redshift fluctuations (ARF,
Hernández-Monteagudo et al., 2019, hereafter HMCMA). Being a 2D observable,
the ARF field can easily be cross-correlated with other 2D observables, such
as the 2D galaxy density filed and the CMB lensing fiels. ARF present other
interesting features, such as being correlated to the cosmic, radial, peculiar
velocity fields (HMCMA, Chaves-Montero et al., 2019), or being particularly
insensitive to additive systematics that remain constant under the redshift
shell subject to analysis.

In this paper we apply the Fisher formalism on the angular galaxy clustering,
the ARF and the CMB lensing convergence observables, and explore their
sensitivity to cosmology in two different observational set ups, mimicking
those expected for the DESI and Euclid surveys. We consider the CMB lensing
convergence field among our observables, since it constitutes an intrinsically
different probe whose dependence on the parameters defining the galaxy sample
is different from that of angular galaxy clustering and ARF. Our scope is to
assess if the ARF field can provide complementary information on the galaxy
density field and on the CMB lensing field.

The paper is organized as follows. We introduce the spectroscopic galaxy
surveys and CMB experiments that we use in our analysis in Sect. 2. In Sect.
3, we present the angular galaxy clustering, the ARF, and the CMB lensing
convergence field. In Sect. 4, we compute the foreseen signal to noise ratios
of these combination of probes, while also introducing the covariance among
those observables. In Sect. 5, we present the predicted constraints on
cosmological parameters in the fiducial $\Lambda$CDM scenario. Finally we
discuss our findings in Sect. 6 and conclude in Sect. 7.

Throughout this paper, we use the Planck 2018 cosmology as our fiducial
cosmology. We take the values given in Table 2, in the column 6 (best-fit with
BAO), of Planck Collaboration et al. (2018a). We use the following naming
conventions: observable refers to a spherical 2D field built on measured
quantities, such as counts, redshifts, or deflection angles, while probe
refers to the combination of one or two observables into a summary statistics.
In practice, our probes will be the two-point angular power spectra
$C_{\ell}$. The redshift due to the Hubble expansion is denoted by $z$, while
$z_{\rm obs}$ is the measured redshift (which includes redshift distortions
induced by radial peculiar velocities);$\Omega_{\rm m,0}$ is the density of
matter at $z=0$ in units of the critical density and $H_{0}$ is the Hubble
constant; $r(z)=\int{\rm d}z\;c\,/\,H(z)$ is the line of sight comoving
distance, and ${\rm d}V_{\Omega}={\rm d}V/{\rm d}\Omega=r^{2}\,{\rm
d}r=r^{2}(z)\,c/H(z)\,{\rm d}z$ is the comoving volume element per solid
angle, with ${\rm d}\Omega$ a differential solid angle element. Vectors are in
bold font and a hat denotes a unit vector.

##  2 Surveys

Survey | Euclid | DESI
---|---|---
Survey area | 15 000 deg2 | 14 000 deg2
Redshift estimation | Slitless spectroscopy | Optical fiber spectroscopy
Targets |  $H_{\alpha}$ emission line | [OII] doublet
Redshift range | $0.9<z<1.8$ | $0.6<z<1.6$
Average number of galaxies | 1950 gal deg-2 | 1220 gal deg-2
Galaxy bias | $b_{\mathrm{g}}(z)=0.79+0.68\,z$ | $b_{\mathrm{g}}(z)=0.84/D(z)$
Reference | Euclid Collaboration et al. (2019) | DESI Collaboration et al. (2016)
Table 1: Specifications for the two galaxy surveys under consideration.

(a) DESI

(b) Euclid

Figure 1: Galaxy density distribution as a function of redshift for the
emission line galaxies (ELG) of DESI (left panel), and the Euclid
spectroscopic sample (right panel). The filled coloured lines show the
Gaussian bins used in our analysis, colour coded as a funciton of bin index.
The orange line shows the CMB lensing efficiency kernel (with arbitrarily
normalization).

Among the wealth of current and upcoming experiments, we choose two
representative cases for spectroscopic large scale structures (LSS) surveys,
namely the DESI and the Euclid experiments. We detail their specifications in
Table 1.

On the CMB side, we consider first a Planck-like experiment, currently the
state-of-the-art data base in terms of multifrequency, full sky CMB data
(Planck Collaboration et al., 2018a). In order to observe the future
sensitivity reachable on the smallest angular scales via ground CMB
experiments, we also consider the Simons Observatory (The Simons Observatory
Collaboration et al., 2019) and the CMB-Stage 4 (Abazajian et al., 2019). Both
shall cover thousands of square degrees of the southern sky ($>40\%$ of the
sky) with extremely high sensitivity ($\leq 2\,$ $\mu$K arcmin) and fine
angular resolution (at the arcmin level)

###  2.1 The DESI experiment

DESI is a ground-based survey that will cover $14,000\leavevmode\nobreak\
\mathrm{deg}^{2}$ on the sky and will measure the redshift of about 30 million
galaxies using optical fibers spectroscopy (DESI Collaboration et al., 2016).
It will target four different classes of galaxies. In this paper, we compute
forecasts for the emission line galaxies (ELG) sample which is the largest
sample of the survey. It ranges from $z=0.6$ up to $z=1.6$. The expected
galaxy distribution $\bar{n}_{\rm g}(z)$ (see Fig. 1a) and the galaxy bias
$b(z)$ are calibrated based on the DEEP2 survey (Newman et al., 2013). The
(linear) bias of the spatial distribution of this galaxy population with
respect to dark matter is a redshift dependent quantity approximated by

$b_{\rm g}(z)=0.84/D(z)\;,$ (1)

with $D(z)$ denoting the growth factor of linear matter density perturbations.

The DESI experiment successfuly achieved the comissioning phase in the first
halo of 2020 and should start observing the sky in the second half of 2020 for
a five year period. The DESI collaboration plans to release annual datasets.

###  2.2 The Euclid spectroscopic survey

The Euclid satellite will observe about $15\,000\leavevmode\nobreak\
\mathrm{deg}^{2}$ of the extragalactic sky (Laureijs et al., 2011). The NISP
instruments will provide slitless spectroscopy, allowing for precise redshift
determinations for about $1950$ gal deg-2. The spectroscopic survey will
target $\rm H_{\alpha}$ emission-line galaxies in the redshift range
$0.9<z<1.8$. We assume the model 3 from Pozzetti et al. (2016) for the
expected number density of galaxies $\bar{n}_{\mathrm{g}}(z)$ (see Fig. 1b).
For the expected galaxy linear bias, we fit a linearly redshift dependent bias
on the values of the Table 3 of Euclid Collaboration et al. (2019), yielding

$b_{\mathrm{g}}(z)=0.79+0.68\,z\;.$ (2)

The Euclid satellite will be launched in 2022 and will be operated for six
years. The data will be released in three batches: 2, 4 and 7 years after the
launch.

###  2.3 Tomography

As already mentioned above, our forecasts are based on a tomographic approach
where the entire redshift range covered by a galaxy survey is sliced into
different redshift bins. Centred at each of these redshift bins, we consider
Gaussian redshift shells of a given width $\sigma_{\rm z}$ centered on
redshifts $z_{i}$.

$W_{i}(z)=\exp{\left(-\frac{(z-z_{i})^{2}}{2\sigma_{\rm z}^{2}}\right)}\;.$
(3)

Provided that a Gaussian shell will be diluting information on radial scales
shorter than the Gaussian width, our choice for $\sigma_{\rm z}$ is a
compromise between maximizing the amount of radial scales under study, and
minimizing the impact of non-linear, radial scales in the analysis (Asorey et
al., 2012; Di Dio et al., 2014). In HMCMA, we found that, at $z\simeq 1$, down
to $\sigma_{\rm z}=0.01$ the impact of radial non-linearities was either
negligible or easily tractable with a Gaussian kernel describing thermal,
stochastic, radial motions. We thus adopt $\sigma_{\rm z}=0.01$ for our
forecasts.

As shown in Asorey et al. (2012), the angular galaxy clustering analysis can
recover the same amount of information as the 3D analysis when the bin size is
comparable to the maximum scale probed by the 3D analysis. This gives
$\sigma_{\mathrm{z}}\,c/H(z)\simeq 2\pi/k_{\rm max}$, so in our case, for
$z=1$ and $k_{\rm max}=0.2$ $h$ Mpc-1 (see Sect. 3), we get
$\sigma_{\mathrm{z}}\simeq 0.01$, corresponding to our choice of bin size.

For the two galaxy survey under consideration we take the same number of 20
redshift bins, and since the overlap between consecutive bins is not zero, we
account for all cross-correlations between shells in the covariance matrix. In
this way, redundant information between different shells is fully accounted
for. The redshift bins sample the range from $z=0.65$ to $z=1.65$ for DESI,
and from $z=0.9$ to $z=1.8$ for Euclid. These redshift bins are displayed in
Figs. 1a and 1b, together with the expected number density of tracers for each
survey.

