We use galaxy-galaxy lensing data to test general relativity and $f(T)$

gravity at galaxies scales. We consider an exact spherically symmetric solution

of $f(T)$ theory which is obtained from an approximate quadratic correction,

and thus it is expected to hold for every realistic deviation from general

relativity. Quantifying the deviation by a single parameter $Q$, and following

the post-Newtonian approximation, we obtain the corresponding deviation in the

gravitational potential, shear component, and effective surface density (ESD)

profile. We used five stellar mass samples and divided them into blue and red

to test the model dependence on galaxy color, and we modeled ESD profiles using

Navarro-Frenk-White (NFW) profiles. Based on the group catalog from the Sloan

Digital Sky Survey Data Release 7 (SDSS DR7) we finally extract

$Q=2.138^{+0.952}_{-0.516}times 10^{-5},$Mpc$^{-2}$ at $1sigma$ confidence.

This result indicates that $f(T)$ corrections on top of general relativity are

favored. Finally, we apply information criteria, such as the AIC and BIC ones,

and although the dependence of $f(T)$ gravity on the off-center effect implies

that its optimality needs to be carefully studied, our analysis shows that

$f(T)$ gravity is more efficient in fitting the data comparing to general

relativity and $Lambda$CDM paradigm, and thus it offers a challenge to the

latter.

Based on the analysis of galaxy-galaxy lensing data, this study examines the validity of general relativity and $f(T)$ gravity at the scale of galaxies. The $f(T)$ theory is a spherically symmetric solution obtained from a quadratic correction, which is expected to hold for realistic deviations from general relativity. By quantifying the deviation using a parameter $Q$ and employing the post-Newtonian approximation, the study investigates the effects of $f(T)$ theory on the gravitational potential, shear component, and effective surface density profiles.

To test the model’s dependence on galaxy color, the study divides the samples into blue and red categories and models the effective surface density profiles using Navarro-Frenk-White (NFW) profiles. Using the group catalog from the Sloan Digital Sky Survey Data Release 7 (SDSS DR7), the study extracts a value of $Q=2.138^{+0.952}_{-0.516}times 10^{-5},$Mpc$^{-2}$ at sigma$ confidence. This result suggests that $f(T)$ corrections on top of general relativity are preferred.

The study further applies information criteria such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC). While acknowledging that the optimality of $f(T)$ gravity requires careful examination due to its dependence on the off-center effect, the analysis demonstrates that $f(T)$ gravity provides a better fit to the data compared to both general relativity and the $Lambda$CDM paradigm. Therefore, it poses a challenge to the latter.

## Future Roadmap: Challenges and Opportunities

The results of this study open up several avenues for future research in the field of gravity theories and cosmology. Here is a roadmap that outlines potential challenges and opportunities:

### 1. Further Investigation of $f(T)$ Theory

The $f(T)$ theory, with its quadratic correction, has shown promise in explaining deviations from general relativity. However, its optimality needs to be thoroughly examined, especially in terms of the off-center effect. Researchers should conduct in-depth studies to understand the limitations and potential improvements of $f(T)$ gravity.

### 2. Testing the Model across Different Galaxy Types

While this study considered the dependence of $f(T)$ gravity on galaxy color by dividing the samples into blue and red, future research should explore the applicability of the model to other galaxy types as well. Investigating the effects of $f(T)$ gravity on a wider range of galaxies can provide valuable insights into its universality and suitability for different astrophysical environments.

### 3. Refining the Measurement of $Q$ Parameter

Improving the accuracy and precision of the measurement of the $Q$ parameter is crucial for a more robust evaluation of $f(T)$ gravity. Researchers should develop innovative observational techniques and data analysis methods to obtain more precise estimates of this parameter.

### 4. Comparison with Other Gravity Theories

While $f(T)$ gravity has shown advantages over general relativity and $Lambda$CDM paradigm based on the current analysis, it is important to compare it with other alternative gravity theories as well. Investigating how $f(T)$ gravity fares against competing theories can provide a comprehensive understanding of its strengths and weaknesses.

### 5. Incorporating Cosmological Observations

Expanding the scope of research to include cosmological observations can enhance our understanding of how $f(T)$ gravity operates on larger scales. By investigating the compatibility of $f(T)$ theory with cosmological data, researchers can assess its validity in a broader cosmological context.

In conclusion, the results of this study indicate that $f(T)$ gravity provides a more efficient fit to galaxy-galaxy lensing data compared to general relativity and $Lambda$CDM paradigm. However, further investigations are needed to fully understand the limitations and potential applications of $f(T)$ gravity. Through ongoing research and the exploration of new avenues, scientists can continue to push the boundaries of our understanding of gravity and its implications for our universe.