In the context of dark energy it is possible to construct viable single-field models based on BD theory. In what follows we discuss cosmological dynamics of dark energy models based on the action (10.10) in the flat FLRW background given by (2.12) (see, e.g., [596, 22, 85, 289, 5, 327, 139, 168] for dynamical analysis in scalar-tensor theories). Our interest is to find conditions under which a sequence of radiation, matter, and accelerated epochs can be realized. This depends upon the form of the field potential . We first carry out general analysis without specifying the forms of the potential. We take into account non-relativistic matter with energy density and radiation with energy density . The Jordan frame is regarded as a physical frame due to the usual conservation of non-relativistic matter (). Varying the action (10.10) with respect to and , we obtain the following equations

where .We introduce the following dimensionless variables

and also the density parameters These satisfy the relation from Eq. (10.12). From Eq. (10.13) it follows that Taking the derivatives of , and with respect to , we find where .Name | ||||

(a) MDE | 0 | |||

(b1) Kinetic 1 | 0 | 0 | ||

(b2) Kinetic 2 | 0 | 0 | ||

(c) Field dominated | 0 | |||

(d) Scaling solution | ||||

(e) de Sitter | 0 | 1 | 0 | |

If is a constant, i.e., for the exponential potential , one can derive fixed points for Eqs. (10.18) – (10.20) by setting (). In Table 1 we list the fixed points of the system in the absence of radiation (). Note that the radiation point corresponds to . The point (a) is the -matter-dominated epoch (MDE) during which the density of non-relativistic matter is a non-zero constant. Provided that this can be used for the matter-dominated epoch. The kinetic points (b1) and (b2) are responsible neither for the matter era nor for the accelerated epoch (for ). The point (c) is the scalar-field dominated solution, which can be used for the late-time acceleration for . When this point yields the cosmic acceleration for . The scaling solution (d) can be responsible for the matter era for , but in this case the condition for the point (c) leads to . Then the energy fraction of the pressureless matter for the point (d) does not satisfy the condition . The point (e) gives rise to the de Sitter expansion, which exists for the special case with [which can be also regarded as the special case of the point (c)]. From the above discussion the viable cosmological trajectory for constant is the sequence from the point (a) to the scalar-field dominated point (c) under the conditions and . The analysis based on the Einstein frame action (10.6) also gives rise to the MDE followed by the scalar-field dominated solution [23, 22].

Let us consider the case of non-constant . The fixed points derived above may be regarded as “instantaneous”
points^{7} [195, 454]
varying with the time-scale smaller than . As in metric f (R) gravity () we are
interested in large coupling models with of the order of unity. In order for the potential to
satisfy local gravity constraints, the field needs to be heavy in the region such that
. Then it is possible to realize the matter era by the point (d) with . Moreover the
solutions can finally approach the de Sitter solution (e) with or the field-dominated
solution (c). The stability of the point (e) was analyzed in [596, 250, 242] by considering
linear perturbations , and . One can easily show that the point (e) is stable for

For the f (R) model (5.19) the field is related to the Ricci scalar via the relation . Then the potential in the Jordan frame can be expressed as

For theories with general couplings we consider the following potential [596] which includes the potential (10.22) in f (R) gravity as a specific case with the correspondence and , , and . The potential behaves as for and in the limits (for ) and (for ). This potential has a curvature singularity at as in the models (4.83) and (4.84) of f (R) gravity, but the appearance of the singularity can be avoided by extending the potential to the regions () or () with a field mass bounded from above. The slope is given byDuring the radiation and deep matter eras one has from Eqs. (10.12) – (10.13) by noting that is negligibly small relative to the background fluid density. From Eq. (10.14) the field is nearly frozen at a value satisfying the condition . Then the field evolves along the instantaneous minima given by

As long as we have that . In this regime the slope in Eq. (10.24) is much larger than 1. The field value increases for decreasing and hence the slope decreases with time.Since around , the instantaneous fixed point (d) can be responsible for the matter-dominated epoch provided that . The variable decreases in time irrespective of the sign of the coupling and hence . The de Sitter point is characterized by , i.e.,

The de Sitter solution is present as long as the solution of this equation exists in the region . From Eq. (10.24) the derivative of in terms of is given by When , we can show that the function is positive and hence the condition is satisfied. This means that the de Sitter point (e) is a stable attractor. When , the function can be negative. Plugging Eq. (10.26) into Eq. (10.27), we find that the de Sitter point is stable for If this condition is violated, the solutions choose another stable fixed point [such as the point (c)] as an attractor.The above discussion shows that for the model (10.23) the matter point (d) can be followed by the stable de Sitter solution (e) for . In fact numerical simulations in [596] show that the sequence of radiation, matter and de Sitter epochs can be in fact realized. Introducing the energy density and the pressure of dark energy as we have done for metric f (R) gravity, the dark energy equation of state is given by the same form as Eq. (4.97). Since for the model (10.23) increases toward the past, the phantom equation of state () as well as the cosmological constant boundary crossing () occurs as in the case of metric f (R) gravity [596].

As we will see in Section 10.3, for a light scalar field, it is possible to satisfy local gravity constraints for . In those cases the potential does not need to be steep such that in the region . The cosmological dynamics for such nearly flat potentials have been discussed by a number of authors in several classes of scalar-tensor theories [489, 451, 416, 271]. It is also possible to realize the condition , while avoiding the appearance of a ghost [416, 271].

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