**1. Introduction**

Several open problems in modern physics at both ultraviolet and infrared scales seem to justify the need to enlarge or revise General Relativity (GR). For example, at astrophysical and cosmological scales, in order for observations to agree with the theoretical predictions of GR, it is necessary to assume the existence of the so-called dark matter and dark energy. But, up to now, at a fundamental level, no experimental evidence has been found to prove the existence of such unknown forms of matter and energy. This fact, together with other shortcomings of GR, represents the signal of a possible breakdown in our understanding of gravity; the possibility of developing extended or alternative theories of gravity is then to be seriously taken into account.

In the last thirty years [1,2], many extensions of GR have been actually proposed; among these, *f*(*R*)-gravity certainly remains one of the most direct and simplest [3–7]: it relies on the idea that the gravitational Lagrangian may depend on the Ricci scalar *R* in a more general way than the linear one as it happens in the Einstein-Hilbert action. Recently, *f*(*R*)-gravity has received great interest in view of its successes in accounting for both cosmic speed-up and missing matter at cosmological and astrophysical scales, respectively (see, for example, [8–10]).

At the same time, including the torsion tensor among the geometrical attributes of space-time is another way to extend GR. Cartan was the first to introduce torsion in the geometrical background; after him, Sciama and Kibble embodied it within the framework of Einstein gravity implementing the idea that spin can be source of torsion as energy does for curvature [11–13]. The resulting theory, known as Einstein–Cartan–Sciama–Kibble (ECSK) theory, has been the first generalization of GR trying to take the spin of elementary fields into account, and it still remains one of the most serious attempts in this direction [14–16].

Following this paradigm, *f*(*R*)-gravity with torsion consists in one of the simplest extensions of the ECSK theory, just as purely metric *f*(*R*)-gravity is with respect to GR. The key idea is again that of replacing the Einstein-Hilbert Lagrangian with a non-linear function of the scalar curvature. A remarkable consequence of the non-linearity of the gravitational Lagrangian is that torsion can be non-zero even without the presence of spin, as long as the trace of the matter stress–energy tensor is not constant [17–21]. This is a noticeable difference with respect to ECSK theory, where instead torsion can exist only coupled to spin. It is known that torsion may give rise to singularity-free and accelerated cosmological models [22], and a torsion arising from the non-linearity of the gravitational Lagrangian function could amplify this effects and make them possible even in the absence of spin. This is a feature that makes *f*(*R*)-gravity with torsion interesting enough to be studied in depth.

Of course, in order for any physical theory to be viable, it has to possess an associated initial value problem correctly formulated in such a way that the dynamical evolution is uniquely determined and consistent with causality requirements. More specifically, the following properties have to hold: (i) small perturbations of the initial data have to generate small perturbations in the subsequent dynamics; (ii) changes of the initial data have to preserve the causal structure of the theory. The initial value problem of the theory is well-posed if both these requests are satisfied.

It is well known that GR has a well-posed initial value problem, so resulting in a stable theory with a robust causal structure [23–26]. In order to be considered as a viable extension of the Einstein theory, *f*(*R*)-gravity should also have such a feature.

About this, by taking advantage of the dynamical equivalence with O'Hanlon theories [27], it is easily seen that purely metric *f*(*R*)-gravity possesses a well-posed Cauchy problem [28] regardless of the explicit form of the function *f*(*R*).

As far as the theory with torsion is concerned, the issue is quite simple whenever the trace of the stress–energy tensor is constant: in this circumstance and in the absence of matter spin sources, in fact, the theory is equivalent to GR with or without a cosmological constant, depending on the explicit expression of the function *f*(*R*). For instance, this is what happens in vacuo and in the case of coupling to electromagnetic or Yang–Mills fields. Instead, the coupling to other kinds of matter sources must be discussed carefully case by case. Here, we face the Cauchy problem in the presence of a perfect fluid or a Klein–Gordon scalar field. Making use of some different techniques, such as conformal transformations and dynamic equivalence with scalar-tensor theories, we formulate sufficient conditions to ensure that the related Cauchy problem is well-posed, also showing that there exist *f*(*R*) functions that actually satisfy these requirements. The so-stated conditions can be adopted as a selection rule for viable *f*(*R*)-models with torsion.

Another important mathematical aspect concerning every theory of gravitation is related to the problem of matching different spacetimes like, for instance, joining together the interior with the exterior region of a relativistic stars. The requirements which have to be fulfilled to solder two different spacetimes are commonly known as junction conditions.

In GR, junction conditions have been investigated by different authors, including Lichnerowicz [29,30], Taub [31], Choquet–Bruhat [32] and Israel [33], and the solution of the problem is now very well known. In [34], the reader can find a very clear discussion about the topic.

On the contrary, at least in the authors' knowledge, very few works deal with junction conditions in ECSK theory: an analysis has been performed by Arkuszewski et al. [35], by means of the formalism of tensor–valued differential forms [36–38], while the same topic has been indirectly addressed by Bressange [39] following the same approach as in [34]. Concerning *f*(*R*)-gravity in purely metric formulation, a discussion of junction conditions has been proposed by Deruelle et al. [40] and Senovilla [41].

In this paper, we address the topic within the theory with torsion, analyzing the junction conditions for *f*(*R*)-gravity with torsion. Borrowing arguments and notations from [34], after formulating the junction conditions, we discuss their explicit form in the case of coupling to a Dirac field and a spin fluid. As we shall see, the resulting junction conditions are very similar to those existing in ECSK theory. However, this close similarity is only formal. Indeed, due to the contributions that the non linearity of the gravitational Lagrangian function *f*(*R*) gives to the contortion tensor, the obtained junction conditions are seen to involve also the trace of the stress–energy tensor and its first derivatives evaluated on the separation hypersurface. This is a remarkable difference with respect to the ECSK theory, which translates into conditions also concerning the function *f*(*R*). Therefore, as in the case of the Cauchy problem, the study of the junction conditions can help to distinguish viable from nonviable *f*(*R*)-models with torsion.

The layout of the paper is the following: In Section 2, we illustrate some generalities about *f*(*R*)-gravity with torsion. In Section 3, we address the Cauchy problem. In Section 4, we discuss the junction conditions. Finally, we devote Section 5 to conclusions. Throughout the paper, we use natural units (¯*h* = *c* = 8*πG* = 1).
