Growth of Structure

Structure growth tests how nonrelativistic matter responds to cosmological gravitational potentials after the expansion history is fixed. The observable is never a bare growth factor: galaxy clustering, redshift-space anisotropy, and peculiar velocities pass through galaxy bias, tracer selection, nonlinear dynamics, baryonic feedback, massive neutrinos, distance conversion, and covariance. A gravity conclusion is warranted only over scales where those ingredients and the perturbation closure are controlled.

Required background. Probability and Statistical Inference supplies the normalized likelihood, nuisance marginalization, covariance, and interval conventions required to infer growth from selected tracers.

Helpful background. Weak-Field Gravity supplies the two scalar potentials and gauge logic; Relativistic Cosmology supplies linear perturbations and transfer functions; and Tests of Gravity supplies the bounded comparison with a GR null.

Matter response to a cosmological potential

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

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

With spatial Fourier kernel $e^{i\mathbf k\cdot\mathbf x}$, $\nabla^2$ acts as $-k^2$. For pressureless matter on subhorizon linear scales, a modified Poisson relation can be written

$$-k^2\Psi =4\pi G a^2\mu(a,k)\rho_m\Delta_m,$$

where $k$ is comoving wavenumber and $\Delta_m$ is the comoving matter density contrast. The late-time GR baseline has $\mu=1$ when other species’ density and anisotropic-stress contributions to this constraint are negligible. Relativistic species, clustering dark energy, or a different choice of matter species require their contributions explicitly; $\mu\ne1$ inferred from an incomplete source term would not establish modified gravity.

The growth equation follows from two familiar statements: mass conservation and acceleration in a potential. In cosmic time $t$, let $\mathbf u=a\,d\mathbf x/dt$ be the peculiar velocity. Neglecting pressure and the subhorizon-suppressed potential time derivatives, the linear equations are

$$\dot\delta_m=-\frac1a\nabla\cdot\mathbf u, \qquad \dot{\mathbf u}+H\mathbf u=-\frac1a\nabla\Psi.$$

Differentiate the first equation, insert the second, and use the Poisson relation with $\Delta_m\simeq\delta_m$ on these scales. This yields $\ddot\delta_m+2H\dot\delta_m-4\pi G\mu\rho_m\delta_m=0$. If matter follows metric geodesics and scale dependence is negligible, write $\delta_m(a)=D(a)\delta_m(1)$, with $D(1)=1$. Since $d/dt=H\,d/d\ln a$, the growth factor obeys

$$\frac{d^2D}{d(\ln a)^2} +\left(2+\frac{d\ln H}{d\ln a}\right) \frac{dD}{d\ln a} -\frac32\Omega_m(a)\mu(a)D=0.$$

Here $\Omega_m(a)=8\pi G\rho_m/(3H^2)$. The $2H\dot\delta_m$ term opposes growth through expansion, while positive $\mu$ strengthens attraction. The equation exposes the degeneracy: growth can change through $H(a)$, effective gravity $\mu$, clustering species, or initial conditions. A growing-mode initial condition selects the physical solution; the differential equation also has a second mode. With scale-dependent gravity, evolve $D(a,k)$ rather than one universal $D(a)$. The growth rate $f=d\ln D/d\ln a$ is sometimes approximated by $f\simeq\Omega_m^\gamma$; this is a fit over a specified regime, not an identity Ishak 2019, §§3.2.1–3.2.3, §5.4, and §6.2.

From tracer anisotropy to growth

In the linear Kaiser limit, the redshift-space tracer power is

$$P_g^s(k,\zeta,z) =\left[b(k,z)+f(k,z)\zeta^2\right]^2P_m(k,z),$$

where $\zeta$ is the cosine to the line of sight and $b$ is galaxy bias. This expression assumes a single line of sight, deterministic linear density bias, negligible velocity bias, and negligible random small-scale motions. It follows because converting peculiar velocity into an apparent radial displacement adds $f\zeta^2\delta_m$ to the galaxy density contrast; squaring its Fourier amplitude gives the power Kaiser 1987, §2.

Surveys commonly constrain combinations such as $f\sigma_8$, not $f$ independently. Here $\sigma_8(z)$ is the root-mean-square linear matter contrast smoothed in comoving spheres of radius $8h^{-1}\,\mathrm{Mpc}$, where $h=H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$. For scale-independent growth, $\sigma_8(z)=D(z)\sigma_8(0)$. Rewriting the amplitude as $(b\sigma_8+f\sigma_8\zeta^2)^2$ times a unit-normalized spectral shape shows what is measured. If growth is scale dependent, a quoted single $f\sigma_8$ is a model-dependent compression. Away from the linear limit, random motions elongate clusters along the line of sight (the “fingers of God” effect); nonlinear bias, redshift errors, fiber assignment, and incorrect distance conversion also alter the anisotropy.

Peculiar-velocity surveys probe the velocity divergence more directly but require distance indicators, Malmquist-bias corrections, and correlated calibration. Photometric clustering covers large volumes yet has redshift-distribution uncertainty. Cross-correlations can constrain bias, but only if overlapping masks, shot noise, and sample covariance are retained.

