The Teleparallel Theory is equivalent to General Relativity, but whereas in

the latter gravity has to do with curvature, in the former gravity is described

by torsion. As is well known, there is in the literature a host of alternative

theories of gravity, among them the so called extended theories, in which

additional terms are added to the action, such as for example in the $f(R)$ and

$f(T)$ gravities, where $R$ is the Ricci scalar and $T$ is the scalar torsion,

respectively. One of the ways to probe alternative gravity is via compact

objects. In fact, there is in the literature a series of papers on compact

objects in $f(R)$ and $f(T)$ gravity. In particular, there are several papers

that consider $f(T) = T + xi T^2$, where $xi$ is a real constant. In this

paper, we generalise such extension considering compact stars in $f (T ) = T +

xi T^beta$ gravity, where $xi$ and $beta$ are real constants and looking

out for the implications in their maximum masses and compactness in comparison

to the General Relativity. Also, we are led to constrain the $beta$ parameter

to positive integers which is a restriction not imposed by cosmology.

## Exploring Compact Objects in Extended Theories of Gravity

In recent years, there has been a surge of interest in alternative theories of gravity that go beyond General Relativity. These extended theories introduce additional terms to the action, offering new ways to describe gravity. One such theory is the Teleparallel Theory, where gravity is described by torsion rather than curvature.

Among the various extended theories, $f(R)$ and $f(T)$ gravities have gained significant attention. These theories involve adding extra terms to the action, involving the Ricci scalar $R$ and scalar torsion $T$, respectively.

A promising avenue for probing alternative gravity theories is through the study of compact objects. Compact stars, in particular, have been extensively explored in the context of $f(R)$ and $f(T)$ gravity. One specific extension that has been investigated is $f(T) = T + xi T^2$, with $xi$ being a real constant.

In this paper, we aim to generalize this extension by considering compact stars in $f(T) = T + xi T^beta$ gravity. Here, $xi$ and $beta$ are real constants that allow us to explore the implications on the maximum masses and compactness of these objects in comparison to General Relativity.

An interesting aspect that arises from our investigation is the restriction imposed on the $beta$ parameter. We find that it must be constrained to positive integers, which is not a restriction enforced by cosmology.

### Roadmap for Future Research:

- Further investigate and refine the generalized extension $f(T) = T + xi T^beta$ gravity theory.
- Explore the implications of different values of $beta$ on the maximum masses and compactness of compact stars in $f(T)$ gravity.
- Compare the results obtained in $f(T)$ gravity with those predicted by General Relativity to identify any deviations.
- Consider the implications of the restricted $beta$ parameter on the overall consistency and validity of the theory.
- Extend the study to other compact objects, such as neutron stars, to gain a more comprehensive understanding of the behavior of $f(T)$ gravity.

Challenges and Opportunities:

While this research presents exciting opportunities to explore alternative theories of gravity and their implications on compact objects, there are several challenges to overcome:

- The complexity of the mathematical formalism involved in $f(T)$ theories requires careful analysis and numerical calculations.
- Validating the predictions of $f(T)$ gravity through observational data from compact objects poses a significant challenge due to the limited availability of precise measurements.
- Ensuring consistency with cosmological observations and constraints while studying compact stars in $f(T)$ gravity is essential to assess the viability of the theory.

In conclusion, the investigation of compact objects in extended theories of gravity, specifically $f(T) = T + xi T^beta$ gravity, offers new avenues for understanding the nature of gravity. By exploring the implications on maximum masses and compactness, we can gain insights into deviations from General Relativity. However, addressing challenges related to mathematical complexity, observational validation, and cosmological consistency will be crucial for advancing our understanding of alternative gravity theories.