Cosmological Tests of Gravity

Cosmology tests gravity by asking whether one set of field equations can explain the background expansion, matter growth, lensing potentials, tensor-wave propagation, and nonlinear return to locally tested behavior. No single probe closes that system. Distances alone can be represented by an effective dark-energy stress; clustering alone is degenerate with bias and matter physics; lensing alone measures an integrated potential; and flexible modified-growth functions need not describe a stable theory.

Helpful background. Relativistic Cosmology supplies FLRW backgrounds, perturbations, distances, and structure formation; Weak-Field Gravity supplies metric potentials and gauge transformations; Probability and Statistical Inference supplies multi-probe likelihoods and correlated nuisance models; and Tests of Gravity supplies the common theory-to-conclusion logic.

Enter the cosmological closure problem

Use conformal Newtonian gauge for scalar perturbations, conformal time $\tau$, comoving coordinates $\mathbf x$, and $a_0=1$:

$$ds^2=a^2(\tau)\left[-(1+2\Psi)d\tau^2 +(1-2\Phi)d\mathbf x^2\right].$$

Background distances constrain $H(z)$ and curvature. Nonrelativistic matter responds chiefly to $\Psi$, while lensing responds to $\Phi+\Psi$. Tensor amplitudes can acquire a propagation distance distinct from the electromagnetic luminosity distance. At nonlinear scales, any modification that survives cosmologically must also satisfy local constraints, often through screening. A complete test therefore links all these regimes with common initial conditions and theory parameters.

A useful first question is what would change if the expansion history stayed fixed. Galaxies could fall into overdensities faster if the effective force strengthened, while light deflection could remain unchanged if the two potentials changed in opposite directions. Distances would miss this difference. Measuring matter motion and light deflection together is therefore more informative than treating either as a direct measurement of a single “strength of gravity.” Start with the first three pages before using the parameterized-theory machinery.

Follow the chapter in this order:

1. Expansion History and Gravity shows why candles, rulers, and clocks tightly determine geometry without choosing modified gravity over dark-energy stress or calibration.
2. Growth of Structure derives matter response and traces $f\sigma_8$ through bias, velocities, nonlinear physics, and survey selection.
3. Lensing, Clustering, and Gravitational Slip compares the two scalar potentials with common-sky cross-correlations and explains why bias cancellation is approximate.
4. Parameterized Cosmological Gravity distinguishes observable $\mu,\Sigma$ functions, effective-field property functions, stability, superhorizon completion, and the quasistatic limit.
5. Standard Sirens and Propagation Friction compares gravitational-wave and electromagnetic distances through host, inclination, lensing, population, and calibration uncertainty.
6. Nonlinear Scales and Screening requires one theory and simulation to recover unscreened cosmology, an environmental transition, and a screened local limit.

What each layer contributes

Ishak’s review separates geometry probes in §4 from the modified-growth functions in §5.2, their time and scale evolution in §5.3, the growth index in §5.4, the $E_G$ statistic in §5.5, and PPF completion in §5.6 Ishak 2019, §§4.1–4.4 and §§5.2–5.7. Ferreira independently organizes the subject around backgrounds, linear perturbations, phenomenological tests, screening, and the domains in which those descriptions can be compared Ferreira 2019, §§2–6. This organization prevents a common error: applying one quasistatic $\mu,\Sigma$ convention to superhorizon CMB modes, nonlinear halos, and local systems.

Layer	Principal observable	Closure requirement
Background	$H(z)$, $D_M/r_d$, $D_H/r_d$, luminosity distance	State curvature, ruler, calibration, and effective stress assumptions
Linear growth	Clustering, redshift-space anisotropy, peculiar velocity	Propagate bias, neutrinos, baryons, initial amplitude, and scale cuts
Weyl potential	Cosmic shear, galaxy–galaxy lensing, CMB lensing	Join shear/redshift calibration, intrinsic alignment, masks, and covariance
Parameter basis	$\mu,\eta,\Sigma$ or effective-field functions	Enforce gauge, initial conditions, stability, causality, and regime
Tensor distance	Standard-siren amplitude versus redshift	Marginalize calibration, inclination, host, lensing, population, and selection
Nonlinear return	Halo, profile, velocity, lensing–dynamics, and environment statistics	Demonstrate convergence and the same theory’s screened and unscreened limits

Current release boundary

For the evidence cutoff 2026-08-19, this chapter uses the named Planck 2018 legacy likelihood as an early-universe reference Planck Collaboration 2020, papers I, V, VI, and VIII. DES Year 3 supplies a reproducible 3-by-2-point weak-lensing and clustering example with shear, redshift, bias, intrinsic-alignment, scale-cut, covariance, and nuisance treatment DES Collaboration 2022, §§II–IV and Tables I–II. Here “3-by-2-point” means shear–shear, galaxy–shear, and galaxy–galaxy correlations.

