Relativistic Structure Formation

Cosmic structure is a sequence of maps, not a single growth equation. Primordial perturbations are processed by radiation, baryons, neutrinos, and horizon entry; gravitational instability amplifies the surviving matter modes; shell crossing ends the single-stream description; halo and galaxy definitions introduce calibrated statistics; and an observer finally samples tracers on a perturbed past light cone. This chapter follows that sequence without mistaking a coordinate density, halo catalog, or compressed survey likelihood for the underlying matter field.

Helpful background. The Einstein Equation supplies Einstein Equations and their constraints; Relativistic Matter and Compact Objects supplies fluid and kinetic matter; Relativistic Cosmology supplies the background and gauge-invariant perturbations; Computational Relativity supplies convergence and benchmark practice; and Probability and Statistical Inference supplies random fields, covariance, selection, and likelihood reasoning.

Before starting, check that you can distinguish physical from comoving wavenumber, explain why a Fourier-space power spectrum needs a normalization convention, and state when a density contrast depends on the time slicing. If the first two are unfamiliar, review Fourier modes and random fields through the linked preparation pages. If the third is unfamiliar, begin with the gauge-invariant density construction in the cosmological-perturbation chapter before using a bias relation.

From primordial modes to observed tracers

The six pages are ordered by the physical loss of simple descriptions.

1. Linear Growth of Matter Perturbations starts from the relativistic density–velocity system, derives the subhorizon growth equation, checks the growing and decaying Einstein–de Sitter modes, and compares exact growth with the growth-index approximation.
2. Transfer Functions and the Matter Power Spectrum separates primordial statistics, horizon-entry processing, equality, baryon acoustic structure, diffusion, neutrino free streaming, and late growth while fixing every power-spectrum normalization.
3. Nonlinear Gravitational Clustering follows mode coupling to the Zel’dovich Jacobian and first shell crossing, then distinguishes analytic collapse models from resolution- and code-dependent simulation calibration.
4. Halo Formation and Mass Functions builds abundance predictions from smoothing, collapse thresholds, first-crossing logic, halo definitions, and a mass-conservation benchmark.
5. Galaxy Bias and Redshift-Space Distortions turns tracer selection, bias operators, peculiar velocities, Alcock–Paczynski conversion, windows, covariance, and nuisance priors into a versioned survey likelihood.
6. Relativistic Corrections to Large-Scale Structure restores the Doppler, lensing, gravitational, evolution, and observer terms required for a gauge-invariant light-cone observable near the Hubble scale.

For an analytic linear route, read pages 1–2, use page 5 for the density-plus-Kaiser limit, then read page 6 to learn its horizon-scale boundary. For a nonlinear and survey route, read page 1, proceed through pages 3–5, and return to page 2 whenever a simulation or likelihood quotes a power-spectrum convention. Neither route permits skipping the selection and coordinate map between matter and an observed catalog.

One convention and equation chain

Cosmic time is $t$, conformal time is $\eta$, and $\mathcal H=a'/a=aH$. Uppercase $K$ denotes background spatial curvature; lowercase $k=\lvert\mathbf k\rvert$ is a comoving Fourier wavenumber under $e^{i\mathbf k\cdot\mathbf x}$. The power spectrum is defined by

$$\left\langle \delta(\mathbf k)\delta^*(\mathbf k') \right\rangle =(2\pi)^3\delta_D^{(3)}(\mathbf k-\mathbf k')P_\delta(k), \qquad \Delta_\delta^2(k)=\frac{k^3P_\delta(k)}{2\pi^2}.$$

For scalar perturbations in Newtonian gauge,

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right],$$

where $\Phi$ is the lapse potential and $\Psi$ the spatial-curvature potential. The comoving matter density, rather than an unspecified coordinate density, sources the Poisson constraint in the late matter–$\Lambda$ limit:

$$\delta\rho_m^{\rm com}=\delta\rho_m-3\mathcal H\delta q_m, \qquad \nabla^2\Psi=4\pi Ga^2\delta\rho_m^{\rm com}.$$

When radiation or another component has appreciable perturbations, its comoving density and momentum must be included in the total constraint. The matter-transfer page uses this comoving density convention to relate $\delta_m$ to primordial $\mathcal R$; Newtonian-gauge density agrees only in its subhorizon limit.

In the pressureless, subhorizon, single-stream limit the growing factor obeys

$$D''+\mathcal H D'-4\pi Ga^2\rho_mD=0,$$

with $D\propto a$ and a decaying $D\propto a^{-3/2}$ mode in Einstein–de Sitter. A transfer function then records scale-dependent earlier processing, while statistical homogeneity and isotropy reduce the two-point function to $P(k)$. Ma and Bertschinger give the relativistic species system, initial conditions, and numerical transfer architecture in Ma and Bertschinger 1995, §§4–8, printed pp. 10–32 (Open PDF); Tong develops the growth, transfer, power, smoothing, and spherical-collapse approximations in Tong 2019, §§3.1–3.3 (Official PDF).

