Parameterized Cosmological Gravity

A cosmological gravity parameterization can summarize measured departures without identifying a theory. To interpret it as a physical model, one must also close the perturbation equations in a declared gauge and regime, specify initial data, and check stability and the domain of validity. Arbitrary functions $\mu(a,k)$ and $\Sigma(a,k)$ need not correspond to a consistent covariant theory. Effective-field functions provide a more structured theory space, but still require stability, scale, and strong-coupling checks.

Required background. Probability and Statistical Inference supplies function priors, parameter measures, posterior geometry, and model comparison for flexible gravity coordinates.

Helpful background. Weak-Field Gravity supplies perturbative gauge transformations; Relativistic Cosmology supplies background and linear evolution; and Tests of Gravity supplies the distinction between a measured coordinate and a complete theory.

Observable two-function bases

Use conformal Newtonian gauge, $a_0=1$, conformal time $\tau$, and comoving Fourier modes $e^{i\mathbf k\cdot\mathbf x}$:

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

one common linear, late-time parameterization is

$$-k^2\Psi=4\pi G a^2\mu(a,k)\rho_m\Delta_m,$$ $$-k^2(\Phi+\Psi) =8\pi G a^2\Sigma(a,k)\rho_m\Delta_m.$$

Here $\Delta_m$ is the comoving matter density contrast and $k$ is comoving wavenumber. GR gives $\mu=\Sigma=1$ when additional species’ perturbations and anisotropic stress can be neglected in these late-time constraints. Dividing the Weyl equation by the $\Psi$ equation gives $1+\Phi/\Psi=2\Sigma/\mu$, so the slip $\eta=\Phi/\Psi$ obeys $\Sigma=\mu(1+\eta)/2$. Here $\eta$ is not the time coordinate, and the ratio requires $\Psi\ne0$. Ishak’s §5.2 gives the $Q,R$ and $\mu,\eta$ formulations in Eqs. (86)–(91), followed by the conversion to $\Sigma$ in Eq. (92) Ishak 2019, §5.2, Eqs. (86)–(92).

These equations do not by themselves specify superhorizon evolution, extra-field initial modes, matter couplings, or nonlinear screening. Their signs also depend on the Fourier and potential conventions. A basis used in one Boltzmann code cannot be transferred to another without checking gauge, normalization, species perturbations, and whether the quasistatic approximation has already been imposed.

Effective-field property functions

For the Horndeski scalar–tensor class with minimally coupled matter treated by Bellini and Sawicki, a specified background, matter sector, effective Planck-mass normalization, and four time-dependent property functions determine linear perturbation equations once initial data are supplied:

• $\alpha_K$ describes scalar kineticity;
• $\alpha_B$ describes kinetic braiding between scalar and metric;
• $\alpha_M=d\ln M_*^2/d\ln a$ describes the Planck-mass run rate;
• $\alpha_T=c_T^2-1$ describes tensor-speed excess.

The mass and its run rate are related, not independent free functions:

$$M_*^2(a)=M_*^2(a_\star) \exp\!\left[\int_{a_\star}^{a}\alpha_M(\tilde a)\,d\ln\tilde a\right].$$

The GR normalization is $M_*^2=1/(8\pi G)$, using the reduced Planck mass. A constant but incorrectly normalized $M_*$ does not recover the same measured gravitational coupling merely because $\alpha_M=0$.

Writing $\widetilde\Omega_m=\rho_m/(3H^2M_*^2)$ in the Bellini–Sawicki normalization, the frequently used ansatz

$$\alpha_i(a)=[1-\widetilde\Omega_m(a)]\widehat\alpha_i$$

is a time basis, not the definition of the functions Bellini and Sawicki 2014, §3.1, Eq. (3.2). Their definitions in terms of the covariant Lagrangian appear in Appendix A, Eqs. (A.7)–(A.10). Different covariant theories can share the same linear property functions, so linear observations cannot reconstruct distinctions that this compression deliberately removes.

Stability is a prior on the physical model, not a posterior robustness option. The quadratic action requires

$$M_*^2\gt0,\qquad D\equiv\alpha_K+\frac32\alpha_B^2\gt0, \qquad c_s^2\gt0,\qquad c_T^2=1+\alpha_T\gt0.$$

For the propagating scalar in the regular variables used there, its kinetic coefficient is $Q_s=2M_*^2D/(2-\alpha_B)^2$. Positive kinetic coefficients avoid ghosts, and positive squared sound speeds avoid high-frequency gradient instabilities Bellini and Sawicki 2014, Eqs. (3.12)–(3.14). The displayed test assumes these variables are regular; $\alpha_B=2$ needs a separate variable or constraint analysis, not division by zero. A vanishing scalar kinetic term also requires interpretation: it may indicate strong coupling in a model that still claims a propagating scalar, whereas exact GR has no additional gravitational scalar to test. The exact GR branch is therefore not rejected solely because $D=0$ when all property functions vanish.