DES Y3 is not the newest DES cosmology analysis. The Y6 analysis was submitted in January 2026; its version 2 distinguishes its cosmological results from the data products promised upon journal acceptance, while noting that the Y6 Gold catalog is public DES Collaboration 2026, §IX. A public imaging catalog, a published constraint, and a released likelihood are different inputs. Reusing the Y3 example does not justify labeling it a current Y6 constraint.

DESI DR2 provides BAO cosmology chains and supporting products from its first three years DESI Collaboration 2025–2026, official DR2 publication list. Its July 2026 Lyα-forest analysis also extracts Alcock–Paczyński geometry from the full correlation shape; its data-availability statement must be read separately from the BAO-chain release DESI Collaboration 2026, §§V–VI and VIII. Neither product is automatically a general galaxy-growth, slip, or screening likelihood. The underlying DR2 spectra and redshifts must not be assumed public merely because derived chains are available. Euclid Consortium 2025, Q1 release overview identifies Q1 as a quick, non-cosmology release.

Every combination must retain release identifiers, calibration, masks, selection and completeness, priors, covariance across overlapping samples, theory and likelihood software versions, scale cuts, emulator domains, and chain diagnostics. Planck, DES, and DESI constraints cannot be multiplied as independent if shared external data or sky modes have already entered a supplied chain.

Reading a cosmological gravity claim

First ask whether the background and perturbation conventions are fixed. Next ask whether a phenomenological function can be realized by a stable theory across the analyzed scales. Then inspect nuisance priors: galaxy bias, baryonic feedback, neutrino mass, photometric redshift, shear calibration, intrinsic alignment, nonlinear emulation, host completeness, and lensing can rotate or imitate the gravity direction. Finally require an independent synthetic-data or simulation control.

The strongest discriminator is a pattern across channels: the same theory should predict geometry, the relative response of matter and light, tensor damping, and an environmental screening transition. Agreement in one channel cannot prove GR uniquely, and a discrepancy in one channel does not identify modified gravity before systematics and matter physics are excluded. A superseding likelihood, calibration, covariance, emulator, or stable theory map can change the corresponding inference.

Review the chapter

Why can two models with exactly the same $H(z)$ yield different lensing and clustering observables?

Solution

The background fixes homogeneous expansion and distance conversion but not the perturbation closure. One model may use GR plus an effective dark-energy stress, while another changes the relation between density and $\Psi$ or produces slip between $\Phi$ and $\Psi$. Matter growth then responds differently to $\Psi$, and lensing responds differently to $\Phi+\Psi$. Therefore equal distances do not imply equal transfer functions, growth rates, or Weyl potentials; calibrated perturbation probes are required to distinguish the models.

References

• DES Collaboration, “Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” Physical Review D 105, 023520 (2022), DOI.
• DES Collaboration, “Dark Energy Survey Year 6 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” submitted manuscript, arXiv:2601.14559v2 (2026), Preprint.
• DESI Collaboration, “DESI DR2 Cosmology Results and Supporting Data Products,” official publication list and cosmology-chain release (2025–2026), DESI Data.
• DESI Collaboration, “DESI DR2 Results IV: Alcock–Paczyński Measurements from the Lyman Alpha Forest and Cosmological Constraints,” submitted manuscript, arXiv:2607.27410v1 (2026), Preprint.
• Euclid Consortium, “Euclid Data Release: Q1,” official release overview (2025), Release page.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Pedro G. Ferreira, “Cosmological Tests of Gravity,” Annual Review of Astronomy and Astrophysics 57, 335–374 (2019), DOI.
• Planck Collaboration, “Planck 2018 Results. I, V, VI, and VIII,” Astronomy & Astrophysics 641, A1, A5, A6, and A8 (2020), official publication list.