# Optimal Reconstruction of Baryon Acoustic Oscillations for DESI 2024

E. Paillas  0000-0002-4637-2868 Z. Ding  0000-0002-3369-3718 X. Chen  H. Seo
0000-0002-6588-3508 N. Padmanabhan  A. de Mattia  A. J. Ross
0000-0002-7522-9083 S. Nadathur  0000-0001-9070-3102 C. Howlett
0000-0002-1081-9410 J. Aguilar  S. Ahlen  0000-0001-6098-7247 O. Alves  U.
Andrade  0000-0002-4118-8236 D. Brooks  E. Buckley-Geer  E. Burtin  S. Chen
0000-0002-5762-6405 T. Claybaugh  S. Cole  0000-0002-5954-7903 K. Dawson  A.
de la Macorra  0000-0002-1769-1640 Arjun Dey  0000-0002-4928-4003 P. Doel  K.
Fanning  0000-0003-2371-3356 S. Ferraro  0000-0003-4992-7854 J. E. Forero-
Romero  0000-0002-2890-3725 C. Garcia-Quintero  0000-0003-1481-4294 E.
Gaztañaga  H. Gil-Marín  0000-0003-0265-6217 S. Gontcho A Gontcho
0000-0003-3142-233X G. Gutierrez  C. Hahn  0000-0003-1197-0902 M. M. S Hanif
0009-0006-2583-5006 K. Honscheid  M. Ishak  0000-0002-6024-466X R. Kehoe  A.
Kremin  0000-0001-6356-7424 M. Landriau  0000-0003-1838-8528 L. Le Guillou
0000-0001-7178-8868 M. E. Levi  0000-0003-1887-1018 M. Manera
0000-0003-4962-8934 P. Martini  0000-0002-4279-4182 L. Medina-Varela  A.
Meisner  0000-0002-1125-7384 J. Mena-Fernández  0000-0001-9497-7266 R. Miquel
J. Moustakas  0000-0002-2733-4559 E. Mueller  A. Muñoz-Gutiérrez  A. D. Myers
J. A. Newman  0000-0001-8684-2222 J. Nie  0000-0001-6590-8122 G. Niz
0000-0002-1544-8946 N. Palanque-Delabrouille  0000-0003-3188-784X W. J.
Percival  0000-0002-0644-5727 C. Poppett  F. Prada  0000-0001-7145-8674 A.
Pérez-Fernández  0009-0006-1331-4035 M. Rashkovetskyi  0000-0001-7144-2349 M.
Rezaie  0000-0001-5589-7116 A. Rosado-Marin  G. Rossi  R. Ruggeri
0000-0002-0394-0896 E. Sanchez  0000-0002-9646-8198 C. Saulder
0000-0002-0408-5633 E. F. Schlafly  0000-0002-3569-7421 D. Schlegel  M.
Schubnell  D. Sprayberry  G. Tarlé  0000-0003-1704-0781 D. Valcin
0000-0003-0129-0620 M. Vargas-Magaña  0000-0003-3841-1836 J. Yu  S. Yuan
0000-0002-5992-7586 R. Zhou  0000-0001-5381-4372 and H. Zou
0000-0002-6684-3997

###### Abstract

Baryon acoustic oscillations (BAO) provide a robust standard ruler to measure
the expansion history of the Universe through galaxy clustering. Density-field
reconstruction is now a widely adopted procedure for increasing the precision
and accuracy of the BAO detection. With the goal of finding the optimal
reconstruction settings to be used in the DESI 2024 galaxy BAO analysis, we
assess the sensitivity of the post-reconstruction BAO constraints to different
choices in our analysis configuration, performing tests on blinded data from
the first year of DESI observations (DR1), as well as on mocks that mimic the
expected clustering and selection properties of the DESI DR1 target samples.
Overall, we find that BAO constraints remain robust against multiple aspects
in the reconstruction process, including the choice of smoothing scale,
treatment of redshift-space distortions, fiber assignment incompleteness, and
parameterizations of the BAO model. We also present a series of tests that
DESI followed in order to assess the maturity of the end-to-end galaxy BAO
pipeline before the unblinding of the large-scale structure catalogs.

##  1 Introduction

Galaxy redshift surveys have shaped our understanding of the origin and
evolution of the Cosmos. Measurements of baryon acoustic oscillations [BAO; 1,
2, 3] in the galaxy distribution have allowed us to measure the expansion
history of the Universe with high precision, establishing $\Lambda$CDM as our
standard cosmological paradigm [4, 5, 6, 7, 8]. These oscillations are a
characteristic pattern of clustering in the large-scale structure (LSS),
corresponding to remnants of acoustic waves that traveled through the photon-
baryon plasma in the early Universe. These sound waves imprinted a
characteristic scale in the distribution of matter, given by the distance the
perturbations traveled before they froze in place once hydrogen recombined and
the baryons stopped being dragged by the photons (the baryon drag epoch). Over
time, these cosmic ripples provided sites where galaxies preferentially
formed, and, as a consequence, this sound horizon scale is also found in the
spatial distribution of galaxies, either as a localized bump in the two-point
correlation function or as wiggles in the power spectrum [9, 10].

Non-linear gravitational evolution broadens the BAO peak in the correlation
function or damps high-k oscillations in the power spectrum. In addition, the
location of the peak can also be shifted due to non-linear evolution and
galaxy biasing [11, 12]. These effects degrade the precision of BAO
measurements but can be partially corrected. A technique known as
reconstruction has been designed to reverse the non-linear effects by shifting
overdensities back to their initial positions, so that the sound horizon scale
can be recovered with greater statistical significance [13]. In addition,
reconstruction has been shown to be effective in removing the shifts in the
acoustic peak caused by non-linearities and galaxy biasing, reducing the
systematic errors in BAO measurements [e.g. 14, 15, 16, 17].

Clustering analyses from the last decade, based on the Sloan Digital Sky
Survey [SDSS; 18], the Baryon Oscillation Spectroscopic Survey [BOSS; 19], the
extended Baryon Oscillation Spectroscopic Survey [eBOSS; 20], and WiggleZ
[21], have routinely adopted reconstruction as a standard to extract
cosmological information from BAO [22, 23, 24, 25, 26, 6, 7]. Recently, [8]
reported the first detection of BAO from the early data of the Dark Energy
Spectroscopic Instrument (DESI) survey [27, 28], showing that reconstruction
effectively increases the statistical significance of BAO detection for
various DESI target samples.

Running reconstruction on galaxy maps requires certain modeling and
algorithmic choices that can potentially affect the performance of the
reconstruction process. Many different reconstruction algorithms have been
proposed in the literature [e.g. 22, 29, 30], and for a given model, several
hyperparameters must be calibrated to optimally recover the linear density
field. With the standard reconstruction algorithms that have been applied to
galaxy surveys, one needs to first assume a fiducial cosmology to convert
galaxy redshifts to distances, as well as a fiducial value for the growth rate
of cosmic structure and the linear bias of the galaxy sample, both of which
are necessary to calculate the Zel’dovich displacement field that is at the
core of the reconstruction formalism [31, 13]. One also needs to make choices
related to the numerical implementation of the reconstruction recipes, such as
how exactly the discrete galaxy positions are mapped to a smoothed overdensity
field on a grid, which can also play a role in determining the displacement
field [32, 33, 23, 34].

Ultimately, the optimal reconstruction settings depend on the specific galaxy
sample that is being analyzed, with considerations for the sample number
density, redshift range, and survey geometry. The goal of this article is to
present a thorough study of BAO reconstruction as applied to the DESI 2024
results, which use the first data release of the DESI cosmological survey
(DESI DR1; [35]). For this purpose, the first half of our paper analyzes mocks
that match the DESI DR1 target samples with different reconstruction schemes
and settings, with the aim of optimizing the pipeline that DESI will use to
perform the main BAO analysis, presented in [36], and the corresponding
cosmological interpretation of the results, presented in [37].

To minimize potential confirmation biases during parameter inference, the LSS
catalogs analyzed by the DESI Collaboration are blinded to cosmology, as
detailed in [38]. In this context, blinding involves the deliberate
concealment or modification of the catalogs, ensuring that the choices and
interpretations in our analysis remain unbiased. The second half of our paper
concerns a series of tests performed on the blinded DESI DR1 data and mocks
that assess the robustness of the BAO constraints against different
measurement and modeling choices in our pipeline, which informed the decision
of unblinding the catalogs to perform the final cosmological analysis from
BAO.

The paper is organized as follows. In Section 2 we describe the observables
used in this work, including the clustering and the reconstruction of the
blinded DESI DR1 catalogs. In Section 3 we present the theoretical aspect,
including the description of the BAO model and parameter inference, as well as
a description of the mock galaxy catalogs that mimic the DESI DR1 selection
and clustering properties. The results are presented in Section 4, where we
present parameter recovery tests on the mock galaxy catalogs and unblinding
tests using the DESI DR1 data.

Throughout this paper, we adopt a fiducial cosmology that is characterized by
$\omega_{\rm cdm}=0.1200$, $\omega_{\rm b}=0.02237$, $h=0.6736$, $N_{\rm
ur}=2.0328$, and one massive neutrino with $\omega_{\nu}=0.00064420$, which
matches the Planck 2018 base-$\Lambda$CDM parameters [39], as well as the
baseline cosmology from the AbacusSummit simulations [40], described in
Section 3.1. A careful study of the impact of this choice of fiducial
cosmology on the BAO analysis is presented in [41].

##  2 Observations

###  2.1 DESI DR1 samples

The Dark Energy Spectroscopic Instrument (DESI; [42]) is a multi-fiber
spectrograph installed on the Nicholas U. Mayall 4-meter telescope at Kitt
Peak National Observatory in Arizona, which can simultaneously take spectra of
5,000 objects using fiber-fed robotic positioners [43]. The DESI data release
1 [DR1; 35] comprises observations from May 14, 2021, following a period of
successful survey validation [27], which resulted in DESI’s first Early Data
Release [44], through June 14, 2022. Depending on the observerving conditions,
DESI dynamically allocates its observing time into a ‘bright-time’ program,
and a ‘dark-time’ program. The bright-time program comprises the Bright Galaxy
Survey [45], while the dark-time program observes luminous red galaxies,
emission-line galaxies, and quasars, as well as a sample of Milky Way stars
[MWS, 46]. In addition, DESI targets Lyman-$\alpha$ forest quasars to trace
the distribution of neutral hydrogen [47].

For this study, we use the large-scale structure (LSS) catalogs presented in
[48], focusing on the BAO measurements of the following DR1 target samples:

  * •

The Bright Galaxy Sample (BGS): This sample consists of 300,017 good redshifts
in the redshift range $0.1<z<0.4$ over an area of 7,473 $\deg^{2}$. Although
the nominal Bright Galaxy Sample is flux-limited and has a number density that
has a strong redshift dependence, the sample used for the DR1 analysis was
engineered to have a roughly constant number density of at $0.1<z<0.4$, which
was achieved by using $r$ band absolute magnitude cuts with a $k$+E correction
[48]. The resulting number density for this BGS sample is roughly constant at
$3\times 10^{-4}\,h^{3}\mathrm{Mpc}^{-3}$ when averaged over the DR1
footprint, having a clustering amplitude similar to that of LRGs (see below),
making them similarly biased with respect to the dark matter density field.

  * •

The Luminous Red Galaxy Sample (LRGs): This sample includes 2,138,600 good
redshifts over 5,840$\,\deg^{2}$ in the redshift range $0.4<z<1.1$. These are
red, passive galaxies that are highly biased with respect to the underlying
matter distribution, making them specially suitable for BAO analyses, similar
to the previous LRG samples from SDSS [19, 20]. It has a roughly constant
number density of $3.5\times 10^{-4}\,h^{3}\mathrm{Mpc}^{-3}$ up to $z=0.8$,
with a decrease in density at $0.8<z<1.1$ (see Figure 1 from [36]). We further
split this sample into three redshift bins when calculating the two-point
clustering: $0.4<z<0.6$ (LRG1), $0.6<z<0.8$ (LRG2), and $0.8<z<1.1$ (LRG3).

  * •

The Emissison-line Galaxy Sample (ELGs): This sample comprises 2,432,022 good
redshifts in the range $0.8<z<1.6$ over an area of 5,914 $\deg^{2}$. These are
[OII] emission-line galaxies that are generally active star-forming galaxies.
They are less clustered with respect to the matter density field compared to
passive galaxies such as the LRGs. We subdivide them into two disjoint
redshift bins for the clustering measurements: $0.8<z<1.1$ (ELG1) and
$1.1<z<1.6$ (ELG2).