Positive sound-speed squares do not prove luminality, global causality, or a Wilsonian ultraviolet completion. Any stronger causality prior must state which characteristic cone is compared with the matter light cone and over which backgrounds. Positivity also does not guarantee that the effective theory remains weakly coupled through the largest observational wavenumber.

Superhorizon completion and quasistatic limits

The quasistatic limit neglects time derivatives relative to spatial gradients. It can yield effective $\mu$ and slip on sufficiently small linear scales, but it turns a propagating scalar equation into a constraint. Bellini and Sawicki show that the strict $k\to\infty$ terms agree while subleading terms and homogeneous modes can differ Bellini and Sawicki 2014, §4.3. Thus the approximation requires $k/(aH)\gg1$ and $c_s k/(aH)\gg1$ for the scalar mode, with any Compton or braiding transition resolved explicitly—not merely a wavelength “inside the Hubble radius.”

A parameterized post-Friedmann (PPF) completion instead matches a superhorizon consistency relation to a quasistatic modified Poisson equation through an auxiliary field. Hu and Sawicki use a spatial metric perturbation with the opposite sign to the convention above and define $\Phi_-^{\rm HS}=(\Phi_{\rm HS}-\Psi_{\rm HS})/2=-(\Phi+\Psi)/2$. In their source convention they write

$$k^2(\Phi_-^{\rm HS}+\Gamma_{\rm HS}) =4\pi G a^2\rho_m\Delta_m$$

and evolve $\Gamma_{\rm HS}$ so that the large- and small-scale limits join continuously Hu and Sawicki 2007, §§II–III, Eqs. (16)–(21). Equivalently, the auxiliary term must be translated together with the potential sign before it is inserted into this page’s $\Phi+\Psi$ equation. Their §IV supplies a nonlinear interpolation ansatz, not a universal derivation of screening. A theory-specific nonlinear calculation is still required.

Physical checks before inference

A defensible parameter draw must pass four independent checks. First, it recovers the chosen GR background and perturbations as $\mu,\Sigma\to1$ or the property functions vanish with constant $M_*^2=1/(8\pi G)$. Second, it evolves the declared superhorizon initial data without introducing a gauge artifact. Third, the full dynamics and quasistatic solution agree where their hierarchy is valid. Fourth, ghost, gradient, strong-coupling, and screening conditions hold over the regimes used by the likelihood. Screening is relevant when extending the model to nonlinear or local observables, rather than a conclusion supplied by the linear property functions alone.

The parameter measure matters: a prior uniform in binned $\mu$ is not uniform in $\alpha_i$, sound speed, or a theory coupling. Causal smoothness kernels and stability rejection change the effective volume. Results must therefore state the basis, knots or functional form, correlations, initial conditions, approximation regime, code implementation, and null limit.

Dimensional and limiting checks remain useful even in a numerical implementation: $\mu$, $\eta$, $\Sigma$, and the $\alpha_i$ are dimensionless, observables must be invariant under a consistent gauge translation, and independently implemented Boltzmann solvers should agree in both the GR and declared modified limits.

Exercises

Take $\mu=1.2$ and $\eta=\Phi/\Psi=0.5$. Compute $\Sigma$, check the GR limit algebraically, and explain why an arbitrary scale-dependent pair $\mu,\eta$ need not define a viable theory.

Solution

Substitution gives $\Sigma=\mu(1+\eta)/2=1.2(1.5)/2=0.9$. Setting $\mu=\eta=1$ instead gives $\Sigma=1$, so the algebraic GR limit is correct. Viability does not follow from this relation because it supplies no evolution equation for an extra degree of freedom, no superhorizon initial condition, and no ghost or gradient test. Therefore the functions describe an observable closure only in their declared regime until a stable covariant model or complete effective system realizes them.

References

• Emilio Bellini and Ignacy Sawicki, “Maximal Freedom at Minimum Cost: Linear Large-Scale Structure in General Modifications of Gravity,” Journal of Cosmology and Astroparticle Physics 07, 050 (2014), DOI.
• Mustapha Ishak, “Testing General Relativity in Cosmology,” Living Reviews in Relativity 22, article 1 (2019), DOI.
• Wayne Hu and Ignacy Sawicki, “Parameterized Post-Friedmann Framework for Modified Gravity,” Physical Review D 76, 104043 (2007), DOI.