# First Detection of the BAO Signal from Early DESI Data

Jeongin Moon,1 David Valcin,2 Michael Rashkovetskyi,3 Christoph Saulder,4
Jessica Nicole Aguilar,5 Steven Ahlen,6 Shadab Alam,7 Stephen Bailey,5 Charles
Baltay,8 Robert Blum,9 David Brooks,10 Etienne Burtin,11 Edmond Chaussidon,11
Kyle Dawson,12 Axel de la Macorra,13 Arnaud de Mattia,11 Govinda Dhungana,14
Daniel Eisenstein,3 Brenna Flaugher,15 Andreu Font-Ribera,16 Jaime E. Forero-
Romero,17 Cristhian Garcia-Quintero,18 Satya Gontcho A Gontcho,5 Julien Guy,5
Malik Muhammad Sikandar Hanif,19 Klaus Honscheid,20,21 Mustapha Ishak,18
Robert Kehoe,22 Sumi Kim,23 Theodore Kisner,5 Anthony Kremin,5 Martin
Landriau,5 Laurent Le Guillou,24 Michael Levi,5 Marc Manera,25 Paul
Martini,26,21 Patrick McDonald,5 Aaron Meisner,9 Ramon Miquel,16,27 John
Moustakas,28 Adam Myers$,^{29}$ Seshadri Nadathur,30 Richard Neveux,31 Jeffrey
A. Newman,32 Jundan Nie,33 Nikhil Padmanabhan,8 Nathalie Palanque-
Delabrouille,5,11 Will Percival,34,35 Alejandro Pérez Fernández,13 Claire
Poppett,36,5 Francisco Prada,37 Anand Raichoor,5 Ashley J. Ross,26,21 Graziano
Rossi,1⋆ Lado Samushia,38 David Schlegel,5 Hee-Jong Seo,2 Gregory Tarlé,19
Mariana Vargas Magana,13 Andrei Variu,39 Benjamin Alan Weaver,9 Martin J.
White,40 Christophe Yèche,11 Sihan Yuan,41 Cheng Zhao,39 Rongpu Zhou,5 Zhimin
Zhou,33 and Hu Zou33
Affiliations are listed at the end of the paper E-mail<EMAIL_ADDRESS>(GR<EMAIL_ADDRESS>(JM)

(Accepted 2023 August 29. Received 2023 August 28; in original form 2023 April
19)

###### Abstract

We present the first detection of the baryon acoustic oscillations (BAO)
signal obtained using unblinded data collected during the initial two months
of operations of the Stage-IV ground-based Dark Energy Spectroscopic
Instrument (DESI). From a selected sample of $261,291$ Luminous Red Galaxies
spanning the redshift interval $0.4<z<1.1$ and covering $1651$ square degrees
with a $57.9\%$ completeness level, we report a $\sim 5\sigma$ level BAO
detection and the measurement of the BAO location at a precision of $1.7\%$.
Using a Bright Galaxy Sample of $109,523$ galaxies in the redshift range
$0.1<z<0.5$, over $3677$ square degrees with a $50.0\%$ completeness, we also
detect the BAO feature at $\sim 3\sigma$ significance with a $2.6\%$
precision. These first BAO measurements represent an important milestone,
acting as a quality control on the optimal performance of the complex
robotically-actuated, fiber-fed DESI spectrograph, as well as an early
validation of the DESI spectroscopic pipeline and data management system.
Based on these first promising results, we forecast that DESI is on target to
achieve a high-significance BAO detection at sub-percent precision with the
completed 5-year survey data, meeting the top-level science requirements on
BAO measurements. This exquisite level of precision will set new standards in
cosmology and confirm DESI as the most competitive BAO experiment for the
remainder of this decade.

###### keywords:

cosmology: large-scale structure of Universe, observations, dark energy –
galaxies: statistics – methods: data analysis, statistical.

††pubyear: 2023††pagerange: First Detection of the BAO Signal from Early DESI
Data– ‣ First Detection of the BAO Signal from Early DESI Data

##  1 Introduction

The precise measurement of the expansion history of the Universe remains one
of the key challenges in modern cosmology, and represents a compelling probe
of the nature of dark energy (DE). The distance-redshift relation over a wide
redshift range tests whether the accelerated expansion is consistent with a
cosmological constant ($\Lambda$) or requires a dynamical explanation. It is
also an important constraint on the growth rate of structures, allowing
precise probes of gravity on cosmological scales, and on the Hubble constant,
shedding light on the source of the “Hubble tension” as coming either from as-
yet unappreciated astrophysical systematics or new physics. Finally, it breaks
cosmological parameter degeneracies in, e.g., neutrino mass measurements.
Recent results from state-of-the-art experiments have provided highly accurate
constraints on the basic parameters of the standard spatially flat
$\Lambda$CDM cosmological model, dominated by collisionless cold dark matter
(CDM) and a DE component in the form of $\Lambda$ (Planck Collaboration et
al., 2020; eBOSS Collaboration et al., 2021; Dark Energy Survey Collaboration
et al., 2022).

The baryon acoustic oscillation (BAO) method is one of the most mature and
robust probes of expansion history. Acoustic oscillations in the early pre-
recombination Universe imprint a feature in the galaxy distribution at a scale
($r_{\rm d}$) set by the sound horizon evaluated at the drag epoch. The
physics of these oscillations and the scale of this feature (which constitutes
a fundamental standard ruler) are exquisitely calibrated by cosmic microwave
background (CMB) measurements. Furthermore, the scale of the sound horizon is
much bigger than the scale of physics of nonlinear structure formation and
galaxy biasing, making it robust to the subsequent evolution of the Universe;
for a review on BAO, see Weinberg et al. (2013) and references therein. The
apparent size of this standard ruler across and along the line of sight
characterizes the angular diameter distance ($D_{\rm A}$) and the Hubble
parameter ($H$) as a function of redshift. Previous surveys have successfully
measured these quantities directly from the BAO feature at different
redshifts. Examples of first BAO detections obtained from multiple tracers
include Eisenstein et al. (2005), Cole et al. (2005), Blake et al. (2012), du
Mas des Bourboux et al. (2020), while the most recent results are reported in
eBOSS Collaboration et al. (2021) and in Dark Energy Survey Collaboration et
al. (2022).

BAO measurements at sub-percent precision are considered primary science
targets for the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration
et al., 2016), along with novel constraints on theories of modified gravity
and inflation, and on neutrino masses. DESI, the only Stage-IV DE experiment
that is currently taking data, aims to provide multiple sub-percent distance
measurements over a broad $0<z<3.5$ redshift range. DESI represents an order-
of-magnitude improvement both in the volume surveyed and in the number of
galaxies measured over previous experiments – e.g., Extended Baryon
Oscillation Spectroscopic Survey (eBOSS; Dawson et al., 2016), a key component
of the fourth generation (SDSS-IV; Blanton et al., 2017) of the Sloan Digital
Sky Survey (SDSS; York et al., 2000). In addition, DESI builds in a number of
internal systematics checks using multiple tracer populations to probe common
volumes.

Given the exquisite precision achievable by the DESI survey, the DESI
collaboration decided to blind the redshift data to avoid any confirmation
biases that can potentially impact all of the cosmological analyses. A general
procedure to blind a modern redshift survey has been discussed in Brieden et
al. (2020), and the exact implementation into the DESI framework will be
described elsewhere. However, for early quality assurance tests and in order
to validate the data processing and analysis pipelines, the first two months
of DESI observations (hereafter referred to as DESI-M2) have been kept
intentionally unblinded.

In this work, we use the DESI-M2 dataset and report the first high-
significance detection of the BAO signal from the initial two months of DESI
operations. As part of testing these early data, we integrate the eBOSS BAO
pipeline into the DESI analysis framework, and apply such pipeline to measure
the BAO scale with updates to accommodate all of the DESI specifics. While the
DESI survey has four primary galaxy tracer populations to measure
clustering,111The DESI survey also has a fifth tracer, i.e., the
Lyman-$\alpha$ (Ly$\alpha$) forest. the survey strategy implies that not all
tracers will have the same completeness in the very early data. The two most
complete samples are the DESI Bright Galaxy Sample (BGS) and the Luminous Red
Galaxies (LRGs). We focus on these two samples here for the BAO measurement,
since our simulations suggest that we would not expect a BAO detection in the
Emission Line Galaxy (ELG) and Quasar (QSO) samples given the number density,
the low completeness, and the volume of this early data. A similar signal-to-
noise (SN) consideration led us to concentrate on the isotropic distance
measurements $\alpha\equiv D_{\rm A}^{2/3}H^{-1/3}/r_{\rm d}$, probed by the
angle-averaged galaxy correlation function (the "monopole"). Future DESI
analyses will present measurements using all four tracer populations, as well
as measurements of $D_{\rm A}H$ from the Alcock-Paczynski effect (Alcock &
Paczynski, 1979).

As we will show in our analysis, even these early data yield a precision in
distance comparable to measurements from previously completed surveys (i.e.,
Anderson et al., 2012; Bautista et al., 2021), highlighting the remarkable
statistical power of the DESI data. In the spirit of the DESI blinding policy,
we restrict ourselves to providing just the statistical precision of the
measurements rather than the actual distance values – which will be presented
instead in a series of DESI Year 1 (Y1) forthcoming cosmological papers. This
work therefore should be seen as an end-to-end quality assurance of the DESI
data management system, as well as an early validation of the DESI
spectroscopic pipeline.

The layout of the paper is organized as follows. In Section 2, we briefly
describe the main aspects of the DESI-M2 sample used in this work, along with
the procedure to build the corresponding large-scale structure (LSS) data
catalogs. In Section 3, we present the approximate and $N$-body-based mocks
adopted in the core analysis, and explain how such synthetic catalogs are
constructed in order to mimic the complex footprint and characteristics of the
DESI-M2. In Section 4, we illustrate all of the analysis tools, namely the
chosen two-point clustering estimator, the density field reconstruction
technique, and the BAO fitting methodology. Section 5 addresses covariance
matrices, and in particular the construction, calibration, and validation on
mock data of semi-analytical semi-empirical covariances for the BAO fitting
procedure. More details on the covariance matrices adopted here are reported
in a companion paper (Rashkovetskyi et al., 2023). The main results are
detailed in Section 6, where we assess the precision and detection statistics
of the BAO feature in the LRG and BGS samples. We then briefly address the
expected precision of the final Year 5 (Y5) DESI LRG sample in Section 7, in
terms of the BAO detection level, based on forecasts obtained from our
promising early results. Finally, we conclude in Section 8, where we summarize
the main findings and highlight the relevance for the upcoming Y1 DESI
dataset. We also leave some additional material in Appendix A.

