A time-dependent gravitational constant will have mainly three effects on the CMB angular power spectrum (see  for discussions in the framework of scalar-tensor gravity in which is considered as a field):
In full generality, the variation of on the CMB temperature anisotropies depends on many factors: (1) modification of the background equations and the evolution of the universe, (2) modification of the perturbation equations, (3) whether the scalar field inducing the time variation of is negligible or not compared to the other matter components, (4) on the time profile of that has to be determine to be consistent with the other equations of evolution. This explains why it is very difficult to state a definitive constraint. For instance, in the case of scalar-tensor theories (see below), one has two arbitrary functions that dictate the variation of . As can be seen, e.g., from [435, 378], the profiles and effects on the CMB can be very different and difficult to compare. Indeed, the effects described above are also degenerate with a variation of the cosmological parameters.
In the case of Brans–Dicke theory, one just has a single constant parameter characterizing the deviation from general relativity and the time variation of . Thus, it is easier to compare the different constraints. Chen and Kamionkowski  showed that CMB experiments such as WMAP will be able to constrain these theories for if all parameters are to be determined by the same CMB experiment, if all parameters are fixed but the CMB normalization and if one uses the polarization. For the Planck mission these numbers are respectively, 800, 2500 and 3200.  concluded from the analysis of WMAP, ACBAR, VSA and CBI, and galaxy power spectrum data from 2dF, that , in agreement with the former analysis of . An analysis  indictates that The ‘WMAP-5yr data’ and the ‘all CMB data’ both favor a slightly non-zero (positive) but with the addition of the SDSS poser spectrum data, the best-fit value is back to zero, concluding that between recombination and today, which corresponds to .
From a more phenomenological prospect, some works modeled the variation of with time in a purely ad-hoc way, for instance  by assuming a linear evolution with time or a step function.
As explained in detail in Section 3.8.1, changing the value of the gravitational constant affects the freeze-out temperature . A larger value of corresponds to a higher expansion rate. This rate is determined by the combination and in the standard case the Friedmann equations imply that is constant. The density is determined by the number of relativistic particles at the time of nucleosynthesis so that nucleosynthesis allows to put a bound on the number of neutrinos . Equivalently, assuming the number of neutrinos to be three, leads to the conclusion that has not varied from more than 20% since nucleosynthesis. But, allowing for a change both in and allows for a wider range of variation. Contrary to the fine structure constant the role of is less involved.
The effect of a varying can be described, in its most simple but still useful form, by introducing a speed-up factor, , that arises from the modification of the value of the gravitational constant during BBN. Other approaches considered the full dynamics of the problem but restricted themselves to the particular class of Jordan–Fierz–Brans–Dicke theory [1, 16, 26, 84, 102, 128, 441, 551] (Casas et al.  concluded from the study of helium and deuterium that when and when .), of a massless dilaton with a quadratic coupling [105, 106, 134, 446] or to a general massless dilaton . It should be noted that a combined analysis of BBN and CMB data was investigated in [113, 292]. The former considered constant during BBN while the latter focused on a nonminimally quadratic coupling and a runaway potential. It was concluded that from the BBN in conjunction with WMAP determination of set that has to be smaller than 20%. However, we stress that the dynamics of the field can modify CMB results (see previous Section 4.4.1) so that one needs to be careful while inferring from WMAP unless the scalar-tensor theory has converged close to general relativity at the time of decoupling.
In early studies, Barrow  assumed that and obtained from the helium abundances that , which implies that , assuming a flat universe. This corresponds in terms of the Brans–Dicke parameter to . Yang et al.  included the deuterium and lithium to improve the constraint to , which corresponds to . It was further improved by Rothman and Matzner  to implying . Accetta et al.  studied the dependence of the abundances of D, 3He, 4He and 7Li upon the variation of and concluded that , which roughly corresponds to . All these investigations assumed that the other constants are kept fixed and that physics is unchanged. Kolb et al.  assumed a correlated variation of , and and got a bound on the variation of the radius of the extra dimensions.
Although the uncertainty in the helium-4 abundance has been argued to be significantly larger that what was assumed in the past , interesting bounds can still be derived . In particular translating the bound on extra relativistic degress of freedom () to a constraint on the speed-up factor (), it was concluded , since , that
The relation between the speed-up factor, or an extra number of relativistic degrees of freedom, with a variation of is only approximate since it assumes that the variation of affects only the Friedmann equation by a renormalization of . This is indeed accurate only when the scalar field is slow-rolling. For instance , the speed-up factor is given (with the notations of Section 5.1.1) by
Living Rev. Relativity 14, (2011), 2
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