At small distances the induced curvature dominates and gravity behaves as in four dimensions, while at large distances the leakage of gravity within the extra dimension weakens the force of gravity. The DGP model is thus a model of modified gravity in the infrared, and as we shall see, the graviton effectively acquires a soft mass, or resonance.
The five-dimensional Einstein equation of motion are then given by can be obtained by integrating this equation over and taking the limit , so that the jump in the extrinsic curvature across the brane is related to the Einstein tensor and stress-energy tensor of the matter field confined on the brane.
In DGP the four-dimensional graviton is effectively massive. To see this explicitly, we look at perturbations about flat spacetime
In this gauge the five-dimensional Einstein tensor is simply
Since there is no source along the or directions (), we can immediately infer that4.2 how to keep track of the brane-bending mode which is partly encoded in .
Using these relations in the five-dimensional de Donder gauge, we deduce the relation for the purely four-dimensional part of the metric perturbation,
Solving this equation with the requirement that as , we infer the following profile for the perturbations along the extra dimension2.2.3 and obtain the expression for the sourced metric fluctuation on the brane 2.57*) in terms of . In particular the polarization tensor includes the standard factor of as opposed to as would be the case in GR. This is again the manifestation of the vDVZ discontinuity which is cured by the Vainshtein mechanism as for Fierz–Pauli massive gravity. See [165*] for the explicit realization of the Vainshtein mechanism in DGP which is where it was first shown to work explicitly.
In Fourier space the propagator for the graviton in DGP is given by2.58*) and 1*. As already emphasized, the graviton in DGP cannot be thought of a single massive mode, but rather as a resonance picked about .
We see that the spectral density is positive for any , confirming the fact that about the normal (flat) branch of DGP there is no ghost.
Notice as well that in the massless limit , we see appearing a representation of the Dirac delta function,2.2.3.)
In Section 4.1.1 we have remained vague about the gauge-fixing and the implications for the brane position. The brane-bending mode is actually important to keep track of in DGP and we shall do that properly in what follows by keeping all the modes.
We work in the five-dimensional ADM split with the lapse , the shift and the four-dimensional part of the metric, . The five-dimensional Einstein–Hilbert term is then expressed as
After fixing the de Donder gauge (4.5*), we can make the addition gauge transformation , and remain in de Donder gauge provided satisfies linearly . This residual gauge freedom can be used to further fix the gauge on the brane (see [389*] for more details, we only summarize their derivation here).
Keeping the brane at the fixed position imposes since we need and should be bounded as (the situation is slightly different in the self-accelerating branch and this mode can lead to a ghost, see Section 4.4 as well as [361*, 98*]).
Using the bulk profile and integrating over the extra dimension, we obtain the contribution from the bulk on the brane (including the contribution from the gauge-fixing term) in terms of the gauge invariant quantity[389*] which arises from the fact that we integrate from to imposing a -mirror symmetry at , rather than considering only one side of the brane as in [389*]. Both conventions are perfectly reasonable. 4.27*) we can see that the most convenient gauge fixing term is [389*] 4.27*) with that on the boundary (4.31*) we end up with the linearized action on the four-dimensional DGP brane [389*] 4.1.1, we had set and the field is then related to the brane bending mode. In either case we see that the extrinsic curvature carries part of this mode.
Omitting the mass terms and other relevant operators, the action is diagonalized in terms of the different graviton modes at the linearized level (which encodes the helicity-2 mode), (which is part of the helicity-1 mode) and (helicity-0 mode),
We will be discussing the meaning of ‘decoupling limits’ in more depth in the context of multi-gravity and ghost-free massive gravity in Section 8. The main idea behind the decoupling limit is to separate the physics of the different modes. Here we are interested in following the interactions of the helicity-0 mode without the complications from the standard helicity-2 interactions that already arise in GR. For this purpose we can take the limit while simultaneously sending while keeping the scale fixed. This is the scale at which the first interactions arise in DGP.
In DGP the decoupling limit should be taken by considering the full five-dimensional theory, as was performed in [389*]. The four-dimensional Einstein–Hilbert term does not give to any operators before the Planck scale, so in order to look for the irrelevant operator that come at the lowest possible scale, it is sufficient to focus on the boundary term from the five-dimensional action. It includes operators of the form4.22*), and in particular the combination , with 4.34*) and (4.35*))
The phenomenology of DGP is extremely rich and has led to many developments. In what follows we review one of the most important implications of the DGP for cosmology which the existence of self-accelerating solutions. The cosmology and phenomenology of DGP was first derived in [159*, 163*] (see also [388*, 385*, 387*, 386*]).