  * •

The Quasar Sample (QSO): This sample consists of $856,652$ good redshifts at
$0.8<z<2.1$, covering an area of 7,249 $\deg^{2}$. This is the sample that has
the largest bias with respect to the underlying dark matter distribution,
since it has a clustering amplitude that is somewhere in between the LRGs and
ELGs, but at a much higher effective redshift.

To avoid confirmation bias in the cosmological analysis, the LSS catalogs of
each target sample were blinded to cosmology. In practice, the redshifts of
the tracers were shifted with a certain prescription, which changes the
position of the BAO feature and the redshift-space distortions (RSD) signal.
The catalogs were only unblinded once 1) the full end-to-end analysis pipeline
was fixed and 2) the data and results run through this pipeline had passed a
series of predefined tests, ensuring that the choices and interpretations in
our analysis remain unbiased. A detailed description of the blinding scheme is
presented in [38], while unblinded measurements are presented in [36]. Beyond
the above samples, the DESI 2024 results will also include BAO in the
Lyman-$\alpha$ forest [49], and both sets will be combined to infer cosmology
[37]. Analyses using clustering information beyond just the BAO scale will be
presented in [50], and will provide additional cosmological constraints,
including on primordial non-Gaussianities [51, 52].

In addition to the tracer LSS catalogs, we also use a series of random
catalogs that follow the footprint and radial selection of the DESI samples
but with no intrinsic clustering [53], which we use to estimate the
overdensity field for reconstruction and the two-point clustering.

###  2.2 Density-field reconstruction

In a Lagrangian framework, we can relate the Eulerian position of a galaxy at
a time $t$, denoted by ${\bf x}({\bf q},t)$, to its initial position ${\bf q}$
in Lagrangian space by

$\displaystyle{\bf x}({\bf q},t)={\bf q}+\bm{\Psi}({\bf q},t),$ (2.1)

where $\bm{\Psi}({\bf q},t)$ is the Lagrangian displacement vector in real
space. The galaxy velocity flows that are sourced by structure formation,
which are captured by the displacement vector, smear and shift the BAO signal,
degrading its detection significance and biasing the distance-redshift
measurements. The goal of BAO reconstruction [11] is to reverse the non-linear
motions in the density field, increasing the accuracy and precision of the BAO
method. Reconstruction achieves this by calculating the displacement field
based on an estimate of the galaxy velocity field, which is in turn estimated
from the observed galaxy density field.

The first-order term in a perturbative expansion of $\bm{\Psi}({\bf q})$ using
Lagrangian perturbation theory, also known as the Zel’dovich approximation
[31], reads

$\bm{\Psi}({\bf q})=\int\frac{d^{3}k}{(2\pi)^{3}}\frac{i{\bf
k}}{k^{2}}\delta({\bf k})e^{i{\bf k}\cdot{\bf q}}\,.$ (2.2)

In addition, the displacement field can be related to the observed redshift-
space galaxy field $\delta_{\text{g}}$ via the linear continuity equation [54,
22],

$\displaystyle\nabla\cdot\bm{\Psi}+\frac{f}{b}\nabla\cdot\left[(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}\right]=-\frac{\delta_{\text{g}}}{b},$
(2.3)

where $f$ is the linear structure growth rate, ${\bf\hat{r}}$ is the line-of-
sight direction, and $b$ is the linear galaxy bias. On linear scales, we can
assume $\bm{\Psi}$ is an irrotational field. Then, we can write the
displacement as the gradient of a potential, $\bm{\Psi}=-\nabla\phi$, and
rewrite Eq. 2.3 as

$\nabla^{2}\phi+\frac{f}{b}\nabla\cdot(\nabla\phi_{r}){\bf\hat{r}}=\frac{\delta_{\rm
g}}{b}$ (2.4)

The solution to Eq. 2.4 can be solved in configuration space using finite-
difference approximations [22] or multigrid relaxation techniques [29].
Alternatively, Eq. 2.3 can be solved in Fourier space, taking advantage of the
computational efficiency of Fast Fourier Transforms (FFTs) [33, 30].

Tracer | Redshift range | Linear bias $b$ | Growth rate $f$ | Smoothing scale $\Sigma_{\text{sm}}$
---|---|---|---|---
BGS | 0.1–0.4 | 1.5 | 0.682 | $15\,h^{-1}\mathrm{Mpc}$
LRGs | 0.4–1.1 | 2.0 | 0.834 | $15\,h^{-1}\mathrm{Mpc}$
ELGs | 0.8–1.6 | 1.2 | 0.900 | $15\,h^{-1}\mathrm{Mpc}$
QSO | 0.8–2.1 | 2.1 | 0.928 | $30\,h^{-1}\mathrm{Mpc}$
Table 1: Fiducial redshift range, tracer linear bias, growth rate of
structure, and smoothing scale assumed when reconstructing each DESI target
sample. The smoothing scale was chosen after testing different values for each
tracer. The growth rate of structure is determined from our fiducial cosmology
and the effective redshift of each sample.

Although the methods mentioned above use different numerical techniques to
solve Eqs. 2.3 and 2.4, their base underlying algorithm can be summarized in
the following steps [29]:

  1. 1.

Smooth the galaxy overdensity field using a kernel $\mathcal{S}$ to filter out
high-k modes that are not well described by linear theory.

  2. 2.

Assuming a value for the structure growth rate $f$ and the linear galaxy bias
$b$, estimate the displacement field $\bm{\Psi}$ by solving Eq. 2.3. The line-
of-sight component of the displacement is multiplied by $1+f$ to account for
RSD.

  3. 3.

Move galaxy positions by $-\bm{\Psi}$ to obtain the displaced density field
$\delta_{d}$.

  4. 4.

Move an initially unclustered catalogue of random particles by $-\bm{\Psi}$ to
obtain the ‘shifted’ field $\delta_{s}$. In this step, one can choose to
include or ignore the $1+f$ factor in the displacement along the line of
sight, which results in two different reconstruction conventions. The RecSym
convention applies the $1+f$ factor to both galaxies and randoms, preserving
linear RSD in the reconstructed field. The RecIso convention does not include
the $1+f$ factor when displacing randoms, and therefore results in a more
isotropically-clustered sample after reconstruction.

The reconstructed density field can then be estimated as
$\delta_{r}\equiv\delta_{d}-\delta_{s}$, with a power spectrum
$P_{r}\propto\langle|\delta^{2}_{r}|\rangle$. In configuration space, the
post-reconstruction correlation function can be calculated using a modified
version of the Landy-Szalay estimator [55], replacing the DR and RR terms in
the numerator by DS and SS, where D and S denote the displaced data catalog
and the shifted random catalogue, respectively.

It should be noted that other improved reconstruction algorithms have been
proposed in the literature [e.g., 56, 57, 58], which follow different recipes.
Although these methods hold significant promise for reconstructing the linear
small-scale density field in the very low shot noise regime, the expected
improvements in the BAO distance measurements are marginal at the galaxy
number densities of DESI DR1.

For this work and the main DESI BAO analysis presented in [36], we adopt the
iterative reconstruction in Fourier space proposed in [33] as our default
reconstruction algorithm. In [59], we show that this implementation gives
consistent results with the multigrid algorithm, but with a reduced
computational cost. Below, we present a brief description of the algorithm,
and refer the reader to [59] for a detailed comparison with reconstruction
algorithms in the context of BAO analyses.

####  2.2.1 Reconstruction based on iterative Fast Fourier Transforms

We start by decomposing $(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}$ into the
gradient of a scalar potential field $A$ and the curl of a vector field ${\bf
B}$ using Helmholtz’s Theorem:

$(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}=\nabla A+\nabla\times{\bf B}\,.$
(2.5)

Substituting this in Eq. 2.3, we obtain

$\nabla(\phi+fA)=-\nabla\nabla^{-2}\frac{\delta_{\rm g}}{b}\,,$ (2.6)

where we have assumed that $\bm{\Psi}$ can be written as the gradient of a
potential, as in Eq. 2.4. The formula above is exact, but cannot be solved
directly using FFTs. The difficulty arises from the second term on the right-
hand side of Eq. 2.5. If we ignore this term by assuming
$(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}$ is irrotational,

$(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}\approx\nabla A\,,$ (2.7)

then we can write

$\bm{\Psi}+\frac{f}{b}(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}=-\nabla\nabla^{-2}\frac{\delta_{\rm
g}}{b}\,.$ (2.8)

The right-hand side of this equation can be computed using FFTs, with a
solution given by

$\bm{\Psi}={\tt IFFT}\left[-\frac{i{\bf k}\delta({\bf
k})}{k^{2}b}\right]-\frac{\beta}{1+\beta}\left\\{{\tt IFFT}\left[\frac{i{\bf
k}\delta({\bf k})}{k^{2}b}\right]\cdot{\bf\hat{r}}\right\\}{\bf\hat{r}}\,$
(2.9)

where IFFT denotes the inverse Fourier Transform, and $\beta=f/b$.

In [30], it has been shown that assuming
$(\bm{\Psi}\cdot{\bf\hat{r}}){\bf\hat{r}}$ is irrotational overcorrects for
RSD in the reconstructed field, which is not desirable. However, they propose
to start from this approximation and then use an iterative scheme to recover
the real-space displacement field, provisional on knowing the value of the
growth rate of structure $f$.

If we start from Eq. 2.9 and completely ignore the RSD component, we can write
the displacement as

$\bm{\Psi}={\tt IFFT}\left[-\frac{i{\bf k}\delta_{\rm g,rsd}({\bf
k})}{k^{2}b}\right]\,$ (2.10)

or, in terms of the potential, as

$\phi_{\rm est,1}={\tt IFFT}\left[-\frac{\delta_{\rm g,rsd}({\bf
k})}{k^{2}b}\right]\,.$ (2.11)

Here, $\delta_{\rm g,rsd}$ is the observed redshift-space galaxy overdensity
field. We can use this first estimate of the potential to estimate the real-
space overdensity from the redshift-space one,

$\frac{\delta_{\rm g,real,1}}{b}=\frac{\delta_{\rm
g,rsd}}{b}+\frac{f}{b}\nabla\cdot(\nabla\phi_{\rm
est,1}\cdot{\bf\hat{r}}){\bf\hat{r}}$ (2.12)

In the next iteration, we build upon this estimate of the real-space
overdensity to re-calculate the potential,

$\phi_{\rm est,2}={\tt IFFT}\left[-\frac{\delta_{\rm real,1}({\bf
k})}{k^{2}b}\right]\,.$ (2.13)

and the operation continues. Once the algorithm has converged, we can estimate
the displacement as

$\Psi_{{\rm FFT},n}=\nabla\nabla^{-2}\left(\frac{\delta_{{\rm
g,real},n}}{b}\right)\,,$ (2.14)

where $\delta_{{\rm g,real},n}$ is the estimation of the real-space galaxy
overdensity field on the $n$-th iteration.

The description above constitutes the IFFT algorithm in its basic form.
However, to speed up convergence, different starting points can be considered
for the first iteration. In pyrecon, during the first iteration, the second
term on the right-hand side of Eq. 2.12 takes an additional factor
$1/(1+\beta)$, motivated by Eq. 2.9.

We have previously run convergence tests to assess the number of iterations
required to obtain an optimal reconstruction of the DESI target samples. Based
on results that are presented in [59], we choose $n=3$ as the default number
of iterations for the IFFT algorithm throughout the rest of this work.

It should be noted that this method differs from the one adopted in the eBOSS
clustering analysis. The eBOSS method [60, 7] iteratively updates the galaxy
and random particle positions, while the IFFT algorithm we adopt here updates
the overdensity on the rectangular meshes, matching the algorithm proposed in
[33, 30]. This is the first time that this algorithm has been implemented in a
BAO data analysis.