##  2 DESI Main Survey Data: First Two Months

In this section, we provide a concise description of the DESI-M2 dataset,
along with several specifics on the LSS catalog construction. A number of
additional technical details can be found in the quoted supporting papers,
many of which are still in a preparatory phase and will be available at the
time of the official Y1 DESI data release.

###  2.1 DESI Early Data: General Aspects

Table 1: Statistics of the four primary DESI targets from the DESI-M2 dataset, including completeness information. Target | $N_{\rm North}$ | $N_{\rm South}$ | $N_{\rm Total}$ |  $z$ range | Area [deg2] | Completeness
---|---|---|---|---|---|---
BGS Bright | 239492 | 390988 | 630480 | 0.1 - 0.5 | 3677 | 0.500
BGS Bright, $M_{r}<-21.5$ | 38472 | 71051 | 109523 | 0.1 - 0.5 | 3677 | 0.500
LRG | 80651 | 180640 | 261291 | 0.4 -1.1 | 1651 | 0.579
ELG | 55383 | 117145 | 172528 | 0.8 - 1.6 | 976 | 0.297
QSO | 70337 | 153453 | 223790 | 0.8 - 3.5 | 2906 | 0.778

DESI began its main program on May 17, 2021. Its commissioning and ‘Survey
Validation’ (SV) phases (DESI Collaboration et al., 2023a) had proved the
instrument (Abareshi et al., 2022) and operations strategy (Schlafly et al.,
2023) to be efficient. The DESI collaboration decided that the first two
months of the observations of DESI main survey data (i.e., DESI-M2) could be
analyzed without the blinding restrictions imposed on the rest of the sample
that will be used for DESI Y1 Key Projects.

The DESI-M2 data were observed on nights in 2021 from May 14th through July
9th on 304 dark time and 342 bright time ‘tiles’. Each tile represents a
specific sky location pointing of the telescope and specific target selection
for each of the 5000 robotic positioners populating the DESI focal plane
(Silber et al., 2022) determined by the DESI fiberassign software (Raichoor et
al. in preparation). The spectra extracted from these observations were
reduced by the DESI spectroscopic pipeline (Guy et al., 2023) and released to
the DESI collaboration as the Guadalupe spectroscopic product. The redshift
measurements in these Guadalupe data are used in this paper and will be made
public with the DESI Y1 data release (DR1), i.e., they are not available in
the DESI early data release (EDR; DESI Collaboration et al. 2023b).222The
analogous spectroscopic data reductions and redshift fits for the EDR are Fuji
and will be publicly available on NERSC here:
https://data.desi.lbl.gov/public/edr/spectro/redux/fuji

The DESI-M2 tiles are primarily first pass tiles that do not overlap each
other. In dark time, the full DESI survey observes tiles in 7 overlapping
passes, with a median overlap of 5 (Schlafly et al., 2023). Thus, the DESI-M2
data are substantially less complete in the area observed than they will be
when the survey is finished. This incompleteness affects all samples, but is
most extreme for the target classes that are given the lowest priority during
the assignment of fibers on a tile (DESI fiber assignment is reported in
Raichoor et al. in preparation). We describe this further next, when
discussing the different DESI target classes.

###  2.2 DESI Targets

DESI divides its observing time into a ‘bright’ and a ‘dark’ time program, for
which the targeting is done independently (Myers et al., 2023). During dark
time, in order of priority for fiber assignment, QSOs (Chaussidon et al.,
2023), LRGs (Zhou et al., 2023), and ELGs (Raichoor et al., 2023) are
observed. QSOs with redshifts greater than $2.1$ are selected for follow-up in
order to increase the SN of the spectra in the Ly$\alpha$ forest region.
During bright time, a BGS (Hahn et al., 2023) is observed, which has a
‘Bright’ and ‘Faint’ component, as well as Milky Way stars (MWS; Cooper et al.
2022). In this work, we only consider the higher priority BGS Bright sample.

For detailed discussions of how these target samples were chosen, we refer the
reader to the individual selection papers previously cited. Table 1 summarizes
key properties of the samples and Figure 1 shows the redshift distribution of
each sample. Combined, they will allow measurements of large-scale clustering
modes at better than the sample variance limit to $z<1.6$ (DESI Collaboration
et al., 2023a). The QSO sample provides this information at a lower sampling
rate all the way to redshifts greater than $3$ and further samples density
fluctuations via the variance of Ly$\alpha$ forest absorption in each
spectrum. The BGS sample is approximately flux limited and thus has a spatial
density that rapidly increases as the redshift gets lower and is approximately
sample variance limited to $z<0.5$. There is also substantial overlap between
the LRG and ELG catalogs, which will allow comparison between results obtained
from the most massive and passive galaxies (LRG) and those that are actively
star forming (ELG).

Figure 1: Redshift distribution of the four primary DESI tracers, from the
DESI-M2 clustering catalogs. Figure 2: Footprints of the DESI-M2 BGS Bright
(top) and LRG (bottom) clustering samples, color-coded by completeness
weights. The total areas highlighted by the pink color represent the final
DESI Y5 expected footprint. The specific DESI-M2 areas covered by the BGS
Bright and LRG samples are respectively 3677 deg2 and 1651 deg2.

###  2.3 LSS Catalog Construction

The construction of the LSS catalogs involves determining the area on the sky
where good observations were possible for each tracer, applying criteria on
the DESI data to select reliable redshifts within a given redshift range, and
providing weights that correct for variations in observing completeness,
target density due to changes in imaging conditions, and relative redshift
success due to variations in DESI observing. The overall process is similar to
that applied to SDSS (most recently eBOSS; Ross et al. 2020), with the
specific details of DESI observations accounted for as we describe here. The
pipeline that was applied to the DESI-M2 sample represents an early version of
the DESI LSS catalog pipeline, which will be fully described (and considerably
improved) in Y1 publications. Many aspects of the pipeline match that applied
to the DESI ‘One Percent Survey’, which is detailed in the overall description
of the DESI EDR (DESI Collaboration et al., 2023b). In what follows, we
provide details on the specific choices applied for DESI-M2.

The ‘randoms’ that populate the sky area where good observations were possible
were produced using the same procedures as applied to the DESI One Percent
Survey LSS catalogs. DESI randoms are produced using a standard such that each
individual (and independent) set has a density of $2500$ deg-2. We use $10$
such sets for the DESI-M2 clustering measurements and thus the total sky
density of the random samples used is $25000$ deg-2. The process of creating
DESI randoms produces significantly different areas for different tracer types
due to the priority masking (e.g., we have no randoms in areas where LRGs
could not have been observed because a higher priority QSO target was assigned
to the fiber positioner associated with that sky location). In order to
determine the effective area occupied by each sample, we simply count the
number of randoms in one of the final LSS random catalogs sets and divide by
$2500$ deg-2. These areas are given in Table 1. While the total effective area
is considerably different per tracer, the footprint of tiles is the same for
all dark/bright time tracers. Figure 2 shows the footprint of tiles for BGS
(bright time; top panel) and LRG (dark time; bottom panel) tracers. The plot
is constructed via a web interface provided by David
Kirkby,333https://observablehq.com/@dkirkby/skymap/ and it is color coded by
the completeness in each tile grouping.

For the data samples, we follow the same procedures as applied to the One
Percent Survey LSS catalogs (described with the EDR) in order to select
targets of the given type that could have been observed. Any unobserved
targets at this stage were not observed because a target of the same type was
instead observed at the given sky location. Each observed target is given a
completeness weight, WEIGHT_COMP, equal to the total number of targets (of the
given type) at the location of the observed target (with unique ‘locations’
determined by the combination observed tile and fiber positioner; see DESI
Collaboration et al. 2023b for more details). In particular, we note that the
fiber patrol radius is at most 89”: it depends on e.g., the focal plane
position due to the optics.

The criteria for the BGS and LRG samples we focus this study on are:

  * •

LRG: $0.4<z<1.1$, ZWARN$=$0, DELTACHI2$>$15

  * •

BGS: $0.1<z<0.5$, ZWARN$=$0, DELTACHI2$>$40.

ZWARN is a bitmask generated by the redshift pipeline (Guy et al., 2023),
where any non-zero value indicates a problem. The criteria on DELTACHI2, which
is the difference in $\chi^{2}$ between the two best-fitting redshift
solutions, were shown to provide pure and complete samples in the respective
targeting papers (Hahn et al. 2023; Zhou et al. 2023). For LRGs, the choice of
redshift range is motivated by the fact that the number density is
approximately constant at $5\times 10^{-4}(h^{-1}{\rm Mpc})^{3}$ between
$0.4<z<0.8$. At $z>0.8$, the LRG density decreases mostly due to the sample’s
minimum flux threshold and is less than $1\times 10^{-4}(h^{-1}{\rm Mpc})^{3}$
for $z>1.1$. Similarly, the density of the BGS sample decreases to less than
$1\times 10^{-4}(h^{-1}{\rm Mpc})^{3}$ for $z>0.5$. For the BGS sample, we
also apply an absolute magnitude cut in the $r$-band $M_{\rm r}<-21.5$. When
obtaining the absolute magnitude, we simply apply the distance modulus, i.e.,
we do not apply any corrections for evolution (‘e’ correction) or the shape of
the spectrum (‘k’ correction). The cut provides a sample with roughly constant
number density at $\sim 8\times 10^{-4}h^{3}$Mpc-3 and clustering amplitude
for $z<0.4$ and is thus sufficient for our preliminary study. Future DESI
studies will likely include k+e corrections, especially for the selection of
BGS samples.

We then add two more weights in order to account for variations in the
selection of the data. The first corrects for fluctuations in the target data
that are due to variation in the imaging data quality. To do so, we apply the
random forest regression method (Chaussidon et al., 2021) available as an
option in the regressis
package,444https://github.com/echaussidon/regressis/releases/tag/1.0.0 given
maps of imaging properties compiled by the DESI targeting team. The data
(after redshift cuts) and randoms are combined to produce a map of the
projected density of the sample at Healpix (Górski et al., 2005) $N_{\rm
side}=256$ and is compared to maps of the depth and PSF size in the $g$, $r$,
$z$, and $W1$ bands, the $E(B-V)$ Galactic extinction according the Schlegel
et al. (1998) dust maps, and the stellar density observed in the Gaia 2nd data
release (Gaia Collaboration et al., 2018). The regressis random forest method
is used to determine a model of the projected density fluctuations as a
function of those map quantities and the inverse of the model is included in
the catalogs as a weight, ‘WEIGHT_SYS’. For our LRG and BGS samples, very
similar clustering results are obtained when instead obtaining the weights
using the linear regression method applied to eBOSS, described in (Ross et
al., 2020).

