We consider a mimetic set up in which the mimetic scalar is coupled to a vector field. It is shown that such a field with a time-like component does not contribute to the background equations and yet produces healthy isocurvature perturbations with respect to ghost and gradient instabilities in spite of the absence of any propagating curvature perturbations at the level of the quadratic action. We then consider a vector field with space-like components which leads to an anisotropic Bianchi universe and show that the ghost and gradient instabilities are absent in the limit of high momenta and that the propagating curvature perturbations have healthy UV behavior.I.

We write down Boltzmann equation for massive particles in a spatially curved FRW universe and solve the approximate line-of-sight solution for evolution of matter density, including the effects of spatial curvature to the first order of approximation. It is shown that memory of early time gravitational potential is affected by presence of spatial curvature. Then we revisit Boltzmann equation for photons in the general FRW background. Using it, we show that how the frequency of oscillations and damping factor (known as Silk damping) changed in presence of spatial curvature. At last, using this modified damping factor in hydrodynamic regime of cosmological perturbations, we find our analytic solution which shows the effects of spatial curvature on growing mode of matter density.

Apart from the assumption that the inflation started at an infinite time in the past, the more realistic initial state of the quantum fluctuations is described by a mixed quantum state imposed at a finite value of the initial time.
One of the most important non-trivial vacua is the $\alpha$-vacuum, which is specified by a momentum cutoff $\Lambda$ \cite{Danielsson:2002kx}. As a consequence, 
the initial condition is imposed at different initial times for the different $k$-modes. This modifies the amplitude of the quantum fluctuations, and thus the corresponding power spectra. 
In this paper, we consider the imprint of the $\alpha$-vacuum state on the power spectrum of scalar perturbations in a generic $f(R)$ gravity by assuming an ultraviolet cutoff $\Lambda$. As a specific model, we consider the Starobinsky model and find the trans-Planckian power spectrum. We find that the leading order corrections to the scalar power spectra in $f(R)$ gravity have an oscillatory behavior as in general relativity \cite{Lim}, and furthermore, the results are in sufficient agreement with the $\Lambda$CDM model.