###  2.4 The Planck experiment

The Planck satellite was launched in 2009 and scanned the full sky until 2013
in CMB frequencies. The satellite hosted two instruments, the HFI operating in
six frequency bands between 100 GHz and 857 GHz, and the LFI instrument
operating in three band between 30 GHz and 77 GHz. The CMB maps were produced
by combining these frequencies to remove the contribution from the galaxy and
other foregrounds sources. The final maps have noise of $27\,\mu
K\,\rm{arcmin}$, and an the effective beam with full width at half maximum of
7 arcmin. The final data release of Planck was published in Planck
Collaboration et al. (2018a).

The CMB lensing field has been estimated with a minimum variance quadratic
estimator, combining temperature and polarization data. It is to date the most
precise map of the integral of the density of matter on the full extra
galactic sky, covering $\sim 70\,\%$ of the sky, which allowed to get an
estimate of the lensing-potential power spectrum over lensing multipoles
$8\leq L\leq 400$ (Planck Collaboration et al., 2018b).

###  2.5 The Simons Observatory

The Simons Observatory consists of four different telescopes placed in the
Atacama Desert in Chile, with the goal of providing an exquisite mapping of
the CMB intensity and polarization anisotropies from a few degrees down to
arcminutes scales. Three of the telescopes have 0.5 m of aperture, and with an
angular resolution close to half a degree, will map 10 % of the sky targeting
the moderate to large angular scales. Their primary goal is to measure large-
scale polarisation from the background of primordial gravitational waves.

Alongside these small telescopes, one 6 m diameter telescope will observe at
27, 39, 93, 145, 225, and 280 GHz, with an angular resolution close to the
arcminute, appropriate to obtain a high resolution map of the lensing
potential of the CMB. It is expected to reach a sensitivity level of
$6\leavevmode\nobreak\ \mu$K arcmin on 40 % of the sky.

###  2.6 CMB Stage 4

The CMB Stage 4 (CMB-S4) experiment will be the successor of the Simons
Observatory, which will combine resources with the successor of the South Pole
telescope and the BICEP/Keck collaborations. Its main scope is to measure the
imprint of primordial gravitational waves on the CMB polarization anisotropy,
but it will also perform a wide survey with a high resolution which will allow
to probe the secondary anisotropies with unprecedented accuracy. Its deep and
wide survey will cover $\sim 60\%$ of the extra-galactic sky and will be
conducted over seven years using two 6 m telescopes located in Chile, each
equipped with $121,760$ detectors distributed over eight frequency bands from
30 GHz to 270 GHz. These observations will provide CMB temperature and
polarization maps with a resolution of $\leq 1.5\,\rm arcmin$ and with a noise
level of $1\leavevmode\nobreak\ \mu$K arcmin. This very-high sensitivity at
small scales both in temperature and polarization, on a large fraction of the
sky, will allow to get an accurate estimation of the CMB lensing potential.

##  3 Observables

In this paper, we consider three different observables, namely the angular
galaxy clustering, the corresponding ARF, and the CMB lensing convergence
field. In order to compute the forecasts, we shall restrict to the linear
scales, where the cosmological linear theory of perturbations apply. In
practice, we ignored all scales above $k_{\rm max}=0.2$ $h$ Mpc-1 at all
redshifts. This is a conservative approach, as one could consider a scale
cutoff which evolves with redshift as in Di Dio et al. (2014). We shall also
assume that our observables are Gaussian distributed, and that the information
content is completely captured by the two-point momenta, in particular the
angular power spectrum, either auto or cross, depending on whether we combine
different observables or not.

In what follows, we describe our model of the observables, so that expressions
for their angular power spectrum can be derived thereafter.

###  3.1 Galaxy Angular Density Fluctuations

The 3D field of the number density of galaxies is noted as
$n_{\mathrm{g}}(z,\@vec{\hat{n}})$, where $\@vec{\hat{n}}$ denotes a direction
on the sky. The average number density of galaxies at a redshift $z$ is
defined by $\bar{n}_{\mathrm{g}}(z)=\left\langle
n_{\mathrm{g}}(z,\@vec{\hat{n}})\right\rangle_{\@vec{\hat{n}}}$. The 3D field
of galaxy density contrast is then given by

$\delta_{\mathrm{g}}^{\rm
3D}(z,\@vec{\hat{n}})=\frac{n_{\mathrm{g}}(z,\@vec{\hat{n}})-\bar{n}_{\mathrm{g}}(z)}{\bar{n}_{\mathrm{g}}(z)}\;.$
(4)

We assume that the galaxy density contrast traces the dark matter density
contrast $\delta_{\mathrm{m}}^{\rm 3D}$ via a scale-independent bias:
$\delta_{\mathrm{g}}^{\rm 3D}(z,\@vec{\hat{n}})=b_{g}(z)\,\delta_{m}^{\rm
3D}(z,\@vec{\hat{n}})$. This bias depends on the properties of the galaxies
used as a tracer for each survey, and are given in Eqs. 1 and 2.

In our analysis, we model the observed redshift of galaxies $z_{\rm obs}$ as a
3D field. It is defined as the sum of the redshift induced by the Hubble flow,
and the redshift due to the peculiar velocity of galaxies:

$z_{\rm
obs}(z,\@vec{\hat{n}})=z+(1+z)\,\frac{\@vec{\mathrm{v}}(z,\@vec{\hat{n}})\cdot\@vec{\hat{n}}}{c}\;,$
(5)

where $\@vec{\mathrm{v}}$ is the peculiar velocity field of galaxies. We
neglect other sources of redshift distortions which are significantly smaller
than those considered here (HMCMA).

The angular galaxy clustering field is then modelled by an integral along the
line of sight in which, at every redshift $z$, only galaxies within the
selection function $W(z_{\rm obs};z_{i})$ are included:

$\begin{split}\delta_{\mathrm{g}}^{i}(\@vec{\hat{n}})\ &=\frac{1}{N^{i}_{\rm
g}}\int_{z=0}^{\infty}\;{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,b_{\mathrm{g}}(z)\,\delta_{\mathrm{m}}^{\rm
3D}(z,\,\@vec{\hat{n}})\,W_{i}\left[z_{\mathrm{obs}}(z,\@vec{\hat{n}})\right],\end{split}$
(6)

where $N^{i}_{\rm g}=\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)$ is the average number of
galaxies per solid angle, under the $i$-th selection function $W_{i}$ centred
on redshift $z_{i}$, and in practice can be computed from an angular average
over the survey’s footprint.

We next expand the selection function, retaining only linear terms in density
and velocity fluctuations, finding:

$\begin{split}\delta_{\mathrm{g}}^{i}(\@vec{\hat{n}})\
&\simeq\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)\\\
&\times\left[b_{\mathrm{g}}(z)\,\delta_{\mathrm{m}}^{\rm
3D}(z,\,\@vec{\hat{n}})+(1+z)\,\frac{{\rm d}\ln W_{i}}{{\rm
d}z}\frac{\@vec{\mathrm{v}}(z,\@vec{\hat{n}})\cdot\@vec{\hat{n}}}{c}\right]\;,\end{split}$
(7)

with the derivative ${\rm d}\ln W_{i}/{\rm
d}z=-(z-z_{i})/\sigma_{\mathrm{z}}^{2}$.

###  3.2 Angular redshift fluctuations

The ARF field represents the spatial variations of the average redshift of
galaxies on the sky. The average redshift of galaxies is given by

$\begin{split}\bar{z}&=\frac{1}{N^{i}_{\rm
g}}\,\left\langle\int_{z=0}^{\infty}{\rm d}V_{\Omega}\;z_{\rm
obs}(z,\@vec{\hat{n}})\,n_{\mathrm{g}}(z,\@vec{\hat{n}})\,W_{i}\left[z_{\mathrm{obs}}(z,\@vec{\hat{n}})\right]\right\rangle_{\@vec{\hat{n}}}\\\
&=\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\,z\,\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)\;.\end{split}$ (8)

We thus define the ARF field as follows,

$\begin{split}\delta_{\mathrm{z}}^{i}(\@vec{\hat{n}})=\frac{1}{N^{i}_{\rm
g}}\int_{z=0}^{\infty}&{\rm d}V_{\Omega}\;(z_{\rm
obs}(z,\@vec{\hat{n}})-\bar{z})\,\bar{n}_{\mathrm{g}}(z)\,\\\
&\times\left[1+b_{\mathrm{g}}(z)\,\delta_{\mathrm{m}}^{\rm
3D}(z,\,\@vec{\hat{n}})\right]\,W_{i}\left[z_{\mathrm{obs}}(z,\@vec{\hat{n}}\,)\right]\;,\end{split}$
(9)

where we again refer to a redshift bin centred upon $z_{i}$. Expanding the
Gaussian selection function at first order and retaining only linear terms in
density and velocity, we find:

$\begin{split}\delta_{\mathrm{z}}^{i}(\@vec{\hat{n}})\simeq\frac{1}{N^{i}_{\rm
g}}&\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)\,\Biggl{[}\left(z-\bar{z}\right)\,b_{\mathrm{g}}(z)\,\delta_{\mathrm{m}}^{\rm
3D}(z,\@vec{\hat{n}})\\\
&\quad+(1+z)\,\frac{\@vec{\mathrm{v}}(z,\@vec{\hat{n}})\cdot\@vec{\hat{n}}}{c}\,\left(1+(z-\bar{z})\,\frac{{\rm
d}\ln W_{i}}{{\rm d}z}\right)\Biggr{]}\;.\end{split}$ (10)

Note that given the small widths adopted ($\sigma_{\rm z}=0.01$), it is safe
to assume that the bias $b(z)$ remains constant within the redshift bin.

