Cosmological Perturbations and Gauge Invariance

A perturbed universe is not described by “small changes” until one says which changes are coordinate relabelings and which are observable. This chapter builds that distinction from the scalar–vector–tensor split, constructs gauge-invariant metric and matter variables, derives their constrained Einstein evolution, classifies regular primordial modes, and closes the chain with particle phase-space transport. The result is the linear framework behind CMB and large-scale-structure transfer functions.

Helpful background. The Einstein equation introduces the field equations and Bianchi identity, while relativistic matter supplies the stress–energy variables. Formulations of general relativity separates gauge, constraints, and evolution; the cosmology overview locates the FLRW background; and PDEs and Green functions explains Fourier modes and inverse operators.

Enter the perturbation system

Before starting, check that you can linearize a tensor equation about a known background, distinguish a constraint from an evolution equation, and use $\nabla^2\to-k^2$ for an $e^{i\mathbf k\cdot\mathbf x}$ Fourier mode. You should also be able to explain why an inverse Laplacian requires a boundary condition and why a homogeneous perturbation can overlap a change of background parameters. Review the linked Einstein-equation, formulation, and PDE pages if any of those steps is unfamiliar.

The six pages support two main routes:

1. For gauge-invariant gravitational dynamics, read Scalar, Vector, and Tensor Decomposition, Gauge Transformations in Cosmological Perturbation Theory, Bardeen Potentials and Curvature Perturbations, and Linearized Einstein and Conservation Equations in order.
2. For initial conditions and numerical transfer, read pages 1–4 for the convention and constraints, then Adiabatic and Isocurvature Initial Modes and Einstein–Boltzmann Evolution.

The first route explains what the variables mean. The second shows how a regular primordial mode becomes a coupled hierarchy of metric, fluid, and radiation moments. The regular mode classification is developed by Bucher, Moodley, and Turok 2000, §II, pp. 6–14 (Open PDF); the controlled hierarchy approximations are described by Blas, Lesgourgues, and Tram 2011, §§2–4, pp. 4–25 (Open PDF).

One convention package

The chapter works at first order about spatially flat FLRW and uses conformal time, $\mathcal H=a'/a$, and $e^{i\mathbf k\cdot\mathbf x}$. Its scalar metric is

$$ds^2=a^2\left[-(1+2\phi)d\eta^2 +2\partial_iB\,d\eta\,dx^i +\left((1-2\psi)\delta_{ij} +2\partial_i\partial_jE\right)dx^i dx^j\right].$$

Under the passive coordinate change

$$\widetilde x^\mu=x^\mu+(T,\,\partial^iL), \qquad \widetilde{\delta Q}=\delta Q-\mathcal L_\xi\bar Q,$$

the combination $\sigma=E'-B$ transforms as $\widetilde\sigma=\sigma-T$. The invariant potentials are therefore

$$\Phi=\phi-\mathcal H\sigma-\sigma', \qquad \Psi=\psi+\mathcal H\sigma.$$

Newtonian gauge sets $B=E=0$ for $k\ne0$, so $\phi=\Phi$ and $\psi=\Psi$. For matter, the chapter defines $\delta T^0{}_i=\partial_i\delta q$ and

$$\zeta=-\psi-\mathcal H\frac{\delta\rho}{\rho'}, \qquad \mathcal R=-\psi+\mathcal H\frac{\delta q}{\rho+p}.$$

These signs travel as one package. In particular, the Ma–Bertschinger Newtonian potentials have the dictionary $\Phi=\psi_{\rm MB}$ and $\Psi=\phi_{\rm MB}$, while their velocity divergence obeys $\theta_{\rm MB}=-k^2\delta q/(\rho+p)$. Their matter shear $\sigma_{\rm MB}$ is not the metric variable $\sigma=E'-B$.

From a decomposition to a prediction

Each page supplies one necessary link:

Step	Mathematical object	Question answered
SVT split	helicity $0,\pm1,\pm2$ sectors	Which components can mix on an isotropic linear background?
gauge transformation	Lie derivative of background fields	Which perturbations change under a relabeling?
invariant variables	$\Phi,\Psi,\zeta,\mathcal R$	Which slicings and physical effects do the variables represent?
Einstein–matter system	constraints, evolution, conservation	Which initial data are admissible and which equations are redundant?
primordial basis	AD, CDI, BI, NID, NIV amplitudes	Which regular relative-density and relative-velocity seeds are independent?
kinetic transfer	Boltzmann multipoles and collision terms	How do initial modes become matter and radiation observables?

