A probabilistic proof of the spherical excess formula
Daniel A. Klain

A triangle $T$ in the unit sphere with inner angles $\theta_{1}$,
$\theta_{2}$, and $\theta_{3}$ has area given by the spherical excess
formula:111This formula was discovered in 1603 by Thomas Harriot [5, p. 65]
and is also known as Girard’s formula [1, p. 95].

(1) $\displaystyle\mathtt{Area}(T)\;=\;\theta_{1}+\theta_{2}+\theta_{3}-\pi.$

$\theta_{1}$$\theta_{2}$$\theta_{3}$ Figure 1. A spherical triangle.

This note offers a probabilistic proof of the angle excess formula (1), based
on the observation that an unbounded cone at the origin in $\mathbb{R}^{3}$
has only three kinds of $2$-dimensional orthogonal projections: a cone in
$\mathbb{R}^{2}$, a half-plane in $\mathbb{R}^{2}$ (this is an outcome of
probability zero), and all of $\mathbb{R}^{2}$. See Figure 2.

?1 vertex2 edges(measure zerooutcome)0 vertices0 edges Figure 2. Projections
of a 3-dimensional cone.

Observe that, if we omit the middle outcome of measure zero, the number of
edges on each projected figure is twice the number of vertices.

Some notation will help to interpret angles as probabilities. Let $\SS$ denote
the unit sphere in $\mathbb{R}^{3}$ centered at the origin, having surface
area $4\pi$.

Suppose that $P$ is a convex polytope in $\mathbb{R}^{3}$, and let $x$ be any
point of $P$. The solid inner angle $a_{P}(x)$ of $P$ at $x$ is given by

$a_{P}(x)=\\{u\in\SS\;|\;\ x+\epsilon u\in P\;\;\hbox{ for some
}\epsilon>0\\}.$

Let $\alpha_{P}(x)$ denote the measure of the solid angle
$a_{P}(x)\subseteq\SS$, given by the usual surface area measure on subsets of
the sphere.

If $F$ is a proper face of a convex polytope $P$, then the solid inner angle
measure $\alpha_{P}(x)$ is the same at every point $x$ in the relative
interior of $F$. This value will be denoted by $\alpha_{P}(F)$.

Consider the case of an unbounded cone $C$ with single vertex at the origin
$o$, as in Figure 2. Specifically, let $v_{1},v_{2},v_{3}$ be three linearly
independent unit vectors in $\mathbb{R}^{3}$, and let $C$ denote all non-
negative linear combinations:

$C=\\{t_{1}v_{1}+t_{2}v_{2}+t_{3}v_{3}\;|\;t_{i}\geq 0\\}.$

The polyhedral cone $C$ has exactly one vertex at $o$ and three (unbounded)
edges $e_{i}$ along the directions of the vectors $v_{i}$. Note that
$\alpha_{C}(o)$ is the area of the spherical triangle with vertices at
$v_{i}$. Denote the inner angles of this triangle by $\theta_{i}$, as in
Figure 1 (where $o$ lies at the center of the sphere in Figure 1).

Given a uniformly random unit vector $u$, let $C_{u}$ denote the orthogonal
projection of $C$ onto the plane $u^{\perp}$. Evidently $C_{u}$ will resemble
one of the outcomes in Figure 2. Specifically, $C_{u}$ will cover the entire
plane $u^{\perp}$ iff $u$ lies in the interior of $\pm a_{C}(o)$. It follows
that $C_{u}=u^{\perp}$ with probability

$\frac{\mathtt{Area}(a_{C}(o))+\mathtt{Area}(-a_{C}(o))}{4\pi}\;=\;\frac{2\alpha_{C}(o)}{4\pi}\;=\;\frac{\alpha_{C}(o)}{2\pi}.$

Since the number of vertices of $C_{u}$ is either $0$ or $1$, the expected
number of vertices of $C_{u}$ is given by the complementary probability

(2) $\displaystyle E(\hbox{\\# of
vertices})\;=\;1-\frac{\alpha_{C}(o)}{2\pi}.$

Meanwhile, an edge $e$ projects to the interior of $C_{u}$ iff $u$ lies in the
interior of $\pm a_{C}(e)$. Taking the complement as before, $e$ projects to a
boundary edge of $C_{u}$ with probability $1-\frac{\alpha_{C}(e)}{2\pi}$.
Observe that each solid angle measure $\alpha_{C}(e_{i})$ is given by
$2\theta_{i}$ (see Figure 3), so that the expected number of edges of $C_{u}$
is

(3) $\displaystyle E(\hbox{\\# of
edges})\;=\;\sum_{i}\left(1-\frac{\alpha_{C}(e_{i})}{2\pi}\right)\;=\;\sum_{i}\left(1-\frac{\theta_{i}}{\pi}\right)\;=\;3-\frac{\theta_{1}+\theta_{2}+\theta_{3}}{\pi}.$

$\theta$$e$ Figure 3. $\alpha_{C}(e)=2\theta$

Since the number of edges in $C_{u}$ is almost surely twice the number of
vertices (see Figure 2), the identities (2) and (3) imply that

(4) $\displaystyle
3-\frac{\theta_{1}+\theta_{2}+\theta_{3}}{\pi}\;=\;E(\hbox{\\# of
edges})\;=\;2E(\hbox{\\# of vertices})\;=\;2-\frac{\alpha_{C}(o)}{\pi}.$

It is now immediate from (4) that

$\alpha_{C}(o)\;=\;\theta_{1}+\theta_{2}+\theta_{3}-\pi,$

as asserted in (1).

❧

While the argument above generalizes easily to the case of a spherical polygon
with more than 3 sides, in higher dimensions there is a proliferation of cases
that makes this approach much more complicated. However, a variation of this
approach was applied in [2] to an $n$-simplex $\Delta$ to obtain the identity

$\displaystyle
p_{\Delta}\;=\;\frac{2}{n\omega_{n}}\sum_{v}\alpha_{\Delta}(v),$

where $p_{\Delta}$ is the probability that a random orthogonal projection
$\Delta_{u}$ is an $(n-1)$-simplex, $\omega_{n}$ denotes the volume of the
unit ball in $\mathbb{R}^{n}$, and where the sum is taken over all vertices
$v$ of the $n$-simplex $\Delta$.

Similar and more general arguments were used even earlier by Perles and
Shephard [4] (see also [3, p. 315a] and [6]) to give a simple proof of the
Gram-Euler identity for convex polytopes,

$\displaystyle\sum_{F\subseteq\partial P}(-1)^{\dim
F}\alpha_{P}(F)\;=\;(-1)^{n-1}n\omega_{n},$

where the sum is taken over all proper faces $F$ of an $n$-dimensional convex
polytope $P$.

## References

  * [1] H. S. M. Coxeter. (1969). _Introduction to Geometry_ , 2nd ed. New York: Wiley.
  * [2] D. V. Feldman, D. Klain. (2009). Angles as probabilites. _Amer. Math. Monthly_. 116(8):732–735.
  * [3] B. Grünbaum. (2003). _Convex Polytopes_ , 2nd ed. New York: Springer.
  * [4] M. A. Perles, G. C. Shephard. (1967). Angle sums of convex polytopes. _Math. Scand._ 21:199–218.
  * [5] J. Stillwell. (1992). _Geometry of Surfaces_. New York: Springer.
  * [6] E. Welzl. (1994). Gram’s equation — a probabilistic proof. In: _Results and Trends in Theoretical Computer Science – Graz, 1994_ , Lecture Notes in Comput. Sci., vol. 812. Berlin: Springer, pp. 422-424.

Department of Mathematical Sciences, University of Massachusetts Lowell,
Lowell, MA 01854 USA
<EMAIL_ADDRESS>


