By contrast, the brane-world model of Dvali–Gabadadze–Porrati [135] (DGP), which was
generalized to cosmology by Deffayet [115], modifies general relativity at low energies.
This model produces ‘self-acceleration’ of the late-time universe due to a weakening of
gravity at low energies. Like the RS model, the DGP model is a 5D model with infinite extra
dimensions.^{3}

The action is given by

The bulk is assumed to be 5D Minkowski spacetime. Unlike the AdS bulk of the RS model, the Minkowski bulk has infinite volume. Consequently, there is no normalizable zero-mode of the 4D graviton in the DGP brane-world. Gravity leaks off the 4D brane into the bulk at large scales, , where the first term in the sum (415) dominates. On small scales, gravity is effectively bound to the brane and 4D dynamics is recovered to a good approximation, as the second term dominates. The transition from 4D to 5D behaviour is governed by the crossover scale . For a Minkowski brane, the weak-field gravitational potential behaves as On a Friedmann brane, gravity leakage at late times in the cosmological evolution can initiate acceleration – not due to any negative pressure field, but due to the weakening of gravity on the brane. 4D gravity is recovered at high energy via the lightest massive modes of the 5D graviton, effectively via an ultra-light meta-stable graviton.The energy conservation equation remains the same as in general relativity, but the Friedmann equation is modified [115]:

The Friedmann equation can be derived from the junction condition or the Gauss–Codazzi equation as in the RS model. To arrive at Equation (418) we have to take a square root which implies a choice of sign. As we shall see, the above choice has the advantage of leading to acceleration – but at the expense of a ‘ghost’ (negative kinetic energy) mode in the scalar graviton sector. The ‘normal’ (non-self-accelerating) DGP model, where the opposite sign of the square root is chosen, has no ghost, and is discussed below.From Equation (418) we infer that at early times, i.e., , the general relativistic Friedman equation is recovered. By contrast, at late times in an expanding CDM universe, with , we have

so that expansion accelerates and is asymptotically de Sitter. The above equations imply In order to achieve self-acceleration at late times, we require since . This is confirmed by fitting supernova observations, as shown in Figures 19 and 20. The dimensionless cross-over parameter is defined as and the LCDM relation, is modified toLCDM and DGP can both account for the supernova observations, with the fine-tuned values and , respectively. When we add further constraints to the expansion history from the baryon acoustic oscillation peak at and the CMB shift parameter, the DGP flat models are in strong tension with the data, whereas LCDM models provide a consistent fit. This is evident in Figures 19 and 20 though this conclusion depends on a choice of light curve fitters in the SNe observations. The open DGP models provide a somewhat better fit to the geometric data – essentially because the lower value of favoured by supernovae reduces the distance to last scattering and an open geometry is able to extend that distance. For a combination of SNe, CMB shift and Hubble Key Project data, the best-fit open DGP also performs better than the flat DGP [402], as shown in Figure 21.

Observations based on structure formation provide further evidence of the difference between DGP and LCDM, since the two models suppress the growth of density perturbations in different ways [296, 295]. The distance-based observations draw only upon the background 4D Friedman equation (418) in DGP models – and therefore there are quintessence models in general relativity that can produce precisely the same supernova distances as DGP. By contrast, structure formation observations require the 5D perturbations in DGP, and one cannot find equivalent quintessence models [251]. One can find 4D general relativity models, with dark energy that has anisotropic stress and variable sound speed, which can in principle mimic DGP [263]. However, these models are highly unphysical and can probably be discounted on grounds of theoretical consistency.

For LCDM, the analysis of density perturbations is well understood. For DGP the perturbations are much more subtle and complicated [251]. Although matter is confined to the 4D brane, gravity is fundamentally 5D, and the 5D bulk gravitational field responds to and back-reacts on 4D density perturbations. The evolution of density perturbations requires an analysis based on the 5D nature of gravity. In particular, the 5D gravitational field produces an effective “dark” anisotropic stress on the 4D universe, as discussed in Section 3.4. If one neglects this stress and other 5D effects, and simply treats the perturbations as 4D perturbations with a modified background Hubble rate – then as a consequence, the 4D Bianchi identity on the brane is violated, i.e., , and the results are inconsistent. When the 5D effects are incorporated [251, 71], the 4D Bianchi identity is automatically satisfied. (See Figure 22.)

