DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

Abstract

We present baryon acoustic oscillation (BAO) measurements from more than 14 million galaxies and quasars drawn from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), based on three years of operation. For cosmology inference, these galaxy measurements are combined with DESI Lyman-α forest BAO results presented in a companion paper. The DR2 BAO results are consistent with DESI DR1 and SDSS, and their distance-redshift relationship matches those from recent compilations of supernovae (SNe) over the same redshift range. The results are well described by a flat ΛCDM model, but the parameters preferred by BAO are in mild, 2.3σ tension with those determined from the cosmic microwave background (CMB), although the DESI results are consistent with the acoustic angular scale θ∗ that is well-measured by Planck. This tension is alleviated by dark energy with a time-evolving equation of state parametrized by w0 and wa, which provides a better fit to the data, with a favored solution in the quadrant with w0 greater than −1 and wa less than 0. This solution is preferred over ΛCDM at 3.1σ for the combination of DESI BAO and CMB data. When also including SNe, the preference for a dynamical dark energy model over ΛCDM ranges from 2.8 − 4.2σ depending on which SNe sample is used. We present evidence from other data combinations which also favor the same behavior at high significance. From the combination of DESI and CMB we derive 95% upper limits on the sum of neutrino masses, finding a sum of neutrino masses less than 0.064 eV assuming ΛCDM and less than 0.16 eV in the w0wa model. Unless there is an unknown systematic error associated with one or more datasets, it is clear that ΛCDM is being challenged by the combination of DESI BAO with other measurements and that dynamical dark energy offers a possible solution.

I. Introduction

Cosmic acceleration remains the most pressing problem in contemporary cosmology, implying a pervasive new form of energy with exotic physical properties, or a breakdown of Einstein gravity on cosmological scales, or perhaps both. To probe the physics of acceleration, cosmologists seek to measure the history of cosmic expansion and the history of gravitational clustering with the greatest achievable precision over a wide span of redshift. Baryon acoustic oscillations (BAO) provide a powerful tool for measuring the expansion history, using a characteristic scale that is imprinted on matter clustering by pressure waves that propagate in the coupled baryon-photon fluid of the pre-recombination Universe. This paper examines the cosmological implications of the BAO measurements from the second data release (DR2) of the Dark Energy Spectroscopic Instrument (DESI), consisting of data from the first three years of operation.

Since the first clear detections of BAO in the Sloan Digital Sky Survey and the Two-Degree Field Galaxy Redshift Survey, BAO measurements have played a central role in observational cosmology. The key technical requirement is a large-volume spectroscopic survey with sufficient sampling density, and previous surveys designed with BAO measurements as a defining goal include WiggleZ, the Baryon Oscillation Spectroscopic Survey (BOSS) of SDSS-III, and its extension eBOSS in SDSS-IV. In addition to galaxy and quasar redshifts, BOSS and eBOSS measured BAO in the Lyα forest absorption spectra of quasars above redshift 2, an approach first proposed elsewhere. Transverse BAO can also be measured in photometric surveys, though the precision obtained for a given number of tracers is much higher with spectroscopic redshifts. DESI is designed specifically to enable a spectroscopic BAO survey of unprecedented power and efficiency, as shown in its survey validation based on the early data release. In its first year of observations, DESI already achieved BAO measurements competitive with those of all previous surveys combined. Now, with redshifts of more than 30 million galaxies and quasars, and Lyα forest spectra of more than 820,000 quasars, the DESI DR2 sample is by far the largest spectroscopic galaxy sample to date.

The BAO technique, which is key to DESI, exploits the enhancement of clustering at the scale of the pre-recombination sound horizon. Here the speed of sound in the photon-baryon fluid enters, and the drag redshift is approximately 1060, the redshift at which acoustic waves stall because photons no longer ‘drag’ the baryons. Assuming standard pre-recombination physics, the sound horizon can be computed given the densities of baryons, cold dark matter (CDM), photons, and other relativistic species; scaled to the best-fit Planck values, and to the energy content of three neutrino species that are fully relativistic before the drag epoch, it is 147.05 Mpc.

A measurement of the BAO scale in the transverse direction at a given redshift constrains the transverse comoving distance, which in the limit of small curvature converges to the flat universe case. A measurement in the line-of-sight direction constrains the expansion rate, or the corresponding Hubble distance. Because the inferred distances are relative to the sound horizon, the directly constrained quantities are the ratios of transverse and line-of-sight distance to the sound horizon.