# The effect of quasar redshift errors on Lyman-$\alpha$ forest correlation
functions

Samantha Youles,1 Julian E. Bautista,1,2 Andreu Font-Ribera,3,4 David Bacon,1
James Rich,5David Brooks,4 Tamara M. Davis,6 Kyle Dawson,7 Axel de la
Macorra,8 Govinda Dhungana,9 Peter Doel,4Kevin Fanning,10 Enrique Gaztañaga,11
Satya Gontcho A Gontcho,12 Alma X. Gonzalez-Morales,13,14Julien Guy,12 Klaus
Honscheid,15,16 Vid Iršič,17,18 Robert Kehoe,9 David Kirkby,19 Theodore
Kisner,12Martin Landriau,12 Laurent Le Guillou,20 Michael E. Levi,12 Paul
Martini,15,21Andrea Muñoz-Gutiérrez,8 Nathalie Palanque-Delabrouille,5,12
Ignasi Pérez-Ràfols,3,20Claire Poppett,12,22,23 César Ramírez-Pérez,3 Michael
Schubnell,10 Gregory Tarlé10 and Michael Walther24,25
1Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama
Building, Portsmouth, PO1 3FX, UK
2Aix Marseille Univ, CNRS/IN2P3, CPPM, Marseille, France
3Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain
4Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
5IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
6School of Mathematics and Physics, University of Queensland, 4072, Australia
7Department of Physics and Astronomy, The University of Utah, 115 South 1400
East, Salt Lake City, UT 84112, USA
8Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México
9Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, TX 75275, USA
10Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
11Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans
s/n, 08913 Bellaterra, Barcelona, Spain
12Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA
13Consejo Nacional de Ciencia y Tecnología, Av. Insurgentes Sur 1582. Colonia
Crédito Constructor, Del. Benito Juárez C.P. 03940, México D.F. México
14Departamento de Física, Universidad de Guanajuato - DCI, C.P. 37150, Leon,
Guanajuato, México
15Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA
16Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA
17Kavli Institute for Cosmology, University of Cambridge, Madingley Road,
Cambridge CB3 0HA, UK
18Cavendish Laboratory, University of Cambridge, 19 J. J. Thomson Ave.,
Cambridge CB3 0HE, UK
19Department of Physics and Astronomy, University of California, Irvine,
92697, USA
20Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France
21Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA
22Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA
23University of California, Berkeley, 110 Sproul Hall #5800 Berkeley, CA
94720, USA
24University Observatory, Faculty of Physics, Ludwig-Maximilians-Universität,
Scheinerstr. 1, 81677 München, Germany
25Excellence Cluster ORIGINS, Boltzmannstrasse 2, D-85748 Garching, Germany
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Using synthetic Lyman-$\alpha$ forests from the Dark Energy Spectroscopic
Instrument (DESI) survey, we present a study of the impact of errors in the
estimation of quasar redshift on the Lyman-$\alpha$ correlation functions.
Estimates of quasar redshift have large uncertainties of a few hundred
$\text{km~{}s}^{-1}\,$due to the broadness of the emission lines and the
intrinsic shifts from other emission lines. We inject Gaussian random redshift
errors into the mock quasar catalogues, and measure the auto-correlation and
the Lyman-$\alpha$-quasar cross-correlation functions. We find a smearing of
the BAO feature in the radial direction, but changes in the peak position are
negligible. However, we see a significant unphysical correlation for small
separations transverse to the line of sight which increases with the amplitude
of the redshift errors. We interpret this contamination as a result of the
broadening of emission lines in the measured mean continuum, caused by quasar
redshift errors, combined with the unrealistically strong clustering of the
simulated quasars on small scales.

###### keywords:

cosmology – large-scale structure of Universe

††pubyear: 2022††pagerange: The effect of quasar redshift errors on
Lyman-$\alpha$ forest correlation functions–The effect of quasar redshift
errors on Lyman-$\alpha$ forest correlation functions

##  1 Introduction

The study of dark energy, as a potential explanation for the accelerated
nature of the expansion of the Universe, demands high precision measurements
of the expansion rate. These measurements are possible with the use of
standard candles or standard rulers, particularly those that are visible out
to large distances, or equivalently, to early cosmic times. Riess et al.
(1998) and Perlmutter et al. (1998) measured the flux and redshift of type-Ia
supernovae (SNIa) which are standardizable candles. This enabled the
calculation of luminosity distance, $D_{L}$, as a function of redshift, which
showed that the expansion of the Universe is accelerating. Since then, both
the quantity and quality of recent SNIa data have contributed to the reduction
of uncertainties on parameters describing dark energy (Scolnic et al., 2018;
Brout et al., 2019).

The accelerated expansion has been confirmed using a completely independent
probe: baryon acoustic oscillations (BAO) as a standard ruler. These acoustic
oscillations in the primordial plasma, prior to recombination, left an imprint
on the large-scale structure of the Universe that corresponds to the size of
the sound horizon, $r_{d}$, at the drag epoch. This scale manifests as a peak
in the matter-density correlation function at comoving separations $\sim$ 147
Mpc, or equivalently, as an oscillatory pattern in the power spectrum. In the
transverse direction, the BAO peak measures the ratio $D_{M}(z)/r_{d}$, where
$D_{M}(z)=(1+z)D_{A}(z)$ is the comoving angular-diameter distance. In the
radial direction, it determines $D_{H}(z)/r_{d}$, where $D_{H}(z)=c/H(z)$ is
the Hubble distance. These observables are used to infer the expansion history
of the Universe and derive cosmological parameters of the models that describe
it.

Since the first BAO measurements (Eisenstein et al., 2005; Cole et al., 2005)
several spectroscopic surveys have been built with the goal of measuring the
BAO scale in the distribution of matter. This distribution has been
traditionally mapped with galaxies as a tracer. As galaxies become relatively
faint above redshifts of $z=1$, quasars have been used to trace the matter
field at those higher redshifts. At $z>2$, a new window has been recently
opened to observe BAO using Lyman-$\alpha$ ( $\text{Ly}\alpha$ ) forests,
features seen in the spectra of quasars, caused by the absorption of light by
neutral hydrogen (Busca et al., 2013; Slosar et al., 2013; Kirkby et al.,
2013; Delubac et al., 2015; Bautista et al., 2017; de Sainte Agathe et al.,
2019) and in cross-correlation with quasars (Font-Ribera et al., 2014; du Mas
des Bourboux et al., 2017; Blomqvist et al., 2019; du Mas des Bourboux et al.,
2020).

An accurate estimate of the redshift of quasars is essential for determining
the 2-point statistics used for cosmological inference. Errors above several
hundred $\text{km~{}s}^{-1}\,$ on the redshift estimates cause the 2-point
functions to be smeared in the line-of-sight direction, reducing the precision
in BAO measurements (even though this smearing is accounted for in the
modelling). The broad emission line centres in the spectra of quasars are not
necessarily good indicators of the host galaxy redshift (also named systemic
redshift) which makes it problematic to obtain precise measurements. Due to
the complex dynamic of the line-emitting regions in quasars, the broad line
centres can be shifted with respect to their expected locations in the quasar
rest-frame. Using spectra with high signal to noise ratio from 32 epochs in
the Sloan Digital Sky Survey Reverberation Mapping project (SDSS-RM), Shen et
al. (2016) calculated the relative velocity shifts between pairs of lines, and
between lines and the systemic redshift when available. The systemic redshifts
of quasars were obtained when narrow emission lines and/or stellar absorption
lines could be observed. Broad high-ionisation lines, such as CIII, are
typically blue-shifted by tens or hundreds of $\text{km~{}s}^{-1}\,$ and may
have a strong luminosity dependence. Conversely the velocity shifts tend to be
smaller for low-ionisation lines such as MgII, with no luminosity dependence.
This study enabled the authors to derive empirical recipes for unbiased
estimation of redshift with uncertainties based on various lines across a
range of redshifts. Template fitting software, such as redrock,
111https://github.com/desihub/redrock uses templates built from a principal
component analysis (PCA), which cannot fully account for all spectral
variations of quasars.

The eBOSS collaboration (Dawson et al., 2016; Alam et al., 2021) used several
different redshift estimators for quasars, all included in the official SDSS
quasar catalogue (Lyke et al., 2020) from the Data Release 16 (Ahumada et al.,
2020). In the clustering analysis of DR16 $\text{Ly}\alpha$ forests, du Mas
des Bourboux et al. (2020) performed a detailed study of the impact of
different redshift estimators on the BAO parameters, using both real data and
mock catalogues. From mocks, they observed that variations in the uncertainty
of BAO best-fit parameters of $\sim 0.5\sigma$ with different redshift
estimators, was consistent with statistical error. Analogously, photometric
redshift uncertainties in galaxy clustering and BAO measurements have been
found to reduce the constraining power on the Hubble parameter (Chaves-Montero
et al., 2018).

The Dark Energy Spectroscopic Instrument (DESI, DESI Collaboration et al.
2016) has recently begun a five-year programme of observations, in which it
will observe three times more $z>2$ quasars than SDSS. They will be used for
clustering measurements with the $\text{Ly}\alpha$ forest. DESI is expected to
produce roughly 1% uncertainties in the BAO parameters from these forests.
DESI is a multi-object optical spectrograph that receives light from 5000
optical fibres mounted at the focal plane of the 4-metre class Mayall
Telescope, in Arizona, USA. The light of each object is split and dispersed
onto three cameras, each one corresponding to blue, red, and infra-red
wavelengths. The resolution of spectra is about 2000 – 3200 in the blue, 3200
– 4100 in the red and 4100 – 5000 in the infrared end. For each sky pointing,
the exposure time is dynamically tuned to the current observing conditions in
order to match the signal to noise ratio obtained for a 1000 second exposure
taken in ideal conditions. Spectra are reduced and calibrated with a fully
automated spectroscopic pipeline. DESI observes simultaneously different types
of targets: emission line galaxies, luminous red galaxies, quasars as tracers
and quasars containing $\text{Ly}\alpha$ forests. At the end of its 5-year
programme, DESI expects to cover about 14k deg2 of the observable sky. The
DESI $\text{Ly}\alpha$ forest survey aims to obtain four observations for more
than 800k quasars with redshifts $z>2.1$, corresponding to a density of 60
deg-2 (Chaussidon et al. in prep.). There are three methods of redshift
estimation being employed: template fitting with redrock and two machine
learning algorithms, quasarNET (Busca & Balland, 2018) and SQUEzE (Pérez-
Ràfols et al., 2020). Routine visual inspection is not a feasible option as
DESI is expected to observe $\mathcal{O}(10^{6})$ quasars.

In this work, we quantify the impact of errors in the redshift estimates on
the clustering of the $\text{Ly}\alpha$ forest. In particular, we looked into
the impact on BAO parameters derived from the $\text{Ly}\alpha$ auto-
correlation and the $\text{Ly}\alpha$ \- quasar cross-correlation, using
synthetic versions of the completed DESI dataset (five years of observations).

