Varying Newton’s constant: a cure for gravitational maladies?

We show that a slowly varying Newton’s constant, consistent with existing bounds, can potentially explain a host of observations pertaining to gravitational effects or phenomena across distances spanning from planetary to the cosmological, relying neither on the existence of dark matter or (and) dark energy, nor on any expected high proportions of either of them in the Universe. It may also have implications at very short distances or quantum gravity scales.

The manuscript has no associated data or the data will not be deposited. Observational datasets used for the statistical estimation are retrieved from well-known references, duly cited in this paper.

• 22 December 2024

  The original online version of this article was revised: In this article in PDF file the affiliations of the two authors were superimposed on one another making them unreadable. The original article has been corrected.

• 20 December 2024

  A Correction to this paper has been published: https://doi.org/10.1140/epjp/s13360-024-05927-0

1. At the level of the cosmological density perturbations though, the quasi-Newtonian formalism is expected to differ significantly from the relativistic one. Nevertheless, while such a perturbative study, in an appropriate covariant formulation for the G-variation, is being carried out in some ongoing works [41, 42], we keep it beyond the scope of this paper.

This work was supported by the Natural Sciences and Engineering Research Council of Canada. SS acknowledges financial support from the Faculty Research Programme Grant—IoE, University of Delhi (Ref.No./ IoE/ 2024-25/12/FRP). The authors are grateful to the anonymous reviewer of the paper, for making several important suggestions and bringing in notice a wealth of valued references.

Authors and Affiliations

1. Theoretical Physics Group, Department of Physics and Astronomy, University of Lethbridge, 4401 University Drive, Lethbridge, AB, T1K 3M4, Canada

   Saurya Das

2. Department of Physics and Astrophysics, University of Delhi, Delhi, 110007, India

   Sourav Sur

Corresponding author

Correspondence to Sourav Sur.

The original online version of this article was revised: In this article in PDF file the affiliations of the two authors were superimposed on one another making them unreadable. The original article has been corrected.

Appendix: Covariant embedding of the varying G—an illustration

Following refs. [17,18,19], consider the nonlocal extension of the teleparallel equivalent of general relativity (TEGR), which we shall refer henceforth as the nonlocal gravity (NLG) theory. The formalism is essentially developed in analogy with the nonlocal electrodynamics of media, wherein the constitutive relations between the electromagnetic excitation \(H^{\alpha \beta } = (\vec {\pmb D},\vec {\pmb H})\) and the field strength \(F_{\alpha \beta } = (\vec {\pmb E},\vec {\pmb B})\) depend on the evolution history and are hence nonlocal:

where g denotes the determinant of the metric tensor \(g_{\alpha \beta }\) and \(\chi ^{\alpha \beta \gamma \delta }\) is a kernel which determines the degree of nonlocality of the electromagnetic configuration.

Teleparallelism, on the other hand, concerns preferred orthonormal tetrad frames, such that

where \(e_\mu ^{~\hat{\alpha }}\) denotes the tetrad field, and \(\eta _{\hat{\alpha }\hat{\beta }}\) is the Minkowski metric of the local tangent space, with \(\hat{\alpha },\hat{\beta },\dots \) being the corresponding coordinate labels, different from \(\alpha ,\beta ,\mu ,\nu ,\dots \), which are the usual curved space coordinate labels. Defining an asymmetric affine connection, known as the Weitzenböck connection

one perceives curvature-freeness, i.e. the space-time is parallelizable, in accord with \(\nabla _\nu e_\mu ^{~\hat{\alpha }} = 0\), and as such, the metricity condition \(\nabla _\nu g_{\alpha \beta } = 0\), where \(\nabla _\nu \) denotes the covariant derivative in terms of \(\Gamma ^\sigma _{\mu \nu }\) given by eq. (37). The space-time nonetheless admits a nonzero torsion

which is nothing but the gravitational field strength, similar to the field strength \(F_{\alpha \beta } = 2 \partial _{[\alpha } A_{\beta ]}\) in electrodynamics (with \(A_\beta \) being the corresponding field potential). On the whole, the teleparallel gravitational formulation is that of a translational gauge theory—an Abelian one, analogous to electrodynamics. Such a formulation is in fact equivalent to general relativity (GR) in the sense that it leads to the same field equations, with the gravitational gauge field strength given by the Einstein tensor expressed in terms of the tetrads and the Weitzenböck connection as

where \(\kappa ^2 = 8 \pi G\) (with G being the usual Newton’s constant), and

In analogy with the electrodynamics of media, Eq. (40) is generally considered as a constitutive relation, with \(\mathcal {H}_{\mu \nu \sigma }\) treated as the gravitational excitation from the gauge theoretic perspective. The nonlocal generalization to this, proposed in a series of papers [17,18,19], is an augmentation of the modified torsion tensor \(\mathcal {C}_{\mu \nu \sigma }\) on the right hand side of Eq. (40) with a nonlocal term \(N_{\mu \nu \sigma }\) whose tangent space projection is expressed as

where \(\chi \) is a causal kernel, and

with \(\widetilde{C}_{\hat{\mu }}\) denoting the tangent space projection of the torsion pseudo-trace vector

and q being a dimensionless constant which determines the extent of the gravitational parity violation.

The Newtonian limit of the NLG theory leads to a modified Poisson’s equation which corresponds precisely to a logarithmic correction term in the gravitational potential we are dealing here in the context of the G-variation. The NLG Newtonian regime has therefore been of interest and extensive studies have been carried out from the point of view of explaining the flat rotation curves of galaxies [72,73,74]. However, beyond linearization, the (fully covariant) NLG theory has no known nonlinear exact solution till date. As such, the focus has shifted in recent times to the study of the local limit of the nonlocal TEGR, construed from the following form of the NLG kernel [75,76,77]:

where S(x) is a dimensionless scalar function of coordinates, which is referred to as the gravitational susceptibility function, in analogy with electrodynamics.

For the spatially homogeneous cosmological space-times, the susceptibility function can be taken to be varying with time only. The nonlocal TEGR equations lead to the modified Friedmann equations:

where \(\rho \) and p denote, respectively, the total energy density and the total pressure, and \(k = 0, \pm 1\) is the spatial curvature constant. In addition, there are two constraints, viz.

Therefore, either (i) \(\, k = \pm 1\) and \(\,S =\) constant, or (ii) \(k = 0\) and S is an arbitrary function of time. While the case (i) implies the same closed and open Friedmann solutions, modulo a numerical scaling of \(\rho \) and p by a factor \((1 + S)^{-1}\), the case (ii) is potentially of much interest as it not only complies with the assumption of the spatial flatness of the universe, which the observations grossly indicate, but also marks the outcome of an effective G-variation, since the above equations (47) and (48) can be recast as the following set:

where

As such, choosing suitably the function S(t), and hence \(G_{\text {eff}}(t)\), one can reproduce the cosmological equations (18) and (21) obtained in the quasi-Newtonian framework (upon replacing G by \(G_0\) and \(G_{\text {eff}}(t)\) by G(t)). Nevertheless, whether such a choice is justifiable from a physical standpoint is presently being examined, among several issues, in the ongoing works [41, 42].

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