11institutetext: 1 Dipartimento di Fisica, Sapienza Università di Roma,
Piazzale Aldo Moro 2, 00185 Roma, Italy
2 INAF-Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monteporzio
Catone, Italy
3 INFN-Sezione di Roma, Piazzale Aldo Moro, 2 - c/o Dipartimento di Fisica,
Edificio G. Marconi, 00185 Roma, Italy
4 Aix-Marseille Université, CNRS/IN2P3, CPPM, Marseille, France
5 Institut d’Estudis Espacials de Catalunya (IEEC), Carrer Gran Capitá 2-4,
08034 Barcelona, Spain
6 Institut de Ciencies de l’Espai (IEEC-CSIC), Campus UAB, Carrer de Can
Magrans, s/n Cerdanyola del Vallés, 08193 Barcelona, Spain
7 Dipartimento di Fisica, Università degli Studi di Torino, Via P. Giuria 1,
10125 Torino, Italy
8 INFN-Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy
9 INAF-Osservatorio Astrofisico di Torino, Via Osservatorio 20, 10025 Pino
Torinese (TO), Italy
10 Institut de Recherche en Astrophysique et Planétologie (IRAP), Université
de Toulouse, CNRS, UPS, CNES, 14 Av. Edouard Belin, 31400 Toulouse, France
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Astroparticle Physics, 24 quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland
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s/n, 08193 Barcelona, Spain
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Orsay, France
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University Street, Montreal, QC H3A 2T8, Canada
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5822, 69622 Villeurbanne, France
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France
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University of Manchester, Oxford Road, Manchester M13 9PL, UK
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Edouard Belin, 31401 Toulouse Cedex 9, France
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16, 69120 Heidelberg, Germany
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Winterthurerstrasse 190, 8057 Zurich, Switzerland
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93/2, 40129 Bologna, Italy
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85 Institute of Space Science, Str. Atomistilor, nr. 409 Măgurele, Ilfov,
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89 Universitäts-Sternwarte München, Fakultät für Physik, Ludwig-Maximilians-
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90 Departamento de Física, FCFM, Universidad de Chile, Blanco Encalada 2008,
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92 Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas
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93 Instituto de Astrofísica e Ciências do Espaço, Faculdade de Ciências,
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94 Universidad Politécnica de Cartagena, Departamento de Electrónica y
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95 Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700
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97 Infrared Processing and Analysis Center, California Institute of
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98 IFPU, Institute for Fundamental Physics of the Universe, via Beirut 2,
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99 Junia, EPA department, 41 Bd Vauban, 59800 Lille, France
100 Instituto de Física Teórica UAM-CSIC, Campus de Cantoblanco, 28049 Madrid,
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101 CERCA/ISO, Department of Physics, Case Western Reserve University, 10900
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Trieste TS, Italy
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108 Dipartimento di Fisica e Scienze della Terra, Università degli Studi di
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109 Istituto Nazionale di Fisica Nucleare, Sezione di Ferrara, Via Giuseppe
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Tiepolo 11, 34131 Trieste, Italy
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di Genova, via Dodecaneso 33, 16146, Genova, Italy
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Spain
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Stockholm University, Stockholm, SE-106 91, Sweden
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Scientifica 1, Roma, Italy
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Arnesano, 73100, Lecce, Italy
133 INFN, Sezione di Lecce, Via per Arnesano, CP-193, 73100, Lecce, Italy
134 Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The
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135 Department of Astrophysical Sciences, Peyton Hall, Princeton University,
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136 Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200
Copenhagen, Denmark

# Euclid preparation

TBD. Forecast impact of super-sample covariance on 3x2pt analysis with Euclid

Euclid Collaboration: D. Sciotti1,2,3<EMAIL_ADDRESS>S. Gouyou
Beauchamps4,5,6 V. F. Cardone2,3 S. Camera7,8,9 I. Tutusaus10,11,12,5 F.
Lacasa11,13 Alexandre Barreira14,15 A. Gorce16 M. Aubert17,18 P. Baratta4 R.
E. Upham19 M. Bonici20 C. Carbone20 S. Casas21 S. Ilić22,23,10 M.
Martinelli2,3 Z. Sakr24,25,10 A. Schneider26 R. Maoli1,2 R. Scaramella2,3 S.
Escoffier4 W. Gillard4 N. Aghanim13 A. Amara27 S. Andreon28 N. Auricchio29 M.
Baldi30,29,31 S. Bardelli29 D. Bonino9 E. Branchini32,33 M. Brescia34,35 J.
Brinchmann36 V. Capobianco9 J. Carretero37,38 F. J. Castander12,5 M.
Castellano2 S. Cavuoti35,39 A. Cimatti40 R. Cledassou23,41 Deceased G.
Congedo42 C. J. Conselice19 L. Conversi43,44 Y. Copin17 L. Corcione9 F.
Courbin45 H. M. Courtois46 M. Cropper47 A. Da Silva48,49 H. Degaudenzi50 J.
Dinis49,48 F. Dubath50 X. Dupac44 S. Dusini51 M. Farina52 S. Farrens53 P.
Fosalba12,5 M. Frailis54 E. Franceschi29 M. Fumana20 S. Galeotta54 B.
Garilli20 B. Gillis42 C. Giocoli29,31 A. Grazian55 F. Grupp56,57 L.
Guzzo58,28,59 S. V. H. Haugan60 W. Holmes61 I. Hook62 F. Hormuth63 A.
Hornstrup64,65 P. Hudelot66 K. Jahnke67 B. Joachimi68 E. Keihänen69 S.
Kermiche4 A. Kiessling61 M. Kunz11 H. Kurki-Suonio70,71 P. B. Lilje60 V.
Lindholm70,71 I. Lloro72 D. Maino58,20,59 O. Mansutti54 O. Marggraf73 K.
Markovic61 N. Martinet74 F. Marulli75,29,31 R. Massey76 S. Maurogordato77 E.
Medinaceli29 S. Mei78 Y. Mellier79,66,80 M. Meneghetti29,31 G. Meylan45 M.
Moresco75,29 L. Moscardini75,29,31 E. Munari54 S.-M. Niemi81 C. Padilla37 S.
Paltani50 F. Pasian54 K. Pedersen82 V. Pettorino83 S. Pires53 G. Polenta84 M.
Poncet23 L. A. Popa85 F. Raison56 R. Rebolo86,87 A. Renzi88,51 J. Rhodes61 G.
Riccio35 E. Romelli54 M. Roncarelli29 R. Saglia89,56 D. Sapone90 B.
Sartoris89,54 M. Schirmer67 P. Schneider73 A. Secroun4 G. Seidel67 S.
Serrano5,6 C. Sirignano88,51 G. Sirri31 L. Stanco51 J.-L. Starck91 P. Tallada-
Crespí92,38 A. N. Taylor42 I. Tereno48,93 R. Toledo-Moreo94 F.
Torradeflot38,92 E. A. Valentijn95 L. Valenziano29,96 T. Vassallo89,54 A.
Veropalumbo28 Y. Wang97 J. Weller89,56 A. Zacchei54,98 G. Zamorani29 J.
Zoubian4 E. Zucca29 A. Biviano54,98 A. Boucaud78 E. Bozzo50 C. Colodro-Conde86
D. Di Ferdinando31 R. Farinelli29 J. Graciá-Carpio56 N. Mauri40,31 C.
Neissner37,38 V. Scottez79,99 M. Tenti96 Y. Akrami100,101,102,103,104 V.
Allevato35,105 C. Baccigalupi106,98,54,107 M. Ballardini108,109,29 F.
Bernardeau110,66 A. Blanchard10 S. Borgani54,111,107,98 A. S. Borlaff112,113
C. Burigana114,96 R. Cabanac10 A. Cappi29,77 C. S. Carvalho93 G.
Castignani75,29 T. Castro54,107,98 G. Cañas-Herrera81,115 K. C. Chambers116 A.
R. Cooray117 J. Coupon50 A. Díaz-Sánchez118 S. Davini33 G. De Lucia54 G.
Desprez119 S. Di Domizio120 J. A. Escartin Vigo56 I. Ferrero60 F. Finelli29,96
L. Gabarra88,51 K. Ganga78 J. Garcia-Bellido100 E. Gaztanaga12,5,27 F.
Giacomini31 G. Gozaliasl70 H. Hildebrandt121 J. Jacobson122 J. J. E.
Kajava123,124 V. Kansal91 C. C. Kirkpatrick69 L. Legrand11 A. Loureiro125,104
J. Macias-Perez126 M. Magliocchetti52 G. Mainetti127 C. J. A. P. Martins128,36
S. Matthew42 L. Maurin13 R. B. Metcalf75 M. Migliaccio129,130 P.
Monaco111,54,107,98 G. Morgante29 S. Nadathur27 A. A. Nucita131,132,133 M.
Pöntinen70 L. Patrizii31 V. Popa85 C. Porciani73 D. Potter26 A.
Pourtsidou42,134 A. G. Sánchez56 E. Sefusatti54,98,107 M. Sereno29,31 P.
Simon73 A. Spurio Mancini47 J. Stadel26 J. Steinwagner56 R. Teyssier135 S.
Toft65,136 M. Tucci50 C. Valieri31 J. Valiviita70,71 M. Viel98,54,106,107

###### Abstract

Context. Deviations from Gaussianity in the distribution of the fields probed
by large-scale structure surveys generate additional terms in the data
covariance matrix, increasing the uncertainties in the measurement of the
cosmological parameters. Super-sample covariance (SSC) is among the largest of
these non-Gaussian contributions, with the potential to significantly degrade
constraints on some of the parameters of the cosmological model under study –
especially for weak lensing cosmic shear.

Aims. We compute and validate the impact of SSC on the forecast uncertainties
on the cosmological parameters for the Euclid photometric survey, and
investigate how its impact depends on the forecast specifics.

Methods. We follow the recipes outlined by the Euclid Collaboration (EC) to
produce $1\sigma$ constraints through a Fisher matrix analysis, considering
the Gaussian covariance alone and adding the SSC term – computed through the
public code PySSC. The constraints are produced both by using Euclid’s
photometric probes in isolation and by combining them in the ‘3$\times$2pt’
analysis.

Results. We meet EC requirements on the forecasts validation, with an
agreement at the 10% level between the mean results of the two pipelines
considered, and find the SSC impact to be non-negligible – halving the Figure
of Merit of the dark energy parameters ($w_{0}$, $w_{a}$) in the 3$\times$2pt
case and substantially increasing the uncertainties on $\Omega_{{\rm
m},0},w_{0}$, and $\sigma_{8}$ for the weak lensing probe. We find photometric
galaxy clustering to be less affected as a consequence of the lower probe
response. The relative impact of SSC does not show significant changes under
variations of the redshift binning scheme, while it is smaller for weak
lensing when marginalising over the multiplicative shear bias nuisance
parameters, which also leads to poorer constraints on the cosmological
parameters. Finally, we explore how the use of prior information on the shear
and galaxy bias changes the SSC impact. It turns out that improving shear bias
priors does not have a significant impact, while galaxy bias must be
calibrated to sub-percent level in order to increase the Figure of Merit by
the large amount needed to achieve the value when SSC is not included.

###### Key Words.:

Cosmology: cosmological parameters – theory – large-scale structure of
Universe – observations

##  1 Introduction

The last decades have witnessed a remarkable improvement in the precision of
cosmological experiments, and consequently in our grasp of the general
properties of the Universe. The $\Lambda$CDM concordance cosmological model
provides an exquisite fit to observational data coming both from the very
early and very late Universe, but, despite its success, the basic components
it postulates are poorly understood. In fact, the nature of the mechanism
responsible for the observed accelerated cosmic expansion (Riess et al. 1998;
Perlmutter et al. 1999), dark energy, and of the component accounting for the
vast majority of the matter content, dark matter, is still unknown. Upcoming
Stage IV surveys like the Vera C. Rubin Observatory Legacy Survey of Space and
Time (LSST, Ivezić et al. 2019), the Nancy Grace Roman Space Telescope
(Spergel et al. 2015), and the Euclid mission (Laureijs et al. 2011) promise
to help deepen our understanding of these dark components and the nature of
gravity on cosmological scales, by providing unprecedented observations of the
large-scale structures (LSS) of the Universe.

Because of their high accuracy and precision, these next-generation
experiments will require accurate modelling of both the theory and the
covariance of the observables under study to produce precise and unbiased
estimates of the cosmological parameters. Amongst the different theoretical
issues to deal with is the super-sample covariance (SSC), a form of sample
variance arising from the finiteness of the survey area. It has been first
introduced for cluster counts in Hu & Kravtsov (2003) – sometimes being
referred to as ‘beat coupling’, see Rimes & Hamilton (2006); Hamilton et al.
(2006) – and has received a lot of attention in recent years (Takada & Hu
2013; Li et al. 2014; Barreira et al. 2018b; Digman et al. 2019; Bayer et al.
2022; Yao et al. 2023). See also Linke et al. (2023) for an insightful
discussion on SSC in real space. From here on Barreira et al. (2018b) will be
cited as B18.

The effect arises from the coupling between ‘super-survey’ modes, with
wavelength $\lambda$ larger than the survey typical size $L=V_{\rm s}^{1/3}$
(where $V_{\rm s}$ is the volume of the survey) and short-wavelength
($\lambda<L$) modes. This coupling is in turn due to the significant nonlinear
evolution undergone by low-redshift cosmological probes (contrary to, for
example, the cosmic microwave background), which breaks the initial
homogeneity of the density field, making its growth position-dependent. In
Fourier space, this means that modes with different wavenumber
$k=2\pi/\lambda$ become coupled. The modulation induced by the super-survey
modes is equivalent to a change in the background density of the observed
region, which affects and correlates all LSS probes. It is accounted for as an
additional, non-diagonal term in the data covariance matrix beyond the
Gaussian covariance, which is the only term that would exist if the random
field under study was Gaussian. Being the most affected by nonlinear dynamics,
the smaller scales are heavily impacted by SSC, where the effect is expected
to be the dominant source of statistical uncertainty for the 2-point
statistics of weak lensing cosmic shear (WL): it has in fact been found to
increase unmarginalised uncertainties up to a factor of about 2 (for a Euclid-
like survey, see Barreira et al. 2018a; Gouyou Beauchamps et al. 2022). In the
case of photometric galaxy clustering (GCph; again, for a Euclid-like survey),
Lacasa & Grain (2019) – hereafter LG19 – found the cumulative signal-to-noise
to be decreased by a factor around $6$ at $\ell_{\rm max}=2000$. These works,
however, either do not take into account marginalised uncertainties or the
variability of the probe responses, do not include cross-correlations between
probes, or do not follow the full specifics of the Euclid survey detailed
below.