This paper is organised as follows. Section 2 describes the data sets used in
the analysis. Section 3 describes the methods used in our analysis, and the
results from the analysis on simulated data are presented in Section 4.
Section 5 contains a description of the model for the contamination of the
correlation functions by redshift errors. In Section 6, we discuss the
implications of our findings and present our conclusions.

In this work, conversion from angular and redshift separations to physical
separation are made using a flat-LCDM model with $\Omega_{m}=0.315$.

##  2 Synthetic Data Sets

This work is based solely on results obtained on synthetic sets of
$\text{Ly}\alpha$ data that mimic properties of the DESI $\text{Ly}\alpha$
survey. In this section we describe the basic principles behind the production
of these synthetic data sets, the particularities of DESI data, some special
sets used to test our hypotheses, and how we mimic the intrinsic errors in
quasar redshift estimates.

###  2.1 Procedure for Ly$\boldsymbol{\alpha}$ mock creation

Synthetic $\text{Ly}\alpha$ forest datasets are used by spectroscopic surveys
such as eBOSS or DESI to test the BAO analysis pipeline and evaluate
systematic effects. They are designed to reproduce the astrophysical and
instrumental characteristics of the survey data. By having a large number of
realisations of synthetic surveys, usually a few hundred, we can test
systematic effects to high precision.

There are several methods to create mock $\text{Ly}\alpha$ data. N-body
hydrodynamical simulations (Borde et al., 2014; Rossi et al., 2014; Walther et
al., 2021; Chabanier et al., 2020) are among the most realistic methods to
create forests but are too computationally expensive for producing hundreds of
realisations containing volumes as large as the real surveys. Hybrid methods,
(e.g., Peirani et al., 2014; Sorini et al., 2016), also rely on N-body
simulations to calibrate a model used to produce quick lognormal mock
catalogues. The method we use in this work to create synthetic data is based
on correlated Gaussian random fields (Coles & Jones, 1991; Le Goff et al.,
2011; Font-Ribera et al., 2012a; Bautista et al., 2015), whose correlations
follow a given input power spectrum. The random field mimics the matter
density field in the Universe. Quasars are placed via Poisson sampling in
regions where the density is larger than a given threshold. One dimensional
density "skewers" along the line-of-sight to each quasar are drawn by
interpolating the same random field. These can then be transformed to a
transmitted flux fraction, representative of a $\text{Ly}\alpha$ forest, by
adding small-scale fluctuations, converting from density to optical depth, and
adding redshift-space distortions (RSD).

We used an implementation of the Gaussian method of synthetic
$\text{Ly}\alpha$ forests, named LyaCoLoRe (Farr et al., 2020a). CoLoRe mock
catalogues use the package CoLoRe222https://github.com/damonge/CoLoRe
(Ramírez-Pérez et al., 2021) to place quasars in a lognormal density field.
Skewers are drawn from each quasar, have small-scale power added to them and
these are transformed into transmitted flux fractions using the LyaCoLoRe
package. 333https://github.com/igmhub/LyaCoLoRe This process produces
noiseless transmission fields that have correct clustering properties on
scales relevant for BAO studies. LyaCoLoRe stores skewers of metal absorption
and a table of high column density systems (HCDs) in its output, which may be
optionally added to the $\text{Ly}\alpha$ skewers during subsequent stages of
the pipeline. These simulated datasets cover the projected 5-year DESI
footprint and contain around one million quasars in the redshift range
$z\in[1.8,3.8]$, of which 700,000 have $z>2.1$.

###  2.2 Simulating DESI spectra of quasars

We generated mock quasar spectra reproducing observational properties of the
Dark Energy Spectroscopic Instrument, as described in Section 1. All of the
mock data sets used in this work have density of forests of 50 deg-2 within
the full DESI footprint, using the methodology described in du Mas des
Bourboux et al. (2020). Prior to Survey Validation, this was the expected
density for five years of DESI observations, but this target has since been
revised upwards to 60 deg-2.

Spectral properties of our mock quasars are simulated with the desisim
package.444https://github.com/desihub/desisim Continuum templates (for the
unabsorbed flux of the quasar with emission lines)555Sometimes in the
literature ”continuum” refers to flux without emission lines. are generated by
using functions from the simqso library (McGreer et al., 2021), which contains
a broad set of tools to generate mock quasar spectra. Templates are composed
by a series of broken power-laws with independent Gaussian slope
distributions, and a set of emission lines defined by their rest frame
wavelength, equivalent width and Gaussian r.m.s. width. The slopes and
emission line profile distribution used for the mocks in this work uses a
modified version of the BOSS DR9 model, which includes some emission lines
from the composite model of BOSS spectra from Table 4 of Harris et al. (2016),
and some adjustment of the equivalent widths so that the mean continuum
resembles better the one obtained in eBOSS Data Release 14. DESI intend to co-
add four observations of $\text{Ly}\alpha$ quasars. We therefore convolve
spectra to the instrumental resolution, and add pixel noise corresponding to
4000 second exposures (i.e. four times the nominal exposure time).

Astrophysical contaminants are often included in synthetic realisations of the
$\text{Ly}\alpha$ forest. Metal absorbers were added to mocks for the first
time in Bautista et al. (2015) and were first modelled in the correlation
function in Bautista et al. (2017), where the change in the BAO peak
parameters caused by metal absorption was found to be less than 1%. Similarly,
Font-Ribera & Miralda-Escudé (2012) simulated the impact of High Column
Density absorbers (HCDs), and discussed its impact on the measured
correlations. It is important to model these contaminants in BAO analyses (du
Mas des Bourboux et al., 2020), but its impact is orthogonal to the effect
discussed in this paper. For this reason, we decided not to include them in
our mocks.

However, we do simulate the impact of non-linear peculiar velocities in the
quasar redshifts, a phenomenon known as Fingers of God (FoG). Since our mocks
are generated from Gaussian fields, we only have access to linear peculiar
velocities. Therefore we simulate FoG by applying a random shift to the quasar
redshifts drawn from a Gaussian distribution with an r.m.s. given by the
parameter $\sigma_{v,\rm{FoG}}$. We use by default a value of
$\sigma_{v,\rm{FoG}}=150$ $\text{km~{}s}^{-1}\,$, similar to the expected
velocity dispersion in halos hosting quasars at $z\sim 2$, but in Section 4 we
also use mocks with an extreme value of $\sigma_{v,\rm{FoG}}=500$
$\text{km~{}s}^{-1}\,$.

###  2.3 Simulating quasar redshift errors

Even though we try to capture the diversity of quasar continua in our mocks,
fitting algorithms on simulated data often perform better than in real data
(Farr et al., 2020b). For this reason, instead of trying to estimate redshifts
from our mock spectra we decided to emulate errors in the pipeline redshift
estimation. We add Gaussian random errors to the quasar redshifts in the mock
catalogues, using by default an r.m.s. of $\sigma_{v,z}=500$
$\text{km~{}s}^{-1}\,$.

It is important to highlight a key difference between how we simulate FoG and
redshift errors. Even though they both add random shifts to the quasar
redshift, the shift emulating FoG is applied before we generate the quasar
continuum. On the other hand, the shift emulating redshift errors only affects
the value of redshift in the quasar catalogue that will be used in the
analysis, but it does not change the simulated spectrum.

###  2.4 No-QSO-clustering and no-forest mock data sets

As described in Section 5, the effect of redshift errors on the
$\text{Ly}\alpha$ clustering depends on both the amplitude of the quasar
clustering and the smoothing of the mean continuum template.

In order to test these assumptions, we created two special types of mock data
sets in addition to the standard mocks:

  * •

No-QSO-clustering mock data sets contain quasars randomly distributed in the
volume, regardless of the local density, so both the auto-correlation of
quasars and the QSO- $\text{Ly}\alpha$ cross-correlation are zero by
construction. The $\text{Ly}\alpha$ auto-correlation is conserved, since
$\text{Ly}\alpha$ forests are constructed from the correlated underlying
density field as in the standard mock sets.

  * •

No-forest mock data sets were constructed assuming that quasar spectra have no
$\text{Ly}\alpha$ absorption, to allow analysis of the effect without
$\text{Ly}\alpha$ clustering. The quasars are placed at the peaks of the
density field as in the standard mock sets, so the QSO auto-correlation is
conserved. These data sets are effectively noiseless, having a simulated
exposure time of $10^{6}~{}$ seconds.

In Section 4 we show how quasar redshift errors impact the observed clustering
of these special mock sets, validating the assumptions of our contamination
model.

##  3 Methods

Current studies of the $\text{Ly}\alpha$ forest correlation function for the
measurement of baryon acoustic oscillations follow a rather simple approach,
summarised in this section. Briefly, the steps are:

  * •

fit of quasar continua,

  * •

estimate of transmission and associated weights,

  * •

estimate of correlation functions,

  * •

estimate of distortion and covariance matrices,

  * •

fit of the BAO model.

Each of these steps can be performed using the publicly available code
"Package for Igm Cosmological Correlation Analyses",
picca.666https://github.com/igmhub/picca (du Mas des Bourboux et al., 2021) We
refer the reader to du Mas des Bourboux et al. (2020) for the full description
of the methodology and its validation.