####  2.2.2 The pyrecon reconstruction toolkit

We reconstruct the DESI DR1 samples using
pyrecon111https://github.com/cosmodesi/pyrecon, a comprehensive reconstruction
toolkit developed by the DESI collaboration, which implements multiple
reconstruction algorithms, accommodates various conventions, and provides the
flexibility to process periodic-box simulations or survey data with non-
uniform geometries. pyrecon begins by drawing two identical rectangular grids
with a given resolution (which we set to $4\,h^{-1}\mathrm{Mpc}$ for this
work222We have performed convergence tests to choose an optimal value for the
grid resolution, given our choice of smoothing scales and the computational
restrictions due to the increase in memory allocation as the grid resolution
increases. We also add some padding to the grid borders so that the size of
the grid is 20% larger on each side than what would be strictly necessary to
perfectly contain the catalogs, which avoids introducing spurious correlations
on the scales of interest when performing FFTs, which inherently assume
periodicity for the grid operations.) that encompasses the survey volume. The
galaxy and random particles are then painted on each grid using triangular-
shaped cloud interpolation [61]. The galaxy and random fields are smoothed in
Fourier space using a Gaussian kernel of width $\Sigma_{\text{sm}}$,

$\mathcal{S}=e^{-(k\Sigma_{\text{sm}})^{2}/2}\,,$ (2.15)

after which they are converted back to configuration space (we test different
$\Sigma_{\text{sm}}$ values for each tracer, and determine the optimal
smoothing scale in Section 4.2). The overdensity field is then estimated by
taking the ratio of the galaxy and random grids. Grid cells that contain fewer
random particles than a given threshold are assigned a density contrast of
zero. We set this threshold to 75% of the mean number of random particles per
cell. This recipe is common for all reconstruction algorithms implemented in
pyrecon, after which the displacement field depends on the specifics of each
algorithm (in our case, the IFFT algorithm).

Throughout this work, we adopt some fiducial values for the linear galaxy bias
and the growth rate of structure when running reconstruction, which are listed
in Table 1. The linear galaxy bias parameters have been determined by [62,
63], and correspond to 1.5 for BGS, 2.0 for LRGs, 1.2 for ELGs, and 2.1 for
QSO. We calculate the linear growth rate of structure from
cosmoprimo333https://github.com/cosmodesi/cosmoprimo as
$f=\sigma_{8}^{vv}(z_{\rm eff})/\sigma_{8}^{dd}(z_{\rm eff})$, where
$\sigma_{8}^{vv}$ and $\sigma_{8}^{vv}$ are the root mean square of velocity
and density perturbations in spheres of $8\,h^{-1}\mathrm{Mpc}$, respectively
[64]. This is calculated using our fiducial cosmology model, at the effective
redshift of each target sample, which is 0.296 for BGS, 0.780 for LRGs, 1.194
for ELGs, and 1.495 for QSO [48]. The impact on reconstruction due to changes
in fiducial cosmology has been studied in [65, 66], and a general study of
fiducial cosmology effects on the DESI BAO analysis is presented in detail in
[41]. In addition, [12] present theoretical calculations showing the expected
(small) impact of assuming an incorrect galaxy bias or growth rate on the BAO
constraints.

###  2.3 Clustering measurements

We study the clustering properties of the DESI DR1 samples by means of
anisotropic two-point statistics in configuration and Fourier space. In
particular, we focus on the multipole moments of the anisotropic galaxy
correlation function and the power spectrum of each sample. We calculate the
correlation function multipoles with
pycorr444https://github.com/cosmodesi/pycorr, which is a wrapper around a
modified version of the corrfunc pair-counting code [67]. Before its
decomposition into multipoles, the correlation function is binned in $s$ and
$\mu$, where $s$ is the redshift-space scalar separation between the pair and
$\mu$ is the cosine of the angle between the vector connecting the pair and
the observer’s line of sight. We use linear bins of width
$4\,h^{-1}\mathrm{Mpc}$ in $s$ from $50\,h^{-1}\mathrm{Mpc}$ to
$150\,h^{-1}\mathrm{Mpc}$. Similarly, the power spectrum multipoles are
calculated with pypower555https://github.com/cosmodesi/pypower., where the
power spectrum estimator takes reference from [68]. We use $k$ bins of width
$0.005\,h\,\mathrm{Mpc}^{-1}$ between $0.02\,h\,\mathrm{Mpc}^{-1}$ and
$0.3\,h\,\mathrm{Mpc}^{-1}$. These scales were identified as optimal for BAO
fitting by [12]. More details of the calculation of these clustering
statistics, including how galaxies are weighed to optimally measure the
clustering signal, are presented in [48].

To estimate the errors associated with the correlation functions, we use semi-
analytical semi-empirical covariance matrices generated with the RascalC code
[69, 70], while the errors associated with the power spectra are based on
CovaPT [71, 72]. Refs. [73, 74, 72] present a comprehensive description of the
construction and validation of these covariance matrices for DESI DR1.

Parameter |  $P(k)$ prior |  $\xi(r)$ prior | Interpretation
---|---|---|---
$\alpha_{\text{iso}}$ | $\mathcal{U}(0.8,1.2)$ | $\mathcal{U}(0.8,1.2)$ | Isotropic BAO dilation
$\alpha_{\text{AP}}^{*}$ | $\mathcal{U}(0.8,1.2)$ | $\mathcal{U}(0.8,1.2)$ | Anisotropic (AP) BAO dilation
$\Sigma{\perp}$ | $\mathcal{N}(\Sigma^{\mathrm{fid}}_{\perp},1.0)$ | $\mathcal{N}(\Sigma^{\mathrm{fid}}_{\perp},1.0)$ | Transverse BAO damping [$\,h^{-1}\mathrm{Mpc}$]
$\Sigma{\parallel}$ | $\mathcal{N}(\Sigma^{\mathrm{fid}}_{\parallel},2.0)$ | $\mathcal{N}(\Sigma^{\mathrm{fid}}_{\parallel},2.0)$ | Line-of-sight BAO damping [$\,h^{-1}\mathrm{Mpc}$]
$\Sigma_{s}$ | $\mathcal{N}(2.0,2.0)$ | $\mathcal{N}(2.0,2.0)$ | Fingers-of-God damping [$\,h^{-1}\mathrm{Mpc}$]
$b_{1}$ | $\mathcal{U}(0.2,4)$ | $\mathcal{U}(0.2,4)$ | Linear galaxy bias
$d\beta^{*}$ | $\mathcal{U}(0.7,1.3)$ | $\mathcal{U}(0.7,1.3)$ | Linear RSD parameter
$a_{0,n}$ | $\mathcal{N}(0,10^{4})$ | N/A | Spline parameters for the monopole
$a_{2,n}^{*}$ | $\mathcal{N}(0,10^{4})$ | $\mathcal{N}(0,10^{4})$ | Spline parameters for the quadrupole
$b_{0,n}$ | N/A | $\mathcal{U}(-\infty,\infty)$ | Unknown large scale systematics
$b_{2,n}^{*}$ | N/A | $\mathcal{U}(-\infty,\infty)$ | Unknown large scale systematics
Table 2: Priors on the parameters that are varied during fits to the power
spectrum (second column) and the correlation function (third column).
$\mathcal{U}$ and $\mathcal{N}$ denote uniform and Gaussian distributions,
respectively. The parameters marked with a star are fixed to their default
values for isotropic BAO fits (BGS, ELG1 and QSO), where the default values
are $d\beta=1$, $a_{2,n}=0$, $b_{2,n}=0$. The mean values of the priors on the
damping parameters for each tracer are shown in Table 3.

##  3 Modeling

###  3.1 DESI mocks

To test the effects of reconstruction in a controlled setting, we use two
collections of mock galaxy catalogs that were calibrated to match the
clustering and selection properties of the DESI DR1 target samples, as
described in [75]: full N-body mocks based on the AbacusSummit suite of
simulations [40], and approximate mocks based on the EZmock algorithm [76].
All mocks were constructed assuming a Planck 2018 base-$\Lambda$CDM cosmology,
more specifically using the mean of the marginalized parameter distributions
from the Planck TT,TE,EE+lowE+lensing likelihood [39]: $\omega_{\rm
cdm}=0.1200$, $\omega_{\rm b}=0.02237$, $h=0.6736$, and $N_{\rm ur}=2.0328$.

The N-body mocks start from halo catalogs from the $2\,h^{-1}\mathrm{Gpc}$
AbacusSummit simulation boxes [40], at snapshots within the redshift range of
each DESI sample. Dark matter haloes are populated with galaxies using the
Halo Occupation Distribution (HOD) model, with parameters that are calibrated
to match the number density and clustering of each galaxy sample (an
assessment of the impact of the HOD modelling on the DESI error budget is
presented in [62, 63]). Mocks with cutsky geometry are then constructed by
replicating and patching the $2\,h^{-1}\mathrm{Gpc}$ boxes from different
snapshots, converting them to sky coordinates and trimming them to match the
DESI footprint and radial selection of each sample. For LRGs, we use the
$z=0.5$ snapshot to construct the $z<0.6$ portion of the cutsky, and the
$z=0.8$ snapshot to cover the $z>0.6$ portion. For ELGs, snapshots at
$z=0.950$ and $z=1.325$ are used to cover $z<1.1$ and $z>1.1$, respetively.
The QSO cutsky reads data from the $z=1.400$ snapshot for all redshifts. We
note that these cutsky mocks differ from the full lightcones described in
[77].

The approximate mocks consist of a series of 1000 galaxy catalogs generated
with the EZmock algorithm [76] with the same volume, number density, and HOD
as the AbacusSummit simulations. The EZmocks are based on the Zel’dovich
approximation, including prescriptions for the missing physical ingredients,
such as stochastic scale-dependent, non-local and non-linear biasing
contributions.

Since DESI measures thousands of galaxy redshifts simultaneously using fiber-
fed robotic positioners, the galaxy clustering can be artificially decreased
at small scales due to the fiber assignment scheme, which is sometimes unable
to simultaneously target galaxy pairs that are positioned too close on the
sky. To ensure that both the AbacusSummit and EZmocks reproduce this
systematic error on the clustering measurements, they are passed through a
pipeline that mimics the fiber assignment scheme of the DESI DR1 samples [48].

###  3.2 Modelling the BAO signal

An extensive discussion on the modeling of the pre- and post-reconstruction
BAO signal is presented in [12]. Here, we present a brief description of the
main constituents of the model in configuration and Fourier space.

####  3.2.1 Fourier space

Our model for the observed galaxy power spectrum is based on the work of [12]
and can be written generically as

$P(k,\mu)=\mathcal{B}(k,\mu)P_{\rm nw}(k)+\mathcal{C}(k,\mu)P_{\rm
w}(k)+\mathcal{D}(k)\,,$ (3.1)

where $P_{\rm nw}(k)$ and $P_{\rm w}(k)$ denote the smooth (no-wiggle) and BAO
(wiggle) components of the linear power spectrum, respectively, which are
obtained following the peak average method from [78]. The linear matter power
spectrum template is predicted from
CLASS666https://github.com/lesgourg/class_public[79] using our fiducial
cosmology.

Following [80, 81], we adopt the following parametric form for
$\mathcal{B}(k,\mu)$:

$\mathcal{B}(k,\mu)=\left(b+f\mu^{2}(1-s(k)\right)^{2}F_{\rm fog}\,,$ (3.2)

where $F_{\rm fog}=\left(1+\frac{1}{2}k^{2}\mu^{2}\Sigma_{s}^{2}\right)^{-2}$
accounts for the ‘Fingers of God’ effect due to halo virialization [82]. For
pre-reconstruction and the RecSym convention, $s(k)=1$, while for RecIso,
$s(k)=\exp\left[-(k\Sigma_{\rm sm})^{2}/2\right]$, where $\Sigma_{\rm sm}$ is
the reconstruction smoothing scale.

$\mathcal{C}(k,\mu)$ captures the anisotropic non-linear damping on the BAO
feature,

$\mathcal{C}(k,\mu)=\left(b+f\mu^{2}\right)^{2}\exp\left[-\frac{1}{2}k^{2}\biggl{(}\mu^{2}\Sigma_{||}^{2}+(1-\mu^{2})\Sigma^{2}_{\perp}\biggl{)}\right]$
(3.3)

where $\Sigma_{||}$ and $\Sigma_{||}$ model the damping for modes along and
perpendicular to the line of sight.

The $\mathcal{D}(k)$ factor captures any deviation from linear theory in the
broadband shape of the power spectrum multipoles. Motivated by [12], we
parameterize it separately for each clustering multipole using a spline basis
with bases separated by $\Delta$,

$\mathcal{D}_{\ell}(k>k_{\rm
min})=\sum_{n=-1}^{7}a_{n}W_{3}\left(\frac{k}{\Delta}-n\right)\,,$ (3.4)

where $W_{3}$ is a piecewise cubic spline kernel [83, 61]. We set
$\Delta=0.06\,h\,\mathrm{Mpc}^{-1}$, which is chosen such that the spline
basis is able to match the broadband shape of the power spectrum, without
reproducing the BAO wiggles themselves.