After, we obtain a weight to account for variations in redshift success based
on the particulars of DESI observations. Zhou et al. (2023) showed that the
LRG redshift success can be modeled as a function of the effective observing
time and the target’s fiber flux in the $z$-band. A similar dependency exists
for BGS, with the $r$-band fiber flux the relevant photometric quantity. The
inverse of the best-fit model for the failure rate is used as ‘WEIGHT_ZFAIL’.
We find that applying these redshift failure weights has very little impact on
the clustering measurements used in this work.

Next, we determine ‘FKP’ weights (Feldman et al., 1994) in order to properly
weight each volume element with respect to how each sample’s number density
changes with redshift. This is simply given by:

$w_{\rm FKP}=\frac{1}{1+n(z)CP_{0}}$ (1)

where $n(z)$ is the weighted number per volume, $C$ is the mean completeness
for the sample, and $P_{0}$ is a fiducial power-spectrum amplitude. We use
$P_{0}=10^{4}(h^{-1}{\rm Mpc})^{3}$ for LRGs and $P_{0}=7\times
10^{3}(h^{-1}{\rm Mpc})^{3}$ for BGS. These values approximately match the
monopole of the power spectrum at $k=0.15h/{\rm Mpc}$ for the respective
samples.

The redshifts and all four weights are then randomly sampled from the data
catalog and attached to the random catalogs in order to match the radial
selection function. Finally, the catalogs are normalized separately (and all
weights are fit for separately) in the North and South photometric regions.

##  3 Mock Catalogs

In our analysis, we utilize DESI mock galaxy catalogs for statistically
testing the performance of the BAO fits, as well as for validating the adopted
covariance matrices in terms of BAO fitting. Here, we briefly describe the
main characteristics of the various sets of mocks, along with the DESI
customization procedure to include survey realism.

###  3.1 DESI Mocks: General Description

We use two different sets of DESI mock galaxy catalogs for the LRG sample: one
type is directly constructed from $N$-body simulations (i.e., AbacusSummit;
Maksimova et al., 2021), while a second type is based on approximated methods
(i.e., EZmocks; Zhao et al., 2021).

The $N$-body-based realizations are part of the first official set of DESI
mock galaxy catalogs (Alam et al. in preparation) which were calibrated based
on an early reduction of the One Percent Survey spectroscopic data for LRG
(see Section 3.3).555The matching data in the final reductions are publicly
released as part of the DESI EDR. This set is made of $25$ cutsky simulations
based on the $2\,h^{-1}{\rm Gpc}$ AbacusSummit
runs.666https://abacussummit.readthedocs.io/en/latest/ The halo occupation
distribution (HOD) model for LRGs is calibrated using small-scale (below
$5\,h^{-1}{\rm Mpc}$) wedges in combination with large-scale bias evolution
where available. The LRG mocks are further subsampled to approximately match
the $n(z)$ distribution of the specific LRG sample (called ‘main’) considered
in this paper (see Figure 16 of Zhou et al., 2023, for the ‘main’ selection in
the One Percent Survey). The mocks implementing the DESI survey geometry and
specifics (denoted as ‘cutsky’ mocks) are generated using the simulation
output near the primary redshift of LRGs that we do not report in this paper.
The $2\,h^{-1}{\rm Gpc}$ box is repeated and then the coordinates are
converted to sky coordinates.777The code used to create the cutsky/lightcones
can be found at https://github.com/Andrei-EPFL/generate_survey_mocks/

The approximate mock realizations consist of 1000 EZmocks for LRGs, and are
built with an elaborated procedure centered on the Zel’dovich approximation
(Zel’dovich, 1970). They do account for stochastic scale-dependent, non-local,
and nonlinear biasing contributions: extensive details on the production
methodology can be found in the original release paper by Chuang et al.
(2015). The EZmocks have accurate clustering properties consistent with the
previously described $N$-body-based AbacusSummit realizations – and nearly
indistinguishable from actual $N$-body solutions – in terms of one-point, two-
point, and three-point statistics.

We note that we have decided not to use any mocks for the BGS sample,
primarily for reasons related to a calibration performed with an earlier DESI
dataset than the one considered in this study.

###  3.2 DESI Mocks: Masking and Customization

Figure 3: [Top] Map of a single LRG EZmock realization with the pixel
probability of the HEALPix mask. [Bottom] Dispersion of $1000$ LRG EZmocks,
after application of the different masks described in Section 3.2, compared
with the actual LRG DESI-M2 redshift distribution.

An important step in the mock-making pipeline is the incorporation of survey
realism, namely the characteristics of the various synthetic realizations need
to accurately match the properties of the DESI-M2 sample. This is achieved via
the application of a succession of masks, as we schematically describe in what
follows.

Survey Masks. We subsample the mocks using a tile mask matching the footprint
of DESI-M2 and match the redshift distribution of each target: for LRGs, the
$n(z)$ distribution is based on DESI-M2 results (Figure 2), while for BGS on
the One Percent Survey distribution. This tile mask cuts the data and random
to the circular region around each tile center.

Intra-tile Geometry. The geometrical area where DESI targets could have been
observed is obtained for the data catalogs following the procedure described
in Section 2.3. In order to analyze all $1000$ mocks, we approximate the
results of the procedure run on the data using a HEALPix (Górski et al., 2005)
map built from the random catalogs. As a reference, we utilize the same random
cuts to the target catalog sky area that were the inputs to the LSS catalog
process, additionally cutting them to be within the tile area previously
defined. In every $N_{\rm side}=1024$ pixel, we count the number of randoms in
both the LSS catalog and in the reference randoms. The ratio of these counts
approximates the small-scale holes in the observed footprint. We apply it by
sub-sampling the mock data and randoms by the fraction in each pixel. In this
way, the effective area of the footprint, as determined by summing random
points assumed to have a constant surface density, matches that of the
observed data. This is shown as an example in the top panel of Figure 3, to be
compared with the bottom panel of Figure 2.

Incompleteness Assignment. At this stage, we still have more simulated
galaxies than those observed in the actual DESI data. This is because, for
most of the locations, only one fiber is available to observe multiple
targets. In order to approximate this effect in terms of number counts, we
simply take the overall assignment completeness of the data in the LSS
catalogs, i.e., $N_{\rm observed}/N_{\rm total}$, where $N_{\rm total}$ is the
number of targets within the DESI-M2 footprint where observations were
possible. This type of completeness should vary strongly as a function of the
number of overlapping DESI tiles, but we simply apply a constant factor (an
average of $0.51$ for the $1000$ realizations) given that over $90\%$ of the
DESI-M2 area is covered by only a single tile. To this end, the bottom panel
of Figure 3 shows that the observed DESI-M2 LRG redshift counts per square
degree match well those obtained from mock data, after applying the assignment
incompleteness factor. This procedure is implemented in the $N$-body based
AbacusSummit realization as well as in the approximate EZmocks. Finally, we
note that the incompleteness will be modeled more rigorously for the
forthcoming DESI Y1 analysis.

###  3.3 DESI Mocks: Calibration

In terms of clustering properties (see Section 4.1), the two sets of mocks
described here have been tuned via survey and completeness masks with an
earlier version of the DESI LRG clustering measurements (i.e., One Percent
Survey data) having a 10% lower amplitude than the DESI-M2 LRG sample
considered in the current study. This can be readily inferred from Figure 4,
where we contrast the observed LRG clustering in the DESI-M2 sample at $r\sim
20\,h^{-1}{\rm Mpc}$ with the average clustering of 1000 LRG EZmocks. For this
reason, in the present analysis mocks are only used for validation purposes,
and we will be adopting semi-analytical semi-empirical covariances rather than
mock-based covariances for our primary BAO fits – as described in Section 5.
In fact, a calibration offset in the two-point clustering (although located
outside of the BAO fitting range) would manifest in a substantial difference
in the covariance between different scales, causing a 17.6% impact on the
resulting BAO precision when pairing such a mock-based covariance matrix with
the actual data clustering.

Figure 4: Monopole of the LRG two-point correlation function before
reconstruction, as measured from the DESI-M2 sample (blue dots) and from the
average of 1000 EZmocks (orange dots). Data errorbars are obtained from a
jackknife covariance directly inferred from DESI-M2 LRGs, while mock errorbars
are drawn from the LRG EZmock sample covariance. As mentioned in the main text
(Section 3.3), the $\sim 10\%$ difference near the $\sim 20~{}\,h^{-1}{\rm
Mpc}$ peak is not surprising, as these mocks were tuned with an earlier
version of the DESI data. Hence, in the present work mocks are only used for
validation purposes.

##  4 Analysis Methods

In this section, we illustrate all of the analysis tools adopted in our work,
from the two-point clustering estimator to the density field reconstruction,
until the BAO fitting methodology. In particular, the DESI team is currently
studying all aspects of the BAO pipeline given the stringent requirements on
theoretical and observational systematics that will be imposed by a dataset as
powerful as we expect by the end of the survey. For the investigation of this
preliminary DESI-M2 data, and to make contact with earlier work on the
subject, we choose to largely follow the analysis choices made by the
BOSS/eBOSS surveys. We highlight these choices in what follows, while
referring the reader to the original papers for more extensive details.

###  4.1 Two-Point Correlation Function Estimator

Figure 5: Two-point correlation function measurements of the four DESI
tracers, obtained from the DESI-M2 sample. Errorbars are derived from the
diagonal of the corresponding covariance matrixes, although we caution the
reader of a significant bin-to-bin correlation in these measurements. Model
curves are simple damped linear theory predictions that indicate the expected
overall clustering amplitude and BAO damping typical at the mean redshift of
the target samples.

