Curved spacetimes in the lab

We present some new ideas on how to design analogue models of quantum fields living in curved spacetimes using ultra-cold atoms in optical lattices. We discuss various types of static and dynamical curved spacetimes achievable by simple manipulations of the optical setup. Examples presented here contain two-dimensional spaces of positive and negative curvature as well as homogeneous cosmological models and metric waves. Most of them are extendable to three spatial dimensions. We mention some interesting phenomena of quantum field theory in curved spacetimes which might be simulated in such optical lattices loaded with bosonic or fermionic ultra-cold atoms. We also argue that methods of differential geometry can be used, as an alternative mathematical approach, for dealing with realistic inhomogeneous optical lattices.