If you want to understand the planet Earth, then why not go back to the beginning of the Universe? The big bang is an event that we do not understand. It is thought to have happened about 13.75 billion years ago. What occurred, as we understand it, is mind blowing. The entire universe as we know it today seemed to have come out of nowhere and very quickly. This is currently described by the theory of inflation, which estimates that within $1 \cdot 10^{-36}$ and $1 \cdot 10^{-32}$ seconds the universe expanded by a factor of $10^{78}$ in volume.

Where did all this energy come from? One way to account for this energy is offered by the cyclic universe theory that basically says that prior to the big bang, there was another universe that contracted down in a “big crunch,” which then gave rise to the big bang. This process could have occurred over and over, where our universe is just one universe in the process. The cyclic universe theory has been studied by Gott and Lin (1998), Steinhard and Turok in many papers using a string theory formulation, and by many others. In the treatment of the cyclic universe theories, it is an open problem to understand how one universe could smoothly be continued into another, since the differential equations that describe the inflation become undefined (singular) at the big bang itself.

Let $t$ be a variable denoting time, where the big bang occurs at $t = 0\ $. For the universe prior to ours, $t < 0\ $, and for our universe, $t > 0\ $. A recent paper published by this author shows how to smoothly extend one universe into another through the big bang by making use of a special transformation of the variables in the differential equations. These differential equations are called the *Friedmann equations*, and under certain assumptions they can be reduced to a system of ordinary differential equations, which are undefined at the big bang. It has been recently proven in a paper by Belbruno (2013), that a special regularization transformation of the position, velocity, and time variables can be made, where the differential equations are smooth at the big bang and a unique solution to these differential equations can be found from one universe to another. This represents a solution to this problem that had not been previous obtained. The method of using a regularizing transformation was a new approach. This paper is entitled *“On the Regularizability of the Big Bang Singularity.”* The method had never been used in cosmology and previously used mainly in classical celestial mechanics. This methodology was used in a previous paper by Belbruno and Pretorius (2011) on dynamics about a black hole, entitled *“A Dynamical Systems Approach to Schwarzschild Null Geodesics.”*

A particularly intriguing result obtained in this paper is that the unique continuation of one universe into another is possible if and only if a key parameter in the problem, called the equation of state, can be written as a ratio of two integers which are relatively prime. Some new work by this author and BingKan Xue has generalized these results by assuming more physically relevant modeling.

The above image features a painting I did in 2006 of the universe immediately after the big bang. It illustrates the microwave background radiation of the universe, inspired by the Wilkinson Anisotropy Probe data. The most intense radiation is in red and the least in black-blue. The painting is entitled, Microwave Radiation of the Universe. (oil on canvas, 30″ x 16″, 2006). Please see my art site. .

Edward Belbruno

Department of Astrophysical Sciences

Princeton University