10.2 Cosmological dynamics of dark energy models based on Brans–Dicke theory

The first attempt to apply BD theory to cosmic acceleration is the extended inflation scenario in which the BD field φ is identified as an inflaton field [374571]. The first version of the inflation model, which considered a first-order phase transition in BD theory, resulted in failure due to the graceful exit problem [37561365]. This triggered further study of the possibility of realizing inflation in the presence of another scalar field [39478]. In general the dynamics of such a multi-field system is more involved than that in the single-field case [71Jump To The Next Citation Point]. The resulting power spectrum of density perturbations generated during multi-field inflation in BD theory was studied by a number of authors [570272156569].

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 (10.10View Equation) in the flat FLRW background given by (2.12View Equation) (see, e.g., [596Jump To The Next Citation Point22Jump To The Next Citation Point852895327139168] 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 U (ϕ ). We first carry out general analysis without specifying the forms of the potential. We take into account non-relativistic matter with energy density ρ m and radiation with energy density ρ r. The Jordan frame is regarded as a physical frame due to the usual conservation of non-relativistic matter (−3 ρm ∝ a). Varying the action (10.10View Equation) with respect to gμν and ϕ, we obtain the following equations

3F H2 = (1 − 6Q2 )Fϕ˙2 ∕2 + U − 3H F˙ + ρm + ρr, (10.12 ) ˙ 2 ˙2 ¨ ˙ 2F H = −[ (1 − 6Q )F ϕ − F +] H F − ρm − 4ρr∕3, (10.13 ) (1 − 6Q2) F ϕ¨+ 3H ˙ϕ + ˙F∕ (2F )ϕ˙ + U,ϕ + QF R = 0, (10.14 )
where F = e −2Qϕ.

We introduce the following dimensionless variables

˙ ∘ ---- ∘ ---- x1 ≡ √-ϕ--, x2 ≡ 1-- U--, x3 ≡ 1-- ρr-, (10.15 ) 6H H 3F H 3F
and also the density parameters
--ρm-- 2 2 2 2 √-- Ωm ≡ 3F H2 , Ωr ≡ x 3, ΩDE ≡ (1 − 6Q )x1 + x2 + 2 6Qx1. (10.16 )
These satisfy the relation Ωm + Ωr + ΩDE = 1 from Eq. (10.12View Equation). From Eq. (10.13View Equation) it follows that
( ) H˙- 1-−-6Q2- 2 2 2 2 2 √ -- 2 H2 = − 2 3 + 3x1 − 3x2 + x3 − 6Q x1 + 2 6Qx1 + 3Q (λx2 − 4Q ). (10.17 )
Taking the derivatives of x1, x2 and x3 with respect to N = lna, we find
dx √6-- √-- ---1 = ---(λx22 − 6x1 ) dN 2√ -- --6Q-[ 2 2 √ -- 2 2 ] -˙H- + 2 (5 − 6Q )x1 + 2 6Qx1 − 3x2 + x3 − 1 − x1H2 , (10.18 ) √ -- ˙ dx2- = --6(2Q − λ)x1x2 − x2 H--, (10.19 ) dN 2 H2 dx3 √ -- H˙ ---- = 6Qx1x3 − 2x3 − x3--2 , (10.20 ) dN H
where λ ≡ − U,ϕ∕U.


Table 1: The critical points of dark energy models based on the action (10.10View Equation) in BD theory with constant λ = − U ∕U ,ϕ in the absence of radiation (x = 0 3). The effective equation of state ˙ 2 we ff = − 1 − 2 H ∕(3H ) is known from Eq. (10.17View Equation).
Name x1 x2 Ωm weff
(a) ϕMDE --√6Q--- 3(2Q2−1) 0 -3−-2Q2-- 3(1− 2Q2)2 --4Q2--- 3(1−2Q2)
(b1) Kinetic 1 1 √6Q+1- 0 0 3− √6Q 3(1+√6Q-)
(b2) Kinetic 2 √6Q1−1 0 0 √- 3(3+1−√66QQ-)
(c) Field dominated -√6(4Q−λ)-- 6(4Q2−Q λ− 1) [6−λ2+8Qλ−-16Q2-]1∕2 6(4Q2− Qλ−1)2 0 20Q2−-9Qλ−3+λ2- − 3(4Q2− Qλ−1)
(d) Scaling solution √- -6- 2λ ∘ ---------- 3+2Q-λ−26Q2- 2λ 1 − 3−12Q22+7Qλ- λ − 2Q- λ
(e) de Sitter 0 1 0 − 1