We compute all of the anisotropic redshift-space correlation functions
$\hat{\xi}$’s with the well-known Landy-Szalay estimator (LS; Landy & Szalay,
1993), namely:

$\hat{\xi}(s,\mu)=\frac{DD(s,\mu)-2DR(s,\mu)+RR(s,\mu)}{RR(s,\mu)},$ (2)

where $DD(s,\mu)$ and $RR(s,\mu)$ are the normalized weighted number of pairs
in the data and random catalogs – respectively – binned as a function of the
separation $s$ between two galaxies, $\mu\in[-1,1]$ is the cosine angle
between the galaxy pair and the line of sight (LOS), and $DR(s,\mu)$ denotes
pair counts between data ($D$) and randoms ($R$). The LS estimator gets
modified when the reconstruction procedure (described in Section 4.2) is
applied. In essence, a shifted random catalog (termed $S$) should be used in
the numerator of Equation (2) in substitution of $R$, and one needs to replace
$DR$ with $DS$ and $RR$ with $SS$, respectively. We use $200$ $\mu$-bins
spanning the interval [$-1,1$] and $4h^{-1}{\rm Mpc}$ $s$-bins for BGS and
LRGs.

The anisotropic correlation function $\hat{\xi}(s,\mu)$ is then integrated
over the Legendre polynomials $\mathcal{L}_{\ell}(\mu)$ to obtain the various
multipoles; in the current analysis, we only use the monopole, i.e., $\ell=0$:

$\displaystyle\hat{\xi}_{\ell}(s)$
$\displaystyle=\frac{2\ell+1}{2}\int_{-1}^{1}d\mu\hat{\xi}(s,\mu)\mathcal{L}_{\ell}(\mu)$
(3)
$\displaystyle\simeq\frac{2\ell+1}{2}\sum_{i}\hat{\xi}(s,\mu_{i})\int_{\Delta\mu_{i}}d\mu\mathcal{L}_{\ell}(\mu).$
(4)

In the last equality, we have made explicit the discrete sum over $\mu$-bins,
weighted by the analytic integral of $\mathcal{L}_{\ell}(\mu)$ over each
$\mu$-bin having width $\Delta\mu_{i}$.888Such a summation scheme, contrary to
weighting $\hat{\xi}(s,\mu_{i})$ by
$\mathcal{L}_{\ell}(\mu_{i})\Delta\mu_{i}$, ensures that the $\ell>0$
multipoles are exactly zero if $\hat{\xi}(s,\mu)$ remains constant as a
function of $\mu$.

All of the two-point correlation function calculations are performed with the
Python package pycorr,999https://github.com/cosmodesi/pycorr which wraps a
modified version101010https://github.com/adematti/Corrfunc of the Corrfunc
package (Sinha & Garrison, 2019,
2020).111111https://github.com/manodeep/Corrfunc

Figure 5 shows the observed two-point correlation functions of the four
tracers discussed in Section 2.2, contrasted with simple damped linear theory
models that indicate the expected overall clustering amplitude and BAO damping
typical at the mean redshift of the target samples. From LRGs and BGS, we
observe a local bump near the expected location of the BAO peak. Moreover,
while BGS observations appear to lie systematically below the model curve at
scales greater than $120h^{-1}{\rm Mpc}$, this is simply because there are
fewer modes at larger separations in these early DESI data. Therefore, they
are highly correlated and thus the amplitude of the two-point correlation
function decreases at those scales.121212 In addition, note that we only fit
up to $150h^{-1}{\rm Mpc}$ for the BAO analysis, where the corresponding
linear theory prediction is still consistent with observations – within
errorbars. For ELGs and QSOs, the amplitude of the observed clustering appears
consistent with the theoretical expectations (within errors), although it is
challenging to identify a clear BAO-like signature. Indeed, we do not expect a
BAO detection from ELGs and QSOs of the DESI-M2 sample, given the small survey
volume in combination with the low completeness (ELGs) and high shot noise
(QSOs).

###  4.2 Density Field Reconstruction

We apply the density field reconstruction technique (Eisenstein et al., 2007)
on the observed galaxy density fields in order to partially recover the BAO
feature that has been degraded due to structure growth and redshift space
distortions (RSD). To do so, we follow the iterative procedure described in
Burden et al. (2015), as implemented in the IterativeFFTReconstruction
algorithm of the pyrecon package131313https://github.com/cosmodesi/pyrecon
with the RecIso convention.141414RecIso is a choice to remove the large-scale
anisotropy due to redshift-space distortions in the process of reconstruction
(Padmanabhan et al., 2012; Seo et al., 2016). The density contrast field is
smoothed by a Gaussian kernel of width $15\,h^{-1}{\rm Mpc}$ and three
iterations are performed, assuming an approximate growth rate and the expected
bias for each sample. The choice of these reconstruction conditions along with
the assumed fiducial cosmology were shown to have a very marginal impact on
BAO measurements in earlier galaxy survey samples – i.e., see Vargas-Magaña et
al. (2018) and Carter et al. (2020).

###  4.3 BAO Fitting Methodology

Table 2: Covariance matrices utilized in this work. Name | Tracer | Notes
---|---|---
DESI-M2-EZ | LRG | Constructed from EZmock clustering
RascalC-EZ | LRG |  RascalC calibrated on EZmock clustering
RascalC-LRG | LRG |  RascalC calibrated on DESI-M2 LRG clustering
RascalC-BGS | BGS |  RascalC calibrated on DESI-M2 BGS clustering

We employ the same BAO fitting pipeline that has been previously applied to a
large number of BOSS and eBOSS analyses (Ross et al., 2017; Ata et al., 2018;
Hou et al., 2020; Raichoor et al.,
2020).151515https://github.com/ashleyjross/BAOfit The accuracy of such
methodology was demonstrated to be sufficient at the precision demanded by
BOSS/eBOSS data, especially for determining the isotropic BAO scale. However,
advances are expected to be necessary in order to meet the exquisite precision
expected for DESI Y5, and thus an improved DESI pipeline is currently under
development and will be presented along with the DESI Y1 analyses.

In this study, we only fit for the monopole of the correlation function, hence
for the isotropic scaling parameter $\alpha$. Our BAO pipeline is ‘template-
based’, and it essentially coincides with the algorithm introduced by Xu et
al. (2012). However, the BAO templates are generated via the formulae defined
by Equations 9-13 of Ross et al. (2017), where the linear power spectrum is
split into a BAO and a no-BAO components, and damping is added solely to the
BAO that depends on the LOS angle.

The templates require a choice of four parameters that are kept fixed during
the fitting process, namely: $\beta$, $\Sigma_{\rm s}$, $\Sigma_{||}$,
$\Sigma_{\perp}$. These determine, respectively, the degree of anisotropy with
respect to the LOS in linear RSD (Kaiser, 1987), the degree of streaming
velocity, the degree of radial BAO damping, and the degree of transverse BAO
damping. Such parameters are fixed separately for different samples and the
pre- or post-reconstruction fits. For galaxy velocities, we set $\beta=0.4$,
$\Sigma_{\rm s}=3\,h^{-1}{\rm Mpc}$ for both samples; such choices guarantee
an approximate match to the anisotropic clustering as measured from the
DESI-M2 data, although their impact in the fitting process is essentially
negligible since we only fit for the monopole. The BAO damping parameters for
post-reconstruction are fixed to $\Sigma_{||}$, $\Sigma_{\perp}$ = 3, 5
$\,h^{-1}{\rm Mpc}$ for both samples, roughly consistent with those
used/determined in previous studies (e.g., Seo et al. 2016; Ross et al. 2017;
Vargas-Magaña et al. 2018; Bautista et al. 2021). For pre-reconstruction, we
set these to 6, 10 $\,h^{-1}{\rm Mpc}$ for the BGS sample (again, roughly
consistent with the pre-reconstruction results from Seo et al. 2016) and
reduce them to 4, 8 $\,h^{-1}{\rm Mpc}$ for pre-reconstruction LRGs. This
evolution in the pre-reconstruction values roughly corresponds to the change
in the linear growth factor between the effective redshifts $z_{\rm eff}$ of
the two samples, noting that the BAO damping is expected to scale with the
amount of non-linear structure growth, which approximately scales with the
linear growth factor (Seo et al., 2016). The post-reconstruction values are
smaller and constant as reconstruction helps to reduce the effect of non-
linearities (hence, smaller damping values), and the degree of remaining non-
linearity does not depend strongly on the initial degree (hence the
independence of redshift). The impact of fixing these choices was already
shown to be negligible at the precision of BOSS DR12 (Ross et al., 2017), and
therefore is also not a concern in the present work.

The procedure just described produces a theory template, $\xi_{0}$.
Subsequently, the data is fit against this template evaluated with a scaling
parameter $\alpha$, a free amplitude, and a polynomial with three nuisance
terms:

$\xi_{\rm mod}(s)=B\xi_{0,\rm t}(s\alpha)+A_{0}+A_{1}/s+A_{2}/s^{2}.$ (5)

The polynomial has been shown to account for any difference between the broad-
band shape of the template $\xi_{0,\rm t}$ and the measured $\hat{\xi}_{0}$;
e.g., due either to cosmology or to observational systematics. The model is
evaluated at the $s$ of the data bin assuming a spherically symmetric
distribution.161616For our binsize of 4$\,h^{-1}{\rm Mpc}$, this corresponds
to a 0.03% effect compared to just using the bin center. The
$\chi^{2}(\alpha)$ is computed on a grid of spacing $0.001$ in $\alpha$, where
the minimum $\chi^{2}$ at each grid point is determined by varying
$B,A_{0},A_{1},A_{2}$. We note that the various $\chi^{2}$ are inferred from
the data vector $\vec{D}$ and covariance matrix $\boldsymbol{C}$ via
$\chi^{2}=\vec{D}\boldsymbol{C}^{-1}\vec{D}^{t}$, as routinely done. Our data
vectors are always selected to have $50<s<150\,h^{-1}{\rm Mpc}$.

Finally, the derived likelihood on the value of $\alpha$ can be used to
constrain cosmological models via

$\alpha=\frac{D_{\rm V}(z)r^{\rm fid}_{\rm d}}{D^{\rm fid}_{\rm V}(z)r_{\rm
d}}$ (6)

and

$D_{\rm V}(z)=\left[cz(1+z)^{2}H(z)^{-1}D^{2}_{\rm A}(z)\right]^{1/3},$ (7)

where $H(z)$ and $D_{\rm A}(z)$ are evaluated at an effective redshift of the
data sample being tested.

