Cosmological implications of the diffeomorphism field

Gravitational dynamics are classically governed by Einstein’s theory of general relativity, which describes how spacetime warps according to matter and energy distributed throughout the universe. At the heart of general relativity is an object called the metric, which determines distances in spacetime. General relativity tells the metric how to change throughout spacetime. Of particular interest to us is general relativity in four dimensions, as it seems our universe has one temporal direction and three spatial directions. Despite this, we can consider the theory of general relativity in two dimensions. Here, the theory cannot tell the metric how it should behave, but this can be altered by considering the theory quantum mechanically. By doing so, we can find a theory of two-dimensional quantum gravity that includes an extra object called the diffeomorphism field.

Fermions are a specific type of quantum mechanical particle, and the most familiar example of a fermion is likely the electron. If we consider a theory that describes a so-called gauge field interacting with fermions, we find in two-dimensions that the gauge field appears analogously to how the diffeomorphism field showed up in two-dimensional gravity. The only difference here is that we have a well-known theory that describes how this gauge field changes, while we have no similar notion for the diffeomorphism field. We recently found a model, built from so-called projective connections, that tells the diffeomorphism field how to behave and allows for its higher-dimensional realization.

In essence, projective connections are understood by considering the path an object takes through space. Visualizing this path, it is clear that the traversal speed does not change the path itself. In spacetime, paths are determined by an object called the connection. The theory of projective connections can then be understood as a theory that considers when two seemingly different connections yield the same path but with possibly different traversal speeds. This theory of projective connections is used to construct a model we call TW Gravity, which tells us how the metric, connection, and diffeomorphism field change.

With TW Gravity, we can elevate the initially two-dimensional diffeomorphism field to four dimensions and understand how it alters our usual understanding of gravitational physics. In this work, we focus on applications to the universe’s largescale structure with a specific interest in the origin of the cosmological constant and an early phase in the universe called inflation. The cosmological constant is a contributor to dark energy, which is the somewhat mysterious object driving space to expand. On the other hand, cosmological inflation is a period of exponential expansion of the universe, which is hypothesized to have occurred in the early universe and have given rise to features we see in the Cosmic Microwave Background radiation.

We show that both of these phenomena are at least partially explained by the introduction of TW Gravity. These applications of TW Gravity to the cosmological constant and inflation take place under certain simplifying assumptions. For this reason, we end by casting TW Gravity in a framework suitable for future studies seeking to understand its application to cosmology.