# DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon
Acoustic Oscillations

DESI Collaboration  A. G. Adame  J. Aguilar  S. Ahlen  0000-0001-6098-7247 S.
Alam  0000-0002-3757-6359 D. M. Alexander  0000-0002-5896-6313 M. Alvarez  O.
Alves  A. Anand  0000-0003-2923-1585 U. Andrade  0000-0002-4118-8236 E.
Armengaud  0000-0001-7600-5148 S. Avila  0000-0001-5043-3662 A. Aviles
0000-0001-5998-3986 H. Awan  0000-0003-2296-7717 B. Bahr-Kalus
0000-0002-4578-4019 S. Bailey  0000-0003-4162-6619 C. Baltay  A. Bault
0000-0002-9964-1005 J. Behera  S. BenZvi  0000-0001-5537-4710 A. Bera  F.
Beutler  0000-0003-0467-5438 D. Bianchi  0000-0001-9712-0006 C. Blake
0000-0002-5423-5919 R. Blum  0000-0002-8622-4237 S. Brieden
0000-0003-3896-9215 A. Brodzeller  0000-0002-8934-0954 D. Brooks  E. Buckley-
Geer  E. Burtin  R. Calderon  0000-0002-8215-7292  R. Canning  A. Carnero
Rosell  0000-0003-3044-5150 R. Cereskaite  J. L. Cervantes-Cota
0000-0002-3057-6786 S. Chabanier  0000-0002-5692-5243 E. Chaussidon
0000-0001-8996-4874 J. Chaves-Montero  0000-0002-9553-4261 S. Chen
0000-0002-5762-6405 X. Chen  T. Claybaugh  S. Cole  0000-0002-5954-7903 A.
Cuceu  0000-0002-2169-0595 T. M. Davis  0000-0002-4213-8783 K. Dawson  A. de
la Macorra  0000-0002-1769-1640 A. de Mattia  N. Deiosso  0000-0002-7311-4506
A. Dey  0000-0002-4928-4003 B. Dey  0000-0002-5665-7912 Z. Ding
0000-0002-3369-3718 P. Doel  J. Edelstein  S. Eftekharzadeh  D. J. Eisenstein
A. Elliott  0000-0001-6537-6453 P. Fagrelius  K. Fanning  0000-0003-2371-3356
S. Ferraro  0000-0003-4992-7854 J. Ereza  0000-0002-0194-4017 N. Findlay
0009-0007-0716-3477 B. Flaugher  A. Font-Ribera  0000-0002-3033-7312 D.
Forero-Sánchez  0000-0001-5957-332X J. E. Forero-Romero  0000-0002-2890-3725
C. S. Frenk  0000-0002-2338-716X C. Garcia-Quintero  0000-0003-1481-4294 E.
Gaztañaga  H. Gil-Marín  0000-0003-0265-6217 S. Gontcho A Gontcho
0000-0003-3142-233X A. X. Gonzalez-Morales  0000-0003-4089-6924 V. Gonzalez-
Perez  0000-0001-9938-2755 C. Gordon  D. Green  0000-0002-0676-3661 D. Gruen
R. Gsponer  0000-0002-7540-7601 G. Gutierrez  J. Guy  B. Hadzhiyska
0000-0002-2312-3121 C. Hahn  0000-0003-1197-0902 M. M. S Hanif
0009-0006-2583-5006 H. K. Herrera-Alcantar  0000-0002-9136-9609 K. Honscheid
C. Howlett  0000-0002-1081-9410 D. Huterer  0000-0001-6558-0112 V. Iršič
0000-0002-5445-461X M. Ishak  0000-0002-6024-466X S. Juneau  N. G. Karaçaylı
0000-0001-7336-8912 R. Kehoe  S. Kent  0000-0003-4207-7420 D. Kirkby
0000-0002-8828-5463 A. Kremin  0000-0001-6356-7424 A. Krolewski  Y. Lai  T.-W.
Lan  0000-0001-8857-7020 M. Landriau  0000-0003-1838-8528 D. Lang  J. Lasker
0000-0003-2999-4873 J.M. Le Goff  L. Le Guillou  0000-0001-7178-8868 A.
Leauthaud  0000-0002-3677-3617 M. E. Levi  0000-0003-1887-1018 T. S. Li
0000-0002-9110-6163 E. Linder  0000-0001-5536-9241 K. Lodha
0009-0004-2558-5655 C. Magneville  M. Manera  0000-0003-4962-8934 D. Margala
0009-0001-5897-1956 P. Martini  0000-0002-4279-4182 M. Maus  P. McDonald
0000-0001-8346-8394 L. Medina-Varela  A. Meisner  0000-0002-1125-7384 J. Mena-
Fernández  0000-0001-9497-7266 R. Miquel  J. Moon  S. Moore  J. Moustakas
0000-0002-2733-4559 E. Mueller  A. Muñoz-Gutiérrez  A. D. Myers  S. Nadathur
0000-0001-9070-3102 L. Napolitano  0000-0002-5166-8671 R. Neveux  J. A. Newman
0000-0001-8684-2222 N. M. Nguyen  0000-0002-2542-7233 J. Nie
0000-0001-6590-8122 G. Niz  0000-0002-1544-8946 H. E. Noriega
0000-0002-3397-3998 N. Padmanabhan  E. Paillas  0000-0002-4637-2868 N.
Palanque-Delabrouille  0000-0003-3188-784X J. Pan  0000-0001-9685-5756 S.
Penmetsa  W. J. Percival  0000-0002-0644-5727 M. M. Pieri  M. Pinon  C.
Poppett  A. Porredon  0000-0002-2762-2024 F. Prada  0000-0001-7145-8674 A.
Pérez-Fernández  0009-0006-1331-4035 I. Pérez-Ràfols  0000-0001-6979-0125 D.
Rabinowitz  A. Raichoor  0000-0001-5999-7923 C. Ramírez-Pérez  S. Ramirez-
Solano  M. Rashkovetskyi  0000-0001-7144-2349 C. Ravoux  0000-0002-3500-6635
M. Rezaie  0000-0001-5589-7116 J. Rich  A. Rocher  0000-0003-4349-6424 C.
Rockosi  0000-0002-6667-7028 N.A. Roe  A. Rosado-Marin  A. J. Ross
0000-0002-7522-9083 G. Rossi  R. Ruggeri  0000-0002-0394-0896 V. Ruhlmann-
Kleider  0009-0000-6063-6121 L. Samushia  0000-0002-1609-5687 E. Sanchez
0000-0002-9646-8198 C. Saulder  0000-0002-0408-5633 E. F. Schlafly
0000-0002-3569-7421 D. Schlegel  M. Schubnell  H. Seo  0000-0002-6588-3508 A.
Shafieloo  0000-0001-6815-0337 R. Sharples  0000-0003-3449-8583 J. Silber
0000-0002-3461-0320 A. Slosar  A. Smith  0000-0002-3712-6892 D. Sprayberry  T.
Tan  0000-0001-8289-1481 G. Tarlé  0000-0003-1704-0781 P. Taylor  S. Trusov
L. A. Ureña-López  0000-0001-9752-2830 R. Vaisakh  0009-0001-2732-8431 D.
Valcin  0000-0003-0129-0620 F. Valdes  0000-0001-5567-1301 M. Vargas-Magaña
0000-0003-3841-1836 L. Verde  0000-0003-2601-8770 M. Walther
0000-0002-1748-3745 B. Wang  0000-0003-4877-1659 M. S. Wang
0000-0002-2652-4043 B. A. Weaver  N. Weaverdyck  0000-0001-9382-5199 R. H.
Wechsler  0000-0003-2229-011X D. H. Weinberg  0000-0001-7775-7261 M. White
0000-0001-9912-5070 J. Yu  Y. Yu  0000-0002-9359-7170 S. Yuan
0000-0002-5992-7586 C. Yèche  0000-0001-5146-8533 E. A. Zaborowski
0000-0002-6779-4277 P. Zarrouk  0000-0002-7305-9578 H. Zhang
0000-0001-6847-5254 C. Zhao  0000-0002-1991-7295 R. Zhao  0000-0002-7284-7265
R. Zhou  0000-0001-5381-4372 T. Zhuang  H. Zou  0000-0002-6684-3997

###### Abstract

We present cosmological results from the measurement of baryon acoustic
oscillations (BAO) in galaxy, quasar and Lyman-$\alpha$ forest tracers from
the first year of observations from the Dark Energy Spectroscopic Instrument
(DESI), to be released in the DESI Data Release 1. DESI BAO provide robust
measurements of the transverse comoving distance and Hubble rate, or their
combination, relative to the sound horizon, in seven redshift bins from over 6
million extragalactic objects in the redshift range $0.1<z<4.2$. To mitigate
confirmation bias, a blind analysis was implemented to measure the BAO scales.
DESI BAO data alone are consistent with the standard flat $\Lambda$CDM
cosmological model with a matter density $\Omega_{\mathrm{m}}=0.295\pm 0.015$.
Paired with a baryon density prior from Big Bang Nucleosynthesis and the
robustly measured acoustic angular scale from the cosmic microwave background
(CMB), DESI requires $H_{0}=(68.52\pm 0.62)\,\,{\rm km\,s^{-1}\,Mpc^{-1}}$. In
conjunction with CMB anisotropies from Planck and CMB lensing data from Planck
and ACT, we find $\Omega_{\mathrm{m}}=0.307\pm 0.005$ and $H_{0}=(67.97\pm
0.38)\,\,{\rm km\,s^{-1}\,Mpc^{-1}}$. Extending the baseline model with a
constant dark energy equation of state parameter $w$, DESI BAO alone require
$w=-0.99^{+0.15}_{-0.13}$. In models with a time-varying dark energy equation
of state parametrized by $w_{0}$ and $w_{a}$, combinations of DESI with CMB or
with type Ia supernovae (SN Ia) individually prefer $w_{0}>-1$ and $w_{a}<0$.
This preference is 2.6$\sigma$ for the DESI+CMB combination, and persists or
grows when SN Ia are added in, giving results discrepant with the $\Lambda$CDM
model at the $2.5\sigma$, $3.5\sigma$ or $3.9\sigma$ levels for the addition
of the Pantheon+, Union3, or DES-SN5YR supernova datasets respectively. For
the flat $\Lambda$CDM model with the sum of neutrino mass $\sum m_{\nu}$ free,
combining the DESI and CMB data yields an upper limit $\sum m_{\nu}<0.072$
$(0.113){\,\rm eV}$ at 95% confidence for a $\sum m_{\nu}>0$ $(\sum
m_{\nu}>0.059){\,\rm eV}$ prior. These neutrino-mass constraints are
substantially relaxed if the background dynamics are allowed to deviate from
flat $\Lambda$CDM.

##  1 Introduction

The discovery of the accelerated expansion of the universe [1, 2] demonstrated
that the dynamics of the universe are presently dominated by dark energy, a
component with negative pressure [for reviews, see, e.g., 3, 4]. Over the past
quarter-century, a wide variety of cosmological measurements have lent further
support for what has become the standard cosmological model: a spatially flat
universe with an energy budget today composed of about 5% baryonic matter, 25%
cold dark matter (CDM), 70% dark energy in the form of Einstein’s cosmological
constant ($\Lambda$) and smaller contributions provided by massive neutrinos
and radiation.

Increasingly accurate data from type Ia supernovae (SN Ia) have confirmed and
significantly strengthened the original ground-breaking results for the
accelerated expansion of the universe (e.g., [5, 6, 7, 8, 9]). Meanwhile,
mapping out the anisotropies in the cosmic microwave background (CMB)
radiation, starting with the ground-breaking COBE results [10], and continuing
with increasingly precise measurements [11, 12, 13] that led to the full-sky
maps by WMAP [14] and Planck [15] experiments, as well as mapping of the CMB
anisotropy on smaller angular scales [16, 17], revolutionised the field of
cosmology by providing percent-level constraints on key cosmological
parameters. Recent cosmological constraints from probes of the large-scale
structure [18, 19, 20, 21], CMB [22, 23], and distance measurements from SN Ia
[24, 25, 26] have largely confirmed and sharpened constraints on the
cosmological model while establishing an increasingly sophisticated
methodology framework focused on stringent control of systematic errors.

Current constraints on key cosmological parameters are largely consistent with
the $\Lambda$CDM cosmological model with cold dark matter and dark energy
described by the cosmological constant $\Lambda$. However, dark energy
dynamics has not been stringently tested. In addition, within the $\Lambda$CDM
model, tensions have appeared at various degrees of statistical significance
(the two most popular being referred to as the “Hubble tension” and the
“$\sigma_{8}$ tension”). Such tensions, if not due to unaccounted systematics,
may indicate new physics beyond $\Lambda$CDM.

The use of galaxies as tracers of large-scale structure has traditionally
played an essential role in cosmology and provides important complementary
information to that from SN Ia and CMB. Galaxy clustering was pioneered half a
century ago and further developed and applied to larger and better galaxy
catalogs in the intervening years [27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37]. It has established itself as a key cosmological measurement that provides
a direct link between observations and properties of dark matter and dark
energy.

The clustering of matter encodes a preferred scale, the sound horizon at the
baryon drag epoch of the early universe [38]. This feature, which is imprinted
on the matter distribution of the early universe by physics around
recombination and earlier, is stretched with the expansion of the universe,
appearing at a comoving galaxy separation of $r_{\mathrm{d}}\sim
150\,\text{Mpc}$. Hence $r_{\mathrm{d}}$ is a standard ruler, whose length is
dictated by early-time physics. In particular, the length of the standard
ruler may be calibrated with high accuracy, e.g., by CMB observations [15].
Since galaxies trace the matter content of the universe, the BAO feature is
transferred into galaxy clustering, where it manifests as a single localised
peak in the galaxy correlation function and an oscillatory signature, or
“wiggles”, in the galaxy power spectrum. Furthermore, the signature is also
visible in other tracers of mass such as fluctuations in the Lyman-$\alpha$
forest — spectral features that indicate the radial distribution of neutral
hydrogen clouds between the observer and distant quasars. Because the BAO
feature has a distinct signature and resides in the linear-clustering regime,
measurements of this scale are relatively free from systematic errors
associated with nonlinear physics. BAO provides a key cosmological probe
sensitive to the cosmic expansion history, with well-controlled systematics.
Using BAO we can test for dark energy dynamics and spatial curvature, and in
combination with other probes constrain the Hubble constant, the sum of
neutrino masses, and the number of light species. Several reviews of the BAO
as a cosmological probe are available [e.g., 39, 40, 41, 42].

The accuracy with which the BAO feature may be measured from a galaxy survey
is principally limited by sample variance and Poisson noise in the galaxy
clustering measurements, necessitating galaxy surveys with effective volume of
at least $1\,h^{-3}\,\text{Gpc}^{3}$ [43, 44, 45].111An exception is the
Ly$\alpha$ BAO measurement, limited by the number of spectra and their signal-
to-noise, with only a minor contribution from cosmic variance [46, 47, 48].
The BAO feature was first detected in 2005 by the Sloan Digital Sky Survey
(SDSS) [49] and the Anglo-Australian Telescope Two-degree Field Galaxy
Redshift Survey [50]. Subsequent measurements, leading to distance
determinations accurate to within a few percent, were performed using the
SDSS-III Luminous Red Galaxy Sample [51, 52], the WiggleZ Dark Energy Survey
[53, 54, 55] and the 6-degree Field Galaxy Survey [56, 57]. Further extensions
of the SDSS yielded more accurate percent-level BAO measurements using the
Baryon Oscillation Spectroscopic Survey [BOSS, 58, 59, 60] at $z<0.7$ and its
extension [eBOSS, 61, 62, 63] at $z>0.7$. BAOs have also been detected within
the Lyman-$\alpha$ forest mapped by quasar surveys, allowing distance and
expansion measurements at $z>2$ [64, 48, 65, 66, 67]. Prior to DESI these
different BAO measurements, considered together, had mapped out the
cosmological distance-redshift relation with $1\%$-$2\%$ accuracy at a series
of redshifts in the range $z<2.5$ [68, 42].

The Dark Energy Spectroscopic Instrument (DESI) is carrying out a Stage IV
survey [69, 70] that was designed to significantly improve cosmological
constraints through measurements of the clustering of galaxies, quasars, and
the Lyman-$\alpha$ forest. DESI is conducting a five-year survey over $14,200$
square degrees in the redshift range $0.1<z<4.2$ with a spectroscopic sample
size that will be ten times that of the previous SDSS surveys. DESI covers six
different classes of tracers, including low redshift galaxies of the bright
galaxy survey (BGS), luminous red galaxies (LRG), emission line galaxies
(ELG), quasars as direct tracers, and Lyman-$\alpha$ (Ly$\alpha$) forest
quasars to trace the distribution of neutral hydrogen. Additionally, a sample
of stellar targets is being observed to a high density in an overlapping Milky
Way Survey [71] to explore the stellar evolution and kinematics of the Milky
Way. For cosmology, DESI is designed to impose stringent constraints on both
the expansion history and the growth rate of large scale structure. Promising
detections of the BAO feature at the few percent level [72] from the DESI
early data release [73] have confirmed that DESI is on target to meet the top-
level science requirements on BAO measurements. Specifically, DESI will
tightly constrain matter density in the universe, the equation of state of
dark energy, spatial curvature, the amplitude of primordial fluctuations, and
neutrino mass. It will also sharply test modifications to the general theory
of relativity proposed to explain the accelerated expansion of the universe
[74, 75, 76, 77].

