DESI DR2 BAO measurements and qualified dark-energy inference

The source reports baryon acoustic oscillation (BAO) distance measurements from more than 14 million DESI Data Release 2 galaxies and quasars observed over three years, together with the companion Lyman-$\alpha$ forest BAO results. DESI is the Dark Energy Spectroscopic Instrument on the 4-m Nicholas U. Mayall Telescope at Kitt Peak. Its robotic focal plane assigns 5000 fibers across a 7-square-degree field; ten spectrographs record target, calibration-star and sky spectra from 360 to 980 nm. The paper analyzes main-survey observations from 14 May 2021 through 9 April 2024. It reports wavelength and flux calibration from this setup; this item does not independently reprocess or recalibrate those spectra.

The paper's Table 2 reports 1,188,526 bright-galaxy-sample (BGS) entries in $0.1<z<0.4$, 4,468,483 luminous-red-galaxy (LRG) entries in $0.4<z<1.1$, 6,534,844 emission-line-galaxy (ELG) entries in $0.8<z<1.6$, and 2,062,839 quasar (QSO) entries in $0.8<z<3.5$. Their assignment completeness and redshift-success fractions are, respectively, BGS $75.5\mathrm{\%}$ and $98.8\mathrm{\%}$, LRG $82.6\mathrm{\%}$ and $99.0\mathrm{\%}$, ELG $53.7\mathrm{\%}$ and $73.9\mathrm{\%}$, and QSO $93.6\mathrm{\%}$ and $68.0\mathrm{\%}$. The paper states that the success rates and completeness are computed without additional redshift cuts; the sample ranges and footprints also differ by tracer. Table 3 gives the BAO fitting bins: BGS $0.1<z<0.4$, LRG1 $0.4<z<0.6$, LRG2 $0.6<z<0.8$, the baseline combined LRG3+ELG1 sample $0.8<z<1.1$, ELG2 $1.1<z<1.6$, and QSO $0.8<z<2.1$. In the overlapping $0.8<z<1.1$ interval, the baseline fit combines LRG3 and ELG1; Table 3 also reports their separate counts. The companion Lyman-$\alpha$ analysis uses forest spectra of quasars above $z=2.09$ and quasar positions above $z=1.77$ for the cross-correlation; its forest BAO sample covers $1.8<z<4.2$. These are selected catalog entries, not a single complete sample with uniform redshift coverage. For this item, a spatially flat homogeneous and isotropic FLRW background means the spacetime $I\times {\mathbb{R}}^3$ with metric $d{s}^2=-c^{2}d{t}^2+a(t{)}^2{\delta }_{ij}d{x}^{i}d{x}^j,$ where $I$ is an open interval measured in seconds, $t\in I$, $\mathbf{x}\in {\mathbb{R}}^3$ has coordinates in metres, $a:I\to (0,\mathrm{\infty })$ is a smooth dimensionless scale factor, and $c$ is the speed of light in metres per second. A comoving observer is the worldline $t\mapsto (t,{\mathbf{x}}_0)$ for fixed ${\mathbf{x}}_0$; the metric makes $t$ its proper-time parameter. The fitted branch is expanding over the sampled redshift range: $\dot{a}>0$. Normalize $a(t_0)=1$ today, define redshift by $1+z=a(t_0)\mathrm{/}a(t)$, and set $H(t)=\dot{a}(t)\mathrm{/}a(t)$ and $H(z)=H(t(z))$, with $H$ in inverse seconds. For nonnegative redshifts in the fit, define the transverse comoving and radial Hubble distances by $D_M(z)=c{\int }_0^z\frac{d{z}^{\mathrm{\prime }}}{H(z^{\mathrm{\prime }})},D_H(z)=\frac{c}{H(z)},$ and the comoving sound horizon at baryon-drag redshift $z_d$ by $r_d={\int }_{z_d}^{\mathrm{\infty }}\frac{c_s(z)}{H(z)}dz,$ where $z_d$ is the redshift at which baryons cease to be efficiently dragged by photons in the adopted early-time model, and $c_s(z)$ is the propagation speed, in metres per second, of its coupled baryon-photon acoustic mode. Require $H(z)>0$ over the integration ranges and finite $r_d$ in each fit. The isotropic BAO distance is $D_V(z)=[z{D}_H(z)D_M(z{)}^2{]}^{1\mathrm{/}3}$. Here $D_M,D_H,D_V,r_d$ have units of length, and the reported ratios $D_M\mathrm{/}{r}_d$, $D_H\mathrm{/}{r}_d$, and $D_V\mathrm{/}{r}_d$ are dimensionless. For the model-level galaxy/quasar correlation, let $\mathrm{\Xi }$ be a set of tracer realizations, let $\mathcal{A}$ be a sigma-algebra on $\mathrm{\Xi }$, and let $\mathbb{P}$ be a countably additive probability measure on $\mathcal{A}$ with $\mathbb{P}(\mathrm{\Xi })=1$. For tracer $t$, let ${\mathrm{\Omega }}_t\subseteq {\mathbb{R}}^3$ be the measurable comoving survey window selected by its sky mask and redshift range. A model number-density field is a jointly measurable map $n_t:\mathrm{\Xi }\times {\mathrm{\Omega }}_t\to [0,\mathrm{\infty })$ on the product of $\mathcal{A}$ and the Borel sigma-algebra of ${\mathrm{\Omega }}_t$, with finite second moments at the pairs used. Its mean intensity is ${\bar{n}}_t(\mathbf{x}):={\mathbb{E}}_{\mathbb{P}}[n_t(\cdot ,\mathbf{x})]$, assumed finite and strictly positive on ${\mathrm{\Omega }}_t$; both fields have units of number per comoving volume. Define the dimensionless model overdensity ${\delta }_t(\omega ,\mathbf{x}):=n_t(\omega ,\mathbf{x})\mathrm{/}{\bar{n}}_t(\mathbf{x})-1$. The redshift-space model two-point function is ${\xi }_t(\mathbf{x},\mathbf{r}):={\mathbb{E}}_{\mathbb{P}}[{\delta }_t(\cdot ,\mathbf{x}){\delta }_t(\cdot ,\mathbf{x}+\mathbf{r})]$ whenever both positions lie in ${\mathrm{\Omega }}_t$. The observed catalogs are finite and discrete, not smooth density functions; this equation defines a model correlation and assumes no smoothness of catalog data. For a forest sightline $q$ and selected spectral pixel $\lambda$, the dimensionless transmitted-flux contrast is ${\delta }_q(\lambda )=\frac{f_{\alpha ,q}(\lambda )}{\bar{F}(\lambda )C_q(\lambda )}-1,$ where $f_{\alpha ,q}$ is the measured spectral flux density, $\bar{F}(\lambda )>0$ is the dimensionless mean transmission, and $C_q(\lambda )>0$ is the unabsorbed quasar continuum in the same flux-density units as $f_{\alpha ,q}$. On the retained pixels, the fitted expected flux is the product $\bar{F}(\lambda )C_q(\lambda )$; the continuum alone is not the denominator. The source's BAO feature is the acoustic-scale excess in its binned redshift-space galaxy/ quasar and forest correlations, fit against the paper's templates. The ratios are inferred fit parameters relative to the model sound horizon, not direct measurements of absolute distances. Distances are reported in Mpc, with factors of $c$ and $H$ converted consistently.