The present article has two aims. First, we intend to validate the forecast
constraints on the cosmological parameters both including and neglecting the
SSC term; these are produced using two independent codes, whose only shared
feature is their use of the public Python module
PySSC111https://github.com/fabienlacasa/PySSC222https://pyssc.readthedocs.io/en/latest/index.html
(LG19) to compute the fundamental elements needed to build the SSC matrix.
Second, we investigate the impact of SSC on the marginalised uncertainties and
the dark energy Figure of Merit (FoM), both obtained through a Fisher forecast
of the constraining power of Euclid’s photometric observables.

The article is organized as follows: Sect. 2 presents an overview of the SSC
and the approximations used to compute it. In Sect. 3 we outline the
theoretical model and specifics used to produce the forecasts, while Sect. 4
provides technical details on the codes’ implementation and validation. Then,
we study in Sect. 5 the impact of SSC on Euclid constraints, for different
binning schemes and choices of systematic errors and priors. Finally, we
present our conclusions in Sect. 6.

##  2 SSC theory and approximations

###  2.1 General formalism

Throughout the article, we will work with 2D-projected observables, namely the
angular Power Spectrum (PS), which in the Limber approximation (Limber 1953;
Kaiser 1998) can be expressed as

$C^{AB}_{ij}(\ell)=\int{\rm
d}V\;W^{A}_{i}(z)W^{B}_{j}(z)P_{AB}(k_{\ell},z)\;,$ (1)

giving the correlation between probes $A$ and $B$ in the redshift bins $i$ and
$j$, as a function of the multipole $\ell$; $k_{\ell}=(\ell+1/2)/r(z)$ is the
Limber wavenumber and $W_{i}^{A}(z),W_{j}^{B}(z)$ are the survey weight
functions (WFs), or ?kernels?. Here we consider as the element of integration
${\rm d}V=r^{2}(z)\frac{{\rm d}r}{{\rm d}z}{\rm d}z$ which is the comoving
volume element per steradian, with $r(z)$ being the comoving distance.

The SSC between two projected observables arises because real observations of
the Universe are always limited by a survey window function
$\mathcal{M}(\boldsymbol{x})$. Taking $\mathcal{M}(\boldsymbol{x})$ at a given
redshift, thus considering only its angular dependence
$\mathcal{M}(\hat{\boldsymbol{n}})$333Here we do not consider a redshift
dependence of $\mathcal{M}(\hat{\boldsymbol{n}})$ but this can happen for
surveys with significant depth variations across the sky. This is discussed in
Lacasa et al. (2018)., with $\hat{\boldsymbol{n}}$ the unit vector on a
sphere, we can define the background density contrast as (Lacasa et al. 2018)

$\delta_{\rm b}(z)=\frac{1}{\Omega_{\rm S}}\int{\rm
d}^{2}\hat{n}\,\mathcal{M}(\hat{\boldsymbol{n}})\,\,\delta_{\rm
m}\left[r(z)\hat{\boldsymbol{n}},z\right]\;,$ (2)

with $r(z)\hat{\boldsymbol{n}}=\boldsymbol{x}$. In this equation, $\delta_{\rm
m}(\boldsymbol{x},z)=\left[\rho_{\rm m}(\boldsymbol{x},z)/\bar{\rho}_{\rm
m}(z)-1\right]$ is the matter density contrast, with $\rho_{\rm
m}(\boldsymbol{x},z)$ the matter density and $\bar{\rho}_{\rm m}(z)$ its
spatial average over the whole Universe at lookback time $z$ and $\Omega_{\rm
S}$ the solid angle observed by the survey.

In other words, $\delta_{\rm b}$ is the spatial average of the density
contrast $\delta_{\rm m}(\boldsymbol{x},z)$ over the survey area:

$\displaystyle\langle\delta_{\rm
m}(\boldsymbol{x},z)\rangle_{\text{universe}}=0\;,$ (3)
$\displaystyle\langle\delta_{\rm
m}(\boldsymbol{x},z)\rangle_{\text{survey}}=\delta_{\rm b}(z)\;.$ (4)

The covariance of this background density contrast is defined as
$\sigma^{2}(z_{1},z_{2})\equiv\langle\delta_{\rm b}(z_{1})\,\delta_{\rm
b}(z_{2})\rangle$ and in the full-sky approximation is given by (Lacasa &
Rosenfeld 2016)

$\sigma^{2}(z_{1},z_{2})=\frac{1}{2\pi^{2}}\int{\rm d}k\;k^{2}\,P_{\rm
mm}^{\,\rm lin}\left(k,z_{12}\right)\,{\rm j}_{0}\left(kr_{1}\right)\,{\rm
j}_{0}\left(kr_{2}\right)\;,$ (5)

with $P_{\rm mm}^{\,\rm lin}(k,z_{12})\equiv D(z_{1})\,D(z_{2})\,P_{\rm
mm}^{\,\rm lin}(k,z=0)$ the linear matter cross-spectrum between $z_{1}$ and
$z_{2}$, $D(z)$ the linear growth factor and ${\rm j_{0}}(kr_{i})$ the first-
order spherical Bessel function, and $r_{i}=r(z_{i})$. The use of the linear
PS reflects the fact that the SSC is caused by long-wavelength perturbations,
which are well described by linear theory. Note that we have absorbed the
$\Omega_{\mathrm{S}}^{-1}$ prefactor of Eq. (2), equal to $4\pi$ in full sky,
in the ${\rm d}V_{i}$ terms, being them the comoving volume element per
steradian.

Depending on the portion of the Universe observed, $\delta_{\rm b}$ will be
different, and in turn the PS of the considered observables
$P_{AB}(k_{\ell},z)$ (appearing in Eq. 1) will react to this change in the
background density through the probe response $\partial
P_{AB}(k_{\ell},z)/\partial\delta_{\rm b}$.

SSC is then the combination of these two elements, encapsulating the
covariance of $\delta_{\rm b}$ and the response of the observables to a change
in $\delta_{\rm b}$; the general expression of the SSC between two projected
observables is (Lacasa & Rosenfeld 2016):

${\rm
Cov_{SSC}}\left[C^{AB}_{ij}(\ell),C^{CD}_{kl}(\ell^{\prime})\right]=\int{\rm
d}V_{1}{\rm d}V_{2}\;W^{A}_{i}(z_{1})\,W^{B}_{j}(z_{1})\,\\\ \times
W^{C}_{k}(z_{2})\,W^{D}_{l}(z_{2})\frac{\partial
P_{AB}(k_{\ell},z_{1})}{\partial\delta_{\rm b}}\,\frac{\partial
P_{CD}(k_{\ell^{\prime}},z_{2})}{\partial\delta_{\rm
b}}\,\sigma^{2}(z_{1},z_{2})\;.$ (6)

We adopt the approximation presented in Lacasa & Grain (2019), which assumes
the responses to vary slowly in redshift with respect to
$\sigma^{2}(z_{1},z_{2})$. We can then approximate the responses with their
weighted average over the $W^{A}_{i}(z)$ kernels (Gouyou Beauchamps et al.
2022):

$\frac{\partial\bar{P}_{AB}(k_{\ell},z)}{\partial\delta_{\rm
b}}=\frac{\int{\rm d}V\;W_{i}^{A}(z)W_{j}^{B}(z)\,\partial
P_{AB}(k_{\ell},z)/\partial\delta_{\rm b}}{\int{\rm
d}V\;W^{A}_{i}(z)W^{B}_{j}(z)}\;,$ (7)

and pull them out of the integral. The denominator on the right-hand side
(r.h.s.) acts as a normalization term, which we call $I^{AB}_{ij}$. We can
further manipulate the above expression by factorising the probe response as

$\frac{\partial P_{AB}(k_{\ell},z)}{\partial\delta_{\rm
b}}=R^{AB}(k_{\ell},z)P_{AB}(k_{\ell},z)\;,$ (8)

where $R^{AB}(k_{\ell},z)$, the response coefficient, can be obtained from
simulations, as in Wagner et al. (2015a, b); Li et al. (2016); Barreira et al.
(2019), or from theory (e.g. via the halo model) as in Takada & Hu (2013);
Krause & Eifler (2017); Rizzato et al. (2019). Following LG19, we can
introduce the probe response of the angular power spectrum $C^{AB}_{ij}(\ell)$
in a similar way, using Eq. (1)

$\displaystyle\frac{\partial C^{AB}_{ij}(\ell)}{\partial\delta_{\rm b}}$
$\displaystyle=\int{\rm d}V\;W_{i}^{A}(z)W_{j}^{B}(z)\,\frac{\partial
P_{AB}(k_{\ell},z)}{\partial\delta_{\rm b}}$ $\displaystyle\equiv
R^{AB}_{ij}(\ell)C^{AB}_{ij}(\ell)\;.$ (9)

Substituting Eq. (8) into the r.h.s. of Eq. (7), using Eq. (2.1) and dividing
by the sky fraction observed by the telescope $f_{\rm
sky}=\Omega_{\mathrm{S}}/4\pi$, we get the expression of the SSC which will be
used throughout this work:

${\rm
Cov_{SSC}}\left[C^{AB}_{ij}(\ell)\,C^{CD}_{kl}(\ell^{\prime})\right]\simeq
f_{\rm sky}^{-1}\left[R^{AB}_{ij}(\ell)\,C^{AB}_{ij}(\ell)\right.\\\
\times\left.R^{CD}_{kl}(\ell^{\prime})\,C^{CD}_{kl}(\ell^{\prime})\,S^{A,B;C,D}_{i,j;k,l}\right]\;.$
(10)

In the above equation, we have defined

$S^{A,B;C,D}_{i,j;k,l}\equiv\int{\rm d}V_{1}{\rm
d}V_{2}\;\dfrac{W^{A}_{i}(z_{1})W^{B}_{j}(z_{1})}{I^{AB}_{ij}}\,\dfrac{W^{C}_{k}(z_{2})W^{D}_{l}(z_{2})}{I^{CD}_{kl}}\,\sigma^{2}(z_{1},z_{2})\,.$
(11)

The $S^{A,B;C,D}_{i,j;k,l}$ matrix (referred to as $S_{ijkl}$ from here on) is
the volume average of $\sigma^{2}(z_{1},z_{2})$, and is a dimensionless
quantity. It is computed through the public Python module PySSC, released
alongside the above-mentioned LG19. A description of the way this code has
been used, and some comments on the inputs to provide and the outputs it
produces, can be found in Sect. 4.

The validity of Eq. (10) has been tested in LG19 in the case of GCph and found
to reproduce the Fisher matrix (FM, Tegmark et al. 1997) elements and signal-
to-noise ratio from the original expression (Eq. 6):

  1. -

within 10% discrepancy up to $\ell\simeq 1000$ for
$R^{AB}_{ij}(k_{\ell},z)={\rm const}$;

  2. -

within 5% discrepancy up to $\ell\simeq 2000$ when using the linear
approximation in scale for $R^{AB}(k_{\ell},z)$ provided in Appendix C of the
same work.

The necessity to push the analysis to smaller scales, as well as to
investigate the SSC impact not only for GCph but also for WL and their cross-
correlation, has motivated a more exhaustive characterization of the probe
response functions, which will be detailed in the next section.
Another approximation used in the literature has been presented in (Krause &
Eifler 2017): the $\sigma^{2}(z_{1},z_{2})$ term is considered as a Dirac
delta in $z_{1}=z_{2}$. This greatly simplifies the computation, because the
double redshift integral ${\rm d}V_{1}{\rm d}V_{2}$ collapses to a single one.
This approximation is used by the other two available public codes which can
compute the SSC: PyCCL (Chisari et al. 2019) and CosmoLike (Krause & Eifler
2017). Lacasa et al. (2018) compared this approximation against the one used
in this work, finding the former to fare better for wide redshift bins (as in
the case of WL), and the latter for narrow bins (as in the case of GCph).

Lastly, we note that in Eq. (10) we account for the sky coverage of the survey
through the full-sky approximation by simply dividing by $f_{\rm sky}$; in the
case of Euclid we have $\Omega_{\mathrm{S}}=14\,700\ \rm{deg}^{2}\simeq
4.4776\ \rm{sr}$. The validity of this approximation has been discussed in
Gouyou Beauchamps et al. (2022), and found to agree at the percent level on
the marginalized parameter constraints with the more rigorous treatment
accounting for the exact survey geometry, when considering large survey areas.
For this test they considered an area of $15\,000\ \rm{deg}^{2}$ and a survey
geometry very close to what Euclid will have, i.e. the full-sky with the
ecliptic and galactic plane removed. Intuitively, the severity of the SSC
decays as $f_{\rm sky}^{-1}$ because larger survey volumes are able to
accommodate more Fourier modes.

Note that we are considering here the maximum sky coverage that Euclid will
reach, i.e. the final data release (DR3). For the first data release (DR1),
the sky coverage will be significantly lower and the full-sky approximation
will not hold. In that case, the partial-sky recipe proposed in Gouyou
Beauchamps et al. (2022) should be considered instead.

###  2.2 Probe response

As mentioned in the previous section, one of the key ingredients of the SSC is
the probe response. To compute this term for the probes of interest, we build
upon previous works (Wagner et al. 2015a, b; Li et al. 2016; Barreira &
Schmidt 2017, B18), and compute the response coefficient of the matter PS as

$R^{\rm mm}(k,z)=\frac{\partial\ln P_{\rm mm}(k,z)}{\partial\delta_{\rm
b}}=1-\frac{1}{3}\frac{\partial\ln P_{\rm mm}(k,z)}{\partial\ln k}+G_{1}^{\rm
mm}(k,z)\;.$ (12)

$G_{1}^{\rm mm}(k,z)$ is called the growth-only response; it is constant and
equal to 26/21 in the linear regime and it can be computed in the nonlinear
regime using separate universe simulations, as done in Wagner et al. (2015b),
whose results have been used in B18 (and in the present work). The latter uses
a power law to extrapolate the values of the response for $k>k_{\rm max}$,
with $k_{\rm max}$ being the maximum wavenumber at which the power spectrum is
reliably measured from the simulations. Further details on this extrapolation,
as well as on the redshift and scale dependence of $R^{\rm mm}$, can be found
respectively in Sect. 2 and in the left panel of Fig. 1 of B18. We note that
$R^{\rm mm}$ is the response coefficient of isotropic large-scale density
perturbations; we neglect the contribution from the anisotropic tidal-field
perturbations to the total response of the power spectrum (and consequently to
the SSC), which has been shown in B18 to be subdominant for WL with respect to
the first contribution (about 5% of the total covariance matrix at
$\ell\gtrsim 300$).

