Embedding cosmology and gravity

I start with a scenario where the universe is an abstract space ℳ having d dimensions. There is a two dimensional surface embedded in it. Embedding is a map from the embedded surface to ℳ that has a field theory described by Sigma model. I take d directions of ℳ to be the generators of a symmetry group SU(n) of the Lagrangian of the embedding. This means embedding has n flavors. Then I introduce spontaneous symmetry breaking in the theory and define the direction along which the symmetry breaking occurs as time. Next I write down the modified Einstein’s equation including the embedding. Then I discuss embedding’s relation to the expansion of the universe. After that I construct an inflationary scenario with embedding as inflaton and discuss its connection to Starobinsky R^2 model. Finally, I discuss the effect of inflation on the non-commutativity of the spacetime.