To get some intuition on how cosmology gets modified in DGP, we first look at de Sitter-like solutions and then infer the full Friedmann equation in a FLRW-geometry. We thus start with five-dimensional Minkowski in de Sitter slicing (this can be easily generalized to FLRW-slicing),
, and the scale factor is given by . The metric (4.43*) is indeed Minkowski in de Sitter slicing if the warp factor is given by
We can now derive the Friedmann equation on the brane by integrating over the -component of the Einstein equation (4.2*) with the source (4.3*) and consider some energy density . The four-dimensional Einstein tensor gives the standard contribution on the brane and so we obtain the modified Friedmann equation
For reasons which will become clear in what follows, the choice corresponds to the stable branch of DGP while the other choice corresponds to the self-accelerating branch of DGP. As is already clear from the higher-dimensional perspective, when , the warp factor grows in the bulk (unless we think of the junction conditions the other way around), which is already signaling towards a pathology for that branch of solution.
This modified Friedmann equation has been derived assuming a constant , which is only consistent if the energy density is constant (i.e., a cosmological constant). We can now derive the generalization of this Friedmann equation for non-constant . This amounts to account for and other derivative corrections which might have been omitted in deriving this equation by assuming that was constant. But the Friedmann equation corresponds to the Hamiltonian constraint equation and higher derivatives (e.g., and higher derivatives of ) would imply that this equation is no longer a constraint and this loss of constraint would imply that the theory admits a new degree of freedom about generic backgrounds namely the BD ghost (see the discussion of Section 7).
However, in DGP we know that the BD ghost is absent (this is ensured by the five-dimensional nature of the theory, in five dimensions we start with five dofs, and there is thus no sixth BD mode). So the Friedmann equation cannot include any derivatives of , and the Friedmann equation obtained assuming a constant is actually exact in FLRW even if is not constant. So the constraint (4.47*) is the exact Friedmann equation in DGP for any energy density on the brane.
Independently of the ghost issue in the self-accelerating branch of the model, there has been a vast amount of investigation on the observational viability of both the self-accelerating branch and the normal (stable) branch of DGP. First because many of these observations can apply equally well to the stable branch of DGP (modulo a minus sign in some of the cases), and second and foremost because DGP represents an excellent archetype in which ideas of modified gravity can be tested.
Observational tests of DGP fall into the following two main categories:
- Tests of the Friedmann equation. This test was performed mainly using Supernovae, but also using Baryonic Acoustic Oscillations and the CMB so as to fix the background history of the Universe [162, 217, 221, 286, 391, 23, 405, 481, 304, 382, 462]. Current observations seem to slightly disfavor the additional term in the Friedmann equation of DGP, even in the normal branch where the late-time acceleration of the Universe is due to a cosmological constant as in CDM. These put bounds on the graviton mass in DGP to the order of , where is the Hubble parameter today (see Ref.  for the latest bounds at the time of writing, including data from Planck). Effectively this means that in order for DGP to be consistent with observations, the graviton mass can have no effect on the late-time acceleration of the Universe.
- Tests of an extra fifth force, either within the solar system, or during structure formation
(see for instance [362, 260, 452, 451, 222, 482] Refs. [453, 337, 442] for N-body simulations
as well as Ref. [17, 441] using weak lensing).
Evading fifth force experiments will be discussed in more detail within the context of the Vainshtein mechanism in Section 10.1 and thereafter, and we save the discussion to that section. See Refs. [388*, 385, 387, 386*, 444] for a five-dimensional study dedicated to DGP. The study of cosmological perturbations within the context of DGP was also performed in depth for instance in [367, 92].
The cosmology of DGP has led to a major conceptual breakthrough, namely the realization that the Universe could be ‘self-accelerating’. This occurs when choosing the branch of DGP, the Friedmann equation in the vacuum reduces to [159*, 163*]
This realization has opened a new field of study in its own right. It is beyond the scope of this review on massive gravity to summarize all the interesting developments that arose in the past decade and we simply focus on a few elements namely the presence of a ghost in this self-accelerating branch as well as a few cosmological observations.
The existence of a ghost on the self-accelerating branch of DGP was first pointed out in the decoupling limit [389*, 411*], where the helicity-0 mode of the graviton is shown to enter with the wrong sign kinetic in this branch of solutions. We emphasize that the issue of the ghost in the self-accelerating branch of DGP is completely unrelated to the sixth BD ghost on some theories of massive gravity. In DGP there are five dofs one of which is a ghost. The analysis was then generalized in the fully fledged five-dimensional theory by K. Koyama in [360*] (see also [263, 361] and [98*]).
