Question

Assume ds²=−c²dt²+a²(t)dx² and that a˙=κa for κ a positive constant. Our universe appears to be moving asymptotically toward such a case (although except with a 3-dimensional space instead of a 1-dimensional space). Determine what we call the "future horizon." If a light signal is sent out at time t1 from x1, in the positive x direction, what value of x2 will it get to, given an infinite amount of time? This difference, x2−x1 is called the future horizon.

Answer 1

In an accelerating universe, there are events which will be unobservable as as signals from future events become redshifted to arbitrarily long wavelengths in the exponentially expanding de Sitter space. This sets a limit on the farthest distance that we can possibly see as measured in units of proper distance today. Or, more precisely, there are events that are spatially separated for a certain frame of reference happening simultaneously with the event occurring right now for which no signal will ever reach us, even though we can observe events that occurred at the same location in space that happened in the distant past. While we will continue to receive signals from this location in space, even if we wait an infinite amount of time, a signal that left from that location today will never reach us. Additionally, the signals coming from that location will have less and less energy and be less and less frequent until the location, for all practical purposes, becomes unobservable. In a universe that is dominated by dark energy which is undergoing an exponential expansion of the scale factor, all objects that are gravitationally unbound with respect to the Milky Way will become unobservable, in a futuristic version of Kapteyn's universe.