**The validity of Scalar Fields with an exponential potential in the Swampland era. **

The project can be downloaded from the following link:

This report was submited as my final project of my degree at the University of Nottingham. Working under the supervision of Prof. Edmund Copeland, I replicated the results from the 1997 paper *Exponential potentials and cosmological scaling solutions*__,__ in which Prof. Copeland was one of the authors.

With he development of General Relativity, it was found that one of the main predictions of this theory was that the universe is not static. This expansion was first parametrized by Friedmann, whose equations show that the evolution of the scale factor depends exclusively on what there is inside the universe.

Therefore, we can study how the pressence of matter and radiation increments the speed of expansion. However, there are more possible things inside the universe apart from what we can see. For instance, the presence of a scalar field also stores energy, and so we shall expect it to also increase the speed of the expansion. Thus, in the paper I replicated, the authors studied how an homogeneous Scalar Field with an exponential potential drives the expansion of a flat universe.

In my project, I started by deriving the equations of motion of a scalar field inside the universe. For this, I minimised the Hilbert-Einstein action and solve the Einstein's equation. From there, I just had to solve the resulting equations to find the evolution of the system. However, things weren't so easy, given that the solution was not analytic. Therefore, I used perturbation methods to find the fixed values of the system, and then calculate their stability to find the global atractor (and so the late time solution).

It was found that depending on the slope of the potential, it contributed in different qualitative ways. For instance, it could mimic the dominating component of the moment (and so be identical to matter or radiation), or it could be the dominating source of expansion, creating an acceleration of the universe (=inflation).

Even though I calculated this algebraically, I wanted to test the solutions by running a simulations and finding where the values of the field ended up depending on their initial conditions. In the report it can be found that they agree with the previous results.

After this, I included another potential into my scalar field. Therefore, its energy could be splited into a Kinetic term and two potential terms, being each one an exponential with different slopes. The same process (but harder) to the one described above was used. The interesting thing from this system is that each potential could couple to different solutions of the ones described before. Thus, the field could start mimicking matter during early times, until the second potential gains enough energy to dominate, and so the field begins the inflation. This could be used to explain the current accelerated expansion of our universe.

As before, I used numerical methods to calculate the evolution of the field. Moreover, I computed another simulation in which we could see the percentage of each component throughout time. An example of what I described in the last paragraph can be found in the figure at the right. As I did not incuded this graph on the final report, there are no labels. However, vertical axis represent the percentage, while horizontal represent time evolution (as it is an example, units are not very important).

Moreover, y_1 and y_2 correspond to each exponential potential. The first one would be mimicking matter, while the second one will be the one driving inflation. As we can tell, it leads to the full dominance of the scalar field (as in our Universe).

Finally, I wanted to test the validity of this against the current knowledge that we have from our universe. In the observational side, I used the data received from distant quasars by __ Risaliti et al.__ (even though there is no official consensus on the method they used, I though it could be interesting to check my results with theirs). From this, I found few constraints to the system, giving a wide variety of possible solutions that matched their data.

Moreover, on the theoretical side, I compared these solutions to the Swampland Conjecture. This conjecture heavily constraints the inflationary picture, given that its criteria sets an upper bound on the change of the field's value and a lower bound on the slope of the potential. For instance, if the conjecture was true, it would imply the non existance of a Cosmological Constant. However, in the project I was able to find few solutions that were *still* valid. This is because they have not crossed the upper bound defined before, but in the distant future they will, leading up to the creation of new physics.

In conclusion, in this project I learned about how to work with the Friedmann's equations to get the qualitative evolution of a scalar field and the scale factor. After finding the stability of the fixed points from the system, I studied the validity of each solution as an explanation for the current expansion rate of the universe. This kind of descriptions are very powerful because of their generality, which can be used to find an effective explanation of the very own nature of Dark Energy.