When perturbing about Minkowski, it was shown that the graviton has an effective mass . When perturbing on top of the self-accelerating solution a similar analysis can be performed and one can show that in the vacuum the graviton has an effective mass at precisely the Higuchi-bound, (see Ref. [307*]). When matter or a cosmological constant is included on the brane, the graviton mass shifts either inside the forbidden Higuchi-region , or outside . We summarize the three case scenario following [360, 98]
- In [307*] it was shown that when the effective mass is within the forbidden Higuchi-region, the helicity-0 mode of graviton has the wrong sign kinetic term and is a ghost.
- Outside this forbidden region, when , the zero-mode of the graviton is healthy but there exists a new normalizable brane-bending mode in the self-accelerating branch8 which is a genuine degree of freedom. For the brane-bending mode was shown to be a ghost.
- Finally, at the critical mass (which happens when no matter nor cosmological constant is present on the brane), the brane-bending mode takes the role of the helicity-0 mode of the graviton, so that the theory graviton still has five degrees of freedom, and this mode was shown to be a ghost as well.
In summary, independently of the matter content of the brane, so long as the graviton is massive , the self-accelerating branch of DGP exhibits a ghost. See also  for an exact non-perturbative argument studying domain walls in DGP. In the self-accelerating branch of DGP domain walls bear a negative gravitational mass. This non-perturbative solution can also be used as an argument for the instability of that branch.
Different ways to remove the ghosts were discussed for instance in  where a second brane was included. In this scenario it was then shown that the graviton could be made stable but at the cost of including a new spin-0 mode (that appears as the mode describing the distance between the branes).
Alternatively it was pointed out in  that if the sign of the extrinsic curvature was flipped, the self-accelerating solution on the brane would be stable.
Finally, a stable self-acceleration was also shown to occur in the massless case by relying on Gauss–Bonnet terms in the bulk and a self-source AdS5 solution . The five-dimensional theory is then similar as that of DGP (4.1*) but with the addition of a five-dimensional Gauss–Bonnet term in the bulk and the wrong sign five-dimensional Einstein–Hilbert term,13), where here the wrong sign kinetic term in five-dimensions is balanced by the Gauss–Bonnet term in such a way that the graviton has the correct sign kinetic term on the self-sourced AdS5 solution. The length scale is related to this AdS length scale, and the self-accelerating branch admits a stable (ghost-free) de Sitter solution with .
We do not discuss this model any further in what follows since the graviton admits a zero (massless) mode. It is feasible that this model can be understood as a bi-gravity theory where the massive mode is a resonance. It would also be interesting to see how this model fits in with the Galileon theories [412*] which admit stable self-accelerating solution.
In what follows, we go back to the standard DGP model be it the self-accelerating branch () or the normal branch ().
One of the main motivations behind modifying gravity in the infrared is to tackle the old cosmological constant problem. The idea behind ‘degravitation’ [211*, 212*, 26*, 216*] is if gravity is modified in the IR, then a cosmological constant (or the vacuum energy) could have a smaller impact on the geometry. In these models, we would live with a large vacuum energy (be it at the TeV scale or at the Planck scale) but only observe a small amount of late-acceleration due to the modification of gravity. In order for a theory of modified gravity to potentially tackle the old cosmological constant problem via degravitation it needs to have the two following properties:
- First, gravity must be weaker in the infrared and effectively massive [216*] so that the effect of IR sources can be degravitated.
- Second, there must exist some (nearly) static attractor solutions towards which the system can evolve at late-time for arbitrary value of the vacuum energy or cosmological constant.
The first requirement is present in DGP, but as was shown in [216*] in DGP gravity is not ‘sufficiently weak’ in the IR to allow degravitation solutions. Nevertheless, it was shown in  that the normal branch of DGP satisfies the second requirement for any negative value of the cosmological constant. In these solutions the five-dimensional spacetime is not Lorentz invariant, but in a way which would not (at this background level) be observed when confined on the four-dimensional brane.
For positive values of the cosmological constant, DGP does not admit a (nearly) static solution. This can be understood at the level of the decoupling limit using the arguments of [216*] and generalized for other mass operators.
Inspired by the form of the graviton in DGP, , we can generalize the form of the graviton mass to[153*], any such modification leads to ghosts, so we do not consider this case here. corresponds to a UV modification of gravity, and so we focus on . 4.51*) when only the term contributes and all the other operators vanish for . This is the case if as shown in [216*]. This explains why in the case of DGP which corresponds to border line scenario , one can never fully degravitate a cosmological constant.