The combination of BAO and cosmic microwave background (CMB) anisotropies is powerful for two reasons. First, the CMB provides tight constraints on the baryon and matter densities, leading to a 0.2% determination of the sound horizon. As a result, the BAO plus CMB combination allows absolute measurements of the transverse and line-of-sight distances. Second, the same physics imprints both the BAO and the acoustic peaks in the CMB power spectrum. The angular scale of these peaks is measured with exquisite precision, a fractional error of about one part in ten thousand, giving a near-perfect measurement of the ratio of the transverse distance to the sound horizon at the end of recombination, at a redshift of about 1089.

Type Ia supernovae (SNe) are standardizable candles which serve as low-redshift probes in addition to BAO, measuring the luminosity distance. The SNe Hubble diagram provided the first direct evidence for cosmic acceleration, and subsequent cosmological surveys have obtained well measured light curves for many hundreds of supernovae out to redshift 1 and beyond. BAO and SNe measurements constrain dark energy and neutrino masses through their impact on the background evolution, thus determining the expansion history. Assuming that general relativity correctly describes the dynamics of expansion, the evolution of the expansion rate is governed by the Friedmann equation, with contributions from baryons and cold dark matter, radiation, curvature, neutrinos and dark energy. The energy densities of baryons and cold dark matter scale as the cube of one plus redshift. The scaling of the neutrino energy density transitions from the cube to the fourth power at high redshifts.

For BAO cosmology, the important characteristic of ‘CDM’ is that its energy density scales as the cube of one plus redshift, both before and after recombination, and that it does not couple non-gravitationally to photons or baryons in a way that affects the scale of the acoustic oscillations. Some variations such as self-interaction or ‘warm’ thermal velocities would affect small scale clustering and galaxy formation but not BAO. Decaying dark matter, on the other hand, would alter that energy scaling even if the dark matter is cold and non-interacting, thus affecting the BAO scale.

The ΛCDM model assumes a cosmological constant dark energy with an energy density that is constant in space and time. If dark energy has an equation-of-state parameter w, the ratio of its pressure to its energy density, then its energy density evolves with redshift according to an integral over one plus w. For constant w that becomes a simple power law, and a cosmological constant corresponds to w equal to −1. A commonly used parametric model expresses w in terms of the expansion factor a as

w(a) = w0 + wa (1 − a),

so that w evolves from a value of about w0 plus wa at high redshift to a present-day value of w0. This parametrization accurately represents the behavior of many physically motivated dark energy models, though more complicated evolution is possible. We refer to models that assume CDM and this form as w0waCDM and models with constant w as wCDM. Through most of this paper we will assume a flat universe, motivated by the tight constraints obtained on curvature when it is allowed to vary freely.

The combination of BAO, CMB, and SNe data has allowed tight constraints on the energy density and equation of state of dark energy, on space curvature, on neutrino masses, and on many possible departures from standard cosmology. The analysis of the DESI DR1 measurements provided tight constraints on parameters of the ΛCDM model and intriguing hints of evolving dark energy, with significances ranging from about 2.5σ to about 3.9σ depending on the combination of datasets used for the analysis. The analysis of the full shape of the power spectrum measured with galaxies and quasars confirmed these findings and added new information on the amplitude of perturbations and modified gravity parameters.

Rapidly evolving dark energy, with a magnitude of wa of order unity, would be an astounding discovery, and these results have inspired both enthusiastic theorizing and healthy skepticism. In our analysis here of the DESI DR2 BAO results, we pay particular attention to the nature and statistical significance of the evidence for evolving dark energy and to how that evidence depends on the choice of datasets. We also examine the constraints on the sum of neutrino masses from the DESI DR2 data in combination with CMB and SNe, for both ΛCDM and w0waCDM. When we refer to ‘DESI’ alone in tables and figure legends, we treat the BAO as an uncalibrated standard ruler. In some of our ΛCDM analyses, we examine constraints that adopt a big bang nucleosynthesis (BBN) prior on the baryon density, with the value of the matter density coming from the model fit itself. We achieve tighter constraints and sharper tests by combining DESI with CMB data that directly constrain the baryon and matter densities and add the precise measurement of the acoustic angular scale.