If the fiducial cosmology used to convert redshifts to distances differs from
the true cosmology of the Universe, it can induce geometric distortions in the
galaxy distribution. Moreover, the sound horizon in the template cosmology
could also differ from the true sound horizon. To account for both effects, we
parameterize the position of the BAO wiggles via the dilation parameters
$\alpha_{\perp}$ and $\alpha_{\parallel}$, which scale the separations in the
directions across and along the line of sight:

$\displaystyle k^{\rm fid}$
$\displaystyle=\frac{k}{\alpha_{\perp}}\left[1+\mu^{2}\left(\frac{1}{F^{2}}-1\right)\right]^{1/2}$
(3.5) $\displaystyle\mu^{\rm fid}$
$\displaystyle=\frac{\mu}{F}\left[1+\mu^{2}\left(\frac{1}{F^{2}}-1\right)\right]^{-1/2}\,,$
(3.6)

where $F=\alpha_{\parallel}/\alpha_{\perp}$. Here, the fid superscript denote
quantities in the fiducial cosmology. The dilation parameters can be
interepreted in terms of cosmology as

$\displaystyle\alpha_{\parallel}$ $\displaystyle=\frac{H^{\rm
fid}(z)}{H(z)}\frac{r_{s}^{\rm fid}}{r_{s}(z_{d})}$ (3.7)
$\displaystyle\alpha_{\perp}$ $\displaystyle=\frac{D_{\rm M}(z)}{D_{\rm
M}^{\rm fid}(z)}\frac{r_{s}^{\rm fid}(z_{d})}{r_{s}(z_{d})}\,,$ (3.8)

where $D_{\rm M}(z)$ is the transverse comoving distance to redshift $z$,
$H(z)$ is the Hubble parameter, and $r_{s}(z_{d})$ is the sound horizon at the
baryon drag epoch $z_{d}$. From here, we can define the following quantities:

$\displaystyle\alpha_{\text{iso}}$
$\displaystyle\equiv(\alpha_{\perp}^{2}\alpha_{\parallel})^{\frac{1}{3}},$
(3.9) $\displaystyle\alpha_{\text{AP}}$
$\displaystyle\equiv\frac{\alpha_{\parallel}}{\alpha_{\perp}}.$ (3.10)

which encode the isotropic and anisotropic BAO dilation, respectively.

####  3.2.2 Configuration space

Our starting point is the anisotropic power spectrum obtained from Eq. 3.1
without the $\mathcal{D}(k)$ term, which are Hankel-transformed to
configuration space as

$\xi_{l}(s)=\frac{i^{\ell}}{2\pi^{2}}\int_{0}^{\infty}k^{2}j_{\ell}(ks)P_{\ell}(k)\mathrm{d}k\,,$
(3.11)

where $j_{\ell}$ are the spherical Bessel functions.

To parameterize the broadband part of the correlation function, we take the
Hankel transform of the $n=[0,1]$ spline bases from Eq. 3.4 for the
quadrupole, and we additionally include even-power polynomial terms for both
monopole and quadrupole:

$\mathcal{\tilde{D}}_{\ell}(s>s_{\rm
min})=\sum_{n=0}^{1}b_{n}\left(\frac{r}{\Delta_{s}}\right)^{2n}\,,$ (3.12)

where $\Delta_{s}$ = $k_{\rm min}/2\pi$.

Parameter | Recon | BGS | LRGs | ELGs | QSO
---|---|---|---|---|---
$\Sigma_{\perp}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Pre | $6.5$ | $4.5$ | $4.5$ | $3.5$
$\Sigma_{\parallel}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Pre | $10.0$ | $9.0$ | $8.5$ | $9.0$
$\Sigma_{\text{s}}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Pre | $2.0$ | $2.0$ | $2.0$ | $2.0$
$\Sigma_{\perp}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Post | $3.0$ | $3.0$ | $3.0$ | $3.0$
$\Sigma_{\parallel}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Post | $8.0$ | $6.0$ | $6.0$ | $6.0$
$\Sigma_{\text{s}}^{\rm fid}\,[\,h^{-1}\mathrm{Mpc}]$ | Post | $2.0$ | $2.0$ | $2.0$ | $2.0$
Table 3: Mean values of the Gaussian priors (Table 2) for the non-linear BAO
damping parameters across and along the line of sight ($\Sigma_{\perp}$ and
$\Sigma_{\parallel}$, respectively) and the Fingers-of-God damping
($\Sigma_{\text{s}}$), depending on whether they are used in pre- or post-
reconstruction BAO fits of each tracer.

###  3.3 Model fitting

We perform the BAO fitting using
desilike,777https://github.com/cosmodesi/desilike which is a python package
that provides a common framework for writing DESI likelihoods. We sample
posterior distributions using the Markov Chain Monte Carlo (MCMC) code emcee
[84], adopting the priors listed in Table 2. Use flat priors in all cases
except for the damping parameters, for which we adopt Gaussian priors centred
on the fiducial values listed in Table 3. These Gaussian priors have been
informed by running fits to data vectors averaged over many realizations of
the DESI mocks, ensuring that the signal-to-noise ratio (SNR) is large enough
to let the damping parameters vary freely during the fit. In later sections,
we show that the recovered BAO parameters from fits to DESI data are largely
insensitive to this choice of priors. In addition to the spline broadband
parameters described in the previous section, we also allow for a linear RSD
parameter scaling $d\beta\equiv f/f^{\rm fid}$, where $f^{\rm fid}$ is the
fiducial growth rate of structure, and a bias parameter $b_{1}$ that regulates
the clustering amplitude.

Apart from MCMC posterior sampling, we perform posterior profiling using
minuit [85], which is incorporated in desilike. In the following, we focus
mainly on the BAO scaling parameters $\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$ while marginalizing over all other nuisance parameters.

Given the SNR of the clustering measurements of the different DESI samples,
only certain tracers and redshift bins allow two-dimensional BAO fits where
$\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$ are varied simultaneously.
These are the LRGs, and ELG2. For BGS, QSO, and ELG1, we fit only the monopole
and vary $\alpha_{\text{iso}}$ during the fit. However, in all cases we model
the full set of clustering multipoles to account for their potential impact
(via convolution with the survey geometry window function) on the clustering
monopole.

##  4 Results

###  4.1 A qualitative exploration of reconstruction

---
Figure 1: Distribution of the displacements vectors from the LRGs
reconstruction, projected along the three Cartesian axes of the reconstruction
grid. The black line shows the blinded DESI DR1 data, and the thinner blue
lines show 10 different realizations of the AbacusSummit mocks. We compare
against a Gaussian with the same mean and dispersion as the DESI data, shown
by the grey-dashed line.

We begin by exploring the distribution of displacement vectors resulting from
reconstructing the BGS sample from the blinded DESI DR1 data, shown by the
thick black line in Fig. 1. In this case, we have used a reconstruction
smoothing scale of $15\,\,h^{-1}\mathrm{Mpc}$, and the fiducial values for the
linear galaxy bias and the growth rate of structure listed in Section 2.2.
Since the galaxy overdensity field is estimated on a rectangular grid that
encapsulates the survey volume (see Section 2.2.2), Fig. 1 shows the
displacement vectors along the three axes of the grid. They are roughly
symmetric around zero in each direction and approximately follow a Gaussian
distribution. Strong non-linear motions cause deviations from Gaussianity for
large displacement values. The AbacusSummit mocks, of which we show ten
independent realizations in the same figure, are largely consistent with the
DESI data. More quantitatively, the DESI displacements are centred at
$(0.39,-0.08,0.07)\,h^{-1}\mathrm{Mpc}$, with a dispersion of
$(4.60,4.29,3.73)\,h^{-1}\mathrm{Mpc}$. The mean of 25 mocks yields a mean of
$(0.03,0.05,0.01)\,h^{-1}\mathrm{Mpc}$, with a dispersion of
$(4.53,4.19,3.68)\,h^{-1}\mathrm{Mpc}$. We have explicitly checked that a
similar level of agreement is found for the other tracers, but we do not
include the figures here for conciseness. Although for BAO analyses the
displacement field is used as a means of recovering the linear density field
and sharpening the acoustic feature, the reconstructed velocity field itself
is also of great importance for other analyses that focus on the imprints of
galaxy velocities on the cosmic microwave background, such as the kinematic
Sunyaev-Zel’dovich effect [86, 87].

|
---|---
Figure 2: Left: The open and filled markers show the monopole of the galaxy two-point correlation function before and after reconstruction, respectively, averaged over 25 realizations of the AbacusSummit mocks for LRGs at $0.4<z<0.6$. The error bars, which represent the error on the mean, are calculated from a semi-analytic/semi-empirical covariance matrix. The solid lines correspond to the best-fit model. The bottom subpanel shows the relative error between the measurements and the best-fit models. Right: $\Delta\chi^{2}$ as a function of the BAO scaling parameter $\alpha_{\text{iso}}$. Solid and dashed lines show results from models with and without BAO wiggles, respectively. The horizontal dashed lines show different thresholds of detection. |
---|---
Figure 3: Left: The circles show the broadband-subtracted monopole of the
anisotropic power spectrum averaged over 25 realizations of the AbacusSummit
mocks, before and after reconstructing the density field. The solid lines show
the best-fit BAO model. Right: Constraints on the isotropic BAO scaling
parameter $\alpha_{\text{iso}}$ with and without reconstruction.

Having estimated the displacement field, we can use it to estimate the
clustering from the reconstructed samples. Fig. 2 compares the monopole of the
anisotropic correlation function for LRG1 before and after reconstruction. The
pre-reconstruction correlation function shows a prominent BAO peak around
$\sim 100\,h^{-1}\mathrm{Mpc}$, which is smeared due to non-linear
gravitational motions. Reconstruction helps to sharpen the BAO peak by
partially restoring the linearity of the density field. The BAO model of
Section 3.2, shown by the solid line, is an excellent fit to the data, giving
a $\chi^{2}$ of 33.4 for 37 degrees of freedom after reconstruction. The
correlation function is averaged over 25 mock realizations, and the covariance
represents the error of the combined volume of all these mocks
($200\,\,h^{-3}\mathrm{Mpc}^{3}$). In the panel on the right side, we also
display $\Delta\chi^{2}$ for the pre- and post-reconstruction cases using
solid curves. The profiles are well centered around $\alpha_{\text{iso}}=1$,
showing that the model recovers unbiased parameter constraints even for this
large volume. The reconstructed case has a sharper distribution due to the
better localized BAO bump in the correlation function, which leads to a better
determination of the scaling parameter. We can contrast these distributions
with the case where we fit a model without BAO wiggles to assess the
significance of the BAO detection. The dashed curves show the $\Delta\chi^{2}$
between the best-fit BAO models and the models without BAO. From this we see
that reconstruction increases the statistical significance of the BAO
detection from approximately $18\sigma$ to $27\sigma$. Such a high detection
level is only achieved, of course, since we are fitting the combined set of
mocks for which the SNR of the data vector is exquisite.

Reconstruction not only sharpens the BAO feature, but can also help removing
systematic shifts in the location of the peak due to non-linear evolution and
galaxy biasing. This is exemplified in Fig. 3, where we compare the pre- and
post-reconstruction power spectrum of the BGS sample from the AbacusSummit
mocks. To better see the BAO wiggles, we have subtracted the broadband
component of the power spectrum from both measurements. It can be noticed that
the pre-reconstruction case not only exhibits damped wiggles, but also a
slight shift in phase towards larger $k$ values. The BAO fit from the pre-
reconstruction data, shown in the right-hand panel, is biased high with
respect to the expectation of $\alpha_{\text{iso}}=1$ at a more than
2-$\sigma$ level. In addition to reducing the error on $\alpha_{\text{iso}}$
by 50%, reconstruction also effectively corrects the bias seen in the mean
value.