The probes considered in the present study are WL, GCph and their cross-
correlation (XC); the corresponding power spectra are given by the following
expressions

$P_{AB}(k,z)=\left\\{\begin{array}[]{ll}\displaystyle{P_{\rm
mm}(k,z)}&\displaystyle{A=B={\rm L}}\\\ &\\\ \displaystyle{b_{(1)}(z)P_{\rm
mm}(k,z)}&\displaystyle{A={\rm L}\,,\ \ B={\rm G}}\\\ &\\\
\displaystyle{b^{2}_{(1)}(z)P_{\rm mm}(k,z)}&\displaystyle{A=B={\rm G},}\\\
\end{array}\right.$ (13)

with $({\rm L,G})$ for (shear, position), $P_{\rm mm}(k,z)$ the nonlinear
matter PS and $b_{(1)}(z)$ the linear, scale-independent and deterministic
galaxy bias. A comment is in order about the way we model the galaxy-matter
and galaxy-galaxy power spectra. We are indeed using a linear bias, but the
nonlinear recipe for the matter power spectrum $P_{\rm mm}(k,z)$. This is
reminiscent of the hybrid 1-loop perturbation theory (PT) model adopted by,
e.g., the DES Collaboration in the analysis of the latest data release (Krause
et al. 2021; Pandey et al. 2022), but we drop the higher-order bias terms.
This simplified model has been chosen in order to be consistent with the IST:F
(Euclid Collaboration: Blanchard et al. 2020, from hereon EC20) forecasts,
against which we compare our results (in the Gaussian case) to validate them.
We are well aware that scale cuts should be performed in order to avoid
biasing the constraints, but we are here more interested in the relative
impact of SSC on the constraints than the constraints themselves. Any
systematic error due to the approximate modelling should roughly cancel out in
the ratio we will compute later on. Note also that we choose to include a
perfectly Poissonian shot noise term in the covariance matrix, rather than in
the signal, as can be seen in Eq. (25). The responses for the different probes
can be obtained in terms444Since we are using the nonlinear matter power
spectrum $P_{\rm mm}(k,z)$, we do not force $R^{\rm mm}(k,z)$ to reduce to its
linear expression, that is to say, we do not set $G_{1}^{\rm mm}=26/21$ in Eq.
(12). of $R^{\rm mm}(k,z)$ by using the relations between matter and galaxy PS
given above

$R^{\rm gg}(k,z)=\frac{\partial\ln P_{\rm gg}(k,z)}{\partial\delta_{\rm
b}}=R^{\rm mm}(k,z)+2b_{(1)}^{-1}(z)\left[b_{(2)}(z)-b_{(1)}^{2}(z)\right],$
(14)

and similarly for $R^{\rm gm}$:

$R^{\rm gm}(k,z)=\frac{\partial\ln P_{\rm gm}(k,z)}{\partial\delta_{\rm
b}}=R^{\rm mm}(k,z)+b_{(1)}^{-1}(z)\left[b_{(2)}(z)-b_{(1)}^{2}(z)\right].$
(15)

Having used the definitions of the first and second-order galaxy bias, i.e.,
$b_{(1)}(z)=(\partial n_{\rm g}/\partial\delta_{\rm b})/n_{\rm g}$ and
$b_{(2)}(z)=(\partial^{2}n_{\rm g}/\partial\delta_{\rm b}^{2})/n_{\rm g}$,
with $n_{\rm g}$ the total angular galaxy number density, in ${\rm
arcmin}^{-2}$. In the following, where there is no risk of ambiguity, we will
drop the subscript in parenthesis when referring to the first-order galaxy
bias – i.e., $b(z)=b_{(1)}(z)$ – to shorten the notation, and we will indicate
the value of the first-order galaxy bias in the $i$-th redshift bin with
$b_{i}(z)$. More details on the computation of these terms can be found in
Sect. 3.6. Note that Eqs. (14)–(15) are obtained by differentiating a PS model
for a galaxy density contrast defined with respect to (w.r.t.) the observed
galaxy number density, and so they already account for the fact that the
latter also “responds” to the large scale perturbation $\delta_{\rm b}$. This
is also the reason why $R^{\rm GG}_{ij}(\ell)$ can have negative values: for
galaxy clustering, the (number) density contrast $\delta_{\rm gal}$ is
measured w.r.t. the observed, local number density ${\bar{n}}_{\rm gal}$:
$\delta_{\rm gal}=n_{\rm gal}/{\bar{n}}_{\rm gal}-1$ . The latter also
responds to a background density perturbation $\delta_{\rm b}$, and it can
indeed happen that ${\bar{n}}_{\rm gal}$ grows with $\delta_{\rm b}$ faster
than $n_{\rm gal}$, which leads to $\delta_{\rm gal}$ decreasing with
increasing $\delta_{\rm b}$ (which also implies $\partial C^{\rm
GG}_{ij}(\ell)/\partial\delta_{\rm b}<0$). We also stress the fact that the
second-order galaxy bias appearing in the galaxy-galaxy and galaxy-lensing
response coefficients is not included in the signal, following EC20. Once
computed in this way, the response coefficient can be projected in harmonic
space using Eq. (2.1), and inserted in Eq. (10) to compute the SSC in the LG19
approximation. The projected $R^{AB}_{ij}(\ell)$ functions are shown in Fig.
1.

Figure 1: Projected response coefficients for the WL and GCph probes and their
cross-correlation, for the central redshift bin ($0.8\lesssim z\lesssim 0.9$)
– the shape and amplitude of the functions for different redshift pairs are
analogous. For WL, the baryon acoustic oscillations wiggles are smoothed out
by the projection, due to the kernels being larger than the GCph ones. The
different amplitude of the response is one of the main factors governing the
severity of SSC.

##  3 Forecasts specifics

In order to forecast the uncertainties in the measurement of the cosmological
parameters, we will follow the prescriptions of the Euclid forecast validation
study (EC20), with some updates to the most recent results from the EC, which
are used for the third Science Performance Verification (SPV) of Euclid before
launch. In particular, the update concerns the fiducial value of the linear
bias, the redshift distribution $n(z)$ and the multipole binning.

Once again, the observable under study is the angular PS of probe $A$ in
redshift bin $i$ and probe $B$ in redshift bin $j$, given in the Limber
approximation by Eq. (1). The $P_{AB}(k_{\ell},z)$ multi-probe power spectra
are given in Eq. (13); in the following, we will refer interchangeably to the
probes (WL, XC, GCph) and their auto- and cross-spectra (respectively, LL, GL,
GG).

###  3.1 Redshift distribution

First, we assume that the same galaxy population is used to probe both the WL
and the GCph PS. We therefore set

$n_{i}^{\rm L}(z)=n_{i}^{\rm G}(z)=n_{i}(z)\;,$ (16)

where $n_{i}^{\rm L}(z)$ and $n_{i}^{\rm G}(z)$ are respectively the
distribution of sources and lenses in the $i$-th redshift bin. Then, the same
equality applies for the total source and lens number density, $\bar{n}^{\rm
L}$ and $\bar{n}^{\rm G}$.

A more realistic galaxy redshift distribution than the analytical one
presented in EC20 can be obtained from simulations. We use the results from
Euclid Collaboration: Pocino et al. (2021), in which the $n(z)$ is constructed
from photometric redshift estimates in a 400 ${\rm deg}^{2}$ patch of the
Flagship 1 simulation (Potter et al. 2017), using the training-based
directional neighbourhood fitting algorithm (DNF, De Vicente et al. 2016).

The training set is a random subsample of objects with true (spectroscopic)
redshifts known from the Flagship simulation. We choose the fiducial case
presented in Euclid Collaboration: Pocino et al. (2021), which takes into
account a drop in completeness of the spectroscopic training sample with
increasing magnitude. A cut in magnitude $I_{\scriptscriptstyle\rm E}<24.5$,
isotropic and equal for all photometric bands, is applied, corresponding to
the optimistic Euclid setting. The DNF algorithm then produces a first
estimate of the photo-$z$, $z_{\rm mean}$, using as metric the objects’
closeness in colour and magnitude space to the training samples. A second
estimate of the redshift, $z_{\rm mc}$, is computed from a Monte Carlo draw
from the nearest neighbour in the DNF metric. The final distributions for the
different redshift bins, $n_{i}(z)$, are obtained by assigning the sources to
the respective bins using their $z_{\rm mean}$, and then taking the histogram
of the $z_{\rm mc}$ values in each of the bins – following what has been done
in real surveys such as the Dark Energy Survey (Crocce et al. 2019; Hoyle et
al. 2018).

As a reference setting, we choose to bin the galaxy distribution into ${\cal
N}_{\rm b}=$ 10 equipopulated redshift bins, with edges

$\displaystyle z_{\rm edges}=\\{$ $\displaystyle
0.001,0.301,0.471,0.608,0.731,0.851,$ $\displaystyle
0.980,1.131,1.335,1.667,2.501\\}\;.$ (17)

The total galaxy number density is $\bar{n}=28.73\,{\rm arcmin}^{-2}$. As a
comparison, this was set to $30\,{\rm arcmin}^{-2}$ in EC20. Note that this
choice of redshift binning will be discussed and varied in Sect. 5.4.

###  3.2 Weight functions

We model the radial kernels, or weight functions, for WL and GCph following
once again EC20. Adopting the eNLA (extended nonlinear alignment) prescription
for modelling the intrinsic alignment (IA) contribution, the weight function
${\cal W}_{i}^{A}(z)$ for the lensing part is given by (see e.g. Kitching et
al. 2017; Kilbinger et al. 2017; Taylor et al. 2018)

${\cal{W}}_{i}^{\rm L}(z)={\cal{W}}_{i}^{\gamma}(z)-\frac{{\cal{A}}_{\rm
IA}{\cal{C}}_{\rm IA}\Omega_{{\rm m},0}{\cal{F}}_{\rm
IA}(z)}{D(z)}{\cal{W}}^{\rm IA}(z)\;,$ (18)

where we have defined555Equation (19) assumes the Universe is spatially flat.
For the general case, one must replace the term in brackets with
$f_{K}(r^{\prime}-r)/f_{K}(r^{\prime})$, with $f_{K}(r)$ the function giving
the comoving angular-diameter distance in a non-flat universe.

${\cal{W}}_{i}^{\gamma}(z)=\frac{3}{2}\left(\frac{H_{0}}{c}\right)^{2}\Omega_{{\rm
m},0}(1+z)r(z)\int_{z}^{z_{\rm
max}}{\frac{n_{i}(z^{\prime})}{\bar{n}}\left[1-\frac{r(z)}{r(z^{\prime})}\right]\;{\rm
d}z^{\prime}},$ (19)

and

${\cal{W}}_{i}^{\mathrm{IA}}(z)=\frac{1}{c}\frac{n_{i}(z)}{\bar{n}}H(z)\;.$
(20)

Finally, in Eq. (18), ${\cal{A}}_{\rm IA}$ is the overall IA amplitude,
${\cal{C}}_{\rm IA}$ a constant, ${\cal{F}}_{\rm IA}(z)$ a function modulating
the dependence on redshift, and $D(z)$ is the linear growth factor. More
details on the IA modelling are given in Sect. 3.5.

The GCph weight function is equal to the IA one, as long as Eq. (16) holds:

${\cal{W}}_{i}^{\rm
G}(z)={\cal{W}}_{i}^{\mathrm{IA}}(z)=\frac{1}{c}\frac{n_{i}(z)}{\bar{n}}H(z)\;.$
(21)

Fig. 2 shows the redshift dependence of Eqs. (18) and (21), for all redshift
bins. Note that we choose to include the galaxy bias term $b_{i}(z)$ in the PS
(see Eq. 13) rather than in the galaxy kernel, as opposed to what has been
done in EC20. This is done to compute the galaxy response as described in
Sect. 2.2. Since the galaxy bias is assumed constant in each bin, however, the
question is of no practical relevance when computing the $S_{ijkl}$ matrix,
since the constant bias cancels out.

We note that the above definitions of the lensing and galaxy kernels (${\cal
W}_{i}^{A}(z),\ A={\rm L,G}$) differ from the ones used in LG19. This is
simply because of a different definition of the $C_{ij}^{AB}(\ell)$ Limber
integral, which is performed in ${\rm d}V$ in LG19 and in ${\rm d}z$ in EC20.
The mapping between the two conventions is simply given by the expression for
the volume element:

${\rm d}V=r^{2}(z)\frac{{\rm d}r}{{\rm d}z}{\rm
d}z=c\,\frac{r^{2}(z)}{H(z)}{\rm d}z\;,$ (22)

and

$W^{A}_{i}(z)={\cal W}^{A}_{i}(z)/r^{2}(z)\;,$ (23)

with $A={\rm L,G}$. In Fig. 2 we plot the values of ${\cal W}^{A}_{i}(z)$ to
facilitate the comparison with EC20. As outlined in Appendix A, when computing
the $S_{ijkl}$ matrix through PySSC, the user can either pass the kernels in
the form used in LG19 or the one used in EC20 – specifying a non-default
convention parameter.