VII. Dark Energy

Probing the behavior and nature of dark energy is the primary goal of DESI. The question of perhaps greatest interest, and the one that BAO measurements can best illuminate, is the value of the equation-of-state parameter w, the ratio of pressure to energy density, and its possible evolution with time. To examine this we will primarily use the so-called Chevallier-Polarski-Linder (CPL) parametrization. While this form does not arise directly from an underlying physical model, it is a flexible parametrization that is capable of matching the predictions for observable quantities obtained in a wide range of models that are physically motivated. The accompanying paper explores various other parametrizations, as well as non-parametric reconstruction methods.

For certain ranges of parameters w0 and wa, this parametrization allows so-called ‘phantom’ behavior of dark energy, in which the equation of state crosses to the regime where w is less than −1 and the null energy condition — which requires that the energy density of dark energy not increase with the expansion of the Universe — is violated. For single scalar-field models of dark energy, this phantom crossing presents severe theoretical difficulties. However, more complex models of dark energy, with multiple fields, other dark energy internal degrees of freedom, or non-minimal coupling, can evade these difficulties, as can some modified gravity models. We therefore adopt wide uniform priors on the parameters, w0 in the range −3 to 1 and wa in the range −3 to 2, together with imposing the condition that w0 plus wa is negative to enforce early matter domination. While other justifiable choices are possible, and the values of Bayesian quantities such as the model evidence will always depend on the particular choice used, we consider this the minimal empirical approach. Whenever the equation of state crosses the w equals −1 boundary we use the parametrized post-Friedmann approach to include dark energy perturbations when calculating CMB power spectra — however, as shown below, the method of accounting for dark energy perturbations does not play a major role, since simply applying an early-Universe CMB prior largely reproduces the same results on w0 and wa.

Our primary measure of the statistical significance of preference for evolving dark energy from a given data combination is based on the difference in the best-fit chi-squared between the ΛCDM and w0waCDM models for that combination. Because ΛCDM is nested within w0waCDM, corresponding to w0 equal to −1 and wa equal to 0, Wilks’ theorem implies that this difference should follow a chi-squared distribution with two degrees of freedom under the assumption the null hypothesis holds, and assuming that errors are Gaussian and correctly estimated. To translate it into familiar terms, we quote the corresponding frequentist significance for a one-dimensional Gaussian distribution. We also compute the Deviance Information Criterion (DIC), which takes into account the Bayesian complexity of the model and penalizes including extra parameters.

A. Results

From DESI DR2 BAO alone, we obtain rather weak constraints on the parameters: w0 equals −0.48 with an upward uncertainty of 0.35 and a downward uncertainty of 0.17, and wa is bounded above by −1.34, which mildly favor the quadrant with w0 greater than −1 and wa less than 0. The upper bound on wa here is the 68% limit, and wa equal to 0 is not excluded at 95%. As was the case in DR1, BAO data alone define a degeneracy direction in the w0-wa plane, but they do not show a strong preference for dark energy evolution: the improvement in the best-fit chi-squared relative to the ΛCDM case is equivalent to a preference of just 1.7σ. Note that the posteriors in this poorly constrained case are cut off by the priors, so the marginalized means and limits quoted above are prior-dependent.

The minimal extension we consider, beyond BAO data alone, is to add a high-redshift constraint from the early universe. This can be achieved by imposing CMB-derived priors on the acoustic angular scale and the baryon and matter densities. These priors are independent of the late-time dark energy, and also marginalize over contributions such as the late integrated Sachs-Wolfe effect and CMB lensing. Therefore, they provide us with an early time physics prior that can help us set the sound horizon and is based solely on early-Universe information. The result from this data combination is w0 equals −0.43 plus or minus 0.22 and wa equals −1.72 plus or minus 0.64. While this is still bounded by the prior at the lower end, the posterior already clearly disfavors ΛCDM. The chi-squared difference decreases to −8.0, indicating a preference for an evolving dark energy equation of state at the 2.4σ level.

Replacing these minimal early-Universe priors with the full CMB information leads to only a small shift in the marginalized posteriors, giving w0 equals −0.42 plus or minus 0.21 and wa equals −1.75 plus or minus 0.58, showing that most of the information that the CMB provides on the equation of state comes from its role in anchoring early-Universe values and thus limiting the freedom for models to fit the low-redshift data without an evolving dark energy component. Nevertheless, when including the full CMB information the chi-squared difference decreases to −12.5, corresponding to a 3.1σ preference for evolving dark energy. This change is driven primarily by the inclusion of CMB lensing, the effect of which is by construction not captured in the minimal early-Universe priors.

