Figure 1:
The field potential (3.16) in the Einstein frame corresponding to the model (3.6). Inflation is realized in the regime . 

Figure 2:
Four trajectories in the plane. Each trajectory corresponds to the models: (i) CDM, (ii) , (iii) with , and (iv) . From [31]. 

Figure 3:
(Top) The potential versus the field for the Starobinsky’s dark energy model (4.84) with and . (Bottom) The inverted effective potential for the same model parameters as the top with . The field value, at which the inverted effective potential has a maximum, is different depending on the density , see Eq. (5.22). In the upper panel “de Sitter” corresponds to the minimum of the potential, whereas “singular” means that the curvature diverges at . 

Figure 4:
Evolution of versus the redshift in the model (4.83) with and for four different values of . For these model parameters the dispersion of with respect to is very small. All the perturbation modes shown in the figure have reached the scalartensor regime () by today. From [589]. 

Figure 5:
The regions (i), (ii) and (iii) for the model (4.84). We also show the bound coming from the local gravity constraints as well as the condition (4.87) coming from the stability of the de Sitter point. From [589]. 

Figure 6:
Comparison between body simulations and the two fitting formulas in the f (R) model (4.83) with . The circles and triangles show the results of body simulations with and without the chameleon mechanism, respectively. The arrow represents the maximum value of by which the perturbation theory is valid. (Left) The fitting formula by Smith et al. [540] is used to predict and . The solid and dashed lines correspond to the power spectra with and without the chameleon mechanism, respectively. For the chameleon case is determined by the perturbation theory with . (Right) The body results in [479] are interpolated to derive without the chameleon mechanism. The obtained is used for the HS fitting formula to derive the power spectrum in the chameleon case. From [371]. 

Figure 7:
(Left) Evolution of the effective gravitational potential (denoted as in the figure) versus the scale factor (with the present value ) on the scale for the CDM model and f (R) models with = 0.5, 1.5, 3.0, 5.0. As the parameter increases, the decay of decreases and then turns into growth for . (Right) The CMB power spectrum for the CDM model and f (R) models with = 0.5, 1.5, 3.0, 5.0. As increases, the ISW contributions to low multipoles decrease, reach the minimum around = 1.5, and then increase. The black points correspond to the WMAP 3year data [561]. From [545]. 

Figure 8:
The evolution of the variables and for the model with , together with the effective equation of state . Initial conditions are chosen to be and . From [253]. 

Figure 9:
The allowed region of the parameter space in the plane for BD theory with the potential (10.23). We show the allowed region coming from the bounds and as well as the the equivalence principle (EP) constraint (10.35). 

Figure 10:
The thinshell field profile for the model with , , , and . This case corresponds to , , and . The boundary condition of at is , which is larger than the analytic value . The derivative is the same as the analytic value. The left and right panels show for and , respectively. The black and dotted curves correspond to the numerically integrated solution and the analytic field profile (11.26) – (11.28), respectively. From [594]. 

Figure 11:
The profile of the field (in units of ) versus the radius (denoted as in the figure, in units of ) for the model (4.84) with , , and (shown as a solid line). The dashed line corresponds to the value for the minimum of the effective potential. (Inset) The enlarged figure in the region . From [43]. 

Figure 12:
The evolution of (multiplied by ) and versus the redshift for the model (12.16) with parameters and . The initial conditions are chosen to be , , and . We do not take into account radiation in this simulation. From [182]. 

Figure 13:
Plot of the absolute errors (left) and (right) versus for the model (12.16) with . The model parameters are and . The iterative method provides the solutions with high accuracy in the regime . From [188]. 

Figure 14:
The convergence power spectrum in f (R) gravity () for the model (5.19). This model corresponds to the field potential (10.23). Each case corresponds to (a) , , (b) , , and (c) the CDM model. The model parameters are chosen to be , , and . From [595]. 
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