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

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 generally have two orthogonal sets of solutions. One is the so-called growing (cosine) mode and the other is the decaying (sine) mode, where the nomenclature is derived from their behaviour on superhorizon (subhorizon) scales. In most cosmological analyses it is assumed that only the growing mode is a viable solution, because on very large scales and early times the decaying solution shows singular behavior and the amplitude of the mode is also highly suppressed in many inflationary models. However, physically interesting models do exist that would allow for decaying solutions, such as models in which the Universe today originates from a bounce. Without singling out a specific model, an interesting and valid question is whether a decaying mode can actually result in a sensible cosmology, and withstand current precision cosmological constraints. The decaying mode is qualitatively different to the growing mode of adiabatic perturbations as it evolves with time on superhorizon scales. The time dependence of this mode on superhorizon scales is analyzed in both the synchronous gauge and the Newtonian gauge to understand the true gauge invariant behavior of these modes. We then provide a gauge invariant procedure of normalizing this mode on subhorizon scales. Then we explore constraints on the amplitude of this mode on scales between k similar to 10(-5) Mpc(-1) and k similar to 10(-1) Mpc(-1) using the temperature and polarization anisotropies from the cosmic microwave background, by computing the Fisher information. Binning the primordial power nonparametrically 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 polarization data makes the decaying mode more constrained. The decaying mode amplitude is thus constrained by similar to 1/l of the growing mode. On superhorizon 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 superhorizon decaying mode constraints. The results presented here can generally be used to nonparametrically constrain any model of the early Universe.