Why is the universe described in terms of Euclidean space and not Minkowski spacetime? The universe is described as an infinite Euclidean space in cosmology. Why isn't it treated as Minkowski spacetime?
 A: In standard cosmology, the spatial part of the universe is described by a flat Euclidean space.  The enveloping spacetime is described by a robertson-walker spacetime (plus small perturbations).  What on Earth is a Robertson-Walker spacetime?
To do this, I'm going to draw an analogy:
Well, imagine that you're constructing a 3-D shape, and you want to build it out of unit circles.  If you're building a cylinder, all you need to do is define the radius of your cylinder as $R$, expand out your cylinder to the proper radius by multiplying lengths on your unit circle by $R$, and then stack the circles on top of each other, and blam!  There is a cylinder.
What if you instead wanted to make a cone?  Well, then you know that the radius of the circle at any height $h$ is given by $R(h)=R_{b}\left(1-\frac{h}{H}\right)$, where $H$ is the height of the cylinder, and $R_{b}$ is the radius of the base.  Then, to construct your cone, you merely need to stack the circles of the appropriate radius on top of each other, and there's your cone!
Now, to make a robertson-walker spacetime, you do the same basic thing.  At every constant $t$, you have a 3-D Euclidean space (there are other options, but the observationally correct one is flat space), and then you stack them on top of each other, expanding your distances by an amount $a(t)$ at a given time.  All you have to do is figure out what the function $a(t)$ is, and then you're done.  It turns out that, quite generally, it's a requirement that, sometime in the past, $a(t)$ must take the value of zero, so there's the Big Bang.  You can get a few other quick results with minimal thinking, too, such as how quickly matter should densify in the past.  
But the important thing to note is that the function $a(t)$ changes the geometry pretty radically--you can get a cone instead of a cylinder, or a lot of other shapes.  We need to do real general relativity to figure out what form $a(t)$ takes, and that is a bit beyond the scope of this question.
