Expansion History and Gravity

Expansion data determine the geometry through which light and standard rulers propagate, but expansion alone does not identify the gravitational field equations. A modified-gravity model, a dark-energy stress tensor, spatial curvature, an early-time ruler shift, or a calibration error can reproduce the same distance–redshift relation. A gravity test emerges only when the background reconstruction is joined to a perturbation or consistency prediction that those explanations cannot all share.

Required background. Probability and Statistical Inference supplies the likelihood, nuisance priors, covariance, and model-comparison rules used for heterogeneous distance data.

Helpful background. Weak-Field Gravity supplies the distinction between background and perturbations; Relativistic Cosmology supplies FLRW distances and the sound horizon; and Tests of Gravity supplies the theory–observable–measurement chain.

Geometry measured by clocks, candles, and rulers

Normalize the scale factor to $a_0=1$ and use cosmic time. The radial comoving distance is

$$\chi(z)=\int_0^z\frac{dz'}{H(z')},$$

and the transverse comoving distance is $D_M=S_K(\chi)$. With spatial curvature $K$ of dimension inverse length squared, $S_K(\chi)=\sin(\sqrt K\chi)/\sqrt K$ for $K\gt0$, $\chi$ for $K=0$, and $\sinh(\sqrt{-K}\chi)/\sqrt{-K}$ for $K\lt0$. For metric light propagation and photon-number conservation,

$$D_A(z)=\frac{D_M(z)}{1+z}, \qquad D_L(z)=(1+z)D_M(z).$$

Supernovae constrain relative luminosity distances after light-curve standardization and an absolute-magnitude calibration. Radial and transverse baryon acoustic oscillations (BAO) constrain $D_H/r_d=1/[H(z)r_d]$ and $D_M/r_d$, where $r_d$ is the drag-epoch sound horizon. Cosmic chronometers estimate $H(z)$ through differential galaxy ages, but stellar-population synthesis enters as a model systematic. None directly measures an Einstein-tensor component without assumptions about rulers, photons, matter, and curvature.

A convenient background reconstruction writes

$$\frac{H^2(a)}{H_0^2} =\Omega_{m0}a^{-3}+\Omega_{r0}a^{-4} +\Omega_{K0}a^{-2}+\Omega_{\rm eff}(a).$$

Here $\Omega_{K0}=-K/H_0^2$, while $\Omega_{\rm eff}(a)$ is normalized to today’s critical density, not the instantaneous one. It can represent material dark energy in GR or an effective term from modified field equations. Thus fitting $w(a)$ does not decide which interpretation is correct. Ishak makes this degeneracy explicit before cataloging geometry probes Ishak 2019, §§3.1.1–3.1.3 and §4.1.

To see the ambiguity directly, suppose matter and radiation are separately conserved and define $\rho_{\rm eff}=3H_0^2\Omega_{\rm eff}/(8\pi G)$. The continuity equation fixes

$$p_{\rm eff}=-\rho_{\rm eff}-\frac13\frac{d\rho_{\rm eff}}{d\ln a}, \qquad w_{\rm eff}=-1-\frac13\frac{d\ln\rho_{\rm eff}}{d\ln a},$$

where the logarithmic form assumes $\rho_{\rm eff}\gt0$. A constant effective density gives $w_{\rm eff}=-1$; a positive density proportional to $a^{-q}$ gives $w_{\rm eff}=-1+q/3$. The pressure formula remains meaningful at a density zero even when $w_{\rm eff}$ diverges. None of these background identities supplies a stable perturbation theory. That missing information is why growth and slip add a different kind of test.

There is a separate calibration degeneracy. In a fixed dimensionless expansion history, $D_L\propto H_0^{-1}$, while a supernova magnitude depends on $M_B+5\log_{10}(D_L/\mathrm{Mpc})+25$. A shift of the absolute magnitude $M_B$ can therefore compensate a shift of $H_0$. BAO likewise measures distances in units of $r_d$. These observables can be precise without independently fixing either absolute scale.

