5.1 Introducing new fields: generalities
5.1.1 The example of scalar-tensor theories
Let us start to remind how the standard general relativistic framework can be extended to make
dynamical on the example of scalar-tensor theories, in which gravity is mediated not only by a massless
spin-2 graviton but also by a spin-0 scalar field that couples universally to matter fields (this
ensures the universality of free fall). In the Jordan frame, the action of the theory takes the form
where
is the bare gravitational constant. This action involves three arbitrary functions (
,
and
) but only two are physical since there is still the possibility to redefine the scalar field.
needs to be
positive to ensure that the graviton carries positive energy.
is the action of the matter fields that
are coupled minimally to the metric
. In the Jordan frame, the matter is universally coupled to the
metric so that the length and time as measured by laboratory apparatus are defined in this
frame.
The variation of this action gives the following field equations
where
is the trace of the matter energy-momentum tensor
. As
expected [183
], we have an equation, which reduces to the standard Einstein equation when
is
constant and a new equation to describe the dynamics of the new degree of freedom while the
conservation equation of the matter fields is unchanged, as expected from the weak equivalence
principle.
It is useful to define an Einstein frame action through a conformal transformation of the
metric
. In the following all quantities labeled by a star (*) will refer to Einstein
frame. Defining the field
and the two functions
and
(see, e.g., [191])
by
the
action (157) reads as
The kinetic terms have been diagonalized so that the spin-2 and spin-0 degrees of freedom of the theory are
perturbations of
and
respectively. In this frame the field equations are given by
with
and
. In this version, the Einstein equations are not modified, but
since the theory can now be seen as the theory in which all the mass are varying in the same way, there is a
source term to the conservation equation. This shows that the same theory can be interpreted as a varying
theory or a universally varying mass theory, but remember that whatever its form the important
parameter is the dimensionless quantity
.
The action (157) defines an effective gravitational constant
. This constant does
not correspond to the gravitational constant effectively measured in a Cavendish experiment. The Newton
constant measured in this experiment is
where the first term,
corresponds to the exchange of a graviton while the second term
is
related to the long range scalar force, a subscript
referring to the quantity evaluated today. The
gravitational constant depends on the scalar field and is thus dynamical.
This illustrates the main features that will appear in any such models: (i) new dynamical
fields appear (here a scalar field), (ii) some constant will depend on the value of this scalar
field (here
is a function of the scalar field). It follows that the Einstein equations will be
modified and that there will exist a new equation dictating the propagation of the new degree of
freedom.
In this particular example, the coupling of the scalar field is universal so that no violation of the
universality of free fall is expected. The deviation from general relativity can be quantified in terms of the
post-Newtonian parameters, which can be expressed in terms of the values of
and
today as
These expressions are valid only if the field is light on the solar system scales. If this is not the case then
these conclusions may be changed [287
]. The solar system constraints imply
to be very small, typically
while
can still be large. Binary pulsar observations [125, 189] impose that
.
The time variation of
is then related to
,
and the time variation of the scalar field today
This example shows that the variation of the constant and the deviation from general relativity quantified
in terms of the PPN parameters are of the same magnitude, because they are all driven by the same new
scalar field.
The example of scalar-tensor theories is also very illustrative to show how deviation from general
relativity can be fairly large in the early universe while still being compatible with solar system constraints.
It relies on the attraction mechanism toward general relativity [130
, 131].
Consider the simplest model of a massless dilaton (
) with quadratic coupling
(
). Note that the linear case correspond to a Brans–Dicke theory with a
fixed deviation from general relativity. It follows that
and
. As long as
, the Klein–Gordon equation can be rewritten in terms of the variable
as
As emphasized in [130], this is the equation of motion of a point particle with a velocity dependent inertial
mass,
evolving in a potential
and subject to a damping force,
. During the cosmological evolution the field is driven toward the minimum of the coupling
function. If
, it drives
toward 0, that is
, so that the scalar-tensor theory becomes
closer and closer to general relativity. When
, the theory is driven way from general relativity
and is likely to be incompatible with local tests unless
was initially arbitrarily close from
0.
It follows that the deviation from general relativity remains constant during the radiation era (up to
threshold effects in the early universe [108, 134
] and quantum effects [85]) and the theory is then attracted
toward general relativity during the matter era. Note that it implies that postulating a linear or inverse
variation of
with cosmic time is actually not realistic in this class of models. Since the theory is fully
defined, one can easily compute various cosmological observables (late time dynamics [348
], CMB
anisotropy [435
], weak lensing [449], BBN [105, 106, 134]) in a consistent way and confront them with
data.
5.1.2 Making other constants dynamical
Given this example, it seems a priori simple to cook up a theory that will describe a varying fine-structure
constant by coupling a scalar field to the electromagnetic Faraday tensor as
so that the fine-structure will evolve according to
. However, such an simple implementation may
have dramatic implications. In particular, the contribution of the electromagnetic binding energy to the
mass of any nucleus can be estimated by the Bethe–Weizäcker formula so that
This
implies that the sensitivity of the mass to a variation of the scalar field is expected to be of the order of
It follows that the level of the violation of the universality of free fall is expected to be of the level of
. Since the factor
typically ranges as
, we deduce that
has to be very small for the solar system constraints to be
satisfied. It follows that today the scalar field has to be very close to the minimum of the coupling function
. This led to the idea of the least coupling mechanism [135
, 136
] discussed in Section 5.4.1. This is
indeed very simplistic because this example only takes into account the effect of the electromagnetic binding
energy (see Section 6.3).
Let us also note that such a simple coupling cannot be eliminated by a conformal rescaling
since
so
that the action is invariant in
dimensions.
This example shows that we cannot couple a field blindly to, e.g., the Faraday tensor to make the
fine-structure constant dynamics and that some mechanism for reconciling this variation with local
constraints, and in particular the university of free fall, will be needed.