###  3.1 Continuum fitting

To compute correlations, we use the contrast, $\delta_{q,i}$, of the
transmission $F_{q}(\lambda^{o}_{i})$ at a given observer-frame wavelength
$\lambda^{o}_{i}$ of pixel $i$ and quasar $q$. The transmission contrast is
defined as

$\delta_{q,i}=\frac{F_{q}(\lambda^{o}_{i})}{\bar{F}(\lambda^{o}_{i})}-1,$ (1)

where $\bar{F}$ is the sample’s mean transmitted fraction at the absorber
redshift, assumed to be only a function of redshift $z$ (or observed
wavelength $\lambda^{o}$ if we assume a single transition, such as
$\text{Ly}\alpha$ ). We can convert between redshift $z$ and observed
wavelength $\lambda^{o}$ by assuming a given rest-frame wavelength of the
absorption. In this work, we focus on the $\text{Ly}\alpha$ absorption, for
which $\lambda_{\alpha}=1216$ Å.

We use a rest-frame spectral region between the $\text{Ly}\alpha$ and
Ly$\beta$ broad emission lines, i.e.,
$\lambda^{r}\in[1040,1200]~{}\text{\AA}$. The observed frame wavelength of the
spectra is $\lambda_{\rm obs}\in[3600,5500]~{}\text{\AA}$.

The transmission $F_{q,i}=F_{q}(\lambda^{o}_{i})$ is defined as the ratio
between the observed flux in a given pixel and quasar, $f_{q,i}$, and the
unabsorbed flux level, commonly referred as the continuum, $C_{q,i}$, such
that the contrast can be written as

$\delta_{q,i}=\frac{f_{q,i}}{C_{q,i}\bar{F}_{i}}-1.$ (2)

In du Mas des Bourboux et al. (2020) and previous studies, the continuum is
assumed to be a universal function of wavelength in the rest-frame of the
quasar, scaled by a per-quasar linear function of log-wavelength,

$\hat{C}_{q,i}=\bar{C}(\lambda^{r}_{i})\left(a_{q}+b_{q}\log{\lambda^{o}_{i}}\right),$
(3)

where $\lambda^{r}_{i}=\lambda^{o}_{i}/(1+z_{q})$, $z_{q}$ is the redshift of
the quasar, $a_{q}$ and $b_{q}$ are fitted parameters for each quasar,
$\bar{C}$ is also referred as the mean continuum.

In practice, due to noise, spectrograph resolution, and the non-Gaussian
nature of the distribution of the transmission, it is hard to break the
degeneracy between the mean continuum $\bar{C}$ and the mean transmission
$\bar{F}$, and estimate them separately. Therefore, in du Mas des Bourboux et
al. (2020) they use Eq. 3 as the model for the product $C_{q,i}\bar{F}_{i}$ in
Eq. 2, so we can re-write it as simply:

$\hat{\delta}_{q,i}=\frac{f_{q,i}}{\hat{C}_{q,i}}-1.$ (4)

Current methods start by assuming a shape for $\bar{C}(\lambda^{r})$, dividing
the observed flux by it, and fitting for $a_{q}+b_{q}\log(\lambda_{i})$,
assuming Gaussian statistics. Once all quasar spectra are fitted, $\bar{C}$ is
computed by stacking the ratio $f_{i}/[a_{q}+b_{q}\log(\lambda_{i})]$ in the
quasar rest-frame. This new mean continuum is then used again to fit for all
parameters $a_{q}$ and $b_{q}$. This process is repeated a few times until
convergence. We highlight that for computing $\bar{C}$, an estimate of the
quasar redshift $z_{q}$ is used. This estimate might be affected by intrinsic
biases, scatter or measurement errors.

If the mean continuum were flat, the redshift smearing would have no effect on
the derived template. However, because of broad emission lines in the mean
continuum that are further broadened by the redshift errors, the derived
template is systematically different from the true mean continuum. As we will
see in Section 5, this fact combined with quasar clustering modifies the
quasar-forest and forest-forest correlations.

###  3.2 Correlation functions

The auto-correlation function of $\text{Ly}\alpha$ transmission fluctuations
is defined as

$\xi^{{\rm Ly}\alpha\times{\rm
Ly}\alpha}(\vec{r}_{A})=\langle\delta(\vec{x})\delta(\vec{x}+\vec{r}_{A})\rangle,$
(5)

where $\vec{r}_{A}$ is the separation vector between two forest pixels in the
volume, which can be decomposed into separations parallel to the line-of-
sight, $r_{\parallel}$, and transverse to the line-of-sight, $r_{\perp}$. The
cross-correlation between quasars and $\text{Ly}\alpha$ transmission
fluctuations is defined as

$\xi^{{\rm QSO}\times{\rm
Ly}\alpha}(\vec{r}_{A})=\langle\delta_{Q}(\vec{x})\delta(\vec{x}+\vec{r}_{A})\rangle\approx\langle\delta(\vec{x}_{Q}+\vec{r}_{A})\rangle,$
(6)

where $\delta_{Q}(\vec{x})$ is the fluctuation of the number density of
quasars. The approximated formula on the right-hand side is valid under the
assumption that quasars are sparse (shot-noise dominated), in which case the
cross-correlation is simply the average $\text{Ly}\alpha$ transmission around
quasars at positions given by $\vec{x}_{Q}$ (see Appendix B in Font-Ribera et
al. (2012b)). The auto-correlation function between quasars is $\xi^{{\rm
QSO}\times{\rm
QSO}}\left(\vec{r}_{A}\right)=\langle\delta_{Q}(\vec{x})\delta_{Q}(\vec{x}+\vec{r}_{A})\rangle$.
The quasar auto-correlation is an important ingredient of our model in Section
5 and is estimated from the mock catalogues.

In practice, the ensemble averages in the above definitions of the correlation
functions are in fact averages over the pairs of objects in a given volume.
Correlation functions are estimated in bins of separation $r_{\parallel}$ and
$r_{\perp}$. The estimator used for the auto-correlation of the
$\text{Ly}\alpha$ forest is

$\hat{\xi}^{{\rm Ly}\alpha\times{\rm Ly}\alpha}_{A}=\frac{\sum_{(i,j)\in
A}w_{i}w_{j}\delta_{i}\delta_{j}}{\sum_{(i,j)\in A}w_{i}w_{j}},$ (7)

where $w_{i}$ is the weight assigned to pixel $i$. The weights used are
described in Eq. 4 of du Mas des Bourboux et al. (2020). The sums are over all
pairs of pixels for which their separation is within the bounds of bin $A$.
Analogously, the cross-correlation estimator is

$\hat{\xi}^{{\rm QSO}\times{\rm Ly}\alpha}_{A}=\frac{\sum_{(q,i)\in
A}w_{q}w_{i}\delta_{i}}{\sum_{(q,i)\in A}w_{q}w_{i}},$ (8)

where $q$ indexes quasars. The auto-correlation of quasars is computed with
the Landy-Szalay estimator (Landy & Szalay, 1993) defined as

$\hat{\xi}_{A}^{{\rm QSO}\times{\rm
QSO}}=\frac{DD_{A}-2DR_{A}+RR_{A}}{RR_{A}},$ (9)

where $DD_{A}=\sum_{(i,j)\in A}w_{i}w_{j}/W_{qq}$ is the weighted number of
quasar pairs in bin $A$, $W_{qq}$ is the weighted total number of available
quasar pairs in the volume. A Poisson sample of unclustered points, named
randoms, is built following the geometry of the survey. $RR_{A}$ is the
normalised number of random pairs in bin $A$, while $DR_{A}$ is the number of
cross pairs between quasars and randoms.

The weights $w_{i}$ assigned to the transmission contrasts $\delta_{i}$ are
the inverse of the total pixel variance, which is assumed to be a combination
of some intrinsic variance (function of observed wavelength only), and
instrumental variance. The intrinsic variance is estimated from the full set
of forests. The weights assigned to both forest pixels and quasars also take
into account the evolution of their clustering with redshift, so
$w_{i}\propto(1+z_{i})^{\gamma}$. The default values used in the standard
mocks are the same as the evolution in the eBOSS data, $\gamma=1.9$ for forest
pixels (du Mas des Bourboux et al., 2017) and $\gamma=0.44$ for quasars (du
Mas des Bourboux et al., 2019). Imperfect weighting is not expected to bias
the results. For the special mocks, such as no-QSO-clustering or no-forest
(see Section 2.4), the redshift evolution is neglected.

###  3.3 Distortion and covariance matrices

Our continuum fitting procedure uses information from the forest itself, so
each $\hat{\delta}_{q,i}$ is a linear combination of all $\delta_{q,j}$ from
the same quasar $q$. When computing the correlation functions, each pair of
pixels brings with it contributions from their whole respective forests. This
distorts the measured correlation function. Bautista et al. (2017) introduced
a method to account for this distortion which was also used in subsequent
analysis, including du Mas des Bourboux et al. (2020). If the difference
between the true $\delta$ and the distorted $\delta$ is a linear function of
$\log\lambda$, the distorted correlation is a matrix times the undistorted
correlation. This matrix is referred to as the distortion matrix. We compute
these distortion matrices for our mock catalogues following the same
procedure.

Real forests contain absorption by elements other than hydrogen, such as
silicon, nitrogen and iron, which creates spurious correlations. None of our
mock catalogues contain these elements, as the effect is unrelated to the
topic treated in this work.

