9.2 ‘Normal’ DGP

The self-accelerating DGP is effectively ruled out as a viable cosmological model by observations and by the problem of the ghost in the gravitational sector. Indeed, it may be the case that self-acceleration generically comes with the price of ghost states. The ‘normal’ (i.e., non-self-accelerating and ghost-free) branch of the DGP [371Jump To The Next Citation Point], arises from a different embedding of the DGP brane in the Minkowski bulk (see Figure 24View Image). In the background dynamics, this amounts to a replacement rc → − rc in Equation (418View Equation) – and there is no longer late-time self-acceleration. Therefore, it is necessary to include a Λ term in order to accelerate the late universe:
∘ --------- 2 K 1 K 8πG Λ H + -2-+ -- H2 + --2 = ----ρ + -. (432 ) a rc a 3 3
(Normal DGP models with a quintessence field have also been investigated [88].) Using the dimensionless crossover parameter defined in Equation (422View Equation), the densities are related at the present time by
-------- ∘ 1 − Ω = − ∘ Ω---+ ∘ Ω--+-Ω---+-Ω--, (433 ) K rc rc m Λ
which can be compared with the self-accelerating DGP relation (424View Equation).
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Figure 24: Left: The embedding of the self-accelerating and normal branches of the DGP brane in a Minkowski bulk. (From [81Jump To The Next Citation Point].) Right: Joint constraints on normal DGP (flat, K = 0) from SNLS, CMB shift (WMAP3) and BAO (z = 0.35) data. The best-fit is the solid point, and is indistinguishable from the LCDM limit. The shaded region is unphysical and its upper boundary represents flat LCDM models. (From [279Jump To The Next Citation Point].)

An interesting feature of the normal branch is the ‘degravitation’ property, i.e., that Λ is effectively screened by 5D gravity effects. This follows from rewriting the modified Friedmann equation (432View Equation) in standard general relativistic form, with

∘ --------- 3 K Λeff = Λ − -- H2 + -2-< Λ. (434 ) rc a
Thus, 5D gravity in normal DGP can in principle reduce the bare vacuum energy significantly [371, 298]. However, Figure 24View Image shows that best-fit flat models, using geometric data, only admit insignificant screening [279]. The closed models provide a better fit to the data [167Jump To The Next Citation Point], and can allow a bare vacuum energy term with ΩΛ > 1, as shown in Figure 25View Image. This does not address the fundamental problem of the smallness of ΩΛ, but is nevertheless an interesting feature. We can define an effective equation-of-state parameter via
˙Λ + 3H (1 + w )Λ = 0. (435 ) eff eff eff
At the present time (setting K = 0 for simplicity),
(Ωm + Ω Λ − 1)Ωm weff,0 = − 1 − ------------------------< − 1, (436 ) (1 − Ωm )(Ωm + Ω Λ + 1)
where the inequality holds because Ωm < 1. This reveals another important property of the normal DGP model: effective phantom behaviour of the recent expansion history. This is achieved without any pathological phantom field (similar to what can be done in scalar-tensor theories [42]). Furthermore, there is no “big rip” singularity in the future associated with this phantom acceleration, unlike the situation that typically arises with phantom fields. The phantom behaviour in the normal DGP model is also not associated with any ghost problem – indeed, the normal DGP branch is free of the ghost that plagues the self-accelerating DGP [81].
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Figure 25: Left: Joint constraints on normal DGP from SNe Gold, CMB shift (WMAP3) and H0 data in the projected curvature-Λ plane, after marginalizing over other parameters. The best-fits are the solid points, corresponding to different values of Ωm. (From [167Jump To The Next Citation Point].) Right: Numerical solutions for the normal DGP density and metric perturbations, showing also the quasistatic solution, which is an increasingly poor approximation as the scale is increased. Compare with the self-accelerating DGP case in Figure 23View Image. (From [71].)

Perturbations in the normal branch have the same structure as those in the self-accelerating branch, with the same regimes – i.e., below the Vainshtein radius (recovering a GR limit), up to the Hubble radius (Brans–Dicke behaviour), and beyond the Hubble radius (strongly 5D behaviour). The quasistatic approximation and the numerical integrations can be simply repeated with the replacement rc → − rc (and the addition of Λ to the background). In the sub-Hubble regime, the effective Brans–Dicke parameter is still given by Equations (425View Equation) and (426View Equation), but now we have ωBD > 0 – and this is consistent with the absence of a ghost. Furthermore, a positive Brans–Dicke parameter signals an extra positive contribution to structure formation from the scalar degree of freedom, so that there is less suppression of structure formation than in LCDM – the reverse of what happens in the self-accelerating DGP. This is confirmed by computations, as illustrated in Figure 25View Image.

The closed normal DGP models fit the background expansion data reasonably well, as shown in Figure 25View Image. The key remaining question is how well do these models fit the large-angle CMB anisotropies, which is yet to be computed at the time of writing. The derivative of the ISW potential ˙ ϕ− can be seen in Figure 25View Image, and it is evident that the ISW contribution is negative relative to LCDM at high redshifts, and goes through zero at some redshift before becoming positive. This distinctive behaviour may be contrasted with the behaviour in f(R ) models (see Figure 26View Image): both types of model lead to less suppression of structure than LCDM, but they produce different ISW effects. However, in the limit r → ∞ r, normal DGP tends to ordinary LCDM, hence observations which fit LCDM will always just provide a lower limit for rc.

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Figure 26: Top: Measurement of the cross-correlation functions between six different galaxy data sets and the CMB, reproduced from [166Jump To The Next Citation Point]. The curves show the theoretical predictions for the ISW-galaxy correlations at each redshift for the LCDM model (black, dashed) and the three nDGP models, which describe the 1 − σ region of the geometry test from Figure 25View Image. (From [167].) Bottom: Theoretical predictions for a family of f(R ) theories compared with the ISW data [166] measuring the angular CCF between the CMB and six galaxy catalogues. The model with B0 = 0 is equivalent to LCDM, while increasing departures from GR produce negative cross-correlations. (From [165].)


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