Variable Gravity Approximation

• Variable Gravity Approximation is a framework that promotes Newton’s constant to a dynamic field, redefining gravitational interactions beyond standard general relativity.
• It employs scalar-tensor models, nonlocal theories, and renormalization-group techniques to generate analytic solutions across inflationary, matter-dominated, and dark energy epochs.
• Observational tests, N-body simulations, and geophysical models constrain the theory, offering insights into cosmic evolution and refined atmospheric parameterizations.

The variable gravity approximation encompasses a diverse set of frameworks in which the effective gravitational strength—typically quantified by Newton's constant $G$ or the squared Planck mass $M_\mathrm{Pl}^2$—becomes a dynamical field or function, often to capture physics beyond general relativity. This concept appears across fully covariant models (scalar-tensor gravity, nonlocal gravity, renormalization-group improvements) as well as in non-relativistic, Newtonian-like regimes, and is integral to modern cosmological model building, laboratory and astrophysical tests of gravity, and precision geophysical modeling.

1. Fundamental Formalism and Theoretical Foundations

At the core, the variable gravity approximation generalizes Einstein's theory by promoting the gravitational "constant" to a field-dependent or space-time-dependent object. The archetype is the scalar-tensor action in the Jordan frame: $$S[\phi,g_{\mu\nu}] = \int d^4x \sqrt{-g}\left[ -\frac{1}{2}F(\phi)R + \frac{1}{2}K(\phi)g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi + V(\phi) \right]$$ where:

• $F(\phi) = M_\mathrm{Pl}^2(\phi)$ captures a dynamically evolving Planck mass.
• $K(\phi)$ is the "kinetial", controlling the scalar's kinetic normalization.
• $V(\phi)$ is the potential (e.g., quadratic, constant).

The effective gravitational coupling is then $G_\mathrm{eff}(\phi) = 1/[8\pi F(\phi)]$. Modifications and key properties arise directly from the scalar dependence in $F$, producing distinctive phenomenology and rich cosmological dynamics (Wetterich, 2013).

In the Newtonian regime, variable $G$ models can be constructed to preserve Galilean invariance and recover standard gravity in suitable limits. Specifically, $\sigma(x,t)$ can be introduced as a dimensionless scalar field such that $G_\mathrm{eff}=G_0\sigma(x,t)$, with $\sigma$ sourced and evolved according to fundamental field equations derived from a generalized Lagrangian (Fabris et al., 2020).

2. Representative Models and Regimes

Several concrete models instantiate the variable gravity paradigm, each with specific parameterizations and physical implications.

Scalar-Tensor Cosmologies: Wetterich's models (A and B) specify $F(\chi)$, $K(\chi)$, and $V(\chi)$ for a cosmon field $\chi$, yielding inflation, radiation, matter, and dark energy eras under a unified scalar degree of freedom. For example, model (A) has $F(\chi)=\chi^2$ and $V(\chi)=\mu^2\chi^2$, facilitating analytic treatment of cosmological epochs and accommodating early dark energy as a persistent subdominant component (Wetterich, 2013).

Renormalization Group Improved Gravity: The gravitational coupling $G(x)$ is promoted to a dynamical field within a modified Einstein–Hilbert action, with a kinetic term for $G$ and a potential for the cosmological "constant" $\Lambda(G)$. Imposing Noether symmetry methods constrains the form of $\Lambda(G)$, rendering the system integrable and yielding analytic Friedmann–Lemaître–Robertson–Walker (FLRW) solutions for $a(t)$, $G(t)$, and $\Lambda(t)$ (Mondal et al., 2019).

Newtonian-like Variable G: Fabris et al. construct a non-relativistic Lagrangian where both the Newtonian potential $\psi$ and a scalar $\sigma$ evolve dynamically. Spherical body solutions reveal interior corrections to the gravitational potential (modified Helmholtz equation for $\psi$), but standard Newtonian behavior in the vacuum. Cosmological extension produces modified Friedmann equations with explicit $G(t)$ dependence (Fabris et al., 2020).

Brans–Dicke–Inspired Manev Gravity: The effective force law acquires a $1/r^3$ correction due to variable $G$, leading to observable consequences such as perihelion precession corrections and negligible impact on tidal limits (Roche limit). Constraints on the free parameter $\omega$ (Brans–Dicke analog) are extracted from solar system tests (Escórcio et al., 2023).

3. Cosmological Dynamics and Unified Evolution

Variable gravity models support a sequence of cosmological epochs within unified scalar field dynamics:

• Inflationary Era: Large positive $K(\phi)$ enables slow-roll inflation; observables such as the scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ are determined by the effective kinetic function $k^2(\varphi)$ after mapping to the Einstein frame. For $N$ e-folds, typical results are $n_s\approx 1-2/N$, $r\approx 8/N$ for quadratic models (Wetterich, 2013, Hossain et al., 2014).
• Radiation and Matter Domination: For $K+6\ll1$, the Universe tracks background fluids, with a small but constant early dark energy fraction $\Omega_h \simeq n/\alpha^2$ ($n=4$ for radiation, $n=3$ for matter).
• Late-Time Dark Energy and Accelerated Expansion: At late epochs, the cosmon potential and/or neutrino mass couplings trigger a transition to dark energy domination, matching observed acceleration without requiring strong fine-tuning of parameters.
• Singularity Avoidance: In the Jordan frame, the big bang singularity is regularized, with smooth curvature invariants, while singularities may reappear as coordinate artifacts in the Einstein frame (Wetterich, 2013).

