So far we have studied modification to the Einstein–Hilbert action via the introduction of a general function of the Ricci scalar. Among the possible modifications of gravity this may be indeed a very special case. Indeed, one could think of a Lagrangian with all the infinite and possible scalars made out of the Riemann tensor and its derivatives. If one considers such a Lagrangian as a fundamental action for gravity, one usually encounters serious problems in the particle representations of such theories. It is well known that such a modification would introduce extra tensor degrees of freedom [635, 283, 284]. In fact, it is possible to show that these theories in general introduce other particles and that some of them may lead to problems.

For example, besides the graviton, another spin-2 particle typically appears, which however, has a kinetic term opposite in sign with respect to the standard one [572, 67, 302, 465, 153, 303, 99]. The graviton does interact with this new particle, and with all the other standard particles too. The presence of ghosts, implies the existence of particles propagating with negative energy. This, in turn, implies that out of the vacuum a particle (or more than one) and a ghost (or more than one) can appear at the same time without violating energy conservation. This sort of vacuum decay makes each single background unstable, unless one considers some explicit Lorentz-violating cutoff in order to set a typical energy/time scale at which this phenomenon occurs [145, 161].

However, one can treat these higher-order gravity Lagrangians only as effective theories, and consider the free propagating mode only coming from the strongest contribution in the action, the Einstein–Hilbert one, for which all the modes are well behaved. The remaining higher-derivative parts of the Lagrangian can be regarded as corrections at energies below a certain fundamental scale. This scale is usually set to be equal to the Planck scale, but it can be lower, for example, in some models of extra dimensions. This scale cannot be nonetheless equal to the dark energy density today, as otherwise, one would need to consider all these corrections for energies above this scale. This means that one needs to re-sum all these contributions at all times before the dark energy dominance. Another possible approach to dealing with the ghost degrees of freedom consists of using the Euclidean-action path formalism, for which, one can indeed introduce a notion of probability amplitude for these spurious degrees of freedom [294, 162].

The late-time modifications of gravity considered in this review correspond to those in low energy scales. Therefore we have a correction which begins to be important at very low energy scales compared to the Planck mass. In general this means that somehow these corrections cannot be treated any longer as corrections to the background, but they become the dominant contribution. In this case the theory cannot be treated as an effective one, but we need to assume that the form of the Lagrangian is exact, and the theory becomes a fundamental theory for gravity. In this sense these theories are similar to quintessence, that is, a minimally coupled scalar field with a suitable potential. The potential is usually chosen such that its energy scale matches with the dark energy density today. However, for this theory as well, one needs to consider this potential as fundamental, i.e., it does not get quantum corrections that can spoil the form of the potential itself. Still it may not be renormalizable, but so far we do not know any 4-dimensional renormalizable theory of gravity. In this case then, if we introduce a general modification of gravity responsible for the late-time cosmic acceleration, we should prevent this theory from introducing spurious ghost degrees of freedom.

12.1 Lovelock scalar invariants

12.2 Ghosts

12.3 gravity

12.3.1 Cosmology at the background level and viable models

12.3.2 Numerical analysis

12.3.3 Solar system constraints

12.3.4 Ghost conditions in the FLRW background

12.3.5 Viability of gravity in the presence of matter

12.3.6 The speed of propagation in more general modifications of gravity

12.4 Gauss–Bonnet gravity coupled to a scalar field

12.2 Ghosts

12.3 gravity

12.3.1 Cosmology at the background level and viable models

12.3.2 Numerical analysis

12.3.3 Solar system constraints

12.3.4 Ghost conditions in the FLRW background

12.3.5 Viability of gravity in the presence of matter

12.3.6 The speed of propagation in more general modifications of gravity

12.4 Gauss–Bonnet gravity coupled to a scalar field

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