12.4 Gauss–Bonnet gravity coupled to a scalar field

At the end of this section we shall briefly discuss theories with a GB term coupled to a scalar field with the action given in Eq. (12.6View Equation). The scalar coupling with the GB term often appears as higher-order corrections to low-energy, tree-level effective string theory based on toroidal compactifications [275Jump To The Next Citation Point276]. More explicitly the low-energy string effective action in four dimensions is given by
∫ [ ] 4 √ --- −ϕ 1- 1- 2 S = d x − ge 2R + 2(∇ ϕ ) + ℒM + ℒc ⋅⋅⋅ , (12.56 )
where ϕ is a dilaton field that controls the string coupling parameter, 2 ϕ gs = e. The above action is the string frame action in which the dilaton is directly coupled to a scalar curvature, R. The Lagrangian ℒM is that of additional matter fields (fluids, axion, modulus etc.). The Lagrangian ℒ c corresponds to higher-order string corrections including the coupling between the GB term and the dilaton. A possible set of corrections include terms of the form [273Jump To The Next Citation Point105Jump To The Next Citation Point147Jump To The Next Citation Point]
1- ′ [ 4] ℒc = − 2 α λζ(ϕ) c𝒢 + d(∇ ϕ) , (12.57 )
where ′ α is a string expansion parameter and ζ(ϕ ) is a general function of ϕ. The constant λ is an additional parameter which depends on the types of string theories: λ = − 1∕4,− 1∕8, and 0 correspond to bosonic, heterotic, and superstrings, respectively. If we require that the full action agrees with the three-graviton scattering amplitude, the coefficients c and d are fixed to be c = − 1, d = 1, and ζ(ϕ ) = − e− ϕ [425].

In the Pre-Big-Bang (PBB) scenario [275] the dilaton evolves from a weakly coupled regime (gs ≪ 1) toward a strongly coupled region (g ≳ 1 s) during which the Hubble parameter grows in the string frame (superinflation). This superinflation is driven by a kinetic energy of the dilaton field and it is called a PBB branch. There exists another Friedmann branch with a decreasing curvature. If ℒc = 0 these branches are disconnected to each other with the appearance of a curvature singularity. However the presence of the correction ℒc allows the existence of non-singular solutions that connect two branches [273105Jump To The Next Citation Point147Jump To The Next Citation Point].

The corrections ℒc are the sum of the tree-level ′ α corrections and the quantum n-loop corrections (n = 1,2,3,⋅⋅⋅) with the function ζ(ϕ ) given by ∑ (n−1)ϕ ζ (ϕ ) = − n=0 Cne, where Cn (n ≥ 1) are coefficients of n-loop corrections (with C0 = 1). In the context of the PBB cosmology it was shown in [105Jump To The Next Citation Point] there exist regular cosmological solutions in the presence of tree-level and one-loop corrections, but this is not realistic in that the Hubble rate in Einstein frame continues to increase after the bounce. Nonsingular solutions that connect to a Friedmann branch can be obtained by accounting for the corrections up to two-loop with a negative coefficient (C2 < 0[105147]. In the context of Ekpyrotic cosmology where a negative potential V (ϕ ) is present in the Einstein frame, it is possible to realize nonsingular solutions by taking into account corrections similar to ℒc given above [588]. For a system in which a modulus field is coupled to the GB term, one can also realize regular solutions even without the higher-derivative term 4 (∇ ϕ) in Eq. (12.57View Equation[3422433633733862312582]. These results show that the GB term can play a crucial role to eliminate the curvature singularity.

In the context of dark energy there are some works which studied the effect of the GB term on the late-time cosmic acceleration. A simple model that can give rise to cosmic acceleration is provided by the action [463Jump To The Next Citation Point]

∫ √ ---[ 1 1 ] S = d4x − g -R − -(∇ ϕ)2 − V (ϕ) − f(ϕ)𝒢 + SM , (12.58 ) 2 2
where V (ϕ ) and f(ϕ) are functions of a scalar field ϕ. This can be viewed as the action in the Einstein frame corresponding to the Jordan frame action (12.56View Equation). We note that the conformal transformation gives rise to a coupling between the field ϕ and non-relativistic matter in the Einstein frame. Such a coupling is assumed to be negligibly small at low energy scales, as in the case of the runaway dilaton scenario [274176]. For the exponential potential −λϕ V(ϕ ) = V0e and the coupling f (ϕ ) = (f0∕μ)eμϕ, cosmological dynamics has been extensively studied in [463360Jump To The Next Citation Point361Jump To The Next Citation Point593Jump To The Next Citation Point] (see also [523452453381]). In particular it was found in [360Jump To The Next Citation Point593Jump To The Next Citation Point] that a scaling matter era can be followed by a late-time de Sitter solution which appears due to the presence of the GB term.

Koivisto and Mota [360] placed observational constraints on the above model using the Gold data set of Supernovae Ia together with the CMB shift parameter data of WMAP. The parameter λ is constrained to be 3.5 < λ < 4.5 at the 95% confidence level. In the second paper [361], they included the constraints coming from the BBN, LSS, BAO and solar system data and showed that these data strongly disfavor the GB model discussed above. Moreover, it was shown in [593] that tensor perturbations are subject to negative instabilities in the above model when the GB term dominates the dynamics (see also [290]). Amendola et al. [25] studied local gravity constraints on the model (12.58View Equation) and showed that the energy contribution coming from the GB term needs to be strongly suppressed for consistency with solar-system experiments. This is typically of the order of Ω ≲ 10−30 GB and hence the GB term of the coupling f(ϕ)𝒢 cannot be responsible for the current accelerated expansion of the universe.

In summary the GB gravity with a scalar field coupling allows nonsingular solutions in the high curvature regime, but such a coupling is difficult to be compatible with the cosmic acceleration at low energy scales. Recall that dark energy models based on f(𝒢) gravity also suffers from the UV instability problem. This shows how the presence of the GB term makes it difficult to satisfy all experimental and observational constraints if such a modification is responsible for the late-time acceleration. This property is different from metric f (R) gravity in which viable dark energy models can be constructed.

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