The DGP model is given by the action
whereThe equations of motion read
where If we consider the flat FLRW brane (), we obtain the modified Friedmann equation [201, 203]
In the DGP model the modification of gravity comes from a scalar-field degree of freedom, usually called
, which is identified as a brane bending mode in the bulk. Then one may wonder if such a field mediates
a fifth force incompatible with local gravity constraints. However, this is not the case, as the Vainshtein
mechanism is at work in the DGP model for the length scale smaller than the Vainshtein radius
, where
is the Schwarzschild radius of a source. The model can evade solar system
constraints at least under some range of conditions on the energy-momentum tensor [204, 285, 496]. The
Vainshtein mechanism in the DGP model originates from a non-linear self-interaction of the scalar-field
degree of freedom.
Although the DGP model is appealing and elegant, it is also plagued by some problems. The first one is
that, although the model does not possess ghosts on asymptotically flat manifolds, at the quantum level, it
does have the problem of strong coupling for typical distances smaller than 1000 km, so that the theory is
not easily under control [401]. Besides the model typically possesses superluminal modes. This may not
directly violate causality, but it implies a non-trivial non-Lorentzian UV completion of the theory [304].
Also, on scales relevant for structure formation (between cluster scales and the Hubble radius), a
quasi-static approximation to linear cosmological perturbations shows that the DGP model contains a ghost
mode [369
]. This linear analysis is valid as long as the Vainshtein radius
is smaller than the cluster
scales.
The original DGP model has been tested by using a number of observational data at the background
level [525, 238, 405, 9, 549
]. The joint constraints from the data of SN Ia, BAO, and the CMB shift
parameter show that the flat DGP model is under strong observational pressure, while the open DGP model
gives a slightly better fit [405, 549]. Xia [622] showed that the parameter
in the modified Friedmann
equation
[221] is constrained to be
(68% confidence level)
by using the data of SN Ia, BAO, CMB, gamma ray bursts, and the linear growth factor of
matter perturbations. Hence the flat DGP model (
) is not compatible with current
observations.
On the sub-horizon scales larger than the Vainshtein radius, the equation for linear matter
perturbations in the DGP model was derived in [400, 369] under a quasi-static approximation:
The index of the growth rate
is approximated by
[395]. This is quite
different from the value
for the
CDM model. If the future imaging survey of galaxies can
constrain
within 20%, it will be possible to distinguish the
CDM model from the DGP model
observationally [624]. We recall that in metric f (R) gravity the growth index today can be as small as
because of the enhanced growth rate, which is very different from the value in the DGP
model.
Comparing Eq. (13.53) with the effective gravitational constant (10.42
) in BD theory with a massless
limit (or the absence of the field potential), we find that the parameter
has the following relation
with
:
There have been studies regarding a possible regularization in order to avoid the ghost/strong coupling
limit. Some of these studies have focused on smoothing out the delta profile of the Ricci scalar on the
brane, by coupling the Ricci scalar to some other scalar field with a given profile [363, 362].
In [516] the authors included the brane and bulk cosmological constants in addition to the
scalar curvature in the action for the brane and showed that the effective equation of state of
dark energy can be smaller than . A monopole in seven dimensions generated by a SO(3)
invariant matter Lagrangian is able to change the gravitational law at its core, leading to a
lower dimensional gravitational law. This is a first approach to an explanation of trapping of
gravitons, due to topological defects in classical field theory [508, 184]. Other studies have
focused on re-using the delta function profile but in a higher-dimensional brane [334, 333, 197].
There is also an interesting work about the possibility of self-acceleration in the normal DGP
branch [
in Eq. (13.50
)] by considering an f (R) term on the brane action [97] (see
also [4]). All these attempts indeed point to the direction that some mechanism, if not exactly
DGP rather similar to it, may avoid a number of problems associated with the original DGP
model.
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