Figure 2: First two plots: weight functions, or kernels, for the two
photometric probes. The analytic expressions for these are, respectively, Eq.
(18) (left, WL) and Eq. (21) (right, GCph). At high redshifts the IA term
dominates over the shear term in the lensing kernels, making them negative.
The rightmost plot shows the sources (and lenses) redshift distribution per
redshift bin, obtained from the Flagship 1 simulation as described in Sect.
3.1

###  3.3 Gaussian covariance

The Gaussian part of the covariance is given by the following expression

$\displaystyle{\rm Cov_{\rm G}}$
$\displaystyle\left[\hat{C}_{ij}^{AB}(\ell),\hat{C}_{kl}^{CD}(\ell^{\prime})\right]=\left[(2\,\ell+1)\,f_{\rm
sky}\,\Delta\ell\right]^{-1}\delta_{\ell\ell^{\prime}}^{\rm K}$
$\displaystyle\times\Bigg{\\{}\left[C_{ik}^{AC}(\ell)+{N}_{ik}^{AC}(\ell)\right]\left[C_{jl}^{BD}(\ell^{\prime})+N_{jl}^{BD}(\ell^{\prime})\right]$
$\displaystyle+\left[C_{il}^{AD}(\ell)+N_{il}^{AD}(\ell)\right]\left[C_{jk}^{BC}(\ell^{\prime})+N_{jk}^{BC}(\ell^{\prime})\right]\Bigg{\\}}\;,$
(24)

where we use a hat to distinguish the estimators from the true spectra. T
noise PS $N_{ij}^{AB}(\ell)$ are, for the different probe combinations

$N_{ij}^{AB}(\ell)=\left\\{\begin{array}[]{ll}\displaystyle{(\sigma_{\epsilon}^{2}/\bar{n}_{i}^{\rm
L})\,\delta_{ij}^{\rm K}}&\displaystyle{A=B={\rm L}\;\;({\rm WL})}\\\ &\\\
\displaystyle{0}&\displaystyle{A\neq B}\\\ &\\\
\displaystyle{(1/\bar{n}_{i}^{\rm G})\,\delta_{ij}^{\rm
K}}&\displaystyle{A=B={\rm G}\;\;({\rm GCph}})\;.\\\ \end{array}\right.$ (25)

In the above equations $\delta_{ij}^{\rm K}$ is the Kronecker delta and
$\sigma_{\epsilon}^{2}$ the variance of the total intrinsic ellipticity
dispersion of WL sources – where
$\sigma_{\epsilon}=\sqrt{2}\sigma_{\epsilon}^{(i)}$, $\sigma_{\epsilon}^{(i)}$
being the ellipticity dispersion per component of the galaxy ellipse. We note
that the average densities used in Eq. (25) are not the total number
densities, but rather those in the $i$-th redshift bin. In the case of ${\cal
N}_{\rm b}$ equipopulated redshift bins, they can be simply written as
$\bar{n}_{i}^{A}=\bar{n}^{A}/{\cal N}_{\rm b}$ for both $A=({\rm L,G})$.
Finally, we recall that $f_{\rm sky}$ is the fraction of the total sky area
covered by the survey, while $\Delta\ell$ is the width of the multipole bin
centered on a given $\ell$. From Sect. 3.1 we have that $\bar{n}=28.73\,{\rm
arcmin}^{-2}$, while we set $\sigma_{\epsilon}=0.37$ (from the value
$\sigma_{\epsilon}^{(i)}=0.26$ reported in Euclid Collaboration: Martinet et
al. 2019) and $f_{\rm sky}=0.356$ (corresponding to
$\Omega_{\mathrm{S}}=14\leavevmode\nobreak\ 700$ deg2). We have now all the
relevant formulae for the estimate of the Gaussian and the SSC terms of the
covariance matrix. To ease the computation of Eq. (3.3) we have prepared an
optimized Python module,
Spaceborne_covg666https://github.com/davidesciotti/Spaceborne_covg, available
as a public repository.

In the context of the present work, we do not consider the other non-Gaussian
contribution to the total covariance matrix, the so-called connected non-
Gaussian (cNG) term. This additional non-Gaussian term has been shown to be
sub-dominant with respect to the Gaussian and SSC terms for WL both in
Barreira et al. (2018a) and in Upham et al. (2022). For what concerns galaxy
clustering, Wadekar et al. (2020) showed that the cNG term was subdominant,
but this was for a spectroscopic sample so (i) they had a much larger
contribution from shot-noise-related terms compared to what is considered here
for the Euclid photometric sample, and (ii) they considered larger and more
linear scales than in the present study. Lacasa (2020) showed that the cNG
term in the covariance matrix of GCph only impacts the spectral index
$n_{\mathrm{s}}$ and HOD parameters, but there are a few differences between
that analysis and the present work, such as the modelling of galaxy bias. Thus
it is still unclear whether the cNG term has a strong impact on cosmological
constraints obtained with GCph. Quantifying the impact of this term for the
3$\times$2pt analysis with Euclid settings is left for future work.

###  3.4 Cosmological model and matter power spectrum

We adopt a flat $w_{0}w_{a}$CDM model, i.e., we model the dark energy equation
of state with a Chevallier–Polarski–Linder (CPL) parametrisation (Chevallier &
Polarski 2001; Linder 2005)

$w(z)=w_{0}+w_{a}\,z/(1+z)\;.$ (26)

We also include a contribution from massive neutrinos with total mass equal to
the minimum allowed by oscillation experiments (Esteban et al. 2020)
$\sum{m_{\nu}}=0.06\ {\rm eV}$, which we do not vary in the FM analysis. The
vector of cosmological parameters is then

$\boldsymbol{\theta}_{\rm cosmo}=\left\\{\Omega_{{\rm m},0},\Omega_{{\rm
b},0},w_{0},w_{a},h,n_{\rm s},\sigma_{8}\right\\}\;,$ (27)

with $\Omega_{{\rm m},0}$ and $\Omega_{{\rm b},0}$ being respectively the
reduced density of total and baryonic matter today, $h$ is the dimensionless
Hubble parameter defined as $H_{0}=100\,h\ \mathrm{km}\ \mathrm{s}^{-1}\
\mathrm{Mpc}^{-1}$ where $H_{0}$ is the value of the Hubble parameter today,
$n_{\rm s}$ the spectral index of the primordial power spectrum and
$\sigma_{8}$ the root mean square of the linear matter density field smoothed
with a sphere of radius 8
${\mathrm{\mathnormal{h}}}^{-1}\text{\,}\mathrm{Mpc}$. Their fiducial values
are

$\boldsymbol{\theta}_{\rm cosmo}^{\,\rm
fid}=\left\\{0.32,0.05,-1.0,0.0,0.67,0.96,0.816\right\\}\;.$ (28)

This is used as input for the evaluation of the fiducial nonlinear matter PS,
which is obtained using the TakaBird recipe, i.e., the HaloFit version updated
by Takahashi et al. (2012) with the Bird et al. (2012) correction for massive
neutrinos. This recipe is implemented in both
CLASS777https://lesgourg.github.io/class_public/class.html (Blas et al. 2011)
and CAMB888https://camb.info/ (Lewis et al. 2000).

###  3.5 Intrinsic alignment model

We use the eNLA model as in EC20, setting ${\cal{C}}_{\rm IA}=0.0134$ and

${\cal{F}}_{\rm IA}(z)=(1+z)^{\,\eta_{\rm IA}}\left[\langle
L\rangle(z)/L_{\star}(z)\right]^{\,\beta_{\rm IA}}\;,$ (29)

where $\langle L\rangle(z)/L_{\star}(z)$ is the redshift-dependent ratio of
the mean luminosity over the characteristic luminosity of WL sources as
estimated from an average luminosity function (see e.g. Joachimi et al. 2015,
and references therein). The IA nuisance parameters vector is

$\boldsymbol{\theta}_{\rm IA}=\\{{\cal{A}}_{\rm IA},\eta_{\rm IA},\beta_{\rm
IA}\\}\;,$ (30)

with fiducial values

$\boldsymbol{\theta}_{\rm IA}^{\,\rm fid}=\\{1.72,-0.41,2.17\\}\;.$ (31)

All of the IA parameters, except for ${\cal{C}}_{\rm IA}$, will be varied in
the analysis.

###  3.6 Linear galaxy bias and multiplicative shear bias

Following EC20 we model the galaxy bias as scale-independent. We move beyond
the simple analytical prescription of EC20 and use the fitting function
presented in Euclid Collaboration: Pocino et al. (2021), obtained from direct
measurements from the Euclid Flagship galaxy catalogue, based in turn on the
Flagship 1 simulation:

$b(z)=\frac{Az^{B}}{1+z}+C\;,$ (32)

setting $(A,B,C)=(0.81,2.80,1.02)$.

The galaxy bias is modelled to be constant in each bin with the fiducial value
obtained by evaluating Eq. (32) at effective values $z^{\rm eff}_{i}$ computed
as the median of the redshift distribution considering only the part of the
distribution at least larger than 10% of its maximum. The $z^{\rm eff}_{i}$
values obtained in this way are

$\displaystyle z^{\rm eff}=\\{$ $\displaystyle 0.233,0.373,0.455,0.571,0.686,$
$\displaystyle 0.796,0.913,1.070,1.195,1.628\\}\;.$ (33)

We therefore have ${\cal N}_{\rm b}$ additional nuisance parameters

$\boldsymbol{\theta}_{\rm gal.\,bias}=\\{b_{1},b_{2},\ldots,b_{{\cal N}_{\rm
b}}\\}\;,$ (34)

with fiducial values

$\displaystyle\boldsymbol{\theta}^{\rm fid}_{\rm gal.\,bias}=\\{$
$\displaystyle 1.031,1.057,1.081,1.128,1.187,$ (35) $\displaystyle
1.258,1.348,1.493,1.628,2.227\\}\;.$

We can take a further step forward towards the real data analysis by including
the multiplicative shear bias parameters, $m$, defined as the multiplicative
coefficient of the linear bias expansion of the shear field
$\boldsymbol{\gamma}$, see e.g. (Cragg et al. 2023):

$\hat{\boldsymbol{\gamma}}=(1+m)\,\boldsymbol{\gamma}+c$ (36)

with $\hat{\boldsymbol{\gamma}}$ the measured shear field,
$\boldsymbol{\gamma}$ the true one, $m$ the multiplicative and $c$ the
additive shear bias parameters (we will not consider the latter in the present
analysis). The multiplicative shear bias can come from astrophysical or
instrumental systematics (such as the effect of the point spread function –
PSF), which affect the measurement of galaxy shapes. We take the $m_{i}$
parameters (one for each redshift bin) as constant and with a fiducial value
of 0 in all bins. To include this further nuisance parameter, one just has to
update the different angular PS as

$\left\\{\begin{array}[]{l}\displaystyle{C_{ij}^{\rm
LL}(\ell)\rightarrow(1+m_{i})(1+m_{j})C_{ij}^{\rm LL}(\ell)}\\\ \\\
\displaystyle{C_{ij}^{\rm GL}(\ell)\rightarrow(1+m_{j})C_{ij}^{\rm
GL}(\ell)}\\\ \\\ \displaystyle{C_{ij}^{\rm GG}(\ell)\rightarrow C_{ij}^{\rm
GG}(\ell)}\;,\\\ \end{array}\right.$ (37)

where $m_{i}$ is the $i$-th bin multiplicative bias, and the GCph spectrum is
unchanged since it does not include any shear term. We will then have:

$\boldsymbol{\theta}_{\rm shear\,bias}=\\{m_{1},m_{2},\ldots,m_{{\cal N}_{\rm
b}}\\}\;,$ (38)

with fiducial values

$\boldsymbol{\theta}_{\rm shear\,bias}^{\,\rm
fid}=\left\\{0,0,\ldots,0\right\\}\;.$ (39)

These nuisance parameters – except the multiplicative shear bias ones, unless
specified – are varied in the Fisher analysis so that the final parameters
vector is

$\boldsymbol{\theta}=\boldsymbol{\theta}_{\rm
cosmo}\cup\boldsymbol{\theta}_{\rm IA}\cup\boldsymbol{\theta}_{\rm
gal.\,bias}\cup\boldsymbol{\theta}_{\rm shear\,bias},$

and

$\boldsymbol{\theta}^{\,\rm fid}=\boldsymbol{\theta}_{\rm cosmo}^{\,\rm
fid}\cup\boldsymbol{\theta}_{\rm IA}^{\,\rm fid}\cup\boldsymbol{\theta}_{\rm
gal.\,bias}^{\,\rm fid}\cup\boldsymbol{\theta}_{\rm shear\,bias}^{\,\rm
fid}\;,$

both composed of ${\cal N}_{\rm p}=7+3+2{\cal N}_{\rm b}=2{\cal N}_{\rm b}+10$
elements.

####  3.6.1 Higher-order bias

In order to compute the galaxy-galaxy and galaxy-galaxy lensing probe response
terms (Eqs. 14 and 15) we need the second-order galaxy bias $b_{(2)}(z)$. To
do this we follow Appendix C of LG19, in which this is estimated following the
halo model999We neglect here the response of $\langle N|M\rangle$ to a
perturbation $\delta_{\rm b}$ in the background density. as (Voivodic &
Barreira 2021; Barreira et al. 2021)

$b_{(i)}(z)=\int{{\rm d}M\;\Phi_{\rm MF}(M,z)b_{(i)}^{\rm h}(M,z)\langle
N|M\rangle/n_{\rm gal}(z)},$ (40)

with

$n_{\rm gal}(z)=\int{{\rm d}M\;\Phi_{\rm MF}(M,z)\langle N|M\rangle},$ (41)

the galaxy number density, $\Phi_{\rm MF}(M,z)$ the halo mass function (HMF),
$b_{(i)}^{\rm h}(M,z)$ the $i$-th order halo bias, and $\langle N|M\rangle$
the average number of galaxies hosted by a halo of mass $M$ at redshift $z$
(given by the halo occupation distribution, HOD). These are integrated over
the mass range $\log M\in[9,16]$, with the mass expressed in units of solar
masses. The expression for the $i$-th order galaxy bias (Eq. 40) is the same
as Eq. (C.2) of LG19, but here we are neglecting the scale dependence of the
bias evaluating it at $k=0$ so that $u(k\,|\,M=0,z)=1$, $u(k\,|\,M,z)$ being
the Fourier Transform of the halo profile. Strictly speaking, this gives us
the large-scale bias, but it is easy to check that the dependence on $k$ is
negligible over the range of interest.

Although Eq. (40) allows the computation of both the first and second-order
galaxy bias, we prefer to use the values of $b_{(1)}(z)$ measured from the
Flagship simulation for the selected galaxy sample; this is to maintain
consistency with the choices presented at the beginning Sect. 3.6. For each
redshift bin, we vary (some of) the HOD parameters to fit the measured
$b_{(1)}(z)$, thus getting a model for $b_{(1)}^{\rm h}(z)$. We then compute
$b_{(2)}^{\rm h}(z)$ using as an additional ingredient the following relation
between the first and second-order halo bias, which approximates the results
from separate universe simulations (Lazeyras et al. 2016) within the fitting
range $1\lesssim b_{(1)}^{\rm h}\lesssim 10$:

$\displaystyle b_{(2)}^{\rm h}(M,z)$ $\displaystyle=0.412-2.143\,b_{(1)}^{\rm
h}(M,z)$ $\displaystyle+0.929\,\left[b_{(1)}^{\rm
h}(M,z)\right]^{2}+0.008\,\left[b_{(1)}^{\rm h}(M,z)\right]^{3}\;.$ (42)

Finally, we plug the $b_{(2)}^{\rm h}$ values obtained in this way back into
Eq. (40) to get the second-order galaxy bias. The details of the HMF and HOD
used and of the fitting procedure are given in Appendix B.

###  3.7 Data vectors and Fisher matrix

Up to now, we have been fully general without making any assumptions about the
data. We now need to set data-related quantities.