The baseline flat $\mathrm{\Lambda }$CDM comparison is the paper's spatially flat Friedmann–Lemaître–Robertson–Walker cosmology in general relativity with a cosmological-constant component whose pressure equals minus its energy density, pressureless cold dark matter and baryons at late times, photons, and the standard baseline neutrino sector ($N_{\mathrm{e}\mathrm{f}\mathrm{f}}=3.044$ and fixed $\sum m_{\nu }=0.06\mathrm{e}\mathrm{V}$ in the source's baseline model). Here $N_{\mathrm{e}\mathrm{f}\mathrm{f}}$ is its effective relativistic-species parameter and $\sum m_{\nu }$ is the total neutrino rest mass; both are fixed inputs in this analysis, not fitted parameters. Their homogeneous components use perfect-fluid stress tensors $T^{\mu }{}_{\nu }=\mathrm{diag}(-\rho ,p,p,p)$ in the stated comoving frame; pressureless matter has $p=0$ and the cosmological-constant component has $p_{\mathrm{\Lambda }}=-{\rho }_{\mathrm{\Lambda }}$. Their densities and the likelihoods are inputs to the fit, not results of this item. The cosmological interpretation combines the BAO estimates with stated cosmic-microwave-background (CMB) likelihoods and, in separate fits, Type Ia supernova samples.

The dynamical-dark-energy extension replaces the cosmological-constant equation of state with that of a homogeneous effective component, $w(a)=w_0+w_a(1-a),$ where $w(a)=p_{\mathrm{d}\mathrm{e}}(a)\mathrm{/}{\rho }_{\mathrm{d}\mathrm{e}}(a)$, with $p_{\mathrm{d}\mathrm{e}}$ and ${\rho }_{\mathrm{d}\mathrm{e}}$ the model pressure and energy density. Since $a(t_0)=1$, $w_0$ is the present equation-of-state parameter; $w_a$ controls its linear dependence on $1-a$. The fits remain spatially flat. The authors use broad uniform priors $w_0\in [-3,1]$, $w_a\in [-3,2]$, and impose $w_0+w_a<0$ to enforce early matter domination. These choices are part of the reported analysis.

DESI Collaboration, abstract, §§II and VII

DESI DR2 Results I, §II.4, Eqs. (1)–(2)

The spectra are reduced and redshifts estimated through the DESI pipelines. The paper reports an original homogeneous reduction (Kibo) and a corrected rerun (Loa) after a co-addition error between separate exposures was found; about 0.1% of redshifts changed significantly between them, and both catalog versions are released. The analysis constructs large-scale-structure catalogs, blinds the BAO peak position until analysis choices are fixed, and applies IterativeFFT density-field reconstruction to galaxy/quasar catalogs. It then estimates the binned redshift-space two-point correlation with the post-reconstruction Landy–Szalay estimator ${\widehat{\xi }}_{\mathrm{r}\mathrm{e}\mathrm{c}}=(DD-2DS+SS)\mathrm{/}RR,$ where $D$ is the displaced data catalog, $S$ is the shifted random catalog, and $R$ is an unshifted random catalog sampling the source-defined survey selection function. Here $DD$, $DS$, $SS$, and $RR$ are the respective source-normalized weighted pair counts in a bin, with $RR>0$. The numerator uses displaced data and shifted randoms; its normalization uses unshifted randoms. The estimator retains the source's reconstruction convention, pair weights and normalizations. The bin coordinates are $s$, the redshift-space pair separation, and $\mu =\cos \theta$, where $\theta$ is the angle between the pair and the line of sight; the source decomposes the binned correlation into Legendre multipoles, focusing on the monopole and quadrupole. These are estimates from finite catalogs, not pointwise evaluations of the model fields. For the Lyman-$\alpha$ forest, the companion source's picca analysis fits the expected-flux product $\bar{F}{C}_q$ using a universal rest-frame template and two diversity parameters per quasar. It forms four correlations: region A ($1040$–$1205\mathring{\text{A}}$) with A, A with region B ($920$–$1020\mathring{\text{A}}$), and each region with quasars. The analysis uses line-of-sight and transverse comoving separations and fits these correlations with the source's stated covariance. For a fixed fiducial cosmology, whose distances and sound horizon carry the superscript $\mathrm{f}\mathrm{i}\mathrm{d}$, define the dimensionless template dilations at positive fitted redshift by ${\alpha }_{\parallel }(z)=\frac{D_H(z)r_d^{\mathrm{f}\mathrm{i}\mathrm{d}}}{D_H^{\mathrm{f}\mathrm{i}\mathrm{d}}(z)r_d},{\alpha }_{\perp }(z)=\frac{D_M(z)r_d^{\mathrm{f}\mathrm{i}\mathrm{d}}}{D_M^{\mathrm{f}\mathrm{i}\mathrm{d}}(z)r_d},{\alpha }_{\mathrm{i}\mathrm{s}\mathrm{o}}=({\alpha }_{\parallel }{\alpha }_{\perp }^2{)}^{1\mathrm{/}3},{\alpha }_{\mathrm{A}\mathrm{P}}={\alpha }_{\parallel }\mathrm{/}{\alpha }_{\perp }\mathrm{.}$ The fit marginalizes over broadband and tracer nuisance parameters and includes tracer-specific systematic-error budgets. The companion Lyman-$\alpha$ analysis fits forest auto-correlations and forest--quasar cross-correlations. Cosmological posteriors are sampled with Cobaya using Metropolis--Hastings Markov-chain Monte Carlo, with theoretical predictions from CAMB and the stated likelihoods. The source's convergence criterion is $R-1<0.01$ for the Gelman--Rubin statistic or effective sample size at least ${10}^3$, whichever requires longer chains. This item records the source's published analysis and does not rerun the pipeline, chains or likelihoods.