###  4.2 Impact of the smoothing scale

---
Figure 4: Effect on the constraints on the BAO scaling parameters for LRG2
(top) and ELG2 (bottom), reconstructing the density field using a smoothing
scale of $10\,h^{-1}\mathrm{Mpc}$ (diamonds) or $15\,h^{-1}\mathrm{Mpc}$
(circles). Left: Best-fit values of the isotropic BAO scaling parameter
$\alpha_{\text{iso}}$ before and after reconstruction. Right: 1-$\sigma$
errors on $\alpha_{\text{iso}}$ before and after reconstruction. The open
markers are individual fits to independent realizations of the AbacusSummit
mocks. The filled markers show the mean of the 25 realizations, and the error
bars show the standard deviation.
---
Figure 5: Similar to Fig. 4, but for the anisotropic BAO scaling parameter $\alpha_{\text{AP}}$. Tracer | Recon | $\Sigma_{\rm sm}$ | $\alpha_{\rm iso}$ | $\alpha_{\rm AP}$ | $\Delta\alpha_{\rm iso}/\sigma$ | $\Delta\alpha_{\rm AP}/\sigma$
---|---|---|---|---|---|---
${\tt BGS}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9943\pm 0.0039$ | — | $-1.46$ | —
${\tt LRG1}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9980\pm 0.0026$ | $0.9912\pm 0.0090$ | $-0.77$ | $-0.98$
${\tt LRG2}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9957\pm 0.0021$ | $1.0040\pm 0.0074$ | $-2.05$ | $0.55$
${\tt LRG3}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9957\pm 0.0018$ | $0.9928\pm 0.0065$ | $-2.34$ | $-1.11$
${\tt ELG1}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9984\pm 0.0041$ | — | $-0.39$ | —
${\tt ELG2}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.9959\pm 0.0029$ | $1.0003\pm 0.0099$ | $-1.39$ | $0.03$
${\tt QSO}$ | Post |  30 $h^{-1}{\rm Mpc}$ | $0.9960\pm 0.0034$ | — | $-1.17$ | —
${\tt BGS}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $0.9937\pm 0.0038$ | — | $-1.64$ | —
${\tt LRG1}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $0.9967\pm 0.0026$ | $0.9932\pm 0.0089$ | $-1.31$ | $-0.77$
${\tt LRG2}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $0.9960\pm 0.0021$ | $1.0033\pm 0.0075$ | $-1.92$ | $0.44$
${\tt LRG3}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $0.9957\pm 0.0019$ | $0.9895\pm 0.0065$ | $-2.28$ | $-1.61$
${\tt ELG1}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $1.0030\pm 0.0044$ | — | $0.68$ | —
${\tt ELG2}$ | Post |  10 $h^{-1}{\rm Mpc}$ | $0.9955\pm 0.0030$ | $0.9978\pm 0.0104$ | $-1.51$ | $-0.21$
${\tt QSO}$ | Post |  20 $h^{-1}{\rm Mpc}$ | $0.9930\pm 0.0039$ | — | $-1.82$ | —
${\tt BGS}$ | Pre | — | $1.0077\pm 0.0070$ | — | $1.10$ | —
${\tt LRG1}$ | Pre | — | $0.9979\pm 0.0044$ | $0.9726\pm 0.0167$ | $-0.47$ | $-1.64$
${\tt LRG2}$ | Pre | — | $1.0000\pm 0.0034$ | $1.0042\pm 0.0130$ | $-0.01$ | $0.33$
${\tt LRG3}$ | Pre | — | $0.9996\pm 0.0026$ | $1.0035\pm 0.0091$ | $-0.16$ | $0.38$
${\tt ELG1}$ | Pre | — | $1.0012\pm 0.0051$ | — | $0.23$ | —
${\tt ELG2}$ | Pre | — | $1.0015\pm 0.0033$ | $0.9968\pm 0.0111$ | $0.45$ | $-0.29$
${\tt QSO}$ | Pre | — | $1.0053\pm 0.0038$ | — | $1.39$ | —
Table 4: Mean values and standard deviations from the marginalized posteriors
of the BAO scaling parameters from a fit to the mean of 25 realizations of the
AbacusSummit mocks. We display pre-reconstruction results, as well as post-
reconstruction fits obtained with two different choices of smoothing scale
$\Sigma_{\text{sm}}$. The last two columns show the offset in the mean values
with respect to the expected value of $\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$, in units of the corresponding standard deviation, i.e.
$\Delta\alpha=(\alpha-1)/\sigma$. We use a covariance representative of the
DESI DR1 volume for each tracer.

One of the key hyperparameters that must be tuned to optimally reconstruct the
galaxy catalogs is the scale $\Sigma_{\text{sm}}$ that is used to smooth the
density field before estimating the displacement. Values of
$\Sigma_{\text{sm}}$ that are too large might degrade the information content
by erasing clustering on linear scales, while values that are too small could
increase the noise in the reconstruction process and potentially lead to
biases due to the inclusion of non-linear modes that break the underlying
assumptions in the modeling.

Previous works in the literature have performed careful studies of the impact
of the smoothing scale on the reconstruction performance for SDSS galaxy
samples [32, 33, 23, 34], showing that the optimal choice of smoothing scale
has an important dependence on the tracer characteristics: low-density tracers
are more affected by shot noise, so larger smoothing scales are usually
preferred. The main BOSS and eBOSS clustering analysis adopted a smoothing
scale of $15\,h^{-1}\mathrm{Mpc}$ for their LRG and ELG samples, obtaining a
consistent improvement in the BAO constraints from reconstruction [6, 7].
Here, we revisit these choices for the DESI DR1 sample and run BAO fits with
different $\Sigma_{\text{sm}}$ values. Given the various tracers and redshift
bins to be explored, we restrict ourselves to only two smoothing scales per
subsample, informed by a more extensive exploration of the parameter space
presented in [59].

Figure 4 compares the constraints on the isotropic BAO scaling parameter,
$\alpha_{\text{iso}}$ before and after reconstruction for LRG3 and ELG2,
setting $\Sigma_{\rm sm}$ to $10\,h^{-1}\mathrm{Mpc}$ or
$15\,h^{-1}\mathrm{Mpc}$. These constraints represent the maximum-posterior
values from individual fits to the AbacusSummit mock realizations (empty
circles and diamonds), with the average fit shown by the filled markers with
error bars. Looking at the left panel, we see that for both smoothing scales,
there is good agreement between the best-fit values before and after
reconstruction. This need not necessarily be the case, since non-linear
motions will tend to systematically shift the BAO in the correlation function
towards smaller distances. This can result in $\alpha_{\text{iso}}>1$ before
reconstruction, even if the fiducial cosmology matches the true one, which can
then be corrected by reconstruction (see Fig. 3 from the previous section).
The right panel reveals that for LRG3, reconstruction significantly improves
the errors on $\alpha_{\text{iso}}$ for all mock realizations. This is
qualitatively consistent with the results of [8], who found that for mocks
that match the properties of the DESI EDR LRG sample, reconstruction tends to
improve the errors on the isotropic BAO scaling parameter in about 90% of the
cases. However, for ELG2, some of the mocks lie above the reference dashed
line, which delineates the region where errors degrade after reconstruction.
However, the $\alpha_{\text{iso}}$ errors decrease for most of the mocks,
obtaining consistent constraints from both smoothing scales.

Figure 5 shows the constraints for the anisotropic scaling parameter
$\alpha_{\text{AP}}$. Both tracers recover $\alpha_{\text{AP}}$ around one and
show consistent values before and after reconstruction. The choice of
smoothing scale has a negligible impact for LRG2 and a very small impact on
the average best-fit value for ELG2, resulting in a change that is smaller
than the error on the mean. For LRG2, reconstruction improves the precision on
$\alpha_{\text{AP}}$ for all mocks in a very similar way for the two smoothing
scales. For ELG2, there is consistency in the $\alpha_{\text{AP}}$ error from
both smoothing scales, but a few mocks show increased errors after
reconstruction.

In Appendix A, we show the equivalent figures for the other redshift bins and
tracers, and Table 4 shows a compilation of the constraints before and after
reconstruction with different smoothing scales. To estimate the constraints
shown in Table 4, we fit the mean of the 25 AbacusSummit realizations, using a
reduced covariance matrix that matches the combined volume of all
realizations. For the QSO, due to the lower tracer number density and shot
noise, we test slightly larger smoothing scales of $20\,h^{-1}\mathrm{Mpc}$
and $30\,h^{-1}\mathrm{Mpc}$, while for BGS, LRG1, LRG2, and ELG2 we test
$\Sigma_{\text{sm}}=10\,h^{-1}\mathrm{Mpc}$ and $15\,h^{-1}\mathrm{Mpc}$. In
terms of the reduction of errors for the scaling parameters, we find a
relatively good consistency between the different choices of scale for most
tracers. For $\alpha_{\text{iso}}$, the reduction in errors from the first
choice of smoothing scale ($30\,h^{-1}\mathrm{Mpc}$ for QSO and
$15\,h^{-1}\mathrm{Mpc}$ for the rest) is largest for BGS, with a 44%. The
smallest improvement is of 11%, for QSO. For LRG1, LRG2, and LRG3, we get a
31%, 38%, and 31% error reduction, respectively, while for ELG1 and ELG2, we
get a 20% and 12% error reduction. The second choice of smoothing scale
($20\,h^{-1}\mathrm{Mpc}$ for QSO and $10\,h^{-1}\mathrm{Mpc}$ for the rest)
leads to an error reduction of 46% and 2.6% for BGS and QSO, respectively. For
LRG1, LRG2 and LRG3, we get a 41%, 38%, and 30% improvement, while for ELG1
and ELG2, we get a 14% and 9% error improvement. For $\alpha_{\text{AP}}$,
both smoothing scales lead to consistent reduction in errors for LRGs, around
46%, 42%, and 28% for LRG1, LRG2, and LRG3, respectively. For ELG2, we see a
larger improvement from $\Sigma_{\text{sm}}=15\,h^{-1}\mathrm{Mpc}$, which
reduces the errors by 11%, compared to the 6.3% reduction in error from
$\Sigma_{\text{sm}}=10\,h^{-1}\mathrm{Mpc}$. Overall, all tracers benefit from
reconstruction when looking at the average clustering from all mocks, and the
first choice of smoothing scale tends to lead to a better error reduction when
comparing the different tracers and scaling parameters.

Focusing on the last two columns of Table 4, we can quantify the bias in the
BAO fits with respect to the expectation of $\alpha_{\text{iso}}=1$ and
$\alpha_{\text{AP}}=1$. More formally, the bias is defined as
$(\alpha-1)//\sigma$, where $\sigma$ is the standard deviation of the
marginalized posterior quoted in the fourth and fifth columns. We remind the
reader that these offsets have been calculated by assuming a covariance matrix
associated with the combination of 25 AbacusSummit realizations, which results
in a volume much larger than the DESI DR1 sample. Overall, we do not find
evidence that the choice of smoothing scale produces statistically significant
differences in the recovered bias from the BAO fits. The largest biases are
found for LRG3, for which $\Sigma_{\text{sm}}=15\,h^{-1}\mathrm{Mpc}$ and
$\Sigma_{\text{sm}}=10\,h^{-1}\mathrm{Mpc}$ produce biases of $-2.34\sigma$
and $-2.28$. Similarly, for LRG2, these two scales produce shifts of
$-2.05\sigma$ and $-1.92\sigma$, respectively. Although none of the tracers
show statistically significant offsets with respect to the truth, it is
interesting that almost all of them appear to be biased low in
$\alpha_{\text{iso}}$, which is not the case for the pre-reconstruction fits.
Therefore, in what follows, we further explore possible systematic errors in
the modeling by looking at constraints derived from the EZmocks.