###  3.3 CMB lensing

The image of the primary CMB, emitted at the moment of recombination at
$z\simeq 1100$, is distorted by the gravitational lensing arising as a
consequence of the (slightly inhomogeneous) mass distribution between us and
the surface of last scattering. This modify the initial anisotropy pattern and
creates statistical anisotropy (see Lewis & Challinor, 2006, for a review).
Assuming that the primordial CMB is Gaussian and statistically isotropic, we
can reconstruct the lensing potential $\phi$ with the so called quadratic
estimator (Hu & Okamoto, 2002; Okamoto & Hu, 2003). The lensing potential is
linked to the convergence field $\kappa_{\mathrm{CMB}}$ by

$\kappa_{\mathrm{CMB}}=-\frac{1}{2}\Delta\phi\;.$ (11)

This convergence field is directly proportional to the surface mass density
along the line of sight. The CMB lensing is as such an unbiased estimation of
the distribution of mass. However it is an integrated estimation, whereas
galaxy surveys can enable tomographic analyses thanks to redshift
measurements.

The CMB lensing has been characterized by the Planck CMB survey (Planck
Collaboration et al., 2018b), and by the ACTPol (Sherwin et al., 2017), SPT-SZ
(Omori et al., 2017) and SPTpol (Wu et al., 2019) collaborations. Next
generation CMB surveys such as Simons Observatory or CMB-S4 will increase the
signal to noise ratios at all $\ell$ by almost one and two orders of magnitude
respectively. These new experiments will make the CMB lensing as a sensitive
probe of the dark matter distribution, and via cross-correlation studies it
will be crucial to constrain the growth rate of structure, the neutrino
masses, or the level of primordial non-Gaussianities during the inflationary
epoch.

###  3.4 Angular power spectra

Our statistical tool to test cosmological models are the angular two-point
power spectra $C_{\ell}$ performed over the three fields defined in Sects.
3.1, 3.2 and 3.3. Assuming that the galaxy bias and the growth factors are
scale independent, one can show that our (cross and auto) angular power
spectra can be expressed as the convolution of two kernels
$\Delta_{\ell}^{\mathrm{A}}(k)$ and $\Delta_{\ell}^{\mathrm{B}}(k)$,
corresponding for the fields A and B (see, e.g. Huterer et al., 2001):

$C_{\ell}^{\mathrm{A},\mathrm{B}}=\frac{2}{\pi}\int{\rm
d}k\;k^{2}\,P(k)\,\Delta_{\ell}^{\mathrm{A}}(k)\,\Delta_{\ell}^{\mathrm{B}}(k),$
(12)

where $P(k)$ is the linear 3D matter power spectrum at $z=0$, function of the
wave number $k$.

To obtain the theoretical prediction of our angular power spectra, we start
from the 2D fields defined in Eqs. 7 and 10. The velocity field is related to
the matter density contrast field via the linearized continuity equation
$\partial\,\delta_{\mathrm{m}}^{\rm 3D}/\partial
t+\nabla\@vec{\mathrm{v}}/a=0$, with $a(z)$ the cosmological scale factor,
$a=1/(1+z)$. We introduce the linear growth rate

$f=\frac{{\rm d}\ln D}{{\rm d}\ln a}=-(1+z)\,\frac{1}{D(z)}\,\frac{{\rm
d}D}{{\rm d}z}\;.$ (13)

We assume $f(z)=\Omega_{\rm m}(z)^{\,\gamma}$, with $\gamma=0.55$ (Lahav et
al., 1991; Linder, 2005). The growth factor $D(z)$ is computed by integrating
the growth rate $f(z)$.

One can show that the angular galaxy clustering kernel is the sum of two
terms, one arising from the density of galaxies and the other from the
peculiar line of sight velocities,
$\Delta_{\ell}^{g}=\Delta_{\ell}^{g}|_{\delta}+\Delta_{\ell}^{g}|_{\rm v}$
(see e.g. Padmanabhan et al., 2007):

$\displaystyle\Delta_{\ell}^{g,i}|_{\delta}(k)$
$\displaystyle=\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)\,b_{\mathrm{g}}(z)\,D(z)\,j_{\ell}(k\,r(z))\,,$
(14) $\displaystyle\Delta_{\ell}^{g,i}|_{\rm v}(k)$
$\displaystyle=\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,H(z)\,f(z)\,D(z)\,\frac{{\rm
d}W_{i}}{{\rm d}z}\,\frac{j^{\prime}_{\ell}(k\,r(z))}{k}\;,$ (15)

where $j_{\ell}(x)$ is the spherical Bessel function of order $\ell$ and
$j^{\prime}_{\ell}(x)$ is its derivative $j^{\prime}_{\ell}(x)\equiv{\rm
d}j_{\ell}/{\rm d}x$.

One can thus write the power spectrum as the sum of the contributions from the
density and from the velocity kernels
$C_{\ell}=C_{\ell}^{\,\delta\,,\delta}+2\,C_{\ell}^{\,\delta\,,\rm{v}}+C_{\ell}^{\,\rm
v\,,v}$.

The ARF kernel can also be separated into two kernels:

$\displaystyle\Delta_{\ell}^{z,\,i}|_{\delta}(k)$
$\displaystyle=\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W_{i}(z)\,b_{\mathrm{g}}(z)\,D(z)\,(z-\bar{z})\,j_{\ell}(k\,r(z))\;,$
(16) $\displaystyle\begin{split}\Delta_{\ell}^{z,\,i}|_{\rm
v}(k)&=\frac{1}{N^{i}_{\rm g}}\int_{z=0}^{\infty}{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,H(z)\,f(z)\,D(z)\,W_{i}(z)\\\
&\qquad\qquad\qquad\times\left[1+(z-\bar{z})\,\frac{{\rm d}\ln W_{i}}{{\rm
d}z}\right]\,\frac{j^{\prime}_{\ell}(k\,r(z))}{k}\;.\end{split}$ (17)

The kernel function of the CMB lensing convergence field is given by:

$\Delta_{\ell}^{\kappa}(k)=\frac{3\Omega_{\rm
m,0}}{2}\,\left(\frac{H_{0}}{c}\right)^{2}\int_{r=0}^{r_{*}}{\rm
d}r\;\frac{r}{a(r)}\,\frac{r_{*}-r}{r_{*}}\,D(z(r))\,j_{\ell}(k\,r),$ (18)

where $r_{*}$ the comoving distance from the observer to the last scattering
surface, and $a$ is the cosmological scale factor.

Figure 2: _Top panel_ : Power spectra of for angular galaxy clustering
($\delta_{\mathrm{g}}$, in blue) and ARF ($\delta_{\mathrm{z}}$, in red), for
a Gaussian redshift bin taken in a DESI-like survey. The bin is centered on
$z_{i}=0.75$ and has a standard deviation of $\sigma_{\mathrm{z}}=0.01$. The
dashed line show the term coming from the density kernel
$C_{\ell}^{\,\delta\,,\delta}$ ; the dotted line show the part coming from the
velocity kernel $C_{\ell}^{\,\rm v\,,v}$ and the dot-dashed line show the
cross term $C_{\ell}^{\,\delta\,,\rm{v}}$. The total $C_{\ell}$ power spectra
(plain lines) correspond to the sum
$C_{\ell}=C_{\ell}^{\,\delta\,,\delta}+2\,C_{\ell}^{\,\delta\,,\rm{v}}+C_{\ell}^{\,\rm
v\,,v}$. _Bottom panel_ : Ratio of the velocity dependence in the power
spectrum ($C_{\ell}^{\,\rm v\,,v}+2\,C_{\ell}^{\,\delta\,,\rm{v}}$) over the
complete power spectrum, for the angular galaxy clustering (blue line) and for
the ARF (red line). This figure shows that ARF are more sensitive to the
peculiar velocity of galaxies than angular galaxy clustering, for the same
redshift shell.

The top panel of Fig. 2 shows the angular power spectra of the angular galaxy
clustering and ARF, for a Gaussian selection function of width
$\sigma_{\mathrm{z}}=0.01$ centered on $z_{i}=0.75$ in a DESI-like survey. We
show in the same figure the terms arising from the density fluctuation kernel
and the peculiar velocity kernel (c.f. Eqs. 14 to 17). We can see that the
peculiar velocity term is relatively more important (compared to the total
power spectrum) in the ARF power spectrum than in the angular galaxy
clustering power spectrum. To better illustrate this fact, we show in the
bottom panel Fig. 2 the ratio of the velocity part of the power spectrum
(which is the sum $C_{\ell}^{\,\rm v\,,v}+2\,C_{\ell}^{\,\delta\,,\rm{v}}$)
over the total power spectrum for both angular galaxy clustering and ARF. For
both fields, the peculiar velocity contribution dominates at low $\ell$, while
it vanishes to zero for $\ell>300$. At $\ell=10$, the velocity-dependent part
in the power spectrum represents around $67\,\%$ of the total contribution for
$C_{\ell}^{\,z,\,z}$, while it represents only $58\,\%$ of
$C_{\ell}^{\,g,\,g}$. The difference between the two is even more visible at
$\ell=60$, where the velocity contribution represents $55\,\%$ of
$C_{\ell}^{\,z,\,z}$ and only $35\,\%$ of $C_{\ell}^{\,g,\,g}$.

