Some ODE and PDE problems arising in cosmology

Cosmology is the study of the structure of our universe on the largest scales which have been observed. The dominant physical force involved is gravity and the appropriate theoretical description is given by general relativity. The basic equations are the Einstein equations coupled to equations describing matter. These form a system of PDE whose character is essentially hyperbolic. Of particular interest from both the physical and mathematical points of view is the spatially homogeneous case where the equations reduce to a system of ODE.

There are various motivations, such as the recent discovery that the cosmological expansion is accelerated, which have led to a proliferation of exotic matter models which go under the name of dark energy. I will explain what is known about these mathematically and sketch the mechanism by which they can lead to homogeneity of the universe. The standard mechanism used to explain the generation of inhomogeneities and hence structures such as galaxies is the Jeans instability and I will present an attempt to put this on a better mathematical footing.