In this paper, we report the constraints derived from the measurements of
baryon acoustic oscillations with the first year of data from DESI as part of
a larger series of papers. The results of this paper build upon the two-point
measurements and validation, as detailed in [78]; the BAO measurement from
galaxies and quasars as summarised in [79]; and the Ly-$\alpha$ forest BAO
measurements and validation described in [80]. In a follow-up paper, the
clustering analysis of the first year of DESI data is also performed over a
wider range of scales than just the BAO feature including the effects of
redshift space distortions (RSD) [81]. The Data Release 1 (DR1) of the Dark
Energy Spectroscopic Instrument [82], comprising spectroscopic data obtained
in the first year of observations, will be made public at a later stage.

The paper is structured as follows: Section 2 summarises our data and
methodology, including a brief overview of the BAO data analysis and the
integration of external datasets. Section 3 presents the results of the BAO
analysis and discusses internal consistencies within the DESI data and
external consistencies with SDSS data. Section 4 presents our constraints on
the $\Lambda$CDM model, including a comparison of our results with external
data. Section 5 discusses DESI constraints on extended dark energy models both
independently and in combination with CMB and SN Ia data. In Section 6, we
explore the constraints on the Hubble parameter for a variety of cosmological
models and dataset combinations. Section 7 summarises our constraints on the
neutrino sector. Lastly, we summarise our findings in Section 8.

##  2 Data and methodology

In this section, we introduce the cosmological quantities relevant for this
analysis, review the basics of the BAO as a cosmological probe, then describe
the DESI data we analyze, the external datasets we combine with DESI data, and
the analysis methods we adopt.

###  2.1 Cosmological background

In a homogeneous and isotropic cosmology (i.e., with the Friedmann-Lemaître-
Robertson-Walker (FLRW) metric), the transverse comoving distance is

$D_{\mathrm{M}}(z)=\frac{c}{H_{0}\sqrt{\Omega_{\mathrm{K}}}}\,\sinh\left[\sqrt{\Omega_{\mathrm{K}}}\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})/H_{0}}\right]\
.$ (2.1)

Here $z$ is redshift, $H_{0}$ is the Hubble constant with $H(z)$ the Hubble
parameter, $\Omega_{\mathrm{K}}$ is the curvature density parameter, and $c$
is the speed of light. In $\Lambda$CDM,
$\Omega_{\mathrm{K}}=1-\Omega_{\mathrm{m}}-\Omega_{\Lambda}-\Omega_{\mathrm{R}}$,
where $\Omega_{\mathrm{m}}$, $\Omega_{\mathrm{R}}$, and $\Omega_{\Lambda}$ are
the energy densities relative to critical in matter, radiation, and the
cosmological constant, respectively, all evaluated at the present time. Since
$\sinh$ is a complete function, the expression above holds for positive,
negative, or zero $\Omega_{\mathrm{K}}$. The Hubble parameter in $\Lambda$CDM
is

$H(z)=H_{0}\sqrt{\Omega_{\mathrm{m}}(1+z)^{3}+\Omega_{\mathrm{R}}(1+z)^{4}+\Omega_{\mathrm{K}}(1+z)^{2}+\Omega_{\Lambda}}.$
(2.2)

The Hubble parameter also contains the contribution by massive neutrinos given
by their energy density relative to critical $\Omega_{\nu}$. They are non-
relativistic at low redshifts that DESI tracers probe and hence contribute to
$H(z)$ like matter, but their contribution becomes that of a relativistic
species at high redshift.

BAO measurements depend on the sound horizon at the drag epoch
$r_{\mathrm{d}}$. This is the distance that sound can travel between the Big
Bang and the drag epoch which indicates the time when the baryons decoupled.
The drag epoch occurs slightly later (at $z_{\mathrm{d}}\simeq 1060$ in the
standard model) than photon decoupling ($z_{\ast}\simeq 1090$) simply because
there are so few baryons relative to the number of photons. The expression of
$r_{\mathrm{d}}$ is given by

$r_{\mathrm{d}}=\int_{z_{\mathrm{d}}}^{\infty}\frac{c_{\mathrm{s}}(z)}{H(z)}dz\,,$
(2.3)

where $c_{\mathrm{s}}$ is the speed of sound which, prior to recombination, is
given by

$c_{\mathrm{s}}(z)=\frac{c}{\sqrt{3\displaystyle\left(1+\frac{3\rho_{\mathrm{B}}(z)}{4\rho_{\gamma}(z)}\right)}}\,$
(2.4)

where $\rho_{\mathrm{B}}$ and $\rho_{\gamma}$ are the baryon and radiation
energy densities, respectively. The speed of sound evaluates to approximately
$c_{\mathrm{s}}\simeq c/\sqrt{3}$ well before recombination, but then
decreases, and finally drops sharply at recombination. For standard early-time
physics assumptions, the drag-epoch sound horizon can be approximated with
[83]

$r_{\mathrm{d}}=\frac{147.05}{\rm
Mpc}\left(\frac{\omega_{\mathrm{m}}}{0.1432}\right)^{-0.23}\left(\frac{N_{\mathrm{eff}}}{3.04}\right)^{-0.1}\left(\frac{\omega_{\mathrm{b}}}{0.02236}\right)^{-0.13},$
(2.5)

where $\omega_{\mathrm{m}}\equiv\Omega_{\mathrm{m}}h^{2}$ and
$\omega_{\mathrm{b}}\equiv\Omega_{\mathrm{b}}h^{2}$ are the matter and baryon
physical energy densities today, respectively, and $N_{\mathrm{eff}}$ is the
effective number of extra relativistic degrees of freedom.

The sound horizon at the drag epoch leaves an imprint in the distribution of
matter that serves as a cosmological standardised ruler [84, 85, 86, 46]. To
illustrate the concept, consider an ensemble of galaxy pairs at a given
redshift $z$: if the separation vectors of the pairs are oriented
perpendicular to the observer’s line-of-sight, a preferred angular separation
of galaxies $\Delta\theta$ may be observed, measuring the comoving distance
$D_{\mathrm{M}}(z)=r_{\mathrm{d}}/\Delta\theta$ to this redshift. With the
separation vector oriented parallel to the line-of-sight, a preferred redshift
separation $\Delta z$ may be observed, measuring a comoving distance interval
that for small intervals gives the Hubble parameter at that redshift,
represented in this paper by the equivalent distance variable
$D_{\mathrm{H}}(z)=c/H(z)=r_{\mathrm{d}}/\Delta z$. BAO measurements hence
constrain the quantities $D_{\mathrm{M}}(z)/r_{\mathrm{d}}$ and
$D_{\mathrm{H}}(z)/r_{\mathrm{d}}$. This interpretation holds true for
standard assumptions and models not too dissimilar from $\Lambda$CDM, given
the statistical power of the data [87]. For those BAO measurements in certain
redshift bins with low signal-to-noise ratio, we instead quote constraints on
the angle-averaged quantity, $D_{\mathrm{V}}(z)/r_{\mathrm{d}}$, where
$D_{\mathrm{V}}(z)$ is the angle-average distance that quantifies the average
of the distances measured along, and perpendicular to, the line of sight to
the observer [49]:

$D_{\mathrm{V}}(z)=\left(zD_{\mathrm{M}}(z)^{2}D_{H}(z)\right)^{1/3}.$ (2.6)

BAO measurements are performed at a series of redshifts, allowing constraints
to be obtained on the cosmological parameters governing the distance-redshift
relation including curvature and dark energy, and the Hubble constant if
external information on the absolute BAO scale is provided.

###  2.2 DESI BAO data from Data Release 1

DESI spectroscopic targets are selected from photometric catalogs of the 9th
public data release of the DESI Legacy Imaging Surveys
[88],222https://www.legacysurvey.org/dr9/ drawn from three optical surveys in
the $grz$ optical bands: DECaLS using the DECam camera [89] (which includes
data from the Dark Energy Survey (DES) [90]) south of declination
$32.375^{\circ}$, and north of this limit the Beijing-Arizona Sky Survey
(BASS) [91], and the Mosaic z-band Legacy Survey (MzLS) [92]. Four different
classes of extragalactic targets are defined: the bright galaxy sample (BGS,
[93]), luminous red galaxies (LRG, [94]), emission line galaxies (ELG [95]),
and quasars (QSO [96]).

Spectroscopic observations of these targets are carried out with the DESI
instrument [70] mounted on the Nicholas U. Mayall Telescope at Kitt Peak
National Observatory in Arizona. Each observation field is covered by a
“tile”, consisting in a set of targets located within that sky area [97] and
assigned to each of the 5000 fibers in the focal plane of the telescope. Each
fiber is placed at the celestial coordinates of its assigned target by a
robotic positioner [98, 99] and guide its light from the focal plane to one of
the ten DESI spectrographs. DESI observing time is dynamically separated into
a “bright” time program (when BGS are observed) and “dark” time observations
(when LRG, ELG, and QSO are targeted) depending on observing conditions.
Redshift distributions, exposure times, calibration and observation procedures
were optimised during a period of survey validation [100] that included a
visual inspection campaign [101, 102]. The DR1 spectroscopic dataset is built
from main survey operations starting from May 14, 2021 through June 14, 2022,
using an observing strategy meant to prioritise depth [103], resulting in
$2,744$ “dark” tiles and $2,275$ “bright” tiles. The covered tile surface area
is of order $7,500\;\deg^{2}$, just over half of the expected final coverage
of $14,200\;\deg^{2}$. However, the completeness within this area will
significantly increase, as one can infer given an expected final number of
9,929 dark and 5,676 bright observed tiles [103]. The combined effective
volume is expected to increase by more than a factor 3 [79]. The observed data
are processed by the DESI spectroscopic pipeline [104] on a daily basis for
immediate quality checks. The redshift catalogs used for this analysis and
released with DESI DR1 are obtained from a spectroscopic reduction run with a
fixed pipeline version internally denoted as “iron”.

Large-scale structure catalogs of galaxy and quasar positions suitable for the
clustering analysis are built from the redshift and parent target catalogs and
their two-point function measurements are obtained with all DR1 specific
details presented in [78]; a technical overview of the general process is
presented in [105]. The selection function is defined and correction weights
are designed to compensate for systematic variations due to the effects of
imaging anisotropies on the input target samples [106, 107], redshift
measurement efficiency [108, 109], and incompleteness in target assignment
[110, 111, 112]. Simulations of the DR1 dataset are presented in [113].

Studies of the Ly$\alpha$ forest are based on a quasar catalog with
alternative redshift estimates to minimize possible biases caused by the same
Ly$\alpha$ absorption [114]. The method to extract Ly$\alpha$ fluctuations
from the quasar spectra is described in [115], including the masking of pixels
contaminated by Broad Absorption Line features [116] and Damped Lyman-$\alpha$
systems [117]. Below, we provide details on each of the target samples studied
and the properties of the particular samples that are used for this analysis.

##### The Bright Galaxy Sample (BGS, $0.1<z<0.4$):

The nominal target selection (BGS Bright) relies on $r$-magnitude cuts tuned
to achieve uniform density across the photometric samples. Gaia catalogs [118]
are additionally used to remove point-like sources. The resulting sample has
854 targets per square degree, which are assigned fibers with high priority
during bright time observations.

The nominal BGS sample is flux-limited and high density, with strong evolution
in redshift. Over $5.5$ million reliable BGS redshifts were measured in DR1.
However, in order to produce a more homogeneous sample, a cut was engineered
to produce a sample of roughly constant comoving number density of $5\times
10^{-4}\;\,h^{3}\,\text{Mpc}^{-3}$ using $k$-corrected $r$-band absolute
magnitudes from [119, 120] and a redshift-dependent correction for evolution
(matching the sample used for [73]). The BGS number density is similar to that
of LRGs at $z=0.4$ and high enough to make shot-noise a minor contribution to
the BAO statistical uncertainty (see [78] for more details). The final BGS
clustering sample used for the BAO measurement comprises $300,017$ redshifts
in $0.1<z<0.4$, with an assignment completeness of $61.6\%$, which is expected
to increase to $\sim 80\%$ in the finalized survey.

##### The Luminous Red Galaxy Sample (LRG, $0.4<z<0.6$ and $0.6<z<0.8$):

The LRG target selection [94] uses photometry from the Wide-field Infrared
Survey Explorer (WISE) [121] to select red objects with a nearly constant
comoving number density of $5\times 10^{-4}\;\,h^{3}\,\text{Mpc}^{-3}$ over
$0.4<z<0.8$. An additional cut on the $z$-magnitude measured within the
aperture of a DESI fiber further selects targets with high S/N spectra. The
obtained $624\;\deg^{-2}$ target sample is assigned fibers with priority
higher than ELG targets but lower than QSO targets.

The DESI DR1 LRG clustering sample used for BAO measurements consists of
$2,138,600$ redshifts in the redshift interval $0.4<z<1.1$. The DR1 assignment
completeness is $69.2\%$, which is expected to increase to $90\%$ in the
completed survey. The lower redshift bound was chosen to separate the sample
from BGS, as most low-redshift LRG targets are also BGS targets, while the
upper bound was designed to ensure a minimum density of
$10^{-4}\;\,h^{3}\,\text{Mpc}^{-3}$. This sample is further split for the
clustering analysis into 3 disjoint redshift ranges, $0.4<z<0.6$, $0.6<z<0.8$,
and $0.8<z<1.1$ (for brevity, we sometimes refer to these as LRG1, LRG2 and
LRG3, respectively), however we do not use any BAO measurement from the
highest-redshift LRG bin alone. It is instead combined with the overlapping
lowest-redshift ELG bin, as described in the combined LRG and ELG sample
section below.

##### The Emission Line Galaxy Sample (ELG, $1.1<z<1.6$):

ELG targets are selected with colour cuts in the $(g-r)$ vs $(r-z)$ plane to
prioritise objects with [OII] emission in the desired redshift range [95] for
secure redshift measurements. Low-redshift objects are filtered out with a
$g$-magnitude cut and another cut on the $g$-magnitude within the aperture of
a DESI fiber further favors targets with high S/N spectra. While a random
$10\%$ fraction of the ELG sample receives the same fiber assignment priority
as LRGs to facilitate small-scale clustering measurements in the redshift
range where the samples overlap, the remaining $90\%$ of ELGs are assigned
fibers at a lower priority than LRGs and QSOs.

The DR1 ELG sample defined for clustering analysis in [78] comprises
$2,432,022$ reliable redshifts in the interval $0.8<z<1.6$. The lower redshift
bound aligns with that of the high redshift LRG sample, while the upper bound
rejects objects whose [OII] doublet falls outside of the spectrograph
wavelength coverage. In DR1, the fiber completeness is only $35\%$, which
should increase to over $60\%$ in the final dataset. The sample is split into
two disjoint redshift ranges for clustering analysis, $0.8<z<1.1$ and
$1.1<z<1.6$ (similarly referred to as ELG1 and ELG2 for brevity). We do not
use any standalone BAO measurement from the low-redshift ELG bin; it is
instead combined with the overlapping high-redshift LRG bin.

##### The combined LRG and ELG Sample (LRG+ELG, $0.8<z<1.1$):

The high-redshift LRG and the low-redshift ELG samples overlap in the range
$0.8<z<1.1$. These two samples are concatenated with inverse-variance weights
to obtain a combined LRG+ELG catalog, following the methodology presented in
[122]. The increased combined density is expected to improve the
reconstruction efficiency, while the inverse-variance weights are designed to
maximize the measurement of the BAO precision. As described in [79], the
obtained combined LRG+ELG BAO measurement is $\sim 10\%$ more precise, while
being fully consistent with the LRG-alone measurement in this redshift range
(and with that of ELG alone, although this has a larger uncertainty). We
therefore use the combined LRG+ELG $0.8<z<1.1$ BAO measurement for the
cosmological inference.

##### The Quasar Sample (QSO, $0.8<z<2.1$):

The QSO target selection relies on identifying a flux excess in the near-
infrared, which is assessed by comparing near-infrared WISE $W1$, $W2$
magnitudes to Legacy Imaging Surveys optical $grz$ magnitudes [96]. Sources
with stellar morphology (PSF type) are selected, and a $r$-magnitude cut is
applied to eliminate bright stars and faint targets. Quasars are then selected
from the $r$ magnitude and 10 colors using $grzW1W2$ with a Random Forest (RF)
algorithm, designed to produce a target sample of density $310\deg^{-2}$. The
resulting QSO target sample is assigned fibers with maximum priority.

The DR1 QSO sample used for BAO measurements in $0.8<z<2.1$ consists of
$856,652$ redshifts. Quasars at $z>2.1$ are not used as part of this sample.
Because of their maximum priority, the assignment completeness of QSO in DR1
is already high, $87.6\%$.

