**1. Introduction**

There are two motivations that lead to the construction of modifications of gravity. The first is purely theoretical, namely, to construct gravitational theories that do not suffer from the renormalizability problems of general relativity and thus are closer to a quantum description [1,2]. The second is cosmological, namely, to construct gravitational theories that at a cosmological framework can describe the early and late accelarating eras [3–7] , as well as to alleviate various observational tensions [8].

There is a rich literature on modified and extended theories of gravity. One may start from the Einstein–Hilbert Lagrangian and add extra terms, resulting in *f*(*R*) gravity [9–11], in *f*(*G*) gravity [12–14], in *f*(*G*, T ) theories [15], in *f*(*P*) gravity [16–18] in Lovelock gravity [19,20], in Weyl gravity [21], in Horndeski/Galileon scalar-tensor theories [22,23], etc. Nevertheless, one can follow a different approach and add new terms to the equivalent torsional formulation of gravity, resulting in *f*(*T*) gravity [24,25], in *f*(*T*, *TG*) gravity [26–28], in *f*(*T*, *B*) gravity [29,30], in scalar-torsion theories [31], etc. Torsional gravity has been proven to exhibit interesting phenomenology, both at the cosmological framework [32–57] and at the level of local, spherically symmetric solutions [58–75].

**Citation:** Asimakis, P.; Saridakis, E.N.; Basilakos, S.; Yesmakhanova, K. Big Bang Nucleosynthesis Constraints on *f*(*T*, *TG*) Gravity. *Universe* **2022**, *8*, 486. https://doi.org/10.3390/ universe8090486

Academic Editor: Luca Fabbri

Received: 25 June 2022 Accepted: 2 September 2022 Published: 14 September 2022

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One crucial test that every modification of gravity should pass that is usually underestimated in the literature is the confrontation with Big Bang Nucleosynthesis (BBN) data [76–80]. Specifically, the amount of modification needed in order to fulfill the late-time cosmological requirements must not at the same time spoil the successes of early-time cosmology, and among them the BBN phase. Hence, whatever are the advantages of a specific modified theory of gravity, if it cannot satisfy the BBN constraints it must be excluded [81–84].

In the present manuscript, we are interested in investigating the BBN epoch in a universe governed by *f*(*T*, *TG*) gravity. In particular, we desire to study various specific models that are known to lead to viable phenomenology and extract constraints on the involved model parameters. The plan of the article is as follows: In Section 2, we briefly present *f*(*T*, *TG*) gravity, extracting the field equations and applying them to a cosmological framework. In Section 3, we summarize the BBN formalism and provide the difference in the freeze-out temperature caused by the extra torsion terms. Then, in Section 4, we investigate five specific *f*(*T*, *TG*) models, confronting them with the observational BBN bounds. Finally, Section 5 is devoted to the Conclusions.
