The quantum fluctuations produced early in the inflationary phase provide a convenient mechanism to generate initial conditions for cosmological perturbations. Consequently, the nature of the quantum fluctuations are imprinted into the primordial perturbations, eventually dictating the statistical properties of the structures in the universe. In this talk, we will explore non-Bunch-Davies initial conditions for inflation, focusing on how they generate statistical inhomogeneities and anisotropies. The signature of statistical inhomogeneity may be hidden in the observed primordial two-point function and CMBR fluctuations. Furthermore, statistical inhomogeneity generates primordial three-point function even for Gaussian theories, and may be constrained from observed primordial bispectra. Therefore, deviation from statistical homogeneity and isotropy can be used as a probe for the initial condition for inflation. Finally, we will discuss the implication for the ergodic hypothesis when statistical homogeneity is broken.

In the second part of the talk, I will briefly summarize part of my PhD work. A conformal transformation can provide alternative but equivalent descriptions of Einstein's gravity through a wide class of modified theories of gravity. The universes in these equivalent descriptions may evolve in drastically different ways from the standard model of cosmology. Since all these universes are connected via transformations of units, physical observables remain the same across these representations; however, their interpretations change. We will explore implications of such conformally dual universes on observables such as redshift and CMB anisotropies.