The dynamical evolution of $G$ introduces additional terms in the effective Friedmann equations, e.g.,

$$\frac{\ddot a}{a} = -\frac{4\pi G_0 \sigma \rho}{3} + \frac{\omega}{6}\left(\frac{\dot\sigma}{\sigma}\right)^2$$

enabling power-law solutions and, depending on model branch and parameters, acceleration, deceleration, or bounce scenarios (Fabris et al., 2020, Mondal et al., 2019).

4. Observational Consequences and Tests

Variable gravity scenarios are tightly constrained by cosmological and astrophysical observations:

• Inflationary observable predictions adhere to slow-roll expectations, with $n_s$ and $r$ values consistent with Planck data for suitable parameter choices, e.g., $n_s\approx0.95$–0.97, $r\approx0.05$–0.13 depending on model and $N$ (Wetterich, 2013, Hossain et al., 2014).
• Time Variation of G: Nonlocal gravity and scalar–tensor variants generically predict $|\dot G/G| \sim H_0$, violating bounds from pulsar timing and lunar laser ranging ($|\dot G/G| \lesssim 10^{-12}$ yr⁻¹). Only models with specific structure, such as the RT nonlocal model, evade these constraints (Tian et al., 2019).
• Post-Newtonian and Celestial-Mechanics Effects: The Manev-like $1/r^3$ correction modifies pericenter precession (e.g., Mercury's perihelion) but leaves the Roche limit practically unchanged ($\Delta r_R / r_R \lesssim 10^{-4}$ for solar system bodies) (Escórcio et al., 2023).
• Weak Equivalence Principle Violations: Some variable G models only avoid equivalence-principle violation by suppressing the relevant couplings or by constraining $\omega$ to unobservably large values, negating observable effects (Escórcio et al., 2023).

5. Numerical Methods and Applicability to Structure Formation

Within large-scale $N$-body simulations and perturbation theory, the variable gravity approximation is often treated as a quasi-static or scale-dependent correction:

• In $f(R)$ gravity, the quasi-static approximation neglects time derivatives of the scalaron, justified by the relative slowness of background evolution compared to spatial variations. High-resolution simulations demonstrate that these omitted terms introduce subpercent-level corrections in matter power spectra, validating the quasi-static (variable gravity) approach for cosmological structure formation modeling (Bose et al., 2014).
• Diagnostic metrics such as the matter spectrum ratio $\Delta P/P(k)$ confirm the adequacy of the approximation across physically relevant scales.

6. Application to Geophysical and Atmospheric Regimes

In global atmospheric modeling for oblate planets, "variable gravity" refers to accounting for the spatial dependence of $g$ due to geometric flattening and planetary rotation:

• Dubos (2018) introduces three closed-form metric approximations (I, II, III) capturing the latitude–height dependence of $g$ and atmospheric geometry, removing leading-order errors from traditional spherical-geoid (SG) and deep-atmosphere (DA) models (Dubos, 2018).
• Approximations I and II are conformal, facilitating analytical and numerical implementation by preserving the form of horizontal operators, while Approximation III (TD14-based) achieves $O(\epsilon)$ accuracy everywhere at the cost of non-conformality.
• For Earth's parameters, variable gravity corrections produce errors well below $10^{-4}\,g$ up to stratospheric altitudes.

Approximation	Accuracy Regime	Conformality	Max Residual Error in $g$
I	$O(\epsilon)$, near-surface	Yes	$\sim 10^{-5}g$
II	$O(1)$, all altitudes	Yes	$\sim 10^{-5}g$
III	$O(\epsilon)$, all altitudes	No	$\sim 10^{-5}g$

This mathematical structure allows atmospheric simulations to incorporate latitude and height dependence of gravity and metric dilation without complicating existing algorithms.

7. Limitations, Extensions, and Outlook

While the variable gravity approximation provides a versatile and unifying theoretical structure, its domain of validity is model-dependent:

• Newtonian-like variable $G$ is strictly nonrelativistic, becoming invalid for relativistic sources, strong fields, or regimes demanding light propagation or gravitational waves (Fabris et al., 2020, Escórcio et al., 2023).
• Cosmological models require careful mapping between Jordan and Einstein frames to separate physical effects from coordinate artifacts (Wetterich, 2013).
• In scalar–tensor and nonlocal gravities, the challenge is to enforce observational bounds on $|\dot G/G|$ and equivalence-principle violation, which severely restrict parameter space (Tian et al., 2019, Escórcio et al., 2023).

A plausible implication is that viable variable gravity models for late-time cosmology must be constructed to suppress $G$-time variation or fine-tune couplings, while applications to inflation and unified dynamical dark energy remain fertile for theoretically consistent and observationally testable scenarios (Wetterich, 2013, Hossain et al., 2014).

The variable gravity approximation continues to inform both phenomenological model building and the design of high-precision simulations from cosmology to planetary science.