The action for metric f (R) gravity can be evaluated on a FLRW background, in terms of the fields
and
, see [125, 124
] (and also [129, 128, 415, 433, 429, 604, 603, 199, 200, 132]). Then the
Lagrangian turns out to be non-singular,
, where
, and
. Its Euler–Lagrange
equation is given by
. Contracting these equations with a vector function
, (where
and
are two unknown functions of the
), we obtain
It can be proved that such do exist [124], and they correspond to
For a general it is not possible to solve the Euler–Lagrange equation and the constraint
equation (13.59
) at the same time. Hence, we have to use the Noether constraint in order to find the subset
of those
which make this possible. Some partial solutions (only when
) were found, but
whether this symmetry helps finding viable models of f (R) is still not certain. However, the
f (R) theories which possess Noether currents can be more easily constrained, as now the original
freedom for the function
in the Lagrangian reduced to the choice of the parameters
and
.
Recently another symmetry, the Galileon symmetry, for a scalar field Lagrangian was imposed on the
Minkowski background [455]. This idea is interesting as it tries to decouple light scalar fields from matter
making use of non-linearities, but without introducing new ghost degrees of freedom [455
]. This symmetry
was chosen so that the theory could naturally implement the Vainshtein mechanism. However, the same
mechanism, at least in cosmology, seems to appear also in the FLRW background for scalar fields which do
not possess such a symmetry (see [539
, 351
, 190
]).
Keeping a universal coupling with matter (achieved through a pure nonminimal coupling with ),
Nicolis et al. [455
] imposed a symmetry called the Galilean invariance on a scalar field
in the
Minkowski background. If the equations of motion are invariant under a constant gradient-shift on
Minkowski spacetime, that is
Nicolis et al. [455] found that there are only five terms with
which can be inserted
into a Lagrangian, such that the equations of motion respect the Galileon symmetry in 4-dimensional
Minkowski spacetime. The first three terms are given by
This result can be extended to arbitrary dimensions [202]. One can find, analogously to the
Lovelock action-terms, an infinite tower of terms that can be introduced with the same property
of keeping the equations of motion at second order. In particular, let us consider the action
In general non-linear terms discussed above may introduce the Vainshtein mechanism to decouple the
scalar field from matter around a star, so that solar-system constraints can be satisfied. However the modes
can have superluminal propagation, which is not surprising as the kinetic terms get heavily
modified in the covariant formalism. Some studies have focused especially on the term
only, as this corresponds to the simplest case. For some models the background cosmological
evolution is similar to that in the DGP model, although there are ghostlike modes depending
on the sign of the time-velocity of the field
[158]. There are some works for cosmological
dynamics in Brans–Dicke theory in the presence of the non-linear term
[539, 351, 190]
(although the original Galileon symmetry is not preserved in this scenario). Interestingly the
ghost can disappear even for the case in which the Brans–Dicke parameter
is smaller
than
. Moreover this theory leaves a number of distinct observational signatures such
as the enhanced growth rate of matter perturbations and the significant ISW effect in CMB
anisotropies.
At the end of this section we should mention conformal gravity in which the conformal invariance forces the gravitational action to be uniquely given by a Weyl action [414, 340]. Interestingly the conformal symmetry also forces the cosmological constant to be zero at the level of the action [413]. It will be of interest to study the cosmological aspects of such theory, together with the possibility for the avoidance of ghosts and instabilities.
http://www.livingreviews.org/lrr-2010-3 | ![]() This work is licensed under a Creative Commons License. Problems/comments to |