If λ is a constant, i.e., for the exponential potential U = U0e −λϕ, one can derive fixed points for Eqs. (10.18View Equation) – (10.20View Equation) by setting dxi∕dN = 0 (i = 1,2,3). In Table 1 we list the fixed points of the system in the absence of radiation (x3 = 0). Note that the radiation point corresponds to (x1,x2,x3 ) = (0, 0,1). The point (a) is the ϕ-matter-dominated epoch (ϕMDE) during which the density of non-relativistic matter is a non-zero constant. Provided that Q2 ≪ 1 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 |Q | ≲ 1). The point (c) is the scalar-field dominated solution, which can be used for the late-time acceleration for weff < − 1∕3. When 2 Q ≪ 1 this point yields the cosmic acceleration for √ -- √ -- − 2 + 4Q < λ < 2 + 4Q. The scaling solution (d) can be responsible for the matter era for |Q | ≪ |λ |, but in this case the condition we ff < − 1∕3 for the point (c) leads to λ2 ≲ 2. Then the energy fraction of the pressureless matter for the point (d) does not satisfy the condition Ω ≃ 1 m. The point (e) gives rise to the de Sitter expansion, which exists for the special case with λ = 4Q [which can be also regarded as the special case of the point (c)]. From the above discussion the viable cosmological trajectory for constant λ is the sequence from the point (a) to the scalar-field dominated point (c) under the conditions Q2 ≪ 1 and √ -- √ -- − 2 + 4Q < λ < 2 + 4Q. The analysis based on the Einstein frame action (10.6View Equation) also gives rise to the ϕMDE followed by the scalar-field dominated solution [2322].

Let us consider the case of non-constant λ. The fixed points derived above may be regarded as “instantaneous” points7 [195454] varying with the time-scale smaller than H −1. As in metric f (R) gravity (√ -- Q = − 1∕ 6) we are interested in large coupling models with |Q| of the order of unity. In order for the potential U (ϕ) to satisfy local gravity constraints, the field needs to be heavy in the region 2 R ≫ R0 ∼ H 0 such that |λ| ≫ 1. Then it is possible to realize the matter era by the point (d) with |Q | ≪ |λ|. Moreover the solutions can finally approach the de Sitter solution (e) with λ = 4Q or the field-dominated solution (c). The stability of the point (e) was analyzed in  [596Jump To The Next Citation Point250242] by considering linear perturbations δx 1, δx 2 and δF. One can easily show that the point (e) is stable for

d-λ- dλ- Q dF (F1 ) > 0 → dϕ (ϕ1) < 0, (10.21 )
where F1 = e−2Qϕ1 with ϕ1 being the field value at the de Sitter point. In metric f (R) gravity (√ -- Q = − 1∕ 6) this condition is equivalent to m = Rf,RR∕f,R < 1.

For the f (R) model (5.19View Equation) the field ϕ is related to the Ricci scalar R via the relation 2ϕ∕√6 −(2n+1) e = 1 − 2n μ(R∕Rc ). Then the potential U = (F R − f)∕2 in the Jordan frame can be expressed as

