## New Article Published in Journal of Geometry and Physics

Our new article was recently published in The Journal of Geometry and Physics. It is shown that under certain conditions, The Einstein Field Equations have the same form as a fold bifurcation seen in Dynamical Systems theory, showing even a deeper connection between General Relativity and Dynamical Systems theory! (You can click the image below to be taken to the article):

## Lectures on Nonlinear Dynamical Systems

Here is a link to my lectures on nonlinear dynamical systems given at York University during the Winter semester of 2017.

These lectures start off with manifold theory, and end with examples in biology, game theory, and general relativity/cosmology.

## The Possible Initial States of The Universe

Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.

What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter $\gamma$. In particular, some examples are:

• $\gamma = 0$: Vacuum energy
• $\gamma = 4/3$: Radiation
• $\gamma = 2$: Stiff Fluid

Note: Click the image below to access the simulation!

In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:

1. dS: de Sitter universe – Inflationary epoch
2. M: Milne universe
3. F: spatially flat FLRW universe – our present-day universe
4. E: Einstein static universe

Note how by changing the value of $\gamma$ , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of $\gamma$ is F a saddle or stable node.

## Attempts at a General Einstein Equation for an Arbitrary FLRW Cosmology

I tried to derive a general Einstein field equation for an arbitrary FLRW cosmology. That is, one that can handle any of the possible spatial curvatures: hyperbolic, spherical, or flat. Deriving the equation was easy, solving it was not! It ends up being a nonlinear, second-order ODE, with singularities at a=0, which turns out to be the Big Bang singularity, which obviously is of physical significance. Anyways, here’s a log of my notebook, showing the attempts. More to follow!