In closing this part, we highlight that while the methodology adopted here is
largely equivalent to the one exploited in previous BOSS/eBOSS analyses, the
version of the BAO pipeline used in this work has been fully updated to be
compatible with DESI code packages assuming generic cosmological backgrounds
and primordial/linear power spectrum calculations: such effort is carried out
within the cosmodesi framework.171717https://github.com/cosmodesi/BAOfit_xs/
To this end, the most significant change specific to the BAO fitting procedure
is how we isolate the BAO feature, namely by splitting the input linear power
spectrum into a smooth function with no-BAO and another one that is pure BAO.
To achieve such splitting, we apply the technique described in Wallisch (2018)
and coded in the bao_filter
module181818https://github.com/cosmodesi/cosmoprimo/blob/main/cosmoprimo/bao_filter.py
of the cosmoprimo package. The impact of this change in filtering the BAO
feature is less than $\sim 0.1\%$ on the measured value of $\alpha$.

##  5 Covariance Matrices

In this section, we briefly address covariance matrices, and in particular the
construction, calibration, and validation of semi-analytical semi-empirical
covariances on mock data – eventually adopted for the BAO fitting procedure.

###  5.1 Covariance Matrices: Types and Conventions

Figure 6: Scatter plots showing the BAO detection significance in units of
standard deviations (left panels), the $\alpha$’s (middle panels), and the
$\sigma_{\alpha}$ values (right panels) related to the validation procedure of
the RascalC-based covariance. BAO best fits are performed on a mock-by-mock
basis, using 1000 LRG EZmocks (open green dots) and 25 AbacusSummit LRG
realizations (filled yellow points). Stars of the same color are averages over
the corresponding entire set of realizations. Top panels refer to pre-
reconstruction measurements, while bottom panels display post-reconstruction
quantities. Individual BAO fits using the RascalC covariance calibrated on
EZmocks (i.e., RascalC-EZ; $x$-axes) are contrasted with those performed
adopting the EZmock covariance (i.e., DESI-M2-EZ; $y$-axes). As evident from
the figure, the narrow scatter along the diagonal implies that both
covariances produce compatible results, validating the usage of RascalC-based
covariances for our primary DESI-M2 BAO fits.

The primary BAO fits performed in our main analysis are obtained with semi-
analytical semi-empirical covariance matrices, generated by the RascalC code
(Philcox et al., 2020). As mentioned in Section 3.3, this choice is mainly
driven by the fact that the galaxy mocks available at the time of this study –
implementing all of the DESI-M2 survey characteristics – have been calibrated
with an earlier version of the DESI LRG clustering (i.e., the One Percent
Survey), rather than with the LRG clustering as measured directly from the
DESI-M2 dataset considered here. Nevertheless, we also construct a numerical
covariance from 1000 LRG EZmocks (termed ‘DESI-M2-EZ’), and use it for
validating the LRG RascalC semi-analytical covariance in terms of BAO fitting.
In order to calibrate the RascalC covariance matrix for LRGs, besides the
previous numerical covariance, we also utilize a covariance directly
constructed from the jackknife estimates of the LRG sample under
consideration. Once the RascalC semi-analytical covariance is validated (in
terms of BAO fits) for the LRG sample, we build a similar semi-empirical
covariance for BGS galaxies and calibrate it using jackknife estimates
obtained from the corresponding BGS dataset. Table 2 reports all of the
covariance matrices utilized in this work. Specifically, in terms of name
conventions, we indicate with ‘RascalC-EZ’ the semi-analytical covariance
based on the EZmocks LRG clustering, with ‘RascalC-LRG’ the semi-empirical
covariance calibrated on the DESI-M2 LRG clustering measurements (including
jackknife), and with ‘RascalC-BGS’ the one based on the DESI-M2 BGS
clustering.

Next, we provide some additional information on RascalC-based covariances, and
then present the BAO fitting validation tests performed on the mock LRG
sample.

###  5.2 RascalC Covariances

The semi-analytical semi-empirical covariance matrices used in our analysis
are obtained via the publicly available code RascalC (Philcox et al., 2020;
Rashkovetskyi et al., 2023).191919https://github.com/oliverphilcox/RascalC The
procedure to construct such covariances only requires a two-point correlation
function as input (along with its optional jackknife estimates), and a random
catalog. The RascalC algorithm integrates the Gaussian terms for the
covariance matrix using importance sampling from the set of random points. It
then progressively changes the amount of shot noise, which has the effect of
empirically rescaling those terms. The optimization of the shot noise level is
performed on separate jackknife covariance estimates. Once the optimal shot
noise level is determined (i.e., its best-fit value), a rescaling based on
such best-fit is applied to finally obtain the full covariance matrix terms.

For the construction of the RascalC-LRG and RascalC-BGS covariances (i.e., the
covariances calibrated on the DESI-M2 dataset), the input correlation
functions are measured directly from the DESI-M2 LRG and BGS samples,
respectively. In addition, 60 jackknife regions are assigned based on data
points with a K-means subsampler. For the pre-reconstruction case, the RascalC
code is run on 10 random catalogs separately, and the integration results are
finally averaged. Building the post-reconstruction covariances calibrated on
the DESI-M2 dataset requires some additional steps, described in detail in
Rashkovetskyi et al. (2023). In essence, such procedure depends upon the usage
of non-shifted random catalogs for normalization, shifted random catalogs for
sampling, and a slightly different two-point correlation function than the
familiar LS estimator (i.e., Equation 2). In this latter case, the code is run
on 20 random catalogs separately, and integration results are eventually
averaged.

In order to build the RascalC-EZ covariance, we use instead the averaged pre-
and post-reconstructed LRG EZmock correlation functions obtained from 1000
EZmocks, without jackknife estimates, and with no shot-noise rescaling
(Gaussian). We run the RascalC code on 10 concatenated random catalogs in pre-
reconstrution, and on 20 randoms for the post-reconstruction case.

As concluding remarks for this section, we note that, by construction, the LRG
RascalC data covariance (i.e., RascalC-LRG) gives larger error bars than the
DESI-M2-EZ sample covariance, while the RascalC covariance based on the EZmock
clustering (i.e., RascalC-EZ) is consistent with the DESI-M2-EZ sample
covariance. A more detailed assessment on the performance of RascalC-based
covariances is presented in Rashkovetskyi et al. (2023).

###  5.3 Validation of RascalC Covariances for BAO Fitting

Before performing BAO fits on the DESI-M2 dataset, we validate our LRG RascalC
covariance against a set of approximate LRG EZmocks and $N$-body based
AbacusSummit realizations. While the following tests are performed on LRGs, we
note that the validation procedure is general and would apply to any tracers.

Specifically, we first compute the two-point clustering statistics of 1000 LRG
EZmocks and 25 AbacusSummit LRG realizations with the estimator presented in
Section 4.1, adopting default FKP weights. We then apply the reconstruction
algorithm detailed in Section 4.2 to all of the mocks, assuming a smoothing
scale of $15h^{-1}{\rm Mpc}$. After, we build the pre- and post-reconstruction
EZmock covariance (i.e., DESI-M2-EZ; Table 2), and a RascalC covariance (i.e.,
RascalC-EZ; Table 2), which is based on the exact average clustering inferred
from the entire set of pre- and post-reconstruction LRG EZmocks. Finally, we
use both covariances to fit 1000 individual EZmocks as well as 25 individual
AbacusSummit realizations with the fitting procedure explained in Section 4.3,
within the spatial range $50-150h^{-1}{\rm Mpc}$.202020Note that the Percival
factor (Percival et al., 2014) has been applied to the EZmocks. In essence, we
quantify the BAO best fits and relative errors on a mock-by-mock basis.

Figure 6 shows the results of such a validation test. Top panels refer to pre-
reconstruction measurements, while bottom panels display post-reconstruction
quantities. From left to right, we report the BAO detection significances in
units of standard deviations, and the $\alpha$ and $\sigma_{\alpha}$ values
for all of the individual fits performed to the two sets of mocks using the
RascalC-EZ covariance ($x$-axes), against the corresponding values obtained
with the DESI-M2-EZ covariance ($y$-axes). Open green dots display LRG EZmocks
measurements, while filled yellow dots are for AbacusSummit LRG synthetic
catalogs. Stars of the same color represent averages over the corresponding
entire set of realizations. As evident from the figure, the scatter along the
diagonal is quite narrow (both for the pre- and post-reconstruction cases),
implying that the two covariances produce compatible results. Moreover, the
average values in the various panels are almost overlapping, strongly
confirming the consistency between RascalC and mock sample covariances.

A further validation test is reported in Figure 7, where we show the
histograms of ($\alpha-\langle\alpha\rangle)/\sigma_{\alpha}$, with
$\langle\alpha\rangle$ the mean of the scaling parameter, measured from the
$\xi(s)$’s of the pre- (top panel) and post-reconstruction (bottom panel) LRG
mocks. This quantity represents an approximation for the signal-to-noise ratio
(SNR) of the BAO measurement. Here, we use the $\chi^{2}$ test to assess the
validity of the covariances. In essence, we compare the observed scatter in
the best-fitting $\alpha$ for the 1000 LRG EZmocks to the $\sigma_{\alpha}$
estimated in each individual fit from the $\Delta\chi^{2}(\alpha)$ curve. Red
lines and histograms refer to measurements performed on the EZmock set using
the RascalC-EZ covariance, while blue lines and histograms correspond to
analogous measurements done assuming the numerically-based DESI-M2-EZ
covariance. Results are compared with Gaussian distributions, showing good
agreement, as confirmed by near-zero Kolmogorov-Smirnov (K-S) $D_{n}$ tests.
Moreover, the corresponding $p$-values imply that our values are drawn from a
Gaussian distribution, and that the values of $\sigma_{\alpha}$ we measure
from the $\chi^{2}$ distribution are faithful descriptors of the error on
$\alpha$ measured by fitting $\xi(s)$. Once again, this test represents
another confirmation of the validity of our semi-analytical semi-empirical
RascalC-EZ covariance, which produces results compatible with the numerical
case.

Figure 7: Histograms of ($\alpha-\langle\alpha\rangle)/\sigma_{\alpha}$, with
$\langle\alpha\rangle$ the mean of the scaling parameter, measured from the
$\xi(s)$’s of the pre- (top panel) and post-reconstruction (bottom panel) LRG
mocks. Measurements are performed on 1000 LRG EZmocks, assuming the RascalC-EZ
covariance (red lines and histograms), or a numerically-based DESI-M2-EZ
covariance (blue lines and histograms). Results are then compared with
Gaussian distributions, showing good agreement, and indicating that the values
of $\sigma_{\alpha}$ we measure from the $\chi^{2}$ distribution are faithful
descriptors of the error on $\alpha$ measured by fitting $\xi(s)$. This test
represents a further validation of our semi-analytical RascalC-EZ covariance,
which produces results compatible with the numerical one.