[ ] μRc 2n + 1 ( 2ϕ∕√6)2n∕(2n+1) U (ϕ ) = -2-- 1 − (2nμ)2n∕(2n+1)- 1 − e . (10.22 )
For theories with general couplings Q we consider the following potential [596Jump To The Next Citation Point]
[ −2Qϕ p] U (ϕ) = U0 1 − C (1 − e ) (U0 > 0, C > 0, 0 < p < 1), (10.23 )
which includes the potential (10.22View Equation) in f (R) gravity as a specific case with the correspondence U0 = μRc ∕2 and 2n∕(2n+1) C = (2n + 1)∕(2nμ ), √ -- Q = − 1∕ 6, and p = 2n∕(2n + 1 ). The potential behaves as U (ϕ) → U0 for ϕ → 0 and U(ϕ ) → U0 (1 − C ) in the limits ϕ → ∞ (for Q > 0) and ϕ → − ∞ (for Q < 0). This potential has a curvature singularity at ϕ = 0 as in the models (4.83View Equation) and (4.84View Equation) of f (R) gravity, but the appearance of the singularity can be avoided by extending the potential to the regions ϕ > 0 (Q < 0) or ϕ < 0 (Q > 0) with a field mass bounded from above. The slope λ = − U,ϕ∕U is given by
2CpQe −2Qϕ(1 − e− 2Q ϕ)p− 1 λ = --------------------------. (10.24 ) 1 − C (1 − e −2Qϕ)p

During the radiation and deep matter eras one has 2 ˙ R = 6 (2H + H ) ≃ ρm ∕F from Eqs. (10.12View Equation) – (10.13View Equation) by noting that U0 is negligibly small relative to the background fluid density. From Eq. (10.14View Equation) the field is nearly frozen at a value satisfying the condition U,ϕ + Q ρm ≃ 0. Then the field ϕ evolves along the instantaneous minima given by

1 ( 2U pC )1∕(1−p) ϕm ≃ --- ---0--- . (10.25 ) 2Q ρm
As long as ρ ≫ 2U pC m 0 we have that |ϕ | ≪ 1 m. In this regime the slope λ in Eq. (10.24View Equation) is much larger than 1. The field value |ϕm | increases for decreasing ρm and hence the slope λ decreases with time.

Since λ ≫ 1 around ϕ = 0, the instantaneous fixed point (d) can be responsible for the matter-dominated epoch provided that |Q| ≪ λ. The variable F = e−2Q ϕ decreases in time irrespective of the sign of the coupling Q and hence 0 < F < 1. The de Sitter point is characterized by λ = 4Q, i.e.,

2(1 − F1)1−p C = -------------. (10.26 ) 2 + (p − 2)F1
The de Sitter solution is present as long as the solution of this equation exists in the region 0 < F1 < 1. From Eq. (10.24View Equation) the derivative of λ in terms of ϕ is given by
dλ 4CpQ2F (1 − F )p− 2[1 − pF − C (1 − F )p] ---= − -------------------------p-2------------. (10.27 ) dϕ [1 − C (1 − F) ]
When 0 < C < 1, we can show that the function g(F) ≡ 1 − pF − C (1 − F )p is positive and hence the condition dλ ∕dϕ < 0 is satisfied. This means that the de Sitter point (e) is a stable attractor. When C > 1, the function g(F ) can be negative. Plugging Eq. (10.26View Equation) into Eq. (10.27View Equation), we find that the de Sitter point is stable for
1 F1 > -----. (10.28 ) 2 − p
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 (10.23View Equation) the matter point (d) can be followed by the stable de Sitter solution (e) for 0 < C < 1. In fact numerical simulations in [596Jump To The Next Citation Point] show that the sequence of radiation, matter and de Sitter epochs can be in fact realized. Introducing the energy density ρDE and the pressure PDE of dark energy as we have done for metric f (R) gravity, the dark energy equation of state wDE = PDE ∕ρDE is given by the same form as Eq. (4.97View Equation). Since for the model (10.23View Equation) F increases toward the past, the phantom equation of state (wDE < − 1) as well as the cosmological constant boundary crossing (w = − 1 DE) occurs as in the case of metric f (R) gravity [596Jump To The Next Citation Point].

As we will see in Section 10.3, for a light scalar field, it is possible to satisfy local gravity constraints for |Q | ≲ 10−3. In those cases the potential does not need to be steep such that λ ≫ 1 in the region R ≫ R0. The cosmological dynamics for such nearly flat potentials have been discussed by a number of authors in several classes of scalar-tensor theories [489451416Jump To The Next Citation Point271Jump To The Next Citation Point]. It is also possible to realize the condition wDE < − 1, while avoiding the appearance of a ghost [416271].


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