Gravitational Constant Variation

• Gravitational constant variation is the study of potential changes in Newton's G over time, space, and environment, informed by scalar-tensor, entropic, and higher-dimensional theories.
• Empirical investigations employ observations from stellar evolution, cosmology, and gravitational waves to set stringent bounds on any variability in G.
• Recent models and data analyses constrain G's rate of change to extremely small values (|dot G/G| ≲ 10⁻¹² yr⁻¹), guiding future research in fundamental physics.

Gravitational constant variation refers to theoretical and observational explorations of whether Newton's gravitational constant, $G$, is truly invariant in space and time or subject to temporal, spatial, or environment-dependent changes. The prospect of a varying $G$ is predicted by numerous extensions of general relativity and unified theories, and offers a unique observational window on fundamental physics at quantum, astrophysical, and cosmological scales. Approaches to this question span multifaceted methodologies: entropy-based gravity with quantum corrections, scalar-tensor and higher-dimensional frameworks, distinct astrophysical and cosmological constraints, and analyses of gravitational wave propagation. The following sections present a comprehensive synthesis based strictly on peer-reviewed arXiv studies and their quantitative findings.

1. Theoretical Frameworks for Gravitational Constant Variation

Multiple theoretical constructs underpin investigations into gravitational constant variation. Scalar-tensor theories, such as Jordan–Brans–Dicke models, generically predict $G\propto1/\Phi$, where $\Phi$ is a dynamical scalar field with evolution tied to cosmological or local environmental factors (Mimoso et al., 2011, Ballardini et al., 2021). In geometric scalar gravity (GSG), $G_N$ emerges from non-minimal coupling of a scalar field via a disformal transformation, leading to variations in both space and time through $$G_N = G_0 \left[Q(\phi)\right]^{-7/2} a^{-2}(a-3)^{-1} V(\phi)^{-3/2}$$ (Bronnikov, 2020).

In the entropic gravity paradigm, gravity is understood as an emergent thermal force; applying the generalized uncertainty principle (GUP) at the Planck scale leads directly to a time-dependent $G$ via corrections to Newton’s second law. The effective gravitational constant runs as $G = G_N e^{-t/\tau}$, with $\tau^{-1} = c\lambda / (2(\delta x)^2)$ set by quantum mechanical parameters (Setare et al., 2010). In multidimensional models with higher-curvature corrections (e.g., Gauss–Bonnet gravity in $D$ dimensions), $G$ depends on the evolving internal volume and can remain constant or vary extremely slowly, depending on the balancing of subspace expansion parameters (Ivashchuk et al., 2015).

Certain modified gravity scenarios trigger environmental variation: the effective $G$ depends on local density via the dynamics of a scalar mediating fifth force, with chameleon or Damour–Polyakov screening suppressing variation in high-density regimes while allowing $G$ to vary in vacuum or intergalactic environments (Brax, 2013). Within Kaluza–Klein frameworks, coupling of a scalar to geomagnetic potentials modulates $G$ in laboratory experiments, producing apparent temporal and spatial variations depending on the local magnetic environment (Mbelek, 2018).

2. Importance of Dimensionless Quantities and Observational Interpretation

Physical significance attaches solely to dimensionless combinations of fundamental constants, as emphasized forcefully in (Moss et al., 2010). For gravity, the relevant combination is $\alpha_g \equiv G m_p^2 / (\hbar c)$. Constraints on $G$ alone are ambiguous, since its value depends on unit conventions; only variations in $\alpha_g$ reflect true changes in fundamental interactions. This principle underpins robust interpretation of experimental and observational bounds, from stellar astrophysics to cosmological recombination.

Table: Representative Dimensionless Gravitational Quantities

Observable	Dimensionless Parameter	Physical Relevance
Stellar Mass Scale	$M \sim \alpha_e \alpha_g^{3/2} m_p$	Mass hierarchy, main sequence
Chandrasekhar Mass	$M_{Ch} \sim \alpha_g^{3/2} m_p$	WD stability threshold
Cosmological effects	$\alpha_g, \alpha_e$	BBN, CMB, structure formation

Rigorous focus on dimensionless parameters ensures constraints are grounded in observable physics, independent of arbitrary measurement units or system histories.

