Embedding Cosmology and Gravity

TL;DR

This paper explores a novel embedding-based framework for cosmology and gravity, linking symmetry breaking, inflation, and non-commutative spacetime effects within a higher-dimensional sigma model approach.

Contribution

It introduces an embedding scenario with SU(n) symmetry, incorporates spontaneous symmetry breaking, and connects these ideas to inflation and modifications of Einstein's equations.

Findings

Embedding can serve as an inflaton in cosmological models.

The approach relates embedding to the Starobinsky R^2 inflation model.

Inflation influences the non-commutativity of spacetime.

Abstract

I start with a scenario where the universe is an abstract space $\mathcal{M}$ having $d$ dimensions. There is a two dimensional surface embedded in it. Embedding is a map from the embedded surface to $\mathcal{M}$ that has a field theory described by Sigma model. I take $d$ directions of $\mathcal{M}$ 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.