DESI 2024 VII: cosmological constraints from the full-shape modeling of clustering measurements

This DESI Collaboration paper asks how well cosmological parameters can be constrained when one goes beyond the baryon acoustic oscillation (BAO) “standard ruler” and instead models the full shape of large-scale structure (LSS) clustering. The central motivation is that the broadband power spectrum shape and its redshift evolution encode both geometry (distances) and growth of structure, making full-shape analyses sensitive to parameters such as the matter density ${\Omega }_m$, the fluctuation amplitude ${\sigma }_8$, the Hubble constant $H_0$, neutrino masses, and possible deviations from general relativity (GR). The paper matters because DESI’s first-year data release (DR1) is already capable of percent-level cosmology, and full-shape modeling can tighten constraints and test fundamental physics in ways BAO-only analyses cannot.

Methodologically, the study uses DESI DR1 clustering measurements of multiple tracers—galaxies and quasars—over a large footprint and redshift range. The dataset comprises over 4.7 million unique galaxies and QSOs over $\sim 7,500$ square degrees, spanning $0.1<z<2.1$. Tracers are split into six redshift bins: BGS (0.1–0.4), three LRG bins (0.4–0.6, 0.6–0.8, 0.8–1.1), ELG (1.1–1.6), and QSO (0.8–2.1). The Lyman- forest is used only for BAO geometry (not growth), so it enters through BAO likelihood rather than full-shape growth.

The full-shape analysis measures the monopole and quadrupole of the redshift-space power spectrum multipoles (hexadecapole is measured but not used in cosmological inference). The analysis restricts wavenumbers to $0.02<k/(h{\mathrm{Mpc}}^{-1})<0.20$ with bin width $\Delta k=0.005h{\mathrm{Mpc}}^{-1}$. Power spectrum estimation uses standard survey techniques: random catalogs with the same selection and weights, window-matrix corrections, and “rotation” of the measured multipoles and covariance to reduce off-diagonal structure. Small-scale effects from fiber assignment and angular cuts are mitigated by removing small angular separations and by using the window matrix and covariance treatment validated in supporting DESI papers. Imaging systematics are corrected via catalog-level weights and an “angular integral constraint” correction estimated from mocks. Covariances are estimated from 1000 fast approximate mocks (EZmocks) and rescaled using RascalC to match semi-empirical covariance predictions.

A key methodological feature is blinding: the BAO component is blinded by modifying observed galaxy redshifts to shift the BAO peak position, and redshift-space distortion information is blinded by applying a shift in the growth rate $f$. This is designed to prevent confirmation bias during analysis development.

The likelihood combines DESI full-shape power spectrum multipoles with post-reconstruction BAO measurements (from the companion BAO analysis) and includes their joint covariance. The total DESI (FS+BAO) likelihood is then optionally combined with external datasets: Planck 2018 PR3 CMB temperature/polarization likelihoods (TT, TE, EE) plus CMB lensing reconstruction using Planck+ACT; BBN-based priors on ${\Omega }_b{h}^2$ (via PRyMordial with reaction-rate marginalization) when CMB is not used; a weak prior on $n_s$ (ns10) when CMB is not used; type Ia supernovae (PantheonPlus, Union3, and DES-SN5YR); and DES Year-3 weak lensing and clustering (3-point and 6-point) datasets, with modified-gravity-specific likelihoods when relevant. Bayesian inference is performed with cobaya using CAMB (and ISiTGR for modified gravity), sampling with Metropolis-Hastings MCMC (four parallel chains) and using velocileptors for perturbation-theory predictions.

In the baseline flat $\Lambda$CDM model, the headline results show that adding full-shape information to BAO yields strong constraints on matter density and growth. For DESI (FS+BAO) combined with BBN and the weak $n_s$ prior, the paper finds - ${\Omega }_m=0.2962\pm 0.0095$ - ${\sigma }_8=0.842\pm 0.034$ - $S_8\equiv {\sigma }_8({\Omega }_m/0.3{)}^{0.5}=0.836\pm 0.035$ - $H_0=68.56\pm 0.75\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$ When CMB data are added (Planck TT/TE/EE plus Planck+ACT lensing), uncertainties shrink substantially to - ${\Omega }_m=0.3056\pm 0.0049$ - ${\sigma }_8=0.8121\pm 0.0053$ - $H_0=68.07\pm 0.38\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. The paper emphasizes that CMB lensing is important for ${\sigma }_8$ and $S_8$, while the combined DESI+BAO+full-shape information improves ${\Omega }_m$ and $H_0$ even relative to CMB alone.

The most comprehensive combination in $\Lambda$CDM uses DESI (FS+BAO) plus CMB plus DESY3 6-point (which includes CMB lensing in the DESY3 analysis, so the CMB lensing reconstruction is removed from the CMB likelihood to avoid double counting). This yields very tight constraints: - ${\Omega }_m=0.3009\pm 0.0034$ - ${\sigma }_8=0.8028_{-0.0045}^{+0.0050}$ - $S_8$ at the $\sim 0.6\%$ level - $H_0=68.40\pm 0.27\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$. The paper also reports that adding DESY3 3-point or 6-point shifts ${\sigma }_8$ and $S_8$ downward by about one standard deviation, consistent with lensing preferring lower fluctuation amplitudes.

