Decay of false vacuum in new inflation

[spaghetti3451]

In new inflation, the time scale for the decay of the false vacuum is controlled by

$- m^{2} = \frac{\partial^{2}V}{\partial\phi^{2}}\bigg|_{\phi=0}.$

This is the negative mass-squared of the scalar field when it is at the top of the hill in the potential diagram. This is an adjustable parameter as far as new inflation is concerned, but $m$ has to be small compared to the Hubble constant or else the model does not lead to enough inflation. So, for parameters that are chosen to make the inflationary model work, the exponential decay of the false vacuum is slower than the exponential expansion.

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1. Why is the time scale for the decay of the false vacuum controlled by $- m^{2} = \frac{\partial^{2}V}{\partial\phi^{2}}\bigg|_{\phi=0}?$

2. Why does $m$ have to be small compared to the Hubble constant or else the model does not lead to enough inflation?

3. Why is the exponential decay of the false vacuum slower than the exponential expansion?

 

[AndyW]

1. The time scale for the decay of the false vacuum is controlled by the mass-squared of the scalar field because it determines the shape of the potential energy curve for the field. In new inflation, the scalar field is initially at the top of the potential energy curve, which is the false vacuum state. As the field rolls down the potential, the false vacuum decays and inflation begins. The rate of this decay is determined by the curvature of the potential energy curve, which is given by the mass-squared of the scalar field.

2. In order for the inflation model to work, the scalar field must roll slowly down the potential energy curve, allowing for a long period of inflation. If the mass-squared of the scalar field is too large, the potential energy curve will be too steep and the field will roll too quickly, leading to a shorter period of inflation. Therefore, in order for the inflation model to produce enough inflation, the mass-squared of the scalar field must be small compared to the Hubble constant.

3. The exponential decay of the false vacuum is slower than the exponential expansion because the mass-squared of the scalar field is small compared to the Hubble constant. This means that the curvature of the potential energy curve is relatively flat, allowing for a slow and gradual decay of the false vacuum. On the other hand, the Hubble constant is much larger, leading to a rapid and exponential expansion of the universe during inflation. Therefore, the decay of the false vacuum is slower compared to the expansion of the universe.