Cosmology Beyond General Relativity

Cosmology turns an alternative gravitational action into a long chain of predictions: a homogeneous branch, constrained perturbations, growth and lensing transfer functions, gravitational-wave propagation, and quantities that can be compared with observations. This chapter develops that chain without identifying a successful expansion history with a successful theory. Background degeneracy, extra degrees of freedom, stability, screening, initial conditions, and EFT validity remain separate questions.

Helpful background. Einstein Equations supplies the FLRW and linearized GR baseline; Gravitational Actions and Boundary Terms supplies the variational origin of modified equations; Formulations of General Relativity supplies constraints, gauges, and initial data; Gravity Beyond General Relativity locates cosmology within the wider theory space; and Causality and Global Structure supplies the hyperbolicity and characteristic-cone tests used in deciding viability.

Enter this chapter

The starting point is always a defined theory: fields, action or equations, matter coupling, and solution branch. Reducing it on FLRW produces modified Friedmann equations, but those equations alone do not reveal all propagating modes. A second variation of the full action and elimination of lapse and shift constraints produce the kinetic and gradient matrices that decide linear stability.

Only then is it safe to compress the perturbations into common response functions. Nonrelativistic matter responds to $\Psi$, lensing responds to $\Phi+\Psi$, and gravitational waves have their own kinetic coefficient, characteristic speed, mass, and possible mixing. Each compression carries a redshift, scale, frequency, and cutoff domain.

The chapter therefore follows the direction of calculation rather than the direction of observational presentation:

$$\text{action and branch} \longrightarrow \text{background} \longrightarrow \text{reduced perturbations} \longrightarrow \text{transfer functions} \longrightarrow \text{measurement model}.$$

The last arrow requires additional assumptions. This volume develops the theoretical predictions and identifies where they fail. Survey likelihoods, detector systematics, population assumptions, and numerical bounds are developed in Tests of Gravity.

Shared cosmological conventions

For flat FLRW,

$$ds^2=-dt^2+a^2(t)d\boldsymbol{x}^2 =a^2(\eta)\left(-d\eta^2+d\boldsymbol{x}^2\right),$$

with $H=\dot a/a$ and $\mathcal H=a'/a=aH$. A dot denotes $d/dt$. On pages where a prime instead denotes $d/d\ln a$, that choice is stated beside the equation.

Scalar perturbations in Newtonian gauge use

$$ds^2=a^2\left[-(1+2\Psi)d\eta^2 +(1-2\Phi)d\boldsymbol{x}^2\right].$$

With negligible material anisotropic stress, the GR reference limit has $\Phi=\Psi$. Fourier spatial derivatives satisfy $\nabla^2\to-k^2$. Matter is minimally coupled to the displayed metric unless a page explicitly says otherwise; this is the metric that defines rulers, clocks, redshift, and electromagnetic propagation.

These conventions are not repeated as a substitute for derivation. A conformal or disformal field change must carry the matter coupling with it, and a phenomenological response function must retain the metric convention under which it was defined.

Route through the six pages

1. Modified Friedmann Dynamics derives metric-$f(R)$ background equations, their scalar constraint, de Sitter branch condition, and GR limit.
2. Cosmological Perturbations and Degrees of Freedom performs the scalar–vector–tensor decomposition and explains why constraints must be eliminated before kinetic eigenvalues and sound speeds are interpreted.
3. Effective Field Theory of Cosmic Acceleration organizes unitary-gauge operators, restores the scalar clock, and separates background functions from perturbation-only coefficients.
4. Growth, Lensing, and Gravitational Slip derives the $\mu$–$\eta$–$\Sigma$ relations and the scale-dependent growth equation, with explicit quasistatic limits.
5. Cosmological Gravitational-Wave Propagation derives tensor friction, speed, mass, mixing, and the propagation contribution to the gravitational-wave luminosity distance.
6. Theoretical Viability and Comparison with Observations combines stability, cutoff, screening, initial-condition, and branch requirements to explain which predictions can be tested, using results available at the stated date.

