The idea of using stellar evolution to constrain the possible value of was originally proposed by
Teller [487
], who stressed that the evolution of a star was strongly dependent on
. The luminosity of a
main sequence star can be expressed as a function of Newton’s gravitational constant and its mass by using
homology relations [224
, 487]. In the particular case that the opacity is dominated by free-free transitions,
Gamow [224] found that the luminosity of the star is given approximately by
. In the
case of the Sun, this would mean that for higher values of
, the burning of hydrogen will
be more efficient and the star evolves more rapidly, therefore we need to increase the initial
content of hydrogen to obtain the present observed Sun. In a numerical test of the previous
expression, Delg’Innocenti et al. [140
] found that low-mass stars evolving from the Zero Age Main
Sequence to the red giant branch satisfy
, which agrees to within 10% of the
numerical results, following the idea that Thomson scattering contributes significantly to the
opacity inside such stars. Indeed, in the case of the opacity being dominated by pure Thomson
scattering, the luminosity of the star is given by
. It follows from the previous
analysis that the evolution of the star on the main sequence is highly sensitive to the value of
.
The driving idea behind the stellar constraints is that a secular variation of leads to a variation of
the gravitational interaction. This would affect the hydrostatic equilibrium of the star and in particular its
pressure profile. In the case of non-degenerate stars, the temperature, being the only control parameter, will
adjust to compensate the modification of the intensity of the gravity. It will then affect the nuclear
reaction rates, which are very sensitive to the temperature, and thus the nuclear time scales
associated to the various processes. It follows that the main stage of the stellar evolution, and in
particular the lifetimes of the various stars, will be modified. As we shall see, basically two
types of methods have been used, the first in which on relate the variation of
to some
physical characteristic of a star (luminosity, effective temperature, radius), and a second in which
only a statistical measurement of the change of
can be inferred. Indeed, the first class of
methods are more reliable and robust but is usually restricted to nearby stars. Note also that they
usually require to have a precise distance determination of the star, which may depend on
.
The first application of these idea has been performed with globular clusters. Their ages, determined for instance from the luminosity of the main-sequence turn-off, have to be compatible with the estimation of the age of the galaxy. This gives the constraint [140]
The effect of a possible time dependence of on luminosity has been studied in the case of globular
cluster H-R diagrams but has not yielded any stronger constraints than those relying on celestial
mechanics
A side effect of the change of luminosity is a change in the depth of the convection zone so that the inner
edge of the convecting zone changes its location. This induces a modification of the vibration modes of the
star and particularly to the acoustic waves, i.e., -modes [141
].
Helioseismology. These waves are observed for our star, the Sun, and helioseismology allows one to
determine the sound speed in the core of the Sun and, together with an equation of state, the central
densities and abundances of helium and hydrogen. Demarque et al. [141] considered an ansatz in which
and showed that
over the last
years, which corresponds to
. Guenther et al. [240] also showed that
-modes could provide even much
tighter constraints but these modes are up to now very difficult to observe. Nevertheless, they concluded,
using the claim of detection by Hill and Gu [251], that
. Guenther et al. [239]
then compared the
-mode spectra predicted by different theories with varying gravitational
constant to the observed spectrum obtained by a network of six telescopes and deduced that
White dwarfs. The observation of the period of non-radial pulsations of white dwarf allows to set similar
constraints. White dwarfs represent the final stage of the stellar evolution for stars with a mass smaller to
about . Their structure is supported against gravitational collapse by the pressure of
degenerate electrons. It was discovered that some white dwarfs are variable stars and in fact
non-radial pulsator. This opens the way to use seismological techniques to investigate their internal
properties. In particular, their non-radial oscillations is mostly determined by the Brunt–Väisälä
frequency
A variation of can influence the white dwarf cooling and the light curves ot Type Ia supernovae.
García-Berro et al. [225] considered the effect of a variation of the gravitational constant on the
cooling of white dwarfs and on their luminosity function. As first pointed out by Vila [518], the energy of
white dwarfs, when they are cool enough, is entirely of gravitational and thermal origin so
that a variation of
will induce a modification of their energy balance and thus of their
luminosity. Restricting to cold white dwarfs with luminosity smaller than ten solar luminosity, the
luminosity can be related to the star binding energy
and gravitational energy,
, as
The late stages of stellar evolution are governed by the Chandrasekhar mass
mainly determined by the balance between the Fermi pressure of a degenerate electron gas and
gravity.
Simple analytical models of the light curves of Type Ia supernovae predict that the peak of
luminosity is proportional to the mass of nickel synthesized. In a good approximation, it is a
fixed fraction of the Chandrasekhar mass. In models allowing for a varying , this would
induce a modification of the luminosity distance-redshift relation [227, 232, 435
]. However,
it was shown that this effect is small. Note that it will be degenerate with the cosmological
parameters. In particular, the Hubble diagram is sensitive to the whole history of
between
the highest redshift observed and today so that one needs to rely on a better defined model,
such as, e.g., scalar-tensor theory [435
] (the effect of the Fermi constant was also considered
in [194]).
In the case of Type II supernovae, the Chandrasekhar mass also governs the late evolutionary stages of
massive stars, including the formation of neutron stars. Assuming that the mean neutron star mass is
given by the Chandrasekhar mass, one expects that . Thorsett [492
]
used the observations of five neutron star binaries for which five Keplerian parameters can be
determined (the binary period
, the projection of the orbital semi-major axis
, the
eccentricity
, the time and longitude of the periastron
and
) as well as the relativistic
advance of the angle of the periastron
. Assuming that the neutron star masses vary slowly as
, that their age was determined by the rate at which
is increasing (so that
) and that the mass follows a normal distribution, Thorsett [492] deduced that, at
,
It has recently been proposed that the variation of inducing a modification of the binary’s binding
energy, it should affect the gravitational wave luminosity, hence leading to corrections in the chirping
frequency [554]. For instance, it was estimated that a LISA observation of an equal-mass inspiral event with
total redshifted mass of
for three years should be able to measure
at the time of merger to
better than 10–11/yr. This method paves the way to constructing constraints in a large band of redshifts as
well as in different directions in the sky, which would be an invaluable constraint for many
models.
More speculative is the idea [25] that a variation of can lead a neutron to enter into the
region where strange or hybrid stars are the true ground state. This would be associated with
gamma-ray bursts that are claimed to be able to reach the level of 10–17/yr on the time variation of
.
http://www.livingreviews.org/lrr-2011-2 |
Living Rev. Relativity 14, (2011), 2
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