3. Astrophysical and Cosmological Constraints on Gravitational Constant Variation

A diverse arsenal of astrophysical and cosmological observations constrains $\dot G / G$ over cosmic time and space:

a. Stellar and White Dwarf Evolution: Full evolutionary calculations with time-varying $G$ in white dwarfs and main sequence stars, using codes such as LPCODE, reveal that even small secular decreases in $G$ have strong effects—shortening main sequence lifetimes (e.g., $\tau_{MS}$) and significantly accelerating white dwarf cooling, especially in higher-mass systems (Althaus et al., 2011, García-Berro et al., 2011). White dwarf luminosity functions in old clusters (e.g., NGC 6791) yield tight bounds: $\dot G / G \lesssim -1.8\times10^{-12}$ yr$^{-1}$ (García-Berro et al., 2011). Pulsating white dwarfs, which are sensitive to cooling rates, provide independent constraints: $|\dot G / G| \lesssim 1.8\times10^{-10}$ yr$^{-1}$ for G117–B15A (Córsico et al., 2013).

b. Asteroseismology: Modeling ancient solar-like oscillators (such as KIC 7970740) with a power-law $G(t) = G_0 (t_0 / t)^\beta$ places constraints at $|\dot G / G| < 5.6 \times 10^{-12}$ yr$^{-1}$ (95% CL), consistent with a constant $G$ over 11 Gyr (Bellinger et al., 2019).

c. Cosmological Probes: Primordial abundances from Big Bang Nucleosynthesis (BBN) tightly restrict $G$ during the first few minutes, improving previous BBN limits by an order of magnitude: $G_{\rm BBN}/G_0 = 0.99^{+0.06}_{-0.05}$ and $\dot G / G_0 = 0.7^{+3.8}_{-4.3}\times10^{-12}$ yr$^{-1}$ (95% CL) (Alvey et al., 2019). Scalar-tensor cosmologies allow $G_{\rm eff}(z=0)/G$ to depart from unity by at most $3\%$ since the radiation era, with future CMB and LSS experiments forecast to reach subpercent sensitivity (Ballardini et al., 2021).

d. Gravitational Wave Observations: Binary neutron star mergers (e.g., GW170817) encode $G$ at emission in the mass limits inferred from waveforms. The current bound: $-1 \lesssim \Delta G / G \lesssim 8$; on average, $-7 \times 10^{-9}$ yr$^{-1} \le \dot G / G \le 5 \times 10^{-8}$ yr$^{-1}$ between merger and present, with future detections expected to improve sensitivity significantly (Vijaykumar et al., 2020). Temporal or spatial variation in $G$ during propagation also modifies GW amplitude and phase, providing complementary constraints described below.

4. Gravitational Constant Variation and Gravitational Wave Propagation

Recent analyses have focused on the imprint of $G$ variation not only on the generation, but particularly on the propagation of gravitational waves (GWs):

• Amplitude and Phase Corrections: Linearized Einstein or Fierz-Pauli action analyses, and Maxwell-like analogy, yield a leading order amplitude correction scaling as $h = \sqrt{G_d/G_s} h_0$ as a GW travels from source (with $G_s$) to detector (with $G_d$). The subleading phase correction $\Delta\Phi = - (3/16\pi f) (G'/G)^2 D_L$ is suppressed as $\mathcal{O}\left[(G'/G)^2\right]$ (Sun et al., 2023, An et al., 29 Aug 2025).
• GW Data Analysis and Detectability: Amplitude bias is entirely degenerate with the unknown luminosity distance $D_L$ unless an electromagnetic counterpart provides $D_L$. For "dark" events, only the phase correction—quadratic in $G'/G$—offers independent information, but with current LIGO-Virgo-KAGRA sensitivities, $|\dot G / G| \lesssim 10^{-7}$/yr remains the upper bound; an improvement by order $n$ requires $2n$ orders increase in $D_L$ or signal-to-noise ratio (An et al., 29 Aug 2025).
• Constraints on Long-Range Modifications: For GW170817, amplitude mismatch between GW-inferred and EM-inferred $D_L$ constrains spatial variation in $G$ to within 80% confidence. When parameterized by Yukawa deviations (e.g., $$G(r) = G_\infty[1 + \alpha (1 + r/\lambda) e^{-r/\lambda}]$$), these analyses provide the first constraints on graviton mass at scales $\lambda \sim 10^{23}$ m, corresponding to $m_g \sim 10^{-31}$ eV (An et al., 2023).