##### The Lyman-$\alpha$ Forest Sample (Ly$\alpha$, $1.77<z<4.16$):

The highest-redshift BAO measurement from DESI DR1 is described in [80], and
is obtained from a combined analysis of correlations of three different
datasets. The first dataset consists of the positions (celestial coordinates
and redshifts) of $709,565$ quasars in the redshift range $1.77<z<4.16$. In
addition, the Ly$\alpha$ forest in the spectra of high-redshift quasars at
$z>2.1$ constitute an alternative tracer of matter density fluctuations. We
use two Ly$\alpha$ forest datasets, consisting of fluctuations in two
different rest-frame wavelength regions of the background quasar spectra:
$428,403$ in region A ($1040<\lambda_{r}<1205$ Å) and $137,432$ in region B
($920<\lambda_{r}<1020$ Å).333Region B is treated separately from region A as
it is also affected by higher order Lyman lines. In order to increase the
signal-to-noise of the spectra, quasars identified as having $z>2.1$ after a
single epoch of observation are prioritised for up to four observations.

Table 1 summarises the properties of the different samples used in this
analysis, as well as the BAO measurements obtained from them using the methods
described below. Overall, this analysis of the DESI Data Release 1 relies on
over $6$ million unique redshifts, more than twice the number of redshifts
considered in the final SDSS cosmological analysis [42]. The aggregate
precision on the BAO isotropic scale is $0.49\%$ to be compared to $0.60\%$
for the final SDSS measurements.

###  2.3 BAO analysis

We now briefly describe how DESI BAO analyses of DR1 data implement the
standard ruler measurement.

BAO analyses usually need to assume a fiducial cosmology although the
resulting distance and expansion measurements remain independent of these
assumptions in a leading-order approximation, unless the assumed fiducial
model differs significantly from the model to be tested, or the truth [123,
124, 125]. First, a fiducial cosmological model is adopted to convert tracer
angular positions and redshifts to comoving coordinates, and the two-point
clustering pattern is measured as a function of comoving separation assuming
this fiducial model. The clustering pattern perpendicular and parallel to the
line-of-sight changes in a predictable way between fiducial and trial
cosmologies according to the Alcock-Paczynski (AP) effect [126, 127], allowing
a wide range of cosmological models to be tested [128].

Second, the theoretical calibration of the BAO scale is encoded by a full
early-universe matter power spectrum generated assuming a fiducial
cosmological model [38, 129]. This matter power spectrum is used to produce a
template tracer clustering model, indexed by the sound horizon at baryon drag,
which is scaled by two free scaling parameters—along and perpendicular to the
line-of-sight—during data fitting. In order to measure the location of the BAO
feature alone, the model also includes “nuisance” parameters for the broad-
band shape of the tracer power spectrum, which are marginalized over in the
analysis [58]. Residual theoretical systematics arising from nonlinear matter
growth, tracer bias, and redshift-space distortions are reduced by this
procedure, and are generally considered to be present at a level of $\sim
0.1\%$ [130, 131, 132, 133]. The modeling and the procedure for performing BAO
galaxy and quasar fits have been updated for DESI from the broadband
polynomial marginalization procedure used in previous surveys, according to
the theoretical framework laid out in [134].

Finally, the amplitude of the BAO feature imprinted in the early universe is
diluted by the bulk-flow motion of matter from its true locations in the
density field, driven by the growth of structure, which changes tracer
separations by a few $h^{-1}\,\text{Mpc}$. This effect may be partially
reversed using density-field reconstruction to estimate these displacements
based on the gravitational field inferred from the observed tracer
distribution [135, 136]. Applying these reversed displacements to tracers
sharpens the BAO signature and mitigates the small ($<0.3\%$) nonlinear shift
of the BAO-scale in late-time clustering, allowing for more accurate
cosmological inferences [137, 138]. The reconstruction algorithm used has been
updated relative to that used for eBOSS [139] and the specific choice of
algorithm and settings were validated as described in [140].

For the galaxy and quasar BAO data points considered in this paper we make use
of the post-reconstruction clustering measurements in configuration space
presented in [79]. Anisotropic BAO measurements are obtained for LRG and ELG
samples, but we restrict to isotropic fits for BGS and QSO samples due to the
lower signal-to-noise ratio of the two-point function measurements. The
uncertainties in these clustering measurements are described by covariance
matrices whose construction and validation are outlined in [141, 142, 143].
Various contributions to the total systematic error budget arising from
theoretical modeling uncertainties (which are at most $0.1\%$ and $0.2\%$ of
the isotropic and anisotropic BAO parameters, respectively [134]),
uncertainties due to the galaxy-halo connection (of $\lesssim 0.2\%$ [144,
145]) and observational systematic effects (which are negligible [106, 108,
79]) have been estimated in a series of supporting papers. These are added
together in quadrature as summarised in [79] to compute the total error budget
for the BAO measurements in the BGS, LRG, ELG, and QSO samples. Together,
these steps represent the state-of-the-art in the measurement of BAO from
galaxy tracers and are described in detail in the companion DESI paper [79].

In addition to measuring the clustering of galaxies and quasars, we also make
use of the Ly$\alpha$ forest. The Ly$\alpha$ BAO measurement is presented in
[80], and it combines a measurement of the auto-correlation of fluctuations in
the Ly$\alpha$ forest and of its cross-correlation with the density of quasars
at $z>1.77$. The method to measure and model the correlations is described in
[146], and it is based on previous Ly$\alpha$ BAO measurements from eBOSS
[67]. In a supporting study [147] we validate the analysis using synthetic
datasets, simulated using the methodology described in [148]. An important
improvement in the methodology is that we now take into account the small
correlation between the measurements of the auto- and the cross-correlation.
Finally, the (minor) impact of correlated noise in the Ly$\alpha$ auto-
correlation is studied and characterised in [149]. Besides the validation
tests using synthetic datasets, in [80] we also present several data splits
and a long list of consistency tests that show that our Ly$\alpha$ BAO
measurement are robust to changes in the measurement methodology or in the
modeling. For instance, we show that adding an ad-hoc smooth component to the
model with up to 48 extra free parameters does not vary the BAO parameters by
more than 0.1%.

The BAO measurements obtained from the various samples used are shown in Table
1. As already mentioned, for the BGS and QSO tracers, we only measure the
angle-averaged $D_{\mathrm{V}}/r_{\mathrm{d}}$ quantity, due to the lower
signal-to-noise achieved. For all other tracers, we quote results in
$D_{\mathrm{M}}/r_{\mathrm{d}}$ and $D_{\mathrm{H}}/r_{\mathrm{d}}$, and
provide the value of the correlation between them, $r$.

tracer | redshift | $N_{\rm tracer}$ | $z_{\rm eff}$ | $D_{\mathrm{M}}/r_{\mathrm{d}}$ | $D_{\mathrm{H}}/r_{\mathrm{d}}$ | $r$ or $D_{\mathrm{V}}/r_{\mathrm{d}}$ | $V_{\rm eff}$
---|---|---|---|---|---|---|---
(Gpc3)
BGS | $0.1-0.4$ | 300,017 | $0.295$ | — | — | $7.93\pm 0.15$ | $1.7$
LRG1 | $0.4-0.6$ | 506,905 | $0.510$ | $13.62\pm 0.25$ | $20.98\pm 0.61$ | $-0.445$ | $2.6$
LRG2 | $0.6-0.8$ | 771,875 | $0.706$ | $16.85\pm 0.32$ | $20.08\pm 0.60$ | $-0.420$ | $4.0$
LRG3+ELG1 | $0.8-1.1$ | 1,876,164 | $0.930$ | $21.71\pm 0.28$ | $17.88\pm 0.35$ | $-0.389$ | $6.5$
ELG2 | $1.1-1.6$ | 1,415,687 | $1.317$ | $27.79\pm 0.69$ | $13.82\pm 0.42$ | $-0.444$ | $2.7$
QSO | $0.8-2.1$ | 856,652 | $1.491$ | — | — | $26.07\pm 0.67$ | $1.5$
Lya QSO | $1.77-4.16$ | 709,565 | $2.330$ | $39.71\pm 0.94$ | $8.52\pm 0.17$ | $-0.477$ | —

Table 1: Statistics for the DESI samples used for the DESI DR1 BAO
measurements used in this paper. For each tracer and redshift range we quote
the number of objects ($N_{\rm tracer}$), the effective redshift ($z_{\rm
eff}$) and effective volume ($V_{\rm eff}$). Note that for each sample we
measure either both $D_{\mathrm{M}}/r_{\mathrm{d}}$ and
$D_{\mathrm{H}}/r_{\mathrm{d}}$, which are correlated with a coefficient $r$,
or $D_{\mathrm{V}}/r_{\mathrm{d}}$. Redshift bins are non-overlapping, except
for the shot-noise-dominated measurements that use QSO (both as tracers and
for Ly$\alpha$ forest).

####  2.3.1 Blinding framework

An imperative of science with modern cosmology experiments, and especially for
a survey with the statistical precision that DESI is able to achieve, is to
mitigate against the possibility of the results being affected by observer
confirmation bias. This can manifest itself through the attribution of
unexpected results, or results which do not match the prior prejudices of the
observer, to systematic effects which must be corrected for in the data
analysis. Thus the observer may—consciously or subconsciously—continue to
search for and correct real or imagined systematic effects in the data only
until they agree with some pre-conceived desired result, and no further.

To mitigate against the possibility of such biases unknowingly entering into
our analyses, we apply a system of blinding our data, to conceal the true
cosmological results during the period where systematic errors were being
investigated and analysis pipelines finalised. This blinding was applied in
two different ways:

  * •

For the discrete galaxy and quasar samples, blinding was applied at the
catalog level [150], according to the procedure described and validated in
[151]. The redshifts of the tracers in the catalogs were shifted in such a way
to change the position of the BAO feature as well as the redshift-space
distortion (RSD) signal, and the weights applied to the tracers were adjusted
in order to blind the measurement of primordial non-Gaussianities from the
large-scale clustering. True redshifts and weights were not made available for
the analysis before unblinding.

  * •

For the Ly$\alpha$ sample, catalog-level blinding was challenging due to sky
lines and Galactic absorption features located at known observed wavelengths
in the spectra. Instead, the apparent position of the BAO peak was shifted by
directly applying an additive component to the measured correlation functions,
as described in [80].

In both cases, all analyses used the blinded data during initial testing and
systematic errors were identified and corrected. Only after the analysis
pipeline had been finalised, the systematic error budget determined, and a
series of strictly prescribed validation tests passed (see [79, 80] for a
detailed description) was the data blinding removed and the unblind results
revealed. For discrete tracers, it was decided prior to unblinding that the
final choice of covariance matrices would be made after unblinding, and that
the LRG and ELG samples would be concatenated in $0.8<z<1.1$ for a combined
BAO measurement in this redshift range.

We mention two unplanned updates in the BAO analysis pipeline for discrete
tracers made after unblinding and decided independently of the actual BAO
measurements: a completeness correction was fixed in the clustering catalogs,
and the implementation of the BAO theoretical model was corrected to fully
match the theoretical framework laid out in [134]. Corrections in the
theoretical model changed BAO measurements by less than $0.3\sigma$
(differences in most redshift bins under $0.1\sigma$), and updates to the
catalogs by at most $0.6\sigma$ (variations in most redshift bins being under
$0.2\sigma$). We also report two minor post-unblinding changes in the
Ly$\alpha$ BAO analysis: a bug fix that led to a $<0.1\sigma$ change in the
BAO measurement, and an update in a bias parameter that had no effect on the
results.

Two additional points are worth noting. For the discrete galaxy and quasar
samples, the analyses related to the full shape of the broadband clustering
signal (described in [81]) and those pertaining to the BAO results presented
in [79] and included in this paper had separate pipelines and robustness
criteria to be satisfied before unblinding, and were not unblinded at the same
time. Additionally, it was decided that cosmological model inference from the
BAO and full shape measurements would remain outside of the scope of the
blinding, and would be performed once BAO and full shape unblinded results are
obtained.444Internal checks of the inference pipeline were performed with
synthetic data prior to unblinding.

####  2.3.2 Summary of DESI DR1 BAO likelihoods

The results used in this paper are derived from the BAO measurements and fits
described above applied to each of the five tracer samples across 7 distinct
redshift bins covering a total redshift range from 0.1 to 4.2.

The BGS and LRG samples chosen cover disjoint redshift ranges. As a result of
the lack of overlap, the correlation between the BAO measurements in these
samples is small, and we assume it to be negligible and do not include any
covariance between them in our cosmological analysis. In the $0.8<z<1.1$
redshift bin, where the LRG and ELG tracers overlap, they are analysed
together using a single multi-tracer approach, described in [122, 79]. The QSO
sample overlaps with this redshift bin, and with the ELGs in the range
$1.1<z<1.6$. In a general case where two tracer samples overlap in volume, the
correlation coefficient between two estimates of the power spectrum obtained
from these tracers may be estimated as:

$C=\frac{\mathrm{Cov}(\hat{P}_{1},\hat{P}_{2})}{\sqrt{\mathrm{Var}(\hat{P}_{1})\mathrm{Var}(\hat{P}_{2})}}=\frac{\int
dV\;\displaystyle\frac{X_{1}X_{2}}{X_{12}}}{(\int dVX_{1})^{1/2}(\int
dVX_{2})^{1/2}}\,,$ (2.7)

where $X_{i}\equiv\left(\frac{n_{i}P_{i}}{1+n_{i}P_{i}}\right)^{2}$ for
$i=1,2$,
$X_{12}\equiv\left(\frac{n_{1}n_{2}P_{1}P_{2}}{n_{12}\sqrt{P_{1}P_{2}}+n_{1}n_{2}P_{1}P_{2}}\right)^{2}$,
$n_{1}$ and $n_{2}$ are the mean tracer number densities of the two samples
and $n_{12}$ is the common sample density, and $P_{1}$ and $P_{2}$ are the
corresponding typical amplitudes of the power spectra of the two tracers at
the BAO scale $k\simeq 0.1\;\,h\,{\rm Mpc^{-1}}$. For different tracer types,
such as the overlapping ELG and QSO samples of relevance in the $1.1<z<1.6$
redshift bin, there are no common tracers so $n_{12}=0$.555In this case and in
the limit of a thin redshift slice, Eq. 2.7 simplifies to
$C=\frac{V_{12}P_{1}P_{2}}{\sqrt{V_{1}V_{2}}(P_{1}+1/\bar{n}_{1})(P_{2}+1/\bar{n}_{2})}$,
as used in [139]. This correlation between power spectrum estimates can be
roughly regarded as an approximation to the correlation between the BAO scale
measurements from the two overlapping tracer samples, although we regard this
only as a guide and caution against using this estimate for any precise
combination of results from different tracers.

For the ELG and QSO tracers in $1.1<z<1.6$, using representative values in Eq.
2.7 gives a very small correlation, $C<0.1$. The correlation between QSOs and
LRGs and ELGs in the lower redshift $0.8<z<1.1$ bin is even smaller. We
therefore ignore this correlation and treat the QSO BAO measurement as
effectively independent of those from LRG and ELG tracers. We also ignore the
correlation between the BAO measurement from the QSO sample (at $z_{\rm
eff}=1.49$) and the Ly$\alpha$ BAO measurement (at $z_{\rm eff}=2.33$). Even
though quasars in the range $1.77<z<2.1$ contribute to both measurements, in
the Ly$\alpha$ BAO measurement they are only used in cross-correlation with
Ly$\alpha$ absorption at $z>2$. Therefore, these measurements would only be
correlated by cosmic variance, but both measurements are dominated by shot-
noise. In this redshift range, the number density of the quasars is less than
$3\times 10^{-5}\,h^{3}\,\text{Mpc}^{-3}$ and given the $P(k)$ at the scales
most relevant for BAO is $<7500\,h^{-3}\,\text{Mpc}^{3}$, $nP\ll 1$ and any
correlation between the measurements will be small.

As described in Section 2.1, the cosmological quantities measured by the BAO
analysis in each redshift bin are either the two (correlated) distance ratios
($D_{\mathrm{M}}/r_{\mathrm{d}}$, $D_{\mathrm{H}}/r_{\mathrm{d}}$) or just a
single one, $D_{\mathrm{V}}/r_{\mathrm{d}}$, depending on the signal-to-noise
ratio. The recovered posteriors in these quantities are in all cases found to
be well approximated by Gaussian distributions. Systematic uncertainties are
added in quadrature to the Gaussian covariance matrices, which we use together
with the mean vectors for the cosmological inference presented hereafter.

###  2.4 External Datasets

In this paper, we compare cosmological results from our data to those from a
number of other recent experiments and provide joint constraints from
different combinations of datasets. We briefly describe these data and
likelihoods below.

####  2.4.1 Big Bang Nucleosynthesis and $\Omega_{\mathrm{b}}h^{2}$

In a flat $\Lambda$CDM background model, the BAO distance ladder results
determine $\Omega_{\mathrm{m}}$ and $H_{0}r_{\mathrm{d}}$. From Eq. 2.5, the
sound horizon depends on the physical matter and baryon densities,
$\omega_{\mathrm{m}}=\Omega_{\mathrm{m}}h^{2}$ and
$\omega_{\mathrm{b}}=\Omega_{\mathrm{b}}h^{2}$. Therefore, assuming standard
neutrino content of the universe ($N_{\mathrm{eff}}=3.044$), we require prior
knowledge of $\Omega_{\mathrm{b}}h^{2}$ to break the degeneracy between
$H_{0}$ and $r_{\mathrm{d}}$.