This difference is caused by the intrinsic different nature of the angular
galaxy clustering and ARF transfer functions: angular galaxy clustering is
sensitive to the average of density and velocity under the Gaussian shell,
whereas ARF is sensitive to radial derivatives of those fields. For narrow
shells, this makes both fields practically uncorrelated (HMCMA), and given the
ratio comparison showed in Fig. 2, one would expect ARF to be more sensitive
than angular galaxy clustering to cosmological parameters impacting peculiar
velocities.

##  4 Signal to noise forecasts

We forecast the expected signal to noise ratio (SNR) for different
combinations of observables. Our data vector $\@vec{D}(\ell)$ contains the
auto- and cross-power spectra between the different observables and between
the redshift bins. In order to compare several combinations of probes, we
define the following data vectors:

$\displaystyle\@vec{D}_{\rm g}(\ell)=\left(C^{\,g_{i},\,g_{j}}_{l}\right)\;,$
(19) $\displaystyle\@vec{D}_{\rm
z}(\ell)=\left(C^{\,z_{i},\,z_{j}}_{l}\right)\;,$ $\displaystyle\@vec{D}_{\rm
g,\,z}(\ell)=\left(C^{\,g_{i},\,g_{j}}_{l},C^{\,g_{i},\,z_{j}}_{l},C^{\,z_{i},\,z_{j}}_{l}\right)\;,$
$\displaystyle\@vec{D}_{\rm
g,\,\kappa_{\mathrm{CMB}}}(\ell)=\left(C^{\,g_{i},\,g_{j}}_{l},C^{\,g_{i},\,\kappa_{\mathrm{CMB}}}_{l},C^{\,\kappa_{\mathrm{CMB}},\,\kappa_{\mathrm{CMB}}}_{l}\right)\;,$
$\displaystyle\@vec{D}_{\rm
g,\,z,\,\kappa_{\mathrm{CMB}}}(\ell)=\biggl{(}C^{\,g_{i},\,g_{j}}_{l},C^{\,g_{i},\,z_{j}}_{l},C^{\,g_{i},\,\kappa_{\mathrm{CMB}}}_{l},C^{\,z_{i},\,z_{j}}_{l},$
$\displaystyle\phantom{xxxxxxxxxxxxxxxxxxxxxx}C^{\,z_{i},\,\kappa_{\mathrm{CMB}}}_{l},C^{\,\kappa_{\mathrm{CMB}},\,\kappa_{\mathrm{CMB}}}_{l}\biggr{)}\,,$
(20)

where $i$ and $j$ are indexes running over the redshift bins. We perform a
tomographic analysis with 20 redshift bins, thus the data vectors containing
only the auto-spectra of angular galaxy clustering and ARF ($\@vec{D}_{\rm g}$
and $\@vec{D}_{\rm z}$) contain 210 $C_{\ell}$ each. The data vector
containing the cross-correlation $\@vec{D}_{\rm g,\,z}$ has 820 $C_{\ell}$ and
the longest data vector $\@vec{D}_{\rm g,\,z,\,\kappa_{\mathrm{CMB}}}$
contains 861 $C_{\ell}$.

In Fig. 3, we display the correlation matrix for the $\@vec{D}_{\rm
g,\,z,\,\kappa_{\mathrm{CMB}}}(\ell=10)$ data vector. We clearly see that, in
the same redshift bin, angular galaxy clustering and ARF are practically un-
correlated (diagonal terms of the top left and lower right blocks close to
zero), but that there is some degree of anti-correlation in neighbouring
redshift bins. We can also observe that the CMB lensing field is almost
uncorrelated with the ARF.

Figure 3: Correlation matrix between our observables, for the 20 redshift bins
in a Euclid-like survey, at $\ell=10$. This matrix corresponds to the
$\@vec{D}_{\rm g,\,z,\,\kappa_{\mathrm{CMB}}}(\ell=10)$ data vector. The value
in each pixel corresponds to $C_{\ell}^{\rm A,B}/\sqrt{C_{\ell}^{\rm
A,A}C_{\ell}^{\rm B,B}}$. All the data vectors considered in Eqs. 19 to 20 are
a subset of this matrix. We see that there is no correlation between
$\delta_{\mathrm{g}}$ and $\delta_{\mathrm{z}}$ inside the same redshift bin
(diagonals of the upper left and lower right blocks), and that there are
opposite and positive correlation for neighboring bins.

We assume that there is no correlation between different multipoles and that
the covariance between the probes is totally captured by a Gaussian
covariance. This assumption is exact on large (linear) scales and if the
survey covers the full sky. On real data, the footprint of the survey and the
presence of masked area will create correlations between multipoles. See e.g.
Krause & Eifler (2017) or Lacasa (2018) for the inclusion the higher-order
(non-Gaussian) terms in the covariance matrix. In this work we neglect these
effects.

Figure 4: Signal to noise ratios of angular galaxy clustering ($\@vec{D}_{\rm
g}$) in blue, ARF ($\@vec{D}_{\rm z}$) in orange and the combinations
$\@vec{D}_{\rm g,\,z}$ in green and $\@vec{D}_{\rm
g,\,z,\,\kappa_{\mathrm{CMB}}}$ in red. We use 20 tomographic Gaussian bins of
width $\sigma_{\mathrm{z}}=0.01$ in a Euclid-like survey, in combination with
a CMB-S4 survey. The total signal to noise ratios for the range of multipoles
$\ell=10$ to $\ell=300$ are shown in the text box on the bottom right.

The SNR of our data vectors as a function of $\ell$, taking into account all
redshift bins and the correlations between them, are given by

$\mathrm{SNR}\left(\@vec{D}(\ell)\right)=\sqrt{\@vec{D}(\ell)^{\,t}
\,\@tens{Cov}_{\ell}^{-1}\,\@vec{D}(\ell)}\;,$ (21)

and the total SNR are

$\mathrm{SNR}\left(\@vec{D}\right)=\sqrt{\displaystyle\sum_{\ell=\ell_{\rm
min}}^{\ell_{\rm
max}}\left[\mathrm{SNR}\left(\@vec{D}(\ell)\right)\right]^{2}}\;.$ (22)

Assuming that there is no correlation between different multipoles, we define
our Gaussian covariance matrix between our data vectors as in Hu & Jain (2004)

$\begin{split}\@tens{Cov}_{\ell}\left(C_{\ell}^{\,\rm{A},\rm{B}},C_{\ell}^{\,\rm{C},\rm{D}}\right)&=\frac{1}{(2\ell+1)\,\Delta\ell\,f_{\rm
sky}}\\\ &\times\left[\left(C_{\ell}^{\,\rm{A},\rm{C}}+\delta^{\rm K}_{\rm
A,C}\,N^{\,\rm{A}}_{\ell}\right)\left(C_{\ell}^{\,\rm{B},\rm{D}}+\delta^{\rm
K}_{\rm B,D}\,N^{\,\rm{B}}_{\ell}\right)\right.\\\
&+\left.\left(C_{\ell}^{\,\rm{A},\rm{D}}+\delta^{\rm K}_{\rm
A,D}\,N^{\,\rm{A}}_{\ell}\right)\left(C_{\ell}^{\,\rm{B},\rm{C}}+\delta^{\rm
K}_{\rm B,C}\,N^{\,\rm{B}}_{\ell}\right)\right]\;,\end{split}$ (23)

with A, B, C, D are the observables
$\left\\{g_{i},\,z_{j},\,\kappa_{\mathrm{CMB}}\right\\}$, $\Delta\ell$ is the
width of the multipole bin, $\delta^{K}_{x,y}$ is the Kronecker delta,
$N_{\ell}$ are the probe specific noise power spectra and $f_{\rm sky}$ is the
sky fraction of the survey considered.

For the sake of simplicity, when combining galaxy surveys with CMB lensing, we
always assume a full overlap of the two. As such, the sky fraction $f_{\rm
sky}$ is always taken to be the one of either DESI or Euclid. Even if not
accurate, this provides a rough estimate of the available constraining power
that the combination of galaxy surveys with CMB lensing will be able to
achieve.

We assume that the noise of the angular galaxy clustering and that of the ARF
are the shot noise arising from the discrete nature of galaxy surveys. We
model it by replacing the power spectrum of dark matter by a Poissonian term,
$P_{\mathrm{shot}}(k,z)=1/\bar{n}_{\mathrm{g}}(z)$, into Eq. 12. From this, we
can derive the following expressions for the shot noise

$\displaystyle N_{\ell}^{\,g_{i},\,g_{j}}$ $\displaystyle=\frac{\delta^{\rm
K}_{i,\,j}}{N_{\mathrm{gal}}^{i}}\;,$ (24) $\displaystyle
N_{\ell}^{\,z_{i},\,z_{j}}$ $\displaystyle=\frac{\delta^{\rm
K}_{i,\,j}}{\left(N_{\mathrm{gal}}^{i}\right)^{2}}\int{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W(z_{i},z)\,\left(z-\bar{z}_{i}\right)^{2}\;,$
(25) $\displaystyle N_{\ell}^{\,g_{i},\,z_{j}}$
$\displaystyle=\frac{\delta^{\,i}_{\,j}}{\left(N_{\mathrm{gal}}^{i}\right)^{2}}\int{\rm
d}V_{\Omega}\;\bar{n}_{\mathrm{g}}(z)\,W(z_{i},z)\,\left(z-\bar{z}_{i}\right)=0\;.$
(26)

We can see here that the shot noise cancels out when computing the cross
correlation between the angular galaxy density and the ARF fields.