The covariance matrix of our measurements are estimated by sub-sampling, i.e.,
we divide the footprint into $p$ sub-regions and compute the correlation
function in each sub-region $\xi_{A}^{p}$. The covariance is written as

$C_{AB}=\frac{1}{\sum_{p}W^{p}_{A}\sum_{p}W^{p}_{B}}\sum_{p}W^{p}_{A}W^{p}_{B}\left(\xi^{p}_{A}\xi^{p}_{B}-\xi_{A}\xi_{B}\right),$
(10)

which assumes that correlations between sub-regions are negligible. The number
of bins $A$ of our correlation functions is usually larger than the number of
the sub-regions $p$, so we smooth the covariance matrix by assuming that its
correlation coefficients are only a function of $\Delta
r_{\parallel}=r_{\parallel}^{A}-r_{\parallel}^{B}$ and $\Delta
r_{\perp}=r_{\perp}^{A}-r_{\perp}^{B}$. We average all correlation
coefficients that have the same $(\Delta r_{\parallel},\Delta r_{\perp})$.

###  3.4 Modelling the correlations

We model the large-scale correlations following the procedure from du Mas des
Bourboux et al. (2020), that we shortly describe here.

A given correlation function (auto or cross) is defined as a sum of a smooth
part and a BAO peak part:

$\xi^{\rm model}_{A}=\xi^{\rm sm}_{A}+\xi^{\rm
peak}_{A}(\alpha_{\parallel},\alpha_{\perp}).$ (11)

Only the peak part depends on the BAO dilation parameters $\alpha_{\parallel}$
and $\alpha_{\perp}$, defined as

$\alpha_{\parallel}=\frac{[D_{H}(\bar{z})/r_{d}]}{[D_{H}(\bar{z})/r_{d}]_{\rm
fid}},~{}\alpha_{\perp}=\frac{[D_{M}(\bar{z})/r_{d}]}{[D_{M}(\bar{z})/r_{d}]_{\rm
fid}},$ (12)

where $r_{d}$ is the comoving size of the sound horizon at drag epoch,
$D_{H}(\bar{z})=c/H(\bar{z})$ is the Hubble distance,
$D_{M}(\bar{z})=(1+\bar{z})D_{A}(\bar{z})$ is the comoving angular diameter
distance assuming a flat universe, and $\bar{z}$ is the mean redshift of the
measurement.

The correlation function model between two tracers is the Fourier transform of
an anisotropic biased power spectrum written as

$P^{\rm
model}(\vec{k})=b_{i}b_{j}(1+\beta_{i}\mu^{2}_{k})(1+\beta_{j}\mu^{2}_{k})P_{\rm
QL}(\vec{k})F_{\rm NL}(\vec{k})G_{\rm bin}(\vec{k}),$ (13)

where $\vec{k}$ is the wavevector, with modulus $k$ and
$\mu_{k}=k_{\parallel}/k$; $b_{i}$ and $\beta_{i}$ are the linear bias and
redshift-space distortions parameters, respectively; $G_{\rm bin}$ accounts
for the binning of the correlation function, $F_{\rm NL}$ is a empirical term
that accounts for the non-linear effects on small scales, and $P_{\rm QL}$ is
the linear matter power spectrum with a empirical anisotropic damping applied
to the BAO peak component. The linear matter power spectrum is computed from a
Boltzmann solver code, such as camb (Lewis et al., 2000).

The non-linear term $F_{\rm NL}$ is only included for the cross-correlation in
this work since our mock forests are built from Gaussian random fields and do
not contain non-linear clustering. Given the Gaussian nature of both FoG and
redshift errors in the simulated datasets, we use a Gaussian kernel with width
$\sigma_{v}$ to model both effects in the cross-correlation:

$F^{{\rm QSO}\times{\rm Ly}\alpha}_{\rm
NL}\left(\vec{k}\right)=\exp{\left[-\frac{\left(k\mu_{k}\sigma_{v}\right)^{2}}{2}\right]}~{}.$
(14)

This is similar to the Gaussian kernel proposed in Percival & White (2009),
with a factor of two difference to take into account that there is only one
quasar field in the cross-correlation.

The final correlation function model accounts for the distortion matrix, as
discussed in Section 3.3. The distorted correlations are written as

$\xi^{\rm dist}_{A}=\sum_{A^{\prime}}D_{AA^{\prime}}\xi^{\rm
model}_{A^{\prime}}.$ (15)

In this work, we only focus on the effect of redshift errors, so we do not add
metals or high-column density systems to the mock catalogues. Therefore, our
theoretical model does not have terms that account for these effects.

###  3.5 Fitting the BAO scale

BAO fits are made over separations of $r\in[10,180]$
$\text{h}^{-1}\text{Mpc}\,$and between directions $\mu\in[0,1]$ and
$\mu\in[-1,1]$ for the auto and cross-correlations respectively. The
correlation function bin size is 4 $\text{h}^{-1}\text{Mpc}\,$, corresponding
to 1590 separation bins for the auto and 3180 bins for the cross. The joint
fit of auto- and cross-correlations uses a total of 4770 measurements. Four
parameters are let free for the fit of the auto-correlations:
$\alpha_{\parallel}$, $\alpha_{\perp}$, $b_{{\rm Ly}\alpha}$ and $\beta_{{\rm
Ly}\alpha}$. For the cross-correlation, an additional parameter $\sigma_{v}$
(Eq. 14) is also fitted. The same five are also let free for combined fits.
Table 1 summarises the parameters used in this work.

Table 1: Summary of free parameters of the model fitted to the auto, cross and joint correlation functions. The last parameter, $\sigma_{v}$, is only used for the cross and joint fits. Parameter | Description
---|---
$\alpha_{\parallel}$ | Radial BAO dilation parameter (Eq. 12)
$\alpha_{\perp}$ | Transverse BAO dilation parameter (Eq. 12)
$b_{Lya}$ | Linear density bias (Eq. 13)
$\beta_{Lya}$ | Linear redshift-space distortions parameter (Eq. 13)
$\sigma_{v}$ | Lorentzian radial dispersion of velocities (Eq. 14)

During the creation of the CoLoRe mocks, a Gaussian smoothing of 2
$\text{h}^{-1}\text{Mpc}\,$is applied to the model power spectrum to account
for the low resolution of the simulation grid. This adds an extra (isotropic)
smoothing to our mocks even before we add FoG or redshift errors. We account
for this by fitting Gaussian smoothing terms for the whole model, similar to
$G(k)$ in Eq. 13, while fixing $\sigma_{v}$ to zero on the fiducial mocks.
These values are subsequently used as fixed parameters when fitting mocks that
do contain FoG and redshift errors.

##  4 Analysis of synthetic data

In this section we present the analysis of the correlations measured from the
simulated data described in Section 2, using the methodology from Section 3.

Figure 1: Correlation functions as a function of separations transverse
($r_{\perp}$) and parallel ($r_{\parallel}$) to the line of sight in standard
mocks. Top panels show the cross-correlation function between
$\text{Ly}\alpha$ forests and quasars while bottom panels show the auto-
correlation of forests. Left panels show the original correlations while mid
panels display correlations with the contamination caused by Gaussian redshift
errors of $\sigma_{v,z}=500$ $\text{km~{}s}^{-1}\,$. The right panels show the
difference between the left and centre panels, isolating the contamination,
which is seen as an oscillating signal at small transverse separations. Note
that negative $r_{\parallel}$ values correspond to $\text{Ly}\alpha$ forest
absorption lying between the neighbouring quasar and the observer. This
scenario results in a larger auto-correlation between the two quasars than the
case where the forest lies behind its neighbouring quasar.

Figure 1 presents the estimated correlation functions versus
$(r_{\parallel},r_{\perp})$ for of the average of ten independent mock
realisations. The top panels show the cross-correlation between
$\text{Ly}\alpha$ forests and quasars while the bottom ones show the auto-
correlation of $\text{Ly}\alpha$ forests. The left panels are mocks without
redshift errors ($\sigma_{v,z}=0$ $\text{km~{}s}^{-1}\,$) while the central
panels contain $\sigma_{v,z}=500$ $\text{km~{}s}^{-1}\,$. The right panels
show the difference between central and left panels, isolating the features in
the correlation function caused by redshift errors. We can see that, for both
the cross- and auto-correlations, these oscillatory features are located at
small transverse separations $r_{\perp}$, decreasing in amplitude as
$r_{\perp}$ increases. Additionally, the cross-correlation signal near zero
separations shows the effect of smearing along the line of sight also caused
by the redshift errors, and the oscillations occur only for negative
$r_{\parallel}$ values. These values occur when the $\text{Ly}\alpha$ forest
pixel is in front of the neighbouring quasar. Where there is a strong cross-
correlation, there is also likely to be a strong correlation between the two
quasars (as a forest is always in front of its own quasar). For positive
$r_{\parallel}$, where the pixel is behind its neighbouring quasar, its own
quasar will be even further behind, so the quasar auto-correlation will be
less strong. The asymmetry in the signal, therefore, strongly suggests a
dependence on the auto-correlation of quasars.

