f(R) gravity is perhaps the most popular modification of gravity used in cosmology, and it is also explored in various other astrophysical scenarios. The field equations in this type of modification are fourth-order, making them difficult to solve for various reasons that we will explore throughout this course. Due to these difficulties, many in the community have misused the equivalence to scalar-tensor theories. In this course, we will review the basic aspects of deriving the field equations from the action and a method that avoids the pathologies associated with the mapping to scalar-tensor theories. We will apply this method to compact objects and cosmology, demonstrating how it is possible to avoid the numerical issues encountered in cosmology when attempting to start our integration at early times, by using parameterizations of the equation of state associated with geometric dark energy.

As we confront the limitations of GR in explaining some of the universe’s most pressing mysteries —dark energy, dark matter, the Hubble tension, and cosmic acceleration— the need for alternative gravity theories has never been more suggestive for exploration. This course offers an overview of the methods used to construct and critically evaluate some of these alternatives, with Cosmological Generalized Quasitopological Gravity (CGQTG) as a successful working example. We will review the construction of the modified gravity named “Cosmological Generalized Quasitopological Gravity” (CGQTG), a theory with the following properties:

1. It has a direct and smooth continuation to General Relativity (GR) when the modification is off.

2. The only propagating mode is the standard graviton for maximally symmetric spacetimes, avoiding massive gravitons, scalars, or phantom modes.

3. The equations of motion are second order in maximally symmetric spacetimes for black holes, and Friedmann-Lemaitre-Robetson-Walker (FLRW) metric.

4. For cosmology FLRW, the geometric sector is the one that provides the acceleration mechanism for early and late-time accelerations, avoiding the need for an inflation field or a cosmological constant.

5. In the same cosmological scenario, the exponential expansion transits naturally to the GR regime, solving the graceful exit problem.

These are the foundations of building theories that stand up to observational and experimental scrutiny. This is a crucial skill set for any researcher in theoretical physics or cosmology today, as the field increasingly looks beyond GR to solve outstanding issues of our current understanding of the universe. 

Extended or modified gravity theories are notoriously difficult to define and formulate at the theoretical level. Their inherently non-linear nature makes them particularly challenging to implement in N-body cosmological simulations. This talk will provide an overview of recent and past progress in the field, with a focus on numerical simulation perspectives. We will explore the advantages of incorporating beyond-GR theories in simulations, as well as the associated caveats and limitations. By shedding light on these challenges, this talk aims to highlight both the promise and the hurdles of advancing our understanding of the cosmos through modified gravity models.