Cosmic bubble and domain wall instabilities I: parametric amplification of linear fluctuations

This is the first paper in a series where we study collisions of nucleated bubbles taking into account the effects of small initial (quantum) fluctuations in a fully 3+1-dimensional setting. In this paper, we consider the evolution of linear fluctuations around highly symmetric though inhomogeneous backgrounds. We demonstrate that a large degree of asymmetry develops over time from tiny fluctuations superposed upon planar and SO(2,1) symmetric backgrounds. These fluctuations arise from zero-point vacuum oscillations, so excluding them by enforcing a spatial symmetry is inconsistent in a quantum treatment. We consider the limit of two colliding planar walls, with fluctuation mode functions characterized by the wavenumber transverse to the collision direction and a longitudinal shape along the collision direction $x$, which we solve for. Initially, the fluctuations obey a linear wave equation with a time- and space-dependent mass $m_{eff}(x,t)$. When the walls collide multiple times, $m_{eff}$ oscillates in time. We use Floquet theory to study the fluctuations and generalize techniques familiar from preheating to the case with many coupled degrees of freedom. This inhomogeneous case has bands of unstable transverse wavenumbers $k_\perp$ with exponentially growing mode functions. From the detailed spatial structure of the mode functions in $x$, we identify both broad and narrow parametric resonance generalizations of the homogeneous $m_{eff}(t)$ case of preheating. The unstable $k_\perp$ modes are longitudinally localized, yet can be described as quasiparticles in the Bogoliubov sense. We define an effective occupation number to show they are created in bursts for the case of well-defined collisions in the background. The transverse-longitudinal coupling accompanying nonlinearity radically breaks this localized particle description, with nonseparable 3D modes arising.