SNe data alone provide a complementary degeneracy direction in the w0-wa plane, as they measure w0 well independently of wa, which is only weakly constrained. The combination of SNe data with DESI BAO can therefore measure w0 and wa without having the posteriors cut off by the prior ranges we assumed. The marginalized posterior results depend on the choice of SNe dataset, with the significances of the preference for the model over ΛCDM ranging from 1.7σ to 3.3σ.

However, the posterior for the combination of DESI BAO and SNe alone allows quite a wide range of posterior values of the matter density. CMB information places extremely tight constraints on the matter density that are largely independent of the late-time background model. Therefore, the full statistical power of the data is achieved through the combination of the BAO, CMB and SNe datasets, giving the marginalized posterior results w0 equals −0.838 plus or minus 0.055 with wa equals −0.62 for the combination with Pantheon-plus; w0 equals −0.667 plus or minus 0.088 with wa equals −1.09 for Union3; and w0 equals −0.752 plus or minus 0.057 with wa equals −0.86 for DESY5. The chi-squared differences are −10.7, −17.4, and −21.0, corresponding to preferences for the w0waCDM model over ΛCDM at the 2.8σ, 3.8σ, and 4.2σ levels, for combination with Pantheon-plus, Union3 and DESY5 respectively. These significances have all increased compared to the values reported based on the DESI DR1 BAO results.

The deviance information criterion values for the combination of DESI plus CMB with Pantheon-plus, Union3 and DESY5 SNe are −6.8, −13.5 and −17.2 respectively. These indicate preferences for the w0waCDM model consistent with those obtained from the chi-squared values above. Again, the changes obtained here with DESI DR2 BAO data compared to the DR1 values show that the preference for w0waCDM has increased with the additional data.

The pivot redshift at which the equation of state in this parametrization is best constrained by the data depends on the particular combination of datasets used. For DESI plus CMB, the pivot redshift is 0.53 and the equation of state there is −1.024 plus or minus 0.043: this is a lower pivot redshift and a tighter constraint than that found for the same combination with DR1 BAO, reflecting the additional constraining power of the DR2 BAO results. For the DESI plus CMB plus DESY5 combination, we find a pivot redshift of 0.31 and an equation of state of −0.954 plus or minus 0.024, indicating a mild preference for a deviation from w equals −1 at the best-measured redshift. As the other two SNe datasets are slightly less constraining than DESY5, the pivot redshifts for combinations with them are slightly larger, as are the uncertainties, but the results for all choices of DESI plus CMB plus SNe are mutually consistent.

These results sharpen the preference already seen in DR1 for an evolving equation of state for dark energy: although the statistical significances from all data depend somewhat on the choice of SNe dataset included, even the weakest of them, the DESI plus CMB plus Pantheon-plus combination, is still nearly 3σ, and the significance is 3.1σ even when excluding all SNe data altogether. In all cases, the favored equation of state shows a phase above −1 at low redshifts and a phantom crossing below −1 above a redshift of about 0.4. Within the w0waCDM model, the necessity of such a crossing and the redshift at which it occurs is determined by the requirement to match the precise CMB measurement of the acoustic angular scale together with the matter density. The details of the recovered form of the equation of state and the w0, wa parameter values naturally depend on the choice of parametrization. The accompanying paper explores various other parametrizations, and non-parametric reconstruction methods, that exhibit a similar behavior. It also performs binned reconstruction of the equation of state without assuming a functional form, and finds a consistent picture. The lowest redshift bin favors a value above −1 at high significance, inconsistent with the ΛCDM expectation of exactly −1.

The apparent preference for phantom crossing at intermediate redshifts, and the consequent violation of the null energy condition, is thus rather independent of parametrization choices made in the analysis. However, the equation of state is not directly observable, and we only observe quantities, such as distances, that depend indirectly on it. In some circumstances it may therefore be possible to construct particular models that provide reasonable fits to the low-redshift data while still respecting an equation of state at or above −1 at all redshifts. The supporting paper examines several such models of dark energy evolution and finds that while they can somewhat outperform ΛCDM, they have low deviance information criterion values relative to w0waCDM for the combination of DESI, CMB and SNe data, indicating a preference for phantom crossing.

