List of Figures

View Image Figure 1:
Schematic of confinement of matter to the brane, while gravity propagates in the bulk (from [75]).
View Image Figure 2:
The RS 2-brane model. (Figure taken from [87].)
View Image Figure 3:
Gravitational field of a small point particle on the brane in RS gauge. (Figure taken from [155].)
View Image Figure 4:
The evolution of the dimensionless shear parameter Ωshear = σ2 ∕6H2 on a Bianchi I brane, for a V = 12m2 ϕ2 model. The early and late-time expansion of the universe is isotropic, but the shear dominates during an intermediate anisotropic stage. (Figure taken from [307].)
View Image Figure 5:
The relation between the inflaton mass m ∕M 4 (M ≡ M 4 p) and the brane tension 4 1∕4 (λ∕M 4) necessary to satisfy the COBE constraints. The straight line is the approximation used in Equation (220View Equation), which at high energies is in excellent agreement with the exact solution, evaluated numerically in slow-roll. (Figure taken from [308].)
View Image Figure 6:
Constraints from WMAP data on inflation models with quadratic and quartic potentials, where R is the ratio of tensor to scalar amplitudes and n is the scalar spectral index. The high energy (H.E.) and low energy (L.E.) limits are shown, with intermediate energies in between, and the 1-σ and 2-σ contours are also shown. (Figure taken from [284].)
View Image Figure 7:
Brane-world instanton. (Figure taken from [154].)
View Image Figure 8:
The evolution of the covariant variable Φ, defined in Equation (304View Equation) (and not to be confused with the Bardeen potential), along a fundamental world-line. This is a mode that is well beyond the Hubble horizon at N = 0, about 50 e-folds before inflation ends, and remains super-Hubble through the radiation era. A smooth transition from inflation to radiation is modelled by w = 13 [(2 − 32𝜖)tanh (N − 50) − (1 − 32𝜖)], where 𝜖 is a small positive parameter (chosen as 𝜖 = 0.1 in the plot). Labels on the curves indicate the value of ρ0∕ λ, so that the general relativistic solution is the dashed curve (ρ0∕λ = 0). (Figure taken from [177].)
View Image Figure 9:
The evolution of Φ in the radiation era, with dark radiation present in the background. (Figure taken from [187].)
View Image Figure 10:
Comparison between typical results of the PS and CI codes for various brane quantities (left); and the typical behaviour of the bulk master variable (right) as calculated by the CI method. Very good agreement between the two different numerical schemes is seen in the left panel, despite the fact that they use different initial conditions. Also note that on subhorizon scales, Δ and ζ undergo simple harmonic oscillations, which is consistent with the behaviour in GR. The bulk profile demonstrates our choice of initial conditions: We see that the bulk master variable Ω is essentially zero during the early stages of the simulation, and only becomes “large” when the mode crosses the horizon. Figure taken from [69].
View Image Figure 11:
The simulated behaviour of a mode on superhorizon scales. On the left we show how the Δ gauge invariant switches from the high-energy behaviour predicted to the familiar GR result as the universe expands through the critical epoch. We also show how the KK anisotropic stress 2 κ4δπℰ steadily decays throughout the simulation, which is typical of all the cases we have investigated. On the right, we show the metric perturbations ψ (= Ψ ) and ℛ (= Φ) as well as the curvature perturbation ζ. Again, note how the GR result Φ ≈ − Ψ ≈ − 2ζ∕3 is recovered at low energy. Figure taken from [69].
View Image Figure 12:
Density perturbation enhancement factors (left) and transfer functions (right) from simulations, effective theory, and general relativity. All of the 𝒬 factors monotonically increase with k∕kc, and we see that the Δ amplitude enhancement due to 𝒪 (ρ∕λ ) effects 𝒬ref is generally larger than the enhancement due to KK effects 𝒬 ℰ. For asymptotically small scales k ≫ k c, the enhancement seems to level off. The transfer functions in the right panel are evaluated at a given subcritical epoch in the radiation dominated era. The T functions show how, for a fixed primordial spectrum of curvature perturbations 𝒫inζf, the effective theory predicts excess power in the Δ spectrum 𝒫 Δ ∝ T 2𝒫iζnf on supercritical/subhorizon scales compared to the GR result. The excess small-scale power is even greater when KK modes are taken into account, as shown by T (k;η) 5D. Figure taken from [69].
View Image Figure 13:
Graviton “volcano” potential around the dS4 brane, showing the mass gap. (Figure taken from [270].)
View Image Figure 14:
The evolution of gravitational waves. We set the comoving wave number to √ -- k = 3 ∕ℓ (𝜖 = 1.0 ∗). The right panel depicts the projection of the three-dimensional waves of the left panel. Figure taken from [199].
View Image Figure 15:
Squared amplitude of gravitational waves on the brane in the low-energy (left) and the high-energy (right) regimes. In both panels, solid lines represent the numerical solutions. The dashed lines are the amplitudes of reference gravitational waves href obtained from Equation (381View Equation). Figure taken from [199].
View Image Figure 16:
The energy spectrum of the stochastic background of gravitational wave around the critical frequency fc in radiation dominated epoch. The filled circles represent the spectrum caused by the non-standard cosmological expansion of the universe. Taking account of the KK-mode excitations, the spectrum becomes the one plotted by filled squares. In the asymptotic region depicted in the solid line, the frequency dependence becomes almost the same as the one predicted in the four-dimensional theory (long-dashed line). Figure taken from [199].
View Image Figure 17:
The CMB power spectrum with brane-world effects, encoded in the dark radiation fluctuation parameter δC∗ as a proportion of the large-scale curvature perturbation for matter (denoted ζ∗ in the plot). (Figure taken from [245].)
View Image Figure 18:
The CMB power spectrum with brane-world moduli effects from the field χ in Equation (414View Equation). The curves are labelled with the initial value of χ. (Figure taken from [361, 51].)
View Image 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].)
View Image 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].)
View Image 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].)
View Image 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 [251].)
View Image 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 [71].) Right: Constraints on DGP (the open model in Figure 21 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 [142].)
View Image Figure 24:
Left: The embedding of the self-accelerating and normal branches of the DGP brane in a Minkowski bulk. (From [81].) 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 [279].)
View Image 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 [167].) 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 23. (From [71].)
View Image Figure 26:
Top: Measurement of the cross-correlation functions between six different galaxy data sets and the CMB, reproduced from [166]. 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 25. (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].)
View Image Figure 27:
Removing a wedge from a sphere and identifying opposite sides to obtain a rugby ball geometry. Two equal-tension branes with conical deficit angles are located at either pole; outside the branes there is constant spherical curvature. From [72].