Modified gravity constraints from the full shape modeling of clustering measurements from DESI 2024

This paper asks whether the growth of cosmic structure and gravitational lensing signals measured by DESI can reveal deviations from general relativity (GR) on large scales, and—if so—how such deviations depend on time and scale. The question matters because cosmic acceleration could reflect either dark energy within GR or modified gravity (MG) that changes how matter clusters and how light propagates. Large-scale structure is particularly powerful for this test because different gravity theories can share the same expansion history yet predict different growth rates and gravitational potentials.

The authors implement a model-agnostic phenomenological strategy: they modify the linearized perturbed Einstein equations by introducing effective functions 1(a,k) and (a,k) (and also (a,k) or (a,k) via relations among potentials). In GR, these functions take unity (or zero in their parameterization offsets). They then confront these MG parameters with DESI 2024 first-year clustering measurements using full-shape modeling, combined with external datasets that constrain geometry and/or lensing and growth: Planck CMB temperature/polarization (with multiple likelihood releases), Planck+ACT CMB lensing, Big Bang Nucleosynthesis (BBN) priors on the physical baryon density when CMB is not used, DES Year 3 weak lensing and clustering (32-pt) from DESY3, and DES Year 5 Type Ia supernovae.

Methodologically, the study is a Bayesian inference pipeline. The DESI input consists of power-spectrum multipoles (monopole and quadrupole; hexadecapole excluded due to systematics) measured in six redshift bins spanning roughly 0.1 < z < 2.1, using the Yamamoto estimator and pypower, with wavenumber cuts 0.02 < k/(h
Mpc) < 0.2 and k = 0.005 h Mpc
. The full-shape modeling uses an effective field theory of large-scale structure approach with perturbation theory (via velocileptors) and includes galaxy/quasar bias and nuisance parameters; BAO information is added from post-reconstruction correlation functions. The combined likelihood includes the full covariance between power-spectrum and BAO measurements.

For MG, the authors explore several parameterizations. First, they use the common
(a,k)–(a,k) framework (and also
(a,k)–(a,k) with
related to gravitational slip). They adopt a functional form where time dependence scales with the dark-energy density DE(a) and scale dependence is encoded by ratios of polynomials in k (with parameters
,
, and additional scale parameters). They also analyze redshift-only and redshift+scale binned schemes with smoothed transitions, and finally they constrain Horndeski theory within the effective field theory (EFT) of dark energy, using both EFT-basis and -basis parameterizations.

Key results: In a 9CDM background, with time-only MG parameterization in the
– framework, DESI alone constrains
(their

) but not
(lensing-sensitive). With the combination DESI(FS+BAO)+BBN+ns10 they obtain
= 0.11^{+0.44}{-0.54}, consistent with GR (zero deviation). When they use the full combination DESI(FS+BAO)+CMB+DESY3+DESY5SN (with the most constraining Planck likelihood choice LoLLiPoP
HiLLiPoP and excluding CMB lensing in some runs to avoid covariance), they find
= 0.05 b1 0.22 and
= 0.008 b1 0.045 in 9CDM. In the same setting they also constrain
and via
–:
= 0.02^{+0.19}{-0.24} and
= 0.09^{+0.36}_{-0.60}. Thus, across multiple dataset combinations and likelihood choices, the MG parameters remain consistent with GR.

A central methodological/interpretive point is the treatment of the Planck PR3 CMB lensing anomaly (the “Alens” issue). The authors report that when using Planck PR3 without CMB lensing reconstruction,
(and similarly
) shows a tension with GR at above 3 significance when combined with DESI. However, this tension largely disappears when using newer Planck likelihoods (Camspec and especially LoLLiPoP
HiLLiPoP) or when adding reconstructed CMB lensing data. They interpret this as evidence that the earlier
tension is driven by the CMB lensing anomaly rather than true MG.

In a dynamical dark-energy background (w0waCDM), the same MG parameters remain consistent with GR: for the combination DESI+CMB(LoLLiPoP
HiLLiPoP)-nl+DESY3+DESY5SN they find
= -0.24^{+0.32}{-0.28} and
= 0.006 b1 0.043. Interestingly, even with MG freedom added, the data still prefer evolving dark energy with w0 = -0.784 b1 0.061 and wa = -0.82^{+0.28}{-0.24} (i.e., w0 > -1 and wa < 0).

The binned MG approach yields more informative constraints on multiple parameters. In 9CDM with two redshift bins (0 z < 1 and 1 z < 2) and assuming GR for z 2, they obtain
= 1.02 b1 0.13,
= 1.04 b1 0.11,
= 1.021 b1 0.029, and
= 1.022^{+0.027}_{-0.023}, all consistent with the GR value of unity in their binning convention. Extending to two redshift bins and two scale bins (k dividing scale kc = 0.01 Mpc
) gives 8 MG parameters (4
and 4
) with 68% credible intervals at the (115%) level for
and (3%) for
, again consistent with GR.

For Horndeski EFT constraints, they parameterize non-minimal coupling via (a) =
a^{s0} (EFT-basis) and map to -basis functions. In 9CDM with DESI(FS+BAO)+DESY5SN+CMB they find
= 0.01189^{+0.00099}{-0.012} and s0 = 0.996^{+0.54}{-0.20}, with a 95% CL upper bound
< 0.0412. In a no-braiding case (B = 0) they report a 95% CL bound cM < 1.14, consistent with GR. When braiding is allowed, they find a mild but consistent preference for cB > 0 (i.e., B b9 0), which they caution could be due to systematics or projection effects.

Limitations: The authors emphasize that their MG parameterizations are defined at the linearly perturbed Einstein-equation level and that they rely on linear/quasi-linear scales through DESI full-shape scale cuts. They also note that their perturbation-theory modeling uses Einstein-de Sitter kernels, which is an approximation that is safe only when deviations from GR are small; they provide kernel-comparison evidence showing sub-percent to few-percent differences for extreme departures (e.g.,

= 0.5) within the fitted k-range. They acknowledge that functional MG forms may not capture all MG models and that binned schemes are more flexible but still limited by the chosen binning. For EFT/ -basis, they highlight that results depend on the assumed time evolution form (e.g., i(a) b9 ci DE(a)) and that extended parameter spaces can introduce projection effects and convergence challenges. Finally, they do not include CMB lensing and DESY3 32-pt simultaneously due to covariance, which constrains the ability to combine all growth and lensing information in one run.

Practical implications: The main practical message is that one year of DESI full-shape clustering, when combined with CMB and large-scale structure lensing/clustering constraints, yields competitive bounds on modified gravity growth and lensing functions, with MG parameters consistent with GR at the (0.25%) to (0.045%) precision level depending on parameter choice. The results also clarify that previously reported tensions in
from Planck PR3 are resolved by updated Planck likelihoods, reducing the risk of misinterpreting CMB lensing systematics as MG. Who should care: cosmologists using DESI clustering to test gravity, analysts interpreting Planck lensing anomalies, and theorists mapping Horndeski/EFT parameters to observables. The mild preference for braiding (cB > 0) motivates follow-up with improved likelihoods, cross-correlations (e.g., ISW-related), and more robust systematics modeling.

Overall, the paper provides a comprehensive, multi-parameter, multi-dataset test of GR vs modified gravity using DESI DR1 full-shape modeling, and it demonstrates both the power of growth measurements and the importance of updated CMB likelihoods for avoiding spurious MG signals.