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# A Flat problem

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No not imaging flats but the spatial flatness of the Universe.

In a thread started by @vlaiv he asked why cosmologists took the Universe to be infinite, with all the problems that poses, rather than finite and closed. While the thread discussed many issues especially about infinity it did not touch on why the Universe being close to spatially flat is surprising.

Well why is it surprising?

The main reason is the Big Bang Model based on GR normal cosmological assumption (homogeneous and isotropic) and the current LCDM model derived from the latest measurement which put it close to being flat.

So what? Well the Friedmann equation (derived from the above) tells us that:

|Ω(t) – 1| = |k|/(a^2 H^2)

Where Ω(t) is the total mass energy density, a = the scale factor and H is Hubble’s constant.

k is a constant that defines of the curvature of spacetime:

k = 0 gives a flat Universe, k < 0 Hyperbolic and k > 0 Hyperspherical

Now if k = 0 the Ω(t) = 1 a constant i.e. the Universe is and always has been spatially flat.

This is just amazing why should it be so? It could be that physics says it has to be like that.

No one has shown this to be the case or even got close.  It was/is a major problem for the original Big Bang theory.

If |k| > 0 then why is it so close to 1 now?

The Universe was initially radiation dominated where |(a^2 H^2) α t^1

Then matter dominated where |(a^2 H^2) α t^2/3

So |Ω(t) – 1|) α t or t^2/3 in the past. (It is now thought to be Dark Energy Dominated)

Again, so what? Well if it has grown like t or t^2/3 what was it like in the past?

At the very worst now 0.5 < Ω < 1.5 so at:

decoupling t ~ 10^13 s we need |Ω(t) – 1| <10^-5

and at nucleosynthesis t ~ 1 s we need |Ω(t) – 1| < 10^-18 !

The same issue as before.

This was why cosmic inflation was introduced to help solve this problem, as well as the horizon problem, with all the fun issues it brings!

Regards Andrew

PS Please forgive the indulgence of this post but it helped me get it straight in my mind.

All Data and equations from “An Introduction to Modern Cosmology” by Andrew Liddle.

Edited by andrew s
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11 minutes ago, andrew s said:

If |k| > 0 then why is it so close to 1 now?

I think this has to do with radius of curvature, and bound on spatial extent. I've read somewhere, but I can't remember where now, nor if it is exact number mentioned - that in case of positive curvature we have lower bound on size of universe being at least x200 in size of our observable patch.

In fact, even if we consider finite spatial extent - it would not be reasonable to expect that ratio of size of observable patch to full extent be a low number such as x1000? There is no firm grounds to base our assumption of this ratio, but let's just for argument sake take usual ratios of size that we encounter in universe - it is not unimaginable for this ratio to be order of 10^100. That is seriously huge universe, but still bounded, on the other hand, in terms of human scales - our observable patch is seriously huge

There in lies the problem with flat universe - almost impossible to measure. One can never precisely measure 0 value - there will always be some error in measurement and therefore we can never be certain.

Could you expand a bit and clarify both reasoning and calculations on density in decoupling era, no need to go beyond that (just to avoid issues with inflation) - how does it impact curvature? From what I can gather, and this is at a glance - due to change in scale factor curvature will behave differently?

On separate note - there is in fact 10 different manifolds that are both flat and finite in extend - maybe we could discuss implications of those?

For example - hypertorus - it is flat, it is bounded, it is multiple connected - which means not isotropic on large scales. However in local patch is is pretty isotripic, what would corresponding spatial bounds need to be in order for us not to observe unisotropy?

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55 minutes ago, vlaiv said:

Could you expand a bit and clarify both reasoning and calculations on density in decoupling era, no need to go beyond that (just to avoid issues with inflation) - how does it impact curvature? From what I can gather, and this is at a glance - due to change in scale factor curvature will behave differently?﻿

Very roughly, The current age of the Universe is about  4 x10^17s and if the current density were about 0.5 then at time t = 10^13 and assuming  |Ω(t) – 1|) α t (I.e. a^2*H^2  α t^-1)

gives  Ω(10^13) ~ 0.5 * 10^13/ 4 *10^17 ~10^-5.

The model excludes inflation. If I recall correctly the topology can't change under GR i.e. the k is constant)

On the constraints on topology the Plank 2015 results put constraints on the nature of the topology see https://arxiv.org/abs/1502.01593 There are lots of papers on trying to constrain the topology but they can only provide limits given the size of the observable Universe.

In a similar discussion on PhysicsForms

"That said, I don't think that it's accurate at all to state that "cosmologist tend to believe that the universe is infinite". A more precise statement would be that cosmologists rarely think about whether the universe is finite or infinite because it's not really something that is answerable. Most cosmologists tend to try to stay within the bounds of answerable questions that have an impact on observations we can potentially make."

I was not trying to reopen the the discussion on what the curvature was but rather point out how close to "flat" it must have been in the past (with or without inflation) to be as close as it is to flat today.

Regards Andrew

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8 hours ago, andrew s said:

Well why is it surprising?

I suppose it could be argued that it is not surprising since we are here talking about it at all. A slightly less flat Universe would have collapsed long ago or would have disallowed the formation of galaxies and stars.

What is truly surprising though is that there are a whole bunch of constants at critical values (Omega being just one of them) that conspire to make a habitable Universe. (cf. 'Just Six Numbers' by Martin Rees or 'The Road to Reality' by Roger Penrose)

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25 minutes ago, Tiki said:

I suppose it could be argued that it is not surprising since we are here talking about it at all. A slightly less flat Universe would have collapsed long ago or would have disallowed the formation of galaxies and stars.

What is truly surprising though is that there are a whole bunch of constants at critical values (Omega being just one of them) that conspire to make a habitable Universe. (cf. 'Just Six Numbers' by Martin Rees or 'The Road to Reality' by Roger Penrose)

The whole range of anthropic principles from hard to weak is indeed thought provoking.

In the case of Omega I think it would be interesting to no what magnitude range (+ve or-ve) is allowable but I have not researched it.

Regards Andrew

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1 hour ago, Tiki said:

I suppose it could be argued that it is not surprising since we are here talking about it at all. A slightly less flat Universe would have collapsed long ago or would have disallowed the formation of galaxies and stars.

I think that in order for the universe to have collapsed long ago, the universe would have to have been substantially less flat. Without dark energy, all closed (homogeneous and isotropic) universes eventually collapse. When there is non-zero dark energy, this is no longer the case., i.e., when there is non-zero dark energy, closed universes can expand forever.

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After reading a bit on the topic, it is indeed surprising.

However, I'm not against inflation - and it does seem to treat this problem well.

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23 hours ago, andrew s said:

In the case of Omega I think it would be interesting to no what magnitude range (+ve or-ve) is allowable but I have not researched it.

Somewhat astonishingly, for a Universe roughly 10 billion years old with a value of Omega not wildly different from 1, the value of Omega when the Universe is just 1 second old could not differ from unity by more than one part in 10^15. ( From 'Just Six Numbers')

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