Scaling for time and mass

I am not even sure if this question makes sense but I will ask it anyway. If we were to scale the universe down to something roughly the size of an atom, what would be the equivalent to 1 billion years. Not looking for any kind of accuracy but just a representative type figure, such as nano seconds or decades just for an example.

No, in my understanding your question doesn’t make sense. It’s apples and oranges.

ETA: even if it made sense, one part of the equation, the factor for scaling down the universe to atom size is unclear because nobody really knows the size of the universe. It’s bigger than what we can observe, that’s for sure.

I might be wrong, but this kind of extreme scale factor seems to bring us back to the theoretical inflationary era, which would be around 10^{-36} to 10^{-33} seconds after the Big Bang.

Which part of the universe? All of it? It’s probably infinite, so that’s right out. Say it’s the visible universe and we set the diameter at 27 billion light years and we shrink it to an atom, say Hydrogen with a diameter of 1.1 ångström. That’s a factor of 2.32×10^27. But you specified time to mass and mass scales like the cube of distance, so that’s 1.25×10^82. So over 40 orders of magnitude shorter than plank time.

How are we even defining “scaled to the size of an atom”? If the entire Universe is scaled down, then presumably so are all of its contents, including all of the atoms. At no point will the Universe be the size of any of its atoms.

What physical constants, if any, are we keeping constant? We can’t keep them all constant, or we wouldn’t be able to scale anything. Or are we not keeping any of them constant, in which case, what are we doing?

I think OP is trying to ask - for a universe with roughly our universe’s properties, are “size” and “time” dependent parameters? In other words, if I “shrink” our universe, must I “shrink” time to keep the general properties of the universe about the same?

Unfortunately, the answer is that it just doesn’t work that way. Neither “size” nor “time” are defining parameters at all - in part because of a misconception that a universe is something that you can just measure against some external reference. There are about 26 independent parameters that define the properties of a universe, and understanding what those parameters mean and how it all works is highly technical.

Note that all of these parameters are dimensionless. Not anything like defining kilograms or seconds.

When you encounter the question of “how much,” you probably think of the force of gravity being determined by a universal gravitational constant, G, and of the “energy of a particle” being determined by its rest mass, such as the mass of an electron, me. You think of the speed of light, c, and for quantum mechanics, Planck’s constant, ħ. But physicists don’t like to use these constants when we describe the Universe, because these constants have arbitrary dimensions and units to them.

But there’s no inherent importance to a unit like a meter, a kilogram or a second; in fact there’s no reason at all to force ourselves to define things like “mass” or “time” or “distance” when it comes to the Universe. If we give the right dimensionless constants (without meters, kilograms, seconds or any other “dimensions” in them) that describe the Universe, we should naturally get out our Universe itself.

So (as @Chronos pointed out):

is not a meaningful thing to suggest for a universe.

That is a beautiful link, thank you!

A beautiful link indeed. I would just add, though, that while new physics might, as they say, result in more constants being needed, it might also result in fewer. For instance, the weak coupling constant, the mass of the W boson, and the mass of the Z boson were once considered to be three different parameters, until a relationship was found between them and they’re now considered to be only two. And we can hope that future developments might uncover other, similar relationships between the various constants.

I wasn’t actually thinking about time like a clock, I was thinking more in terms of decay and the kind of things that would happen inside that time frame that are typical of a universe.

What’s the distinction between those things?

I can’t make a complete distinction because the clock would be my frame of reference.

Yeah, those constants are independent, as far as we can tell with our current knowledge. A substantial amount of effort in physics is in trying to find relationships between those constants.

It may be that one day we can derive all the constants from any other, or we may find that it is impossible to do so, or we may never actually know one way or another.

A clock in physics terms is anything that changes regularly with time. So, atomic decay would be a clock (even though on the individual particle level, it’s random).

One simple interpretation of the OP is: What is (1 billion years) * (radius of [observable] universe) / (radius of atom)? There isn’t any physics in that scaling, but it’s sort of like those “What if all of earth’s history were inside of a year?” types of “visualization” scalings. Although in this case, it doesn’t produce any intuitive time scales for thinking about. The above gives 10-21 s, or a thousandth of a millionth of a millionth of a millionth of a second.

You general point is, of course, a good one, but this particular example is historically incorrect. While something like the W boson was postulated for some time, the Z boson entered our story of physics through the development of electroweak theory first, and thus this boson was born with all the interesting electroweak interplay (couplings, masses, etc.) already present. In fact, it was a startling prediction of the theory that such a boson should be out there. Z-mediated neutral current interactions were subsequently detected in 1973, and the Z boson itself was detected in 1983, confirming the electroweak predictions and raining Nobel Prizes down.

In other words, there was never a period of time when those three parameters were discussed yet didn’t have an understood relationship.

Oops. Obviously intended:
(1 billion years) * (radius of atom) / (radius of [observable] universe)

Although your other formula was interesting too: what would happen if we scaled the universe up to where one new-scale hydrogen atom became the size of the current observable universe?

Sounds like a “Big Rip” scenario; however, it is not clear that this can happen in the real universe: even though space and time themselves are expanding, objects like atoms and solar systems remain bound.