There are three regimes governing structure formation in DGP models:

- On small scales, below the Vainshtein radius (which for cosmological purposes is roughly the scale of clusters), the spin-0 scalar degree of freedom becomes strongly coupled, so that the general relativistic limit is recovered [256].
- On scales relevant for structure formation, i.e., between cluster scales and the Hubble radius, the spin-0 scalar degree of freedom produces a scalar-tensor behaviour. A quasi-static approximation (as in the Newtonian approximation in standard 4D cosmology) to the 5D perturbations shows that DGP gravity is like a Brans–Dicke theory with parameter [251] At late times in an expanding universe, when , it follows that , so that . (This is a signal of the ghost pathology in DGP, which is discussed below.)
- Although the quasi-static approximation allows us to analytically solve the 5D wave equation for the bulk degree of freedom, this approximation breaks down near and beyond the Hubble radius. On super-horizon scales, 5D gravity effects are dominant, and we need to solve numerically the partial differential equation governing the 5D bulk variable [71].

On subhorizon scales relevant for linear structure formation, 5D effects produce a difference between and [251]:

so that there is an effective dark anisotropic stress on the brane: The density perturbations evolve as The linear growth factor, (i.e., normalized to the flat CDM case, ), is shown in Figure 22. This illustrates the dramatic suppression of growth in DGP relative to LCDM – from both the background expansion and the metric perturbations. If we parametrize the growth factor in the usual way, we can quantify the deviation from general relativity with smooth dark energy [293]:Observational data on the growth factor [188] are not yet precise enough to provide meaningful constraints on the DGP model. Instead, we can look at the large-angle anisotropies of the CMB, i.e., the ISW effect. This requires a treatment of perturbations near and beyond the horizon scale. The full numerical solution has been given by [71], and is illustrated in Figure 23. The CMB anisotropies are also shown in Figure 23, as computed in [142] using a scaling approximation to the super-Hubble modes [376] (the accuracy of the scaling ansatz is verified by the numerical results [71]).

It is evident from Figure 23 that the DGP model, which provides a best fit to the geometric data (see Figure 21), is in serious tension with the WMAP5 data on large scales. The problem arises because there is a large deviation of in the DGP model from the LCDM model. This deviation, i.e., a stronger decay of , leads to an over-strong ISW effect (which is determined by ), in tension with WMAP5 observations.

As a result of the combined observations of background expansion history and large-angle CMB anisotropies, the DGP model provides a worse fit to the data than LCDM at about the 5 level [142]. Effectively, the DGP model is ruled out by observations in comparison with the LCDM model.

In addition to the severe problems posed by cosmological observations, a problem of theoretical consistency arises from the fact that the late-time asymptotic de Sitter solution in DGP cosmological models has a ghost. The ghost is signaled by the negative Brans–Dicke parameter in the effective theory that approximates the DGP on cosmological subhorizon scales: The existence of the ghost is confirmed by detailed analysis of the 5D perturbations in the de Sitter limit [246, 175, 81, 246]. The DGP ghost is a ghost mode in the scalar sector of the gravitational field – which is more serious than the ghost in a phantom scalar field. It effectively rules out the DGP, since it is hard to see how an ultraviolet completion of the DGP can cure the infrared ghost problem. However, the DGP remains a valuable toy model for illustrating the kinds of behaviour that can occur from a modification to Einstein’s equations – and for developing cosmological tools to test modified gravity and Einstein’s theory itself.

http://www.livingreviews.org/lrr-2010-5 |
This work is licensed under a Creative Commons License. Problems/comments to |