Big Bang Nucleosynthesis (BBN) theory predicts the abundances of light
elements including D and 4He in the early universe, and these abundances
depend on the baryon to photon ratio $\eta_{10}$. Thus the observational
determination of the primordial deuterium abundance D/H [152] and the helium
fraction $Y_{\rm P}$ [153, 154] can be used to deduce
$\Omega_{\mathrm{b}}h^{2}$.

The resulting constraints on $\Omega_{\mathrm{b}}h^{2}$ depend on the details
of the theoretical predictions and their treatment of underlying nuclear
interaction cross-sections, in particular for the deuterium burning reactions.
For a review of the details we refer interested readers to [155]. A recent
analysis [156] makes use of the new PRyMordial code [157] to recompute the
predictions while correctly marginalizing over uncertainties in the reaction
rates, and reports the conservative constraints

$\Omega_{\mathrm{b}}h^{2}=0.02218\pm 0.00055$ (2.8)

in the standard $\Lambda$CDM model, and

$\Omega_{\mathrm{b}}h^{2}=0.02196\pm 0.00063$ (2.9)

when allowing for additional light relics ($\Lambda$CDM+$N_{\mathrm{eff}}$
model), where in the latter case there is also a covariance between
$\Omega_{\mathrm{b}}h^{2}$ and $N_{\mathrm{eff}}$ which we account for.666This
covariance is available from https://github.com/schoeneberg/2024_bbn_results.
We adopt these values as BBN priors in the following sections. Note that these
BBN priors are more conservative than some previously proposed in the
literature (e.g., [158]); this is a direct result of the marginalization over
reaction rate uncertainties performed in [156].

####  2.4.2 Cosmic Microwave Background

The power spectra of anisotropies in the cosmic microwave background (CMB)
have been exquisitely measured by Planck [159, 15]. A fundamental quantity
that is determined from the oscillations in these power spectra is the
measured sky angular scale of acoustic fluctuations, i.e.
$\theta_{\ast}=r_{\ast}/D_{\mathrm{M}}(z_{\ast})$, where $r_{\ast}$ is the
comoving sound horizon at recombination and $D_{\mathrm{M}}(z_{\ast})$ is the
transverse comoving distance to the redshift of recombination. This angular
scale is a very sharp feature and can be measured precisely independently of
possible systematic effects affecting the high-$\ell$ part of the CMB power
spectrum (in particular the $A_{\mathrm{L}}$ parameter).

Within the $\Lambda$CDM model Planck reports [15]

$100\theta^{\textit{Planck}}_{\ast}=1.04109\pm 0.00030,$ (2.10)

a $0.03\%$ precision measurement through the combination of temperature and
polarization data alone. The central value of
$\theta^{\textit{Planck}}_{\ast}$ is almost entirely independent of the
specific cosmological model assumed, although the uncertainties can increase
significantly in some model extensions affecting early-time physics. This is a
consequence of the simple geometrical interpretation: $\theta_{\ast}$ is the
BAO scale imprinted in the CMB anisotropies at recombination. DESI, on the
other hand, measures the same BAO features imprinted in the galaxy
distributions at lower redshifts. It is therefore natural to combine DESI BAO
results with the constraint on $\theta_{\ast}$ from Planck, which can be
regarded as a minimal and highly robust purely geometric version of the full
CMB information. To achieve this, we implement a Gaussian external prior on
the quantity $100\theta_{\ast}$ with mean $1.04110$ and variance
$0.00053^{2}$. The mean here matches the reported value in [15] when including
CMB lensing, and is very close to that in Eq. 2.10. We have conservatively
increased the width of the prior by $\sim 75\%$ relative to the baseline
result in order to accommodate the increased uncertainties in models beyond
standard flat $\Lambda$CDM, especially in models that allow for additional
light relics ($\Lambda$CDM+$N_{\mathrm{eff}}$), based on the results for that
model from [15].

Of course, the CMB power spectra also contain vastly more information than
just $\theta_{\ast}$, and we also explore the consequences for cosmological
models from the combination of this full CMB information with DESI BAO. We use
as our baseline the temperature (TT) and polarisation (EE) auto-spectra, plus
their cross-spectra (TE), as incorporated in the Commander (for multipoles
$\ell<30$) and plik (for $\ell>30$) likelihoods from the official PR3 release
[160]. Subsequently, a new Planck data release PR4 has been made available,
which involves a consistent reprocessing of the data from both the LFI and HFI
instruments on Planck using the new common pipeline NPIPE, leading to slightly
more data, lower noise, and better consistency between frequency channels
[161]. Various teams have released additional high-$\ell$ likelihood packages
using PR4 [162, 163, 164], which are updated versions of likelihoods included
in earlier Planck data releases, and use alternative methods to derive
information from the high-$\ell$ power. However, in most scenarios we consider
in this paper, results from these updated likelihoods are very similar to
those obtained from the official PR3 plik likelihood. Therefore, we choose to
keep plik as our baseline, and note any variations in the results due to
differences between this and the CamSpec likelihood of [162, 163] in Appendix
B.777CamSpec is bundled as an additional likelihood in the cobaya sampling
code [165, 166], at
https://cobaya.readthedocs.io/en/latest/likelihood_planck.html.

In addition to the primary temperature and polarization anisotropy power
spectra, modern CMB experiments are also able to measure the power spectrum of
the gravitational lensing potential, $C_{L}^{\phi\phi}$, from the connected
4-point function [167, 168]. The latest and most precise CMB lensing data
comes from the combination of NPIPE PR4 Planck CMB lensing reconstruction
[169] and the Data Release 6 of the Atacama Cosmology Telescope (ACT) [22,
170, 171].888The likelihood is available from
https://github.com/ACTCollaboration/act_dr6_lenslike; we use the
actplanck_baseline option. We adopt this combined CMB lensing likelihood from
both experiments as our baseline. For the sake of brevity, in the following
text and figures, we will denote results obtained using temperature and
polarisation information from Planck, and CMB lensing information from the
Planck+ACT combination, simply as “CMB”. Where occasionally necessary, we will
explicitly label results that do not use CMB lensing as “CMB (no lensing)”.

####  2.4.3 Type Ia supernovae

Type Ia supernovae (SN Ia) serve as standardizable candles which offer an
alternative way to measure the expansion history of the universe.
Historically, SN Ia led to the discovery of the accelerating expansion [1, 2],
following earlier, more complex arguments for $\Lambda$-dominated models based
on observations of large-scale structure [172, 173]. Within the $\Lambda$CDM
model SN Ia have lower statistical power than modern BAO measurements, but
provide important information on dark energy when analyzing the less
restricted models considered in Section 5.

In this paper, we make use of three different SN Ia datasets. The Pantheon+
compilation [174] consists of 1550 spectroscopically-confirmed SN Ia in the
redshift range $0.001<z<2.26$. We use the public likelihood from [24]
incorporating the full statistical and systematic covariance, imposing a bound
$z>0.01$ in order to mitigate the impact of peculiar velocities in the Hubble
diagram [175]. More recently, the Union3 compilation of 2087 SN Ia, many (1363
SN Ia) in common with Pantheon+, was presented in [25],999Data provided by the
Union3 team, private communication. and includes a different treatment of
systematic errors and uncertainties based on Bayesian Hierarchical Modelling.
Finally, the Dark Energy Survey, as part of their Year 5 data release,
recently published results based on a new, homogeneously selected sample of
1635 photometrically-classified SN Ia with redshifts $0.1<z<1.3$, which is
complemented by 194 low-redshift SN Ia (which are in common with the Pantheon+
sample) spanning $0.025<z<0.1$ [26].101010Data available at
https://github.com/des-science/DES-SN5YR. The contribution of SN Ia in
general, and the differences between the different datasets111111We note that
these SN Ia data sets are not independent of each other and we therefore do
not combine their results: Pantheon+ and Union3 share $\sim 1360$ supernovae
but differ in their analysis methodology and marginalization over
astrophysical and systematic parameters, while DES contributes a new high-$z$
data set of $\sim 1500$ photometrically-classified SN Ia but still uses about
194 historical SN Ia at $z<0.1$ in common with the other two. are discussed in
Section 5. In the rest of the paper, we will denote the Pantheon+ dataset as
PantheonPlus, and the DES-SN5YR dataset as DESY5, in order to provide a
consistent, concise set of labels suitable for tables and figure legends, and
to avoid ambiguities with the ‘+’ symbol used to denote the combinations of
datasets.

parametrization | parameter | default | prior
---|---|---|---
background-only | $\Omega_{\mathrm{m}}$ | — | $\mathcal{U}[0.01,0.99]$
no $r_{\mathrm{d}}$ calibration | $r_{\mathrm{d}}h\;(\,\text{Mpc})$ | — | $\mathcal{U}[10,1000]$
with $r_{\mathrm{d}}$ calibration | $H_{0}\;(\,{\rm km\,s^{-1}\,Mpc^{-1}})$ | — | $\mathcal{U}[20,100]$
| $\omega_{\mathrm{b}}$ | — | $\mathcal{U}[0.005,0.1]$
CMB | $\omega_{\mathrm{cdm}}$ | — | $\mathcal{U}[0.001,0.99]$
| $\omega_{\mathrm{b}}$ | — | $\mathcal{U}[0.005,0.1]$
| $100\theta_{\mathrm{MC}}$ | — | $\mathcal{U}[0.5,10]$
| $\ln(10^{10}A_{s})$ | — | $\mathcal{U}[1.61,3.91]$
| $n_{s}$ | — | $\mathcal{U}[0.8,1.2]$
| $\tau$ | — | $\mathcal{U}[0.01,0.8]$
extended | $\Omega_{\mathrm{K}}$ | $0$ | $\mathcal{U}[-0.3,0.3]$
|  $w_{0}$ or $w$ | $-1$ | $\mathcal{U}[-3,1]$
| $w_{a}$ | $0$ | $\mathcal{U}[-3,2]$
| $\sum m_{\nu}\;({\,\rm eV})$ | $0.06$ | $\mathcal{U}[0,5]$
| $N_{\mathrm{eff}}$ | $3.044$ | $\mathcal{U}[0.05,10]$
Table 2:  Parameters and priors used in the analysis. All of the priors are
flat in the ranges given. We consider two parametrizations,“background-only”
when using BAO data only, and “CMB” where data from Planck and ACT are
involved. In both cases, the same priors are used for parameter extensions. A
single massive neutrino of mass $\sum m_{\nu}=0.06\;{\,\rm eV}$ is assumed,
except in the $\Lambda$CDM+$m_{\nu}$ model, for which we consider three
degenerate massive neutrino species ($N_{\nu}=3$ in CAMB). In addition to the
flat priors on $w_{0}$ and $w_{a}$ listed in the table, we also impose the
requirement $w_{0}+w_{a}<0$ in order to enforce a period of high-redshift
matter domination.

###  2.5 Cosmological inference

We included in the cosmological inference code cobaya [166, 165] the
PantheonPlus, Union3 and DESY5 SN Ia likelihoods, as well as our new DESI BAO
likelihoods. The CMB likelihoods used are based on public packages that are
either included in the public cobaya version or available directly from the
respective teams.

When running cobaya, we rely on the Boltzmann code CAMB [176, 177] for
theoretical cosmology calculations. When using the combined Planck+ACT lensing
likelihood, we use higher precision settings as recommended by ACT.

All Bayesian inference is performed using the Metropolis-Hastings MCMC sampler
[178, 179] in cobaya. Table 2 summarises the cosmological parameters that are
sampled over in different runs, and the priors that are placed on them. In the
base $\Lambda$CDM model, for data combinations that only probe the background
evolution (i.e., BAO and SN Ia), we either sample in the parameters
$\Omega_{\mathrm{m}}$ and $r_{\mathrm{d}}h$ or—when using external data such
as from BBN in order to help calibrate the $r_{\mathrm{d}}$ value and break
the $r_{\mathrm{d}}h$ degeneracy in BAO data—in $\Omega_{\mathrm{m}}$, $H_{0}$
and the physical baryon density $\omega_{\mathrm{b}}$. When also including CMB
likelihoods, we sample instead in the standard six-parameter basis
$\left(\omega_{\rm cdm},\omega_{\mathrm{b}},100\theta_{\rm
MC},\ln{(10^{10}A_{s})},n_{s},\tau\right)$, where $\theta_{\rm MC}$ is an
approximation to the acoustic angular scale $\theta_{\ast}$, $A_{s}$ is the
amplitude of the primordial scalar power spectrum and $n_{s}$ is its spectral
index, and $\tau$ is the reionization optical depth. We also consider
extensions of the minimal $\Lambda$CDM model in which other parameters are
allowed to vary: the spatial curvature $\Omega_{\mathrm{K}}$, the sum of
neutrino masses $\sum m_{\nu}$, the single constant dark energy equation of
state parameter $w$ or the two parameters $(w_{0},w_{a})$ when allowing a
time-varying equation of state $w(a)$ [180, 181], and the effective number of
relativistic species $N_{\mathrm{eff}}$. Priors on these extended parameters
are shown in the bottom section of Table 2; in any instance where one of these
parameters is not varied, it is held fixed at the default value listed.

When including CMB data, we take advantage of the hierarchy between (fast)
nuisance parameters and (slow) cosmological parameters by taking more steps
along the latter, setting the oversampling parameter oversample_power to
$0.4$. We also use the so-called dragging method that enables taking larger
steps in the slow parameter subspace with the help of intermediate transitions
of fast parameters [182]. For each dataset and model combination, we run four
chains in parallel, starting from proposal covariance matrices built from
preliminary runs. Chains are stopped when the Gelman-Rubin [183] criterion
${\rm R}-1<0.01$ is satisfied, where ${\rm R}$ is the largest eigenvalue of
the ratio of the inter- to intra-chain covariance matrices. We further require
the effective sample size of the chains to be $\gtrsim 10^{3}$ to achieve
percent precision on the moments of the marginal posteriors. In the case of
symmetric 1D marginalized posteriors we report the mean and standard deviation
of the samples, while the $68\%$ minimal credible interval is quoted in other
cases, except when otherwise stated. We use
getdist121212https://github.com/cmbant/getdist [184] to derive the constraints
presented in this paper. Where appropriate, we also compute the best-fit with
the iminuit [185, 186] algorithm, as implemented in cobaya, starting from the
four maximum a posteriori (MAP) points of the corresponding four MCMC chains.
In the following, we note
$\Delta\chi_{\mathrm{MAP}}^{2}=-2\Delta\log{\mathcal{L}}$ the difference
(times $-2$) in log-posteriors at the maximum posterior points.

##  3 DESI Distance Measurements

###  3.1 Distance-Redshift Results

Our summary Table 1 provides the BAO distance scales measured from each of the
DESI tracers and redshift bins. As noted above, for the BGS and QSO samples,
we report a single result for the angle-averaged quantity
$D_{\mathrm{V}}/r_{\mathrm{d}}$ [79]. For all other redshift bins, we quote
marginalized constraints on both $D_{\mathrm{M}}/r_{\mathrm{d}}$ and
$D_{\mathrm{H}}/r_{\mathrm{d}}$ individually; note however that these two
quantities are correlated with each other, with a correlation factor that
depends on redshift, also provided in Table 1 and fully accounted for in the
BAO likelihoods.

Figure 1 shows a summary of these DESI BAO results in the form of a Hubble
diagram. In order to conveniently display the results from all redshift bins
together, we consistently convert from the
$\left(D_{\mathrm{M}}/r_{\mathrm{d}},D_{\mathrm{H}}/r_{\mathrm{d}}\right)$
basis to
$\left(D_{\mathrm{V}}/r_{\mathrm{d}},D_{\mathrm{M}}/D_{\mathrm{H}}\right)$ for
all tracers as labelled.131313In the
$\left(D_{\mathrm{V}}/r_{\mathrm{d}},D_{\mathrm{M}}/D_{\mathrm{H}}\right)$
basis the measurements from discrete tracers are much more independent of each
other (though not totally so). The Ly$\alpha$ BAO measurements still retain a
significant correlation in this basis. The top row then shows
$D_{\mathrm{V}}/r_{\mathrm{d}}$ (scaled by an arbitrary factor of $z^{-2/3}$
for visualisation purposes) in the left panel, and
$D_{\mathrm{M}}/D_{\mathrm{H}}$ (similarly arbitrarily scaled by $z^{-1}$) in
the right panel. The solid and dashed grey lines in each panel indicate the
corresponding model predictions for the $\Lambda$CDM model that best fit the
DESI data (Section 4.1), and the Planck best-fit $\Lambda$CDM model,
respectively. The lower panel shows the same data again but now as the ratio
of the $D_{\mathrm{V}}/r_{\mathrm{d}}$ and $F_{\mathrm{AP}}\equiv
D_{\mathrm{M}}/D_{\mathrm{H}}$ values to those for the best-fit $\Lambda$CDM
model to DESI data. The solid and dashed grey lines in these panels therefore
represent the same two models as in the top row.