For dynamical dark energy in the $w_0{w}_a$CDM parameterization $w(a)=w_0+w_a(1-a)$, the authors find that DESI full-shape information improves the precision of $w_0$ and $w_a$ when combined with CMB and type Ia supernovae, and that the preference for departures from $\Lambda$CDM persists. With DESI (FS+BAO)+CMB+PantheonPlus, they report - $w_0=-0.858\pm 0.061$ - $w_a=-0.68_{-0.23}^{+0.27}$ and similar results with Union3 and DES-SN5YR. They quantify the preference using $\Delta {\chi }_{\mathrm{MAP}}^2$ between the best-fit $w_0{w}_a$ model and the constrained $\Lambda$CDM case: $\Delta {\chi }_{\mathrm{MAP}}^2=-8.8$ (2.5), $-14.5$ (3.4), and $-17.5$ (3.8) for PantheonPlus, Union3, and DES-SN5YR respectively.

Neutrino constraints are a major payoff of full-shape modeling because neutrino masses affect both expansion and growth. Under $\Lambda$CDM with three degenerate neutrinos and a minimal prior $\sum m_{\nu }>0$, DESI (FS+BAO)+BBN+ns10 gives - $\sum m_{\nu }<0.409\mathrm{eV}$ at 95% confidence. Tightening the $n_s$ prior to match Planck improves this to $\sum m_{\nu }<0.300\mathrm{eV}$. Adding CMB yields the strongest baseline limit: - $\sum m_{\nu }<0.071\mathrm{eV}$ at 95% confidence for DESI (FS+BAO)+CMB. The paper notes this is about 15% stronger than DESI BAO-only + CMB, and that the posterior peaks near $\sum m_{\nu }\approx 0$ but is consistent with oscillation lower bounds at roughly the $\sim 2\sigma$ level for normal ordering. Robustness tests using alternative Planck PR4 high- likelihoods (CamSpec, LoLLiPoP/HiLLiPoP) yield slightly weaker bounds (e.g., $0.069\mathrm{eV}$ and $0.081\mathrm{eV}$). In $w_0{w}_a$CDM, the neutrino mass bound relaxes substantially (e.g., $\sum m_{\nu }<0.196\mathrm{eV}$ with DES-SN5YR).

For modified gravity, the authors use a parameterization in which two functions modify the linearized Einstein equations: $\mu (a)=1+{\mu }_0{\Omega }_{\mathrm{DE}}(a)/{\Omega }_{\Lambda }$ and $\Sigma (a)=1+{\Sigma }_0{\Omega }_{\mathrm{DE}}(a)/{\Omega }_{\Lambda }$, with GR corresponding to ${\mu }_0={\Sigma }_0=0$. DESI (FS+BAO) alone constrains ${\mu }_0$ but not ${\Sigma }_0$: - ${\mu }_0=0.11_{-0.54}^{+0.45}$ (consistent with GR). Combining DESI with CMB and DESY3 3-point yields tight constraints on both: - ${\mu }_0=0.04\pm 0.22$ - ${\Sigma }_0=0.044\pm 0.047$. The authors discuss that Planck PR3-based constraints show a known tension in ${\Sigma }_0$ linked to the CMB lensing anomaly, but that using PR4 likelihoods alleviates this.

Limitations are addressed both explicitly and implicitly. The full-shape modeling uses perturbation theory with effective-field-theory counterterms and a large nuisance-parameter set (42 nuisance parameters across six redshift bins). This introduces potential “projection effects” in models with weak constraints and strong degeneracies, particularly in $w_0{w}_a$CDM; the authors therefore avoid reporting DESI-only $w_0{w}_a$ results and rely on combinations with supernovae to break degeneracies. The analysis is also restricted to quasi-linear scales ( $k<0.2h{\mathrm{Mpc}}^{-1}$) to keep nonlinear modeling uncertainties under control. Dominant systematic uncertainties are identified as imaging systematics and the galaxy-halo connection (HOD modeling), with systematic contributions propagated into the covariance. Finally, neutrino and modified-gravity constraints depend on the assumed cosmological background model, and neutrino bounds are sensitive to the CMB lensing likelihood choice.

Practically, the results matter for multiple audiences. Cosmologists and LSS analysts should care because the paper demonstrates that DESI DR1 full-shape modeling is validated and can deliver percent-level constraints on ${\Omega }_m$ and sub-percent constraints on $H_0$ when combined with CMB and DESY3. Particle-physics phenomenologists should care because the neutrino mass upper limit reaches $0.071\mathrm{eV}$ in $\Lambda$CDM, approaching the regime relevant for neutrino ordering tests (though still above the oscillation lower bound for normal ordering). Gravity theorists should care because the modified-gravity parameters are constrained consistently with GR, with DESI full-shape providing sensitivity to the clustering-growth modification parameter ${\mu }_0$.

Overall, the paper’s core contribution is that DESI DR1 full-shape clustering, when combined with BAO and validated perturbation-theory modeling, unlocks growth-sensitive cosmology and tightens constraints on dark energy, neutrinos, and modified gravity beyond BAO-only analyses.