First, we assume to measure $C_{ij}^{AB}(\ell)$ in 10 equipopulated redshift
bins over the redshift range $(0.001,2.5)$. When integrating Eq. (1) in ${\rm
d}z$, $z_{\rm max}$ must be larger than the upper limit of the last redshift
bin to account for the broadening of the bin redshift distribution due to
photo-$z$ uncertainties. We have found that the $C_{ij}^{AB}(\ell)$ stop
varying for $z_{\rm max}\geq 4$, which is what we take as the upper limit in
the integrals over $z$. This also means that we need to extrapolate the bias
beyond the upper limit of the last redshift bin; we then take its value as
constant and equal to the one in the last redshift bin, that is,
$b(z>2.501)=b_{10}$.

Second, we assume the same multipole limits as in EC20, hence examining two
scenarios, namely

  * -

pessimistic:

$(\ell_{\rm min},\ell_{\rm max})=\left\\{\begin{array}[]{ll}(10,1500)&{\rm
for\ WL}\\\ &\\\ (10,750)&{\rm for\ GCph\ and\ XC}\end{array}\right.\;,$

  * -

optimistic:

$(\ell_{\rm min},\ell_{\rm max})=\left\\{\begin{array}[]{ll}(10,5000)&{\rm
for\ WL}\\\ &\\\ (10,3000)&{\rm for\ GCph\ and\ XC}\end{array}\right.\;.$

Then, for the multipole binning, instead of dividing these ranges into ${\cal
N}_{\ell}$ (logarithmically equispaced) bins in all cases as is done in EC20,
we follow the most recent prescriptions of the EC and proceed as follows:

  * •

we fix the centers and edges of 32 bins (as opposed to 30) in the $\ell$ range
$[10,5000]$ following the procedure described in Appendix C. This will be the
$\ell$ configuration of the optimistic WL case.

  * •

The bins for the cases with $\ell_{\rm max}<5000$, such as WL pessimistic,
GCph or XC, are obtained by cutting the bins of the optimistic WL case with
$\ell_{\rm center}>\ell_{\rm max}$. This means that instead of fixing the
number of bins and having different bins’ centers and edges as done in EC20,
we fix the bins’ centers and edges and use a different number of bins,
resulting in, e.g., ${\cal N}_{\ell}^{\,\rm WL}>{\cal N}_{\ell}^{\,\rm GCph}$.

The number of multipole bins is then ${\cal N}_{\ell}^{\,\rm WL}=26$ and
${\cal N}_{\ell}^{\,\rm GCph}={\cal N}_{\ell}^{\,\rm XC}=22$ in the
pessimistic case and ${\cal N}_{\ell}^{\,\rm WL}=32$ and ${\cal
N}_{\ell}^{\,\rm GCph}={\cal N}_{\ell}^{\,\rm XC}=29$ in the optimistic case.
In all these cases, the angular PS are computed at the center of the $\ell$
bin. As mentioned, we will consider the different probes in isolation, as well
as combine them in the ‘3$\times$2pt’ analysis, which includes three 2-point
angular correlation functions (in harmonic space): $C_{ij}^{\rm
LL}(\ell),C_{ij}^{\rm GL}(\ell)$ and $C_{ij}^{\rm GG}(\ell)$. The $\ell$
binning for the 3$\times$2pt case is the same as for the GCph one.

The covariance matrix and the derivatives of the data vector w.r.t. the model
parameters are the only elements needed to compute the FM elements. The one-
dimensional data vector $\boldsymbol{C}$ is constructed by simply compressing
the redshift and multipole indices (and, in the 3$\times$2pt case, the probe
indices) into a single one, which we call $p$ (or $q$). For Gaussian-
distributed data with a parameter-independent covariance, the FM is given by:

$F_{\alpha\beta}=\frac{\partial\boldsymbol{C}}{\partial\theta_{\alpha}}\,{\rm
Cov}^{-1}\,\frac{\partial\boldsymbol{C}}{\partial\theta_{\beta}}=\sum_{pq}\frac{\partial
C_{p}}{\partial\theta_{\alpha}}\,{\rm Cov}^{-1}_{pq}\,\frac{\partial
C_{q}}{\partial\theta_{\beta}}\;,$ (43)

We note that the size of the 3$\times$2pt covariance matrix quickly becomes
large. For a standard setting with ${\cal N}_{\rm b}=10$ redshift bins there
are respectively (55, 100, 55) independent redshift bin pairs for (WL, XC,
GCph), to be multiplied by the different ${\cal N}_{\ell}$. In general, ${\rm
Cov}$ will be a ${\cal N}_{C}\times{\cal N}_{C}$ matrix with

$\displaystyle{\cal N}_{C}$ $\displaystyle=\bigg{[}{\cal N}_{\rm b}({\cal
N}_{\rm b}+1)/2\bigg{]}\bigg{[}{\cal N}_{\ell}^{\,\rm WL}+{\cal
N}_{\ell}^{\,\rm GCph}\bigg{]}+{\cal N}_{\rm b}^{2}{\cal N}_{\ell}^{\,\rm XC}$
$\displaystyle=\bigg{[}{\cal N}_{\rm b}({\cal N}_{\rm b}+1)+{\cal N}_{\rm
b}^{2}\bigg{]}{\cal N}_{\ell}^{\rm 3{\times}2pt},$ (44)

for the 3$\times$2pt – where the second line represents the case with the same
number of $\ell$ bins for all probes, which is the one under study – and

${\cal N}_{C}=\big{[}{\cal N}_{\rm b}({\cal N}_{\rm b}+1)/2\big{]}\,{\cal
N}_{\ell}^{\,\rm WL/GCph}\;.$ (45)

for the WL and GCph cases. As an example, we will have ${\cal N}_{C}^{\rm
3{\times}2pt,\,opt}=6090$.

Being diagonal in $\ell$, most elements of this matrix will be null in the
Gaussian case. As shown in Fig. 3, this is no longer true with the inclusion
of the SSC contribution, which makes the matrix computation much more
resource-intensive. The use of the Numba JIT
compiler101010https://numba.pydata.org can dramatically reduce the CPU time
from about $260\,{\rm s}$ to about $2.5\,{\rm s}$ for the Gaussian + SSC
3$\times$2pt covariance matrix (the largest under study) on a normal laptop
working in single-core mode.

Figure 3: Correlation matrix in log scale for all the statistics of the
3$\times$2pt data-vector in the G and GS cases. The positive and negative
elements are shown in red and blue, respectively. The Gaussian covariance is
block diagonal (i.e., it is diagonal in the multipole indices, but not in the
redshift ones; the different diagonals appearing in the plot correspond to the
different redshift pair indices, for $\ell_{1}=\ell_{2}$). The overlap in the
WL kernels makes the WL block in the Gaussian + SSC covariance matrix much
more dense than the GCph one.

Given the highly non-diagonal nature of the Gaussian + SSC covariance, we can
wonder whether the inversion of this matrix (which is needed to obtain the FM,
see Eq.43) is stable. To investigate this, we compute the condition number of
the covariance, which is defined as the ratio between its largest and smallest
eigenvalues and in this case of order $10^{13}$. This condition number,
multiplied by the standard numpy float64 resolution ($2.22\times 10^{-16}$),
gives us the minimum precision that we have on the inversion of the matrix, of
about $10^{-3}$. This means that numerical noise in the matrix inversion can
cause, at most, errors of order $10^{-3}$ on the inverse matrix. Hence, we
consider the inversion to be stable for the purpose of this work.

##  4 Forecast code validation

In order to validate the SSC computation with PySSC, we compare the $1\sigma$
forecast uncertainties (which correspond to a 68.3% probability, due to the
assumptions of the FM analysis) obtained using two different codes
independently developed by two groups, which we call A and B. To produce the
FM and the elements needed for its computation (the observables, their
derivatives and the covariance matrix), group A uses a private111111Available
upon request to the author, Davide Sciotti code fully written in Python and
group B uses CosmoSIS121212https://bitbucket.org/joezuntz/cosmosis/wiki/Home
(Jennings et al. 2016). As stated in the introduction, the only shared feature
of the two pipelines is the use of PySSC (to compute the $S_{ijkl}$ matrix).
For this reason, and because the SSC is not considered in isolation but added
to the Gaussian covariance, we compare the forecast results of the two groups
both for the Gaussian and Gaussian + SSC cases.

Following EC20, we consider the results to be in agreement if the discrepancy
of each group’s results with respect to the median – which in our case equals
the mean – is smaller than 10%. This simply means that the A and B pipelines’
outputs are considered validated against each other if

$\left|\frac{\sigma_{\alpha}^{i}}{\sigma^{m}_{\alpha}}-1\right|<0.1\quad{\rm
for}\quad i={\rm
A,B};\quad\sigma^{m}_{\alpha}=\frac{\sigma_{\alpha}^{A}+\sigma_{\alpha}^{B}}{2}\;,$
(46)

with $\sigma_{\alpha}^{A}$ the $1\sigma$ uncertainty on the parameter $\alpha$
for group A. The above discrepancies are equal and opposite in sign for A and
B.

The marginalised uncertainties are extracted from the FM $F_{\alpha\beta}$,
which is the inverse of the covariance matrix ${\rm C}_{\alpha\beta}$ of the
parameters: $(F^{-1})_{\alpha\beta}={\rm C}_{\alpha\beta}$. The
unmarginalised, or conditional, uncertainties are instead given by
$\sigma_{\alpha}^{\rm unmarg.}=\sqrt{1/F_{\alpha\alpha}}$. We then have

$\sigma_{\alpha}=\sigma_{\alpha}^{\rm
marg.}=\sqrt{(F^{-1})_{\alpha\alpha}}\;.$ (47)

The uncertainties found in the FM formalism constitute lower bounds, or
optimistic estimates, on the actual parameters’ uncertainties, as stated by
the Cramér-Rao inequality.

In the following, we normalize $\sigma_{\alpha}$ by the fiducial value of the
parameter $\theta_{\alpha}$, in order to work with relative uncertainties:
$\bar{\sigma}^{i}_{\alpha}=\sigma_{\alpha}^{i}/\theta_{\alpha}^{\,\rm fid};\
\bar{\sigma}^{m}_{\alpha}=\sigma^{m}_{\alpha}/\theta_{\alpha}^{\,\rm fid}$,
again with $i={\rm A,B}$. If a given parameter has a fiducial value of 0, such
as $w_{a}$, we simply take the absolute uncertainty. The different cases under
examination are dubbed ‘G’, or ‘Gaussian’, and ‘GS’, or ‘Gaussian + SSC’. The
computation of the parameters constraints differs between these two cases only
by the covariance matrix used in Eq. (43) to compute the FM

$\rm{Cov}=\begin{cases}\rm Cov_{\rm G}&\rm{Gaussian}\\\ \rm Cov_{\rm
GS}=Cov_{\rm G}+Cov_{SSC}&\rm{Gaussian+SSC}\;.\end{cases}$ (48)

As mentioned before, we repeat the analysis for both Euclid’s photometric
probes taken individually, WL and GCph, as well as for the combination of WL,
GCph and their cross-correlation XC, the 3$\times$2pt.

For the reader wanting to validate their own code, we describe the validation
process in Appendix A. Here we sketch the results of the code validation: in
Fig. 4, we show the percent discrepancy as defined in Eq. (46) for the
3$\times$2pt case. Similar results have been obtained for the GCph and WL
cases, both for the optimistic and pessimistic settings specified in Sect.
3.7. The constraints are all found to satisfy the required agreement level
(less than $10\%$ discrepancy with respect to the mean). In light of these
results, we consider the two forecasting pipelines validated against each
other. All the results presented in this paper are the ones produced by group
A.

Figure 4: Percent discrepancy of the normalized $1\sigma$ uncertainties with
respect to the mean for the WL probe, both in the G and GS cases (optimistic
settings). The index $i={\rm A,B}$ indicates the two pipelines, whilst
$\alpha$ indexes the cosmological parameter. The desired agreement level is
reached in all cases (WL, GCph probes and pessimistic case not shown).

##  5 SSC impact on forecasts

We investigate here how the inclusion of SSC degrades the constraints with
respect to the Gaussian case. To this end, we will look in the following at
the quantity

$\mathcal{R}(\theta)=\sigma_{\rm GS}(\theta)/\sigma_{\rm G}(\theta)\;,$ (49)

where $\sigma_{\rm G}(\theta)$ and $\sigma_{\rm GS}(\theta)$ are the usual
marginalised uncertainties on the parameter $\theta$ computed, as detailed
above, with Gaussian or Gaussian + SSC covariance matrix. We run $\theta$ over
the set of cosmological parameters listed in Eq. (27), i.e.,
$\theta\in\\{\Omega_{{\rm m},0},\Omega_{{\rm b},0},w_{0},w_{a},h,n_{\rm
s},\sigma_{8}\\}$.

In addition we examine the Figure of Merit (FoM) as defined in Albrecht et al.
(2006), a useful way to quantify the joint uncertainty on several parameters.
We parameterize the FoM following EC20 to focus on the joint uncertainty on
the dark energy equation of state parameters $w_{0}$ and $w_{a}$, such that

${\rm FoM}=\sqrt{{\rm det}(\tilde{F}_{w_{0}w_{a}})}\;.$ (50)

This quantity is inversely proportional to the area of the 2$\sigma$
confidence ellipse in the plane spanned by the parameters $(w_{0},w_{a})$.
$\tilde{F}_{w_{0}w_{a}}$ is the Fisher sub-matrix obtained by marginalising
over all the parameters but $w_{0}$ and $w_{a}$, and is computed by inverting
$F_{\alpha\beta}$ (that is, taking the parameters’ covariance matrix),
removing all the rows and columns but the ones corresponding to $w_{0}$ and
$w_{a}$ and re-inverting the resulting $2\times 2$ matrix.

We will also use the notation ${\cal{R}}({\rm FoM})$ as a shorthand for ${\rm
FoM}_{\rm GS}/{\rm FoM}_{\rm G}$. We note that, since we expect the
uncertainties to be larger for the GS case, we will have
${\cal{R}}(\theta)>1$, and the FoM being inversely proportional to the area of
the uncertainty ellipse, ${\cal{R}}({\rm FoM})<1$.

