A time-dependent gravitational constant will have mainly three effects on the CMB angular power spectrum
(see [435] 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 [94] 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. [2]
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 [378]. An analysis [549] 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 [89] 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. [84]
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 [455]. 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 [26] 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. [551] included
the deuterium and lithium to improve the constraint to
, which corresponds to
. It was further improved by Rothman and Matzner [441] to
implying
. Accetta et al. [1] 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. [295
] 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 [401], interesting bounds can still be derived [117]. In particular translating
the bound on extra relativistic degress of freedom (
) to a constraint on the speed-up
factor (
), it was concluded [117], 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 [105
], the speed-up factor is given (with the notations of Section 5.1.1)
by
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Living Rev. Relativity 14, (2011), 2
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