A normalized multi-probe likelihood

Let the data vector contain supernova distance moduli, BAO distance ratios, and any direct expansion estimates. A Gaussian approximation is

$$-2\ln\mathcal L =(\mathbf d-\mathbf m)^{T}C^{-1}(\mathbf d-\mathbf m) +\ln\det(2\pi C),$$

where $\mathbf m$ depends on $H(z)$, curvature, $r_d$, calibration, and selection nuisance parameters. Cross-survey covariance and shared calibrators must appear in $C$ or as common latent variables. Priors on $H_0$, $r_d$, curvature, supernova absolute magnitude, neutrino sector, and early-universe physics can rotate the apparent dark-energy or gravity direction.

Selection includes supernova detection and spectroscopic follow-up, galaxy/quasar target selection, redshift quality, survey masks, and BAO reconstruction choices. Completeness is not a cosmetic detail: a redshift-dependent supernova selection can mimic distance evolution, while a contaminated tracer sample can shift the recovered BAO scale. Reproduction pins the release, covariance, fiducial-coordinate conversion, reconstruction algorithm, likelihood software, nuisance priors, scale cuts, and chain convergence.

At the evidence cutoff 2026-08-19, the named CMB background example is the Planck 2018 legacy likelihood and cosmological-parameter products Planck Collaboration 2020, papers I, V, and VI. DESI DR2 BAO results use its first three years, with public cosmology chains and posterior-maximization products DESI Collaboration 2025–2026, official DR2 publication and product list. The July 2026 Lyα-forest full-shape analysis additionally measures Alcock–Paczyński geometry, comparing transverse and radial scales through $D_MH$; it should not be described as BAO-only. Its analysis data have a separate release statement DESI Collaboration 2026, §§V–VI and VIII. The underlying DR2 spectra and redshift catalog are not silently treated as a fully public general-purpose release.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or competing interpretation
Planck legacy acoustic scale and likelihood	Anchors early geometry and the ruler within a declared early-universe model	Foregrounds, calibration, neutrinos, recombination, and model extension affect the inferred ruler
DESI DR2 BAO distances and chains	Sharpen the late-time distance–redshift reconstruction	BAO measures ratios to $r_d$ and does not by itself choose gravity over dark energy
DESI DR2 Lyα full-shape geometry analysis	Adds an Alcock–Paczyński comparison beyond the BAO feature	Geometry and absorption-model information do not by themselves determine the gravitational field equations
Cross-calibrated candles and clocks	Add relative distances or direct expansion information	Absolute calibration, population drift, dust, selection, and stellar models can imitate evolution

The conservative conclusion is that current geometry data tightly restrict viable expansion histories, not that expansion alone selects GR. A competing interpretation may favor evolving dark energy, modified gravity represented as an effective fluid, curvature, or a change in early-time calibration. Their background predictions can be nearly identical, so the issue is unresolved and model-dependent until growth and lensing close the system.

An adversarial independent check constructs two models with the same $H(z)$—one GR model with effective dark-energy stress and one modified-gravity model—but different perturbation growth or slip. A pipeline that claims to distinguish them using distances alone is falsely using priors or hidden perturbation information. Adding blinded growth and lensing data would distinguish them only if calibration, covariance, and nonlinear assumptions are propagated.

What could change the conclusion. The interpretation could change when a superseding Planck likelihood, DESI cosmology product, supernova calibration, or independent distance release materially changes $H(z)$; it could also change when a perturbation analysis breaks or restores the background degeneracy under consistent priors.

Exercises

Show why an exact measurement of $H(a)$ can always be represented in GR by an effective dark-energy density once matter, radiation, and curvature densities are chosen.

Solution

Rearrange the background equation to define $\Omega_{\rm eff}(a)=H^2(a)/H_0^2-\Omega_{m0}a^{-3}-\Omega_{r0}a^{-4}-\Omega_{K0}a^{-2}$. Therefore any measured $H(a)$ supplies a function that can be interpreted as an effective GR source at the homogeneous level. If it is represented as a separately conserved effective fluid, its continuity equation then determines the effective pressure. This construction may be unstable or unphysical as material dark energy, but expansion data alone cannot tell whether the function originated from stress–energy or modified field equations.

References

• 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.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Planck Collaboration, “Planck 2018 Results. I, V, and VI,” Astronomy & Astrophysics 641, A1, A5, and A6 (2020), official publication list.