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Figure 6: BAO fits to synthetic mock galaxy catalogues generated with the EZmock algorithm. The data vector corresponds to the post-reconstruction correlation function multipoles, averaged over 1000 independent mock realizations. The left panels show results using a covariance matrix is representative of the DESI DR1 volume, while the right panel displays results using a reduced covariance associated to the suite of 1000 mocks. For tracers for which we only perform isotropic BAO fits (BGS, ELG1, QSO), we only display the $\alpha_{\text{iso}}$ marginalized posterior in the top subpanels.  Tracer | Recon | $\Sigma_{\rm sm}$ | $\alpha_{\rm iso}$ | $\alpha_{\rm AP}$ | $\Delta\alpha_{\rm iso}/\sigma$ | $\Delta\alpha_{\rm AP}/\sigma$
---|---|---|---|---|---|---
${\tt BGS}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $1.00027\pm 0.00063$ | — | $0.43$ | —
${\tt LRG1}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.99951\pm 0.00046$ | $0.99864\pm 0.00161$ | $-1.06$ | $-0.84$
${\tt LRG2}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.99889\pm 0.00034$ | $0.99923\pm 0.00120$ | $-3.23$ | $-0.64$
${\tt LRG3}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $0.99983\pm 0.00033$ | $0.99850\pm 0.00113$ | $-0.51$ | $-1.33$
${\tt ELG1}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $1.00045\pm 0.00062$ | — | $0.73$ | —
${\tt ELG2}$ | Post |  15 $h^{-1}{\rm Mpc}$ | $1.00023\pm 0.00053$ | $0.99723\pm 0.00173$ | $0.43$ | $-1.60$
${\tt QSO}$ | Post |  30 $h^{-1}{\rm Mpc}$ | $0.99802\pm 0.00054$ | — | $-3.64$ | —
${\tt BGS}$ | Pre | — | $1.00081\pm 0.00108$ | — | $0.75$ | —
${\tt LRG1}$ | Pre | — | $1.00180\pm 0.00078$ | $0.99807\pm 0.00298$ | $2.30$ | $-0.65$
${\tt LRG2}$ | Pre | — | $1.00265\pm 0.00057$ | $1.00159\pm 0.00216$ | $4.63$ | $0.74$
${\tt LRG3}$ | Pre | — | $1.00289\pm 0.00046$ | $1.00012\pm 0.00173$ | $6.29$ | $0.07$
${\tt ELG1}$ | Pre | — | $1.00093\pm 0.00086$ | — | $1.08$ | —
${\tt ELG2}$ | Pre | — | $1.00055\pm 0.00057$ | $1.00309\pm 0.00199$ | $0.97$ | $1.55$
${\tt QSO}$ | Pre | — | $1.00143\pm 0.00064$ | — | $2.25$ | —
Table 5: Mean values and standard deviations from the marginalized posteriors
of the BAO scaling parameters from a fit to the mean of 1000 realizations of
the EZmocks. We display pre-reconstruction results, as well as post-
reconstruction fits obtained with a smoothing scale
$\Sigma_{\text{sm}}=15\,h^{-1}\mathrm{Mpc}$. The last two columns show the
offset in the mean values with respect to the expected value of
$\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$, in units of the corresponding
standard deviation, i.e. $\Delta\alpha=(\alpha-1)/\sigma$. We use a reduced
covariance matrix representative of the combined volume of all EZmocks.

In Fig. 6 we present fits to the post-reconstruction data vectors averaged
over 1000 realizations of the EZmocks, using a covariance matrix that is
representative of the volume of the DESI DR1 samples (left) or a reduced
covariance associated to the suite of 1000 mocks. The constraints for the
latter case are summarized in Table 5, where we have also included the pre-
reconstruction results. Given the computational restriction of reconstructing
this large number of mocks, we do it only for a single smoothing scale for
each tracer, as detailed in the table.

As the left panel of Fig. 6 shows, for a DESI DR1 volume, we obtain parameter
constraints that are largely unbiased and consistent across the various
redshift ranges. The relative constraining power is driven by the effective
volume of each sample, with the highest post-reconstruction precision coming
from the LRG3 and the lowest from the BGS and ELG1. The improvement in errors
compared to pre-reconstruction is consistent with what was observed for
AbacusSummit, with improvements on $\alpha_{\text{iso}}$ ranging from a 7% for
ELG2, to a 41% for LRG1. Similarly, for $\alpha_{\text{AP}}$, the improvement
ranges from 13% for ELG2 to 44% for LRG2.

Focusing on the constraints from Table 5, which are also illustrated in the
right panel of Fig. 6, we see that dividing the covariance matrix by a factor
of 1000, we find two statistically significant biases in the mean values of
the $\alpha_{\text{iso}}$ posteriors: a $-3.23\sigma$ bias for LRG2, and a
$-3.64\sigma$ bias for QSO. Interestingly, when looking at the collection of
all tracers, the post-reconstruction biases are not as coherent as seen in
AbacusSummit, as they can be biased high (BGS, ELG1, ELG2) or low (LRGs and
QSO). In terms of percentage, these offsets are much milder than in
AbacusSummit. For example, the $-3.24\sigma$ bias in the EZmocks LRG2
corresponds to $0.11$%, while in AbacusSummit, the $-2.59\sigma$ bias is of
$0.52$%. Furthermore, for the same tracers and redshift bins, the offsets can
point in different directions, indicating that some fraction of these biases
could be attributed to noise fluctuations given the small number of
AbacusSummit realizations to perform this test. However, we cannot fully
discard the possibility that there are systematic effects in the mocks that
are currently unaccounted for. In the next section, we assess the level at
which fiber assignment incompleteness affects the BAO constraints, showing
that no statistically significant detections of systematics are found due to
this effect.

Overall, given the small sensitivity of the BAO constraints to the smoothing
scales we have tested here as seen in AbacusSummit, and considering the
robustness of the BAO constraints when validated against the EZmocks, we opt
for $\Sigma_{\rm sm}=15\,h^{-1}\mathrm{Mpc}$ for BGS, LRG, and ELG, and
$\Sigma_{\rm sm}=30\,h^{-1}\mathrm{Mpc}$ for QSO. In later sections, we show
that the blinded DESI data are also insensitive to this choice. Furthermore,
these tests not only inform the decision about the smoothing scale, but also
indirectly and approximately reveal the size of the systematic error budget in
the BAO modeling that can be inferred from these mocks. A detailed description
of the different components of the systematic error budget and how this is
added to the statistical error of the unblinded galaxy BAO analysis is
presented in [36].

###  4.3 Impact of the fiber assignment

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Figure 7: Difference in the mean value of the marginalized posterior of the isotropic BAO scaling parameter $\alpha_{\text{iso}}$ between the complete (no fiber assignment incompleteness) and altmtl (realistic fiber assignment incompleteness) mocks, derived from fits to inidividual realizations of the AbacusSummit mocks. The legend shows the mean and dispersion of this difference estimated from all realizations, as well as the standard error on the mean. |  |
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Figure 8: Similar to Fig. 7, but showing results for the anisotropic BAO scaling parameter, $\alpha_{\text{AP}}$. Tracer | Recon | $\langle\alpha^{\rm complete}_{\rm iso}\rangle$ | $\langle\alpha^{\rm complete}_{\rm AP}\rangle$ | $\langle\alpha^{\rm complete}_{\rm iso}-\alpha^{\rm altmtl}_{\rm iso}\rangle$ | $\langle\alpha^{\rm complete}_{\rm AP}-\alpha^{\rm altmtl}_{\rm AP}\rangle$
---|---|---|---|---|---
${\tt BGS}$ | Post | $0.9998\pm 0.0038$ | — | $0.00288\pm 0.00229$ | —
${\tt LRG1}$ | Post | $0.9996\pm 0.0025$ | $0.9934\pm 0.0091$ | $-0.00182\pm 0.00163$ | $0.00374\pm 0.00455$
${\tt LRG2}$ | Post | $0.9967\pm 0.0022$ | $1.0091\pm 0.0075$ | $0.00207\pm 0.00144$ | $0.00096\pm 0.00392$
${\tt LRG3}$ | Post | $0.9958\pm 0.0019$ | $0.9854\pm 0.0063$ | $0.00240\pm 0.00161$ | $-0.00407\pm 0.00512$
${\tt ELG1}$ | Post | $1.0015\pm 0.0035$ | — | $0.00118\pm 0.00422$ | —
${\tt ELG2}$ | Post | $0.9988\pm 0.0030$ | $1.0025\pm 0.0100$ | $0.00387\pm 0.00270$ | $0.00387\pm 0.00676$
${\tt QSO}$ | Post | $0.9968\pm 0.0033$ | — | $0.00207\pm 0.00361$ | —
Table 6: Constraints on the BAO scaling parameters averaged over individual
fits to post-reconstruction correlation functions from 25 realizations of the
complete AbacusSummit mocks, which do not include any fiber assignment
incompleteness. We also show the mean of the differences in the $\alpha$
values between the complete mocks and the altmtl mocks, which mimic the fiber
assignment incompleteness of the DESI target samples.

The assignment of fibers that are used to measure the redshifts of DESI
tracers plays an important role in their clustering properties. To capture
these properties in the mock catalogs, the DESI FiberAssign code mimics the
effect of assignment incompleteness by trying to reproduce the DESI fiber
assignment from real targets. Here, we distinguish between two flavors of
mocks to study the impact of this effect on the BAO constraints after
reconstruction: ‘complete’ mocks which do not include any assignment
incompleteness, and thus can be treated as a baseline for the assessment of
the incompleteness, and ‘altmtl’ mocks, which represent our most realistic
simulations of the DR1 fiber assignment scheme. They apply the algorithm
described in [88], using the FiberAssign code perfectly reproduce the
assignment of DESI fibers on real targets, without approximations. Our results
from AbacusSummit in previous section were derived from the altmtl mocks.

In Table 6, we show the constraints on $\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$ derived from the complete mocks. These constraints have
been obtained by averaging individual fits to the post-correlation functions
from all 25 mock realizations. We note that even in the complete case, there
are deviations from $\alpha_{\text{iso}}=1$ for all tracers, which, although
not statistically significant, point in the same direction as the altmtl mocks
that were used to derive the constraints from the previous section. This
suggests that the fiber assignment process is not the driver of the biases
seen in the AbacusSummit fits, and reinforces the idea that they could be
partially attributed to statistical fluctuations given the small number of
mock realizations.

The fifth and sixth columns show the mean of the differences in
$\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$, respectively, between the
complete and altmtl mocks, as averaged over all mock realizations. The
distribution of individual fits are also shown in Figs. 7 and 8. Overall, we
do not observe any statistically significant shift in the constraints. For
$\alpha_{\text{iso}}$, the largest offset is of $1.49\sigma$ for LRG3. For
$\alpha_{\text{AP}}$, it is for LRG1 at $0.82\sigma$.

---
Figure 9: A comparison of the constraints on the isotropic BAO scaling
parameter $\alpha_{\text{iso}}$ from fits to the mean of 25 AbacusSummit mock
realizations, after having passed the mocks through a code that realistically
mimics the DESI DR1 fiber assignment (altmtl) or using mocks that do not
include any fiber assignment incompleteness (complete). We have used a reduced
covariance matrix associated to the combined volume of all 25 mock
realizations.

In Fig. 9, we show a fit to the mean of 25 mocks for complete and altmtl
mocks, where we have also included pre-reconstruction results. For all these
fits, we use a reduced covariance matrix that is representative of the
combined volume of all mock realizations. For pre-reconstruction, we find that
the inclusion of fiber assignment incompleteness makes little effect on the
constraints for most tracers. Overall, we do not find statistically
significant features that could suggest that the fiber assignment scheme
affects pre- and post-reconstruction measurements differently. In general, our
findings corroborate that the effect of fiber assignment incompleteness on BAO
measurements is relatively mild, since we restrict the analysis to fairly
large scales. Moreover, we are isolating the BAO feature in the modeling, and
potential large-scale systematics are partially absorbed by the broadband
nuisance parameters. However, we need to keep in mind that the small number of
mock realizations available restricts the precision at which we can perform
this test. Although the offsets we observed in this and previous sections are
much smaller than the statistical error from the DESI-DR1 samples, subsequent
analyses with more data will demand stringent tests based on a larger
collection of mock catalogs to match the increase level of precision in the
BAO constraints.

###  4.4 Consistency between the DESI data and mocks

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Figure 10: Measured multipoles from the post-reconstruction correlation
functions of the blinded DESI DR1 data. Filled and open circles show the
monopole and quadrupole moments, respectively, with their corresponding best-
fit models shown by the solid and dashed lines, respectively. For BGS, ELG1
and QSO, we only fit the monopole, due to the lower SNR of the data. The lower
sub-panels show the residuals between the best-fit models and the data in each
case.

Having explored the BAO constraints from the mocks, the main question we wish
to answer in this section is are the BAO constraints obtained from the blinded
DESI data consistent with the results obtained from the distribution of mocks?
The tests presented here constitute some of the criteria that the DESI
Collaboration used to decide whether we were ready to fix the end-to-end
analysis pipeline, unblind the LSS catalogs and go ahead with the main BAO
cosmological analysis.

In Fig. 10, we display the post-reconstruction correlation function multipoles
of the blinded DESI DR1 data, where we have adopted the fiducial smoothing
scales determined in the previous section. For the cases where we only perform
1D BAO fits (BGS, ELG1 and QSO), only the monopole moment is shown. We find
that the BAO model accurately fits the data for all scales considered in the
fit, with residuals between the model and the data that are within $2\sigma$.
In addition to potential systematic errors due to non-linear evolution/biasing
and large-scale systematics, the observed position of the BAO feature can be
affected by Alcock-Paczynski distortions arising due to a potential mismatch
between our fiducial cosmology and the true cosmology of the Universe, as well
as due to the blinding scheme that can change the position of the BAO feature
and the RSD signal. The error bars come from the semi-analytic semi-empirical
RascalC covariances that were tuned to match the clustering of the DESI DR1
samples. These constraints are summarized in Table 7, along with the pre-
reconstruction constraints from the same data.

