If the universe is 13.8 billion years old how come the diameter of the observable universe is 93 billion light-years? The diameter should be 27.6 billion light-years, right? The answer is no. We have to consider the expansion of space since the first light.
The “expansion of space” is more like the “stretching of space“. The “space between galaxies is stretching”.
Speed of light in vacuum (universal speed limit) does not change with the stretching of space. The wavelengths are stretched. Wavelengths get longer (red-shifted). But remember, the speed of light in vacuum does not depend on its wavelength. The speed of light in vacuum does not change with the stretching of space.
This creates an interesting dilemma. The spatial distance defined by 13.4 billion years of light travel from the first stars is much smaller than the radius of the observable universe.
We know the current expansion rate* of space but we don’t know much about the expansion rate in the past. The estimates are complicated by many unknowns. Currently the expansion (stretching) rate is accelerating. Universe is currently expanding faster and faster. Was it always like this in the past? Another unknown is the overall curvature of space. Planck satellite measurements indicate that the overall spatial curvature of the universe is very close to zero. This means that in the absence of matter photons travel on a straight line. If the overall curvature is non-zero the estimates for the size of the observable universe would be different.
The best estimate for the diameter of the observable universe is 28.5 gigaparsecs (93 billion light-years). This is estimated by assuming constant expansion rate for space since the first light.
In the image above there is a phase known as the “cosmic inflation”. Do we take “cosmic inflation” into account when we estimate the size of the observable universe? NO. The “cosmic inflation” relates to the size of the universe. The size of the universe has nothing to do with the size of the observable universe.
What is the size of the universe? We don’t know.
What is beyond the observable universe? We don’t know and we will never know.
We have no idea what is happening across the observable universe at this moment either. The light/radiation we observe from the farthest stars were emitted billions of years ago.
PS: we are not at the center of the universe but we are certainly at the center of the observable universe (by definition).
* There is a puzzling situation for the current expansion rate. See this article by NASA as of April 25, 2019.