The noise of the CMB lensing field reconstructed from Planck is taken from
Planck Collaboration et al. (2018b). For the forecasted Simons Observatory CMB
lensing noise, we take the publicly available noise curves provided by The
Simons Observatory Collaboration et al. (2019)111We use the version 3.1.0 of
the noise curves available on https://github.com/simonsobs/so_noise_models. In
practice, we use the noise curves obtained with the internal linear
combination (ILC) component separation method, assuming the baseline analysis
for a sky fraction of $f_{\mathrm{sky}}=0.4$. For CMB-S4, the lensing noise
curve is taken as the minimum variance N0 bias, which is computed using the
code quicklens222https://github.com/dhanson/quicklens. We assume that CMB-S4
will have a beam size (full width at half maximum) of $1\,\rm arcmin$, a
temperature noise of $\Delta T=1\,\mu\rm K\,arcmin$ and a polarization noise
of $\Delta P=\sqrt{2}\,\mu\rm K\,arcmin$ (Abazajian et al., 2019).

For both Simons Observatory and CMB-S4, $\ell=40$ is the minimum multipole
which will be accessible. We assume that these measurements will be combined
with the Planck lensing signal for lower multipoles. As a result, we use the
lensing noise of Planck for mutipoles below $\ell=40$ when forecasting
constraints with the Simons Observatory and CMB-S4.

We use the linear matter power spectrum $P(k)$ computed with the CLASS
software (Blas et al., 2011). In order to focus on the linear regime we
restrict our analysis to a maximum multipole of $\ell_{\rm max}=300$. Assuming
the Limber approximation $k=(\ell+1/2)/\chi(z)$, this $\ell_{\rm max}$
corresponds to $k=0.18\leavevmode\nobreak\ h\leavevmode\nobreak\
\rm{Mpc}^{-1}$ at a redshift of $z=0.65$. Given that we will sample higher
redshifts, we will probe larger scales ($k$ lower than
$0.18\leavevmode\nobreak\ h\leavevmode\nobreak\ \rm{Mpc}^{-1}$). We hence
expect little impact from non-linear physics in our observables (these are
expected to become relevant on $k<0.2\leavevmode\nobreak\
h\leavevmode\nobreak\ \rm{Mpc}^{-1}$ at $z=0$, and yet shorter at higher
redshifts). Again this is a conservative approach as one could consider a
multipole cuttoff evolving with redshift as in Di Dio et al. (2014). We stress
that, in our computations, we do not use the Limber approximation but the full
computation of spherical Bessel functions.

Our minimum multipole is chosen to be $\ell_{\rm min}=10$. To reduce numerical
noise and to speed up Fisher matrix computations, we perform a linear binning
of the multipoles. In each multipole bin $[\ell_{i},\ell_{i+1}[$, the binned
$C_{\ell}$ is the average of the $C_{\ell}$’s that fall in the bin, and the
binned multipole is taken has $\ell=(\ell_{i}+\ell_{i+1})/2$. We choose a bin
size of $\Delta_{\ell}=3$, which is applied on the full $\ell$ range. We check
that this binning does not impact the constraints from the Fisher matrix by
comparing with the case where we do not perform any binning of the multipoles.

We show in Fig. 4 the SNR for a Euclid-like survey combined with a CMB-S4
survey, for four combination of probes: $\@vec{D}_{\rm g}$, $\@vec{D}_{\rm
z}$, $\@vec{D}_{\rm g,\,z}$ and $\@vec{D}_{\rm g,\,z,\,\kappa_{\mathrm{CMB}}}$
following the redshift binning shown in Fig. 1b. The total signal to noise for
this four data vectors is respectively 544, 545, 778 and 786. This shows that
the tomographic analysis of angular galaxy clustering and ARF have a similarly
high SNR. Moreover, the combined analysis $\@vec{D}_{\rm g,\,z}$ brings more
information than measuring the angular galaxy clustering alone $\@vec{D}_{\rm
g}$, as the SNR is increased by $40\%$.

##  5 Fisher forecasts

We use the Fisher formalism to compute _a priori_ how well our data vectors
defined in Sect. 4 will constrain cosmological parameters in the context of
future surveys. As we assumed that there is no correlation between different
multipoles, the Fisher matrix can be summed over the multipoles and is given
by

$F_{\,i,\,j}=\sum_{\ell_{\rm min}}^{\ell_{\rm
max}}\;\frac{\partial\@vec{D}(\ell)}{\partial\lambda_{i}}\,\@tens{Cov}_{\ell}^{-1}\,\frac{\partial\@vec{D}(\ell)}{\partial\lambda_{j}},$
(27)

with $\@vec{D}$ one of the data vectors defined in Eqs. 19 to 20,
$\left\\{\lambda_{i}\right\\}_{i}$ is the set of free parameters of our model,
and $\@tens{Cov}_{\ell}$ the covariance matrix given in Eq. 23.

The derivatives $\partial\@vec{D}(\ell)/\partial\lambda_{i}$ are computed as
the two-point variation with a 1% step around the fiducial value. We have
checked that our derivatives are numerically stable when changing the step
size.

We compute forecasts for two cosmological models. The first one assumes the
standard $\mathrm{\Lambda CDM}$ model, and the parameters we vary are
$\left\\{\Omega_{\rm m},\Omega_{\rm
baryon},\sigma_{8},n_{\mathrm{s}},h,\right\\}$. The fiducial values of these
parameters are given by Planck Collaboration et al. (2018a). The second model
assumes an evolving dark energy equation of state, with the so-called CPL
parametrization (Chevallier & Polarski, 2001; Linder, 2003):
$w(z)=w_{0}+w_{\mathrm{a}}\,z/(1+z)$. Our second set of free parameters is
then $\left\\{\Omega_{\rm m},\Omega_{\rm
baryon},\sigma_{8},n_{\mathrm{s}},h,w_{0},w_{\mathrm{a}}\right\\}$. In both
cases, we assume a flat universe ($\Omega_{k}=0$) with massless neutrinos
($\sum m_{\nu}=0$). We show in Tab. 2 the fiducial values of the free
parameters.

We also consider a bias parameter assumed constant within each redshift bin,
thus adding one free parameter for each redshift shell, over which we
marginalise in the Fisher analysis. The fiducial values of the galaxy bias
depend on the survey considered and are given in Eqs. 1 and 2. We take the
value at $z_{i}$, the center of the Gaussian shell for each bin.

$\Omega_{\rm b}$ | $\Omega_{\rm m}$ | $n_{\rm s}$ | $h$ | $\sigma_{8}$ | $w_{0}$ | $w_{\rm a}$
---|---|---|---|---|---|---
0.04897 | 0.3111 | 0.9665 | 0.6766 | 0.8102 | -1 | 0
Table 2: Fiducial values of the free parameter of our fiducial cosmological
model. We first consider only parameters in the standard $\mathrm{\Lambda
CDM}$ model, and later we include the $w_{0}$, $w_{\mathrm{a}}$ parameters
from the CPL parametrization of Dark Energy.

###  5.1 Results for the $\mathrm{\Lambda CDM}$ model

Figure 5: Ratio of $1\sigma$ confidence interval relative to the $1\sigma$
value from angular galaxy clustering ($\@vec{D}_{\rm g}$) for $\mathrm{\Lambda
CDM}$ parameters. Constraints are marginalised over the 20 galaxy bias
parameters. Plain lines are for a DESI-like survey while dashed lines are for
a Euclid-like survey. Blue line shows $\@vec{D}_{\rm g}$ (our reference here),
orange lines show $\@vec{D}_{\rm z}$ and green lines show $\@vec{D}_{\rm
g,\,z}$. We see that for most parameters (except $\sigma_{8}$) confidence
intervals shrink by $\sim 50\%$ when using $\@vec{D}_{\rm z}$ instead of
$\@vec{D}_{\rm g}$. When using the combination $\@vec{D}_{\rm g,\,z}$,
$1\sigma$ intervals are shrinked by at least $60\%$ for all parameters. Figure
6: Foreseen constraints ($1\sigma$ contours) for a set of five
$\mathrm{\Lambda CDM}$ parameters, plus three galaxy bias parameters (out of a
total of 20) for a DESI-like survey. We assume 20 tomographic Gaussian bins of
size $\sigma_{\mathrm{z}}=0.01$. The blue line are the constraints for angular
galaxy clustering alone $\@vec{D}_{\rm g}$, the orange line are for the ARF
alone $\@vec{D}_{\rm z}$, and the green line is a joint analysis of both
fields $\@vec{D}_{\rm g,\,z}$. The figures above the 1-D PDFs give the
marginalised $1\sigma$ uncertainty of the parameter for each data vector. We
show here only 3 galaxy bias parameters even if we marginalised upon the 20
bias parameters. Figure 7: Same as Fig. 6 for a Euclid-like survey.

