Dynamical Systems in Cosmology Lectures

In the final two lectures of my differential equations class , I discussed how Dynamical Systems theory can be used to understand and describe the dynamics of cosmological solutions to Einstein’s field equations. Videos and lecture notes posted below:

Lecture Notes:





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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! 

New paper published on cosmological singularities

My new paper has now been published in Annalen der Physik, which is a great honour, because 100 years ago, Einstein’s General Theory of Relativity was also published in the same journal.

This paper describes a method by which one is able to determine whether a given spatially flat cosmological model produces finite-time singularities, and also gives some examples of interesting cosmological model configurations. 

The paper can be accessed by clicking the image below: 


The preprint can be accessed here on the arXiv. 

Making a Cosmological Model

What goes into making a cosmological model? Here is a presentation (that was part of my Ph.D. dissertation) that I have reproduced and embedded here to describe what actually goes into the making of a cosmological model. After describing some general properties, I describe specifically a early-universe model that contains a viscous fluid and a magnetic field. 

The background mathematics can be found in this old presentation of mine here:

Equations Published in a Cosmology Textbook

One of my earliest works was deriving equations which themselves were forms of Einstein’s field equations that described the state of the early universe, which may have had dominant viscous effects. I was delighted to learn that these equations were published in Springer’s Handbook of Spacetime Cosmology textbook.

Here is a snapshot of the textbook page citing these equations:

image1

 

New Paper on Stochastic Eternal Inflation

Our new paper was accepted for publication in Physical Review D. The goal of the paper was to calculate the probability that a multiverse could emerge from a more general background spacetime, in this case, Bianchi Type I coupled to a chaotic inflaton potential. Basically, we found that a multiverse being generated from such a scenario has a small probability of occurring. Further, the fine-tuning problem that the multiverse / eternal inflation is supposed to solve doesn’t actually occur, because fine-tuning is still required of the geometry of the background spacetime, the initial conditions, and most importantly, the amount of anisotropy.

prdabstract

The preprint can be read on the arXiv here.

 

On The Acausality of Heat Propagation

In many physics and chemistry courses, one is typically taught that heat propagates according to the heat equation, which is a parabolic partial differential equation:

\boxed{u_t = \alpha u_{xx}},

where \alpha is the thermal diffusivity and is material dependent. Note also, we are considering the one-dimensional case for simplicity.

Now, let f(x-at) be a solution to this problem, which represents a wave travelling at speed a. We get that

\boxed{-a f' = \alpha f''}.

This implies that

\boxed{u(x,t) = -\frac{\alpha c_1}{a} \exp\left[-\frac{a (x-at)}{\alpha}\right] + c_{2}},

where c_{1}, c_{2} are constants determined by appropriate boundary conditions. We can see that as a \to \infty, u(x,t) < \infty! That is, that even under an infinite propagation speed (greater than the speed of light), the solution to the heat equation remains bounded. PDE folks will also say that solutions to the heat equation have characteristics that propagate at an infinite speed. Thus, the heat equation is fundamentally acausal, indeed, all such distribution propagations from Brownian motions to simple diffusions are fundamentally acausal, and violate relativity theory.

Some efforts have been made, and it is still an active area of mathematical physics research to form a relativistic heat conduction theory, see here, for more information.

What we really need are hyperbolic partial differential equations to maintain causality. That is why, Einstein’s field equations, Maxwell’s equations, and the Schrodinger equation are hyperbolic partial differential equations, to maintain causality. This can be seen by considering an analogous methodology to the wave equation in 1-D:

\boxed{u_{tt} = c^2 u_{xx}}.

Now, consider a travelling wave solution as before f(x-at). Substituting this into this wave equation, we obtain that

\boxed{a^2 f'' = c^2 f'' \Rightarrow a^2 = c^2 \Rightarrow a = \pm c}.

That is, all solutions to the wave equation travel at the speed of light, i.e., a = c! Therefore, wave equations are fundamentally causal, and all dynamical laws of nature, must be given in terms of hyperbolic partial differential equations, as to be consistent with Relativity theory.