Figure 1: Top row: DESI measurements of the BAO distance scales at different
redshifts, parametrized as (left) the ratio of the angle-averaged distance
$D_{\mathrm{V}}\equiv(zD_{\mathrm{M}}^{2}D_{\mathrm{H}})^{1/3}$ to the sound
horizon at the baryon drag epoch, $r_{\mathrm{d}}$, and (right) the ratio of
transverse and line-of-sight comoving distances $F_{\mathrm{AP}}\equiv
D_{\mathrm{M}}/D_{\mathrm{H}}$, from all tracers and redshift bins as labeled.
For visual clarity and to compress the dynamic range of the plot, an arbitrary
scaling of $z^{-2/3}$ has been applied on the left, and $z^{-1}$ on the right.
The solid and dashed grey lines show model predictions from, respectively, the
flat $\Lambda$CDM model that best fits this data, and from a $\Lambda$CDM
model with parameters matching the Planck best-fit cosmology. The BGS and QSO
data points appear only in the left panel and not the right one because the
signal-to-noise ratio of the data is not yet sufficient to measure both
parameters for these tracers. Bottom row: The same data points and models as
in the top row, but now shown as the ratio relative to the predictions for the
best-fit flat $\Lambda$CDM model.

###  3.2 Internal consistency of DESI results

Figure 1 shows visually that the flat $\Lambda$CDM model provides a good fit
to the DESI BAO results: quantitatively, the $\chi^{2}$ value for this fit is
$12.66$ for 10 degrees of freedom (dof), as we have 12 data points and 2 free
parameters, namely $\Omega_{\mathrm{m}}$ and $H_{0}r_{\mathrm{d}}$ (Table 2).
These two parameters have a direct relationship to the BAO data points shown
in Figure 1, since in the flat $\Lambda$CDM model $\Omega_{\mathrm{m}}$ fully
determines $F_{\mathrm{AP}}(z)$ and fixes the shape of
$D_{\mathrm{V}}/r_{\mathrm{d}}$ as a function of redshift, while
$H_{0}r_{\mathrm{d}}$ sets a redshift-independent constant normalization term
for $D_{\mathrm{V}}/r_{\mathrm{d}}$. The single most anomalous result is the
measurement of $F_{\mathrm{AP}}(z=0.51)$, which is mildly offset (at the $\sim
2\sigma$ significance level) from the $\Lambda$CDM expectation. This appears
to be a simple statistical fluctuation and we do not regard it as significant.
In Appendix A we explicitly confirm that this fluctuation does not alter any
of the conclusions drawn about any cosmological models in Sections 4, 5, 6 and
7 below.

It is also worth confirming that the parameters inferred from each individual
redshift bin are consistent with each other. This is shown in Figure 2, where
in the left panel we show the 68% and 95% credible intervals on
$\Omega_{\mathrm{m}}$ and $H_{0}r_{\mathrm{d}}$ obtained from fitting a flat
$\Lambda$CDM model to each DESI tracer type individually, using the priors
described in Table 2. The slightly anomalous $F_{\mathrm{AP}}(z=0.51)$ value
results in a mild shift in the contour for the $0.4<z<0.6$ LRG1 bin, but we do
not regard this as significant. The individual contours have characteristic
degeneracy directions that depend on redshift [187], and show a strong degree
of overlap, confirming the internal consistency of the DESI data. A
consequence of the change in the degeneracy directions with redshift is that
the combination of all tracers provides a tight final constraint, shown in the
right panel of Figure 2.

Figure 2: Left panel: 68% and 95% credible-interval contours for parameters
$\Omega_{\mathrm{m}}$ and $r_{\mathrm{d}}h$ obtained for a flat $\Lambda$CDM
model from fits to BAO measurements from each DESI tracer type individually,
as labeled. Results from all tracers are consistent with each other and the
change in the degeneracy directions arises from the different effective
redshifts of the samples. Right panel: the corresponding results in flat
$\Lambda$CDM for fits to BAO results from all DESI redshift bins (blue), the
final SDSS results from [139] (orange), and the combination of these two as
described in the text (green). The corresponding result from the CMB
(including CMB lensing) is shown in pink.

###  3.3 Comparison to BAO results from SDSS

The region of the sky and the range of redshifts observed by DESI partially
overlap with those observed by the previous generation of BOSS [188] and eBOSS
[189] survey programs of SDSS [190], whose final BAO results were presented in
[139]. It is therefore pertinent to compare our BAO results to those from
SDSS141414[79] also shows a comparison to BAO measurements from 6dFGS [191]
and WiggleZ [192], although these have lower precision. As a rough guide,
$\sim 70\%$ of the DESI DR1 footprint was covered by BOSS; conversely, $\sim
65\%$ of the BOSS footprint has been covered by DESI DR1. Therefore the input
catalog data used in the DESI and SDSS BAO analyses, while different due to
the details of the instrument performance and observing strategy, nevertheless
have a fraction of shared objects, introducing a correlation that depends on
the redshift and tracer type.

This correlation is greatest at low redshifts, where the DESI BGS and LRG
samples overlap the SDSS Main Galaxy Sample and LRGs from BOSS (LOWZ and
CMASS) and eBOSS. There is also a substantial overlap in volume between the
QSO samples of DESI and eBOSS in the redshift range $0.8<z<2.1$. Despite this,
however, fewer than 15% of the DESI quasars are also present in the eBOSS
catalog. As the measurements of QSO power spectra are shot-noise dominated in
both DESI and eBOSS, this results in a negligible overall correlation between
the results.

For redshifts below $z=0.6$, the DESI DR1 data currently covers a smaller
effective volume than SDSS (compare the $V_{\rm eff}$ values in Table 1 to
those reported in [139]), but this will change in future data releases. The
redshift binning used differs between DESI and SDSS, which makes a direct
comparison of BAO results complicated at $z<0.4$, but the $0.4<z<0.6$ and
$0.6<z<0.8$ redshift bins closely match the redshift ranges used by BOSS and
eBOSS.151515eBOSS actually used the redshift range $0.6<z<1.0$ to define the
target sample, but the number density of LRGs in the catalog fell of very
sharply beyond $z>0.85$ so this approximation holds. The SDSS results in these
bins are

$\left.\begin{aligned} D_{\mathrm{M}}/r_{\mathrm{d}}(z=0.51)=13.36\pm 0.21,\\\
D_{\mathrm{H}}/r_{\mathrm{d}}(z=0.51)=22.33\pm 0.58,\end{aligned}\ \right\\}\
\ \mbox{\text{\parbox{85.35826pt}{\begin{flushleft} SDSS~{}LRG $0.4<z<0.6$,
\end{flushleft}}}}$ (3.1)

and

$\left.\begin{aligned} D_{\mathrm{M}}/r_{\mathrm{d}}(z=0.70)=17.86\pm 0.33,\\\
D_{\mathrm{H}}/r_{\mathrm{d}}(z=0.70)=19.33\pm 0.53,\end{aligned}\ \right\\}\
\ \mbox{\text{\parbox{85.35826pt}{\begin{flushleft} SDSS~{}LRG $0.6<z<0.85$,
\end{flushleft}}}}$ (3.2)

respectively, which can be compared to the DESI results in Table 1. While the
results at effective redshift $z=0.51$ are in good agreement, a larger
difference can be seen in the $0.6<z<0.8$ redshift bin, particularly in
comparison to the DESI result $D_{\mathrm{M}}/r_{\mathrm{d}}(z=0.71)=16.85\pm
0.32$.

To gauge the significance of this disagreement, we can estimate the degree of
correlation between power spectrum measurements from DESI and SDSS from Eq.
2.7, using approximate values for the total volume overlap, the number density
of galaxies in common between the two surveys, and assuming a representative
amplitude of the power spectrum $P_{0}=10^{4}\;(h^{-1}{\rm Mpc})^{3}$ for both
samples. This gives indicative values of $C=0.35$ for $0.4<z<0.6$ and $C=0.21$
for $0.6<z<0.8$. Assuming that this degree of correlation also applies to BAO
results derived from the power spectra, and accounting for the predicted
changes in $D_{\mathrm{M}}/r_{\mathrm{d}}$ and $D_{\mathrm{H}}/r_{\mathrm{d}}$
between $z=0.70$ and $z=0.71$ in the DESI fiducial cosmology, the discrepancy
between the DESI and SDSS results in the $0.6<z<0.8$ redshift range is at
roughly the $\sim 3\sigma$ level.

The cause of this difference is not clear. As discussed in [79], numerous
improvements in the data processing and BAO fitting methods have been
introduced for DESI, and reprocessing the raw BOSS/eBOSS catalog data using
the DESI BAO pipeline gives small shifts compared to the results published in
[139]. However, this accounts for a shift of at most a small fraction of the
published uncertainties in $D_{\mathrm{M}}/r_{\mathrm{d}}$ and
$D_{\mathrm{H}}/r_{\mathrm{d}}$. We note that in this redshift bin the SDSS
catalog was a composite formed from the BOSS CMASS galaxy sample extending to
$z\simeq 0.75$ and the deeper eBOSS LRG sample extending to $z=1$ over a much
smaller sky footprint; in contrast the DESI LRG sample is much more uniform.
The cause of the difference may simply an unlucky sample variance
fluctuation—if so, this will soon become clear with later DESI data releases,
which will have a much larger $V_{\rm eff}$ in this redshift range.

While the degree of correlation between BAO results from the galaxy and quasar
samples for DESI and SDSS has only been approximately estimated as described
above, in the Ly$\alpha$ forest BAO analysis, the degree of correlation has
been more thoroughly quantified in [80] and shown to agree very well. [80]
also provides results for a combined “DESI+SDSS” Ly$\alpha$ BAO measurement:

$D_{\mathrm{M}}/r_{\mathrm{d}}(z=2.33)=38.80\pm 0.75,$ (3.3)

and

$D_{\mathrm{H}}/r_{\mathrm{d}}(z=2.33)=8.72\pm 0.14,$ (3.4)

with an anticorrelation of $\rho=-0.48$ between
$D_{\mathrm{M}}/r_{\mathrm{d}}$ and $D_{\mathrm{H}}/r_{\mathrm{d}}$.

The $D_{\mathrm{M}}/r_{\mathrm{d}}(z=0.71)$ measurement from LRGs yields the
greatest difference between SDSS and DESI at any redshift. Nevertheless, using
the BAO measurements from all redshifts together, the BAO distance ladders
obtained from DESI and SDSS result in consistent inference of cosmological
parameters. The right panel of Figure 2 shows the posterior credible intervals
for the parameters $\Omega_{\mathrm{m}}$ and $r_{\mathrm{d}}h$ in a flat
$\Lambda$CDM cosmology from fitting to the DESI (blue) and SDSS (orange) BAO
data. The DESI values are described in Section 4 and Table 3. The results are
clearly in very good agreement with each other, with no significant difference
in $\Omega_{\mathrm{m}}$ and a shift of just $\sim 1\sigma$ in
$r_{\mathrm{d}}h$.

In this paper our primary focus will be on the cosmological consequences of
the DESI DR1 BAO results alone, as a homogeneously analysed dataset across
redshift and tracer type. However, it is also possible to construct a dataset
of BAO distance measurements from the combination of DESI and SDSS results in
order to maximise measurement precision across the entire redshift range
$0.1<z<4.2$. Bearing in mind that the degree of correlation between BAO
results from discrete galaxy and quasar tracers in the two surveys has not
been precisely quantified—to avoid double-counting information—this combined
sample should be selected by choosing the result from the survey covering the
larger effective volume $V_{\rm eff}$ at a given redshift. Thus, the composite
BAO dataset can be constructed as follows:

  * •

at $z<0.6$ where SDSS currently has a larger $V_{\rm eff}$, we use the SDSS
results at $z_{\rm eff}=0.15,0.38$ and $0.51$ in place of the DESI BGS and
lowest-redshift LRG points;

  * •

at $z>0.6$ where DESI has $V_{\rm eff}$ larger than that of SDSS, we use the
DESI results from LRGs over $0.6<z<0.8$, the LRG+ELG combination over
$0.8<z<1.1$, and ELGs and QSOs at higher redshifts; and

  * •

for the Ly$\alpha$ BAO we use the combined DESI+SDSS result from Eqs. 3.3 and
3.4 above.

We use the label “DESI+SDSS” to refer to this composite BAO dataset, while
reiterating that this is not the same as simply combining the likelihoods from
each survey individually due to the overlap in volumes. The posterior in
parameters $(\Omega_{\mathrm{m}},r_{\mathrm{d}}h)$ inferred from fitting a
flat $\Lambda$CDM model to this dataset is shown in green in the right panel
of Figure 2. In Appendix A we present constraints using the DESI+SDSS data
combination in various cosmological models, specifically comparing them to
constraints using DESI data alone.

##  4 Cosmological constraints in the $\Lambda$CDM model

model/dataset | $\Omega_{\mathrm{m}}$ | $H_{0}$ | $10^{3}\Omega_{\mathrm{K}}$ | $w$ or $w_{0}$ | $w_{a}$
---|---|---|---|---|---
[$\,{\rm km\,s^{-1}\,Mpc^{-1}}$]
Flat $\bm{\Lambda}$CDM |  |  |  |  |
DESI | $0.295\pm 0.015$ | — | — | — | —
DESI+BBN | $0.295\pm 0.015$ | $68.53\pm 0.80$ | — | — | —
DESI+BBN+$\theta_{\ast}$ | $0.2948\pm 0.0074$ | $68.52\pm 0.62$ | — | — | —
DESI+CMB | $0.3069\pm 0.0050$ | $67.97\pm 0.38$ | — | — | –
$\bm{\Lambda}$CDM+$\bm{\Omega_{\mathrm{K}}}$ |  |  |  |  |
DESI | $0.284\pm 0.020$ | — | $65^{+68}_{-78}$ | — | —
DESI+BBN+$\theta_{\ast}$ | $0.296\pm 0.014$ | $68.52\pm 0.69$ | $0.3^{+4.8}_{-5.4}$ | — | —
DESI+CMB | $0.3049\pm 0.0051$ | $68.51\pm 0.52$ | $2.4\pm 1.6$ | — | —
$\bm{w}$CDM |  |  |  |  |
DESI | $0.293\pm 0.015$ | — | — | $-0.99^{+0.15}_{-0.13}$ | —
DESI+BBN+$\theta_{\ast}$ | $0.295\pm 0.014$ | $68.6^{+1.8}_{-2.1}$ | — | $-1.002^{+0.091}_{-0.080}$ | —
DESI+CMB | $0.281\pm 0.013$ | $71.3^{+1.5}_{-1.8}$ | — | $-1.122^{+0.062}_{-0.054}$ | —
DESI+CMB+Panth. | $0.3095\pm 0.0069$ | $67.74\pm 0.71$ | — | $-0.997\pm 0.025$ | —
DESI+CMB+Union3 | $0.3095\pm 0.0083$ | $67.76\pm 0.90$ | — | $-0.997\pm 0.032$ | —
DESI+CMB+DESY5 | $0.3169\pm 0.0065$ | $66.92\pm 0.64$ | — | $-0.967\pm 0.024$ | —
$\bm{w_{0}w_{a}}$CDM |  |  |  |  |
DESI | $0.344^{+0.047}_{-0.026}$ | — | — | $-0.55^{+0.39}_{-0.21}$ | $<-1.32$
DESI+BBN+$\theta_{\ast}$ | $0.338^{+0.039}_{-0.029}$ | $65.0^{+2.3}_{-3.6}$ | — | $-0.53^{+0.42}_{-0.22}$ | $<-1.08$
DESI+CMB | $0.344^{+0.032}_{-0.027}$ | $64.7^{+2.2}_{-3.3}$ | — | $-0.45^{+0.34}_{-0.21}$ | $-1.79^{+0.48}_{-1.0}$
DESI+CMB+Panth. | $0.3085\pm 0.0068$ | $68.03\pm 0.72$ | — | $-0.827\pm 0.063$ | $-0.75^{+0.29}_{-0.25}$
DESI+CMB+Union3 | $0.3230\pm 0.0095$ | $66.53\pm 0.94$ | — | $-0.65\pm 0.10$ | $-1.27^{+0.40}_{-0.34}$
DESI+CMB+DESY5 | $0.3160\pm 0.0065$ | $67.24\pm 0.66$ | — | $-0.727\pm 0.067$ | $-1.05^{+0.31}_{-0.27}$
$\bm{w_{0}w_{a}}$CDM+$\bm{\Omega_{\mathrm{K}}}$ |  |  |  |  |
DESI | $0.313\pm 0.049$ | — | $87^{+100}_{-85}$ | $-0.70^{+0.49}_{-0.25}$ | $<-1.21$
DESI+BBN+$\theta_{\ast}$ | $0.346^{+0.042}_{-0.024}$ | $65.8^{+2.6}_{-3.5}$ | $5.9^{+9.1}_{-6.9}$ | $-0.52^{+0.38}_{-0.19}$ | $<-1.44$
DESI+CMB | $0.347^{+0.031}_{-0.025}$ | $64.3^{+2.0}_{-3.2}$ | $-0.9\pm 2$ | $-0.41^{+0.33}_{-0.18}$ | $<-1.61$
DESI+CMB+Panth. | $0.3084\pm 0.0067$ | $68.06\pm 0.74$ | $0.3\pm 1.8$ | $-0.831\pm 0.066$ | $-0.73^{+0.32}_{-0.28}$
DESI+CMB+Union3 | $0.3233^{+0.0089}_{-0.010}$ | $66.45\pm 0.98$ | $-0.4\pm 1.9$ | $-0.64\pm 0.11$ | $-1.30^{+0.45}_{-0.39}$
DESI+CMB+DESY5 | $0.3163\pm 0.0065$ | $67.19\pm 0.69$ | $-0.2\pm 1.9$ | $-0.725\pm 0.071$ | $-1.06^{+0.35}_{-0.31}$

Table 3:  Cosmological parameter results from DESI DR1 BAO data in combination
with external datasets and priors, in the baseline flat $\Lambda$CDM model and
extensions including spatial curvature and two parametrizations of the dark
energy equation of state, as listed. Results are quoted for the marginalized
means and 68% credible intervals in each case, including for upper limits.
Note that DESI data alone measures $r_{\mathrm{d}}h$ and not $H_{0}$, but for
reasons of space this result is omitted from the table and provided in the
text instead. In this and other tables, the shorthand notation “CMB” is used
to denote the addition of temperature and polarisation data from Planck and
CMB lensing data from the combination of Planck and ACT.