The order prevents two common shortcuts. One cannot decide the mode count from the background equations, and one cannot apply a linear cosmological $\mu(a,k)$ inside a screened nonlinear environment without a matching calculation.

A layered prediction test

Layer	Calculation	A positive result does not yet establish
Definition	action, fields, matter frame, branch, boundary data	existence of a cosmological solution
Background	constraint plus evolution equations and GR limit	perturbative stability
Linear dynamics	reduced kinetic, gradient, mass, and characteristic matrices	nonlinear health or screening
Response	scalar and tensor transfer functions with initial conditions	validity beyond their scale and frequency range
Nonlinear closure	screened solution, matching, or validated simulation	a survey likelihood or detector bound
Comparison with observations	prediction arrays, normalizations, units, and failure masks	robustness to measurement systematics

Several checks recur. Dimensions and the GR limit catch normalization errors. Differentiating a Friedmann constraint and using matter conservation tests the Bianchi identity. Positive eigenvalues must be checked over the full stated domain, not at a single epoch. Physical wavenumbers and frequencies must remain below the EFT cutoff. A special background on which a kinetic coefficient vanishes requires a fresh constraint analysis rather than optimistic continuation.

What remains model dependent

No universal pair of functions captures every alternative cosmology. Horndeski theories with one scalar admit an efficient linear basis Bellini and Sawicki 2014, § 3.1 and Appendix A, but extra vectors, several metrics, nonlocal boundary prescriptions, parity violation, and multiple scalars enlarge the state space. Even within one basis, stable function choices need not have a unique covariant completion.

The quasistatic approximation is likewise conditional. It requires separation from the Hubble and scalar sound horizons, absence of rapid transients or nearby kinetic degeneracies, and control below the cutoff. Koyama develops its relation to screening and structure formation in Koyama 2016, §§ 4–6.

The final two pages distinguish theory statements that are settled within their assumptions from broader claims that remain unresolved. These discussions reflect the literature and competing interpretations assessed through 2026-08-19.

Review the chapter

A scalar–tensor model is supplied as four smooth $\alpha$ functions and a background $H(a)$. A code returns finite $\mu(a,k)$ and $\Sigma(a,k)$ on a grid. List the minimum theoretical checks needed before those arrays can be handed to a likelihood analysis.

Solution

First identify the theory class, matter frame, branch, and initial conditions represented by the $\alpha$ functions; smooth functions alone do not define those choices. Verify that $H(a)$ satisfies the background equations and that the effective Planck mass remains positive.

Next reduce the scalar and tensor quadratic actions throughout the grid. Check positive kinetic eigenvalues, positive short-wavelength gradient eigenvalues, acceptable mass timescales, and nonsingular constraint denominators. Confirm that the evolution is hyperbolic and that every $k/a$ remains below the time-dependent cutoff.

Then state whether $\mu$ and $\Sigma$ came from the full evolution or a quasistatic reduction. For the latter, mask points near the Hubble or scalar sound horizon, rapid transients, resonances, and screening transitions. Supply the gauge and sign conventions, units, normalizations, interpolation rule, and failure mask.

Finally keep nonlinear screening, source generation, survey systematics, and likelihood choices separate. If nonlinear scales are included, provide a theory-specific matched calculation or validated simulation rather than extending the linear arrays. With those items documented, the empirical analysis can vary measurement assumptions without silently changing the theory.

References

• Bellini, Emilio, and Ignacy Sawicki. “Maximal Freedom at Minimum Cost: Linear Large-Scale Structure in General Modifications of Gravity.” Journal of Cosmology and Astroparticle Physics 2014 (07), 050 (2014). DOI. Open PDF
• Koyama, Kazuya. “Cosmological Tests of Modified Gravity.” Reports on Progress in Physics 79, 046902 (2016). DOI. Open PDF