Scalar Field Dynamics in Inflation

[AHSAN MUJTABA]

TL;DR Summary

We know how inflation ends classically in a usual quadratic scalar potential case; ##1/2m^{2}\phi^{2}##., i.e. ##\phi ## starts oscillating towards ##0## magnitude.

I am facing a problem while wanting ##\phi## dynamics in a cubic potential; ##g\phi^{3}##. The equation of motion I get for my case is(this follows from the usual Euler-Lagrange equations for ##\phi## in cosmology--Briefly discussed in Carol's Spacetime Geometry, inflation chapter):,
$$\ddot{\phi}+3\sqrt{\frac{8 \pi G}{3}\Bigg(\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}m^{2}\phi^{2}+g\phi^{3} \Bigg)}\dot{\phi}+\Bigg(m^{2}\phi+3g\phi^{2}\Bigg)=0$$
Take ##G=1##. I tried to plot their phase portrait, but I got errors when plotting the equation's actual solutions for ##m=0.5## and ##g=5##. depicting no solutions. I am using Python. Does that mean for cubic potentials(non-symmetric), inflation might happen at some special initial conditions? I am also attaching phase portraits of cubic and quadratic cases. In phase portrait, the attractor represents the equilibrium position of ##\phi## meaning inflation has ended. If I add a cubic term to potential, then there must be two attractors. What do they represent? I am a bit confused.

 

[member38644]

First of all, it is important to note that the equation of motion you have written is not specific to inflation. It is a general equation of motion for a scalar field in a cubic potential, and can be used in various contexts in cosmology, not just for inflation.

Regarding your specific problem, it is possible that there are some issues with the implementation of your code in Python, which is causing the errors in plotting the solutions. I would recommend double checking your code and making sure that it is correctly implementing the equation of motion.

In terms of the phase portrait, it is true that adding a cubic term to the potential will result in two attractors, as opposed to one in the case of a quadratic potential. These two attractors represent the two possible equilibrium positions for the scalar field. In the case of inflation, the attractor at a higher value of the field corresponds to the inflationary phase, while the attractor at a lower value of the field corresponds to the end of inflation.

It is possible that for certain initial conditions, the scalar field will settle at the lower attractor, indicating the end of inflation. This could happen even with a cubic potential, as long as the initial conditions are such that the field does not roll up to the higher attractor.

Overall, it is important to keep in mind that the dynamics of a scalar field in a potential is a highly non-linear system, and it is not always easy to predict the behavior of the field without numerical simulations. It is possible that for certain initial conditions and parameters, the behavior of the field may not be intuitive, and it is important to carefully analyze the solutions and phase portrait to understand the dynamics.