For binned clustering and velocity data $\mathbf d$, a normalized likelihood is

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

with $\mathbf m$ computed from $H$, the initial power spectrum, $\mu$, neutrino masses, bias, nonlinear and baryonic prescriptions, and survey response. Priors on these nuisance functions can dominate the apparent gravity information. Scale cuts are part of the model domain, not a later robustness decoration.

Releases, selection, and reproducibility

At the evidence cutoff 2026-08-19, the named Planck 2018 legacy likelihood anchors the initial spectrum and early transfer function under its declared cosmology Planck Collaboration 2020, papers V–VI. DES Year 3 supplies a public weak-lensing and clustering likelihood with masks, redshift and shear calibration, scale cuts, bias, intrinsic-alignment model, covariance, and nuisance priors DES Collaboration 2022, §§II–IV and Table I. It is a reproducible example here; the newer Y6 analysis and its distinct data-release statement are identified in the chapter overview.

DESI DR2 BAO distances and associated chains sharpen the background entering the growth equation. The release list also includes a July 2026 Lyα full-shape Alcock–Paczyński analysis, so “DR2” must not be used as a synonym for BAO alone DESI Collaboration 2025–2026, official DR2 publication list and products. Its geometrical result is nevertheless not interchangeable with a galaxy $f\sigma_8$ measurement or a general modified-growth likelihood. Derived cosmology chains are also not equivalent to access to the underlying DR2 spectra and redshift catalog.

A reproducible growth analysis records catalog/release identifiers, angular and redshift masks, target and quality selection, completeness weights, redshift failures, fiducial coordinate conversion, window matrix, covariance, theory-code commit, nonlinear emulator or perturbation prescription, neutrino and baryon model, priors, and chain diagnostics. Mock catalogs must pass the identical selection. Independent implementations should recover the GR prediction on synthetic data before a residual is interpreted.

Results and limitations through 2026-08-19

Evidence or result	Position supported	Limitation or counterevidence
Planck legacy initial conditions	Provides an early-time amplitude and transfer-function anchor	Model extensions, optical depth, foregrounds, and neutrino assumptions propagate into late growth
DES Y3 clustering and lensing likelihood	Constrains late structure with explicit calibration and nuisance parameters	Photo-$z$, shear, bias, intrinsic alignment, baryons, nonlinear modeling, and scale cuts remain limiting
DESI DR2 BAO products	Improve the expansion background used in growth predictions	BAO-only products do not independently measure the public DR2 growth likelihood

The conservative position is that current data restrict deviations in specified growth models but do not isolate gravity from every matter and survey effect. A competing explanation of a low or high growth amplitude may invoke modified gravity, neutrino mass, baryonic suppression, galaxy bias, calibration, or initial amplitude. Their relative support depends on model space and priors, so the interpretation remains unresolved and model-dependent.

An adversarial independent check injects a known scale-dependent $\mu(a,k)$ into survey mocks, then analyzes them with alternate bias, baryon, neutrino, redshift, covariance, and nonlinear models. It must recover the injection when in domain, recover $\mu=1$ for GR mocks, and reveal bias when unsafe scales are restored. Cross-code agreement of spectra and likelihood normalization is a separate discriminator.

What could change the conclusion. The interpretation could change after a superseding Planck, DES, DESI full-shape, peculiar-velocity, simulation, or nonlinear-emulator release; it could also change if calibrated mocks change coverage or the safe scale boundary.

Exercises

Assume an Einstein–de Sitter background, so $\Omega_m=1$ and $H\propto a^{-3/2}$, with constant positive $\mu$. Insert $D\propto a^p$ into the growth equation and find the growing exponent. Treat this as a perturbation-closure example, not a claim that every $\mu$ is realized by a complete theory with that background.

Solution

Here $d\ln H/d\ln a=-3/2$. Because $D\propto a^p$, each derivative with respect to $\ln a$ brings down one factor of $p$, so substitution gives $p^2+\tfrac12p-\tfrac32\mu=0$. Therefore $p=[-1+\sqrt{1+24\mu}]/4$ for the growing branch. At the GR value $\mu=1$, this gives $p=1$, hence $D\propto a$ as expected. The discarded root is the decaying mode. The result also shows why expansion and gravity must be specified together: changing the friction term through $H(a)$ would change the inferred $p$ even at fixed $\mu$.

References

• DES Collaboration, “Dark Energy Survey Year 3 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing,” Physical Review D 105, 023520 (2022), DOI.
• DESI Collaboration, “DESI DR2 Cosmology Results and Supporting Data Products,” official publication list and cosmology-chain release (2025–2026), DESI Data.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Nick Kaiser, “Clustering in Real Space and in Redshift Space,” Monthly Notices of the Royal Astronomical Society 227, 1–21 (1987), DOI.
• Planck Collaboration, “Planck 2018 Results. V and VI,” Astronomy & Astrophysics 641, A5 and A6 (2020), official publication list.