## 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:

## On Gravitational Waves

Since I expect the concept of gravitational waves to once become very popular in the next few days, I wrote some quick notes on them, I.e., where they come from. They are handwritten, as I didn’t have time to LaTeX them, but, hopefully, they’ll be useful to interested readers!

Also note that, gravitational waves are not necessarily evidence of inflation. I wrote a paper a few years ago, describing a anisotropic early universe that had an epoch of plane waves that isotropized to our present-day universe. It can be seen here. It was subsequently published in Physical Review D.

Anyways, here are the notes (Interested readers should see the classic texts by Misner, Thorne, Wheeler, Landau and Lifshitz, or Stephani for more details).

## 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:

## 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.

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.

## Mathematical Origins of Life

The purpose of this post is to demonstrate some very beautiful (I think!) mathematics that arises form Darwinian evolutionary theory. It is a real shame that most courses and discussions dealing with evolution never introduce any type of mathematical formalism which is very strange, since at the most fundamental levels, evolution must also be governed by quantum mechanics and electromagnetism, from which chemistry and biochemistry arise via top-down and bottom-up causation. See this article by George Ellis for more on the role of top-down causation in the universe and the hierarchy of physical matter. Indeed, my personal belief is that if some biologists and evolutionary biologists like Dawkins, Coyne, and others took the time to explain evolution with some modicum of mathematical formalism to properly describe the underlying mechanics instead of using it as an opportunity to attack religious people, the world would be a much better place, and the dialogue between science and religion would be much more smooth and intelligible.

In this post today, I will describe some formalism behind the phenomena of prebiotic evolution. It turns out that there has been a very good book by Claudius Gros and understanding evolution as a complex dynamical system (dynamical systems theory is my main area of research), and the interested reader should check out his book for more details on what follows below.

We can for simplicity consider a quasispecies as a system of macromolecules that have the ability to carry information, and consider the dynamics of the concentrations of the constituent molecules as the following dynamical system:

$\boxed{\dot{x}_{i} = W_{ii}x_{i} + \sum_{j \neq i}W_{ij}x_{j} - x_{i} \phi(t)}$,

where $x_{i}$ are the concentrations of $N$ molecules, $W_{ii}$ is the autocatalytic self-replication rate, and $W_{ij}$ are mutation rates.

From this, we can consider the following catalytic reaction equations:

$\boxed{\dot{x}_i = x_{i} \left(\lambda_{i} + k_i^j x_j - \phi \right)}$,

$\boxed{\phi = x^{k}\left(\lambda_{k} + \kappa_k^j x_j\right) }$,

$x_i$ are the concentrations, $\lambda_i$ are the autocatalytic growth rates, and $\kappa_{ij}$ are the transmolecular catalytic rates. We choose $\phi$ such that

$\boxed{\dot{C} = \sum_i \dot{x}_i = \sum_i x_i \left(\lambda_i + \sum_j \kappa_{ij}x_{j} \right) - C \phi = (1-C)\phi}$.

Clearly:

$\lim_{C \to 1} (1-C)\phi = 0$,

that is, this quick calculation shows that the total concentration $C$ remains constant.

Let us consider now the case of homogeneous interactions such that

$\kappa_{i \neq j} = \kappa$, $\kappa_{ii} = 0$, $\lambda_i = \alpha i$,

$\boxed{\dot{x}_{i} = x_{i} \left(\lambda_i + \kappa \sum_{j \neq i} x_{j} - \phi \right)}$,

which becomes

$\boxed{\dot{x}_i = x_i \left(\lambda_i + \kappa - \kappa x_i - \phi\right)}$.

This is a one-dimensional ODE with the following invariant submanifolds:

$\boxed{x_{i}^* = \frac{\lambda_i + \kappa - \phi}{\kappa}}$,

$\boxed{x_i^* = 0, \quad \lambda_i = N \alpha}$.

With homogeneous interactions, the concentrations with the largest growth rates will dominate, so there exists a $N^*$ such that $1 \leq N^* \leq N$ where

$\boxed{x_i^* = \frac{\lambda_i + \kappa - \phi}{\kappa}, \quad N^* \leq i \leq N}$,

$\boxed{0, \quad 1 \leq i < N^*}$.

The quantities $N^*$ and $\phi$ are determined via normalization conditions that give us a system of equations:

$\boxed{1 = \frac{\alpha}{2\kappa} \left[N(N+1) - N^*(N^* - 1)\right] + \left[\frac{\kappa - \phi}{\kappa}\right] \left(N + 1 - N^*\right)}$,

$\boxed{0 = \frac{\lambda_{N^*-1} + \kappa - \phi}{\kappa} = \frac{\alpha(N^* - 1)}{\kappa} + \frac{\kappa - \phi}{\kappa} }$.

For large $N, N^*$, we obtain the approximation

$\boxed{N - N^* \approx \sqrt{\frac{2 \kappa}{\alpha}}}$,

which is the number of surviving species.

Clearly, this is non-zero for a finite catalytic rate $\kappa$. This shows the formation of a hypercycle of molecules/quasispecies.

These computations clearly should be taken with a grain of salt. As pointed out in several sources, hypercycles describe closed systems, but, life exists in an open system driven by an energy flux. But, the interesting thing is, despite this, the very last calculation shows that there is clear division between molecules $i = N^*, \ldots N$ which can be considered as a type of primordial life-form separated by these molecules belonging to the environment.