13.6 DGP model

The Dvali–Gabadadze–Porrati (DGP) [220] braneworld model has been considered as a model which could modify gravity because of the existence of the extra-dimensions. In the DGP model the 3-brane is embedded in a Minkowski bulk spacetime with infinitely large 5th extra dimensions. The Newton’s law can be recovered by adding a 4-dimensional (4D) Einstein–Hilbert action sourced by the brane curvature to the 5D action [219]. While the DGP model recovers the standard 4D gravity for small distances, the effect from the 5D gravity manifests itself for large distances. Remarkably it is possible to realize the late-time cosmic acceleration without introducing an exotic matter source [201Jump To The Next Citation Point203Jump To The Next Citation Point].

The DGP model is given by the action

1 ∫ ∘ --- 1 ∫ √--- ∫ √ --- S = ---2- d5X −g&tidle;&tidle;R + ---2- d4x − gR + d4x − gℒbMrane, (13.46 ) 2κ (5) 2κ (4)
where &tidle;gAB is the metric in the 5D bulk and A B gμν = ∂ μX ∂νX &tidle;gAB is the induced metric on the brane with A c X (x ) being the coordinates of an event on the brane labeled by c x. The first and second terms in Eq. (13.46View Equation) correspond to Einstein–Hilbert actions in the 5D bulk and on the brane, respectively. Note that κ2(5) and κ2(4) are 5D and 4D gravitational constants, respectively, which are related with 5D and 4D Planck masses, M (5) and M (4), via 2 3 κ (5) = 1∕M (5) and 2 2 κ(4) = 1∕M (4). The Lagrangian brane ℒ M describes matter localized on the 3-brane.

The equations of motion read

(5) G AB = 0, (13.47 )
where G(5) AB is the 5D Einstein tensor. The Israel junction conditions on the brane, under which a Z2 symmetry is imposed, read [304Jump To The Next Citation Point]
-1 2 Gμν − rc(K μν − gμνK ) = κ(4)T μν, (13.48 )
where K μν is the extrinsic curvature [609] calculated on the brane and T μν is the energy-momentum tensor of localized matter. Since ∇ μ(K μν − gμνK ) = 0, the continuity equation ∇ μTμν = 0 follows from Eq. (13.48View Equation). The length scale rc is defined by
2 2 -κ(5) -M-(4) rc ≡ 2 κ2 = 2M 3 . (13.49 ) (4) (5)

If we consider the flat FLRW brane (K = 0), we obtain the modified Friedmann equation [201203]

2 H2 − 𝜖-H = κ(4)ρ , (13.50 ) rc 3 M
where 𝜖 = ±1, H and ρM are the Hubble parameter and the matter energy density on the brane, respectively. In the regime −1 rc ≫ H the first term in Eq. (13.50View Equation) dominates over the second one and hence the standard Friedmann equation is recovered. Meanwhile, in the regime rc ≲ H −1, the second term in Eq. (13.50View Equation) leads to a modification to the standard Friedmann equation. If 𝜖 = 1, there is a de Sitter solution characterized by
HdS = 1∕rc. (13.51 )
One can realize the cosmic acceleration today if rc is of the order of the present Hubble radius − 1 H 0. This self acceleration is the result of gravitational leakage into extra dimensions at large distances. In another branch (𝜖 = − 1) such cosmic acceleration is not realized.

In the DGP model the modification of gravity comes from a scalar-field degree of freedom, usually called π, 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 r∗ = (rgr2c)1∕3, where rg 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 [204285496]. 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 [401Jump To The Next Citation Point]. 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 [369Jump To The Next Citation Point]. This linear analysis is valid as long as the Vainshtein radius r∗ is smaller than the cluster scales.

The original DGP model has been tested by using a number of observational data at the background level [525238405Jump To The Next Citation Point9549Jump To The Next Citation Point]. 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 [405549]. Xia [622] showed that the parameter α in the modified Friedmann equation H2 − H α∕r2c− α= κ2(4)ρM ∕3 [221] is constrained to be α = 0.254 ± 0.153 (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 (α = 1) is not compatible with current observations.

On the sub-horizon scales larger than the Vainshtein radius, the equation for linear matter perturbations δm in the DGP model was derived in [400369] under a quasi-static approximation:

¨ ˙ δm + 2H δm − 4πGe ffρmδm ≃ 0, (13.52 )
where ρm is the non-relativistic matter density on the brane and
( ) ( 1 ) H˙ Ge ff = 1 + --- G, β(t) ≡ 1 − 2Hrc 1 + ---2- . (13.53 ) 3β 3H
In the deep matter era one has Hrc ≫ 1 and hence β ≃ − Hrc, so that β is largely negative (|β| ≫ 1). In this regime the evolution of δm is similar to that in GR (δm ∝ t2∕3). Since the background solution finally approaches the de Sitter solution characterized by Eq. (13.51View Equation), it follows that β ≃ 1 − 2Hrc ≃ − 1 asymptotically. Since 1 + 1∕(3β) ≃ 2∕3, the growth rate in this regime is smaller than that in GR.

The index γ of the growth rate fδ = (Ωm )γ is approximated by γ ≈ 0.68 [395]. This is quite different from the value γ ≃ 0.55 for the ΛCDM model. If the future imaging survey of galaxies can constrain γ within 20%, it will be possible to distinguish the ΛCDM model from the DGP model observationally [624]. We recall that in metric f (R) gravity the growth index today can be as small as γ = 0.4 because of the enhanced growth rate, which is very different from the value in the DGP model.

Comparing Eq. (13.53View Equation) with the effective gravitational constant (10.42View Equation) in BD theory with a massless limit (or the absence of the field potential), we find that the parameter ωBD has the following relation with β:

ωBD = 3(β − 1 ). (13.54 ) 2
Since β < 0 for the self-accelerating DGP solution, it follows that ωBD < − 3∕2. This corresponds to the theory with ghosts, because the kinetic energy of a scalar-field degree of freedom becomes negative in the Einstein frame [175]. There is a claim that the ghost may disappear for the Vainshtein radius r∗ of the order of −1 H 0, because the linear perturbation theory is no longer applicable [218]. In fact, a ghost does not appear in a Minkowski brane in the DGP model. In [370] it was shown that the Vainshtein radius in the early universe is much smaller than the one in the Minkowski background, while in the self accelerating universe they agree with each other. Hence the perturbative approach seems to be still possible for the weak gravity regime beyond the Vainshtein radius.

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 [363362]. 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 − 1. 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 [508184]. Other studies have focused on re-using the delta function profile but in a higher-dimensional brane [334333197]. There is also an interesting work about the possibility of self-acceleration in the normal DGP branch [𝜖 = − 1 in Eq. (13.50View Equation)] 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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