Where does the observable universe end?

The cosmic microwave background gives an age of about 13.8 billion years, inferred from the pattern of hot and cold spots in the oldest light there is.

Light from an object further away than light could travel in that time has not arrived. That sphere is called the particle horizon, and it is centred on the observer, which means every observer has a different one.

The photons arriving now from the most distant material we can see have been travelling for almost the whole age of the universe. While they travelled, the space between us and their source kept expanding.

So the source is now much further away than the distance the light covered: roughly 46 billion light years rather than 14. The radius of the observable universe is a present-day distance to something we are seeing as it was in the past.

This is also why galaxies can be receding from us faster than light without anything violating relativity. Recession from expansion is not motion through space, and a 2004 paper written specifically to clear up these confusions showed that we routinely observe galaxies whose recession has always exceeded the speed of light.

For its first 380,000 years the universe was hot enough to be opaque: free electrons scattered light, and no photon travelled far. When it cooled enough for atoms to form, light was released and has been travelling ever since.

That release is the microwave background, and it is a surface rather than a place. It is the furthest anyone can see with light of any wavelength, which is why it is the effective particle horizon rather than the true one. The true one lies beyond it, and by how much is hard to say, because it depends on how long inflation lasted, if there was inflation.

Because the expansion is accelerating, the set of objects we could ever receive a signal from is not growing without limit. Galaxies currently visible are crossing a second, closer boundary, the event horizon, beyond which anything they emit from now on will never arrive. Davis and Lineweaver put the galaxies crossing it now at a redshift of about 1.8.

How firm that is depends on whether the acceleration is a constant of nature or a changing field. The DESI collaboration reports that its measurements, combined with the microwave background, prefer a dark energy that changes over time at 3.1 sigma, and at 2.8 to 4.2 sigma with supernovae added, depending on the sample. The sample that gave 4.2 gives 3.2 after recalibration, which its own team calls a weak preference, and George Efstathiou has argued that an offset between supernova samples could bring the most discrepant one into line with a constant. So the question is open in a way it was not a few years ago.

Nothing observed puts a boundary on the universe as a whole. Planck looked for the matched circles a universe that wraps around on itself would print on the microwave sky and, for the shapes it tested, found no compact topology at any scale below the diameter of the last scattering surface. That is a floor under the size, not a measurement of it, and the measurements are equally consistent with infinite. Nor is the floor the same for every shape: later work showed that in all but the simplest ones the shortest loop around the universe can miss us and be two to six or more times shorter than that diameter.

This is a permanent limit rather than a temporary one. A question about what lies beyond every possible signal is not a question that better instruments answer.