###  4.1 Flat background

In this section, we present cosmological constraints for the flat $\Lambda$CDM
cosmological model in which $\Omega_{\mathrm{K}}=0$,
$\Omega_{\Lambda}=1-\Omega_{\mathrm{m}}$. As discussed in Sections 3.2 and 3.3
above, in this model BAO distance scales alone constrain only two free
parameters: the matter density parameter $\Omega_{\mathrm{m}}$ and the
combination $H_{0}r_{\mathrm{d}}$ of the Hubble constant $H_{0}$ and the sound
horizon $r_{\mathrm{d}}$. The 68% credible-interval results for these
parameters are

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.295\pm 0.015,\\\
r_{\mathrm{d}}h&=(101.8\pm 1.3)\,\text{Mpc},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI~{}BAO,
\end{flushleft}}}}$ (4.1)

where to simplify units and notation we quote results for
$r_{\mathrm{d}}h\equiv H_{0}r_{\mathrm{d}}/(100\;\,{\rm
km\,s^{-1}\,Mpc^{-1}})$ instead of $H_{0}r_{\mathrm{d}}$. The posterior
constraints are shown in the right panel of Figure 2, and it is clear that
they are in very good agreement with the previously reported SDSS values of
$\Omega_{\mathrm{m}}=0.299\pm 0.016$ and $r_{\mathrm{d}}h=(100.4\pm
1.3)\,\text{Mpc}$ [139].

It is apparent from Figure 2 that DESI BAO prefer a somewhat larger value of
$r_{\mathrm{d}}h$ compared to the value $r_{\mathrm{d}}h=(98.82\pm
0.82)\,\text{Mpc}$ obtained from the combination of CMB temperature,
polarisation and lensing (or $r_{\mathrm{d}}h=(98.9\pm 1.0)\,\text{Mpc}$ from
the CMB when excluding CMB lensing). It is therefore of interest to quantify
the level of consistency between these two data sets. We do so by calculating
the relative $\chi^{2}$ between the two datasets as

$\chi^{2}=(\mathbf{p}_{A}-\mathbf{p}_{B})^{T}(\mathrm{Cov}_{A}+\mathrm{Cov}_{B})^{-1}(\mathbf{p}_{A}-\mathbf{p}_{B}),$
(4.2)

and converting it to the probability-to-exceed between two datasets (e.g.
[193, 194]). Here $\mathbf{p}_{A}$ and $\mathbf{p}_{B}$ refer to
($\Omega_{\mathrm{m}},r_{\mathrm{d}}h$) parameter vectors obtained from fits
to the two datasets, and $\mathrm{Cov}_{A}$ and $\mathrm{Cov}_{B}$ to the
corresponding $2\times 2$ covariances. Applying this statistic to DESI DR1 BAO
and CMB, we find that the differences in this parameter space amount to a mild
$1.9\sigma$-level discrepancy when including CMB lensing, or $1.6\sigma$
without. We thus consider that DESI BAO data are consistent with the CMB in
this parameter space.

Since BAO distance measurements alone are sensitive to the combination
$H_{0}r_{\mathrm{d}}$ [195], an external calibration of the sound horizon
$r_{\mathrm{d}}$ is required in order to break the $H_{0}$–$r_{\mathrm{d}}$
degeneracy and obtain a constraint on the Hubble constant $H_{0}$. This method
of calibrating the BAO distance scales using the sound horizon at early times
is known as the “inverse distance ladder” approach [196, 197]. Directly
calibrating the BAO standard ruler using the value $r_{\mathrm{d}}=147.09\pm
0.26\,\text{Mpc}$ obtained from using all CMB and CMB lensing information [15]
gives

$H_{0}=(69.29\pm 0.87)\,{\rm km\,s^{-1}\,Mpc^{-1}}\qquad(\mbox{DESI
BAO\,+\,$r_{\mathrm{d}}$ from CMB}).$ (4.3)

An alternative approach that is more independent of CMB information is to use
a prior on $\Omega_{\mathrm{b}}h^{2}$ obtained from BBN (Section 2.4.1), which
is sufficient to determine $r_{\mathrm{d}}$ and break the degeneracy [198,
68]. Within flat $\Lambda$CDM, DESI DR1 BAO + BBN give

$H_{0}=(68.53\pm 0.80)\,{\rm
km\,s^{-1}\,Mpc^{-1}}\qquad(\mbox{DESI~{}BAO\,+\,BBN}).$ (4.4)

This constraint is 20% more precise than the equivalent one obtained from SDSS
BAO data, $H_{0}=(67.35\pm 0.97)\,{\rm km\,s^{-1}\,Mpc^{-1}}$ [139], which is
a result of the improved precision of the DESI BAO measurements at higher
redshifts, leading to narrower posterior constraints perpendicular to the main
degeneracy direction in the $\Omega_{\mathrm{m}}$–$r_{\mathrm{d}}h$ plane
(Figure 2). The central value is also shifted by about $\sim 1\sigma$ to
higher $H_{0}$, a direct consequence of the similar shift in
$H_{0}r_{\mathrm{d}}$ shown in Figure 2. However, the two results are fully
consistent with each other.

We can also use a conservative model-independent prior on the acoustic angular
scale $\theta_{\ast}$ seen from the CMB (Section 2.4.2), without including
additional CMB information. While on its own knowledge of $\theta_{\ast}$ is
insufficient to break the $r_{\mathrm{d}}$–$h$ degeneracy (because the baryon
density $\Omega_{\mathrm{b}}h^{2}$ remains unknown, so $r_{d}$ cannot be
fixed), it is extremely robust and, in combination with BBN information,
further tightens the $H_{0}$ result to

$H_{0}=(68.52\pm 0.62)\;\,{\rm km\,s^{-1}\,Mpc^{-1}}\qquad(\mbox{DESI
BAO\,+\,BBN\,+\,$\theta_{\ast}$}),$ (4.5)

a 0.9% measurement of the Hubble constant. As discussed further in Section 6
below, this more precise constraint can also be extended to be robust to many
early-universe extensions of the base $\Lambda$CDM model, in particular to
assumptions about the effective number of relativistic species
$N_{\mathrm{eff}}$.

It is interesting to note that the direct $r_{\mathrm{d}}$ calibration from
the CMB produces the weakest constraint on $H_{0}$ despite apparently using
the most external information and being the most model-dependent. This is
because BBN and $\theta_{\ast}$ information affect $\Omega_{\mathrm{m}}$ and
are applied consistently in the posterior sampling, and thus leverage the
degeneracy between $\Omega_{\mathrm{m}}$ and $r_{\mathrm{d}}h$ from BAO data,
while a prior directly applied on $r_{\mathrm{d}}$ cannot do so.

Figure 3: Left panel: marginalized posterior constraints on matter density
$\Omega_{\mathrm{m}}$ and the Hubble constant $H_{0}$, obtained from combining
DESI BAO data with external data used to calibrate the sound horizon
$r_{\mathrm{d}}$, in a flat $\Lambda$CDM cosmological model. The combinations
shown use a prior on $\omega_{\mathrm{b}}$ determined from BBN (blue), the
combination of a BBN $\omega_{\mathrm{b}}$ prior and measurement of the
acoustic angular scale $\theta_{\ast}$ (orange), and $r_{\mathrm{d}}$ directly
calibrated from CMB results from Planck (green). The pink contour shows the
corresponding constraints from the combination of CMB and CMB lensing. Right
panel: The marginalized 1D posteriors on $\Omega_{\mathrm{m}}$ in flat
$\Lambda$CDM, from DESI BAO, CMB and the three SN datasets, as labelled.

The left panel of Figure 3 summarises the constraints in the
$\Omega_{\mathrm{m}}$–$H_{0}$ plane obtained from the combination of DESI BAO
data with each of the external priors discussed so far, and compares them to
the combined CMB result from Planck and ACT. All combinations including DESI
data favour somewhat higher values of $H_{0}$ and lower values of
$\Omega_{\mathrm{m}}$ than the mean values for the CMB posterior. However, the
results are not in serious tension: using the metric outlined in Eq. 4.2
above, the biggest discrepancy is at the $\sim 2.1\sigma$ level, for
DESI+BBN+$\theta_{\ast}$ compared to CMB. Given this level of agreement, there
is no issue with combining DESI and CMB data to obtain joint constraints;
doing so we find

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.3069\pm 0.0050,\\\
H_{0}&=(67.97\pm 0.38)\,{\rm km\,s^{-1}\,Mpc^{-1}}\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{56.9055pt}{\begin{flushleft} DESI~{}BAO+ CMB.
\end{flushleft}}}}$ (4.6)

These results are summarised in Table 3, which also shows parameter
constraints obtained in other extended models.

A final instructive comparison within the context of the flat $\Lambda$CDM is
between the constraints on the matter density $\Omega_{\mathrm{m}}$ offered by
DESI and SN Ia. These are shown in the right panel of Figure 3. PantheonPlus
reports $\Omega_{\mathrm{m}}=0.331\pm 0.018$, Union3 gives
$\Omega_{\mathrm{m}}=0.359^{+0.025}_{-0.028}$, and DESY5 gives
$\Omega_{\mathrm{m}}=0.353\pm 0.017$. There is therefore a moderate variation
in both central values and quoted uncertainties across different SN Ia
compilations, but all of them prefer higher values of $\Omega_{\mathrm{m}}$
than DESI and the CMB. The statistical significance of the differences
compared to DESI, calculated as described above, stands at $1.6\sigma$ for
PantheonPlus, $2.0\sigma$ for Union3, and $2.6\sigma$ for DESY5. While they do
not meet a $3\sigma$ threshold for significant tension, these numbers indicate
a degree of disagreement between these datasets and DESI results when
interpreted in the flat $\Lambda$CDM model. Should these mismatches persist
and become more significant when more data is acquired, they will require
further investigation.

###  4.2 $\Lambda$CDM model with free spatial curvature

Figure 4: 68% and 95% marginalized posterior constraints on
$\Omega_{\mathrm{m}}$–$\Omega_{\Lambda}$ plane (left) and
$\Omega_{\mathrm{m}}$–$\Omega_{\mathrm{K}}$ (right) in the one-parameter
extension of the $\Lambda$CDM model with free curvature,
$\Lambda$CDM$+\Omega_{\mathrm{K}}$. In the left panel the supernova contours
are truncated at the lower-left by the $\mathcal{U}[-0.3,0.3]$ prior on
$\Omega_{\mathrm{K}}$.

Relaxing the condition of spatial flatness in the $\Lambda$CDM model, we allow
the curvature parameter $\Omega_{\mathrm{K}}$ to vary. In an FLRW background,
this is equivalent to allowing the dark energy density
$\Omega_{\Lambda}=1-\Omega_{\mathrm{m}}-\Omega_{\mathrm{K}}$ to vary
independently from the matter density $\Omega_{\mathrm{m}}$, while still
keeping the dark-energy equation of state fixed at $w=-1$. Because DESI
provides relative measures of the BAO scale at multiple redshifts, it can
determine the expansion rate as a function of redshift and thus measure
$\Omega_{\Lambda}$ independent of any calibration of the sound horizon from
external data [199]. In this model DESI alone thus measures the parameters
$\left(\Omega_{\mathrm{m}},\Omega_{\Lambda},r_{\mathrm{d}}h\right)$, and finds
the following 68% credible-interval constraints

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.284\pm 0.020,\\\
\Omega_{\Lambda}&=0.651^{+0.068}_{-0.057},\\\ r_{\mathrm{d}}h&=(100.9\pm
1.6)\,{\rm Mpc},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{99.58464pt}{\begin{flushleft} DESI~{}BAO.
\end{flushleft}}}}$ (4.7)

Expressed in terms of curvature, the DESI result is
$\Omega_{\mathrm{K}}=0.065^{+0.068}_{-0.078}$. Figure 4 shows the
corresponding 2D credible intervals in both
$(\Omega_{\mathrm{m}},\Omega_{\Lambda})$ and
$(\Omega_{\mathrm{m}},\Omega_{\mathrm{K}})$ plane.

The addition of model-independent information on the acoustic angular scale
$\theta_{\star}$ effectively extends the redshift range of the BAO distance
ladder, reaching all the way to last scattering at $z\sim 1090$. This
dramatically improves the curvature constraints, giving
$\Omega_{\mathrm{K}}=0.0108^{+0.015}_{-0.0056}$. Adding a BBN prior this is
further tightened to

$\left.\Omega_{\mathrm{K}}=0.0003^{+0.0048}_{-0.0054}\quad\mbox{\text{\parbox{128.0374pt}{\begin{flushleft}
(DESI~{}BAO\,+\,BBN\,+\,$\theta_{\ast}$). \end{flushleft}}}}\right.$ (4.8)

The corresponding curvature constraint from the CMB is

$\left.\Omega_{\mathrm{K}}=-0.0102\pm
0.0054\quad\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} (CMB).
\end{flushleft}}}}\right.$ (4.9)

This value is $\sim 2\sigma$ away from $\Omega_{\mathrm{K}}=0$, and the
difference is even larger (over $3\sigma$) when not including CMB lensing.
This is a well-known issue in the Planck PR3 likelihood (see discussion in
[15], as well as, e.g., [200, 201]) caused by the combination of the geometric
degeneracy limiting CMB measurements of curvature, and the observed slight
excess of smoothing of high-$\ell$ peaks in the temperature power spectrum.
Later Planck PR4 likelihoods somewhat alleviate, but do not completely remove,
this CMB preference for $\Omega_{\mathrm{K}}<0$ [162, 163, 164]. Thus BAO
results are crucial in providing a curvature constraint that is independent of
the CMB. The combined result from DESI BAO and all CMB information is

$\left.\Omega_{\mathrm{K}}=0.0024\pm
0.0016\quad\mbox{\text{\parbox{99.58464pt}{\begin{flushleft} (DESI~{}BAO+CMB).
\end{flushleft}}}}\right.$ (4.10)

##  5 Dark energy

Over the past quarter century, an impressive variety of cosmological
observations have confirmed and vastly strengthened the evidence that the
expansion of the universe is accelerating. The standard cosmological model has
a flat background geometry, but a sub-critical total (cold and baryonic)
matter density. While the measurements thus far are in good agreement with the
simplest flat $\Lambda$CDM scenario—a vacuum energy described by the
cosmological constant dominating the energy density and responsible for the
late-time acceleration—much effort is dedicated to measuring the expansion and
growth history of the universe and looking for any departures from this model.
In this regard, constraining dark energy models beyond the simplest
$\Lambda$CDM model is a principal goal of DESI.

###  5.1 Flat $w$CDM model

Figure 5: Constraints on $\Omega_{\mathrm{m}}$ and $w$ in the flat $w$CDM
model. The constraints from DESI BAO alone are shown in blue, those from the
CMB in pink, and different SN Ia compilations in solid and dashed green. The
orange contour shows the combined constraint from DESI, CMB and PantheonPlus
SN Ia. All contours show 68% and 95% credible intervals. Note the remarkable
complementarity of cosmological probes in this plane.

Although a cosmological constant fits existing data well, the tiny observed
value of $\Lambda$ relative to typical scales in particle physics poses great
theoretical challenges [202, 203, 204]. Acceleration physics beyond $\Lambda$
necessarily has dynamics – time dependence (and spatial perturbations, though
these have diminishing effect the closer the time dependence is to $\Lambda$,
i.e. constant).

At the background cosmology level entering cosmic distances, the acceleration
physics can be treated as an effective dark energy density and pressure. Thus
we have a dark energy equation of state parameter, or pressure to energy
density ratio, $w(a)$, and a current dark energy density value, $\Omega_{\rm
de}$, to describe the dark energy component. In the early 2000s, with the time
variation $w(a)$ inaccessible to observations, analyses often fixed
$w=\,$const ($w$CDM), but this is insensitive to crucial dynamics that might
be indicated by data. $w$CDM can however still be useful as an alert if the
recovered constraint on constant $w$ has statistically significant deviation
from $w=-1$. Note the converse is not true: measuring $w=-1$ assuming $w$ is
constant does not indicate $\Lambda$ is correct (known as the “mirage of
$\Lambda$” [205]).

