Surely, if the entire universe consists of literally nothing but two individual grains of sand, the concept of an expanding universe is rather meaningless, no? It would manifest itself as nothing more than a lessened force of gravity.
Could we call it a “teeny-weeny pop”?
Clearly, watching these sand grains isn’t going to be Must See TV. They make the Pitch Drop Experiment look like A Day At The Races.
Even if the protons decay, the mass+energy total will remain constant, the center of gravity will remain the same, and they’ll fall together at the same rate. But when they meet it might be more like clouds passing than grains colliding!
This is an unanswered question. I’ve seen serious experts disagree. The simplest formulation is, in a universe with one planet, could it be spinning? If it were, it’d have lower gravity at the equator than at the poles. Sadly, this simplistic formulation doesn’t quite work for technical reasons, but it illustrates the point. Many do believe that with just one planet in the universe, it by definition would not be spinning. It’s the existence of all that other stuff out there that makes a spinning planet spin relative to all that other stuff, and that other stuff actually creates the centrifugal force. (And I’m sure I didn’t state that quite correctly, as CF isn’t a force, but that other stuff bends space to cause the apparent CF. Or something.)
But others disagree, or say that the equations can be interpreted this way but we’re not sure of certain conditions or something like that. Hopefully someone with more knowledge will pipe up and make this clearer. Suffice it to say that your intuition and mine aren’t really up to the challenge here. Ya gots to do the math, and those who can do the math, say that the math either isn’t quite solvable, or that it gives different answers depending on assumptions.
Plus red-shift of any radiation coming from the particles, perhaps?
I’m not saying you’re wrong, but it’s not that simple. In any reference frame, their relative velocity can be broken into a component towards (or away from) each other and a component perpendicular to the line connecting them. The first doesn’t contribute to angular momentum, the second can be removed by changing the frame of reference to remove the velocity of the center of mass. Any remaining rotation can be removed by changing to a rotating frame of reference, But I don’t see you know it’s rotating with nothing else in the universe to compare it to. This is certainly a general relativity question, but I don’t have the math.
But, even in special relativity (or heck, Newtonian), a rotating frame of reference is not invariant, (unlike one moving at constant velocity). So the rotating frame has all kinds of screwy forces (or pseudo-forces or whatever you want to call them) that would prevent the two sand grains from falling directly toward each other.
What is their velocity relative to each other at that time?
Laden or unladen?
Assuming these are spherical sand particles made of SiO[sub]2[/sub], each has a radius of 100 microns*. The speed of one particle relative to the center of mass of the pair at the time of impact would be 6x10[sup]-8[/sup] m/s. If you take them as arbitrarily small, you get arbitrarily large speeds in a classical calculation at the separation goes to zero.
*to enough decimal places that I’ll hazard a guess that the OP started with this size and worked out the mass quoted in the original question. Eh, panache45?
Technical question. Would the values be any different if the two objects were black holes with a mass of 12 billion suns? If so, why?
Yes, quite. In the Newtonian calculation, the gravitational potential energy released during the infall is proportional to the product of the masses of the two objects. The kinetic energy this turns into is proportional to the sum of the masses and the square of the velocity. In the end, then, the final velocity is proportional to the square root of the mass.
Perhaps this question came up by your thinking about how a marble and a cannonball dropped from a height reach the Earth’s surface with the same speed. In that case, the mass of the small object cancels out* and leaves a final velocity that is proportional to the square root of the Earth’s mass, which is naturally the same for the marble and cannonball cases.
- under the good approximation that the Earth is much, much more massive than the small object
What I think actually happens is that the Law of Gravity turns up, says “Only two grains of sand in the whole universe? Five hundred million light-years apart? Bugger it, I’m not clocking on for that” and knocks off early to go down the pub. 
Not gonna be much of a pub if there’s only two grains of sand to build it from…