Where analytic control ends

Three boundaries must remain distinct. First, pressure, radiation, massive neutrinos, or horizon-scale metric evolution make growth scale dependent, so a single $D(a)$ no longer closes the system. Second, shell crossing makes the collisionless distribution multistream: the density remains meaningful, but one pressureless velocity field does not. Third, a halo or galaxy catalog is a new statistical object whose definition includes a smoothing convention, finder, mass boundary, tracer selection, and observing procedure.

The dimensionless power $\Delta^2(k)$ becoming order unity is a warning about mode coupling, not a universal shell-crossing time. Zel’dovich evolution and spherical collapse are complementary controlled models; neither is a substitute for a converged $N$-body calculation after multistreaming. Likewise, Press–Schechter normalization is an analytic mass-accounting check, not a precision universal mass function. Simulation-calibrated fits must travel with their cosmology, volume, resolution, initial conditions, halo finder, mass definition, and redshift range.

The bias expansion makes the same lesson algebraic. At lowest order, $\delta_g=b_1\delta_m+\epsilon$, but symmetry also permits tidal, higher-derivative, nonlinear, and stochastic operators. Desjacques, Jeong, and Schmidt organize these terms and their renormalization in Desjacques, Jeong, and Schmidt 2018, §§2–5 and §§7–9 (Open PDF). Coefficients are tied to a declared source population; they are not properties of matter alone. Detailed baryonic astrophysics is therefore routed to its dedicated treatments rather than hidden inside an unqualified “galaxy bias.”

From theory field to survey likelihood

In a small, plane-parallel patch, number conservation under the velocity map gives the Kaiser control limit

$$P_g^s(k,\mu)=(b_1+f\mu^2)^2P_m(k), \qquad f=\frac{d\ln D}{d\ln a},$$

where $\mu$ is the cosine to the line of sight. A production analysis adds nonlinear bias and velocities, fiducial-distance conversion, mask and window convolution, fiber and redshift completeness, covariance, nuisance priors, and mock coverage. The DESI example in the galaxy-bias page reproduces a pinned compressed Gaussian with its full correlations, then states why that object cannot test alternative scale cuts or velocity models without the full multipole pipeline.

On scales $k\sim\mathcal H$, even the complete density-plus-Kaiser model is not the full observable. Doppler terms scale locally as $\mathcal H/k$ relative to density and potentials as $(\mathcal H/k)^2$; lensing and other path integrals have additional geometric leverage. Gauge invariance requires source, volume, selection, and observer terms as one package. Current claims are kept conditional through the 2026-08-19 evidence cutoff on the two frontier leaves. Dedicated gravity tests belong to the gravity-test volume; this chapter establishes the large-scale-structure observable and its modeling uncertainties without turning a survey fit into a theory verdict.

Scope and onward routes

The chapter stops before detailed baryonic feedback, galaxy-formation microphysics, or modified-gravity consistency tests. It also does not equate a constant-time spatial statistic with a past-light-cone observation. The Cosmic Microwave Background supplies the earlier transfer constraints. Inhomogeneity and the Observed Light Cone derives exact distance, number-count, lensing, and velocity observables and treats spatial averaging separately. Any executable growth and transfer integration must retain the same gauge conventions, initial conditions, limiting cases, and convergence checks.

Review the chapter

In an Einstein–de Sitter interval, a growing mode evolves from $a_i=0.25$ to $a_f=1$. For a tracer with $b_1=2$ and $f=1$, compare the real-space matter power with the transverse and line-of-sight Kaiser powers. Then estimate the relative local Doppler and potential orders at $k=10\mathcal H$. State why this does not estimate the lensing correction.

Solution

Because $D\propto a$, the density amplitude grows by $D_f/D_i=a_f/a_i=4$, so its power grows by $4^2=16$. The Kaiser factors at the final epoch are

$$\frac{P_g^s(\mu=0)}{P_m}=b_1^2=4, \qquad \frac{P_g^s(\mu=1)}{P_m}=(b_1+f)^2=9.$$

Thus the line-of-sight tracer power is $9/4$ times the transverse tracer power in this ideal limit. At $k=10\mathcal H$, an undifferentiated Doppler term is parametrically of relative order $\mathcal H/k=0.1$, while a local potential term is of order $(\mathcal H/k)^2=0.01$. These are scaling estimates, not fitted amplitudes: their coefficients depend on redshift and selection. Lensing is a transverse, line-of-sight integral, so path length, geometry, and source distribution prevent its reduction to the same local power count.

References

• Desjacques, Vincent, Donghui Jeong, and Fabian Schmidt. 2018. “Large-Scale Galaxy Bias.” Physics Reports 733: 1–193. DOI.
• Ma, Chung-Pei, and Edmund Bertschinger. 1995. “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges.” Astrophysical Journal 455: 7–25. DOI.
• Tong, David. 2019. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes. Official PDF.