Einstein’s General Relativity is a beautiful physical theory describing classical gravity in geometrical terms and with an impressive record of empirical verification. However, many attempts to modify it have been made in recent years for a variety of reasons. Among these reasons we can find the non-renormalizability of GR, the hope that dark energy and dark matter can be explained by modified gravity, or just the realization that GR can be at most an effective low-energy theory.

### Higher-order theories of Gravity: *a review*

In our review [Belenchia *et al., *Rept.Prog.Phys. 81 (2018) no.3, 036001], we consider a vast family of gravity theories which are called *Higher Order Theories of gravity.* Essentially these are theories in which additional terms are added to Einstein’s action of GR respecting diffeomorphism invariance. We critically review methods to extract the physical degrees of freedoms of such theories warning the reader on the downfalls of the different methods and collecting a vast amount of material scattered in the extant literature.

### Analogue Gravity

Analogies have always played an important role in physics. Analogue gravity, as the name suggest, is the study of systems (usually condensed matter ones, but not only) which share some feature with gravitational ones. In particular, it has been noted that perturbations (e.g. in certain fluids) can propagate like scalar fields in curved spacetime in an apt regime. Thus, analogues of QFT in curved spacetime are possible to realize in the lab and allow for the investigation of otherwise elusive effects, like Hawking radiation. For a nice review see [Barceló, Liberati, Visser, Living Rev. Relativ. (2011) 14: 3].

While the vast majority of research in analogue gravity has been focused on kinematical analogues of QFT in curved spacetime, some works have tackled the issue of dynamics. A dynamical analogue is a system in which also the backreaction of the analogue-matter-fields on the analogue-geometry is taken into account in the hope to come as close as possible to a GR-like dynamics.

In [Belenchia, Liberati, Mohd, Phys.Rev. D90 (2014) no.10, 104015 ] we find that a relativistic Bose-Einstein condensate (rBEC) can give rise to a dynamical analogue model. In particular, the system allows for the emergence of something similar to Nordstrom scalar gravity with a cosmological constant which is naturally small. While still far away from GR dynamics, this works shows how an emergent paradigm is viable for gravitational theories and represents an example in which from a discrete substratum something similar to a continuum, relativistic, and dynamical spacetime can possibly emerge.