We study "classical" bouncing and Genesis models in beyond Horndeski theory. We give an example of spatially flat bouncing solution that is non-singular and stable throughout the whole evolution. We also provide an example of stable geodesically complete Genesis with similar features. The model is arranged in such a way that the scalar field driving the cosmological evolution initially behaves like full-fledged beyond Horndeski, whereas at late times it becomes a massless scalar field minimally coupled to gravity.

We study whether it is possible to design a "classical" spatially flat bouncing cosmology or a static, spherically symmetric asymptotically flat Lorentzian wormhole in cubic Galileon theories interacting with an extra scalar field. We show that bouncing models are always plagued with gradient instabilities, while there are always ghosts in wormhole backgrounds. * While, in the cosmological context, the energy density of Galileons can indeed increase in time in a healthy way, constructing a complete bouncing or Genesis cosmology (full evolution from t = −∞ to t = +∞) is a challenge. For example, one can construct a spatially flat bouncing model without pathologies at or near the bounce [21,30], yet, in known examples, the gradient instabilities occur at some later or earlier epoch [22,[24][25][26]. Although these instabilities have been argued to remain under control due to higher derivative terms [31], it would be interesting to design an example of a complete "classical" bouncing cosmological model without ghosts and gradient instabilities.Another potential application of the NEC violation is a putative construction of stable asymptotically flat Lorentzian wormholes. However, previous attempts to design a wormhole supported by Galileon have failed [32,33].It is worth noting that there exist bouncing models with nonzero spatial curvature [34,35]. Likewise, there are Lorentzian wormholes which are not asymptotically flat [35]. These solutions employ scalar fields with fairly conventional kinetic terms that do not violate the NEC. On the contrary, we are interested in spatially flat bouncing cosmologies and asymptotically flat wormholes, which necessarily require NEC violation (hence our interest in Galileons).Recently, two no-go theorems have been proven in the Galileon context [33,36,37]. Both apply to general relativity with the Galileon field and no other matter. One theorem shows that spatially flat bouncing cosmological solutions are always plagued with gradient instabilities [36,37]. The other states that, in cubic Galileon theory, static spherically symmetric Lorentzian wormholes are always plagued with ghosts [33].One might hope that these problems can be overcome by adding extra non-Galileonic matter. This matter, if it satisfies the NEC, must interact with Galileon directly; otherwise, the above theorems remain valid [33,36,37]. The simplest option is to add a scalar field with first-derivative terms in the Lagrangian. This is precisely the system studied in this paper. Somewhat surprisingly, we show that, at least for cubic Galileon, the above theorems are still at work: there are always gradient instabilities about bouncing cosmological solutions, and there always exist ghosts in backgrounds of static spherically symmetric Lorentzian wormholes.Concerning wormholes, our spherically symmetric setting is not general. That is, we do not consider a cross term in a metric characteristic of Newman-Unti-Tamburino (NUT) spacetimes. In view of recent interesting results on NUT wormholes [35,38], this generalizat...

We discuss the approaches by Deffayet et al. (DPSV) and Kobayashi et al. (KYY) to the analysis of linearized scalar perturbations about a spatially flat FLRW background in Horndeski theory. We identify additional, potentially important terms in the DPSV approach. However, these terms vanish upon a judicious gauge choice. We derive a gauge invariant quadratic action for metric and Galileon perturbations in $\mathcal{L}_3$ and $\mathcal{L}_3+\mathcal{L}_4$ theories and show that actions obtained in the DPSV and KYY approaches follow from this gauge invariant action in particular gauges.Comment: This is a journal version of the article. The proof of the statement made in Section 3 has been considerably revised. The paper consists of 13 page