Figure 5 shows the constraints on $\Omega_{\mathrm{m}}$ and $w$ from a variety
of different data and combinations. In this plane the DESI contours are close
to vertically aligned, providing a tight constraint on $\Omega_{\mathrm{m}}$
that is largely independent of $w$, in contrast to SN Ia and CMB probes, which
show distinctive degeneracy directions corresponding to the transverse
comoving distances that each of these probes constrains [206]. We find

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.293\pm 0.015,\\\
w&=-0.99^{+0.15}_{-0.13},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI~{}BAO,
\end{flushleft}}}}$ (5.1)

from DESI alone, while combining DESI BAO with BBN and $\theta_{\ast}$
significantly tightens the constraint on $w$ to $w=-1.002^{+0.091}_{-0.080}$.
Adding CMB data shifts the contours slightly along the CMB degeneracy
direction, giving

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.281\pm 0.013,\\\
w&=-1.122^{+0.062}_{-0.054},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{99.58464pt}{\begin{flushleft} DESI~{}BAO+CMB.
\end{flushleft}}}}$ (5.2)

Finally, the tightest constraints are obtained from the combination of these
data with SN Ia. For example for the PantheonPlus SN Ia dataset:

$\left.\begin{aligned} \Omega_{\mathrm{m}}&=0.3095\pm 0.0069,\\\ w&=-0.997\pm
0.025,\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{99.58464pt}{\begin{flushleft}
DESI+CMB\allowbreak+PantheonPlus. \end{flushleft}}}}$ (5.3)

Similar constraints are obtained when substituting PantheonPlus SN Ia for
DESY5 or Union3 (though with slightly larger uncertainties in the latter
case). These results are summarised in Table 3. In summary, DESI data, both
alone and in combination with other cosmological probes, do not show any
evidence for a constant equation of state parameter different from $-1$ when a
flat $w$CDM model is assumed.

###  5.2 Flat $w_{0}w_{a}$CDM model

Figure 6: Left panel: 68% and 95% marginalized posterior constraints in the
$w_{0}$–$w_{a}$ plane for the flat $w_{0}w_{a}$CDM model, from DESI BAO alone
(black dashed), DESI + CMB (pink), and DESI + SN Ia, for the PantheonPlus
[24], Union3 [25] and DESY5 [26] SNIa datasets in blue, orange and green
respectively. Each of these combinations favours $w_{0}>-1$, $w_{a}<0$, with
several of them exhibiting mild discrepancies with $\Lambda$CDM at the
$\gtrsim 2\sigma$ level. However, the full constraining power is not realised
without combining all three probes. Right panel: the 68% and 95% marginalized
posterior constraints from DESI BAO combined with CMB and each of the
PantheonPlus, Union3 and DESY5 SN Ia datasets. The significance of the tension
with $\Lambda$CDM ($w_{0}=-1$, $w_{a}=0$) estimated from the
$\Delta\chi_{\mathrm{MAP}}^{2}$ values is $2.5\sigma$, $3.5\sigma$ and
$3.9\sigma$ for these three cases respectively.

Taking into account the physical dynamics of dark energy, the parametrization
$w(a)=w_{0}+w_{a}\,(1-a)$ was derived and has been demonstrated to match the
background evolution of distances arising from exact dark energy equations of
motion to an accuracy of $\sim 0.1\%$ for viable cosmologies over a wide range
of physics – scalar fields, modified gravity, phase transitions [181, 207]. In
this section, we present constraints on this model, referred to as
$w_{0}w_{a}$CDM, which reduces to $\Lambda$CDM for $w_{0}=-1$, $w_{a}=0$.
Constraining the $w_{0}$–$w_{a}$ parameter space and its corresponding
behavior as well as distinguishing it from $\Lambda$ is a key science goal of
DESI.

We adopt wide flat priors on $w_{0}$ and $w_{a}$ (Table 2), together with the
condition $w_{0}+w_{a}<0$ imposed to enforce a period of high-redshift matter
domination. Since the parameter space we explore includes models whose
equation of state crosses the $w=-1$ boundary, we use the parametrized post-
Friedmann approach [208] to compute the dark energy perturbations when
calculating the CMB angular power spectrum. Figure 6 shows the marginalized
posteriors in the $w_{0}$–$w_{a}$ plane from DESI and combinations with other
external datasets. DESI alone does not have sufficient power to break the
degeneracy between $w_{0}$ and $w_{a}$ and thus the results are cut off by our
priors (see Table 2),

$\left.\begin{aligned} w_{0}&=-0.55^{+0.39}_{-0.21},\\\
w_{a}&<-1.32,\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI~{}BAO,
\end{flushleft}}}}$ (5.4)

with the upper bound on $w_{a}$ referring to the 68% limit. This represents a
mild pull away from the $\Lambda$CDM value, with a
$\Delta\chi_{\mathrm{MAP}}^{2}$ between the maximum a posteriori of the
$w_{0}w_{a}$CDM model and the maximum of the posterior fixing
$(w_{0},w_{a})=(-1,0)$ of just $-3.7$ for 2 additional degrees of freedom. The
cause of this preference for $w_{0}>-1$ is primarily due to the
$F_{\mathrm{AP}}$ measurement from the $0.4<z<0.6$ LRG bin, which lies
slightly higher (at the $\sim 2\sigma$ level) than the best-fit $\Lambda$CDM
model can accommodate, as shown in Figure 1. In order to better fit this data
point, the equation of state $w(z\lesssim 0.5)$ of the best-fit
$w_{0}w_{a}$CDM model prefers to be high, thus preferentially pulling $w_{0}$
to less negative values than $-1$. On the other hand, to fit the other DESI
points which are all fairly close to the $\Lambda$CDM predictions, the
parameter $w_{a}$ prefers to be strongly negative in order to compensate the
integrated effect of $w(z)$ for those quantities at higher redshift.

Given the small $\Delta\chi_{\mathrm{MAP}}^{2}$ it is clear there is no
statistical preference for $w_{0}w_{a}$CDM from DESI alone. CMB data alone
also gives $\Delta\chi_{\mathrm{MAP}}^{2}=-3.7$ for the MAP $w_{0}w_{a}$CDM
model compared to fixing $(w_{0},w_{a})=(-1,0)$, again showing no statistical
preference. Nevertheless, given the overlap of the CMB and DESI contours in
the $\left(w_{0}>-1,w_{a}<0\right)$ quadrant, the combined results give

$\left.\begin{aligned} w_{0}&=-0.45^{+0.34}_{-0.21},\\\
w_{a}&=-1.79^{+0.48}_{-1.0},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI~{}BAO+CMB,
\end{flushleft}}}}$ (5.5)

and the $\Delta\chi_{\mathrm{MAP}}^{2}$ decreases to $-9.5$, indicating a
preference for an evolving dark energy equation of state at the $\sim
2.6\sigma$ level. The contours in this scenario are however still impacted by
the priors we have assumed, thus care is required in interpreting these
shifts.

SN Ia data alone allow for $w_{a}<0$ and, as shown in the left panel of Figure
6, in combination with DESI BAO they also marginally favor $w_{0}>-1$ although
the statistical significance of this preference depends on the particular SN
Ia dataset and is not overwhelming in any case. In order to break the
degeneracy in the $w_{0}$–$w_{a}$ plane it is necessary to look at the joint
constraints from the combination of DESI, CMB and SN Ia probes, shown in the
right panel of Figure 6. These constraints are now not prior-dominated in
either parameter. We find that the results and the associated uncertainties
again vary depending on the choice of supernova dataset, particularly when
comparing PantheonPlus to the other two. However, in all cases, the results
are compatible with each other. We find marginalized posterior means

$\left.\begin{aligned} w_{0}&=-0.827\pm 0.063,\\\
w_{a}&=-0.75^{+0.29}_{-0.25},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft}
DESI+CMB\allowbreak+PantheonPlus, \end{flushleft}}}}$ (5.6)

from combination with PantheonPlus,

$\left.\begin{aligned} w_{0}&=-0.64\pm 0.11,\\\
w_{a}&=-1.27^{+0.40}_{-0.34},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI+CMB\allowbreak+Union3,
\end{flushleft}}}}$ (5.7)

with Union3 SN Ia, and

$\left.\begin{aligned} w_{0}&=-0.727\pm 0.067,\\\
w_{a}&=-1.05^{+0.31}_{-0.27},\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI+CMB\allowbreak+DESY5,
\end{flushleft}}}}$ (5.8)

when using the DESY5 SN Ia data. The $\Delta\chi_{\mathrm{MAP}}^{2}$ values
between the maximum a posteriori of the $w_{0}w_{a}$CDM model and the maximum
of the posterior fixing $(w_{0},w_{a})=(-1,0)$ are $-8.7$, $-15.2$ and $-18.1$
for the combinations of DESI and CMB with PantheonPlus, Union3 and DESY5
respectively. These correspond to preferences for a $w_{0}w_{a}$CDM model over
$\Lambda$CDM model at the $2.5\sigma$, $3.5\sigma$ and $3.9\sigma$
significance levels, respectively.161616Based on several independent MAP
estimates, the uncertainty on these significance levels is of order
$0.2\sigma$.

A Bayesian model-selection analysis gives Bayes factors of $|\ln
B_{21}|=0.65$, $2.4$, and $2.8$ in favor of the $w_{0}w_{a}$CDM model over
$\Lambda$CDM for the combinations of DESI+CMB with PantheonPlus, Union3 and
DESY5, respectively. On Jeffreys’ scale [209, 210], these indicate a weak
preference for $w_{0}w_{a}$CDM over $\Lambda$CDM by the first of these data
combinations, and moderate preference by the latter two.

For this analysis, we ran additional nested-sampling chains using the
PolyChord sampler [211] from which we derived the Bayesian evidence and Bayes
factors using the anesthetic software [212]. We have verified that the
posteriors derived from these auxiliary chains match those derived from the
main MCMC chains described in Section 2.5, and that the Bayes factors reported
here provide consistent interpretations with those estimated from the main
MCMC chains through the Savage-Dickey density ratio approximation [210]. Not
unexpectedly given the current constraints, we find the exact values of the
Bayesian evidence (hence the Bayes factors) mildly depend on the specific
prior ranges for $w_{0}$ and $w_{a}$ (see Table 2).

The Deviance Information Criterion (DIC) for model selection [213] indicates
similar preferences:
$\Delta(\mathrm{DIC})=\mathrm{DIC}_{w_{0}w_{a}\mathrm{CDM}}-\mathrm{DIC}_{\Lambda\mathrm{CDM}}$
of $-2.0$, $-7.4$ and $-7.7$ for the same combinations, respectively (see
Table 1 of [214] for a DIC reference scale).

Further useful information in the $w_{0}w_{a}$CDM analysis is given by the
pivot redshift, $z_{\mathrm{p}}$, and the pivot equation of state,
$w_{\mathrm{p}}\equiv w(z_{\mathrm{p}})$, which inform us about the redshift
and equation-of-state value at which $w(z)$ is best constrained by a given
data set or combination of data sets, see e.g. [215, 216]. We use here the
formalism and definitions of [216] and find

$(z_{\mathrm{p}},w_{\mathrm{p}})=\left\\{\begin{array}[]{cl}(0.57,-1.094\pm
0.070)&\quad\mbox{(DESI+CMB)}\\\\[5.69046pt] (0.26,-0.982\pm
0.028)&\quad\mbox{(DESI+CMB+PantheonPlus)}\\\\[5.69046pt] (0.33,-0.960\pm
0.038)&\quad\mbox{(DESI+CMB+Union3)}\\\\[5.69046pt] (0.26,-0.946\pm
0.026)&\quad\mbox{(DESI+CMB+DESY5).}\end{array}\right.$ (5.9)

Figure 7 provides a summary of the constraints on the expansion history from
the flat cosmological models that we have explored. Here, we show the quantity
$r_{\mathrm{d}}/D_{\mathrm{H}}=H(z)r_{\mathrm{d}}$ as a function of redshift
$z$, rescaled for convenience by $1+z$. The top, middle, and bottom panels
respectively show derived constraints on the temporal evolution of the Hubble
parameter in $\Lambda$CDM, $w$CDM and $w_{0}w_{a}$CDM models from fitting to
DESI data, with the shaded regions indicating the 68% and 95% credible
regions. The same dashed line in all panels indicates the best fit Planck
$\Lambda$CDM model. Because $H(z)/(1+z)$ is directly proportional to
$\dot{a}$, i.e. the expansion rate, its slope with redshift is directly
proportional to minus the cosmic expansion acceleration, $-\ddot{a}$. The
range of the variation of $H(z)$ shows that in each of the three cosmological
models, a period of accelerated expansion (i.e. a negative slope) is required
at $z\lesssim 0.7$.

Figure 7: Model-dependent constraints on the redshift dependence of the Hubble
parameter times the sound horizon, $H(z)r_{\mathrm{d}}\equiv
r_{\mathrm{d}}/D_{\mathrm{H}}$ (scaled by $1/100/(1+z)$ for visual clarity) in
three different classes of models fit to all DESI data from Figure 1. From top
to bottom, the panels show the 68% and 95% credible regions in the
$\Lambda$CDM, $w$CDM and $w_{0}w_{a}$CDM models respectively in the coloured
bands. The dashed line in each panel shows the behaviour in the best fit
Planck $\Lambda$CDM model. For convenience, the scale on the right-hand axis
shows $(r_{\mathrm{d}}/r_{\mathrm{d}}^{\rm Planck})H(z)/(1+z)$.

###  5.3 $w_{0}w_{a}$CDM model with free spatial curvature

Figure 8: Marginalized posteriors on $w_{0}$, $w_{a}$ and
$\Omega_{\mathrm{K}}$ in a model with a time-varying dark energy equation of
state and free spatial curvature, from DESI and CMB data combined with SN Ia
from PantheonPlus, Union3 and DESY5 in blue, orange and green respectively.
All combinations provide tight limits on $\Omega_{\mathrm{K}}$. Constraints on
$w_{0}$ and $w_{a}$ in each case broaden a little compared to those shown in
the flat case (Figure 6) but the overall trend remains the same.

Finally, we also explore the most general case of a time-varying equation of
state $w(a)$ together with allowing free spatial curvature, in the so-called
“$\Omega_{\mathrm{K}}w_{0}w_{a}$CDM model”. Even in this broad parameter
space, the combination of DESI, Planck and SN Ia is able to provide quite
tight constraints. These are shown in Figure 8 for each of the PantheonPlus,
Union3 and DESY5 SN Ia datasets. As seen before, the resultant central values
and uncertainties vary depending on the choice of SN Ia data, but all three
are still broadly consistent with each other, and all combinations place very
tight constraints on the allowed range of deviations from spatial flatness. We
find

$\left.\begin{aligned} w_{0}&=-0.831\pm 0.066,\\\
w_{a}&=-0.73^{+0.32}_{-0.28},\\\ \Omega_{\mathrm{K}}&=0.0003\pm
0.0018,\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft}
DESI+CMB\allowbreak+PantheonPlus, \end{flushleft}}}}$ (5.10)
$\left.\begin{aligned} w_{0}&=-0.64\pm 0.11,\\\
w_{a}&=-1.30^{+0.45}_{-0.39},\\\ \Omega_{\mathrm{K}}&=-0.0004\pm
0.0019,\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI+CMB\allowbreak+Union3,
\end{flushleft}}}}$ (5.11)

and

$\left.\begin{aligned} w_{0}&=-0.725\pm 0.071,\\\
w_{a}&=-1.06^{+0.35}_{-0.31},\\\ \Omega_{\mathrm{K}}&=-0.0002\pm
0.0019,\end{aligned}\ \right\\}\ \
\mbox{\text{\parbox{42.67912pt}{\begin{flushleft} DESI+CMB\allowbreak+DESY5.
\end{flushleft}}}}$ (5.12)

In comparison to the results for the flat $w_{0}w_{a}$CDM model in Section
5.2, the addition of an extra degree of freedom in $\Omega_{\mathrm{K}}$ leads
to a slight broadening of the constraints in the $w_{0}$–$w_{a}$ plane, thus
marginally reducing the significance of the tension with $\Lambda$CDM.
Nevertheless, the trend observed remains the same, with all combinations of
DESI + CMB + SN Ia preferring $w_{0}>-1$ and $w_{a}<0$.

##  6 Hubble constant

The determination of the Hubble constant has been contentious for many decades
[217, 218, 219]. By the turn of the century, a consensus value of around
$70\,\,{\rm km\,s^{-1}\,Mpc^{-1}}$ had emerged [220]. However, since the first
Planck results [221], a growing tension has emerged between $H_{0}$
determinations based on physics of the early universe, which tend to cluster
close to the Planck preferred value of $67\,\,{\rm km\,s^{-1}\,Mpc^{-1}}$
(e.g., [15, 139]), and local distance-ladder measurements based on Cepheids or
other anchors, which mostly prefer larger values around $73\,\,{\rm
km\,s^{-1}\,Mpc^{-1}}$ (e.g., [222, 223, 224, 225]). Although distance ladder
measurements based on the Tip of the Red Giant Branch (TRGB) method [226]
prefer a lower $H_{0}$ than Cepheid-based calibrations, they currently have
larger uncertainties, hence do not yet provide a conclusive assessment. This
tension between the CMB and Cepheid-based determinations stands at the $\sim
4$–$5\sigma$ level in $\Lambda$CDM and, if not due to some unidentified
residual systematics, may indicate deficiencies in the standard cosmological
model. For reviews of the $H_{0}$ tension, see [227, 228, 229].