Table 1: Ratio between the GS and G constraints for all cosmological parameters and the FoM in the reference scenario, for both pessimistic and optimistic assumptions. We remind the reader that in the reference case, we marginalize over the galaxy bias nuisance parameters while holding the multiplicative shear bias ones fixed. $\mathcal{R}(x)$ | $\Omega_{{\rm m},0}$ | $\Omega_{{\rm b},0}$ | $w_{0}$ | $w_{a}$ | $h$ | $n_{\rm s}$ | $\sigma_{8}$ | FoM
---|---|---|---|---|---|---|---|---
WL, Pessimistic | 1.998 | 1.001 | 1.471 | 1.069 | 1.052 | 1.003 | 1.610 | 0.475
WL, Optimistic | 1.574 | 1.013 | 1.242 | 1.035 | 1.064 | 1.001 | 1.280 | 0.451
GCph, Pessimistic | 1.002 | 1.002 | 1.003 | 1.003 | 1.001 | 1.001 | 1.001 | 0.996
GCph, Optimistic | 1.069 | 1.016 | 1.147 | 1.096 | 1.004 | 1.028 | 1.226 | 0.833
3$\times$2pt, Pessimistic | 1.442 | 1.034 | 1.378 | 1.207 | 1.028 | 1.009 | 1.273 | 0.599
3$\times$2pt, Optimistic | 1.369 | 1.004 | 1.226 | 1.205 | 1.018 | 1.030 | 1.242 | 0.622

###  5.1 Reference scenario

Let us start by considering the case with ${\cal N}_{\rm b}=10$ equipopulated
redshift bins, which we will take in the following as a reference. Table 1
gives the values of the ${\cal{R}}$ ratios for the different parameters and
the FoM in both the pessimistic and optimistic scenarios, for the single or
combined probes.

In accordance with previous results in the literature (see e.g. Barreira et
al. 2018a; Upham et al. 2022), we find that the WL constraints are
dramatically affected by the inclusion of SSC. The impact is so severe that
the FoM is reduced by a factor of about $2$ in both the pessimistic and
optimistic scenarios. The marginalised uncertainties worsen by a large factor
for those parameters which correlate the most with the amplitude of the
signal: indeed, the largest ${\cal{R}}(\theta)$ values are obtained for
$(\Omega_{{\rm m,0}},\sigma_{8})$, while ${\cal{R}}(\theta)$ does not
meaningfully deviate from unity for $\theta=(w_{a},h,n_{\rm s})$, and $w_{0}$
sits in between the two extreme cases. This is because the SSC effect is
essentially an unknown shift, or perturbation, in the background density.

The results in Table 1 also show that GCph is not as strongly affected by SSC.
This is an expected result, being the GCph probe response coefficients lower
(in absolute value) than the WL ones, as can be seen in Fig. 1. This is due to
the additional terms that account for the response of the galaxy number
density $n_{\rm g}$ (see Eq. 14), which is itself affected by the super-survey
modes. Moreover, the constraints from GCph alone are obtained by marginalising
over a larger number of nuisance parameters than WL – the galaxy bias
parameters, which are strongly degenerate with the amplitude of the signal.
This works as a sort of effective systematic covariance which makes the SSC
less dominant than in the WL case. Lastly, as can be seen from Fig. 2, all WL
kernels have non-zero values for $z\rightarrow 0$, contrary to the GCph ones.
In this limit, the effective volume probed by the survey tends to 0, hence
making the variance of the background modes $\sigma^{2}$ tend to infinity. We
thus have a larger $S_{ijkl}$ matrix, which is one of the main factors driving
the amplitude of the SSC.

We nevertheless note a $17\%$ decrease of the FoM in the GCph optimistic case,
which is related to the inclusion of nonlinear modes that are more sensitive
to the SSC, as we discuss later.

The full 3$\times$2pt case sits in between the two extremes as a consequence
of the data vector containing the strongly affected WL probe, and the less
affected GCph one. The contribution from the XC probe is again an intermediate
case because of its lower response coefficient, so the final impact on the FM
elements will be intermediate between the WL and GCph cases, as the
${\cal{R}}(\theta)$ values in Table 1 indeed show.

Figure 5: Contour plot for the G and GS constraints, considering the full
3$\times$2pt analysis in the optimistic case, in the reference scenario. For
clarity, the nuisance parameters are shown separately in Fig. 6.

Comparing the optimistic and the pessimistic cases for the two individual
probes, we can see that there is a different behaviour of the SSC as a
function of the maximum multipole. Indeed, for WL the $\mathcal{R}(\theta)$
ratio for the most affected131313This is not the case for the unconstrained
parameters, but the small difference is likely related to numerical artifacts.
parameters is larger in the pessimistic than in the optimistic case. This is
consistent with the results of Upham et al. (2022) showing that the diagonal
elements of the WL total covariance matrix are more and more dominated by the
Gaussian term as we move to higher $\ell$. This is because of the presence of
the scale-independent shape noise in the Gaussian covariance (see Eq. 3.3 for
$A=B={\rm L}$), which largely dominates over the SSC on small scales. As such,
the relative importance of off-diagonal correlations decreases at large $\ell$
which is precisely what happens when moving from the pessimistic to the
optimistic case. This causes the SSC impact to be smaller in the optimistic
case, although we note that the ${\cal{R}}(\theta)$ are still remarkably
large. Indeed, the ${\cal{R}}$ values for the FoM are roughly the same,
pointing to the importance of SSC in both scenarios.

As also seen in Lacasa (2020), we observe the opposite behaviour for the GCph
probe, which is more impacted by the SSC in the optimistic case. This is
because the impact of the shot noise at these scales is lower than the shape
noise for WL, so the SSC still dominates in that multipole range.

In Fig. 5 we show the comparison of the 2D contours for all cosmological
parameters between G and GS in the case of the 3$\times$2pt analysis, in the
optimistic case. Again, we can clearly see that the most impacted parameters
are $\theta=(\Omega_{{\rm m},0},w_{0},\sigma_{8})$. In addition, this shows
that SSC does not seem to strongly affect the correlations between
cosmological parameters.

To conclude this section, it is also worth looking at the impact of SSC on the
astrophysical nuisance parameters. Indeed, although an issue to be
marginalised over when looking at cosmological ones, the IA and the galaxy
bias parameters are of astrophysical interest. We show the impact of SSC on
the constraints on these quantities in Fig. 6, and, as an anticipation of the
next section, we also show the constraints for other WL-related nuisance
parameters, the multiplicative shear bias parameters $m_{i}$.

Figure 6: Percent increase of the marginalised $1\sigma$ uncertainty of the
nuisance parameters, for all probe choices, in the optimistic case and for the
reference scenario.

For IA-related nuisance parameters, the uncertainty increase due to SSC is
lower than 0.5%. The uncertainty on $b_{i}$ and $m_{i}$ in each of the ten
redshift bins is however significantly affected by SSC, showing an increase
between 1 and 14% for $b_{i}$ and between 1 and 18% for $m_{i}$, depending on
the probe combination choice. This is because both of these nuisance
parameters simply act as a multiplicative factor on the power spectrum and are
thus highly degenerated with the effect of SSC. Again, this is because the
first-order effect of SSC is to modulate the overall clustering amplitude
because of a shift in the background density $\delta_{\rm b}$. As mentioned,
this cross-talk between SSC and linear galaxy bias could also explain why the
GCph probe seems less affected by SSC: some of the difference between G and GS
is absorbed by the $b_{i}$ in the marginalisation. This will be also confirmed
for WL in the next section, showing a reduced relative impact of SSC in the
presence of multiplicative shear bias. Note that going beyond the linear
approximation for the modelling of the galaxy bias will add more nuisance
parameters, thus degrading the overall constraints on cosmological parameters
and further reducing the relative degradation of constraints due to SSC.

Finally, comparing how uncertainties on $b_{i}$ and $m_{i}$ react to the
addition of SSC, we can see that surprisingly the $b_{i}$ are more affected in
the 3$\times$2pt case than in the GCph case, while it is the contrary for
$m_{i}$, the uncertainty increase is larger for WL than for 3$\times$2pt. This
difference in the behaviour of the uncertainty increase might come from the
numerous degeneracies existing between these nuisance parameters and the most
constrained cosmological parameters in each case. Though it is not easy to
exactly understand this behaviour, we note that in all cases the
$\mathcal{R}(\theta)$ for these parameters are of the same order of magnitude
and are never completely negligible.

Table 2: Same as Table 1 but removing the flatness prior. $\mathcal{R}(x)$ | $\Omega_{{\rm m},0}$ | $\Omega_{{\rm DE,0}}$ | $\Omega_{{\rm b},0}$ | $w_{0}$ | $w_{a}$ | $h$ | $n_{\rm s}$ | $\sigma_{8}$ | FoM
---|---|---|---|---|---|---|---|---|---
WL, Pessimistic | 2.561 | 1.358 | 1.013 | 1.940 | 1.422 | 1.064 | 1.021 | 1.433 | 0.514
WL, Optimistic | 2.113 | 1.362 | 1.004 | 1.583 | 1.299 | 1.109 | 1.038 | 1.559 | 0.631
GCph, Pessimistic | 1.002 | 1.001 | 1.002 | 1.002 | 1.003 | 1.001 | 1.000 | 1.001 | 0.996
GCph, Optimistic | 1.013 | 1.020 | 1.006 | 1.153 | 1.089 | 1.004 | 1.039 | 1.063 | 0.831
3$\times$2pt, Pessimistic | 1.360 | 1.087 | 1.043 | 1.408 | 1.179 | 1.021 | 1.009 | 1.040 | 0.677
3$\times$2pt, Optimistic | 1.572 | 1.206 | 1.013 | 1.282 | 1.191 | 1.013 | 1.008 | 1.156 | 0.756

###  5.2 Non-flat cosmologies

In the previous section, we investigated the SSC on the cosmological
parameters under the assumption of a flat model. Actually, the requirement on
the FoM assessed in the Euclid Red Book (Laureijs et al. 2011) refers to the
case with the curvature as an additional free parameter to be constrained,
i.e., the non-flat $w_{0}w_{a}$CDM model. This is why in EC20 are also
reported the marginalised uncertainties for the parameter $\Omega_{{\rm
DE,0}}$, with a fiducial value $\Omega_{{\rm DE,0}}^{\rm fid}=1-\Omega_{{\rm
m,0}}$ to be consistent with a flat universe. It is worth wondering which is
the impact of SSC in this case too. This is summarised in Table 2, where we
now also include the impact on $\Omega_{{\rm DE,0}}$.

A comparison with the results in Table 1 is quite hard if we look at the
single parameters. Indeed, opening up the parameter space by removing the
flatness assumption introduces additional degeneracy among the parameters
controlling the background expansion, which are thus less constrained whether
SSC is included or not. We can nevertheless note again that WL is still the
most impacted probe, while GCph is less affected, and the 3$\times$2pt sits in
between. The difference between pessimistic and optimistic scenarios is now
less evident with ${\cal{R}}(\theta)$ increasing or decreasing depending on
the parameter and the probe.

Once more, the most affected parameters for WL are $(\Omega_{{\rm
m,0}},\sigma_{8})$, the uncertainties on which are now further degraded by the
fact that they correlate with the parameter $\Omega_{{\rm DE,0}}$ which is
also affected. Although $(w_{0},w_{a})$ are also degraded by the SSC, a sort
of compensation is at work, so that the overall decrease in the FoM is similar
to the case with the flatness prior. The motivations that make GCph much less
affected still hold when dropping the flatness prior, explaining the
corresponding $\mathcal{R}(\theta)$ values.

We also note an increase of ${\cal{R}}({\rm FoM})$ in the 3$\times$2pt case,
meaning a smaller degradation of the FoM due to SSC. The FoM indeed degrades
by $24\%$ $(32\%)$ in the non-flat case vs. $38\%$ $(40\%)$ for the flat case
in the optimistic (pessimistic) scenario. This can be qualitatively explained
by noting that the decrease of both FoM(G) and FoM(GS) is related to a
geometrical degeneracy which is the same on all scales, whether or not they
are affected by the increase in uncertainty due to the SSC inclusion.

Table 3: Same as Table 2 but adding multiplicative shear bias nuisance parameters. $\mathcal{R}(x)$ | $\Omega_{{\rm m},0}$ | $\Omega_{{\rm DE,0}}$ | $\Omega_{{\rm b},0}$ | $w_{0}$ | $w_{a}$ | $h$ | $n_{\rm s}$ | $\sigma_{8}$ | FoM
---|---|---|---|---|---|---|---|---|---
WL, Pessimistic | 1.082 | 1.049 | 1.000 | 1.057 | 1.084 | 1.034 | 1.025 | 1.003 | 0.917
WL, Optimistic | 1.110 | 1.002 | 1.026 | 1.022 | 1.023 | 1.175 | 1.129 | 1.009 | 0.976
3$\times$2pt, Pessimistic | 1.297 | 1.087 | 1.060 | 1.418 | 1.196 | 1.021 | 1.030 | 1.035 | 0.674
3$\times$2pt, Optimistic | 1.222 | 1.136 | 1.010 | 1.300 | 1.206 | 1.013 | 1.009 | 1.164 | 0.745

###  5.3 Role of nuisance parameters

We can now open up the parameter space by letting the shear bias parameters
introduced in Sect. 3.6 free to vary. We expand the FM by adding these
additional parameters and recompute the ratios of uncertainties with and
without SSC obtaining the results shown141414We do not report here the results
for GCph since they are the same as the ones shown in Table 1, given that
$C_{ij}^{\rm GG}(\ell)$ is unaffected by multiplicative shear bias. in Table
3. We remind the reader that the number of nuisance parameters depends on
which probe (WL or 3$\times$2pt) one is considering. For the WL case, the
${\cal N}_{\rm b}$ multiplicative shear bias parameters add up to the 3 IA
ones leading to the result that the SSC has a very minor impact on the
constraints and on the FoM. The values in Table 3 are actually easily
explained. We recall that ${\cal{R}}(\theta)$ is a ratio between the
constraints with and without the SSC. Adding $m_{i}$ to the cosmological
parameters introduces a degeneracy between $m_{i}$ itself and the parameters
$(\Omega_{\rm m,0},\sigma_{8})$ which set the overall amplitude of
$C_{ij}^{\rm LL}(\ell)$. Such a degeneracy is a mathematical one present on
the whole $\ell$ range, similar to the galaxy bias parameters for GCph. As a
consequence, the constraints on all the parameters and the FoM are strongly
degraded in a way that is independent of the presence of SSC. This is shown in
Fig. 7 and 8, which exhibits the relative uncertainty $\bar{\sigma}$ and the
dark energy FoMs in the G and GS cases for each parameter, if we marginalise
or not on over nuisance parameters. Letting the nuisance parameters free to
vary, i.e. marginalising over them, tends to increase the uncertainty on
cosmological parameters way more than including SSC and this is even more true
when these nuisance parameters are simply multiplicative such as $b_{i}$ and
$m_{i}$.

