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We present results of numerical simulations of the Chern-Simons inflation model proposed by Alexander, Marciano, and Spergel. According to this model, inflation begins with a fermion condensate interacting with a gauge field. Crucial to the success of this mechanism is the assumption that the Chern-Simons interaction would drive energy from the initial random spectrum into a narrow band of frequencies at superhorizon scales. In this work, we numerically confirm this expectation. These gauge fields and currents, when combined with the Friedmann equations, were broken into a system of hyperbolic equations and numerically simulated. It was found in our simulation that, by including the effects of the chiral anomaly for the axial vector current, inflation can end satisfactorily after approximately 60 e-folds.

Our understanding of the early universe is based on the phenomenon of cosmic inflation [

Alexander et al. propose a model in which the early universe is dominated by a gauge field that interacts with a fermion current. This interaction results in an equation of state with consistently negative pressure, the condition needed for inflation [

This code attempts to achieve an end to inflation by modeling the Adler-Bell-Jackiw (ABJ) chiral anomaly [

Another goal of this project is to establish the robustness of the model. One criticism of inflation is that specific theories can focus too much on formalism and lack a clear connection to physical processes that actually could have occurred [

In Section

Because scalar field models are widely found and difficult to distinguish from one another, Alexander et al. suggest an alternate model in which a gauge field interacts with fermions in the early universe to produce an effective scalar field that generates inflation [

In this gauge field model, we have an energy density

We will consider the equation of motion of the gauge field in this case. Its action is as follows:

By varying this action with respect to the gauge field, we find the equation of motion. The gauge field’s equation of motion, in terms of Fourier modes, is then [

We model the flow of current with the ABJ anomaly. It was shown by Adler [

The divergence of the axial fermion current in the ABJ anomaly is defined by

The

Because the universe was essentially a plasma before last scattering, we look to plasma physics for an analogy of how our system should behave. In particular, we consider magnetohydrodynamics (MHD), the study of electrically conducting fluids. Here in the Chern-Simons system, the gauge field is considered analogous to the field part of the MHD equations, while the velocity terms are considered constant and the density is free to vary like in a compressible MHD system. When fluids in motion are electrically conducting, currents in the fluid produce magnetic fields that induce forces and thus change the dynamics of the system when it becomes turbulent [

Utilizing the coupled partial differential equations that describe the very early universe could allow us to see the resulting exponential increase and then “leveling off” of the scale factor in agreement with current cosmological models. Additionally, a simulation of this type will allow us to see directly the physical effects of fine-tuning the initial conditions for these coupled differential equations. We can input the equations of motion for the fields and currents that are interacting to generate inflation, along with the Friedmann equations, which describe the expansion of space, in order to create a system similar to the MHD equations. Here the Friedmann equations will allow us to relate the evolving pressure and density of the system to the evolution of the scale factor.

For the numerical calculation, we use natural units but later evaluate the data in terms of SI units so that the results can be easily compared to the established values. In this simulation, we allow the gauge field to vary, so the most relevant equation from [

The initial gauge field was composed of a random (white noise) spectrum. In order to generate the initial gauge field, we used a random number generator to create a random spectrum with amplitude up to the calculated maximum amplitude,

Because the initial units were entered as Planck units, we assumed that the physical grid (horizon) size corresponded to Planck lengths and the timing output could be interpreted as Planck time. The output was analyzed using several tools including ygraph and Pro Fit (

Using the definition that inflation is active during regions where

The scale factor for the inflationary period (log-log scale).

The Hubble parameter in Figure

The Hubble parameter throughout the inflationary period (log-log scale).

The (a) charge density and (b) magnitude of the gauge field throughout inflation (log-log scales).

The power spectrum (Figure

Power spectral density for the gauge field throughout inflation. The vertical axis is the log of the deviation from the mean. The horizontal axis is frequency in Plank units.

The simulation presented here clearly supports much of the hypotheses proposed by Alexander et al.: the evolution equations given drive energy into a narrow band of modes and cause the universe to expand exponentially. Though the theory does not predict an initial start time for inflation, our value of about

The power spectrum in Figure

It is interesting that exponential growth does not occur for horizon sizes that are too small. This could be due to the unavailability of very long wavelength (low frequency) modes for the gauge field. In the future, accurate measurements of these frequencies could allow us to verify that they are, in fact, superhorizon modes. A more detailed study of the effect of grid resolution on the dynamics of the system could lead to some interesting revelations about the role of low and high-frequency modes in the system. By understanding the spectral dependence on the dynamics of the system, we can better understand how our choice of initial conditions affects the system. We can then better determine what initial conditions lead to the universe which we observe today. Other future work on this project will be to incorporate the product of the charge density and temporal part of the gauge field into the energy density to determine if they also have a significant impact on inflation and the dynamics of the system.

The results presented here only work for the grid size and grid resolutions discussed. This speaks to the level of fine tuning necessary in this model to get the required 60 e-folds of inflation. The value of

The authors would like to thank the University of Houston’s Texas Learning and Computing Center for access to and use of the Xanadu computing cluster. They also thank Stephon Alexander and Annie Preston for many useful conversations.