Figure 9:  $68\%$ credible-interval constraints on the Hubble constant,
assuming the flat $\Lambda$CDM model. The blue, bold whiskers show DESI BAO
measurements in combination with an external BBN prior on
$\Omega_{\mathrm{b}}h^{2}$ and measurement of the acoustic angular scale
$\theta_{\ast}$, the BBN prior alone, or with the CMB measurement of the sound
horizon, $r_{\rm d}^{Planck}$. The thin blue whiskers show the corresponding
results from SDSS BAO and the combination of DESI+SDSS BAO results, as
labelled. The orange whiskers show the results from CMB anisotropy
measurements from Planck and ACT, while the green whiskers show measurements
of $H_{0}$ from the distance ladder with either Cepheids or TRGB.

Figure 10: Left panel: The 68% and 95% credible-interval contours for
$\Omega_{\mathrm{m}}$ and $H_{0}r_{\mathrm{d}}$ obtained from fitting DESI DR1
BAO data in the base flat $\Lambda$CDM model and in four extension models
which modify the background geometry or late-time expansion history. Right
panel: A summary of the tension in the $H_{0}$ measurements obtained from the
DESI BAO results combined with other data, and the SH0ES result of [225],
assuming different cosmological models.

As discussed in Sections 2.4.1 and 4.1, BAO alone cannot provide an absolute
distance measurement, but can constrain $H_{0}$ when combined with some
external information capable of calibrating the sound horizon scale and
breaking the $r_{\mathrm{d}}$–$h$ degeneracy. Within the flat $\Lambda$CDM
cosmological model, calibration of the DESI DR1 BAO with primordial deuterium
abundances and BBN and the acoustic angular scale measurement leads to very
precise $H_{0}$ determinations summarised in Eqs. 4.4 and 4.5, the tightest of
which is $H_{0}=68.52\pm 0.62\;\,{\rm km\,s^{-1}\,Mpc^{-1}}$. As shown in
Figure 9, while this DESI+BBN+$\theta_{\ast}$ constraint is slightly higher
than the Planck CMB value, it remains in tension (at the $3.7\sigma$ level)
with the SH0ES Cepheid-based distance-ladder result $H_{0}=73.04\pm
1.04\;\,{\rm km\,s^{-1}\,Mpc^{-1}}$ (see Figure 10).

When the assumption of the flat $\Lambda$CDM model is relaxed, the $H_{0}$
constraints obtained when adding DESI BAO measurements (in combination with
external priors) are loosened too. There are two qualitatively different ways
in which the more general cosmological models allow more freedom and lead to
weaker $H_{0}$ constraints.

First, the calibration of the sound horizon using BBN relies on assumptions
about the physics at the time of BBN (as well as on the CMB temperature
$T_{0}$, which however is measured very accurately [230]). In particular, if
the effective number of relativistic degrees of freedom $N_{\mathrm{eff}}$ is
allowed to vary from its value $N_{\mathrm{eff}}=3.044$ in the base
$\Lambda$CDM model, the value of $\omega_{\mathrm{b}}$ inferred from light
element abundances is altered. When accounting for this in the
$\omega_{\mathrm{b}}$ prior as discussed in Section 2.4.1 using the results of
[156], but otherwise keeping the late-time geometry unchanged, the $H_{0}$
result from Eq. 4.5 relaxes slightly to

$H_{0}=(68.5\pm 1.4)\;\,{\rm
km\,s^{-1}\,Mpc^{-1}}\qquad(\mbox{DESI\,+\,BBN\,+$\theta_{\ast}$,
free~{}$N_{\mathrm{eff}}$}),$ (6.1)

a $2\%$ measurement. Keeping the BBN prior but dropping the $\theta_{\ast}$
information changes the direction of the contours in the 2D
$(\Omega_{\mathrm{m}},H_{0})$ parameter space but the marginalized 1D
constraint remains very similar, $H_{0}=(68.5\pm 1.5)\;\,{\rm
km\,s^{-1}\,Mpc^{-1}}$. When adding the full CMB information to DESI BAO, this
result slightly shifts to

$H_{0}=(68.3\pm 1.1)\;\,{\rm
km\,s^{-1}\,Mpc^{-1}}\qquad(\mbox{DESI~{}BAO+CMB,~{}free~{}$N_{\mathrm{eff}}$}).$
(6.2)

The second type of additional freedom that leads to weakened $H_{0}$
constraints is allowed by beyond-$\Lambda$CDM models which change the geometry
or the late-time expansion of the universe. This directly affects the BAO
measurement of $r_{\mathrm{d}}h$, thus degrading the precision with which
$H_{0}$ can be determined, even when the early-universe physics remains the
same. The left panel of Figure 10 shows the joint constraint in the
$\Omega_{m}$\- $r_{\mathrm{d}}h$ plane as the amount of freedom in the
parameterization of the expansion history is increased (for the models
considered in the previous section). Clearly, the uncertainty on
$r_{\mathrm{d}}h$ increases dramatically as more model freedom is allowed.
Nevertheless, the central values do not move much, and the results remain
consistent with each other across models: we find $r_{\mathrm{d}}h=(101.9\pm
1.3)\,\text{Mpc}$ ($\Lambda$CDM), $r_{\mathrm{d}}h=(101.0\pm 1.6)\,\text{Mpc}$
($\Lambda$CDM$+\Omega_{\mathrm{K}}$),
$r_{\mathrm{d}}h=(101.7^{+2.9}_{-3.5})\,\text{Mpc}$ ($w$CDM),
$r_{\mathrm{d}}h=(96.4^{+3.3}_{-5.3})\,\text{Mpc}$ ($w_{0}w_{a}$CDM) and
$r_{\mathrm{d}}h=(98.2^{+3.9}_{-6.1})\,\text{Mpc}$
($w_{0}w_{a}$CDM$+\Omega_{\mathrm{K}}$). As a consequence, the $H_{0}$ values
determined from combining DESI BAO with BBN or BBN+$\theta_{\ast}$ priors also
remain stable or slightly decrease, as shown in Table 3. This conclusion
remains true in most cases when combining DESI BAO with the CMB, or with the
CMB and SN Ia information (the one exception is in the $w$CDM model, where the
CMB pulls to higher $H_{0}$ in the absence of SN Ia), i.e., independent of the
late-time modifications studied, BAO data in combination with other probes
always prefer low central values of $H_{0}$. However, as the $H_{0}$
uncertainties increase in these models, the level of tension with SH0ES still
decreases, as summarised in the right panel of Figure 10. Further results on
the value of $H_{0}$ in models which combine extensions to the background
expansion history with changes to the neutrino sector are listed in Table 4.

##  7 Neutrinos

A generic prediction of the hot Big Bang model is a relic neutrino background
which leaves detectable imprints on cosmological observations. Neutrinos are
the only known particles to behave as radiation in the early universe and as
dark matter at late times, so they affect both the acoustic oscillations in
the primordial plasma as well as the background evolution and structure
formation. Cosmological observations are sensitive to both the number of
neutrino species and their total mass (e.g., [69]), making cosmology
constraints complementary to terrestrial neutrino experiments.

###  7.1 Sum of neutrino masses

Figure 11: Left panel: The marginalized 1D posterior constraints on $\sum m_{\nu}$ from different combinations of datasets, in the single parameter extension flat $\Lambda$CDM+$\sum m_{\nu}$ model. As explained in the text, here we use a model with 3 degenerate mass eigenstates and with a minimal prior $\sum m_{\nu}>0$ eV. The minimal masses for the normal or inverted mass ordering scenarios correspond to $\sum m_{\nu}>0.059$ eV and $\sum m_{\nu}>0.10$ eV respectively, shown by the vertical dashed lines and the shaded regions. Right panel: Joint marginalized 68% and 95% credible intervals on $\sum m_{\nu}$ and $H_{0}$ from Planck, CMB and DESI+CMB data, illustrating the degeneracy between these parameters from the CMB, and how DESI BAO data contribute to breaking it. The vertical shaded regions indicate the 68% and 95% constraints on $H_{0}$ from DESI BAO data combined with knowledge of $\theta_{\ast}$ and $\Omega_{\mathrm{b}}h^{2}$ in the $\Lambda$CDM+$\sum m_{\nu}$ model. This shows how DESI BAO breaks the primarily geometric degeneracy to place an upper limit on $\sum m_{\nu}$. model / dataset | $\Omega_{\mathrm{m}}$ |  $H_{0}$ [$\,{\rm km\,s^{-1}\,Mpc^{-1}}$] | $\Sigma m_{\nu}\,[\mathrm{eV}]$ | $N_{\mathrm{eff}}$
---|---|---|---|---
$\mathbf{\Lambda}$CDM+$\bm{\sum m_{\nu}}$ |  |  |  |
DESI+CMB | $0.3037\pm 0.0053$ | $68.27\pm 0.42$ | $<0.072$ | —
$\mathbf{\Lambda}$CDM+$\bm{N_{\mathrm{eff}}}$ |  |  |  |
DESI+CMB | $0.3058\pm 0.0060$ | $68.3\pm 1.1$ | — | $3.10\pm 0.17$
$\bm{w}$CDM+$\bm{\sum m_{\nu}}$ |  |  |  |
DESI+CMB | $0.282\pm 0.013$ | $71.1^{+1.5}_{-1.8}$ | $<0.123$ | —
DESI+CMB+Panth. | $0.3081\pm 0.0067$ | $67.81\pm 0.69$ | $<0.079$ | —
DESI+CMB+Union3 | $0.3090\pm 0.0082$ | $67.72\pm 0.88$ | $<0.078$ | —
DESI+CMB+DESY5 | $0.3152\pm 0.0065$ | $67.01\pm 0.64$ | $<0.073$ | —
$\bm{w}$CDM+$\bm{N_{\mathrm{eff}}}$ |  |  |  |
DESI+CMB | $0.281\pm 0.013$ | $71.0^{+1.6}_{-1.8}$ | — | $2.97\pm 0.18$
DESI+CMB+Panth. | $0.3090\pm 0.0068$ | $67.9\pm 1.1$ | — | $3.07\pm 0.18$
DESI+CMB+Union3 | $0.3097\pm 0.0084$ | $67.8\pm 1.2$ | — | $3.06\pm 0.18$
DESI+CMB+DESY5 | $0.3163\pm 0.0067$ | $67.2\pm 1.1$ | — | $3.09\pm 0.18$
$\bm{w_{0}w_{a}}$CDM+$\bm{\sum m_{\nu}}$ |  |  |  |
DESI+CMB | $0.344^{+0.032}_{-0.026}$ | $64.7^{+2.1}_{-3.2}$ | $<0.195$ |
DESI+CMB+Panth. | $0.3081\pm 0.0069$ | $68.07\pm 0.72$ | $<0.155$ | —
DESI+CMB+Union3 | $0.3240\pm 0.0098$ | $66.48\pm 0.94$ | $<0.185$ | —
DESI+CMB+DESY5 | $0.3165\pm 0.0069$ | $67.22\pm 0.66$ | $<0.177$ | —
$\bm{w_{0}w_{a}}$CDM+$\bm{N_{\mathrm{eff}}}$ |  |  |  |
DESI+CMB | $0.346^{+0.032}_{-0.026}$ | $63.9^{+2.2}_{-3.3}$ | — | $2.89\pm 0.17$
DESI+CMB+Panth. | $0.3093\pm 0.0069$ | $67.5\pm 1.1$ | — | $2.93\pm 0.18$
DESI+CMB+Union3 | $0.3245\pm 0.0098$ | $65.9\pm 1.3$ | — | $2.91\pm 0.18$
DESI+CMB+DESY5 | $0.3172\pm 0.0067$ | $66.6\pm 1.1$ | — | $2.92\pm 0.18$
Table 4: Cosmological parameter results from DESI DR1 BAO data in combination
with external datasets when considering extensions to the baseline
$\Lambda$CDM model in the neutrino sector. Results with two-sided error bars
refer to the marginalized means and 68% credible intervals; upper bounds on
$\sum m_{\nu}$ refer to the 95% limits. Note that the label “CMB” includes CMB
lensing from the combined Planck+ACT likelihood. All constraints on $\sum
m_{\nu}$ assume a model with 3 degenerate mass eigenstates and a minimal prior
$\sum m_{\nu}>0$ eV. (See Eq. 7.3 and Eq. 7.4 for the results using other
priors.) The empty $\sum m_{\nu}$ and $N_{\mathrm{eff}}$ fields indicate that
the fixed respective values of $\sum m_{\nu}=0.06{\,\rm eV}$ and
$N_{\mathrm{eff}}=3.044$ were adopted.

The base model we have adopted so far assumes the sum of neutrino masses to be
$\sum m_{\nu}=0.06$ eV, with a single massive eigenstate and two massless
ones. This is motivated by the lower bound for $\sum m_{\nu}$ from neutrino
oscillation experiments. In this section, we consider a single-parameter
extension beyond this minimal model in which $\sum m_{\nu}$ is allowed to
freely vary, in order to explore the constraining power on $\sum m_{\nu}$ of
DESI data. Amongst terrestrial experiments directly measuring the neutrino
mass, KATRIN [231] has produced the tightest constraints to date, from
measuring the endpoint of the tritium $\beta$-decay spectrum. This gives an
upper bound on the effective electron-neutrino mass that is independent of the
cosmological model of $m_{\beta}<0.8{\,\rm eV}$ (90% CL) [232], equivalent to
$\sum m_{\nu}\lesssim 2.4{\,\rm eV}$ (90% CL).

Neutrino oscillation experiments have also shown that at least two of the
three active neutrino masses are non-zero, but the ordering of these masses is
not known. In the normal ordering or normal hierarchy (NH), the two lowest
mass neutrino eigenstates have the smallest mass splitting, implying that the
total neutrino mass must be $\sum m_{\nu}\geq 0.059{\,\rm eV}$, while in the
inverted hierarchy (IH), however, the smallest mass splitting occurs between
the two highest mass eigenstates, necessitating a total mass of $\sum
m_{\nu}\geq 0.10{\,\rm eV}$ [233]. When allowing $\sum m_{\nu}$ to be a free
parameter, we adopt a model with three degenerate neutrino mass eigenstates.
This degenerate mass model does not exactly correspond to either of the
physically expected NH or IH scenarios; however, it produces a very good
approximation of the observable effects of both [234]. In the event that a
positive detection of non-zero neutrino mass is possible, an analysis using
the degenerate mass model with a prior $\sum m_{\nu}>0$ will recover the
correct value of $\sum m_{\nu}$ for both NH and IH scenarios with little
reconstruction bias [235]. On the other hand, when a positive detection is not
possible, using the degenerate model with appropriately modified priors on
$\sum m_{\nu}$ as above will also recover the correct upper bounds for both
the NH and IH scenarios [236].

The sum of the neutrino masses affects cosmology in two ways. First, the high-
velocity dispersion of neutrinos implies that they free-stream over large
distances, thus suppressing the late-time clustering and power spectrum of
matter at small scales, below the free streaming length. However, if $\sum
m_{\nu}$ is varied with other cosmological parameters along the CMB
degeneracy, the net effect of neutrino masses is an overall almost-scale-
independent suppression of the amplitude of the matter power spectrum
(together with a very small shift in the BAO scale). Second, at the low
redshifts of relevance to DESI, neutrinos are non-relativistic ($\sum
m_{\nu}\gg T\simeq 10^{-3}{\,\rm eV}$) and therefore contribute to the total
non-relativistic matter density
$\omega_{m}=\omega_{b}+\omega_{c}+\omega_{\nu}$, where $\Omega_{\nu}=\sum
m_{\nu}/(93.14{\,\rm eV}h^{2})$, see e.g. [234]. $\sum m_{\nu}$ thus affects
the background evolution and in particular the redshift of matter-$\Lambda$
equality. Forthcoming DESI analyses of the full shape of the galaxy power
spectrum are sensitive to neutrino masses through the first effect, and
constraints from them will be described in [81, 237]. The BAO data used in
this paper only distinguish the background geometry. However, BAO constraints
on the expansion history, when combined with the CMB and CMB lensing, are very
helpful in improving neutrino mass constraints. This is because the CMB is
sensitive to the neutrino mass via the angular diameter distance to
recombination and via the effects of $\sum m_{\nu}$ on the CMB lensing, and
both of these effects can be mimicked in the CMB by varying other cosmological
parameters such as $H_{0}$ and $\omega_{\mathrm{m}}$.

Thus while BAO data are not directly sensitive to the suppression effects of
neutrinos on the power spectrum, by determining the background geometry at low
redshifts they help to break the CMB degeneracy and constrain $H_{0}$, thus
greatly tightening the upper limit on $\sum m_{\nu}$. This effect is
illustrated in the right panel of Figure 11. In extensions of flat
$\Lambda$CDM, the power of BAO to pin down the late-time expansion history
diminishes, resulting in notably weaker $\sum m_{\nu}$ constraints.

For the Planck CMB alone, and assuming flat $\Lambda$CDM, the constraints are

$\left.\sum m_{\nu}<0.21{\,\rm