It is worth noting that allowing the equation of state to vary does not help resolve the so-called Hubble tension, since in w0waCDM the recovered Hubble constant is lower than the Planck ΛCDM value. Although not discussed here, it has been shown that allowing evolving dark energy also does not affect the value of the clustering amplitude parameter determined from DESI data, which remains consistent with values from the CMB.

B. The nature of the evidence for evolving dark energy

The Hubble diagrams illustrate the nature of the evidence for evolving dark energy and its dependence on the adopted datasets. They show the isotropic, perpendicular, and parallel BAO measurements, normalized to the predictions of the Planck ΛCDM cosmology, together with the distance modulus relative to the fiducial Planck ΛCDM prediction for the three SNe datasets. Because the fiducial SNe absolute magnitude is unknown, all data points are free to move up or down together by the same amount, and we have chosen the normalization such that the error-weighted mean of the residual is equal to zero. Equivalently, any model curve can be shifted up or down by a constant, and we have normalized them to match the weighted mean of the data.

While the statistical significance of disagreement cannot be judged accurately from these plots alone, the DESI measurements clearly prefer lower distances, by 1 to 2 percent, than the Planck ΛCDM prediction at redshifts at or below 1. There is a ΛCDM model that fits the DESI data well, but it has a lower matter density than the Planck model, 0.297 against 0.317. The joint-fit model has an intermediate matter density of 0.303, and consequently has a worse fit to both DESI BAO and the CMB and also fails to describe the SNe data. Similarly, the DESY5 data exhibit a tension with Planck ΛCDM, primarily because of the contrast between the low redshift data and the points at higher redshift. The story is similar for Union3, but for Pantheon-plus the low-redshift residual is smaller.

As is the case for BAO, there exist ΛCDM models that can reasonably fit the SNe data, but these have large matter densities that do not well match CMB constraints and are also strongly inconsistent with DESI data, which prefer a matter density that is lower than Planck, not higher. Conversely, the conflict with the DESI-constrained ΛCDM models and DESY5 SNe at low redshifts is worse than the offset from the Planck fiducial model. All these observations point to the fact that the ΛCDM model struggles to consistently fit all three datasets: BAO, CMB, and SNe.

One way to address this tension is to adopt a model with more flexibility in the background expansion. However, the wCDM model with a constant equation of state lacks sufficient flexibility. The best-fitting constant-w model to the DESI plus CMB plus DESY5 combination has w equal to −0.971 and a matter density of 0.310. The high-redshift anchor does not allow enough redshift evolution for this model to provide a good fit to the BAO and SNe data at low redshifts, performing only slightly better than Planck ΛCDM. If instead the wCDM model were chosen to fit the DESI plus DESY5 data, it would necessarily have a matter density that would fail to match the CMB constraints.

On the other hand, the w0waCDM model does have sufficient flexibility to simultaneously achieve good fits to all three datasets. The predictions for w0waCDM models with parameters matching the best fits to DESI plus DESY5 and to DESI plus CMB plus DESY5 are barely distinguishable in the plots, showing that the w0waCDM model that best fits the BAO and SNe automatically also provides a good fit to the CMB. Over the range of redshifts covered by DESI BAO, these model predictions provide a better fit to BAO data than the best such ΛCDM model — in particular, in the fit to the parallel BAO distances — while also simultaneously resolving the mismatch in matter density between DESI and CMB in the ΛCDM framework. They are also equally good at fitting the offsets between low-redshift and high-redshift DESY5 data. These observations qualitatively help to explain why these models are strongly preferred over ΛCDM.

While the qualitative picture is the same for Union3 as for DESY5, the low-redshift residual is smaller for Pantheon-plus and so using this dataset does not strengthen the preference for evolving dark energy relative to that already provided by the joint fit to DESI and the CMB.

C. Are alternative explanations possible?

Given the surprising results from our analysis — our best-fit evolving dark energy models apparently imply phantom dark energy at some redshifts, and order unity changes in the equation of state over the redshift range spanned by the observations considered — it is worth considering alternative explanations for the data.

