How can we show that the very early universe can be considered to be flat?

Let me confess that it's a homework problem. I was working on a heuristic proof of the following problem.

How can we prove that the very early universe can be considered to be flat.

It was asked in my postgraduate course and general theory of relativity is merely introduced. At, present I am struggling with GTR and don't have any great Idea how to prove this.

If anyone can give a rather simple proof, not extensive one.

• How early is very early? Do you want to prove, or provide evidence? If you want to prove, given which mathematically expressible premises are you proving?
– g s
Nov 28 '21 at 9:59
• By very early I think it means the cosmic microwave background of early universe which provide evidence as to how universe can be considered to flat! Nov 28 '21 at 16:58
• At the moment, and since about 2006, the GTR-based cosmological models, including the "standard" one, have been faced with a barrage of inconsistencies that are convincingly described, by a collaborator of the well-known relativist Lee Smolin, at backreaction.blogspot.com/2021/09/…: That's why I've suggested a heuristic model as an alternative to them, which specifically mentions an initial trajectory outlining a disc-shaped section of space. Discs are flat. Nov 29 '21 at 6:16

This is referring to the flatness problem. Specifically it is referring to the value of the parameter $$\Omega$$, which is the ratio of the density to the critical density. For a positively curved (closed) universe $$\Omega>1$$, for a negatively curved (open) universe $$\Omega<1$$ and for a flat universe $$\Omega=1$$. The question is inviting you to show that in the early universe $$\Omega$$ must have been almost exactly equal to one and therefore almost exactly flat.

From the first Friedmann equation we can derive an expression for $$\Omega$$:

$$\Omega(t) = \frac{1}{1 - \frac{3kc^2}{8\pi G\rho(t) a^2(t)}}$$

where $$\rho(t)$$ is the density of matter/energy in the universe and $$a(t)$$ is the scale factor. Both of these are functions of time so $$\Omega(t)$$ is a function of time. The parameter $$k$$ is a constant that describes the curvature of the universe.

The relationship between the scale factor and the energy density is complicated but if assume that the universe contains only matter then $$\rho \propto 1/a^3$$ and the equation above becomes:

$$\Omega(t) = \frac{1}{1 - Ka(t)}$$

for some constant $$K$$. The scale factor $$a(t)$$ increases with time from the Big Bang, and indeed has increased by many orders of magnitude since the Big Bang. This means that if $$\Omega$$ starts out of order unity then with time it evolves away from unity. If $$K > 0$$ (closed universe) $$\Omega$$ increases with time, or if $$K < 0$$ (open universe) $$\Omega$$ decreases with time. If $$K = 0$$ then this is the flat universe and $$\Omega = 1$$ and does not change with time.

The problem is that observation of the universe now, 14 billion or so years after the Big Bang, suggests that the current value of $$\Omega$$ is close to unity. Since $$\Omega$$ evolves away from unity with time, and since the scale factor $$a(t)$$ has changed by many orders of magnitude in the last 14 billion years, this means that a few seconds after the Big Bang the value of $$\Omega$$ must have been extraordinarily close to unity.

How can we prove that the very early universe can be considered to be flat.

The short answer is we can't prove that.

Given the theory of inflation is correct then one can show (see "Flatness problem" linked by @John Rennie) that after the increase of the scale factor by many orders during that period the value of $$\Omega$$ (the ratio of actual to critical density) comes very close to one in the very early universe after inflation has ended.

So the answer to your question depends on what you mean saying "considered to be flat". If you mean euclidean flatness then no, that can't be considered because euclidean flatness requires $$\Omega=1$$.

If the theory of inflation holds then it follows that in the very early universe the value of $$\Omega$$ was very close to $$1$$ meaning that the spatial geometry of the universe was very close to euclidean flatness which doesn't disprove though that the shape of the universe is large 3-sphere.

As you used only verbiage in your question, I think the most heuristic (basically understandable) answer's offered by Einstein-Cartan Theory, which was developed by them in 1929, after the fact that subatomic particles have intrinsic angular momentum (nicknamed "spin", but more complex than the spin of macroscopic objects) had been discovered.

In the cosmological model using ECT, the formation of a black hole occurs in the gravitational collapse of any large rotating star, after expenditure of its nuclear fuel leaves it without radiation pressure sufficient to resist that collapse, which proceeds outward from the star's center, materializing virtual particles by separating them from their paired antiparticles. In ECT (unlike 1915's General Relativity), the fermions (matter particles) have a tiny spatial extent, and their interaction with the stellar fermions (which are exponentially larger) "spins" the newly-materialized particles outward to form a new "local universe", analogous in shape to the skin of a basketball, with the orbit of the first of them outlining the shape of a disc, whose rotation ("inherited" from the parenting star) combines with rapidly successive occurrences of the same process to form that new, closed universe. The causal separation of its particles from those in the local universe that had contained the parenting star occurs when they reach that speed of light which had prevailed in that earlier LU.

Repeated within each local universe, the eventual result is a "multiverse" of local universes (including our own) on sequentially decreasing scales. Because spatial expansion is not the same as the motion of particles relative to each other, it can occur at a rate much faster than the speed of light, so that this model provides the simplest explanation for the possibility that the universe is infinite and filled with stars, even though most of the night sky is black.

The model is described by its originator, the relativist Nikodem Poplawski, in many articles, posted on Cornell University's Arxiv website between 2010 and 2021, that can be found by his name.

Although it might appear remarkably simple, this model depends on math by Cartan, who had (in 1913) introduced the spinor to geometry. Einstein-Cartan Theory is more complex than General Relativity, and, perhaps partly for that reason and partly because the cosmological model developed through the use of it is potentially eternal to the past as well as to the future, it's not taught in many of the private and state universities in regions where a "creation event" is favored, perhaps for cultural reasons amplified by the political reasons which opportunistically derive from them.

• In an Oct. 2020 report of his attendance at a physics conference in Lima, Peru, Poplawski was described as having explained how torsion, apparently thru Dirac's "four-fermion spin-spin interaction", discretizes space thru the contact between ECT's spatially-extended fermions, which establishes a minimum spatial unit (within the "bouncing" cosmology I've sketched) that cannot be duplicated thru GR's interpretation of fermions as "point-like". This "discretization" of space forms the basis for the long-sought link between relativity and quantum mechanics. Dec 6 '21 at 20:28