inflationary dynamics

CQW celebrates its first anniversary with a guest post by Shaun Hotchkiss (from the blog trenches of discovery): the amplitude of the power spectrum of primordial fluctuations arising from inflation is proportional to \(1/\dot{\phi}^2\). Therefore, as \(\dot{\phi}\) decreases, the power spectrum increases. However, in the limit of a massless scalar field (i.e. a perfectly flat potential), \(V(\phi)\) is constant and thus so is the energy density. What reconciles this apparent problem?

bonus question: can you show that \(\Delta \phi = 0\) is solved by the Newton-potential and \((\Delta+m^2)\phi = 0\) is solved by the Yukawa-potential (both in 3d spherical coordinates)? What’s the length scale of the Yukawa-potential in this example? why is there no length scale in the Newton-potential? why are both solutions isotropic?