The results for the $\mathrm{\Lambda CDM}$ model are summarized in Fig. 5,
where we show the ratio of the $1\sigma$ marginalised uncertainties when
including ARF compared to using only angular galaxy clustering, for a DESI-
like and a Euclid-like surveys. Fig. 6 and Fig. 7 show the $1\sigma$
uncertainty ellipses for the $\mathrm{\Lambda CDM}$ parameters and three out
of the 20 galaxy bias parameters for a DESI-like survey and a Euclid-like
survey respectively. Error ellipses for $\@vec{D}_{\rm g}$, $\@vec{D}_{\rm
z}$, and $\@vec{D}_{\rm g,\,z}$ are given by blue, orange, and green curves,
respectively, while marginalised 1$\sigma$ uncertainties for each parameter
are quoted, for these three sets of observables, above the panels containing
the one dimensional probability density distributions (PDFs).

For both types of LSS surveys, we can see in Fig. 5 that ARF ($\@vec{D}_{\rm
z}$) are significantly more sensitive than angular galaxy clustering
($\@vec{D}_{\rm g}$), reducing by a factor two the marginalised uncertainties
of all cosmological parameters but $\sigma_{8}$, to which both observables are
similarly sensitive. For the combined analysis $\@vec{D}_{\rm g,\,z}$,
marginalised uncertainties are reduced by more than $60\,\%$ for all
parameters (including $\sigma_{8}$), compared to the angular galaxy clustering
probe alone $\@vec{D}_{\rm g}$. We find that using ARF in combination with
angular galaxy clustering provides almost the same improvement on the
constraints on cosmological parameter for both surveys, although the
improvement is on average slightly better for our Euclid-like survey.

We see in Figs. 6 and 7 that while the degeneracy direction between different
cosmological parameter pairs seems very similar for both angular galaxy
clustering and ARF, this is again different for $\sigma_{8}$. For
$\@vec{D}_{\rm z}$ this parameter seems rather independent from other
cosmological parameters, while its degeneracy with bias parameters is slightly
tilted with respect to that of $\@vec{D}_{\rm g}$. As a consequence the joint
$\@vec{D}_{\rm g,\,z}$ ellipses show little degeneracy with other parameters,
including bias. We also find that the marginalised constraints from both
experiments are very close, although the Euclid-like experiment provides
slightly more sensitive forecasts.

Fig.13 in Appendix shows the correlation matrix between our free parameters
(including galaxy bias parameters) and illustrates the opposite degeneracies
that both $\sigma 8$ and bias parameters have with the other parameters when
comparing ARF and angular galaxy clustering.

Even for those parameters for which both angular galaxy clustering and ARF
show a similar direction of degeneracy, the combination of the two observables
yields significantly reduced error ellipses. This is mostly due to the lack of
correlation between the ARF and angular galaxy clustering for narrow widths
used in this work ($\sigma_{\mathrm{z}}\leq 0.01$), as noted in HMCMA and
shown here in Fig. 3.

###  5.2 Extension to CPL Dark Energy parametrization

We repeat the analysis detailed above including two new parameters describing
the equation of state of dark energy following the CPL parametrization:
$w_{0}$ and $w_{\rm a}$. We show in Fig. 8 the improvement on the marginalised
uncertainties of the ARF with respect to angular galaxy clustering alone. We
see that $\@vec{D}_{\rm z}$ improves the constraints by $20\,\%$ to $50\,\%$
on this set of free $\mathrm{wCDM}$ parameters, for both surveys. The combined
analysis $\@vec{D}_{\rm g,\,z}$ reduces the uncertainties by at least $50\,\%$
and up to $80\,\%$ for $\Omega_{\rm m}$, $\sigma_{8}$, $w_{0}$ and $w_{\rm
a}$.

The error ellipses are given in Appendix in Fig. 14 for the DESI-like and
Euclid-like experiments, displaying a pattern similar to what was found for
$\mathrm{\Lambda CDM}$, together with the correlation matrices (Fig. 15)

In our idealized case, the combination of ARF with angular galaxy clustering
greatly improves the sensitivity of these surveys to dark energy. As shown in
Fig. 9, the figure of merit of $w_{0}$-$w_{rma}$ increases by more than a
factor of 10 when ARF are combined to angular galaxy clustering. It increases
from 17 to 189 for our DESI-like survey and from 19 to 345 for our Euclid-like
survey.

Figure 8: Ratios of the $1\sigma$ marginalised uncertainties relative to the
$1\sigma$ marginalised uncertainty for $\@vec{D}_{\rm g}$. We assume
$\mathrm{wCDM}$ model and marginalise on 20 galaxy bias parameters (one for
each redshift bin). Orange lines show the ratio for $\@vec{D}_{\rm z}$ and
green lines show the ratio for $\@vec{D}_{\rm g,\,z}$. Solid lines are for a
DESI-like survey while dashed lines are for a Euclid-like survey. We see that
$\@vec{D}_{\rm z}$ improves constraints by up to $50\,\%$ compared to
$\@vec{D}_{\rm g}$ and the combined analysis $\@vec{D}_{\rm g,\,z}$ improves
constraints by up to $80\,\%$. Figure 9: Marginalised constraints ($1\sigma$
contours) on the dark energy equation of state parameters for the DESI-like
(solid lines) and for the Euclid-like (dashed lines) surveys, assuming 20
tomographic Gaussian bins of size $\sigma_{\mathrm{z}}=0.01$. The blue lines
are the constraints for angular galaxy clustering alone, the orange lines are
for ARF alone, and the green lines are a joint analysis of both fields,
$\@vec{D}_{\rm g,\,z}$. These contours are marginalised over the set of
cosmological parameters as before, and over the galaxy bias in the 20 redshift
bins. We display the figure of merit (FoM) of this pair of parameters in the
upper right box for each combination of observables and for each survey.

###  5.3 Combining with CMB lensing

Figure 10: Ratio of the $1\sigma$ constraints for $\mathrm{\Lambda CDM}$
parameters from $\@vec{D}_{\rm g,\,z,\,\kappa_{\mathrm{CMB}}}$ over the
$1\sigma$ constraints from $\@vec{D}_{\rm g,\,z}$ for the Euclid-like
spectroscopic survey. We show combinations with CMB lensing from Planck
(brown), Simons Observatory (pink) and CMB Stage-4 (grey). Constraints are
marginalised over the 20 galaxy bias parameters. Figure 11: Same as Fig. 10
for $\mathrm{wCDM}$ parameters.

We show in Fig. 10 and Fig. 11 the improvements on the constraints of the
$\mathrm{\Lambda CDM}$ and $\mathrm{wCDM}$ parameters for a Euclid-like
survey, when combined with CMB lensing from Planck, Simons Observatory and
CMB-S4, marginalised over the galaxy bias parameters. We see that including
CMB lensing from Planck improves the constraints by maximum of $10\,\%$ in
both cosmologies. The improvement is more significant when combining with
Simons Observatory or CMB-S4. For the Simons Observatory and CMB-S4, in the
$\mathrm{\Lambda CDM}$ model, marginalised uncertainties on $\Omega_{\rm m}$
and $\sigma_{8}$ are decreased by up to $30\,\%$. Other parameters are
improved by $5\,\%$ to $10\,\%$ . For the $\mathrm{wCDM}$ model, the
improvement is of $\sim 15\,\%$ for most parameters, with the most significant
for $\Omega_{\rm m}$ and $w_{\rm a}$, with uncertainties decreased by up to
$30\,\%$. We see that the combination with CMB lensing help decreasing
uncertainties on the $\mathrm{wCDM}$ cosmology.

Since the CMB lensing is an unbiased probe of the distribution of matter, one
of the main interests of combining it with galaxy surveys is to produce tight
constraints on the galaxy bias parameter. We show in Fig. 12 the $1\sigma$
marginalised uncertainties on the galaxy bias parameters for each of the 20
redshift bins in a Euclid-like survey combined with CMB-S4 lensing, for the
$\mathrm{\Lambda CDM}$ model. We compare the constraints obtained for angular
galaxy clustering alone ($\@vec{D}_{\rm g}$), with the ones obtained when
combined with CMB lensing ($\@vec{D}_{\rm g,\,\kappa_{\mathrm{CMB}}}$), with
ARF ($\@vec{D}_{\rm g,\,z}$), and the full combination ($\@vec{D}_{\rm
g,\,z,\,\kappa_{\mathrm{CMB}}}$).

We see that the combination of angular galaxy clustering with ARF provides
better constraints on the galaxy bias than the combination with CMB lensing.
For instance, at a redshift of 1.06, the marginalised uncertainties for the
galaxy bias parameter $b_{3}$ is of 0.025 for the angular galaxy clustering,
it decreases to 0.020 when combined with CMB lensing, and down to 0.013 when
combined with ARF. The combination of the three results in marginalised
uncertainties of 0.08. We can see that the CMB lensing improves constraints by
$\sim 20\,\%$ only, while ARF improves constraints by $\sim 50\,\%$ (a factor
2 improvement). We argue that this is due to the importance of the velocity
term in the ARF kernel (see Fig 2), which does not depend on galaxy bias as it
is sensitive to the full matter distribution.