Figure 2:  $\text{Ly}\alpha$ -quasar cross-correlation function from different
sets of realisations of the mocks. The green bands show the measurement from
the fiducial mocks, with a small value of $\sigma_{v,\rm{FoG}}=150$
$\text{km~{}s}^{-1}\,$and no redshift errors; the pink bands
(indistinguishable from the green in this plot) correspond to the mocks
generated with an extreme value of $\sigma_{v,\rm{FoG}}=500$
$\text{km~{}s}^{-1}\,$; the blue bands shows the results for mocks with large
redshift errors of $\sigma_{v,z}=500$ $\text{km~{}s}^{-1}\,$. These
measurements are computed from the average of 10 realisations of the complete
5-year DESI survey, and the width of the bands corresponds to the scatter
between realisations. Note that $\mu>0$ ($\mu<0$) corresponds to
configurations where the $\text{Ly}\alpha$ pixels are behind (in front of) the
neighbouring quasar. Figure 3: Same as Fig. 2 but for the $\text{Ly}\alpha$
auto-correlation. As expected, Fingers of God do not impact the auto-
correlation. However, the impact of redshift errors $\sigma_{v,z}$ is clearly
visible.

Figures 2 and 3 show respectively the cross-correlation of the
$\text{Ly}\alpha$ forest with quasars and the auto-correlation of
$\text{Ly}\alpha$ forests, measured from different sets of mocks. Different
panels in these figures show different _wedges_ of the correlations, i.e.,
averages over ranges of $\mu=r_{\parallel}/r$. The green bands correspond to
the average of ten fiducial realisations (with different cosmic variance and
instrumental noise), using the standard value of $\sigma_{v,\rm{FoG}}=150$
$\text{km~{}s}^{-1}\,$and ignoring redshift errors ($\sigma_{v,z}=0$); the
pink bands are from ten mocks with a larger value for
$\sigma_{v,\rm{FoG}}=500$ $\text{km~{}s}^{-1}\,$; the blue bands show
measurements on the fiducial mocks, after adding redshift errors to the quasar
catalogues ($\sigma_{v,z}=500$ $\text{km~{}s}^{-1}\,$). The width of the bands
correspond to the standard deviation between the ten realisations of each
dataset, i.e., an estimate of the errors for one realisation. The dashed black
lines show the best-fit model obtained when analysing the fiducial set of
mocks.

###  4.1 Impact of non-linear peculiar velocities (FoG)

Adding FoG to the quasar spectra has a negligible effect in the
$\text{Ly}\alpha$ auto-correlation, making it impossible to distinguish the
green and pink bands in Fig. 3. This is expected since the redshifts of
absorption lines do not depend on the quasar redshift. On the other hand, the
cross-correlation with quasars is smoothed out by the random shifts of the
quasar position along the line of sight. This causes small differences on BAO
scales, but the impact is clearly seen on scales below 25
$\text{h}^{-1}\text{Mpc}\,$in Figure 2.

The impact of FoG on the best-fit parameters from the BAO fits can be seen in
Table 2, where the results from the average of ten realisations are compared.
Here again FoG have a negligible impact on the $\text{Ly}\alpha$ auto-
correlation, but the uncertainties on $\alpha_{\parallel}$ from the cross-
correlation are 15% larger (from $0.42$ to $0.48\%$).

Table 2: Best-fit parameters from ten combined realisations of standard mocks, with simulated Fingers of God velocities of $\sigma_{v,\rm{FoG}}=150$ or $500$ $\text{km~{}s}^{-1}\,$, and simulated redshift errors of $\sigma_{v,z}=0$ or $500$ $\text{km~{}s}^{-1}\,$. The fourth and fifth columns show the BAO parameters. The last column $\sigma_{v}$ shows the best-fit value of the parameter describing the line-of-sight (Gaussian) smoothing affecting the cross-correlation, which includes FoG and other sources of error on the redshift measurement. The model used to fit the correlations does not attempt to account for the new effect discussed in this work. The value of $\sigma_{v}$ can be expressed in units of $\text{h}^{-1}\text{Mpc}\,$by simply dividing by $H_{0}=100h$ km s-1 Mpc-1.  Data Set | $\sigma_{v,\rm{FoG}}~{}[{\rm km/s}]$ | $\sigma_{v,z}~{}[{\rm km/s}]$ | $(\alpha_{\parallel}-1)\times 10^{3}$ | $(\alpha_{\perp}-1)\times 10^{3}$ | $\sigma_{v}~{}[{\rm km/s}]$ |
---|---|---|---|---|---|---
Ly$\alpha$ x Ly$\alpha$ | 150 | 0 | 1.9 $\pm$ 4.6 | 4.6 $\pm$ 5.9 | - |
Ly$\alpha$ x Ly$\alpha$ | 500 | 0 | 1.5 $\pm$ 4.6 | 4.8 $\pm$ 5.9 | - |
Ly$\alpha$ x Ly$\alpha$ | 150 | 500 | -3.8 $\pm$ 5.1 | 7.7 $\pm$ 6.4 | - |
Ly$\alpha$ x QSO | 150 | 0 | -3.2 $\pm$ 4.2 | -0.1 $\pm$ 4.5 | 0 $\pm$ 6 |
Ly$\alpha$ x QSO | 500 | 0 | -3.5 $\pm$ 4.8 | -1.0 $\pm$ 4.6 | 575 $\pm$ 3 |
Ly$\alpha$ x QSO | 150 | 500 | -5.0 $\pm$ 5.2 | 1.8 $\pm$ 4.8 | 612 $\pm$ 3 |
Combined | 150 | 0 | -1.2 $\pm$ 3.2 | 2.1 $\pm$ 3.6 | 0 $\pm$ 4 |
Combined | 500 | 0 | -0.7 $\pm$ 3.3 | 1.1 $\pm$ 3.6 | 561 $\pm$ 2 |
Combined | 150 | 500 | -4.4 $\pm$ 3.6 | 4.1 $\pm$ 3.9 | 588 $\pm$ 2 |

###  4.2 Impact of quasar redshift errors

The blue bands in Figures 2 and 3 show the impact of redshift errors. As
discussed in Section 2.3, we add these redshift errors to the catalogues after
the quasar spectra have been simulated, i.e., the redshifts listed are
different than the redshifts that were used to generate the quasar continua.

Figure 4: QSO- $\text{Ly}\alpha$ cross-correlation (left) and
$\text{Ly}\alpha$ auto-correlation (right) measured from stacks of 10 mock
realisations, each stack with different values of Gaussian redshift errors:
$\sigma_{v,z}=0$, $250$, $500$, and $750$ $\text{km~{}s}^{-1}\,$. We only show
the line-of-sight wedges that show spurious correlations caused by redshift
errors. These significantly alter the broad-band shape, which will affect the
fitting of the correlation functions.

For correlations that are not along the line of sight ($|\mu|<0.8$), the
impact of FoG and redshift errors are indistinguishable: the cross-correlation
is smoothed on small scales, and the auto-correlation is not affected.
However, clear differences appear in the $-1<\mu<-0.8$ cross-correlation wedge
(top left panel of Fig. 2) and more surprisingly in the $0.8<\mu<1$ auto-
correlation wedge (left panel of Fig. 3). As shown in Table 2, redshift errors
seem to degrade the BAO performance not only in the cross-correlations, but
also in the $\text{Ly}\alpha$ auto-correlation.

We believe that this is the first time that these features are detected and
discussed.

In Fig. 4 we show the contaminated wedges for realisations with different
redshift errors $\sigma_{v,z}=0$, $250$, $500$ and 750 $\text{km~{}s}^{-1}\,$.
It is clear that the amplitude of the spurious correlations grows
monotonically with the amplitude of the redshift errors, and importantly,
changes the broad-band shape of the correlation functions which has an impact
on the fitting procedures.

Table 3 shows results of BAO fits to mock realisations with increasing values
of $\sigma_{v,z}$. We report the average best-fit parameters of ten
independent realisations for each case. The reported errors are therefore
$\sqrt{10}\sim 3.16$ smaller than the expected errors of the full 5-year DESI
survey. The best-fit dilation parameters from the auto-correlation function do
not present significant changes when increasing $\sigma_{v,z}$, while
uncertainties do increase slightly for larger redshift errors. The best-fit
$\sigma_{v}$ values correlate well with the input $\sigma_{v,z}$, but are not
in agreement, likely due to the fact that $\sigma_{v}$ also accounts for FoG,
or that our model (Eq. 14) is not necessarily a good match to the Gaussian
errors added to mocks.

These results show that BAO measurements are not biased, even when considering
large values for quasar redshift errors $\sigma_{v,z}$. Only the estimated
errors on $\alpha_{\parallel}$ and $\alpha_{\perp}$ are increased for larger
values of $\sigma_{v,z}$, likely due to the lack of modelling of the effect
caused by redshift errors. Properly accounting for these features in the model
could potentially help to reduce these errors. In the next section we discuss
a potential way to achieve this goal.

