Comment on “Varying-G Cosmology with Type Ia Supernovae” [Am. J. Phys. 79, 57–62 (2011)] Free

The idea of variability of the gravitational constant G is not a novel one. It was first suggested by Dirac¹ who claimed that G ∼ 1/t, in light of his Large Number Hypothesis (LNH). Following Dirac's claim came the Brans-Dicke theory of gravity,² in which the gravitational constant G is replaced with a scalar field $ϕ$ that couples to gravity via a constant ω. This theory was later generalized by Nordtvedt³ to what are now called scalar-tensor theories, which use different variable coupling parameters $ω(ϕ)$⁠. The variability of G has also been accounted for in observations, with various constraints on $|Ġ/G|$ obtained from lunar laser ranging,⁴ the Viking Lander,⁵ the spin rate of pulsars,⁶ distant type Ia supernova,⁷ helioseismological data,⁸ and cosmological nucleosynthesis.⁹ Working in the framework of general relativity, Lau¹⁰ proposed a modification by introducing a variation of G and the cosmological constant Λ while preserving the form of Einstein's field equations. Since then, there have been numerous cosmological models with G and/or Λ as a function of time or scale factor (for a review of some of these models see Ref. 11). For example, it has been shown that a variable G can account for dark matter or some of its effects,¹² and recent dynamical dark-energy models utilizing a time dependent cosmological term¹³ or a variable equation of state parameter ω(z)¹⁴ have also been considered. It is worth mentioning that many theoretical models with extra dimensions, such as string theory, contain a built in mechanism for possible time variation of the couplings.

The Einstein's field equations with variable G and Λ are given by

For the FLRW metric

the field equations are

which have the same form as those in standard GR with constant G and Λ. In a spatially flat (k = 0) model with a matter distribution having an energy momentum tensor T^ij= (ρ + p/c²)uⁱu^j+ pg^ij, where uⁱ= (c, 0, 0, 0) is the four velocity vector in comoving coordinates and p = wc²ρ, 0 ≤ w ≤ 1 is the equation of state relating the pressure p and energy density ρ, the Bianchi identities lead to

where the dots indicate differentiation with respect to the comoving time t. If one assumes the usual energy momentum conservation equation $T ;bab=0$ for the matter distribution, which leads to

then Eq. (5) implies a relation between the variation of G and Λ, given by

Note that Eq. (7) is valid when the energy momentum tensors of the matter distribution and the cosmological term (vacuum energy) are conserved separately; i.e., there is no interchange of energy and momentum between the two components, which therefore requires that a variation in G is accompanied by a variation in the cosmological term Λ, as shown above. Cosmological models with a constant G and variable Λ (or a constant Λ and variable G) have been considered in the literature.^15,16 In this case, the vanishing of the divergence of the total energy momentum tensor leads to Eq. (5) with $Ġ=0$ (or $Λ̇=0$⁠), so that the respective energy momentum tensors cannot be conserved separately.

However, if besides the variable Λ and G parameters one also considers cosmological models in General Relativity with a time dependent speed of light c(t) (see, for example, the perfect fluid cosmological models in Ref. 17), then the above Bianchi identity in Eq. (5) becomes

so that in this case the energy momentum tensor of the matter distribution can be separately conserved, even though Λ (or G) is taken as a constant. This situation is also possible in alternative theories of gravitation, such as Brans-Dicke (BD) theory,² where the time dependent G is represented by a scalar field G = 1/ψ, and the field equations for the FRW metric in Eq. (2) with Λ = 0 and c = 1 are given by¹⁸ 

and

Here, ω is a constant that determines the coupling between the scalar field and gravity, $T=Tμμ$⁠, and $□≡∇i∇i$⁠. The scalar field ψ = 1/G satisfies

In this case it can be shown¹⁹ that the Bianchi identities together with the above scalar field equation implies the conservation of the energy momentum tensor of the matter distribution $Tj;ii=0$⁠, so that one can have BD cosmological models with variable G and Λ = 0 with the energy density of the matter distribution satisfying Eq. (6). Indeed Garcia-Berro et al.²⁰ have used such a model (see also Refs. 21 and 22 for other cosmological models in a generalized BD-theory) with G(z) = G₀(1 – 0.01z + 0.34z² – 0.17z³), where G₀ = G(0) and redshift z = (1 – a)/a, to fit the observational Hubble diagram of SNeIa.