This is why the ${\cal{R}}$ values drop down to values close to unity when
$m_{i}$ are varied, in contrast to what we have found up to now for WL.
Introducing more nuisance parameters degenerated with the amplitude of the
signal dilutes the SSC effect in a larger error budget; because of this, it is
the relative rather than the absolute impact of SSC that decreases. Indeed,
marginalising over nuisance parameters is formally equivalent to having
additional covariance. Note that this does not mean that adding nuisance
parameters improves the constraints. Indeed, the marginalised uncertainties on
all parameters increase (hence the FoM decreases) with respect to the case
when the multiplicative shear bias is fixed. The degradation is, however, the
same with and without SSC so the ${\cal{R}}(\theta)$ values stay close to
unity.

On the contrary, the results for the 3$\times$2pt case show that the SSC still
matters. The additional information carried by the GCph and XC data allows the
partial breaking of the mathematical degeneracy among $(m_{i},\Omega_{\rm
m,0},\sigma_{8})$ hence making again the scale-dependent increase of the
uncertainties due to the inclusion of SSC important. However, the larger
number of nuisance parameters (from 13 to 23) still introduces additional
degeneracies with the cosmological ones hence alleviating the impact of SSC.
The overall effect is, however, small with the ${\cal{R}}$ values being close
to the ones in Table 2. In particular, the FoM degradation is essentially the
same in both the pessimistic and optimistic cases.

Figure 7: Marginalised and unmarginalised 1$\sigma$ uncertainties on the
cosmological parameters, relative to their corresponding fiducial values, in
both the G and GS cases for 3$\times$2pt, GCph and WL. For WL we show the
results in the case where the shear multiplicative biases are either varied or
fixed, in other words whether we marginalise over all nuisance parameters or
only over the IA ones. Figure 8: Dark energy FoM for marginalised and
unmarginalised constraints in both the G and GS cases, 3$\times$2pt, GCph, WL,
and WL with fixed shear multiplicative biases.

Overall, these results suggest a dependence of the SSC significance on both
the number and type of parameters to be constrained. Qualitatively, we can
argue that SSC is more or less important depending on whether the additional
parameters (with respect to the reference case of a flat model with fixed
shear bias) introduce degeneracies which are or not scale-dependent and how
strong is the degeneracy between these parameters and the amplitude of the
power spectrum. In future works lens magnification effects should be included
in the analysis as it was shown to have a significant impact on cosmological
constraints (Unruh et al. 2020). But from our results we can anticipate that
the inclusion of magnification-related nuisance parameters will further dilute
the impact of SSC.

###  5.4 Dependence on redshift binning

The results summarised in Tables 1–3 have been obtained for a fixed choice of
number and type of redshift bins. We investigate here how they depend on these
settings given that we expect both the G and GS constraints to change as we
vary the number and type of bins. We will consider the case of non-flat
models, fixing the multiplicative shear bias parameters in order to better
highlight the impact of SSC. For this same reason, we will only consider the
WL and 3$\times$2pt cases, since SSC has always a modest impact on GCph.

Figure 9: Ratio between WL and 3$\times$2pt marginalised uncertainties
computed by including or neglecting the SSC contribution, as a function of the
number of redshift bins, for the pessimistic and optimistic cases.

Let us first consider changing the number of redshift bins ${\cal N}_{\rm b}$.
We show the scaling of ${\cal{R}}(\theta)$ as a function of ${\cal N}_{\rm b}$
for the WL and 3$\times$2pt probes, respectively, in Fig. 9 – for both the
pessimistic and optimistic assumptions. The most remarkable result is the weak
dependence of ${\cal{R}}({\rm FoM})$ on ${\cal N}_{\rm b}$ as can be inferred
from the small range spanned by the curves in the bottom right panel. The
scaling of ${\cal{R}}(\theta)$ with ${\cal N}_{\rm b}$ depends, instead, on
the parameter and the probe one is looking at. It is quite hard to explain the
observed trends because of the interplay of different contrasting effects. For
instance, a larger number of bins implies a smaller number density in each
bin, hence a larger shot noise. As a consequence, the SSC contribution to the
total covariance for the diagonal elements will likely be more and more
dominated by the Gaussian component because of the larger shot and shape noise
terms. However, this effect also depends on the scale so that, should the SSC
be the dominant component on the scales to which a parameter is most
sensitive, the impact should still be important. On the other hand, a larger
number of bins also comes with a larger number of nuisance parameters which,
as shown above, leads to a reduction of the SSC impact. Quantifying which
actor plays the major role is hard which explains the variety of trends in the
different panels.

Figure 10: FoM ratio vs the number of EP and ED redshift bins for WL (left)
and 3$\times$2pt (right) in the optimistic scenario.

Figure 11: Left. 3$\times$2pt FoM in the optimistic scenario with and without
SSC as a function of the percentage prior $\varepsilon_{b}$ on the galaxy bias
parameters for $\sigma_{m}=(5,50,100)\times 10^{-4}$ (solid, dashed, dotted
lines). Right. FoM ratio as function of $\varepsilon_{b}$ for the three
$\sigma_{m}$ values in the left panel.

As a further modification to the reference settings, we can change how the
redshift bins are defined. We have up to now considered equipopulated (EP)
bins so that the central bins cover a smaller range in $z$, because of the
larger source number density. As an alternative, we divide the full redshift
range into ${\cal N}_{\rm b}$ bins with equal length (ED), and recompute the
FM forecasts with and without SSC. We show the FoM ratio as a function of the
number of bins for EP and ED bins considering WL (left) and 3$\times$2pt
(right) probes in the optimistic scenario in Fig. 10. Note that finding the
exact number and type of redshift bins used to maximize the constraining power
of Euclid is outside the scope of this paper; this effort is indeed brought
forward in the context of the SPV exercise.

In order to qualitatively explain these results, let us first consider the WL
case. Given that the bins are no longer equipopulated, the number density of
galaxies will typically be larger in the lower redshift bins than in the
higher ones. As a consequence, the larger the number of bins, the higher the
shape noise in the higher redshift bins so that the SSC will be subdominant in
a larger number of bins, which explains why its impact decreases (i.e.,
${\cal{R}}({\rm FoM})$ increases) with ${\cal N}_{\rm b}$. Nevertheless, the
impact of SSC will be larger than in the EP case since SSC will dominate in
the low redshift bins which are the ones with the largest S/N. This effect is,
however, less important, so although ${\cal{R}}({\rm FoM})$ is smaller for ED
than for EP bins, the difference is no larger than 3–5%.

When adding GCph and XC into the game, the impact of SSC is determined by a
combination of contrasting effects. On one hand, we can repeat the same
qualitative argument made for WL also for GCph thus pointing at
${\cal{R}}({\rm FoM})$ increasing with ${\cal N}_{\rm b}$. No shape or shot
noise is included in the XC Gaussian covariance, which is then only determined
by how much shear and position are correlated. The larger the number of bins,
the narrower they are, the smaller the cross-correlation between them hence
the smaller the Gaussian covariance. This in turn increases the number of
elements in the data vector whose uncertainty is dominated by the SSC. Should
this effect dominate, we would observe a decrease of ${\cal{R}}({\rm FoM})$
with ${\cal N}_{\rm b}$ with the opposite trend if it is the variation of the
shape and shot noise to matter the most. This qualitative argument allows us
then to roughly explain the non-monotonic behaviour of ${\cal{R}}({\rm FoM})$
we see in the right panel of Fig. 10. It is worth remarking, however, that the
overall change of ${\cal{R}}({\rm FoM})$ for ED bins over the range in ${\cal
N}_{\rm b}$ is smaller than $\sim 12\%$ which is also the typical value of the
difference between ${\cal{R}}({\rm FoM})$ values for EP and ED bins once
${\cal N}_{\rm b}$ is fixed.

The analysis in this section, therefore, motivates us to argue that the
constraints and FoM degradation due to SSC are quite weakly dependent on the
redshift binning.

Figure 12: ${\rm FoM_{GS}}$ contours in the $(\varepsilon_{b},\sigma_{m})$
plane for ${\rm FoM_{GS}/FoM_{\rm ref}}$ going from 0.8 to 1.1 in steps of
0.05 (from right to left).

###  5.5 Requirements on prior information

The results in the previous paragraph show that the SSC may dramatically
impact the constraints on the cosmological parameters. As a consequence, the
3$\times$2pt FoM is reduced by up to $\sim 24\%$ with respect to the case when
only the Gaussian term is included in the total covariance. This decrease in
the FoM should actually not be interpreted as a loss of information due to the
addition of the SSC. On the contrary, one can qualitatively say that removing
SSC from the error budget is the same as adding information that is not
actually there. It is nevertheless interesting to ask which additional
information must be added to recover the Gaussian FoM, which is usually taken
as a reference for gauging the potential of a survey. This information can
come from priors on the nuisance (or cosmological) parameters. In the
following section, we will investigate the former option by adding Gaussian
priors on the galaxy and multiplicative shear bias parameters. This is easily
done in the FM formalism, by adding $(\sigma_{\alpha}^{\rm p})^{-2}$ to the
appropriate diagonal elements of the G and GS FMs ($\sigma_{\alpha}^{\rm p}$
being the value of the prior uncertainty on parameter $\alpha$).

To this end, we consider the realistic case of a non-flat model plus the
galaxy bias and multiplicative shear bias as nuisance parameters. As a
simplifying assumption, we will assume that all the ${\cal N}_{\rm b}$ bias
values $b_{i}$ are known with the same percentage uncertainty
$\varepsilon_{b}=\sigma_{b}/b_{\rm fid}$, while we put a prior $\sigma_{m}$ on
all the $m_{i}$ parameters (having set the fiducial value $m_{\rm fid}$ to 0).
We then compute the FoM with and without SSC for the 3$\times$2pt probe in the
optimistic scenario and investigate how the ratio ${\cal{R}}({\rm FoM})$
scales with $(\varepsilon_{b},\sigma_{m})$ obtaining the results shown in Fig.
11.

A prior on the nuisance parameters increases both the Gaussian and Gaussian +
SSC FoM so that one could expect their ratio to be independent of the prior
itself. This is not exactly the case since the correlation between different
multipoles introduced by SSC alters the way the prior changes the FM elements.
As a result, we find a non-flat scaling of ${\cal{R}}({\rm FoM})$ as can be
seen from the right panel of Fig. 11. When a strong prior is set on the galaxy
bias (i.e., $\varepsilon_{b}\ll 1$), there is not much gain in improving the
knowledge of the multiplicative shear bias so that the solid, dashed, and
dotted lines (corresponding to three $\sigma_{m}$ values) are quite close to
each other. This is no longer the case for larger $\varepsilon_{b}$ values
(i.e., weak or no prior on the bias): lowering $\sigma_{m}$ has now a larger
impact on ${\cal{R}}({\rm FoM})$. The non-monotonic behaviour of
${\cal{R}}({\rm FoM})$ with $\varepsilon_{b}$ tells us that ${\rm FoM}_{\rm
GS}$ increases with decreasing $\varepsilon_{b}$ faster (slower) than ${\rm
FoM}_{\rm G}$ when the galaxy bias is known with an uncertainty smaller
(higher) than the sub-percent level. Another way to interpret it is that the
information gained in the FoM saturates faster when SSC is included: better
constraints on $\varepsilon_{b}$ do not bring more information as the SSC now
dominates the error budget. However, it is worth stressing that, even for a
strong prior on the multiplicative shear bias, the FoM ratio can actually be
improved by less than a few percent under the (likely unrealistic) assumption
of a sub-percent prior on the galaxy bias.

The need for such strong priors comes from the attempt to retrieve the same
FoM as a Gaussian case. Alternatively, one can also wonder which additional
information must be added through priors to retrieve the idealised FoM value
obtained in forecasts that neglect the SSC. In other words, we look for the
requirements that must be put on the priors $(\varepsilon_{b},\sigma_{m})$ in
order to make ${\rm FoM}_{\rm GS}/{\rm FoM}_{\rm ref}=1$, where ${\rm
FoM}_{\rm ref}=295$ is the FoM computed for a non-flat reference case without
SSC and with no priors on galaxy bias, but a fiducial prior
$\sigma_{m}=5\times 10^{-4}$ on the shear bias. The answer to this question is
shown in Fig. 12 for the optimistic scenario and 10 equipopulated redshift
bins. Some numbers help to better understand how priors can indeed supply the
additional information to retrieve the FoM one would obtain in an ideal case
where SSC is absent. Solving

${\rm FoM}_{\rm GS}(\varepsilon_{b},\sigma_{m})=f\,{\rm FoM}_{\rm ref}$

with respect to $\varepsilon_{b}$, we get

$\varepsilon_{b}=\left\\{\begin{array}[]{ll}\displaystyle{(2.34,1.19,0.86)\,\%}&\displaystyle{{\rm
for}\;\sigma_{m}=0.5\times 10^{-4}}\\\ &\\\
\displaystyle{(2.27,1.18,0.85)\,\%}&\displaystyle{{\rm
for}\;\sigma_{m}=5\times 10^{-4}}\\\ &\\\
\displaystyle{(1.40,0.93,0.72)\,\%}&\displaystyle{{\rm
for}\;\sigma_{m}=100\times 10^{-4}}\;,\\\ \end{array}\right.$

where the three values refer to $f=(0.8,0.9,1.0)$. These numbers (and the
contours in Fig. 12) show that it is indeed possible to compensate for the
degradation due to SSC by adding strong priors on the galaxy bias, which have
a much larger impact on the (G and GS) FoM than strong priors on the
multiplicative shear bias. However, it is worth noticing that it is actually
easier to obtain priors on the multiplicative shear bias provided a sufficient
number of realistic image simulations are produced and fed to the shear
measurement code to test its performance. It is therefore worth wondering how
much the FoM is restored by improving the prior on $m$ for a fixed one on the
bias. We find

$\frac{{\rm FoM_{GS}}}{{\rm
FoM_{ref}}}=\left\\{\begin{array}[]{ll}\displaystyle{(2.87,2.86,2.64)}&\displaystyle{{\rm
for}\;\varepsilon_{b}=0.1\%}\\\ &\\\
\displaystyle{(0.95,0.95,0.88)}&\displaystyle{{\rm
for}\;\varepsilon_{b}=1\%}\\\ &\\\
\displaystyle{(0.76,0.76,0.70)}&\displaystyle{{\rm
for}\;\varepsilon_{b}=10\%}\;,\\\ \end{array}\right.$

with the three values referring to $\sigma_{m}=(0.5,5.0,100)\times 10^{-4}$.
As expected, improving the prior on the multiplicative bias with respect to
the fiducial one (which, we remind, is included in ${\rm FoM_{\rm ref}}$) does
not help a lot in recovering the constraining power. However, a $1\%$ prior on
the galaxy bias can almost fully recover the reference FoM thanks to the
additional information compensating for the presence of SSC.