Figure 12: Marginalised 1$\sigma$ confidence values for the galaxy bias
parameters, with a Euclid-like survey alone (plain lines) or in combination
with CMB-S4 lensing survey (dashed lines). We marginalised on the 5 free
parameters of the $\mathrm{\Lambda CDM}$ model. We show constraints with
angular galaxy clustering (blue and brown lines) and in combination with ARF
(green and red lines). The mean redshift of each shell is shown at the top. We
see that the ARF combined with angular galaxy clustering $\@vec{D}_{\rm
g,\,z}$ provides better constraints on galaxy bias than the combination of
angular galaxy clustering with CMB lensing $\@vec{D}_{\rm
g,\,\kappa_{\mathrm{CMB}}}$.

##  6 Discussion

One could argue several reasons why angular observables might be preferred
over standard 3D ones. Probably the main one is the lack of assumption of any
fiducial cosmological model to analyse the data. This means that angular
observables may be directly compared with theoretical predictions without any
intermediate data manipulations that hinge on an assumption whose implications
in the analysis may not always be clear. Moreover, this type of angular
analysis are conducted tomographically in moderately narrow redshift shells,
thus avoiding the assumption that the universe remains effectively frozen in
relatively long time spans, as it may occur in 3D clustering analysis where an
effective redshift must be defined for the entire volume under analysis (see,
e.g., Cuesta et al., 2016). Asorey et al. (2012) and Di Dio et al. (2014) have
shown that when using a large number of narrow redshift slices, a 2D
clustering analysis can produce the same constraints on cosmological
parameters than a 3D clustering analysis, provided that the width of the
redshift slices is comparable to the minimum scale probed in the 3D analysis.
By including the redshift information in a 2D field, the ARF observable keeps
some information about the distribution of galaxies along the line of sight,
which normally disappear when projecting the 3D galaxy density field on a 2D
observable. As we have shown, ARF are improving the usual 2D galaxy clustering
analysis.

Another major interest of using angular observables is that they can easily be
cross-correlated with other 2D observables. Indeed the combination of 3D
probes with 2D probes is not straightforward, especially when one has to
properly take into account the covariances between them (see e.g. Passaglia et
al., 2017; Camera et al., 2018). In this work we used the CMB lensing field
and its cross-correlation with our tomographic analysis of angular galaxy
clustering and ARF. We have shown that this cross-correlations improve the
constraints, especially on the galaxy bias. Chaves-Montero et al. (2019) have
shown that the cross-correlation of the ARF field with the CMB temperature
field can detect the kinematic Sunyaev-Zel’dovich (kSZ) effect at the
$10\sigma$ level.

The point of this paper is not a detailed comparison between 2D and 3D
clustering analysis, but rather an exploration of the added value of including
ARF in cosmological studies of the large scale structures, on top of the
traditional angular galaxy clustering. By its intrinsic different sensitivity
to the cosmic density and velocity fields under the redshift shells, the ARF
change the degeneracies between cosmological parameters, especially with
respect to $\sigma_{8}$ and the galaxy bias, compared to the angular galaxy
clustering. This is due, as claimed in HMCMA, to the fact that angular galaxy
clustering is sensitive to the first moment (the average) of matter density
and velocity under the redshift shells, whereas ARF are sensitive to the
second moment (the variance) of matter density and velocity along the line of
sight, inside these redshift shells. Moreover, we have shown that the ARF and
the angular galaxy clustering inside the same tomographic redshift bin are
almost uncorrelated. Due to this absence of correlation, by combining both we
are able to break degeneracies and give tighter constraints on all the
cosmological parameters we have considered.

The results we obtained in our work can be considered as an optimistic setting
for both galaxy and CMB surveys. We restricted our analysis to the linear
regime and we did not include any systematic effects which could impact our
results and worsen the constraints. It was found in Hernández-Monteagudo et
al. (2020) that the impact of non-linear physics is more severe in angular
galaxy clustering than in ARF. They found that a linear bias was sufficient to
describe the ARF on scales larger than $60\leavevmode\nobreak\ h^{-1}$ Mpc,
while it was not the case for angular galaxy clustering. Indeed, ARF are built
upon the average observed redshift along the line of sight in a redshift
selection function. This is intrinsically different to counting the number of
galaxies in a given region in the universe, and consequently systematics and
non-linearities are affecting differently each observable. In future works we
plan to address systematics and non-linearities, aiming to model more
realistic settings. We expect that the impact of both systematics and non-
linearities will depend on the survey and on the targeted galaxy sample, as
ongoing work on existing galaxy surveys is indicating.

We do not provide a detailed comparison with the forecasted constraints of the
Euclid survey published in Euclid Collaboration et al. (2019). Indeed our
analysis consider a simplistic, linear model of the galaxy clustering. In this
context, our findings indicate that ARF brings significant cosmological
information on top of the traditional angular galaxy clustering. At best, our
results with the angular galaxy clustering probe ($\@vec{D}_{\rm g}$) could be
compared with the linear setting shown in the Table 9 of Euclid Collaboration
et al. (2019) (first line). In that case, their probe is the 3D linear galaxy
power spectrum, with a cutoff value at $k_{\rm max}=0.25\leavevmode\nobreak\
h$ Mpc-1, in four different redshift bins. Their Fisher analysis account for
more parameters describing the anisotropies in the power spectrum and the shot
noise residuals. This 3D probe is intrinsically different to the (2D) angular
power spectrum tomography used in our work, in 20 Gaussian bins, for which we
limit to $k_{\rm max}=0.20\leavevmode\nobreak\ h\leavevmode\nobreak\ $Mpc-1.
Our forecasts with $\@vec{D}_{\rm g}$ for the errors on some parameters are
tighter than theirs (by a factor of $\sim 2$ for $\sigma_{8}$), while for
others we find the opposite situation (e.g., the reduced Hubble parameter $h$,
whose uncertainty in Euclid Collaboration et al. 2019 is roughly one third of
ours).

##  7 Conclusion

We showed that the ARF are a promising cosmological observable for next
generation spectroscopic surveys. We found that for our choice of binning, the
tomographic analysis of ARF retrieves more information than the tomographic
analysis of the angular galaxy clustering. We showed that the joint analysis
of both fields helps in breaking degeneracies between cosmological parameters,
due to their lack of correlation and their different sensitivities to
cosmology. The improvement appears to be particularly significant for the
$\mathrm{wCDM}$ model. We have shown that the figure of merit for the
$w_{0}$-$w_{\rm a}$ parameters was increased by a factor of more than ten when
combining angular galaxy clustering with ARF.

Finally, we have seen that combining angular galaxy clustering with ARF
provides tighter constraints on the galaxy bias parameters compared to the
combination of angular galaxy clustering with CMB lensing. This shows that ARF
are a very powerful probe of the distribution of matter, as it allows to break
the degeneracy between $\sigma_{8}$ and the galaxy bias. For future galaxy
surveys, errors on the cosmological figure of merit will be dominated by
systematic uncertainties and non-linearities, and ARF might provide a novel,
complementary view on those issues.

In our analysis we did not consider massive neutrinos. As the growth rate is
particularly sensitive to them, we expect ARF to be a powerful tool to
constrain the mass of neutrinos. We defer this detailed analysis to an
upcoming work.

Simultaneously from the LSS and CMB fronts, the coincidence in the acquisition
of data sets of tremendous quality and huge size should enable the combination
of standard analyses with new, alternative ones like the one introduced in
this paper. The combination of techniques and observables should work jointly
in the efforts of identifying and mitigating systematics, and pushing our
knowledge of cosmological physics to its limits.

###### Acknowledgements.

The authors acknowledge useful discussions with G. Ariccò, G. Hurier, and J.
Kuruvilla. LL acknowledges financial support from CNES’s funding of the Euclid
project. C.H.-M. acknowledges the support of the Spanish Ministry of Science
and Innovation through project PGC2018-097585-B-C21. NA acknowledges support
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation programme grant agreement ERC-2015-AdG 69556.

