## Can I calculate the size of the Universe by converting light years to kilometres? (Intermediate)

My bright teenage son, after considerable calculation, has concluded that the universe is approximately 162 sextillion miles wide. He based his calculation on the basic 186K mi/sec speed of light x the estimated 13.8 billion year age of the universe. When I pointed out that 13.8 billion years of expansion is not the same as 13.8 billion LIGHT years of expansion, he asserted that I was in fundamental error on that point. I don't mind being in error, but do mind that one of us, now, has clearly gone astray in his basic understanding. If it is me, please set me straight!

From the current rate of expansion of the Universe, astronomers infer that the age of the observable Universe is about 13.8 billion years. In other words, if we assume that the Universe has been expanding at a constant rate since the Big Bang, then the rate of expansion tells us how far back in time the expansion started, which we take to be the beginning of the Universe. If the Universe is 13.8 billion years old, then light has had 13.8 billion years to propagate, and so the statements "13.8 billion years old" and "13.8 billion light years apart" are completely equivalent.

The catch is going from light-years to miles. In the local Universe, we know the conversion, since for all intents and purposes we live in a locally flat, spatially "euclidean" Universe ("euclidean" just means that the three angles of a triangle on a surface add to 180 degrees; this is true for a sheet of paper (which is flat), but not on the surface of a sphere or a saddle (which are both curved)). However, when we look at large distances we have to take the 4-dimensional curvature of the Universe into account. In essence, your son has calculated an accurate "radius" for the observable Universe provided that the Universe is flat (a sort of 4-dimensional sheet in spacetime in which light travels in straight lines), and that the rate of expansion of the Universe has remained constant.

Today, we think that half of your son's assumptions are right. Observations indicate that the Universe is either flat, or so big that the curvature is negligible. However, there is recent evidence that the rate of expansion of the Universe is increasing with time; that is, galaxies are moving away from each other *faster* today than they were in the past. This means that the observable Universe is *more* than 13.8 billion years old. It also means that the energy density of the Universe at present is dominated by "dark energy", a substance with "negative mass" that pushes the Universe apart rather than pulling it together like regular matter does (sound like science fiction? It still is, for the most part, since scientists don't yet have any idea what dark energy is...). The presence of dark energy also affects the curvature of the Universe in the past, which then throws off the conversion from light-years to miles. This is perhaps the best reason why cosmologists avoid using actual distances altogether, unless they are trying to figure out precisely what that conversion factor is.

After 13.8 billion years of expansion, is the universe 13.8 or 27.2 billion years "wide"??? My son asserts that because the expansion is one of space rather than matter, its total dimension = its time of expansion. This logic escapes me. If is is "expanding," surely it is doing so in all directions at once, thus yielding, to my (admittedly fallible) logic the necessity of its "furthest limits" moving diametrically away from each other. I.e., being two years separated in one year's expansion. Am I confusing time and distance here?

Note that in the above paragraphs I have been careful to use the term "observable Universe" rather than Universe. The Universe itself, or the maximum amount of space that we will eventually be able to see given an infinite amount of time, may well be infinite. In quoting a size of the Universe we infer how far we can see in one direction (13.8 billion light years), and how far we can see in the other direction (13.8 billion light years) and add the two to get a size (27.2 billion light years). An age of 13.8 billion light years in each direction therefore leads us to infer that we are at the centre of a sphere with radius 13.8 billion light-years, and hence that the Universe is 27.2 billion light-years "across". The trick, however, is that because the Universe is homogeneous and isotropic, every observer must measure a size of the Universe that is 27.2 billion light years... even ones that are at the "edge" of our observable Universe! This means that either the Universe is sufficiently curved that space doubles back on itself (like on the surface of a sphere), or that the actual Universe is much larger than the observable one. We currently think that the latter possibility is the case.