Initial condition for scalar perturbation in cosmology

Question

I'm learning from this lecture notes https://arxiv.org/abs/0907.5424. On page 58, eq.(192), the author conclude that the initial condition for the field $v$ (Mukhanov variable, introduced on page 56, q.(182)) satisfies the equation, under the condition $k\gg aH$: $$ v_k''+k^2v_k=0. $$ Then he conclude that we should require the vacuum to be the minimum energy state, so $v$ satisfies the initial condition $$ \lim_{\tau\to-\infty}v_k=\frac{e^{-ik\tau}}{\sqrt{2k}}. $$

I don't know how this initial condition comes, and what does this to do with "minimum energy state". Could someone give some explanations?

Answer 1

This is the Bunch-Davies vacuum state. It is de Sitter invariant and corresponds to the zero-particle state as seen by a geodesic (free-falling) observer. In other words, at high momentum, this state is just the Minkowski vacuum which is a physically sensible result: such modes simply do not "feel" any cosmological expansion.

As the vacuum, it is by definition a "minimum energy state".