Initial conditions of the Universe: A sign of the sine mode

Abstract

In the standard big bang model the universe starts in a radiation dominated era, where the gravitational perturbations are described by second order differential equations, which will generally have two orthogonal set of solutions. One is the so called {\it growing(cosine)} mode and the other is the {\it decaying(sine)} mode, where the nomenclature is derived from their behaviour on super-horizon(sub-horizon) scales. The decaying mode is qualitatively different to the growing mode of adiabatic perturbations as it evolves with time on \emph{super-horizon} scales. The time dependence of this mode on super-horizon scales is analysed in both the synchronous gauge and the Newtonian gauge to understand the true gauge invariant behaviour of these modes. We then explore constraints on the amplitude of this mode on scales between $k \sim 10^{-5}$ Mpc$^{-1}$ and $k \sim 10^{-1}$ Mpc$^{-1}$ using the temperature and polarization anisotropies from the cosmic microwave background, by computing the Fisher information. Binning the primordial power non-parametrically into 100 bins, we find that the decaying modes are constrained at comparable variance as the growing modes on scales smaller than the horizon today using temperature anisotropies. Adding polrisation data makes the decaying mode more constrained. The decaying mode amplitude is thus constrained by $\sim 1/l$ of the growing mode. On super-horizon scales, the growing mode is poorly constrained, while the decaying mode cannot substantially exceed the scale-invariant amplitude. This interpretation differs substantially from the past literature, where the constraints were quoted in gauge-dependent variables, and resulted in illusionary tight super-horizon decaying mode constraints. The results presented here can generally be used to non-parametrically constrain any model of the early universe.