Dungan and Prosper²³ assumed a spatially flat universe with Λ = 0 and presented two variable-G cosmological models in General Relativity with G(a) = G₀f(a), such that (i) f(a) = e^b(a–1) and (ii) f(a) = 2/(1 + e^–b(a–1)), where b is a dimensionless parameter, a is the scale factor, and G₀ corresponds to the value of G at the present time when a = 1. In these models, the strength of gravity increases with cosmic time, unlike most of the variable-G models found in the literature that contain a negative value of $Ġ/G$⁠, in line with Dirac's LNH and the observational constraints cited above. After obtaining the expression for the distance modulus μ in terms of red shift z for the above two models, the authors showed that these models fit nicely the Type Ia supernova data compiled by Kowalski et al.²⁴ on 307 supernovae. This led them to the conclusion that the chosen models are consistent (on the basis of the supernovae data used) with the standard ΛCDM model having Ω_Λ ≈ 0.7, and therefore they argued that the supernovae data alone are insufficient to distinguish between the standard and variable-G cosmological models.

In their paper, Dungan and Prosper write down the general Friedmann equation (3) in the form

where H₀ is the current value of the Hubble parameter, Ω_M(a) = ρ(a)/ρ_c0 with $ρc0=3H02/8πG0$ being the current value of the critical density, Ω_Λ = ρ_Λ/ρ_c0 with ρ_Λ = Λc²/8πG₀, and Ω₀ = Ω_M(1) + Ω_Λ such that $−kc2=H02(1−Ω0)$⁠. So in the field equations they assumed a constant value of Λ (which was later taken to be zero, along with the spatial curvature k) and also a constant vacuum density parameter Ω_Λ. They also made an assumption that the matter density parameter Ω(a) = Ω₀a^–3, which from the above definition implies that the density ρ(a) = ρ₀a^–3, i.e., the matter energy density satisfies the conservation law in Eq. (6). However, as shown above, unless one also allows a time dependent c in GR or uses an alternative theory of gravity such as Brans-Dicke theory, the separate conservation of the vacuum and matter components of the energy momentum tensor requires that the variation of G is accompanied by a corresponding variation in Λ, given by Eq. (7). Hence, the assumptions made by the authors that Λ = constant = 0 and G = G₀f(a) in the presence of conservation of the matter component of the energy momentum tensor are inconsistent.

In the conclusion to their paper, the authors make a stronger statement by saying that all varying-G FRW models with accelerated expansion are ruled out by current observational constraints on $Ġ/G$⁠. They base their argument on the form of the Friedmann equation in Eq. (12), which, for a matter dominated universe, gives H²∼ G/a³ such that $Ġ/G≥H0∼7×10−11 yr−1$⁠. This value is about two orders of magnitude greater than the available observational constraints. However, in this case, as already pointed out above, the effect of a varying cosmological term Λ in the Friedmann equation cannot be ignored. The combined effects from the matter and vacuum terms in the Friedmann equation may even push the value of $Ġ/G$ below H₀, thus rendering the model compatible with observations. A case in point is given by the FRW cosmological model presented by Štefančić,²⁵ which contains a decreasing G and a growing cosmological term Λ, assuming separate conservation of the matter and vacuum components of the energy momentum tensor. This model exhibits accelerated expansion with the universe ending up in a “big rip” scenario represented by a de-Sitter spacetime with a constant asymptotic Λ and vanishing G, and it gives values of $Ġ/G$ that are consistent with observations.

So to conclude, although as suggested by the authors in their paper, a “mathematics first” approach to general relativity followed by applications is sometimes less desirable than a “physics” first approach, one still has to make sure that any conceptual approximations and assumptions used are mathematically correct and consistent with the underlying theory.