In the Pre-Big-Bang (PBB) scenario [275] the dilaton evolves from a weakly coupled regime ()
toward a strongly coupled region (
) 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
these branches
are disconnected to each other with the appearance of a curvature singularity. However the
presence of the correction
allows the existence of non-singular solutions that connect two
branches [273, 105
, 147
].
The corrections are the sum of the tree-level
corrections and the quantum
-loop
corrections (
) with the function
given by
, where
(
) are coefficients of
-loop corrections (with
). In the context of the PBB cosmology it
was shown in [105
] 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 (
) [105, 147].
In the context of Ekpyrotic cosmology where a negative potential
is present in the
Einstein frame, it is possible to realize nonsingular solutions by taking into account corrections
similar to
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
in
Eq. (12.57
) [34, 224, 336, 337, 338, 623, 12, 582]. 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 [463]
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 is constrained to
be
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 (12.58
) 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
and hence the GB term
of the coupling
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 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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