# Could the spatially flat universe start small?

The global space has been measured flat within a small margin of error. According to FLRW, the flat universe has always been infinite. At the time of the Big Bang the universe was infinitely large with the infinite energy density at every "point". This means that at the beginning any finite "volume", no matter how small, had an infinite total energy.

I understand that "volume" and other descriptive properties cannot be applied directly to the singularity. What I am actually referring to is limits. E.g., the total volume of the observable universe becomes arbitrary small, as we trace it back in time arbitrary close to time zero:

$$\lim_{t\to 0}{V}=0 \tag{1}$$

Obviously, the total energy of the observable universe is not infinite and has never been infinite in its lifetime. This means that the observable universe stated from an infinitely small "volume" (as described above), essentially from a "point".

Please note that this description is different from the naive view that "the Big Bang happened at a point", as explained here:

Did the Big Bang happen at a point?

While the Big Bang did not happen at a point, out observable universe indeed started from a "point" defined by $$(1)$$ above.

I realize that the content of the observable universe changes in time with the space expansion. This however is irrelevant to my question. The only relevant condition is that the energy of the observable universe is always finite, but obviously not constant.

If the observable universe started from a "point" (as defined) in an infinitely large "space", then any other "point" in this "space" is not in our past light cone, is causally disconnected from our observable universe, and cannot influence us in any way other than by contributing to the global space curvature being flat.

If this is correct, then there seems to be no tangible difference between the universe starting infinitely large or infinitely small. If our observable universe started from a "point" in an infinitely large "space" and any other "point" is causally disconnected from us, then why do we need to consider these other "points" as "existing" in the first place? What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?

Is there anything wrong with this line of thinking? Thanks for your expert insight!

• I didn't understand the idea of "infinitely large or infinitely small". If something is infinite, it is infinite. How can you define "small" infinite and "large" infinite ? Or how do you define them? Jan 25, 2019 at 10:08
• @Reign An example of something infinitely large could be the universe. An example of something infinitely small is a geometric point. Does this help? Jan 25, 2019 at 10:13
• Now I think I did not understand.. Jan 25, 2019 at 10:40
• How can universe start from "small" ? Jan 26, 2019 at 6:12
• A flat universe can be finite, safesphere. The story goes that in days of old, people could not conceive of a world that was curved. They could only conceive of a world with an edge. Nowadays we have cosmologists who cannot conceive of a world that is not curved. They cannot conceive of a world with an edge. Jul 30, 2019 at 8:18

When you write "point" in scare quotes, what you're essentially doing is reinventing the notion of boundary constructions. A couple of good surveys on this topic are:

Sanchez, "Causal boundaries and holography on wave type spacetimes," http://arxiv.org/abs/0812.0243

Ashley, "Singularity theorems and the abstract boundary construction," https://digitalcollections.anu.edu.au/handle/1885/46055

The main thing to realize about boundary constructions in GR is that attempts to apply them to general spacetimes have failed. They are very convenient in the context of Penrose diagrams, but we don't have a useful general theory of them.

Your point about the nonfalsifiability of the existence of unobservable regions of spacetime is fine, but it has nothing to do with cosmology. You can take Minkowski space and do silly things like removing a point from it, or removing everything except for a certain region. This has no consequences for an observer whose past light cone avoids the missing points, but it's a silly thing to do, and we have no laws of physics that would help us to decide what the removed parts of the spacetime should be. This is why relativists only usually want to discuss maximal extensions of spacetimes.

• Thanks for your insight, I appreciate it. While removing arbitrary regions from Minkowski spacetime is silly indeed, the meaning of the proposed procedure is to indicate that the regions outside of our light cone that starts at the Big Bang do not exist and never existed in any meaningful sense of existence. This also does away with a rather unphysical idea of the Big Band being infinite in space with an infinite energy at every point. I do realize this procedure is not rigorous, but just an idea that needs a better definition and may even lead to a geometry different from FLRW. Oct 7, 2019 at 22:14

If this is correct, then there seems to be no tangible difference between the universe starting infinitely large or infinitely small

If you assume the universe is infinite, it has to be infinite at any given time. and only at the initial time, there is big bang singularity.

If the universe is infinite it was always infinite. At $$t=10^{-10000000}$$ it was still infinite. It has to be geometrically. But at $$t=0$$, we have a singularity.

You wrote that infinitely small means like a point, But the universe cannot be squeezed into a point, as you know. So even it's infinitely small it's still infinite.

If our observable universe started from a "point" in an infinitely large "space" and any other "point" is causally disconnected from us, then why do we need to consider these other "points" as "existing" in the first place?

Because the universe has to be infinite at any given time. So these points exist by mathematical definition.

What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?

The universe started from a singularity. If its flat, it has to be infinite again. No matter how small it is.

If you mean "Why we cannot think our universe started like as an observable universe" my answer would be this.

1-Universe is the thing that encounters everything. So it still has to start from a singularity. And if its flat it has to be infinite

2- CMBR radiation shows that there is no preferred direction in the universe. Which points out that there cannot be any -away from point type- expansion. So even the observable universe seems to start from a point, It actually did not start from a point. Observable universe has no "real" center, There is just the universe and we have a limit on what we can see.