Tracer | Redshift | Recon | $\alpha_{\rm iso}$ | $\alpha_{\rm AP}$ | $r$ | $\chi^{2}/{\rm dof}$
---|---|---|---|---|---|---
${\tt BGS}$ | 0.1–0.4 | Post | $0.98416\pm 0.01601$ | — | — | 24.5/19
${\tt LRG1}$ | 0.4–0.6 | Post | $0.98416\pm 0.00988$ | $0.99679\pm 0.03201$ | -0.12 | 40.1/39
${\tt LRG2}$ | 0.6–0.8 | Post | $0.98416\pm 0.01169$ | $0.99679\pm 0.04577$ | -0.09 | 40.2/39
${\tt LRG3}$ | 0.8–1.1 | Post | $0.98416\pm 0.00891$ | $0.99679\pm 0.03279$ | -0.03 | 35.0/39
${\tt ELG1}$ | 0.8–1.1 | Post | $0.98416\pm 0.02450$ | — | — | 23.4/19
${\tt ELG2}$ | 1.1–1.6 | Post | $0.98416\pm 0.01376$ | $0.99679\pm 0.04727$ | -0.31 | 45.7/39
${\tt QSO}$ | 0.8–2.1 | Post | $0.98416\pm 0.02509$ | — | — | 16.9/19
${\tt BGS}$ | 0.1–0.4 | Pre | $0.98416\pm 0.02650$ | — | — | 13.9/19
${\tt LRG1}$ | 0.4–0.6 | Pre | $0.98416\pm 0.01877$ | $0.99679\pm 0.06980$ | 0.44 | 32.6/39
${\tt LRG2}$ | 0.6–0.8 | Pre | $0.98416\pm 0.01869$ | $0.99679\pm 0.07317$ | 0.46 | 51.0/39
${\tt LRG3}$ | 0.8–1.1 | Pre | $0.98416\pm 0.01162$ | $0.99679\pm 0.04469$ | 0.15 | 42.9/39
${\tt ELG1}$ | 0.8–1.1 | Pre | $0.98416\pm 0.05708$ | — | — | 23.6/19
${\tt ELG2}$ | 1.1–1.6 | Pre | $0.98416\pm 0.01813$ | $0.99679\pm 0.06523$ | -0.06 | 34.0/39
${\tt QSO}$ | 0.8–2.1 | Pre | $0.98416\pm 0.02304$ | — | — | 9.7/19
Table 7: Mean values and standard deviations of the marginalized posteriors on the BAO scaling parameters, from fits to the blinded DESI DR1 correlation functions. We display pre- and post-reconstruction constraints, where the reconstruction adopts the fiducial settings determined in Section 4.2. We additionally show the cross-correlation coefficient between $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$, labeled as $r$, and the $\chi^{2}$ per degree of freedom at the best fit. Figure 11: BAO detection level signicance from fits to the correlation functions from the blinded DESI DR1 data. The solid and dashed lines in different colors show the $\Delta\chi^{2}$ for models with and without BAO wiggles, respectively. We display horizontal dotted lines at various levels of detection significance for reference. |  |
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Figure 12: A comparison of the BAO detection level from the blinded DESI DR1
data (vertical dashed line) and individual realizations of the AbacusSummit
simulations (colored histograms).

To assess the statistical significance of the BAO detection from the blinded
DR1 data, we run $\alpha_{\text{iso}}$ fits and compare the difference in
$\chi^{2}$ at the best fit using models with and without BAO wiggles, where
the square root of the $\Delta\chi^{2}$ is quoted as the detection
significance. We display the results in Fig. 11 for all the different tracers.
We obtain the strongest detection level from LRG3 at $8.4\sigma$, while the
weakest detection occurs for ELG1 at $2.1\sigma$. To verify that these
detection levels are consistent with the mocks, in Fig. 12 show the
distribution of detection significance values from fits to individual
realizations of the AbacusSummitmocks. Overall, we observe great consistency
between the mocks and the data for all tracers except ELG2, for which the
blinded DR1 data gives a higher detection level than all mock realizations.

Figure 13: BAO fits of the anisotropic correlation function of the
AbacusSummit mocks and the blinded DESI DR1 for LRG3. Open and filled markers
show pre- and post-reconstruction fits, respectively. The first and second
panels show constraints on $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$,
respectively, fitting the mean of the 25 mocks (first column, where we have
scaled the errors to match that of a single DESI DR1 volume), individual mock
realizations (consecutive 25 columns), and the blinded DESI DR1 data (last
column). The third and fourth panels show the errors of these fits (equivalent
to the size of the error bars from the first two panels). The bottom panel
shows the $\chi^{2}$ value for each fit, where the number of degrees of
freedom is specified in the label as a reference. The shaded red region in
each panel shows the scatter from the 25 mock fits post-reconstruction. Figure
14: Similar to Fig. 24, but showing fits to BGS. Similar to the QSO and ELG1,
we only perform an isotropic BAO fit for this sample, where
$\alpha_{\text{iso}}$ is the only scaling parameter that is varied.

In Fig. 13, we compare the constraints from the blinded DESI LRG3 (black
stars) with the AbacusSummitsimulations (colored circles), where we
distinguish between pre- and post-reconstruction constraints by the empty and
solid markers, respectively. We find excellent agreement between the average
post-reconstruction error on $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$
(red bands) from AbacusSummit, and the blinded DESI data. The average
$\chi^{2}$ value per degree of freedom from the mocks also closely follows
that of the data, from which we get $\chi^{2}=35$ for 39 degrees of freedom,
corresponding to a p-value of 34.7%. Results for BGS are displayed in Fig. 14,
which shows that the $\alpha_{\text{iso}}$ errros and the $\chi^{2}$ at the
best fit from the blinded data are consistent with the mocks at the $1-\sigma$
level. The $\chi^{2}$ is 24.5 for 19 degrees of freedom, corresponding to a
p-value of 17.8%. We show equivalent figures for the remaining tracers, where
we find similar results in most cases. A noteworthy case are the QSO, for
which the blinded DESI DR1 error on $\alpha_{\text{iso}}$ is worse than all
mocks after reconstruction, although they are still consistent to within
$2\sigma$. For LRG1, LRG3, and ELG2, the constraints $\alpha_{\text{AP}}$ are
not displayed for a few or the AbacusSummit mock realizations for which noise
fluctuations shifted the quadrupole BAO features to scales that are not
supported by the data, resulting in fits that hit the prior limit on
$\alpha_{\text{AP}}$. In these cases, we have instead resorted to 1D BAO fits,
thus only providing constraints for $\alpha_{\text{iso}}$.

###  4.5 Robustness of the blinded DESI constraints

Figure 15: Response of the constraints on the isotropic BAO scaling parameter,
$\alpha_{\text{iso}}$, to changes in the data or theory vector, priors, or
parametrization. All constraints come from fits to the blinded DESI-DR1 data.
The baseline configuration that is used for the main BAO analysis is shown at
the top, while the other rows single-variations around the baseline (see main
text for a description of each variation). The circles show the best-fit
values and the error bars correspond to 68% confidence levels. Figure 16: Same
as Fig. 15, but showing constraints for the anisotropic BAO scaling parameter,
$\alpha_{\text{AP}}$.

Figs. 15 and 16 show how the constraints on $\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$ from the blinded DESI 2024 data respond to different
variations in the analysis configuration. To speed-up calculations for this
test, we have used posterior maximization with Minuit rather than MCMC chains,
but we expect the two methods to give comparable results under the assumption
of a Gaussian likelihood. For the baseline case, we fit the post-
reconstruction correlation function multipoles. For LRGs and ELG2, we
simultaneously vary $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$ and fit the
monopole and quadrupole moments. For the remaining samples, we only vary
$\alpha_{\text{iso}}$ and exclusively fit the monopole. The BAO damping
parameters have Gaussian priors with means and widths as detailed in Table 2.
The broadband of the correlation function is parameterized with a piecewise
cubic spline basis (Section 3.2).

The second row shows the pre-reconstruction constraints. Reconstruction
improves the precision on $\alpha_{\text{iso}}$ by 47%, 37% and 23% for LRG1,
LRG2, and LRG3, respectively, by 24% for ELG2, and by 40% for BGS. For ELG1,
the pre-reconstruction errors appear smaller than the chain values reported in
Table 7. We have verified that this is due to a noise fluctuation that causes
the pre-reconstruction marginalized posterior on $\alpha_{\text{iso}}$ to
appear highly non-Gaussian, making the standard deviation from the chains an
unsuitable measure of the error. Measuring the region at which
$\Delta\chi^{2}=1$, we obtain $\alpha_{\text{iso}}=0.987\pm 0.031$, while
reconstruction improves the error by 29%. For QSO, reconstruction ends up
slightly degrading the precision on $\alpha_{\text{iso}}$, by 8.9%. The tests
on the mocks from Fig. 21 in Appendix A show that even though reconstruction
tends to help improve the errors on the scaling parameters for QSO-like
samples on average, there are a few mock realizations for which the errors get
worse, which is consistent with what is found for the DESI blinded data. This
is also in line with previous analyses by eBOSS, which found that
reconstruction did not improve the BAO constraints from their QSO sample due
to the high shot noise contamination [89]. For $\alpha_{\text{AP}}$, the
errors after reconstruction improve by 54%, 37%, 27%, and 28% for LRG1, LRG2,
LRG3, and ELG2, respectively.

The Fourier-space constraints from the power spectrum multipoles, which are
displayed on the third row, show a good agreement with the configuration-space
results, with comparable error bars and offsets that are never larger than
$1\sigma$. In the fourth row, we display the constraints obtained by only
fitting data from the northern galactic cap (NGC), in contrast with the
baseline scenario where the northern and southern galactic caps are combined
at the clustering level. In most cases, we only see a small degradation in the
parameter constraints. We note that for dark-time tracers, the DR1 NGC has a
factor of $\sim 1.74$ larger sky area than the SGC, while for bright-time
tracers, the ratio is $\sim 2.38$ [48].

The fifth row shows the resulting parameter constraints when we adopt uniform
priors $\mathcal{U}(0,10)$ for the BAO damping parameters. In general, we see
that the results are almost equivalent to the baseline case, where we adopt
Gaussian priors. Tests on mock galaxy catalogs by [12] have shown that using
non-informative priors on the damping parameters for data with a low SNR can
potentially lead to weaker constraints and even potential biases. Indeed, as
we have previously discussed, some tracers and redshift bins, such as the
ELG1, do not have a sufficiently large SNR to reliably perform an anisotropic
BAO fit, and we have explicitly checked that when doing so, the priors on the
damping parameters have a strong impact on the recovered best-fit values and
precision of the BAO scaling parameters. However, these low-SNR cases are not
included in the main DESI BAO analysis, and the constraints from the remaining
target samples are robust against this choice.

Previous BAO analyses from BOSS [6] and eBOSS [7] parameterized the broadband
component of the multipole correlation function / power spectrum using
polynomials of slightly varying degrees and functional forms, with
coefficients that vary freely during the fit. In this work, we have adopted
the new spline-basis parameterization for the broadband proposed by [12],
which gives constraints that are in excellent agreement with those obtained
with the (e)BOSS polynomial parameterization, shown in the fifth row of Figs.
15 and 16.

One of the main differences with respect to previous applications of
reconstruction in SDSS in the use of the RecSym convention. As discussed in
Section 2.2, RecSym shifts galaxies and randoms in the same way, using the
full displacement vector that includes the $(1+f)$ factor along the line of
sight, which preserves large-scale RSD in the post-reconstruction clustering.
The convention used in (e)BOSS was RecIso, where the $(1+f)$ factor is not
applied when displacing the randoms, which has the effect of erasing RSD post-
reconstruction. This is exemplified in Fig. 17, where we see how the
quadrupole moment of the LRG2 is strongly suppressed in RecIso. The right-hand
side panel shows how the distribution of displacements along the line of sight
is larger for RecSym due to the additional $(1+f)$ factor. Despite this
important difference, it can be seen that the BAO features in the correlation
function are largely preserved. The last row in Figs. 15 and 16 show the
constraints derived using the RecIso convention. Since the spline-basis
broadband parametrization has only been validated for RecSym, we run this test
using the BOSS polynomial parameterization instead, which was shown to give
consistent results with the spline method for RecSym. The BAO scaling
parameters are largely unaffected by the reconstruction mode, with the largest
shift being less than $0.09\sigma$ for the QSO.

|
---|---
Figure 17: Density-field reconstruction applied to the blinded DESI LRG2 data
using the RecSym or RecIso conventions. In the left, we show the monopole and
quadrupole moments of the correlation function, and in the right, we show the
distribution of displacement vectors for the random catalogue along the radial
direction, given by the difference in the redshift of the random particles
before and after reconstruction.