The paper fits the resulting distance-redshift constraints under flat $\mathrm{\Lambda }$CDM and $w_0{w}_a$CDM. It compares best-fit likelihoods using $\mathrm{\Delta }{\chi }_{\mathrm{M}\mathrm{A}\mathrm{P}}^2$, and also reports the Deviance Information Criterion (DIC).

Under the Spiegelhalter et al. convention, for observed data $y$, parameter vector $\theta$, the article's likelihood $L(y\mid \theta )$ and posterior $p(\theta \mid y)$ under its stated prior, define the deviance $d(\theta ):=-2\log L(y\mid \theta )$, $\bar{d}:={\mathbb{E}}_{\theta \mid y}[d(\theta )]$, $\bar{\theta }:={\mathbb{E}}_{\theta \mid y}[\theta ]$, $p_D:=\bar{d}-d(\bar{\theta })$, and $\mathrm{D}\mathrm{I}\mathrm{C}:=\bar{d}+p_D$. The DESI paper reports DIC differences; this item does not reproduce or rely on those values.

For the reported $3.1\sigma$ DESI+CMB comparison, the paper has $\mathrm{\Delta }{\chi }_{\mathrm{M}\mathrm{A}\mathrm{P}}^2=-12.5$. It maps this improvement to a two-degree-of-freedom ${\chi }^2$ tail using Wilks' theorem, under the stated assumptions that $\mathrm{\Lambda }$CDM is the null model and errors are Gaussian and correctly estimated. The quoted one-dimensional Gaussian-equivalent significance is therefore conditional on that calibration; it is not a probability that either cosmological model is true.