The constraint core is visible in Newtonian gauge:

$$k^2\Psi+3\mathcal H(\Psi'+\mathcal H\Phi) =-4\pi Ga^2\delta\rho, \qquad \Psi'+\mathcal H\Phi=-4\pi Ga^2\delta q.$$

Combining them gives

$$-k^2\Psi =4\pi Ga^2\left(\delta\rho-3\mathcal H\delta q\right),$$

so the relativistic source of the potential is the comoving density perturbation. On large scales,

$$\zeta-\mathcal R =-\frac{k^2\Psi}{3(\mathcal H^2-\mathcal H')}.$$

Thus $\zeta\simeq\mathcal R$ requires a regular gradient limit and a non-vacuum background scale $\mathcal H^2-\mathcal H'=O(\mathcal H^2)$; it is not an identity at exact $k=0$ or near $\rho+p=0$. Conservation of either curvature variable further requires adiabatic pressure and regular momentum, stress, and transfer terms.

Regular initial data then specify an amplitude for each allowed primordial mode. The Einstein–Boltzmann system evolves those amplitudes into species transfer functions. Free-streaming angular structure is expanded as

$$F(\mu)=\sum_{\ell=0}^{\infty} (-i)^\ell(2\ell+1)F_\ell P_\ell(\mu),$$

and neighboring multipoles couple under transport. Collisions make the photon–baryon subsystem stiff before recombination; free streaming makes the angular hierarchy infinite afterward. Tight-coupling expansions and high-$\ell$ closures are therefore controlled numerical approximations, not new physical assumptions about the primordial mode.

The gauge-invariant and conservation structure is developed in Malik and Wands 2009, §§6–9, printed pp. 17–58 (Open PDF). The species equations, normalization dictionary, and constraint checks are given by Ma and Bertschinger 1995, §§4–8, printed pp. 10–32 (Open PDF).

Boundaries of the chapter

The Fourier projectors, Newtonian-gauge fixing, momentum potential, and hierarchy direction all assume $k\ne0$. Exact homogeneous perturbations, harmonic zero modes, curved spatial slices, and bounces require separate treatment. The usual five primordial modes also assume the standard photon–baryon–CDM–neutrino content and exclude independently prepared higher neutrino multipoles; extra species or interactions require a new regularity analysis.

All calculations are linear. Nonlinear mode coupling, late-time bias and redshift-space inference, modified-gravity field equations, defects and other active sources, and current data constraints lie in later chapters or other volumes. The executable example on the final page is a closure unit test, not a production transfer calculation.

Review the chapter

Consider two scalar descriptions related by $\widetilde\psi=\psi+\mathcal HT$, $\widetilde{\delta\rho}=\delta\rho-\rho'T$, and $\widetilde{\delta q}=\delta q+(\rho+p)T$. First verify that $\zeta$ and $\mathcal R$ are invariant. Then suppose two separately conserved fluids have the same local time shift, $\delta\rho_1/\rho_1'=\delta\rho_2/\rho_2'$. Show that their relative entropy $S_{12}=3(\zeta_1-\zeta_2)$ vanishes, and explain why this statement alone does not determine the free-streaming quadrupole.

Solution

For $\zeta$,

$$\widetilde\zeta =-(\psi+\mathcal HT) -\mathcal H\frac{\delta\rho-\rho'T}{\rho'} =\zeta.$$

For $\mathcal R$, the background continuity equation $\rho'=-3\mathcal H(\rho+p)$ is not even needed: direct substitution gives

$$\widetilde{\mathcal R} =-(\psi+\mathcal HT) +\mathcal H\frac{\delta q+(\rho+p)T}{\rho+p} =\mathcal R.$$

For the two fluids,

$$S_{12} =-3\mathcal H\left( \frac{\delta\rho_1}{\rho_1'} -\frac{\delta\rho_2}{\rho_2'} \right)=0.$$

This fixes the relative density slicing but not the whole phase-space distribution. A collisionless quadrupole is an independent angular moment unless the declared primordial regularity and thermal/causal assumptions set it, after which free streaming generates it dynamically.

References

• Blas, D., J. Lesgourgues, and T. Tram, “The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation Schemes,” Journal of Cosmology and Astroparticle Physics 2011(07), 034 (2011), DOI, Open PDF.
• Bucher, M., K. Moodley, and N. Turok, “The General Primordial Cosmic Perturbation,” Physical Review D 62, 083508 (2000), DOI, Open PDF.
• Ma, C.-P., and E. Bertschinger, “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges,” The Astrophysical Journal 455, 7–25 (1995), DOI, Open PDF.
• Malik, K. A., and D. Wands, “Cosmological Perturbations,” Physics Reports 475, 1–51 (2009), DOI, Open PDF.