List of Figures

View Image Figure 1:
The field potential (3.16View Equation) in the Einstein frame corresponding to the model (3.6View Equation). Inflation is realized in the regime κ ϕ ≫ 1.
View Image Figure 2:
Four trajectories in the (r,m ) plane. Each trajectory corresponds to the models: (i) ΛCDM, (ii) f(R) = (Rb − Λ)c, (iii) f (R) = R − αRn with α > 0,0 < n < 1, and (iv) 2 m (r) = − C (r + 1)(r + ar + b). From [31].
View Image Figure 3:
(Top) The potential 2 2 V (ϕ) = (F R − f)∕(2κ F ) versus the field ∘ -------- ϕ = 3∕(16 π)mpl ln F for the Starobinsky’s dark energy model (4.84View Equation) with n = 1 and μ = 2. (Bottom) The inverted effective potential − Veff for the same model parameters as the top with ρ ∗ = 10Rcm2pl. The field value, at which the inverted effective potential has a maximum, is different depending on the density ∗ ρ, see Eq. (5.22View Equation). In the upper panel “de Sitter” corresponds to the minimum of the potential, whereas “singular” means that the curvature diverges at ϕ = 0.
View Image Figure 4:
Evolution of γ versus the redshift z in the model (4.83View Equation) with n = 1 and μ = 1.55 for four different values of k. For these model parameters the dispersion of γ with respect to k is very small. All the perturbation modes shown in the figure have reached the scalar-tensor regime (M 2 ≪ k2∕a2) by today. From [589].
View Image Figure 5:
The regions (i), (ii) and (iii) for the model (4.84View Equation). We also show the bound n > 0.9 coming from the local gravity constraints as well as the condition (4.87View Equation) coming from the stability of the de Sitter point. From [589].
View Image Figure 6:
Comparison between N-body simulations and the two fitting formulas in the f (R) model (4.83View Equation) with n = 1∕2. The circles and triangles show the results of N-body simulations with and without the chameleon mechanism, respectively. The arrow represents the maximum value of k(= 0.08h Mpc −1) by which the perturbation theory is valid. (Left) The fitting formula by Smith et al. [540] is used to predict Pnon− GR and PGR. The solid and dashed lines correspond to the power spectra with and without the chameleon mechanism, respectively. For the chameleon case cnl(z) is determined by the perturbation theory with cnl(z = 0) = 0.085. (Right) The N-body results in [479] are interpolated to derive Pnon−GR without the chameleon mechanism. The obtained P non−GR is used for the HS fitting formula to derive the power spectrum P in the chameleon case. From [371].
View Image Figure 7:
(Left) Evolution of the effective gravitational potential Φeff (denoted as Φ− in the figure) versus the scale factor a (with the present value a = 1) on the scale −1 3 k = 10 Mpc for the ΛCDM model and f (R) models with B0 = 0.5, 1.5, 3.0, 5.0. As the parameter B0 increases, the decay of Φe ff decreases and then turns into growth for B0 ≳ 1.5. (Right) The CMB power spectrum ℓ(ℓ + 1)C ℓ∕(2 π) for the ΛCDM model and f (R) models with B0 = 0.5, 1.5, 3.0, 5.0. As B 0 increases, the ISW contributions to low multipoles decrease, reach the minimum around B0 = 1.5, and then increase. The black points correspond to the WMAP 3-year data [561]. From [545].
View Image Figure 8:
The evolution of the variables y1 and y2 for the model f(R ) = R − β∕Rn with n = 0.02, together with the effective equation of state w eff. Initial conditions are chosen to be − 40 y1 = 10 and −5 y2 = 1.0 − 10. From [253].
View Image Figure 9:
The allowed region of the parameter space in the (Q, p) plane for BD theory with the potential (10.23View Equation). We show the allowed region coming from the bounds Δns (tΛ) < 0.05 and fδ < 2 as well as the the equivalence principle (EP) constraint (10.35View Equation).
View Image Figure 10:
The thin-shell field profile for the model V = M 6ϕ −2 with Φc = 0.2, Δ &tidle;rc∕&tidle;rc = 0.1, mA r&tidle;c = 20, and Q = 1. This case corresponds to 4 &tidle;ρA∕ρ&tidle;B = 1.04 × 10, −3 ϕA = 8.99 × 10, − 1 ϕB = 1.97 × 10 and −1 𝜖th = 1.56 × 10. The boundary condition of φ = ϕ∕ϕA at xi = &tidle;ri∕r&tidle;c = 10− 5 is φ (xi) = 1.2539010, which is larger than the analytic value φ (xi) = 1.09850009. The derivative φ′(xi) is the same as the analytic value. The left and right panels show φ (&tidle;r) for 0 < &tidle;r∕&tidle;r < 10 c and 0 < &tidle;r∕&tidle;r < 2 c, respectively. The black and dotted curves correspond to the numerically integrated solution and the analytic field profile (11.26View Equation) – (11.28View Equation), respectively. From [594].
View Image Figure 11:
The profile of the field ∘ ---- ϕ = 3∕2 ln F (in units of Mpl) versus the radius &tidle;r (denoted as r in the figure, in units of −1∕2 Mpl ρcenter) for the model (4.84View Equation) with n = 1, R∞ ∕Rc = 3.6, and v0 = 10− 4 (shown as a solid line). The dashed line corresponds to the value ϕmin for the minimum of the effective potential. (Inset) The enlarged figure in the region 0 < &tidle;r < 2.5. From [43].
View Image Figure 12:
The evolution of μ (multiplied by 104) and w eff versus the redshift z = a ∕a − 1 0 for the model (12.16View Equation) with parameters α = 100 and − 4 λ = 3 × 10. The initial conditions are chosen to be x = − 1.499985, y = 20, and Ωm = 0.99999. We do not take into account radiation in this simulation. From [182].
View Image Figure 13:
Plot of the absolute errors 2 2 log10(|H i − H Λ − Ci|) (left) and [|H2i−H2Λ−Ci|] log10 |H2i−H2Λ+Ci| (right) versus N = lna for the model (12.16View Equation) with i = 0,1, ⋅⋅⋅,6. The model parameters are α = 10 and λ = 0.075. The iterative method provides the solutions with high accuracy in the regime N ≲ − 4. From [188].
View Image Figure 14:
The convergence power spectrum P κ(ℓ) in f (R) gravity (√ -- Q = − 1∕ 6) for the model (5.19View Equation). This model corresponds to the field potential (10.23View Equation). Each case corresponds to (a) p = 0.5, C = 0.9, (b) p = 0.7, C = 0.9, and (c) the ΛCDM model. The model parameters are chosen to be (0) Ω m = 0.28, n Φ = 1, and 2 − 10 δH = 3.2 × 10. From [595].