Separately, for its flat-$\mathrm{\Lambda }$CDM BAO–CMB consistency comparison, set $H_0:=H(z=0)$, let ${\mathrm{\Omega }}_m$ be the present matter energy density divided by the critical energy density ${\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}:=3H_0^2{c}^2\mathrm{/}(8\pi G)$, where $c$ is the fixed SI speed-of-light constant and $G$ is the positive Newton gravitational coupling in the Einstein equations; neither is estimated by this analysis. Define $h:=H_0\mathrm{/}(100\mathrm{k}\mathrm{m}{\mathrm{s}}^{-1}\mathrm{M}\mathrm{p}{\mathrm{c}}^{-1})$. In this expression ${\rho }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$ and the matter density have units of energy per volume, $H_0$ has units of inverse time, $G$ has units of ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and $h,{\mathrm{\Omega }}_m$ are dimensionless. The paper forms ${\chi }_{\mathrm{r}\mathrm{e}\mathrm{l}}^2=(p_{\mathrm{B}\mathrm{A}\mathrm{O}}-p_{\mathrm{C}\mathrm{M}\mathrm{B}}{)}^T(C_{\mathrm{B}\mathrm{A}\mathrm{O}}+C_{\mathrm{C}\mathrm{M}\mathrm{B}}{)}^{-1}(p_{\mathrm{B}\mathrm{A}\mathrm{O}}-p_{\mathrm{C}\mathrm{M}\mathrm{B}}),p=({\mathrm{\Omega }}_m,h{r}_d),$ where $C_{\mathrm{B}\mathrm{A}\mathrm{O}}$ and $C_{\mathrm{C}\mathrm{M}\mathrm{B}}$ are the corresponding two-parameter posterior covariance matrices. The stated sum uses no cross-covariance term, so the discrepancy calculation treats the BAO and CMB parameter estimates as independent for this comparison. It converts this statistic to a probability-to-exceed (the calibrated tail probability) and reports the equivalent Gaussian significance as $2.3\sigma$ with CMB lensing and $2.0\sigma$ when CMB lensing is excluded. This is a separate covariance-based discrepancy calculation, not the Wilks-calibrated $\mathrm{\Delta }{\chi }_{\mathrm{M}\mathrm{A}\mathrm{P}}^2$ comparison above.

DESI Collaboration, §§III.1–III.2 for the galaxy/quasar and forest correlation analyses, §VI Eq. (18) for the BAO–CMB covariance comparison, and §VII Eq. (22) for the Wilks-calibrated model comparison

Padmanabhan et al., §2.2, steps 7–9 and Eq. (1), for the reconstructed estimator

DESI DR2 Results I, §II.4, Eq. (2), for the fitted expected-flux product

Landy and Szalay, abstract and unreconstructed estimator

Spiegelhalter et al., §7.3, Eqs. (36)–(37)

In the flat $\mathrm{\Lambda }$CDM fit, the DESI DR2 BAO parameters are reported in $2.3\sigma$ tension with CMB-derived parameters ($2.0\sigma$ without CMB lensing). The paper also states that flat $\mathrm{\Lambda }$CDM gives a good fit to the DESI BAO data themselves.

For the $w_0{w}_a$CDM extension, the authors report a $3.1\sigma$ preference over $\mathrm{\Lambda }$CDM for DESI+CMB. Adding Type Ia supernova samples gives $2.8\sigma$ with Pantheon+, $3.8\sigma$ with Union3, and $4.2\sigma$ with DESY5. These are different dataset combinations and are reported model-comparison results, not direct measurements of a fundamental dark-energy law.

DESI Collaboration, abstract, §VII and conclusion

The source's tracer table reports assignment completeness and redshift success rates. Its BAO systematic-error budget increases the total uncertainty in ${\alpha }_{\mathrm{i}\mathrm{s}\mathrm{o}}$ by 1--9% and in ${\alpha }_{\mathrm{A}\mathrm{P}}$ by 0.1--2%, while stating that statistical errors from finite survey volume and sampling density dominate. The paper identifies the $\mathrm{\Delta }{\chi }_{\mathrm{M}\mathrm{A}\mathrm{P}}^2$-to-significance mapping as a Wilks/Gaussian approximation with two additional degrees of freedom. The reported result depends on the cosmological models, CMB likelihood, supernova sample and adopted priors. The authors report that the DESI+CMB preference falls from $3.1\sigma$ to $2.7\sigma$ without CMB lensing and to $2.4\sigma$ when only an early-Universe CMB prior is imposed. The supernova combinations span $2.8$–$4.2\sigma$; this spread is retained rather than summarized as a single significance. The analysis is recorded from the publication and is not independently rerun here.

DESI Collaboration, §VII and robustness discussion

The observations support the stated cosmological distance constraints and conditional model comparisons. They do not prove that dark energy evolves, exclude $\mathrm{\Lambda }$CDM, or establish a string-theory prediction. The significances are statistical inferences for the specified data combinations, model parameterizations and calibrations; they are not probabilities that a theory is true.

DESI Collaboration, abstract, §VII and conclusion