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 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.
