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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 scalar-tensor 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 3-year 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 thin-shell 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]. |