Quote by ohwilleke
I think the distinction you are coming to is a semantic one. In other words, how the phrase "volume of the universe" defined determines the answer, and I believe that at least two different definitions of that phrase are being used in this case.
I would intuitively define "volume of the universe" operationally as something on the order of: "(1) select two points at which matter or energy arising from the Big Bang are present, which are as distant or more distant from each other than any other two points in the universe; (2) call the magnitude of the distance between them d; and (3) the volume of the universe in the spacelike dimensions is then defined to equal pi*d/6".

But in the FriedmannRobertsonWalker model of a flat or open universe, there would be no upper limit on d. Matter and energy are distributed evenly throughout all of space in these models, so in a flat or open universe, for
any finite distance d you can find two bits of matter/energy which are separated by a distance greater than d (although if the distance is too large there will be no possibility of causal interaction between these points since the big bang).
Quote by ohwilleke
n a conventional Big Bang scenario with radiation emitting in all directions from day one and outpacing everything else, one would expect that d would be approximately equal to 2*c*t, or in speed of light units simply 2t, so long as the universe is not contracting. Hence, in a Big Bang scenario, the 3D volume of the universe, if this definition is adopted, is a function of the time elapsed since the Big Bang (defined as t=0 and hence the volume of the universe overall through point t in four dimensions would be the integral from zero to t of f(t) with respect to t. Hence, this definition would produce a finite 3D volume of the universe at any given time t, and a 4D volume of the universe that is infinite or finite depending on the form of f(t) (which depends on the values you put into the FriedmanRobertsonWalker equation in standard GR). (Of course, one would have to be quite clever in defining "t" in the equations above in a way that makes sense).

I think you are fundamentally misunderstanding how the "conventional Big Bang scenario" (ie the FriedmannRoberstonWalker model) actually works. There is no central position in space where the explosion originated, so the notion of radiation "outpacing everything else" doesn't make senserather, at all finite times matter and energy are evenly distributed throughout all of space, so there is no empty region where radiation hasn't reached yet.