Table 3: Average best-fit parameters of 10 independent realisations of standard mocks, with simulated redshift errors of $\sigma_{v,z}=0$, 250, 500 and 750 $\text{km~{}s}^{-1}\,$. All these mocks contain simulated FoG with $\sigma_{v,\rm{FoG}}=150$ $\text{km~{}s}^{-1}\,$. The reported errors are the estimated errors of the mean of 10 realisations so they are $\sqrt{10}$ smaller than the expected errors from the full 5-year DESI survey. The model used to fit the correlations does not attempt to account for the new effect discussed in this work. The value of $\sigma_{v}$ output by the fitter originally in units of $\text{h}^{-1}\text{Mpc}\,$and was converted to $\text{km~{}s}^{-1}\,$by multiplying it by $H(z)/(1+z)=103.9h$ km s-1 Mpc-1 for $z=2.3$. Data Set | $\sigma_{v,z}~{}[{\rm km/s}]$ | $(\alpha_{\parallel}-1)\times 10^{3}$ | $(\alpha_{\perp}-1)\times 10^{3}$ | $\sigma_{v}~{}[{\rm km/s}]$
---|---|---|---|---
Ly$\alpha$ x Ly$\alpha$ | 0 | $1.8\pm 3.6$ | $8.0\pm 5.3$ | -
Ly$\alpha$ x Ly$\alpha$ | 250 | $1.0\pm 3.7$ | $9.5\pm 5.3$ | -
Ly$\alpha$ x Ly$\alpha$ | 500 | $1.6\pm 4.1$ | $9.4\pm 5.6$ | -
Ly$\alpha$ x Ly$\alpha$ | 750 | $5.4\pm 4.8$ | $5.7\pm 6.0$ | -
Ly$\alpha$ x QSO | 0 | $-4.3\pm 3.6$ | $1.0\pm 4.2$ | $0\pm 5$
Ly$\alpha$ x QSO | 250 | $-6.0\pm 3.8$ | $2.7\pm 4.1$ | $188\pm 6$
Ly$\alpha$ x QSO | 500 | $-8.0\pm 4.5$ | $1.3\pm 4.4$ | $646\pm 3$
Ly$\alpha$ x QSO | 750 | $0.7\pm 6.2$ | $-3.3\pm 5.0$ | $1012\pm 3$
Combined | 0 | $-0.9\pm 2.6$ | $3.7\pm 3.3$ | $0\pm 4$
Combined | 250 | $-1.9\pm 2.7$ | $5.1\pm 3.3$ | $150\pm 6$
Combined | 500 | $-2.0\pm 3.1$ | $4.2\pm 3.5$ | $624\pm 2$
Combined | 750 | $4.3\pm 3.8$ | $0.4\pm 3.8$ | $987\pm 2$

##  5 Model for the contamination

In this section we describe theoretically the expected impact of redshift
errors in the correlation functions and we will show that the two main
elements that are responsible for the contamination are:

  1. 1.

a systematic error in the mean continuum estimate, $C_{q,i}$ (Eq. 3), and

  2. 2.

a non-zero quasar-quasar correlation function.

The first effect of redshift errors, if assumed to be reasonably distributed
randomly around some central value (which is not necessarily the true value),
is to smooth the estimate of the mean continuum $\bar{C}(\lambda^{r})$.
Emission lines present in the mean continuum are smeared. Figure 5 shows how
the estimated mean continuum from mock catalogues is modified when increasing
redshift errors. Known emission lines in the forest region include SIV 1063,
1073, FeII 1082, OIII 1084, PV 1118, 1128, and CIII* 1175. The smoothing of
these lines in the continuum is clearly visible in the bottom panel, which
shows the relative difference between mean continua with and without errors.
Redshift errors of this magnitude introduce biases in the mean continuum of
the order of half a percent. These biases have the typical shape of the
subtraction of two line profiles with different widths.

Figure 5: Effect of redshift errors on the mean continuum. The top panel shows
the mean continua for no-forest mocks without errors, and with increasingly
large redshift errors: $\sigma_{v,z}=0,250,500$ and $750~{}\text{km s}^{-1}$.
The positions of some of the strongest emission lines are annotated. The lower
panel shows the $\gamma(\lambda_{r})$ function defined in Eq. 16. The top axis
on the top panel shows the radial separations between pixels and the quasar,
assuming $z_{q}=2.3$.  Figure 6: Correlation function of quasars in real space
(no RSDs) in CoLoRe boxes (at z=2.05), compared with AbacusSummit-based mocks
(Alam et al., in prep.) at z=1.4, with the best-fit linear model, fitted on
scales larger than 25 $\text{h}^{-1}\text{Mpc}$. The deviations from linear
theory in the CoLoRe mocks are significantly stronger than in AbacusSummit
mocks. The high clustering at small scales makes these mocks particularly
sensitive to the contamination from redshift errors.

If we assume a systematic error $\gamma(\lambda^{r})$ in the estimate of the
mean continuum relative to the true mean continuum $\bar{C}_{q}(\lambda^{r})$,
as shown in the bottom panel of Figure 5, such that

$\bar{C}^{*}_{q}(\lambda^{r}_{i})=\bar{C}_{q}(\lambda^{r}_{i})\left[1+\gamma(\lambda^{r}_{i})\right],$
(16)

then the biased transmission contrast can be written from Eq. 4 as

$\hat{\delta}^{*}_{i}=\frac{(1+\hat{\delta}_{i})}{(1+\gamma_{i})}-1\approx\hat{\delta}_{i}-\gamma_{i},$
(17)

where second-order terms have been neglected in the approximation.

When computing correlations (auto- or cross- with quasars) with this biased
contrast field, we perform averages over the distribution of quasars and
observed/rest-frame wavelengths. Given that quasars are not uniformly
distributed over the volume, i.e., they are clustered, the average of $\gamma$
will not average to zero. This gives the intuition for the second element in
our model: the clustering of the quasars.

###  5.1 Impact on quasar-Ly$\boldsymbol{\alpha}$ cross-correlation

We consider two quasars placed at redshifts $z_{q1}$ and $z_{q2}$. Their
separation vector is $\vec{r}_{Q}$. We will study the cross-correlation
between quasar and the forest with a given separation vector $\vec{r}_{A}$
(corresponding to a given separation bin $A$ in a binned estimation of the
correlation function), that can be decomposed into parallel separation
$r^{A}_{\parallel}$ and transverse separation $r^{A}_{\perp}$. Note that given
the small angles involved, we can assume that the transverse component of
$\vec{r}_{Q}$ is $r^{Q}_{\perp}=r^{A}_{\perp}$. For a given
$r^{A}_{\parallel}$ and $z_{q1}$, the forest pixel will be at the observed
wavelength $\lambda^{o}_{A}$ given by
$\lambda^{o}_{A}=\lambda_{\alpha}(1+z_{A})$, where
$z_{A}=z_{q1}+r^{A}_{\parallel}/D_{H}(z_{q1})$. The corresponding quasar rest-
frame wavelength of the same pixel is given by
$\lambda^{r}_{A}=\lambda^{o}_{A}/(1+z_{q2})$.

The cross-correlation $\hat{\xi}_{A}$ is the stack of all
$\hat{\delta}^{*}_{i}$ (Eq. 17) lying at separation $\vec{r}_{A}$ from the
quasars.

$\hat{\xi}^{*}_{A}=\langle\hat{\delta}^{*}\rangle_{A}=\langle\hat{\delta}\rangle_{A}-\langle\gamma\rangle_{A}=\hat{\xi}_{A}-\langle\gamma\rangle_{A}$
(18)

The average $\langle\gamma\rangle_{A}$ can be written as a double integral
over $z_{q1}$ and $z_{q2}$. The first integral is weighted by the distribution
of quasars versus redshift $P(z_{q1})$, while the second is weighted by the
distribution of quasars around $q1$ at a given transverse separation
$r_{\perp}^{Q}=r_{\perp}^{A}$, which we denote $P(z_{q2}\mid
z_{q1},r^{Q}_{\perp})$. The excess probability of finding a quasar $q_{2}$ at
a distance $\vec{r}_{Q}$ from quasar $q_{1}$ is given by the quasar-quasar
correlation function $\xi^{qq}(\vec{r}_{Q})$, so $P(z_{q2}\mid
z_{q1},r_{\perp})=P(z_{q2})\left[1+\xi^{qq}(\vec{r}_{Q})\right]$.

$\displaystyle\langle\gamma\rangle_{A}$ $\displaystyle=\int{\rm
d}z_{q1}P(z_{q1})\int{\rm d}z_{q2}P(z_{q2}\mid
z_{q1},r_{\perp})~{}\gamma(\lambda^{r}_{A})$ (19) $\displaystyle=\int{\rm
d}z_{q1}P^{2}(z_{q1})\int{\rm
d}z_{q2}\left[1+\xi^{qq}(\vec{r}_{Q})\right]~{}\gamma(\lambda^{r}_{A}).$

Note that for a given set of $z_{q1}$, $z_{q2}$ and $\vec{r}_{A}$, there is
only a single possible value for $\lambda^{r}_{A}$. Also,
$\gamma(\lambda^{r})$ is only defined in the forest region, usually between
1040 and 1200Å.

Given that $P(z_{q})$ is a slow-varying function of $z_{q}$ compared to
$\gamma(\lambda_{A}^{r})$, the first term in the square brackets is washed out
by the integrals over $z_{q}$, leaving

$\langle\gamma\rangle_{A}=\int{\rm d}z_{q1}P^{2}(z_{q1})\int{\rm
d}z_{q2}~{}\xi^{qq}(\vec{r}_{Q})~{}\gamma(\lambda^{r}_{A}).$ (20)

The level of contamination therefore depends on the clustering of quasars
$\xi^{qq}$ and on the amplitude of $\gamma$, which is determined by the
amplitude of redshift errors. In mock catalogues, we can estimate $\xi^{qq}$
to use in the calculation of the contamination.

Figure 6 shows the auto-correlation function of quasars from CoLoRe and
AbacusSummit simulations (Garrison et al., 2021; Maksimova et al., 2021;
Hadzhiyska et al., 2021; Bose et al., 2021), without redshift space
distortions. The AbacusSummit mocks are tailored for quasars from DESI SV
(Survey Validation) (Alam et al., in prep). It can be seen that the small-
scale clustering in the CoLoRe mocks is significantly stronger than in
AbacusSummit. This makes the CoLoRe mocks particularly suitable for testing
this model.