• Reign the question is not on establishing what the universe is, but how things works in a possible scenario. How to reconcile the idea that space came to existence every where if space wasn't in place at first? There is no inflation that geometrically overcome this, isn't even problem of causality or not. Moreover, if U doesn't need to be infinite at start, this make meaningless the common assertion "big bang" happened everywhere that we all find immensely of help explaining to people or user that big bang is not a common explosion. Fine with that, of course. Just confused about the whole. Jan 25, 2019 at 15:09
• @Alchimista I know but OP cliams as a fact that the universe is flat. But actually thats not the fact, We cannot know the universe is flat or not. Its hard for me to understand "If U doesnt need to be infinite at start " ? whats that mean ? Jan 25, 2019 at 15:32
• Big bang was never a point type explosion. Universe was infinite even it was in small scales. Jan 25, 2019 at 15:35
• It is what you wrote " assumption started by universe being infinite, but it does not have to be infinite.". For the rest not OP nor me are here proposing a classical explosion. That should be clear. At least to me the source of confusion could be "space came to existence". For me something that was always infinite always had space in itself, whatever shrinked to small. Jan 25, 2019 at 16:51
• "OP claims as a fact that the universe is flat" - No, my claim is that the U can be viewed as finite regardless of the space curvature. Infinite regions causally disconnected from us have no physical meaning. Jan 25, 2019 at 22:08

Could the spatially flat universe start small?

As you are mentioning the observational data we have indicate spatial flatness of the universe. Given this is true there are two possibilities regarding its shape.

1. the shape is like a plane. Then the universe was infinite at the big bang
2. the shape is like 3-Torus. Then the universe is finite and was finite at the big bang. Indeed the WMAP data of the CMB seemed to show a signature suggesting this possibility. That wasn't confirmed however by the Plank mission.

In both cases the observable universe was finite at the big bang.

The question is what do we mean if we say big bang? The hot and dense state of the early universe, often called Planck era, or the state before the "slow-roll inflation" started? I tend to prefer the former because the latter is under investigation and not well understood.

What would stop us from simply postulating that the entire universe started flat, but small, while initially coinciding with the observable universe?

Well the flat 3-Torus is still not ruled out even though the CMB doesn't reveal a signature. If the shape of our universe is a 3-Torus which is much much larger than our observable universe then we can't expect such a signature. So it seems we will never know for sure. Cosmologists mainly believe that the universe is infinite because the 3-Torus is a non-trivial solution.
But in case the universe is a 3-torus then the observable universe would be a tiny fraction of it at the big bang (as interpreted above).

• Thanks for pointing out the possibility of a flat space to be finite as a 3-torus. Indeed my question is not about his case. The assumption in my question is, as you've said, "Cosmologists mainly believe that the universe is infinite because the 3-Torus is a non-trivial solution". I argue however that even in the infinite case the universe can be logically assumed infinitely small at the Big Bang, because any part beyond that is causally disconnected from us and can be renderd non existent by a simple additional initial assumption. Thanks for the answer, but it doesn't address my question. Jan 23, 2019 at 14:20
• If you say " infinitely small at the Big Bang" you mean the universe as a whole, not the observable universe, right? If I understand you correctly you don't agree that causal disconnection doesn't prevent things to exist. Why? Also remember that a finite quantity can't grow and become infinite in finite time.
– timm
Jan 24, 2019 at 8:00
• 1. Yes. 2. Non-falsifiable "existence" has no physical meaning. Science is falsifiable. If we cannot communicate with a "parallel universe", then not only it does not exist for us in any meaningful way, but the very question or concept of its existence is meaningless. 3. If we postulate the entire universe to coincide with our observable universe asymptotically at time zero, then the entire universe is not infinite, but is within the initial light cone of the observable universe. Anything "beyond" it is non-falsifiable. Jan 24, 2019 at 8:38
• @timm the situation can be indeed seen as such: causally disconnected regions can exists. The hard to see it is as you can roll back to bring them in contact. Or in more standard mathematics how to bring an existing infinite down to a point as defined in the question. Jan 24, 2019 at 10:17
• Nice mentioning the thorus, by the way. I was thinking of it as a merely geometric possibility and not as something somehow considered in on going research. Jan 24, 2019 at 10:20

A point could not remain a point while retaining some dimension larger than another, although a sphere or cube of nearly infinite smallness might be indistinguishable from a point, without magnification that might require an inaccessible amount of energy.

I believe that some curvature of spacetime is essential for any temporal repetition, including the operation of clocks. One side of any inflexible object following a curved path necessarily covers more distance in the same amount of time than the side closer to the focus of the curve, and curved objects are more adaptable to the wobbling or precession required for such motion to proceed smoothly, so that symmetrically curved objects are much more common in nature than symmetrically rectangular solids, which are usually rendered asymmetric rather quickly by collisions and friction. (No inflexibility can suffice to prevent the distortion of any object by Lorentz contraction during its acceleration to relativistic speeds, although the passage of light around it will cause that distortion to appear as a change in the object's scale or angle of approach, rather than a change in its shape.) Relativity is formulated to reflect natural formations, rather than artificial ones, and the cubic torus analogized to the pac-man game in some literary representations of spacetime is usually represented schematically as the inner surface of an hourglass or doughnut hole, even though the volume it contains may have been mathematically "cubed".

Although the current CMB data shows space to be very nearly flat, its complete flatness would interfere with such interchangeability of space and time as our recognition that we do not see stars as they are now, but more-or-less as those same stars were when whatever light from them that we are seeing left them.

I say "more or less" because even that "special relativistic" effect would not provide for gravitational distortion of the light rays, which might only be taken into account through General Relativity, and would be further complicated by the possibility that some of the objects causing such distortions (such as dark energy, dark matter, or black holes formed by the gravitational collapse of non-binary stars) might themselves remain invisible to us. (The elliptical orbits of the more common stars whose binary partners would've become black holes would indicate the presence of those BHs to us.)