Finally, in the last rows we include the alternative choice of smoothing scale
that was tested in Section 4.2. Similar to what was found with the mocks, the
resulting BAO constraints are only mildly affected by this choice, with
changes in the recovered best-fit values on $\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$ that are smaller than $0.2\sigma$. We note, however, than
this is due to the fact that in this paper we have narrowed down our options
of smoothing scales to only two choices per tracer with values that are
relatively close to each to other, based on more extensive tests that are
presented in [59]. More extreme values of the smoothing scale are expected to
have stronger impacts on the BAO constraints, as has been also shown in
previous works [32, 33, 23, 34]

###  4.6 Gaussianity of the BAO posterior

|
---|---
|
Figure 18: A comparison of the BAO posterior form the blinded LRGs and ELG2 (filled contours in color), and a multivariate Gaussian distribution with the same mean and covariance (black contours in dashed line style). We sample the parameter space in the $\alpha_{\text{iso}}$-$\alpha_{\text{AP}}$ basis, and report $\alpha_{\parallel}$ and $\alpha_{\perp}$ as derived parameters. |  |
---|---|---
Figure 19: Similar to Fig. 18, but showing results for the 1D BAO fits on BGS,
ELG1, and QSO.

As a final sanity check, we assess whether the posterior distribution of BAO
parameters from the blinded DESI data follows a Gaussian distribution, which
would guarantee that the mean and covariance of the posterior capture all the
information from the fit.

In Fig. 18, we show the posterior of the BAO scaling parameters from fits to
the post-reconstruction correlation functions of LRGs and ELG2, comparing them
with a multivariate Gaussian distribution with the same mean and covariance.
The contours, which correspond to 1-$\sigma$, 2-$\sigma$, and 3-$\sigma$
levels, show an excellent consistency, indicating that the BAO posterior
closely follows the Gaussian approximation. For LRGs, we notice that there is
almost no correlation between $\alpha_{\text{iso}}$ and $\alpha_{\text{AP}}$,
which is also indicated by the small cross-correlation coefficients seen in
Table 7, which are -0.12, -0.09, and -0.03 for LRG1, LRG2, and LRG3,
respectively. For ELG2, we observe a higher correlation coefficient of -0.31.
We have also derived constraints for $\alpha_{\parallel}$ and
$\alpha_{\perp}$, following Eqs. 3.9 and 3.10. We find that the constraints
are much more correlated on this basis, with a correlation coefficient between
$\alpha_{\parallel}$ and $\alpha_{\perp}$ of -0.45 for all tracers. However,
the posterior appears equally Gaussian under both parameterizations.

For completeness, we also show the equivalent figures for the 1D BAO fits
performed on BGS, ELG1 and QSO in Fig. 19, which shows that the
$\alpha_{\text{iso}}$ posteriors are also consistent with a Gaussian
distribution in all cases.

Although we have performed some standardized tests of multivariate normality,
such as the Shapiro-Wilk test [90], we found the results to be very sensitive
to the number of chain points in the fits. In fact, in the limit of infinite
chain points, any test of normality will reject the null hypothesis, except
for a problem that is truly Gaussian by construction. Although the assumption
of a Gaussian likelihood has been the standard for the cosmological
interpretation of the BAO constraints [6, 7], the ever-increasing level of
precision from spectroscopic galaxy surveys will require more quantitative
assessments of the impact of this assumption at the cosmological parameter
level. We plan to explore this further in future work.

##  5 Summary and Conclusions

We have studied the impact of the density-field reconstruction technique on
the DESI DR1 galaxy BAO analysis. Reconstruction is a method that aims at
improving the signal-to-noise ratio of the BAO signal from galaxy clustering
data, improving its detection significance, and therefore increasing the
precision of cosmological constraints derived from BAO. In addition,
reconstruction helps to remove systematic shifts in the observed location of
the BAO feature that are caused by non-linear structure growth and biasing.

To assess the effectiveness of reconstruction in a controlled setting, we have
used two suites of mock galaxy catalogs that match the clustering and
selection properties of the DESI DR1 target samples: the AbacusSummit mocks,
which correspond to high-fidelity HOD catalogs generated from the suite of
N-body simulations presented in [40], and a collection of 1000 approximate
mocks generated with the EZmock algorithm [76]. We have also performed a
series of tests on the DESI DR1 blinded data, assessing the robustness of the
BAO constraints against different assumptions in our analysis configuration.
In this context, the blinding of the data refers to the deliberate
modification of the large-scale structure catalogs that are analyzed while the
BAO pipeline is being tested and calibrated, in order to avoid confirmation
bias in the cosmological analysis. A detailed description of the blinding
scheme in DESI can be found in [38].

Our main findings can be summarized as follows.

  * -

Except for the QSO sample, reconstruction ubiquitously helps to increase the
signal-to-noise ratio of the BAO feature for the blinded DESI data, increasing
the precision of the recovered on the isotropic BAO scaling parameter
$\alpha_{\text{iso}}$ from 23% to 47%, depending on the tracer and redshift
range. This increase in precision is consistent with results obtained from the
mock galaxy catalogs.

  * -

For QSO, while the lack of improvement from the reconstruction of the blinded
DESI data is not unusual due to the high shot noise contamination of this
sample, we find that in the majority of the cases, reconstruction increases
the recovered precision on $\alpha_{\text{iso}}$ from the QSO mocks. This
motivates the decision to adopt reconstruction as the default for all samples
of the DESI galaxy BAO analysis, including QSO.

  * -

We calibrated the optimal smoothing scale at which we smooth the density field
for reconstruction, in order to null out small-scale modes that are highly
non-linear. Among the options we tested, we find that the reconstruction of
the DESI DR1 samples is not particularly sensitive to the choice of smoothing
scale, and values close to those adopted by previous SDSS BAO analyses are
suitable in most cases. The QSO sample is an exception due to the higher shot
noise contribution, and larger smoothing scales are required to optimally
reconstruct the density field. We adopt
$\Sigma_{\text{sm}}=15\,h^{-1}\mathrm{Mpc}$ for BGS, LRGs, and ELGs, and
$\Sigma_{\text{sm}}=30\,h^{-1}\mathrm{Mpc}$ for the QSO, as our fiducial
smoothing scales for the main galaxy BAO analysis.

  * -

Using mocks that have been passed through a pipeline that mimics the fiber
assignment scheme of DESI DR1, we have tested the impact of the fiber
assignment incompleteness on the BAO constraints before and after
reconstruction. Overall, we do not find statistically significant signatures
from this effect on the recovered constraints on the BAO scaling parameters,
regardless of whether the galaxy catalogs are reconstructed or not.

  * -

We have performed a series of unblinding tests that the DESI Collaboration
used to assess whether we were ready to fix the galaxy BAO pipeline, unblind
the LSS catalogs, and perform the official DR1 BAO analysis. These tests
analyzed the consistency between the BAO statistics measured from the blinded
DESI data and the AbacusSummit mocks, as well as the robustness of the
constraints from the blinded data to different configurations in our pipeline.

  * -

The BAO constraints from the DESI blinded data are robust against a wide
variety of measurement and model choices, including the parameterization of
the broadband component of the power spectrum, the priors on the parameters
that capture the BAO non-linear damping, and different reconstruction
conventions. We also find excellent consistency between the constraints
derived from configuration space (the correlation function) and Fourier space
(the power spectrum).

  * -

After reconstruction, the posteriors of the BAO scaling parameters closely
follow a multivariate Gaussian distribution, regardless of whether the scaling
parameters are expressed in the basis that isolates the isotropic and
anisotropic scaling components ($\alpha_{\text{iso}}$ and
$\alpha_{\text{AP}}$), or the perpendicular and parallel to the line of sight
components ($\alpha_{\parallel}$ and $\alpha_{\perp}$).

In this work, we have focused on a single reconstruction algorithm that uses
an iterative Fast Fourier Transforms scheme to recover the real-space
displacement field, as introduced in [33, 30]. In [59], we present a thorough
comparison of different reconstruction algorithms in the context of BAO
analyses. Our companion papers also present detailed studies of various
potential sources of uncertainty in our BAO constraints, including: the choice
of fiducial cosmology [41], the galaxy-halo connection modeling [63, 62], the
calibration of the covariance matrix [72, 74], and the theory model [12]. The
unblinded galaxy BAO measurements from DESI are presented in [36], while the
Lyman-$\alpha$ forest BAO measurements are described in [49]. Both of these
sets are combined to infer cosmology in [37].

The tests presented in this work and in our companion papers are a further
confirmation that BAO remain as one of the most robust probes of large-scale
structure cosmology to date, even under the unprecedented statistical
precision of the DESI DR1 sample. However, future data releases from DESI, as
well as other on-going and upcoming galaxy surveys, such as Euclid [91] and
the Nancy Grace Roman Space Telescope [92], will require ever more stringent
tests and validation, which are likely to push the limits of our understanding
of cosmology and large-scale structure science.

##  6 Data Availability

The data will be publicly available once the paper is accepted for
publication.

## Acknowledgments

ZD acknowledges support from the National Key R&D Program of China
(2023YFA1607800, 2023YFA1607802), the National Science Foundation of China
(grant numbers 12273020 and 11621303), and the science research grant from the
China Manned Space Project with NO. CMS-CSST-2021-A03. H-JS acknowledges
support from the U.S. Department of Energy, Office of Science, Office of High
Energy Physics under grant No. DE-SC0023241. H-JS also acknowledges support
from Lawrence Berkeley National Laboratory and the Director, Office of
Science, Office of High Energy Physics of the U.S. Department of Energy under
Contract No. DE-AC02-05CH1123 during the sabbatical visit. SN acknowledges
support from an STFC Ernest Rutherford Fellowship, grant reference
ST/T005009/2.

This material is based upon work supported by the U.S. Department of Energy
(DOE), Office of Science, Office of High-Energy Physics, 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 was provided by the U.S. National Science
Foundation (NSF), Division of Astronomical Sciences under Contract No.
AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research
Laboratory; the Science and Technology 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 Humanities, Science and Technology of Mexico (CONAHCYT);
the Ministry of Science and Innovation of Spain (MICINN), and by the DESI
Member Institutions: https://www.desi.lbl.gov/collaborating-institutions. Any
opinions, findings, and conclusions or recommendations expressed in this
material are those of the author(s) and do not necessarily reflect the views
of the U. S. National Science Foundation, the U. S. Department of Energy, or
any of the listed funding agencies.

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.

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##  Appendix A Additional Figures

In this section, we show additional figures for some of the tests carried out
in Section 4, including the results for those tracers that we did not add to
the main body of the manuscript. These additional results are still discussed
in the main text, but we reserve the placement of the figures to this appendix
for formatting convenience. This includes the AbacusSummit
$\alpha_{\text{iso}}$ constraints for different reconstruction smoothing
scales for LRG1 and LRG3 (Fig. 20), and for BGS, ELG1, and QSO (Fig. 21). Fig.
22 shows the AbacusSummit constraints on $\alpha_{\text{AP}}$ for different
smoothing scales from fits to the LRG1 and LRG3 samples. Figs. 23, 24, 25, 26
and 27 show the summary of pre- and post-reconstruction constraints from the
AbacusSummit mocks and the DESI DR1 blinded data, for the LRG1, LRG2, ELGs,
and QSO samples.

---
Figure 20: Similar to Fig. 4, but showing results for LRG1 and LRG3.
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Figure 21: Similar to Fig. 4, but showing results for BGS, QSO, and ELG1.
---
Figure 22: Similar to Fig. 5, but showing results for LRG1 and LRG3. Figure
23: Similar to Fig. 13, but for LRG1. Figure 24: Same as Fig. 13, but for
LRG2. Figure 25: Similar to Fig. 14, but for ELG1. Figure 26: Similar to Fig.
13, but for ELG2. Figure 27: Similar to Fig. 14, but for QSO.