Figure 8: Scatter plots for the BAO detection significance in units of
standard deviations (left panel), $\alpha$’s (middle panel), and
$\sigma_{\alpha}$’s (right panel), addressing the effect of reconstruction on
the BAO fitting procedure for the LRG sample. Results are obtained from
individual correlation function fits performed to 1000 LRG EZmocks (open
purple dots) and 25 AbacusSummit LRG synthetic catalogs (filled orange dots).
Stars of identical color represent averages over the corresponding entire set
of realizations. Pre- ($x$-axes) and post-reconstruction ($y$-axes)
measurements are compared, adopting the RascalC-EZ covariance. While
reconstruction increases considerably the BAO detection significance for the
mocks, the same procedure applied to the DESI-M2 LRG sample produces only
marginal effects: the cyan stars in the left and right panels are DESI-M2 LRG
measurements obtained via a RascalC covariance calibrated directly on LRGs,
pointing to a ‘lucky’ realization in the upper right corner of the
significance scatter plot (left panel), and in the lower left corner of the
$\sigma_{\alpha}$ plot (right panel), respectively. This situation is similar
to those reported for BOSS CMASS LRGs (Anderson et al., 2012), and for the
eBOSS LRG sample (Bautista et al., 2021).

In summary, the tests performed in this section clearly prove that using a
RascalC-based covariance returns unbiased and consistent estimates when
compared to results obtained with the numerical EZmock covariance. We can then
safely proceed to tune our RascalC covariance to match the clustering inferred
from the DESI-M2 datasets, and perform the key BAO fits on the LRG and BGS
samples – as we describe in the next section.

##  6 Key Results: BAO Signal Detection

In this section, we present the main results of our DESI-M2 analysis, and
assess the precision and detection statistics of the BAO feature both in the
LRG and BGS samples. We do not report here the best fit BAO scales as inferred
from actual data, since the cosmology is intentionally kept blinded. Our focus
is primarily on LRGs, as they are characterized by the highest SNR among the
four DESI-M2 tracers.

###  6.1 BAO Reconstruction Efficiency

Before performing BAO fits to the DESI-M2 dataset, we first address the effect
of BAO reconstruction on the BAO fitting procedure – focusing on the LRG
sample. In Figure 8, we report the BAO detection significances expressed in
units of standard deviations, as well as the $\alpha$ and $\sigma_{\alpha}$
values from all of the individual correlation function fits performed to the
two sets of LRG mocks previously considered. Specifically, we compare pre-
($x$-axes) and post-reconstruction ($y$-axes) measurements, obtained by
adopting the RascalC-EZ covariance. Open purple dots display EZmocks LRG
results, while filled orange dots are for the AbacusSummit LRG synthetic
catalogs. Similarly to Figure 6, stars of identical color represent averages
over the corresponding entire set of realizations. As evident from the scatter
plot, the BAO detection significance (reported in the left panel) increases
considerably after reconstruction, and the $\alpha$’s of the mocks are closer
to unity after reconstruction (central panel), as expected.212121See also
Table 4 for additional BAO fitting details. Moreover, the errors tend to
improve significantly after reconstruction for about 90% of the cases (i.e.,
right panel). Hence, the reconstruction procedure appears to be efficient on
the mocks.

Reconstruction applied to LRG data appears instead to produce only marginal
effects. To this end, in Figure 8 we overplot the DESI-M2 LRG measurements
(cyan stars in the left and right panels), obtained by using the RascalC-LRG
covariance calibrated directly on LRGs. As evident from the figure, the
DESI-M2 LRG measurements are located in the upper right corner of the
significance scatter plot (left panel), and in the lower left corner of the
$\sigma_{\alpha}$ plot (right panel), respectively: hence, we are in a
similarly ‘lucky’ situation as those reported for the BOSS CMASS LRG sample by
Anderson et al. (2012), and also for eBOSS LRGs (Bautista et al., 2021). Table
4 provides a quantification of the ‘lucky’ realization of the observational
data point, showing that both its detection significance and precision are
consistent with those obtained via mock averages. In particular, focusing on
post-reconstruction results, the detection significance of the data point is
5.050 (in units of standard deviations) with a precision of 1.7%, while the
average EZmocks results yield a detection significance of 4.138 with a
precision of 2.1%, and from the average of the AbacusSummit mocks we obtain
4.242 with a 2.0% precision. Note that a small $\sigma_{\alpha}$ implies a
better BAO detection, thus a higher significance. In essence, while generally
reconstruction improves errors on $\alpha$, this may not happen if the
starting (pre-reconstruction) point already has a low error to begin with
(i.e., a ‘lucky’ realization). In such a situation, reconstruction does not
tend to produce much improvement, as shown in the $\sim 10\%$ of the mocks in
our analysis. This seems to be the case for the DESI-M2 LRG sample data
volume: our recovered $\sigma_{\alpha}$ for data is much smaller than the mean
expected from the mocks (right panel), and our BAO detection significance is
high (left panel), showing a strong and well-defined acoustic peak.

###  6.2 BAO Detection from the LRG Sample

Table 3: BAO key fitting results for DESI-M2 LRGs and BGS. Sample | Reconstruction | BAO Detection Significance | $\alpha+\Delta\alpha$ | min$(\chi^{2})$/dof
---|---|---|---|---
DESI-M2 LRG | Pre-recon | 5.170 | 0.987 $\pm$ 0.016 | 15.619 / 20
| Post-recon | 5.050 | 1.000 $\pm$ 0.017 | 13.463 / 20
DESI-M2 BGS | Pre-recon | 2.337 | 0.980 $\pm$ 0.040 | 13.172 / 20
| Post-recon | 2.963 | 1.001 $\pm$ 0.026 | 16.724 / 20

Figure 9: BAO feature and its significance, as detected in the large-scale
correlation function of DESI-M2 LRGs. [Left] Pre- (lighter pink dots with
errorbars) and post- (red dots with errorbars) reconstruction two-point
clustering statistics inferred from the LRG sample, clearly displaying the BAO
peak. The gray and black lines are respectively pre- and post-reconstruction
fits to $\xi(s)$ in the spatial range $50-150h^{-1}{\rm Mpc}$ over 25 points
with 20 dof, obtained using the RascalC-LRG covariance. Errorbars are the
square root of its diagonal elements. [Right] Detection significance of the
DESI-M2 LRG BAO feature before (lighter gray lines) and after (black lines)
reconstruction. Dotted lines with similar colors are corresponding fits to the
data using a model without BAO. We have shifted each value of $\alpha$ by
$\Delta\alpha$ both in pre- and post-reconstruction, such that the minimum
$\chi^{2}$ of the post-reconstruction result is at 1. We note that an
identical $\Delta\alpha$ was introduced for post-reconstruction LRGs and BGS
(Figure 10), to demonstrate the coherence in terms of cosmological
implications from the two tracers at the two different redshifts, while being
blinded. The BAO peak is detected at $\sim 5\sigma$ confidence with a $1.6\%$
and $1.7\%$ precision in the pre- and post-reconstruction DESI-M2 LRG sample,
respectively, with the reconstruction procedure playing only a marginal role.
Such a remarkable detection level, obtained with just two months of DESI
operations, is comparable to the one reported for the BOSS CMASS sample
(Anderson et al., 2012), and it is quite reassuring – given the high
complexity of the DESI instrument and of the DESI spectroscopic reduction
pipeline.

Figure 10: BAO feature and its significance, as detected in the large-scale
correlation function of DESI-M2 BGS. Line styles and conventions same as in
Figure 9. [Left] Pre- and post-reconstruction two-point clustering statistics
inferred from the BGS sample. BAO fits to $\xi(s)$ are performed in the
spatial range $50-150h^{-1}{\rm Mpc}$, adopting the RascalC-BGS covariance.
Errorbars are the square root of its diagonal elements. [Right] Detection
significance of the DESI-M2 BGS BAO feature before (lighter gray lines) and
after (black lines) reconstruction. Dotted lines with indentical colors are
corresponding fits to the data using a model without BAO. We have shifted each
value of $\alpha$ by the same $\Delta\alpha$ both in pre- and post-
reconstruction, as done for the LRG analysis. The acoustic feature is detected
at $\sim 2.5\sigma$ significance with a $4.0\%$ precision in pre-
reconstruction, and at $\sim 3.0\sigma$ with a $2.6\%$ precision in post-
reconstruction. Clearly, for this galaxy sample, reconstruction plays a more
substantial effect in sharpening the acoustic peak. This BAO detection
represents another relevant milestone of our DESI-M2 analysis.

Figure 9 displays the BAO fit to the DESI-M2 LRG two-point correlation
function, along with its significance: this measurement represents one of the
key results of our analysis. The left panel shows the pre- and post-
reconstruction clustering statistics computed with the LS estimator (points
with errorbars), and the best-fit model (curves). The gray and black curves
are respectively pre- and post-reconstruction fits to $\xi(s)$ in the spatial
range $50-150h^{-1}{\rm Mpc}$,222222Since we assume a bin size of $4h^{-1}{\rm
Mpc}$ for characterizing the $\xi(s)$ clustering statistics, we therefore fit
over 25 points using five parameters, leaving us 20 degrees-of-freedom (dof).
obtained with the procedure detailed in Section 4.3 and using the RascalC-LRG
covariance matrix: errorbars in the plot show the square root of its diagonal
elements. The BAO peak is clearly detected, and well matched to the best-
fitting model. This is confirmed quantitatively: we find $\chi^{2}_{\rm
min}=15.6$ and $\chi^{2}_{\rm min}=13.5$ for the pre- and post-reconstruction
cases assuming 20 dof, respectively. Table 3 reports the specifics of these
BAO fits.