Eq. 20 was used to calculate a model for the contamination in the cross-
correlation function introduced by redshift errors. This equation requires two
inputs, first, the systematic bias in the mean continuum estimate and, second,
the quasar-quasar correlation function. We compare our model with measurements
performed on mock catalogues in the following section.

Figure 7: Average correlation functions (left column: QSO x $\text{Ly}\alpha$
cross-correlation, right column: $\text{Ly}\alpha$ auto-correlation) at
$r_{\perp}<8$ $\text{h}^{-1}\text{Mpc}\,$as a function of radial separations
$r_{\parallel}$ for standard mocks (blue), no-forest (orange) and no-QSO-
clustering (green) mocks. Each panel shows the correlations for
$\sigma_{v,z}=0$ (dashed) and 500 $\text{km~{}s}^{-1}\,$(solid). The
oscillatory features caused by redshift errors can be seen in the standard and
no-forest mocks, but as expected, they are not present in the no-QSO-
clustering mocks.

The model for the contamination in the auto-correlation of $\text{Ly}\alpha$
forests could be constructed as an extension of the cross-correlation model.
However, results on mock catalogues show that the case of the auto-correlation
is more complex than the cross correlation, indicating that there must be more
ingredients to be taken into account in order to correctly reproduce the
contamination. We leave this exploration for future work and we focus on
results for the cross-correlation in the next section.

###  5.2 Contamination in special mocks

The contamination due to redshift errors on the correlations was further
investigated by repeating the analysis on two special sets of mocks. These are
the _no-forest_ and the _no-QSO-clustering_ mocks, described in detail in
Section 2.4. These are meant to test our hypothesis that the contamination
arising from redshift errors depends on both the amplitude of the quasar
clustering and the systematic error in the mean continuum estimate. This means
that the effect should not depend in principle on the clustering of the
$\text{Ly}\alpha$ forest itself.

Figure 7 shows the average cross and auto-correlation functions at
$r_{\perp}<8$ $\text{h}^{-1}\text{Mpc}\,$, plotted as functions of
$r_{\parallel}$. Three sets of mocks are displayed: a stack of 10 realisations
of standard mocks (blue), no-forest (orange) and a single realisation of no-
QSO-clustering mocks (green). Each panel compares the radial correlations for
mocks with and without redshift errors of $\sigma_{v,z}=500$
$\text{km~{}s}^{-1}\,$. The oscillatory features caused by redshift errors are
clearly seen in standard and no-forest mocks, while as expected, no-QSO-
clustering mocks do not present any visible effect.

Figure 8: Cross-correlation functions at small transverse separations
($r_{\perp}<8$ $\text{h}^{-1}\text{Mpc}\,$) as functions of radial separations
$r_{\parallel}$ for mocks with $\sigma_{v,z}=500$ $\text{km~{}s}^{-1}\,$minus
the correlations when $\sigma_{v,z}=0$. Two sets of mocks are shown: standard
mocks in blue and no-forest in orange. The model derived in Section 5,
indicated by the red dashed line, is in good qualitative agreement with the
contamination by redshift errors.

Figure 8 displays the difference $\Delta\xi=\xi_{500}-\xi_{0}$ between the
cross-correlations with and without errors. We observe a very good agreement
between $\Delta\xi$ from standard and no-forest mocks, confirming our
hypothesis. Figure 8 also shows the $\Delta\xi=\xi_{500}-\xi_{0}$ for the
model described in Section 5 as a dashed red line. This model was computed
using the bias in the mean continuum as shown in Fig. 5 and the quasar auto-
correlation displayed in Fig. 6. In the case of the cross-correlation, the
agreement between standard, no-forest and the model is very good, both in
terms of the shape of the features and their amplitudes.

The results on these special mocks and the qualitative agreement of our model
for the contamination demonstrate that the effect of redshift errors in the
cross-correlation depend both on the amplitude of the quasar clustering and on
the systematic errors in the continuum shape.

##  6 Discussion and Conclusions

In this paper, we have described the effect of errors in estimating quasar
redshifts on the $\text{Ly}\alpha$ forest correlation functions, and the
consequent impact on the BAO parameters. We used mock $\text{Ly}\alpha$
forests with redshifts of $z\in[2.1,3.8]$ designed to simulate five years of
the DESI program, to which we added various amplitudes of both Gaussian random
peculiar velocities (Fingers of God) and Gaussian errors in the quasar
redshift estimates.

The two-point correlation functions exhibited unexpected systematic
correlations at separations close to the line of sight when redshift errors
were introduced (but which were absent when only large astrophysical FoG
values were included). These features decrease in amplitude for increasing
transverse separations, similar behaviour to the contamination caused by
metals in the $\text{Ly}\alpha$ forest. We found that these systematic
correlations increase when increasing the amplitude of the redshift errors
added to mocks. We believe it is the first time that this type of
contamination has been observed and studied.

We analysed the impact of such correlations on fits of the baryon acoustic
oscillations. We found that BAO parameters are not significantly shifted by
the addition of Gaussian random redshift errors, for the three cases: cross,
auto, and joint fits of auto+cross-correlations. Redshift errors also cause
the uncertainties in the BAO parameters to increase. We believe that if the
model accounts for the systematic correlations it could reduce these
uncertainties, which is important for current $\text{Ly}\alpha$ surveys. These
results are based on averages of ten realisations of the full 5-year DESI
survey.

We derived a simplified model for the contamination to correlations from
redshift errors, based on two main ingredients: the quasar auto-correlation
function and the systematic bias in the mean continuum. These hypotheses were
tested using special sets of mock catalogues, with either no-QSO-clustering or
no $\text{Ly}\alpha$ absorption. Mocks with no-QSO-clustering do not exhibit
the features, while mocks with no-forest do contain them, confirming our
hypothesis. The amplitudes and shapes of the contamination in these special
mocks well describe the systematic correlations in the more realistic mock
sets in the case of the cross-correlation between $\text{Ly}\alpha$ and
quasars. Modelling the contamination for the $\text{Ly}\alpha$ auto-
correlation is left for future work.

A detailed study of the effects analysed here for the first time is necessary
to optimise the constraining power of the $\text{Ly}\alpha$ forest sample of
DESI. Mock catalogues will need to properly include redshift errors with more
realistic models, not necessarily Gaussian, and consider how different
emission lines have different velocity shifts (e.g., Hewett & Wild, 2010; Shen
et al., 2016). The correlations between emission line velocity shifts could
produce less smoothing of the forest continuum than implied by our
prescription, which assumed random errors. This is because while the quasar
redshift may be in error by some amount $\sigma_{v}$ relative to the systemic
redshift, the quasar redshift determined from emission lines redward of
Ly$\alpha$ may be a better predictor of the locations of the emission lines in
the forest region than of the systemic redshift.

We did not study the issue of systematic biases in redshift estimates for
quasars. Depending on which broad lines are present in the observed spectrum,
systematic shifts may be introduced by spectral templates not accounting for
them. The issue of systematic biases, and how they impact BAO constraints has
been studied in Font-Ribera et al. (2013) and Glanville et al. (2021).

The methodology developed in this paper will be extremely useful in
combination with a better understanding of the DESI data, to help develop
fitting templates to account for these contaminations in the BAO analysis.

Finally, the contamination discussed in this paper could be an important
systematic for studies that want to extract cosmological information from the
full shape of the 3D correlations in the $\text{Ly}\alpha$ forest (Cuceu et
al., 2021). While the BAO measurements seem to be robust against them, it is
possible that other cosmological inference might be biased if the impact of
redshift errors is not taken into account.

## Acknowledgements

SY was supported by a Science and Technology Facilities Council (STFC)
studentship. 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-20-CE-02 - DARKUNI). AFR
acknowledges funds through the program Ramon y Cajal (RYC-2018-025210) of the
Spanish Ministry of Science and Innovation. IFAE is partially funded by the
CERCA program of the Generalitat de Catalunya. IPR was supported by funding
from the European Union’s Horizon 2020 research and innovation programme under
the Marie Skłodowska-Curie grant agreement No. 754510.

This research is supported by the Director, Office of Science, Office of High
Energy Physics of the U.S. Department of Energy under Contract No.
DE–AC02–05CH11231, and by the National Energy Research Scientific Computing
Center, a DOE Office of Science User Facility under the same contract;
additional support for DESI is provided by the U.S. National Science
Foundation, Division of Astronomical Sciences under Contract No. AST-0950945
to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the
Science and Technologies Facilities Council of the United Kingdom; the Gordon
and Betty Moore Foundation; the Heising-Simons Foundation; the French
Alternative Energies and Atomic Energy Commission (CEA); the National Council
of Science and Technology of Mexico (CONACYT); the Ministry of Science and
Innovation of Spain (MICINN), and by the DESI Member Institutions:
https://www.desi.lbl.gov/collaborating-institutions.

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

This research used resources of the National Energy Research Scientific
Computing Center (NERSC), a U.S. Department of Energy Office of Science User
Facility located at Lawrence Berkeley National Laboratory, operated under
Contract No. DE-AC02-05CH11231.

## Data Availability

The analysis code picca is publicly available at
https://github.com/igmhub/picca.

Data points for all plots are available at https://zenodo.org/record/6543559.

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