The simplest of these alternatives is simply a statistical fluctuation. Based on the chi-squared values presented above, the statistical preference for the best-fit w0waCDM model over ΛCDM ranges from around 3σ to over 4σ, depending on the combination of datasets considered. This level of significance is not trivially dismissed, even if it does not yet rise to the 5σ threshold commonly accepted for establishing new physics. The variations in the statistical significance within this range with the choice of SNe sample also highlight the importance of calibrating the low-to-intermediate redshift SNe distance scale.

The constraining power of SNe in measuring the equation of state comes primarily from the comparison of supernovae below and above redshift 0.1. For supernovae above redshift 0.1, which partially overlap the redshift range of DESI, the ΛCDM model that best fits the DESI data is also a good fit to the SNe data. Relative to models that best fit each of the DESY5, Union3 and Pantheon-plus SNe samples alone, over the full redshift range, the DESI best-fit model gives only small shifts in the quality of the fit to the SNe data. Unfortunately, no SNe compilation yet exists in which objects from both redshift regimes were collected as part of a uniform observational program. The Pantheon-plus and Union3 datasets are compilations of SNe observed in many individual datasets, each with different calibrations and selection functions. Even for the DESY5 dataset, while 1635 SNe above redshift 0.1 come from the DES survey program with homogeneous calibration, 194 SNe below redshift 0.1 are drawn from a mix of historical observational programs with the best-controlled calibrations. The consistent calibration and processing of these data is therefore a delicate operation. Recently an offset was noted in the differences between the standardized brightnesses of low-redshift and high-redshift SNe for objects in common between the DESY5 and Pantheon-plus samples; however, others explain the causes for this and argue that the differences are well justified. If we exclude the low-redshift sample entirely and take only the DESY5 SNe, which is the most uniformly calibrated sample, naturally the constraining power and thus the statistical significance of the preference for evolution is reduced, but the best-fit values of w0 and wa remain far from ΛCDM. Even if SNe at all redshifts are excluded altogether, the statistical significance of the preference from DESI plus CMB alone still exceeds 3σ. Future cosmology analyses of homogeneous SNe samples from ZTF and LSST may shed further light on the relative calibration across redshifts.

Using the simple early-Universe prior that marginalizes over information dependent on late-time models in addition to DESI gives very similar posterior constraints on w0 and wa to the full DESI plus CMB combination, and the central values are very stable. However, the tension with ΛCDM drops somewhat, from 3.1σ to 2.4σ, when using these simple priors instead of the full CMB: this is because the CMB lensing likelihood makes a sizable contribution to the chi-squared calculation.

Although recent independent results from SPT and ACT find cosmological parameter values that are consistent with those from Planck, another possibility is that the CMB constraints used here suffer from some unknown systematic error. It is therefore interesting to consider the constraints that can be obtained entirely independent of the high-redshift CMB anchor. As argued above, the primary role of this CMB anchor is that it limits the freedom to vary the matter density when fitting to the low-redshift BAO and SNe distance-redshift measurements. This can be achieved instead by replacing the CMB with the Dark Energy Survey Year 3 three-by-two-point information to obtain a constraint coming entirely from low-redshift cosmological probes. This combination favors the same region of parameter space, although with somewhat larger uncertainties.

We also consider the possibility of an undetected systematic error in our BAO measurements that would have evaded the many internal checks we have performed. For instance, a coherent systematic shift of 1.5% to all the measured isotropic BAO scales, applied in a redshift-independent manner to all tracers, would decrease the tension between DESI and Planck in the ΛCDM model, and thus decrease the evidence for evolving dark energy. However, the required magnitude of such a hypothetical shift exceeds the total systematic error budget indicated by our tests by a factor of 6 for quasars to 10 for bright galaxies and luminous red galaxies, and we consider this highly unlikely.

Physical models that can satisfactorily explain the data without requiring dark energy evolution are more challenging. Allowing the curvature to vary freely, while increasing uncertainties, does not shift the posterior in w0 and wa towards ΛCDM. While we have fixed the sum of neutrino masses at 0.06 eV for our fiducial analysis, it is allowed to vary elsewhere in the paper, and we find that this has almost no effect on the dark energy constraints. This is because fitting the CMB power spectrum imposes a positive correlation between the neutrino mass sum and the matter density, so given a physical prior that the sum is positive, in most of the available parameter space neutrino masses can only increase the CMB value of the matter density, which exacerbates rather than relieves the tension with DESI BAO.