The right panel of Figure 9 displays the likelihood for the DESI-M2 LRG BAO
scale, as represented by $\Delta\chi^{2}=\chi^{2}-\chi^{2}_{\rm min}$, before
and after reconstruction (solid curves). Dotted lines having identical colors
represent corresponding fits to the data using a model without BAO. This
provides two crucial results: the uncertainty on the measurement, and the
significance of the BAO feature. Assuming a Gaussian likelihood, the 1$\sigma$
confidence region is represented by the width of the curve with
$\Delta\chi^{2}<1$. We estimate the 1$\sigma$ uncertainty to be 0.016 in pre-
reconstruction and 0.017 in post-reconstruction, respectively. The BAO
detection significance can be simply determined by comparing results obtained
from a fit to the data using a model without BAO (displayed via dotted lines
in the figure), and once again subtracting the $\chi^{2}_{\rm min}$ from the
BAO fit. This indicates how much better a model containing BAO fits the LRG
data (i.e., actual existence of the BAO peak in the galaxy sample). The
$\Delta\chi^{2}_{\rm min,noBAO}$ is greater than 25 both in pre- and post-
reconstruction. Hence, we report a detection of the BAO feature in the DESI-M2
LRG sample at a significance greater than 5$\sigma$.

We note that we have shifted each $\alpha$ by the corresponding value of
$\Delta\alpha$ in the right panel of Figure 9, such that the minimum
$\chi^{2}$ is at 1 for the post-reconstruction case. The magnitude of the
required shift for the post-reconstruction result was less than 1$\sigma$.
Thus, while we do not reveal the precise value of the BAO scale in this
analysis, we are consistent with the fiducial cosmology.

As illustrated in Figure 9, the BAO detection in the DESI-M2 LRG sample is
highly significant. Such a remarkable detection level, obtained with only two
months of DESI operations, is comparable to the one reported for the BOSS
high-$z$ LRG sample (i.e., CMASS; Anderson et al., 2012), comprised of 264 283
galaxies in the redshift interval $0.43<z<0.7$. Notice also that the
reconstruction procedure has practically no impact on the BAO peak inferred
from the DESI-M2 LRG clustering, as evident from the right panel of Figure 9
(compare the gray and black curves). As pointed out in the previous section,
this is due to the ‘lucky’ starting point of the pre-reconstruction LRG
measurement, which happens to be located in the lower left corner of the
$\sigma_{\alpha}$ plot in Figure 8, yielding already a very low error to begin
with, and hence carrying a high BAO detection significance (i.e., left panel
of the same figure, see the cyan star in the upper right corner of the
significance scatter plot).

In closing this section, we emphasize that the first BAO measurement obtained
with DESI-M2 LRGs represents an important milestone, and its high detection
level is quite reassuring – considering the complexity of the DESI instrument
and of the spectroscopic reduction pipeline. It also constitutes an important
early validation and quality-control of the data management system, as well as
a confirmation of the successful survey design strategy adopted for DESI
targets. Next, we move to the BGS sample and carry out a similar BAO analysis.

###  6.3 BAO Detection from the BGS

Figure 10 contains another central result of our analysis. Here, we report the
BAO feature as detected in the large-scale clustering of the DESI-M2 BGS
BRIGHT sample characterized by a magnitude cut of $-21.5$ (see Section 2),
together with its significance. In detail, the left panel displays two-point
correlation function measurements from those galaxies. Following similar
conventions as in Figure 9, the gray and black curves are respectively the
pre- and post-reconstruction best-fit models to $\xi(s)$ in the spatial range
$50-150h^{-1}{\rm Mpc}$, obtained using the RascalC-BGS covariance matrix.
Errorbars in the plot show the square root of its diagonal elements. Also in
this case, the BAO peak is clearly detected and well matched to the best-
fitting model. Specifically, $\chi^{2}_{\rm min}=13.2/20~{}{\rm dof}$ in pre-
reconstruction, and $\chi^{2}_{\rm min}=16.7/20~{}{\rm dof}$ in post-
reconstruction. See again Table 3 for details on these BAO fits.

The right panel of Figure 10 shows the likelihood for the DESI-M2 BGS BRIGHT
BAO scale before and after reconstruction (solid lines). Dotted lines with
identical colors represent the corresponding fits to the data using a model
without BAO. Similarly to the LRG analysis, we determine the 1$\sigma$
confidence region of the measurements based on the width of the curve with
$\Delta\chi^{2}<1$. This yields an uncertainty of $0.040$ in pre-
reconstruction, and of $0.026$ in post-reconstruction. As evident by comparing
the gray and black curves from both panels of Figure 10, here reconstruction
plays a more substantial effect in sharpening the acoustic peak and in
partially removing the BAO smearing caused by non-linear structure growth. By
comparing the $\chi^{2}$ of the data fits against a model without BAO, we
determine the significance of the BAO feature. We find $\Delta\chi^{2}$ to be
5.5 in pre-reconstruction and 8.8 in post-reconstruction, corresponding to a
BAO detection significance of $\sim 2.3\sigma$ and $\sim 3.0\sigma$,
respectively.

Even for the DESI-M2 BGS sample, we have shifted the individual $\alpha$’s by
the same $\Delta\alpha$ factor applied to the LRG sample, both in pre- and
post-reconstruction. Since the BGS minimum $\chi^{2}$ values are within
$1\sigma$ of $\alpha+\Delta\alpha=1$, we can conclude that the BGS BAO results
are fully consistent with the LRG ones.

Along with the first BAO measurement from the DESI-M2 LRG sample, this first
BAO detection obtained using the DESI-M2 BGS represents another relevant
milestone, as well as an additional early validation of the DESI pipeline and
data management system for the bright time survey.

##  7 Outlook for Future Data Releases

Based on the promising BAO results presented in Section 6, obtained with the
DESI-M2 dataset collected over just two months of operations, we now proceed
to forecast the expected BAO detection significance and accuracy with the
completed survey data, focusing on the final Y5 DESI LRG sample.

To this end, we compute Fisher matrix forecasts of the LRG isotropic BAO
scale. We perform such calculations by adopting covariance matrices
constructed using post-reconstruction EZmocks for the two survey
configurations (i.e., DESI-M2 LRGs and DESI Y5 LRGs, respectively) and the
best-fit monopole model for DESI-M2 – while marginalizing over the same free
parameters as in the actual data fit. We then take the ratio of the two Fisher
estimates to rescale the DESI-M2 constraint, in order to account for the
specific calibration of the LRG EZmocks, tuned on the One Percent Survey
clustering rather than on DESI-M2 (see again Section 3.3 for details). These
Fisher estimates return a factor of $\sim 5.8$ between the BAO SN inferred
from Y5 LRGs and that of DESI-M2 LRGs. When we rescale our data best-fit LRG
estimate of $1.7\%$ accounting for this factor, we then predict a $\sim
0.29\%$ precision on the BAO scale from the final Y5 LRG sample over
$0.4<z<1.1$.

We note that such estimate should be taken as an approximate projection for
DESI Y5, since we have simply assumed the same BAO signal from DESI-M2 while
only changing the covariance. Nevertheless, the projected BAO precision for
DESI Y5 LRGs agrees well with the more accurate DESI LRG Y5 forecasts based on
GoFish232323https://github.com/ladosamushia/GoFish – i.e., 0.25% precision for
the DESI Y5 LRG sample, presented in DESI Collaboration et al. (2023a). This
exquisite level of sub-percent precision on the BAO scale (even from a single
tracer) will confirm DESI as the most competitive BAO experiment for the
remainder of this decade.

##  8 Conclusions

The BAO scale represents a key standard ruler that provides a direct way to
measure the expansion history of the Universe and infer robust cosmological
constraints. Hence, BAO measurements are considered primary DESI science
targets, and a major deliverable at any stage of the survey. Precision on the
expansion history of the Universe constitutes a compelling probe of the nature
of DE. DESI is expected to deeply impact the current understanding of DE,
along with providing unprecedented constrain on theories of modified gravity
and inflation, and on neutrino masses (DESI Collaboration et al., 2016). In
this respect, DESI plans to conduct a series of BAO analyses throughout its
five-year survey time with blinded catalogs: the Y1 sample would be the first
of such rigorously blinded BAO analyses.

The remarkable complexity of the DESI instrument, along with the adopted
survey design and the elaborated DESI spectroscopic pipeline and data
management system, pose a potential challenge to all BAO analyses. It is then
of utmost importance to test such pipelines well in advance, and it has been
the main goal of the current study: this is a crucial aspect in order to
guarantee the success of future BAO investigations, and for confirming the
optimal performance of the DESI spectrograph and the quality level of the data
reduction pipeline. Precisely for this reason, the first two months of DESI
operations were intentionally kept unblinded (i.e., DESI-M2 sample).242424Note
that DESI-M2 is approximately 1/5 of the DESI Y1 sample.

To this end, we have used the DESI-M2 dataset and reported the first high-
significance detection of the BAO signal from the LRG and BGS samples.
Specifically, our primary results are:

  * •

$\sim 5\sigma$ level BAO detection in the DESI-M2 LRG sample at a precision of
$1.7\%$.

  * •

$\sim 3\sigma$ level BAO detection in the DESI-M2 BGS sample at a precision of
$2.6\%$.

In particular, our LRG BAO measurement is comparable to the $5-6\sigma$ BAO
detection obtained with the BOSS high-$z$ LRG sample (i.e., CMASS; Anderson et
al., 2012), comprised of 264,283 galaxies in the redshift interval
$0.43<z<0.7$. Moreover, the BOSS and eBOSS BAO measurements made with LRGs
between $0.4<z<1.0$ (with $N_{\rm gal}=1{,}063{,}828$) returned an aggregate
precision of 0.77% on $D_{\rm V}$ (Bautista et al., 2021), which is only a
factor of $2.2$ times better (in terms of precision) than our quoted result
with DESI-M2 LRGs (having just $N_{\rm gal}=266,269$). This latter aspect is
rather remarkable, considering that the DESI-M2 dataset has been collected
simply during the initial two months of DESI operations.

Based on these results, we forecasted that DESI is on target to achieve a
high-significance BAO detection at a $\sim 0.29\%$ precision with the
completed Y5 LRG sample over $0.4<z<1.1$, meeting the DESI top-level science
requirements on BAO measurements. This exquisite level of precision will set
novel standards in cosmology and confirm DESI as a highly accurate and precise
Stage-IV BAO experiment.

Additional relevant aspects of our investigation on these preliminary DESI-M2
data that are worth highlighting are summarized as follows:

  * •

Although the catalogs we used are unblinded, we presented a blinded cosmology
analysis – in that we do not report here the best fit BAO scale. In fact, we
only presented the precision and detection level of the BAO measurements. We
plan to provide a full cosmological interpretation with the Y1 data release in
the near future, after additional rigorous systematic tests.

