# Mathematicians Attempt to Glimpse Past the Big Bang

## Introduction

About 13.8 billion years ago, the entire cosmos consisted of a tiny, hot, dense ball of energy that suddenly exploded.

That’s how everything began, according to the standard scientific story of the Big Bang, a theory that first took shape in the 1920s. The story has been refined over the decades, most notably in the 1980s, when many cosmologists came to believe that in its first moments, the universe went through a brief period of extraordinarily fast expansion called inflation before settling into a lower gear.

That brief period is thought to have been caused by a peculiar form of high-energy matter that throws gravity into reverse, “inflating” the fabric of the universe exponentially quickly and causing it to grow by a factor of a million billion billion in less than a billionth of a billionth of a billionth of a second. Inflation explains why the universe appears to be so smooth and homogeneous when astronomers examine it at large scales.

But if inflation is responsible for all that can be seen today, that raises the question: What, if anything, came before?

No experiment has yet been devised that can observe what happened before inflation. However, mathematicians can sketch out some possible scenarios. The strategy is to apply Einstein’s general theory of relativity — a theory that equates gravity with the curvature of space-time — as far back into time as it can go.

That’s the hope of three researchers: Ghazal Geshnizjani of the Perimeter Institute, Eric Ling of the University of Copenhagen, and Jerome Quintin of the University of Waterloo. The trio recently published a paper in the *Journal of High Energy Physics* in which, Ling said, “we mathematically showed that there might be a way to see beyond our universe.”

Robert Brandenberger, a physicist at McGill University who was not involved with the study, said the new paper “sets a new standard of rigor for the analysis” of the mathematics of the beginning of time. In some cases, what appears at first to be a singularity — a point in space-time where mathematical descriptions lose their meaning — may in fact be an illusion.

## A Taxonomy of Singularities

The central issue confronting Geshnizjani, Ling and Quintin is whether there is a point prior to inflation at which the laws of gravity break down in a singularity. The simplest example of a mathematical singularity is what happens to the function 1/*x* as *x* approaches zero. The function takes a number *x* as an input, and outputs another number. As *x* gets smaller and smaller, 1/*x* gets larger and larger, approaching infinity. If *x* is zero, the function is no longer well defined: It can’t be relied upon as a description of reality.

Sometimes, however, mathematicians can get around a singularity. For example, consider the prime meridian, which passes through Greenwich, England, at longitude zero. If you had a function of 1/longitude, it would go berserk in Greenwich. But there’s not actually anything physically special about suburban London: You could easily redefine zero longitude to pass through some other place on Earth, and then your function would behave perfectly normally when approaching the Royal Observatory in Greenwich.

Something similar happens at the boundary of mathematical models of black holes. The equations that describe spherical nonrotating black holes, worked out by the physicist Karl Schwarzschild in 1916, have a term whose denominator goes to zero at the event horizon of the black hole — the surface surrounding a black hole beyond which nothing can escape. That led physicists to believe that the event horizon was a physical singularity. But eight years later the astronomer Arthur Eddington showed that if a different set of coordinates is used, the singularity disappears. Like the prime meridian, the event horizon is an illusion: a mathematical artifact called a coordinate singularity, which only arises because of the choice of coordinates.

At a black hole’s center, by contrast, the density and curvature go to infinity in a way that can’t be eliminated by using a different coordinate system. The laws of general relativity start spewing out gibberish. This is called a curvature singularity. It implies that something is taking place that’s beyond the ability of current physical and mathematical theories to describe.

Geshnizjani, Ling and Quintin studied whether the onset of the Big Bang is more like the center of a black hole, or more like an event horizon. Their investigation builds upon a theorem proved in 2003 by Arvind Borde, Alan Guth (one of the first people to propose the idea of inflation) and Alexander Vilenkin. This theorem, known by the authors’ initials as BGV, says that inflation must have had a beginning — it can’t have been going on ceaselessly into the past. There must have been a singularity to kick things off. BGV establishes the existence of this singularity, without saying what kind of singularity it is.

As Quintin puts it, he and his colleagues have worked to figure out if that singularity is a brick wall — a curvature singularity — or a curtain that can be pulled back — a coordinate singularity. Eric Woolgar, a mathematician at the University of Alberta who was not involved in the study, said that it clarifies our picture of the Big Bang singularity. “They can say whether the curvature is infinite at the initial singularity or whether the singularity is milder, which might allow us to extend our model of the universe to times before the Big Bang.”

To classify possible pre-inflationary scenarios, the three researchers used a parameter called the scale factor that describes how the distance between objects has changed over time as the universe expands. By definition, the Big Bang is the time when the scale factor was zero — everything was squeezed into a dimensionless point.

During inflation, the scale factor increased with exponential speed. Before inflation, the scale factor could have varied in any number of ways. The new paper provides a taxonomy of singularities for different scale-factor scenarios. “We show that under certain conditions, the scale factor will produce a curvature singularity, and under other conditions it does not,” Ling said.

Researchers already knew that in a universe with so-called dark energy, but without matter, the start of inflation identified in the BGV theorem is a coordinate singularity that can be eliminated. But the real universe has matter, of course. Might mathematical tricks also make it possible to get around its singularity? The researchers showed that if the amount of matter is negligible compared to the amount of dark energy, then the singularity can be eliminated. “Light rays can actually go through the boundary,” Quintin said. “And in that sense, you can see beyond the boundary; it’s not like a brick wall.” The universe’s history would extend beyond the Big Bang.

However, cosmologists think that the early universe had more matter than energy. In this case, the new work shows that the BGV singularity would be a real physical curvature singularity, at which the laws of gravity stop making sense.

A singularity hints at the fact that general relativity can’t be a complete description of the basic rules of physics. Efforts to form such a description, which would require reconciling general relativity with quantum mechanics, are ongoing. Ling said he sees the new paper as a steppingstone to such a theory. In order to make sense of the universe at the highest energy levels, he said, “we first need to understand classical physics as well as we can.”