5. Environmental and Spatial Variation of Gravitational Constant

Scalar-tensor models with environmental screening mechanisms (chameleon, Damour–Polyakov types) render $G$ dependent on local density. In dense environments, screening suppresses scalar-mediated fifth forces and $G$ variation, limiting fractional deviations in parameters such as the proton-to-electron mass ratio to $<10^{-6}$ locally (Brax, 2013). In geometric scalar gravity, spatial gradients in $G$ remain sufficiently small for nearly circular orbits, but may be detectable for highly eccentric trajectories—e.g., $\sim 4$ cm displacement for the lunar orbit, kilometers over asteroid scales—thus subject to high-precision Solar System measurements (Bronnikov, 2020).

Furthermore, laboratory measurements of $G$ are affected by Earth's variable geomagnetic field, producing apparent temporal drifts as the local scalar (KK) field responds to changes in the magnetic potential. Empirical analysis of data from the HUST lab shows a linear correlation between $G$ measurements and time-dependent geomagnetic indices, with a coupling parameter $F^{-1} = (3.25 \pm 0.35) \times 10^{-14}$ m/J (Mbelek, 2018). True measurement of $G$ independent of local environmental contamination may require space-based experiments.

6. Implications for Cosmology and Fundamental Physics

Time-varying $G$ offers alternative explanations to the cosmological constant for the observed acceleration of universal expansion. By introducing a dynamical $G(t)$ and an additional tensor $S^{\mu\nu}$ in Einstein's equations, cosmological models can reproduce late-time acceleration consistent with supernovae and BAO data, while keeping $G$ variation within 10% over observed redshifts and compatible with stringent local constraints (Hanımeli et al., 2019). In certain Gauss–Bonnet higher-dimensional settings, accelerated expansion is obtained with vanishing or minuscule $G$ variation, depending on the configuration of extra dimensions (Ivashchuk et al., 2015).

In emergent, entropic gravity frameworks with GUP corrections, the running $G$ also provides a holographic realization of Dirac's large numbers hypothesis, with $G \propto t^{-1}$ emerging in the early-time approximation (Setare et al., 2010). This establishes a conceptual bridge between quantum gravitational corrections, cosmic time, and the scale of fundamental couplings.

7. Synthesis and Outlook

The body of research converges on the following points:

• No empirical evidence currently supports substantial spatial or temporal variation of $G$ at the level of $|\dot G / G| > 10^{-12}$/yr over cosmic or local timescales, with even tighter constraints in many astrophysical and laboratory contexts (Alvey et al., 2019, García-Berro et al., 2011, Bellinger et al., 2019).
• Dimensionless couplings such as $\alpha_g$ are the only physically meaningful objects for testing the hypothesis of varying gravity (Moss et al., 2010).
• Detecting a fractional variation in $G$ far above $10^{-12}$ appears disfavored by BBN, stellar, GW, and local measurements.
• Theoretically, most frameworks that permit $G$ variation (e.g., scalar-tensor, emergent gravity, higher-dimensional models) are now strongly constrained by a multi-pronged array of empirical data.
• Attempts to explain cosmic acceleration via $G(t)$ rather than $\Lambda$ demand fine-tuned or carefully parameterized models to avoid conflict with existing local and cosmological measurements (Hanımeli et al., 2019).
• Future improvements in GW observation (higher SNR, increased $D_L$), next-generation CMB/LSS surveys, and space-based $G$ measurements may further tighten constraints or reveal minute signatures of gravitational coupling evolution.

Collectively, these results suggest that if $G$ varies at all, its rate and amplitude must be extremely small, and new physics seeking to incorporate variable gravity must survive stringent multi-scale and multi-epoch observational scrutiny.