Investigating whether the priors proposed here can be achieved in practice
(e.g., through theoretical bias models tailored to galaxy clustering data or
N-body hydrodynamic simulations) is outside the aim of this work. We refer the
interested reader to, e.g., Barreira et al. (2021) and Zennaro et al. (2022)
for some preliminary results.

##  6 Conclusions

Precision cosmology asks for precision computation too: previously neglected
theoretical contributions must therefore now be taken into account. Motivated
by this consideration, we have here computed and studied the impact of SSC on
the Euclid photometric survey, exploring how the different probes and their
combination are affected by this additional, non-Gaussian term in the
covariance matrix. The analysis of the impact of SSC on the spectroscopic
survey, which has been shown to be small in Wadekar et al. (2020) for the
Baryon Oscillation Spectroscopic Survey (BOSS) data, is left for future work.
We employed a FM analysis, producing forecasts of the $1\sigma$ marginalised
uncertainties on the measurement of the cosmological parameters of the flat
and non-flat $w_{0}w_{a}$CDM cosmological models. We validated two different
forecast pipelines against the results of EC20, taking as reference survey the
one specified therein, and then updated the galaxy bias and the source
redshift distributions according to the most recent versions presented in
Euclid Collaboration: Pocino et al. (2021). The SSC was computed relying on
the analytical approximations and numerical routines presented in LG19,
interfacing the public code PySSC with two distinct forecast pipelines to
validate the constraints. As a further step forward, we build upon the work of
LG19 by computing the scale and redshift dependence of the response functions
of the different probes, starting from the results of Wagner et al. (2015b)
and Barreira et al. (2018b).

We find the severity of the impact, quantified by the ratio $\sigma_{\rm
GS}/\sigma_{\rm G}$ between the marginalised uncertainties with and without
SSC, to vary substantially between different parameters and probes. For both
WL and GCph, the most affected parameters are $(\Omega_{{\rm
m,0}},w_{0},\sigma_{8})$, while the constraints on $(\Omega_{{\rm
b,0}},h,n_{\rm s})$ are only weakly degraded by SSC. However, there is a great
difference between the two probes in how much the constraints worsen because
of SSC. In agreement with previous results (Upham et al. 2022; Barreira et al.
2018a), we found the WL case to be dramatically impacted by SSC so that the
corresponding FoM is reduced by as much as $55\%$, while GCph is less affected
with the FoM decrease being about $17\%$. The 3$\times$2pt case sits in
between these two since it receives contributions from both extreme cases.
These results are the consequence of a complicated interplay among three
factors. First, SSC originates from the uncertainty in the determination of
the background mean density when measuring it over a finite region. This
prevents determining the overall amplitude of the matter power spectrum hence
increasing the uncertainty on those parameters that concur in setting its
amplitude, mainly $\Omega_{{\rm m,0}}$ and $\sigma_{8}$. Secondly, the
elements of the SSC matrix depend on the amplitude of the response functions.
Thirdly, the impact depends on how large a contribution the signal receives
from the low-$z$ region, where the effective volume probed is smaller, making
the variance of the background modes larger. Both the last two factors are
more severe for WL than for GCph, hence causing the former probe to be more
affected than the latter.
Finally, the deviation of a given element of the GS FM from the Gaussian one
depends also on its correlations: in other words, the degradation of the
constraints on a given parameter can be large if this is strongly correlated
with a parameter severely degraded by SSC. Quantifying the impact of SSC on a
single parameter is therefore quite hard in general, and must be investigated
on a case-by-case basis taking care of the details of the probe and the way it
depends on the parameter of interest.

Nuisance parameters to be marginalised over act as a sort of additional
contribution to the covariance. As such, the importance of both the Gaussian
and SSC contribution to the overall effective covariance becomes less
important when the number of nuisance parameters increases. In order to
consider cases that mimic the most future Euclid data, we have opened up the
parameter space by adding $\Omega_{{\rm DE,0}}$ (i.e., removing the flatness
prior), and the multiplicative shear bias. It turns out that, as long as the
additional parameters have a scale-independent degeneracy with the most
impacted ones, the relative impact of SSC decreases. We stress, however, that
this reduction in the SSC impact does not come for free. On the contrary, the
marginalised uncertainties on the parameters are definitely worsened, but the
degradation is roughly the same whether the SSC is included or not, hence
making the ratio $\sigma_{\rm GS}/\sigma_{\rm G}$ closer to unity for all
parameters and probes. This result can be taken as a warning against investing
too much effort in refining the estimate of the computationally expensive SSC
when no approximations are done. For a Euclid-like survey, the main concern
would indeed be the number of nuisance parameters, which makes less relevant
the impact of the SSC itself.

We furthermore note, in light of the recent theoretical developments presented
in Lacasa et al. (2023), it appears feasible to include the effect of SSC in
the form of nuisance parameters which would be the value of the density
background $\delta_{\mathrm{b}}$ in each redshift bin. This approach is
interesting as it would reduce the complexity of the data covariance matrix
and would allow for a simpler interpretation of the effect of SSC and how it
is correlated to the other cosmological and nuisance parameters.

Variations in the $z$ binning strategy have contrasting effects: a larger
number of bins means a larger number of nuisance parameters (either galaxy
bias or multiplicative shear bias for each bin), which leads to a loss of
constraining power. Moreover, the larger the number of bins, the larger the
Gaussian contribution to the covariance, making the shot and shape noise
dominate over SSC for diagonal elements. On the downside, a larger number of
bins leads to larger data vectors, thus adding information that can partially
compensate for the increase in the covariance. The contrasting effects at play
conspire in such a way that the degradation of the FoM due to SSC ends up
being approximately independent of the number of redshift bins (cfr. Fig. 10).

An interesting development in this sense is to leverage the SSC dependence on
the low-$z$ contribution to investigate whether its impact could be mitigated
by the use of the BNT (Bernardeau-Nishimichi-Taruya) transform (Bernardeau et
al. 2014), which transforms redshift bins in such a way as to increase the
separation between the WL kernels. This will be investigated in a forthcoming
work.

An alternative strategy is to increase the constraining power by adding
information through informative priors, hence recovering the FoM when SSC is
incorrectly neglected. We investigate this possibility by quantifying the
requirements on the prior information needed to recover the Gaussian FoM. Our
results show that the main role is played here by the priors on galaxy bias
parameters, while the FoM recovery quickly saturates with the prior on the
multiplicative shear bias. However, the galaxy bias must be known to sub-
percent level in order to recover $\sim 90\%$ of the Gaussian FoM.
Investigating whether this is possible is outside the scope of this paper. We
nevertheless note that such remarkable prior information is the same as
stating we are able to model the evolution of the bias with redshift. This is
actually quite difficult based on the current knowledge of galaxy formation
processes. Alternatively, one could investigate whether an empirical fitting
formula can be found as a compromise between the need for strong priors on
bias and the number of nuisance parameters.

Although some more work is needed to make the results more robust, e.g. by
comparing the different approximations presented in the literature, we can
conclude that the effect of including the SSC term in the total covariance
matrix of Euclid photometric observables is definitely non-negligible,
especially for WL and 3$\times$2pt. However, the degradation of the
constraints on cosmological parameters depends on the particular probe and the
number and kind of parameters to constrain. The FoM is nevertheless reduced by
$32\%$ ($25\%$) for the 3$\times$2pt probe in the pessimistic (optimistic)
scenario in the case all cosmological (including $\Omega_{\rm DE,0}$) and
nuisance (multiplicative shear bias) parameters are left free to vary. Mining
most of the gold from the actual Euclid photometric data taking into account
the presence of SSC is a daunting task which we will report on in a
forthcoming publication.

###### Acknowledgements.

The computational part of the work has been performed using the Python
programming language, interfaced with scientific packages like astropy
(Astropy Collaboration: Robitaille et al. 2013; Astropy Collaboration: Price-
Whelan et al. 2018) for cosmological calculations, Numba (Lam et al. 2015) for
code speedup, NumPy (Harris et al. 2020) for matrix manipulation, SciPy
(Virtanen et al. 2020) for numerical integration and Matplotlib (Hunter 2007)
for data visualization. DS would like to thank Raphael Kou for the fruitful
discussion on the SSC impact on GCph. SGB was supported by CNES, focused on
Euclid mission. The project leading to this publication has received funding
from Excellence Initiative of Aix-Marseille University -A*MIDEX, a French
”Investissements d’Avenir” programme (AMX-19-IET-008 -IPhU). SC acknowledges
support from the ‘Departments of Excellence 2018-2022’ Grant (L. 232/2016)
awarded by the Italian Ministry of University and Research (mur). IT
acknowledges funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (Grant
agreement No. 863929; project title “Testing the law of gravity with novel
large-scale structure observables” and acknowledges support from the Spanish
Ministry of Science, Innovation and Universities through grant ESP2017-89838,
and the H2020 programme of the European Commission through grant 776247. The
Euclid Consortium acknowledges the European Space Agency and a number of
agencies and institutes that have supported the development of Euclid, in
particular the Academy of Finland, the Agenzia Spaziale Italiana, the Belgian
Science Policy, the Canadian Euclid Consortium, the French Centre National
d’Etudes Spatiales, the Deutsches Zentrum für Luft- und Raumfahrt, the Danish
Space Research Institute, the Fundação para a Ciência e a Tecnologia, the
Ministerio de Ciencia e Innovación, the National Aeronautics and Space
Administration, the National Astronomical Observatory of Japan, the
Netherlandse Onderzoekschool Voor Astronomie, the Norwegian Space Agency, the
Romanian Space Agency, the State Secretariat for Education, Research and
Innovation (SERI) at the Swiss Space Office (SSO), and the United Kingdom
Space Agency. A complete and detailed list is available on the Euclid web site
(http://www.euclid-ec.org).

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##  Appendix A Details of the code validation

In the following Appendix, we provide an overview of the steps undertaken to
compare and validate the codes used in this work, and some of the lessons
learnt in the process.

In order to compute and validate the results we adopt the scheme sketched in
Fig. 13, which highlights the dependency of each main element of the forecast
computation on the others. In particular, we have that:

  1. 1.

The $1\sigma$ constraints are obtained from the FM through Eq. (47), and the
FM is built in turn from the (inverse) covariance matrix and the derivatives
of the angular PS $C_{ij}^{AB}(\ell)$ as indicated in Eq. (43).

  2. 2.

The Gaussian covariance depends on the $C_{ij}^{AB}(\ell)$ through Eq. (3.3)
(and the noise PS, Eq. 25). The SSC also depends on the $C_{ij}^{AB}(\ell)$,
with the added contribution of the $R_{ij}^{AB}(\ell)$ terms and the output of
the PySSC module, the $S_{ijkl}$ matrix – following Eq. (10).

  3. 3.

The $C_{ij}^{AB}(\ell)$ are constructed by convolving the (nonlinear) matter
PS with the lensing and galaxy weight functions, as in Eq. (1). The $S_{ijkl}$
matrix also depends on the weight functions (see Eq. 11), which are in fact
the main external input needed by PySSC, and on the linear matter PS through
the $\sigma^{2}(z_{1},z_{2})$ term (Eq. 5). It is to be noted, however, that
PySSC computes this PS internally, needing only the specification of a
dictionary of cosmological parameters with which to call the Boltzmann solver
CLASS through the Python wrapper classy. This means that we also have to make
sure that the fiducial value of the parameters used to compute the PS of Eq.
(1) are the same ones passed to PySSC (this time to compute the linear PS), in
order to work with the same cosmology.

While to compute the constraints we follow the scheme from right to left,
starting from the basic ingredients to arrive at the final result, the general
trend of the validation is the opposite: we begin by comparing the final
results, then work our way back whenever we find disagreement.

We then start the comparison from the $\sigma_{\alpha}$. If a discrepancy
larger than $10\%$ is found, we check the quantities they depend on, which in
this case are the covariance matrices (see Eq. 43). If these agree, we check
the codes directly. If these disagree, we iterate the process by checking the
subsequent element in the scheme (in this case the $S_{ijkl}$ matrix and the
$C_{ij}^{AB}(\ell)$), until agreement is found. Essentially, this means that
the disagreement in the outputs of the codes at each step can either come from
the inputs, or from the codes themselves. Once the cause of the discrepancy is
found and fixed, the computation is repeated and the process can start again.

The pipelines under comparison are both written in the Python language. One of
them requires as external inputs the weight functions, the angular PS
$C^{AB}_{ij}(\ell)$ and their derivatives with respect to the cosmological
parameters; whilst the other produces these through the use of
CosmoSIS151515https://bitbucket.org/joezuntz/cosmosis/wiki/Home (Jennings et
al. 2016), and hence needs no external inputs but the vectors of fiducial
cosmological and nuisance parameters.

Figure 13: Some of the most important elements examined in the comparison. The
arrows show the ordering followed to produce the parameters constraints, which
is opposite to the one followed to validate the code. The derivatives of the
PS with respect to the cosmological parameters, entering the final step of the
computation, are not shown.

For the reader wishing to repeat the validation, we list below some of the
lessons learnt in the code comparison process.

  * •

PySSC needs as input the WL and GCph kernels of Eqs. (18) and (21), as well as
their argument, the redshift values. The code then uses this redshift array to
perform the necessary integrals in ${\rm d}V$ through Simpson’s rule. The user
is responsible for sampling the kernels on a sufficiently fine $z$ grid
[$O(10^{4})$ values have been found to be sufficient in the present case] to
make sure these integrals are performed accurately.

  * •

The latest version of PySSC accepts a convention parameter. This specifies
whether the kernels are in the form used in LG19 (convention = 0) or the one
prescribed in EC20 (convention = 1). The two differ by a $1/r^{2}(z)$ factor,
as shown in Eq. (23). Passing the kernels in the EC20 form without changing
the parameter’s value from 0 – the default – to 1 will obviously yield
incorrect results.

  * •

The ordering of the $S_{ijkl}$ matrix’s elements depends on the ordering
chosen when passing the input kernels to PySSC – whether $W_{i}^{\rm L}(z)$
first and $W_{i}^{\rm G}(z)$ second or vice versa. This must be kept in mind
when implementing Eq. (10).

  * •

The GCph constraints can show a discrepancy greater the 10% for the dark
energy equation of state parameters $w_{0}$ and $w_{a}$ even when the
corresponding covariance is found to be in good agreement. This discrepancy is
due to GCph being less numerically stable because of the lower constraining
power compared to the other probes, and because the bias model considered has
a strong degeneracy with $\sigma_{8}$, making the numerical derivatives