## References

  * Abazajian et al. (2019) Abazajian, K., Addison, G., Adshead, P., et al. 2019, arXiv e-prints, arXiv:1907.04473
  * Abbott et al. (2018) Abbott, T. M. C., Abdalla, F. B., Alarcon, A., et al. 2018, Phys. Rev. D, 98, 043526
  * Asorey et al. (2012) Asorey, J., Crocce, M., Gaztañaga, E., & Lewis, A. 2012, MNRAS, 427, 1891
  * Benitez et al. (2014) Benitez, N., Dupke, R., Moles, M., et al. 2014, arXiv e-prints, arXiv:1403.5237
  * Blas et al. (2011) Blas, D., Lesgourgues, J., & Tram, T. 2011, J. Cosmology Astropart. Phys., 2011, 034
  * Bonifacio et al. (2016) Bonifacio, P., Dalton, G., Trager, S., et al. 2016, in SF2A-2016: Proceedings of the Annual meeting of the French Society of Astronomy and Astrophysics, ed. C. Reylé, J. Richard, L. Cambrésy, M. Deleuil, E. Pécontal, L. Tresse, & I. Vauglin, 267–270
  * Camera et al. (2018) Camera, S., Fonseca, J., Maartens, R., & Santos, M. G. 2018, MNRAS, 481, 1251
  * Cenarro et al. (2019) Cenarro, A. J., Moles, M., Cristóbal-Hornillos, D., et al. 2019, A&A, 622, A176
  * Chaves-Montero et al. (2019) Chaves-Montero, J., Hernández-Monteagudo, C., Angulo, R. E., & Emberson, J. D. 2019, arXiv e-prints, arXiv:1911.10690
  * Chevallier & Polarski (2001) Chevallier, M. & Polarski, D. 2001, International Journal of Modern Physics D, 10, 213
  * Cuesta et al. (2016) Cuesta, A. J., Vargas-Magaña, M., Beutler, F., et al. 2016, MNRAS, 457, 1770
  * de Jong & 4MOST Consortium (2015) de Jong, R. S. & 4MOST Consortium. 2015, in IAU General Assembly, Vol. 29, 2255843
  * DESI Collaboration et al. (2016) DESI Collaboration, Aghamousa, A., Aguilar, J., et al. 2016, arXiv e-prints, arXiv:1611.00036
  * Di Dio et al. (2014) Di Dio, E., Montanari, F., Durrer, R., & Lesgourgues, J. 2014, J. Cosmology Astropart. Phys., 2014, 042
  * Doré et al. (2014) Doré, O., Bock, J., Ashby, M., et al. 2014, arXiv e-prints, arXiv:1412.4872
  * Euclid Collaboration et al. (2019) Euclid Collaboration, Blanchard, A., Camera, S., et al. 2019, arXiv e-prints, arXiv:1910.09273
  * Hernández-Monteagudo et al. (2019) Hernández-Monteagudo, C., Chaves-Montero, J., & Angulo, R. E. 2019, arXiv e-prints, arXiv:1911.12056
  * Hernández-Monteagudo et al. (2020) Hernández-Monteagudo, C., Chaves-Montero, J., Angulo, R. E., & Ariccò, G. 2020, arXiv e-prints, arXiv:2005.06568
  * Hu & Jain (2004) Hu, W. & Jain, B. 2004, Phys. Rev. D, 70, 043009
  * Hu & Okamoto (2002) Hu, W. & Okamoto, T. 2002, ApJ, 574, 566
  * Huterer et al. (2001) Huterer, D., Knox, L., & Nichol, R. C. 2001, ApJ, 555, 547
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111
  * Krause & Eifler (2017) Krause, E. & Eifler, T. 2017, MNRAS, 470, 2100
  * Lacasa (2018) Lacasa, F. 2018, A&A, 615, A1
  * Lahav et al. (1991) Lahav, O., Lilje, P. B., Primack, J. R., & Rees, M. J. 1991, MNRAS, 251, 128
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, arXiv e-prints, arXiv:1110.3193
  * Lewis & Challinor (2006) Lewis, A. & Challinor, A. 2006, Phys. Rep, 429, 1
  * Linder (2003) Linder, E. V. 2003, Phys. Rev. Lett., 90, 091301
  * Linder (2005) Linder, E. V. 2005, Phys. Rev. D, 72, 043529
  * Martí et al. (2014) Martí, P., Miquel, R., Castander, F. J., et al. 2014, MNRAS, 442, 92
  * Moles et al. (2008) Moles, M., Benítez, N., Aguerri, J. A. L., et al. 2008, AJ, 136, 1325
  * Newman et al. (2013) Newman, J. A., Cooper, M. C., Davis, M., et al. 2013, ApJS, 208, 5
  * Okamoto & Hu (2003) Okamoto, T. & Hu, W. 2003, Phys. Rev. D, 67, 083002
  * Omori et al. (2017) Omori, Y., Chown, R., Simard, G., et al. 2017, ApJ, 849, 124
  * Padmanabhan et al. (2007) Padmanabhan, N., Schlegel, D. J., Seljak, U., et al. 2007, MNRAS, 378, 852
  * Passaglia et al. (2017) Passaglia, S., Manzotti, A., & Dodelson, S. 2017, Phys. Rev. D, 95, 123508
  * Pérez-González et al. (2013) Pérez-González, P. G., Cava, A., Barro, G., et al. 2013, ApJ, 762, 46
  * Planck Collaboration et al. (2018a) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018a, arXiv e-prints, arXiv:1807.06209
  * Planck Collaboration et al. (2018b) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2018b, arXiv e-prints, arXiv:1807.06210
  * Pozzetti et al. (2016) Pozzetti, L., Hirata, C. M., Geach, J. E., et al. 2016, A&A, 590, A3
  * Scoville et al. (2007) Scoville, N., Aussel, H., Benson, A., et al. 2007, ApJS, 172, 150
  * Sherwin et al. (2017) Sherwin, B. D., van Engelen, A., Sehgal, N., et al. 2017, Phys. Rev. D, 95, 123529
  * The Simons Observatory Collaboration et al. (2019) The Simons Observatory Collaboration, Ade, P., Aguirre, J., et al. 2019, J. Cosmology Astropart. Phys., 2, 056
  * Wu et al. (2019) Wu, W. L. K., Mocanu, L. M., Ade, P. A. R., et al. 2019, ApJ, 884, 70

##  Appendix A Correlation matrices

The correlation matrices in Fig. 13 provide an alternative view of our
results. It shows the correlation matrices for the 5 $\mathrm{\Lambda CDM}$
parameters and the 20 galaxy bias parameters for a DESI-like survey. We see
the opposite correlation of the cosmological parameters $\Omega_{\rm b}$,
$\Omega_{\rm m}$, $n_{\rm s}$ and $h$ with $\sigma_{8}$ for angular galaxy
clustering and ARF. This opposite correlation is mirrored in the correlations
of those three cosmological parameters with galaxy bias parameters. This is
expected as $\sigma_{8}$ and bias are tightly correlated. The different nature
of the correlation of $\sigma_{8}$ and bias with the other cosmological
parameters for angular galaxy clustering and ARF is critical for (partially)
breaking degeneracies when combining angular galaxy clustering with ARF.

(a) $\@vec{D}_{\rm g}$

(b) $\@vec{D}_{\rm z}$

(c) $\@vec{D}_{\rm g,\,z}$

Figure 13: Correlation between parameters of the $\mathrm{\Lambda CDM}$ model,
for a DESI like survey. The top panel is for the angular galaxy clustering
alone, the central panel is for ARF alone, and the bottom panel is when
combining both observables. We see that the angular galaxy clustering and ARF
have opposite correlation coefficients between cosmological paameters and the
galaxy bias. The combination of both helps greatly in breaking degeneracies
with the galaxy bias.

##  Appendix B Results for $\mathrm{wCDM}$ model

We show the ellipses obtained with our Fisher analysis for the $\mathrm{wCDM}$
model in Fig. 14, for a DESI-like and a Euclid-like suvey. For many parameter
pairs, the degeneracy direction (or ellipse orientation) for angular galaxy
clustering and ARF are similar, although the resulting error ellipse in the
joint $\@vec{D}_{\rm g,\,z}$ probe shrink very significantly in all cases. As
a result, foreseen uncertainties in the parameters are divided by a factor of
at least two for all parameters.

We show in Fig. 15 the correlation matrices for the $\mathrm{wCDM}$ model. It
turns out that for $\@vec{D}_{\rm z}$ the new parameters $w_{0},\,w_{\rm a}$,
together with $\sigma_{8}$, constitute an almost separate (or largely un-
correlated) box with respect to all other parameters (see Fig. 15b). This does
not seem to be the situation for angular galaxy clustering $\@vec{D}_{\rm g}$
(Fig. 15a), although this character remains (to great extent) for the joint
observable set ($\@vec{D}_{\rm g,\,z}$, Fig. 15c).

(a) DESI

(b) Euclid

Figure 14: Foreseen constraints ($1\sigma$ contours) in the CPL cosmological
extension $\mathrm{wCDM}$ for a DESI-like survey (top) and a Euclid-like
survey (bottom), assuming 20 tomographic Gaussian bins of size
$\sigma_{\mathrm{z}}=0.01$. The blue line are the constraints for angular
galaxy clustering alone ($\@vec{D}_{\rm g}$), the orange line are for the ARF
alone ($\@vec{D}_{\rm z}$), and the green line is a joint analysis of both
fields ($\@vec{D}_{\rm g,\,z}$). These contours are marginalised over the
galaxy bias in the 20 redshift bins. The numbers above the parameter PDFs give
the marginalised $1\sigma$ uncertainty of each parameter for each data-vector.

(a) $\@vec{D}_{\rm g}$

(b) $\@vec{D}_{\rm z}$

(c) $\@vec{D}_{\rm g,\,z}$

Figure 15: Correlation between parameters of the $\mathrm{wCDM}$ model,
marginalised over the galaxy bias for a DESI-like survey. The top panel is for
the angular galaxy clustering ($\@vec{D}_{\rm g}$), the central panel refers
to ARF alone ($\@vec{D}_{\rm z}$), and the bottom panel is when combining both
observables ($\@vec{D}_{\rm g,\,z}$). We find that $\@vec{D}_{\rm z}$ show
different correlations compared to $\@vec{D}_{\rm g}$. The combination of both
greatly helps in breaking degeneracies as we can see in panel (c).