This realization has motivated the search for theories of massive gravity with , and especially the extension of DGP to higher dimensions where the parameter can get as close to zero as required. This is the main motivation behind higher dimensional DGP [359, 240*] and cascading gravity [135*, 148*, 132*, 149*] as we review in what follows. (In  it was also shown how a regularized version of higher dimensional DGP could be free of the strong coupling and ghost issues).
Note that corresponds to a hard mass gravity. Within the context of DGP, such a model with an ‘auxiliary’ extra dimension was proposed in [235, 133*] where we consider a finite-size large extra dimension which breaks five-dimensional Lorentz invariance. The five-dimensional action is motivated by the five-dimensional gravity with scalar curvature in the ADM decomposition , but discarding the contribution from the four-dimensional curvature . Similarly as in DGP, the four-dimensional curvature still appears induced on the brane[138*] that the theory is free of ghost and its decoupling limit is that of a Galileon.
Furthermore, it was shown in  that it satisfies both requirements presented above to potentially help degravitating a cosmological constant. Unfortunately at higher orders this model is plagued with the BD ghost  unless the boundary conditions are chosen appropriately . For this reason we will not review this model any further in what follows and focus instead on the ghost-free theory of massive gravity derived in [137*, 144*]
It is well known that a tension on a cosmic string does not cause the cosmic strong to inflate but rather creates a deficit angle in the two spatial dimensions orthogonal to the string. Similarly, if we consider a four-dimensional brane embedded in six-dimensional gravity, then a tension on the brane leads to the following flat geometry2*.
This interesting feature has lead to many potential ways to tackle the cosmological constant by considering our Universe to live in a -dimensional brane embedded in two or more large extra dimensions. (See Refs. [4, 3, 408, 414, 80, 470, 458, 459, 86, 82, 247, 333, 471, 81, 426, 409, 373, 85, 460, 155] for the supersymmetric large-extra-dimension scenario as an alternative way to tackle the cosmological constant problem). Extending the DGP to more than one extra dimension could thus provide a natural way to tackle the cosmological constant problem.
Furthermore in -extra dimensions the gravitational potential is diluted as . If the propagator has a Källén–Lehmann spectral representation with spectral density , the Newtonian potential has the following spectral representation1*.
Working back in terms of the spectral representation of the propagator as given in (4.19*), this means that the propagator goes to in the IR as when (as we know from DGP), while it goes to a constant for . So for more than one extra dimension, the theory tends towards that of a hard mass graviton in the far IR, which corresponds to in the parametrization of (4.50*). Following the arguments of [216*] such a theory should thus be a good candidate to tackle the cosmological constant problem.
Both the spectral representation and the fact that codimension-two (and higher) branes can accommodate for a cosmological constant while remaining flat has made the field of higher-codimension branes particularly interesting.
However, as shown in [240*] and [135*, 148*, 132, 149*], the straightforward extension of DGP to two large extra dimensions leads to ghost issues (sixth mode with the wrong sign kinetic term, see also [290, 70]) as well as divergences problems (see Refs. [256, 131, 130, 423, 422, 355, 83]).
To avoid these issues, one can consider simply applying the DGP procedure step by step and consider a -dimensional DGP brane embedded in six dimension. Our Universe would then be on a -dimensional DGP brane embedded in the one, (note we only consider one side of the brane here which explains the factor of 2 difference compared with (4.1*))
Performing the same linearized analysis as in Section 4.1.1 we can see that the four-dimensional theory of gravity is effectively massive with the soft mass in Fourier space
In this six-dimensional model, there are effectively two new scalar degrees of freedom (arising from the extra dimensions). We can ensure that both of them have the correct sign kinetic term by
- Either smoothing out the brane [240, 148] (this means that one should really consider a six-dimensional curvature on both the smoothed and on the -dimensional branes, which is something one would naturally expect9).
- Or by including some tension on the brane (which is also something natural since the setup is designed to degravitate a large cosmological constant on that brane). This was shown to be ghost free in the decoupling limit in  and in the full theory in .
As already mentioned in two large extra dimensional models there is to be a maximal value of the cosmological constant that can be considered which is related to the six-dimensional Planck scale. Since that scale is in turn related to the effective mass of the graviton and since observations set that scale to be relatively small, the model can only take care of a relatively small cosmological constant. Nevertheless, it still provides a proof of principle on how to evade Weinberg’s no-go theorem [484*].
The extension of cascading gravity to more than two extra dimensions was considered in [149*]. It was shown in that case how the brane remains flat for arbitrary values of the cosmological constant on that brane (within the regime of validity of the weak-field approximation). See Figure 3* for a picture on how the scalar potential adapts itself along the extra dimensions to accommodate for a cosmological constant on the brane.