In higher-dimensional theories, a graviton propagating in the bulk can follow
a shorter path, known as a shortcut, compared to a photon propagating in a
$4$-dimensional spacetime. Thus by combining the observations of gravitational
waves and their electromagnetic counterparts, one can gain insights into the
structure and number of extra dimensions. In this paper, we construct a
braneworld model that allows the existence of shortcuts in a
$D(=4+d)$-dimensional spacetime. It has been proven that the equations for
modelling brane cosmology recover the standard Friedmann equations for the late
universe. We derive analytically the graviton and photon horizon radii on the
brane under the low-energy limit. With the event GW170817/GRB 170817A, we find
that the number of extra dimensions has an upper limit of $dleq9$. Because of
the errors in the source redshift and time delay, this upper limit can be
shifted to $dleq4$ and $dleq12$. Considering various astrophysical processes,
the upper limit of $dleq4$ is the most robust.
Conclusions:
- In higher-dimensional theories, a graviton can follow a shorter path compared to a photon in a 4-dimensional spacetime.
- Combining observations of gravitational waves and their electromagnetic counterparts can provide insights into the structure and number of extra dimensions.
- A braneworld model that allows the existence of shortcuts in a D-dimensional spacetime is constructed.
- The equations for modeling brane cosmology recover the standard Friedmann equations for the late universe.
- Under the low-energy limit, the graviton and photon horizon radii on the brane are derived analytically.
- The event GW170817/GRB 170817A suggests an upper limit of d≤9 for the number of extra dimensions.
- Errors in source redshift and time delay can shift the upper limit to d≤4 and d≤12.
- The most robust upper limit based on various astrophysical processes is d≤4.
Future Roadmap:
Based on the conclusions of the study, there are several potential opportunities and challenges that lie ahead in the field of higher-dimensional theories and braneworld models:
Potential Opportunities:
- Further exploration of gravitational waves and their electromagnetic counterparts to gain deeper insights into the structure and number of extra dimensions. This can lead to a better understanding of the fundamental nature of spacetime.
- The development of more sophisticated models for braneworld cosmology, taking into account the shortcuts in higher-dimensional spacetime. This can provide a new framework for explaining the behavior of the universe.
- The utilization of analytical techniques to derive the graviton and photon horizon radii on the brane. This can aid in the development of experimental tests and predictions for future observations.
Potential Challenges:
- Addressing the uncertainties and errors in source redshift and time delay measurements, which can impact the determination of the upper limit for the number of extra dimensions.
- Overcoming technical and computational challenges in constructing and studying braneworld models, as they involve higher-dimensional spacetime and complex gravitational interactions.
- Exploring and understanding the physical implications and consequences of the upper limit of d≤4 for the number of extra dimensions, and how it aligns with other theories and observations.
Overall, the findings of this study open up new avenues for research and exploration in higher-dimensional theories, gravitational waves, and cosmology. The roadmap for the future involves further investigations into extra dimensions, refinement of braneworld models, and experimental tests to confirm or refine the upper limit for the number of extra dimensions.