Musings

All That’s Old is New Again

Dvali and Kachru have written a paper, in which they try to resuscitate “Old Inflation.” In Old Inflation, the inflaton is caught in a false minimum of the scalar potential, leading to inflation. Eventually, it tunnels through the barrier, and ends up in the true minimum. Unfortunately, the bubbles of true vacuum don’t end up percolating the universe. The alternative, “New Inflation,” dispenses with the false minimum, and simply has the inflaton rolling down the scalar potential towards the true minimum. To achieve enough inflation, it must roll very slowly, which requires the potential to be fine-tuned to be ridiculously flat.

This is a bit distasteful; one would prefer a mechanism which would work for a more robust range of parameters of the potential. Dvali and Kachru’s idea is to have the inflaton dynamically trapped at a saddle-point of the potential. Consider the following potential for two scalar fields $$V(\phi,\psi)= \frac{\lambda_1}{2} \left( \phi^2 (\phi - m_1)^2 +m_2^2 \phi^2\right) +\lambda_2 \phi^2\psi^2 +\frac{\lambda_3}{4} (\psi^2 - m_3^2)^2$$ This potential has a global minimum at $\phi=0$, $\psi=\pm m_3$ and a saddle-point at $\phi=\psi=0$. For a reasonable range of parameters, it has a local minimum at $$\phi = \textstyle{\frac{m_1}{4}}\left(3 + \sqrt{1 - 8m_2^2/m_1^2}\right)\sim m_1,\quad \psi=0$$ We’re particularly interested in the limit $$m_1^2 \gg m_2^2,\quad \lambda_1 m_1^2 \gg \lambda_3 m_3^2$$

The scalar potential, $V(\phi,\psi=0)$. The universe starts in the false vacuum, tunnels through the barrier and starts oscillating about the saddle-point.

We imagine the scalar field starts out in the false vacuum, $\phi\sim m_1,\psi=0$. The universe inflates, but eventually, through bubble nucleation, the scalar field tunnels through the barrier, and $\phi$ starts oscillating about $\phi=0$. So far, this is just like Old Inflation. But $\phi=\psi=0$ is not the true vacuum. It’s actually a saddle-point. $\psi$ would like to roll down the hill towards the true minimum. But, since $\phi$ is oscillating very rapidly (we’ve arranged for the frequency of oscillation to be much greater than the scale given by the negative mass-squared in the $\psi$-direction), $\psi$ is prevented from rolling down the hill by the $\lambda_2 \phi^2 \psi^2$ term in the potential. Thus the universe continues to inflate, and the bubble grows to a size larger than the current horizon.

Eventually, the amplitude of the $\phi$ oscillation is red-shifted away, and $\psi$ rolls down the hill, ending inflation. Or so Dvali and Kachru would have us believe.

My problem with this scenario is that I see no reason to believe that the initial conditions for this second stage of inflation (immediately after the bubble forms) have $\psi=\dot{\psi}=0$. If the initial conditions are not very precisely tuned, you miss the saddle-point, and the scalar field rocks its way down to the bottom of the potential, too rapidly to save you from the evils of Old Inflation. It seems to me that small quantum fluctuations will always ruin the Dvali-Kachru scenario.

Update (9/25/2003): Here’s a more careful (and pessimistic) analysis.