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] I believe we can only see 13.7 billion light-years in any direction. Or did they change that when I wasn't looking? That would make for a diameter of roughly 27.4 light-years.

The universe is 13.7 billion years old but that doesn't translate into us being able to see 13.7 billion light years in any direction because the universe has been expanding throughout that time. Instead you need to do a calculation involving a scale factor, a(t), that tells us how much bigger the universe is today compared to some time in the past (i.e. a factor that takes into account the continuing growth of the universe). You'd end up integrating from the beginning of the universe to the present (from t'=0 to t'=t=13.7 billion years) the expression c dt'/ a(t') and multiply it by the scale factor a(t). For a flat universe that scale factor is proportional to t^2/3 so you end up with an expression that looks like D = t^2/3 * c * int(t'^(-2/3)) dt'. That shakes out to be 3ct or 3*13.7 billion light years--3 times what you'd naively expect. A simple calculation like that tells you that we can see over 41 billion light years in every direction in a flat expanding universe (or more, depending on the proportionality constant). This is what the original poster was referring to.

The scale factor used in those calculations comes from a set of very important equations in cosmology, the Friedman equations. The assumption we made there was that we're living in a flat

*matter-dominated*universe which, until about 9 years ago, appeared to be the case All that stuff about the shape of a universe being related to its fate (closed universes with spherical geometry eventually collapsing, open universes with hyperbolic geometries expanding forever, and flat universes slowing their expansions asymptotically to zero as time goes on) is derived under the assumption of a universe where matter is the dominant component. Now, however, we're increasingly confident that some component characterized by negative pressure (called "dark energy," though a cosmologist named Sean Carroll has suggested that "smooth tension" would be a much better name for it) is the dominant element in the universe, which changes things a little.