(For an introduction to cosmology, please click here.)

One of my areas of study is the very early universe--in particular, what was happening at about 10-32 second after the Big Bang. At around this time, most cosmologists think that the universe was expanding approximately exponentially. We can write this as 

a = a0 eHt,

where a is the scale factor, H is the Hubble parameter, and t is the time. (For a discussion of the scale factor, please see The Expansion of the Universe section above.) This is the cosmic inflation hypothesis, and it was proposed in the early 1980s to explain the homogeneity and isotropy of the universe; this is another way of saying that it solves the horizon and flatness problems. We study this hypothesis, and particularly how this process may have happened and its cosmological predictions.

If inflation occurred, the majority of the energy density of the universe at that time must have come from a source that was dominated by potential energy. This is similar to the type of energy that a book has, for instance, when it is on a shelf. It has gravitational potential energy, so that, if it is nudged off the shelf, it will fall of its own volition to the ground. As it falls, it will pick up speed. Inflation is similar, except that, in order for inflation to occur, we need the potential energy to be greater than the kinetic energy. We can refer to this as slow-roll, since this is equivalent to saying that this energy changes, or rolls, slowly. We can understand this a bit more precisely from the relationship between the type of energy that dominates the universe and the evolution of the universe. When the universe inflates, the energy should have the form

 (KE - PE)/(KE + PE) < -1/3,

where KE is kinetic energy and PE is potential energy. In order for the left side to be negative, the potential energy must be significantly larger than the kinetic energy. The standard approach is to approximate the kinetic energy as constant and far smaller than the potential energy; this is known as the slow-roll approximation (SRA), and it can be used to approximate solutions to inflationary models even when it is impossible, as in the vast majority of cases, to find analytical (symbolic and exact) solutions. When this approximation is exact, H will be constant and the left-hand side of the above equation will be -1. As an example, an exponential potential energy gives rise to such inflation. 

The slow-roll approximation, however, is not exact for the vast majority of models. In our group, we research novel methods for computing and removing the error associated with the SRA. We have shown [4] that, in one particular model, the SRA produces errors around 10 or 20%. This was accomplished through a reparameterization which allows the model to be solved analytically. This procedure, however, cannot be generalized as it relies upon finding analytical solutions to a nonlinear differential equation, which is impossible or impractical in the vast majority of models--regardless of how it is parameterized. Our current research focuses on novel methods to circumvent this problem, which we discuss in the next section.

To solve inflationary models without the error produced by the SRA, one could solve via the SRA first and then compute the error of this solution. To do this, one needs a way to connect the SRA with the actual inflationary dynamics. In our recent work [5], we derive a differential equation to accomplish this. This gives us a well-defined algorithm that is pictorially represented below.

This algorithm allows us to 

1. algorithmically remove SRA errors for each solution, and therefore to find exact solutions, up to numerical error; and, 

2. qualitatively categorize inflation models by examining their most basic inflationary predictions, thereby predicting which models will produce particularly large errors and which will not. 

These are powerful results. Our algorithm can reduce the errors produced by the SRA by at least an order of magnitude, as we show by applying it to two popular models of inflation--natural and quartic hilltop inflation. We also apply our qualitative method to five inflationary models, including the two aforementioned ones, to examine their pathogenicity (the extent to which they give large errors via the SRA). Ultimately, this means that one can begin with an inflationary potential and solve the model as per the usual SRA techniques; by looking at the most essential predictions of the model as per the SRA, one can determine if the model is pathogenic. If it is, one can then apply our algorithm to produce exact solutions.

Over the past few decades, we have entered in a period that has often been dubbed 'precision cosmology', characterized by vast improvements in the precision of cosmological observations. As this precision continues to improve, we argue that the SRA may not be sufficient to produce theoretical predictions with concomitant precision. Our work allows models to be qualitatively sorted by their pathogenicity, and, for a general pathological model, it allows the SRA errors to be algorithmically removed. 

We are currently interested in applying these methods to modern and popular models, and in examining their applications to particle physics. 

Below we reproduce a research poster highlighting this research in more mathematical detail.