IX. Conclusions

We have presented BAO measurements from over 14 million discrete galaxy and quasar tracers drawn from the first three years of operation of DESI and which will be included in the second data release (DR2). These results use samples of nearby bright galaxies, luminous red galaxies, emission line galaxies and quasars over the redshift range 0.1 to 2.1, and cover a cumulative effective volume of over 42 cubic gigaparsecs. Complementary BAO measurements from correlations of the Lyα forest and high-redshift quasars at effective redshift 2.33 are presented in the companion paper.

Our BAO analysis largely follows the methods used for the previous DESI DR1 analysis, but with improved statistical precision as the effective volume of the data has increased by more than a factor of two. Some particular differences include the use of a fainter limiting magnitude cut to define the bright galaxy sample, resulting in a higher number density, and the inclusion of quadrupole information in BAO fits to the quasar sample. As for the DR1 analysis, we applied a strict catalog-level blinding to our data while initial data checks were carried out and the analysis pipeline was being finalized. The validation tests and the criteria that were required to be met before the data were unblinded are described in detail in the supporting publication.

The final BAO results provide a precision on the isotropic distance scale measurement that ranges from 1.54% for the quasars down to just 0.45% for our most constraining composite luminous-red-galaxy and emission-line-galaxy sample; other than for the quasars, the precision is sub-percent for every tracer and redshift range. We also obtained a precision of a few percent on the measurement of the Alcock-Paczyński distance ratio for every tracer except the lowest redshift bright galaxy sample. Together these results are the most precise BAO measurements ever made at all redshifts covered, including below redshift 0.8 where DESI DR2 now greatly exceeds the precision of the Sloan Digital Sky Survey.

The DR2 results are very consistent with those previously reported from the DR1 data that form a subset of DR2. The p-value determined from the Kolmogorov-Smirnov test for the distribution of the differences is 0.40. The DR2 results are also consistent with SDSS. A roughly 3σ discrepancy previously reported in the measurement of the transverse BAO scale in one luminous-red-galaxy sample compared to the equivalent result from SDSS has decreased in significance for DR2, lying within the range 1.5σ to 2.6σ, depending on assumptions about the degree of correlation between the two samples.

The combination of all BAO measurements from DR2 is well fit by a flat ΛCDM cosmological model with matter density parameter 0.2975 plus or minus 0.0086 and product of the scaled Hubble constant and sound horizon at the drag epoch of 101.54 plus or minus 0.73 Mpc. This represents a 40% improvement in precision compared to the equivalent results from DR1, but with excellent consistency between the two. However, while a ΛCDM model provides a good fit to DESI data, the model parameters obtained from this fit are now in 2.3σ tension with those derived from the CMB, increased from 1.9σ in DR1. This tension is present despite DESI being consistent with the acoustic angular scale measured by the CMB.

When calibrated with an external big bang nucleosynthesis prior on the baryon density, our BAO measurements correspond to a Hubble constant of 68.50 plus or minus 0.58 kilometres per second per megaparsec in ΛCDM, a value that is independent of any information on CMB anisotropies. In the plane of matter density against Hubble constant, the results from BBN-calibrated BAO are now more discrepant with those from CMB; the offset of the results is again along the degeneracy direction of the CMB that is determined by the very precisely measured acoustic angular scale. Combining DESI BAO with BBN and the Planck acoustic angular scale gives a Hubble constant of 68.45 plus or minus 0.47 kilometres per second per megaparsec, a 0.7% precision measurement competitive with that from the CMB itself, and in strong disagreement with SH0ES.

Within the ΛCDM framework, DESI results are also somewhat in tension with the high matter densities preferred by SNe datasets, which — contrary to DESI — prefer larger matter densities than Planck. While not individually rising to the 3σ significance threshold, these results point to an incompatibility between different cosmological datasets when interpreted in the ΛCDM model. Interestingly, the relative levels of matter density and Hubble constant currently measured by SNe, DESI and CMB datasets match what would be expected if data from a true evolving dark energy model were analyzed in the restrictive ΛCDM model, as pointed out recently.

