Mergers of binary compact objects, accompanied with electromagnetic (EM)

counterparts, offer excellent opportunities to explore varied cosmological

models, since gravitational waves (GW) and EM counterparts always carry the

information of luminosity distance and redshift, respectively. $f(T)$ gravity,

which alters the background evolution and provides a friction term in the

propagation of GW, can be tested by comparing the modified GW luminosity

distance with the EM luminosity distance. Considering the third-generation

gravitational-wave detectors, Einstein Telescope and two Cosmic Explorers, we

simulate a series of GW events of binary neutron stars (BNS) and

neutron-star-black-hole (NSBH) binary with EM counterparts. These simulations

can be used to constrain $f(T)$ gravity (specially the Power-law model

$f(T)=T+alpha(-T)^beta$ in this work) and other cosmological parameters, such

as $beta$ and Hubble constant. In addition, combining simulations with current

observations of type Ia supernovae and baryon acoustic oscillations, we obtain

tighter limitations for $f(T)$ gravity. We find that the estimated precision

significantly improved when all three data sets are combined ($Delta beta

sim 0.03$), compared to analyzing the current observations alone ($Delta

beta sim 0.3$). Simultaneously, the uncertainty of the Hubble constant can be

reduced to approximately $1%$.

Mergers of binary compact objects, such as binary neutron stars and neutron-star-black-hole binaries, with electromagnetic counterparts provide a unique opportunity to explore cosmological models. Gravitational waves and electromagnetic counterparts carry information about the luminosity distance and redshift, respectively, allowing us to test theories such as $f(T)$ gravity.

$f(T)$ gravity modifies the background evolution and introduces a friction term in the propagation of gravitational waves. By comparing the modified gravitational wave luminosity distance with the electromagnetic luminosity distance, we can constrain $f(T)$ gravity and other cosmological parameters such as the Power-law model $f(T)=T+alpha(-T)^beta$ and the Hubble constant.

To investigate $f(T)$ gravity, we can utilize the next-generation gravitational-wave detectors: Einstein Telescope and two Cosmic Explorers. Through simulations of binary neutron stars and neutron-star-black-hole binaries with electromagnetic counterparts, we can obtain constraints on $f(T)$ gravity. Combined with current observations of type Ia supernovae and baryon acoustic oscillations, we can further refine these limitations.

By combining all three data sets (gravitational waves, type Ia supernovae, and baryon acoustic oscillations), we can significantly improve the precision of our estimations for $f(T)$ gravity. The uncertainty in $beta$ decreases from $Delta beta sim 0.3$ when analyzing only current observations, to $Delta beta sim 0.03$ when combining all data sets together. Additionally, the uncertainty in the Hubble constant can be reduced to approximately %$.

## Future Roadmap

### 1. Gather observational data

- Continue observing binary compact object mergers and their electromagnetic counterparts
- Collect data on type Ia supernovae and baryon acoustic oscillations

### 2. Simulate gravitational-wave events

- Create simulations of binary neutron stars and neutron-star-black-hole binaries with electromagnetic counterparts
- Use the simulations to analyze the gravitational wave luminosity distance and compare it to the electromagnetic luminosity distance
- Constrain $f(T)$ gravity and other cosmological parameters

### 3. Combine data sets for tighter constraints

- Combine the simulated gravitational-wave events with the observational data from type Ia supernovae and baryon acoustic oscillations
- Analyze the combined data set to refine the limitations on $f(T)$ gravity

### 4. Evaluate precision improvements

- Assess the precision improvements in estimating $beta$, the Hubble constant, and other cosmological parameters
- Compare the results obtained from analyzing current observations alone to those obtained from combining all three data sets
- Determine the level of uncertainty reduction achieved in each case

### 5. Explore applications and implications

- Analyze the implications of tighter constraints on $f(T)$ gravity and its effects on cosmological models
- Investigate potential applications of $f(T)$ gravity in understanding the nature of dark energy and the expansion of the universe

### 6. Further developments and challenges

- Continued improvements in observational techniques and gravitational-wave detection technology can provide more precise data for future analyses
- Accounting for systematic uncertainties and potential biases in the data sets is crucial for accurate constraints
- Exploring alternative theories and models beyond $f(T)$ gravity that can be tested using similar methodologies