Musings

Let’s expand the potential

near the saddle point, $\varphi =\psi =0$,

where ${\omega }^2={\lambda }_1(m_1^2+m_2^2)$. If the amplitude of the $\varphi$ oscillations is large enough, the anharmonicities of the potential in $\varphi$ might be important, but they would not significantly change the conclusion.

Write

The Hubble expansion damps these oscillations. In an inflating universe, we just get

But — and this is a crucial point — in order for the universe to be inflating, we need the $\frac{{\lambda }_3}{4}{m}_3^4$ term in the potential to dominate the energy density. If the energy stored in the $\varphi$ oscillations dominates, then — instead of $p=-\rho$ and inflation— we get $\mid p\mid \ll \rho$ and ordinary matter-dominated expansion. So, regardless of how you start, the universe starts inflating only when $A_0$ satisfies

In that case, we have

As I said before, we need the oscillations to be underdamped, which means we should also require

Putting these together, we have $A_0^2{H}^2\ll {\lambda }_3{m}_3^4$, or

Following Dvali and Kachru, let’s plug this slowly-decaying simple harmonic motion for $\varphi$, $\varphi =A\cos \omega t$, into the potential and treat it as a time-dependent effective potential for $\psi$ (if the amplitude of the $\psi$ motion is small enough, the back-reaction is negligible).

This is the vertically-forced harmonic oscillator. At first, I was convinced that parametric resonance would destabilize $\psi$ much earlier. But forget about parametric resonance. Let’s just ask: when does ${\lambda }_{2}A(t{)}^2={\lambda }_3{m}_3^2$? Dvali and Kachru want this to happen after $N=50$ e-foldings. That is, $t=N/H$. This comes down to

Using $A_0^2{\omega }^2\ll {\lambda }_3{m}_3^4$, this implies

Let’s be as optimistic as possible, setting $m_3=M_P$ and $\omega =1\text{TeV}$. To get $N=50$, we would still need

That’s wildly implausible.

And parametric resonance would only make things worse…

Unfortunately, the result of this little collaboration was frightfully dull (anyone still reading this post?). But it was a lot of fun to see the possibilities inherent in blogging about physics. Konstantin and I never would have hooked up, but for my post. This is why I took up blogging in the first place.

Besides, pre-blog, this depressing little calculation would just have ended up buried in my notebooks, unpublished. Now it’s on the web for all to read.

Some would call that progress.

Update (9/25/2003): Typo in penultimate formula corrected (Thanks, Sonia!).