Assuming a ΛCDM background, the combination of DESI and CMB data give the tightest upper bound on the neutrino mass sum to date, less than 0.064 eV at the 95% limit in our baseline analysis. Although this relaxes to less than 0.078 eV when using an alternative Planck likelihood, both results are approaching the lower bound set by terrestrial neutrino oscillation experiments of at least 0.059 eV. Indeed, if the model is extended to allow for negative ‘effective’ neutrino masses, most of the posterior mass and the peak of the likelihood lie in the negative region, another possible sign of growing tensions within the ΛCDM model with DESI DR2.

The evidence for a departure from ΛCDM in the form of evolving dark energy has increased with the DR2 BAO data. Comparing the evolving dark energy model to ΛCDM, we find a 3.1σ evidence in favor of dynamical dark energy from DESI plus CMB alone. When we add the recent Pantheon-plus, Union3 or DESY5 SNe datasets to this combination, the preference for w0waCDM over ΛCDM is 2.8σ, 3.8σ or 4.2σ respectively, with all three giving results for w0 and wa consistent with each other within their 68% credible regions. The preferences for w0waCDM under these dataset combinations have increased compared to our DR1 results. All combinations of available datasets we study favor mutually consistent results with w0 greater than −1 and wa less than 0, apparently indicating a weakening dark energy today and a phantom crossing at some point in the past in this parametrization. The supporting paper examines this behavior using a wider range of models and non-parametric methods. We note that the degeneracy direction for the constraints in the w0-wa plane approximately points towards the ΛCDM solution, although this is neither exact nor consistent between fits to different combinations of datasets. This constitutes a weak coincidence as there is no a priori reason for the best-fit values of w0 and wa to have the values that lead to this observation.

We examined the contributions to the preference for evolving dark energy in detail. For DESI plus CMB, the discrepancy in preferred matter densities in ΛCDM is resolved in w0waCDM, leading to the 3.1σ preference for the latter. If we omit CMB lensing the best-fit w0waCDM parameters are unchanged, though the significance drops to 2.7σ. If we also marginalize over any late-time dependence in the CMB and only impose an early-Universe prior, there is a further slight reduction to 2.4σ. The w0waCDM and ‘compromise’ ΛCDM models predict different evolution of the distance-redshift relation below redshift 0.4, with a roughly 2% difference in distances below redshift 0.1. BAO do not provide high precision in this regime, but supernovae do, and the DESY5 and Union3 samples clearly favor the w0waCDM prediction. However, the distance-redshift evolution in the Pantheon-plus analysis is closer to the ΛCDM prediction, so adding Pantheon-plus to DESI plus CMB does not strengthen the evidence for w0waCDM. Our results are robust to changes in the choice of CMB likelihood. We also show that replacing the CMB likelihood with the weak gravitational lensing constraints from the Dark Energy Survey Year 3 three-by-two-point analysis retains a clear preference for evolving dark energy, with, for example, a preference of at least 3σ for w0waCDM from DESI plus that lensing analysis plus DESY5 SNe. Removing the preference for w0waCDM through changes to the DESI measurements themselves would require systematic errors to be far larger than any found in our tests.

In the w0waCDM background model, when the neutrino mass scale is allowed to vary with physical non-negative prior bounds, the upper limit on the neutrino mass sum is significantly relaxed to less than 0.16 eV at 95% from DESI plus CMB, or less than 0.13 eV from DESI plus CMB plus DESY5, while the constraints on w0 and wa do not materially change from the case where the sum is fixed. These limits are entirely consistent with neutrino oscillation experiment results. The supporting paper shows that even when allowing for effective masses with a wide prior that allows negative values, the marginalized posterior in w0waCDM from DESI plus CMB is consistent with the oscillation lower limit and its peak lies in the positive region.

We previously characterized the results from DESI DR1 and external datasets as providing “a tantalizing suggestion of deviations from the standard cosmological model”. The DR2 data presented here have sharpened this evidence, although significance levels still vary depending on the external data used — particularly the choice of SNe sample — and no combination exceeds 4.2σ. Nevertheless, it is becoming clear that unless some unidentified systematic error affects one or several of the different cosmological datasets used, the challenge to the ΛCDM model has increased. Sharper measurements from future DESI analyses and from other experiments will show whether these challenges to the standard cosmological model herald yet another radical transformation in our understanding of the evolution and energy content of the Universe.

(Sections II to VI and VIII, the appendices, the acknowledgements and the reference list are omitted for length; display equations, figures and tables are not reproduced. The complete text is at the source.)