9.1 ‘Self-accelerating’ DGP

Most brane-world models modify general relativity at high energies. The Randall–Sundrum models discussed up to now are a typical example. At low energies, H ℓ ≪ 1, the zero-mode of the graviton dominates on the brane, and general relativity is recovered to a good approximation. At high energies, H ℓ ≫ 1, the massive modes of the graviton dominate over the zero-mode, and gravity on the brane behaves increasingly 5-dimensional. On the unperturbed FRW brane, the standard energy-conservation equation holds, but the Friedmann equation is modified by an ultraviolet correction, (G ℓρ)2. At high energies, gravity “leaks” off the brane and 2 2 H ∝ ρ.

By contrast, the brane-world model of Dvali–Gabadadze–Porrati [135] (DGP), which was generalized to cosmology by Deffayet [115Jump To The Next Citation Point], 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

[ ∫ ∫ ] − 1 1 5 ∘ ------(5) 4 √ --- 16πG-- r- d x − g(5)R + d x − gR . (415 ) c bulk brane
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, r ≫ r c, where the first term in the sum (415View Equation) 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 rc. For a Minkowski brane, the weak-field gravitational potential behaves as
{ r−1 for r ≪ rc ψ ∝ r−2 for r ≫ rc . (416 )
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.
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Figure 19: Joint constraints [solid thick (blue)] from the SNLS data [solid thin (yellow)], the BAO peak at z = 0.35 [dotted (green)] and the CMB shift parameter from WMAP3 [dot-dashed (red)]. The left plot shows LCDM models, the right plot shows DGP. The thick dashed (black) line represents the flat models, ΩK = 0. (From [306].)
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Figure 20: The constraints from SNe and CMB/BAO on the parameters in the DGP model. The flat DGP model is indicated by the vertical dashed-dotted line; for the MLCS light curve fit, the flat model matches to the date very well. The SALT-II light curve fit to the SNe is again shown by the dotted contours. The combined constraints using the SALT-II SNe outlined by the dashed contours represent a poorer match to the CMB/BAO for the flat model. (From [401].)

The energy conservation equation remains the same as in general relativity, but the Friedmann equation is modified [115]:

˙ρ +∘3H-(ρ-+-p) = 0, (417 ) 2 K 1 K 8πG H + -2-− -- H2 + -2 = -----ρ. (418 ) a rc a 3
The Friedmann equation can be derived from the junction condition or the Gauss–Codazzi equation as in the RS model. To arrive at Equation (418View Equation) 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 (418View Equation) we infer that at early times, i.e., Hrc ≫ 1, the general relativistic Friedman equation is recovered. By contrast, at late times in an expanding CDM universe, with ρ ∝ a−3 → 0, we have

H → H = 1-, (419 ) ∞ rc
so that expansion accelerates and is asymptotically de Sitter. The above equations imply
[ ] ˙H − K--= − 4πG ρ 1 + ∘-------1-------- . (420 ) a2 1 + 32πGr2cρ ∕3
In order to achieve self-acceleration at late times, we require
rc ≳ H −01, (421 )
since H0 ≲ H∞. This is confirmed by fitting supernova observations, as shown in Figures 19View Image and 20View Image. The dimensionless cross-over parameter is defined as
Ωr = ----1----, (422 ) c 4 (H0rc )2
and the LCDM relation,
Ωm + Ω Λ + ΩK = 1, (423 )
is modified to
∘ ---∘ -------- Ωm + 2 Ωrc 1 − ΩK + ΩK = 1. (424 )

LCDM and DGP can both account for the supernova observations, with the fine-tuned values 2 Λ ∼ H 0 and rc ∼ H −0 1, respectively. When we add further constraints to the expansion history from the baryon acoustic oscillation peak at z = 0.35 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 19View Image and 20View Image 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 Ωm 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 [402Jump To The Next Citation Point], as shown in Figure 21View Image.

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 (418View Equation) 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 [251Jump To The Next Citation Point]. 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.

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Figure 21: The difference in χ2 between best-fit DGP (flat and open) and best-fit (flat) LCDM, using SNe, CMB shift and H0 Key Project data. (From [402].)

For LCDM, the analysis of density perturbations is well understood. For DGP the perturbations are much more subtle and complicated [251Jump To The Next Citation Point]. 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., ∇ νG μν ⁄= 0, and the results are inconsistent. When the 5D effects are incorporated [251Jump To The Next Citation Point, 71Jump To The Next Citation Point], the 4D Bianchi identity is automatically satisfied. (See Figure 22View Image.)

There are three regimes governing structure formation in DGP models:

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Figure 22: The growth factor g(a ) = Δ (a)∕a for LCDM (long dashed) and DGP (solid, thick), as well as for a dark energy model with the same expansion history as DGP (short dashed). DGP-4D (solid, thin) shows the incorrect result in which the 5D effects are set to zero. (From [251Jump To The Next Citation Point].)
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Figure 23: Left: Numerical solutions for DGP density and metric perturbations, showing also the quasi-static solution, which is an increasingly poor approximation as the scale is increased. (From [71Jump To The Next Citation Point].) Right: Constraints on DGP (the open model in Figure 21View Image that provides a best fit to geometric data) from CMB anisotropies (WMAP5). The DGP model is the solid curve, QCDM (short-dashed curve) is the GR quintessence model with the same background expansion history as the DGP model, and LCDM is the dashed curve (a slightly closed model that gives the best fit to WMAP5, HST and SNLS data). (From [142Jump To The Next Citation Point].)

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

( ) k2ϕ = 4 πGa2 1 − -1- ρΔ, (427 ) 3β ( 1 ) k2ψ = − 4πGa2 1 + --- ρΔ, (428 ) 3β
so that there is an effective dark anisotropic stress on the brane:
2 8πGa2-- k (ϕ + ψ) = − 3β2 ρΔ. (429 )
The density perturbations evolve as
( ) 1 Δ¨ + 2H Δ˙− 4πG 1 − 3β- ρΔ = 0. (430 )
The linear growth factor, g(a) = Δ(a )∕a (i.e., normalized to the flat CDM case, Δ ∝ a), is shown in Figure 22View Image. 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]:
{ f := d-ln-Δ- = Ωm (a)γ, γ ≈ 0.55 + 0.05[1 + w(z = 1 )] GR, smooth DE (431 ) dlna 0.68 DGP

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 [71Jump To The Next Citation Point], and is illustrated in Figure 23View Image. The CMB anisotropies are also shown in Figure 23View Image, as computed in [142Jump To The Next Citation Point] using a scaling approximation to the super-Hubble modes [376] (the accuracy of the scaling ansatz is verified by the numerical results [71Jump To The Next Citation Point]).

It is evident from Figure 23View Image that the DGP model, which provides a best fit to the geometric data (see Figure 21View Image), is in serious tension with the WMAP5 data on large scales. The problem arises because there is a large deviation of ϕ− = (ϕ − ψ )∕2 in the DGP model from the LCDM model. This deviation, i.e., a stronger decay of ϕ−, leads to an over-strong ISW effect (which is determined by ϕ˙−), 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 5σ 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 [246Jump To The Next Citation Point, 175, 81Jump To The Next Citation Point, 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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