The Known Universe (Video)

The Friedmann-Lemaitre-RobertsonWalker (FLRW) metric contains a discrete parameter k = {-1, 0, 1}. For k = 1 the spatial surface is a three dimensional sphere. In this case the spacetime consists of a foliation of three dimensional spheres S^3 which expand and depending on the value of the cosmological constant will continue to do so or recollapse. For k = 0 the spatial surface is a flat space R^3, and these foliate out a spacetime. Each point on a particular R^3 is connected to other points on a preceding R^3 by a time “arrow” t^a = N^a + Ne^a. N is the so called lapse function which tells how far to move up to the next R^3 surface, and N^a is a shift function which tells each point how to slide relative to other points. This holds for the k = 1 case as well, but is more interesting in this case. So you have points on a spatial surface which are pushed forwards to the next surface and then slide along some spatial direction on that next surface. In the case of the observable universe all points slide away from each other, which is the expanding universe, and further this expansion accelerates according to a cosmological constant /\ — called Lambda. The k = -1 case is for a universe with a negative curvature, sometimes referred to as the saddle shaped universe. This space and its ensuing spacetime has certain strange properties which make it unlikely. For k = 0 the space might be flat, but this evolution of sliding points to successive spatial surfaces means these flat spaces are embedded in a spacetime with curvature. Here the curvature is associated with the time direction.

The FLRW equations of motions describe the evolution of a region of space with a radius R. If we assume that k = 0 then this differential equation of motion assumes a particularly simple form

R” = c^2/\R/3. ‘ = time derivative d/dt, “ = d^2/dt^2

The cosmological horizon, something which my mention of here cause some umbrage, occurs at a distance r = sqrt(3//\}. So if we assume that the region of dynamic evolution we are considering is the region contained in this volume then /\/3 = 1/R^2. This results in the simple equation R” = c^2/R, which is the same equation for the “gravity” involved with the production of Hawking-Unruh radiation for a stationary observer a distance R from the event horizon. The other spacetimes with k = 1 or -1 deviate from this, and the gravity associated with the expansion of the universe is not commensurate with principles of quantum fields in curved spacetime. It is also the condition which permits quantum holography of quantum fields or strings interacting with black holes. This matter is a very strange topic, and to understand it you need to suspend certain ideas about sequences of events and the uniqueness of trajectories. I am not going to delve into this as any more length, for you have to do as Louis Carroll’s Red Queen advises and think of several impossible things at the same time.

The data does also suggest that the spatial surface of the universe is flat as well. The COBE and WMAP data on the distant CMB indicates no spatial curvature across that vast distance. Further, the data indicate that the cosmological constant has a particular structure according to a vacuum energy density and its pressure. This condition is where the pressure is equal to the negative of the density (with factors of c) .

There is another strange issue, for if spacetime has a constant energy density, called dark energy, then the expansion of the universe should be effectively creating energy by its expansion. The metric has time dependent metric elements, which means there is no Killing isometry which defines energy conservation as a symmetry of the spacetime. Now again this might punch some people’s conceptions of things, but cosmological spacetimes are such that energy can be generated globally and there is no global conservation law of energy. Yet for the flat R^3 while regions might expand the global space does not, and this (with some care I will not discuss here) avoids the problem of creating energy out of nothing.

In the end there are strong theoretical reasons to think k = 0 or space is flat, and the data strongly indicates this is the case as well.

LC