By contrast, the brane-world model of Dvali–Gabadadze–Porrati  (DGP), which was generalized to cosmology by Deffayet , 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 energy conservation equation remains the same as in general relativity, but the Friedmann equation is modified :
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
LCDM 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 , 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 . One can find 4D general relativity models, with dark energy that has anisotropic stress and variable sound speed, which can in principle mimic DGP . 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 . 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 subhorizon scales relevant for linear structure formation, 5D effects produce a difference between and ::
Observational data on the growth factor  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 , and is illustrated in Figure 23. The CMB anisotropies are also shown in Figure 23, as computed in  using a scaling approximation to the super-Hubble modes  (the accuracy of the scaling ansatz is verified by the numerical results ).
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 . 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.
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