  * •

We focused on the isotropic BAO scale exploiting only the monopole of the LRG
and BGS samples.

  * •

We applied the nominal BAO pipeline that has been previously well-tested with
BOSS and eBOSS data. In particular, we utilized the early version of the
pipeline package cosmodesi that the DESI collaboration team has been
developing, which wraps both existing and new cosmological galaxy survey
analysis pipelines from the literature.

  * •

We constructed, calibrated, and used semi-analytical semi-empirical covariance
matrices based on the RascalC code, and validated those covariances in terms
of BAO fitting procedures (in pre- and post-reconstruction) using a set of
mocks – as detailed in Section 5 and in Rashkovetskyi et al. (2023).

  * •

We also found that the LRG BAO signal from the DESI-M2 data is stronger than
the typical BAO signal present in the LRG mocks. Partly for this reason, the
reconstruction procedure performed on actual LRG data is less effective than
the one performed on LRG mocks. On the other hand, we found that the BGS
sample shows a factor of $\sim 1.5$ precision improvement after
reconstruction. These results are consistent with the typical behavior we find
on DESI-M2 mocks; less scatter is expected for the more complete DESI Y1 and
Y5 samples.

This work represents the first step towards the analysis techniques that will
lead to the key cosmological results from DESI Y1 data. While the BAO results
presented here constitute an important milestone and are quite reassuring in
terms of consistency in the clustering amplitude (considering the complexity
of the DESI instrument and of the spectroscopic reduction pipeline), we
anticipate that the DESI Y1 analysis alone will surpass the cosmological
information from all of the BAO analyses performed to date. This will require
going beyond the legacy BAO analysis setting that has been well-tested using
BOSS and eBOSS data (and also mainly adopted here), with an unprecedented
level of BAO systematic tests and by developing an optimal BAO pipeline –
given the stringent requirements on theoretical and observational systematics
that are imposed by a dataset as powerful as we expect by the end of the
survey. The DESI team is currently working in this direction, and presenting
all these novel technical aspects will be the subject of many forthcoming DESI
Y1 cosmology papers.

## Acknowledgements

JM and GR acknowledge support from the National Research Foundation of Korea
(NRF) through grant No. 2020R1A2C1005655 funded by the Korean Ministry of
Education, Science and Technology (MoEST) and from the faculty research fund
of Sejong University in 2022/2023. DV acknowledges support from the U.S.
Department of Energy, Office of Science, Office of High Energy Physics under
grant No. DE-SC0019091 and No. DE-SC0023241. MR is supported by U.S.
Department of Energy grant DE-SC0007881 and by the Simons Foundation
Investigator program. CS acknowledges support from the National Research
Foundation of Korea (NRF) through grant No. 2021R1A2C101302413 funded by the
Korean Ministry of Education, Science and Technology (MoEST). CS was also
supported by the high performance computing cluster Seondeok at the Korea
Astronomy and Space Science Institute.

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.

This work was led under the supervision of the DESI Y1 catalog generation Key
Project conveners (Arnaud de Mattia and Ashley Ross) and the DESI Y1 BAO Key
Project conveners (Nikhil Padmanabhan and Hee-Jong Seo).

We thank Antonella Palmese, Eric Linder, and Benjamin Weaver for constructive
comments and useful feedback.

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.

## Data Availability

The redshift measurements used in this paper – derived from the DESI Guadalupe
dataset – will be made public with the DESI Y1 data release (DR1). The
analogous spectroscopic data reductions and redshift fits for the DESI EDR
(i.e., Fuji) will be publicly available on NERSC at this url:
https://data.desi.lbl.gov/public/edr/spectro/redux/fuji. All data points shown
in the published graphs are available in machine-readable form in Zenodo at
https://doi.org/10.5281/zenodo.7835433.

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##  Appendix A Supporting Material

Table 4: BAO key fitting results for LRGs. For DESI-M2 LRGs, we use the RascalC-LRG covariance, while for the two sets of mocks we applied instead the RascalC-EZ covariance – see Section 5 and Table 2 for details. We do not show here the $\alpha$ values for the LRG data fits, as the cosmology is kept blinded. Moreover, we do not report DESI-M2 fits with the EZmock numerical covariance, since EZmocks are calibrated with the One Percent Survey data (as explained in Section 3.3). | Reconstruction | BAO results |  RascalC cov |  EZmock cov
---|---|---|---|---
| Pre-recon | Detection significance | 5.170 | –
DESI-M2 LRG |  | Precision | 1.6% | –
| Post-recon | Detection significance | 5.050 | –
|  | Precision | 1.7% | –
|  | Detection significance | 3.423 | 3.597
| Pre-recon | $\alpha$ | 1.006 | 1.006
EZmock LRG |  | Precision | 2.8% | 2.7%
|  | Detection significance | 4.138 | 4.091
| Post-recon | $\alpha$ | 1.000 | 1.001
|  | Precision | 2.1% | 2.3%
|  | Detection significance | 3.623 | 3.801
| Pre-recon | $\alpha$ | 1.001 | 0.997
AbSmock LRG |  | Precision | 2.8% | 2.5%
|  | Detection significance | 4.242 | 4.209
| Post-recon | $\alpha$ | 0.992 | 0.994
|  | Precision | 2.0% | 2.1%

In support of the primary BAO analysis carried out in the main text, in Table
4 we provide some further technical details related to the various BAO fits
performed to the DESI-M2 LRG sample, as well as to the corresponding LRG mocks
adopted in this work – namely, pre- an post-reconstruction results of the BAO
detection significance and precision, along with the $\alpha$ values for the
mock fits. Specifically, as reported in the main text, the DESI-M2 LRG sample
provides a $\sim 5\sigma$ BAO detection significance at a 1.6% and 1.7%
precision in pre- and post-reconstruction, respectively. From the two sets of
mocks considered (AbacusSummit and EZmocks), we also find a significant BAO
detection at more than 3.4 $\sigma$ in pre-reconstruction, and exceeding a 4.0
$\sigma$ detection in post-reconstruction, with a corresponding precision
better than 2.8% (pre-reconstruction) or 2.3% (post-reconstruction).
Remarkably, the $\alpha$-values inferred from the mocks are close to unity,
indicating that the fiducial cosmology is very-well recovered. This also
implies that the mock production pipeline is working properly. Additionally,
from Table 4 one can readily infer that the RascalC-based LRG covariance is
compatible with the EZmock LRG covariance (as we also reported in Section 5),
simply by comparing all the corresponding fitting results obtained with the
two sets of covariances (i.e., last two columns).

##

1Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea
2Department of Physics & Astronomy, Ohio University, Athens, OH 45701, USA
3Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA
4Korea Astronomy and Space Science Institute, 776 Daedeok-daero, Yuseong-gu,
Daejeon 34055, Republic of Korea
5Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720,
USA
6Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215,
USA
7Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005,
India
8Physics Department, Yale University, P.O. Box 208120, New Haven, CT 06511,
USA
9NSF’s NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA
10Department of Physics & Astronomy, University College London, Gower Street,
London, WC1E 6BT, UK
11IRFU, CEA, Université Paris-Saclay, F-91191 Gif-sur-Yvette, France
12Department of Physics and Astronomy, The University of Utah, 115 South 1400
East, Salt Lake City, UT 84112, USA
13Instituto de Física, Universidad Nacional Autónoma de México, Cd. de México
C.P. 04510, México
14Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, TX 75275, USA
15Fermi National Accelerator Laboratory, PO Box 500, Batavia, IL 60510, USA
16Institut de Física dAltes Energies (IFAE), The Barcelona Institute of
Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain
17Departamento de Física, Universidad de los Andes, Cra. 1 No. 18A-10,
Edificio Ip, CP 111711, Bogotá, Colombia
18Department of Physics, The University of Texas at Dallas, Richardson, TX
75080, USA
19University of Michigan, Ann Arbor, MI 48109, USA
20Department of Physics, The Ohio State University, 191 West Woodruff Avenue,
Columbus, OH 43210, USA
21Center for Cosmology and AstroParticle Physics, The Ohio State University,
191 West Woodruff Avenue, Columbus, OH 43210, USA
22Department of Physics, Southern Methodist University, 3215 Daniel Avenue,
Dallas, TX 75275, USA
23Natural Science Research Institute, University of Seoul, 163 Seoulsiripdae-
ro, Dongdaemun-gu, Seoul, South Korea
24Sorbonne Université, CNRS/IN2P3, Laboratoire de Physique Nucléaire et de
Hautes Energies (LPNHE), FR-75005 Paris, France
25Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain
26Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 W 18th Avenue, Columbus, OH 43210, USA
27Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís
Companys, 23, 08010 Barcelona, Spain
28Department of Physics and Astronomy, Siena College, 515 Loudon Road,
Loudonville, NY 12211, USA
29Department of Physics & Astronomy, University of Wyoming, 1000 E.
University, Dept. 3905, Laramie, WY 82071, USA
30Institute of Cosmology & Gravitation, University of Portsmouth, Dennis
Sciama Building, Portsmouth, PO1 3FX, UK
31Institute for Astronomy, University of Edinburgh, Royal Observatory,
Blackford Hill, Edinburgh EH9 3HJ, UK
32Department of Physics & Astronomy and Pittsburgh Particle Physics,
Astrophysics, and Cosmology Center (PITT PACC), University of Pittsburgh, 3941
O’Hara Street, Pittsburgh, PA 15260, USA
33National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun
Rd., Chaoyang District, Beijing, 100012, P.R. China
34Waterloo Centre for Astrophysics, University of Waterloo, 200 University Ave
W, Waterloo, ON N2L 3G1, Canada
35Perimeter Institute for Theoretical Physics, 31 Caroline St. North,
Waterloo, ON N2L 2Y5, Canada
36Space Sciences Laboratory, University of California, Berkeley, 7 Gauss Way,
Berkeley, CA 94720, USA
37GMT RPG - Campus de Excelencia Internacional CIE/UAM + CSIC, Universidad
Autónoma de Madrid
38Department of Physics, Kansas State University, 116 Cardwell Hall,
Manhattan, KS 66506, USA
39Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
40Department of Physics, University of California, Berkeley, 366 LeConte Hall
MC 7300, Berkeley, CA 94720-7300, USA
41SLAC National Accelerator Laboratory, Menlo Park, CA 94305, USA