Is the Universe Flat and What Does That Mean for Its Geometry?

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The discussion centers on the concept of a flat universe, which suggests there is just enough matter and energy to potentially halt its expansion. While current observations indicate the observable universe is nearly flat, they could also support some closed models on larger scales. The term "flat" refers to a Euclidean geometry where light rays are not distorted, meaning angles in triangles sum to 180° and parallel lines remain parallel. In the absence of dark energy, a flat universe would mean eternal expansion, while a closed universe would eventually recollapse. However, with dark energy, the relationship between the universe's fate and its geometry becomes less clear, allowing for various expansion scenarios.
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Some questions:

How sure are we that the Universe is flat?

A flat Universe implies that there is precisely enough matter/energy in the universe to eventually halt the expansion?

What is really meant by "flat"? We theorize that space-time has 4 dimensions, so does "flat" simply mean that light rays are not distorted by the shape of the universe?
 
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Holocene said:
How sure are we that the Universe is flat?
The observable universe is nearly flat, but the observations could be also consistent with some closed models on very large scales.

Holocene said:
A flat Universe implies that there is precisely enough matter/energy in the universe to eventually halt the expansion?
A flat universe without dark energy implies that there is exactly the right amount of matter to have eternal expansion. If the matter density is greater the universe will be closed and will recollapse. This is only valid without dark energy. With dark energy there is no clear relation between fate of the universe and geometry of space; a closed model may also expand forever.

Holocene said:
What is really meant by "flat"? We theorize that space-time has 4 dimensions, so does "flat" simply mean that light rays are not distorted by the shape of the universe?
Flat space is euclidean (three-dimensional) space in which the angles of a triangle sum 180° or in which initially parallel lines remain parallel forever. May be this thread will be interesting for you.
 
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Thread 'Matter density right after the decoupling'

https://en.wikipedia.org/wiki/Recombination_(cosmology) Was a matter density right after the decoupling low enough to consider the vacuum as the actual vacuum, and not the medium through which the light propagates with the speed lower than ##({\epsilon_0\mu_0})^{-1/2}##? I'm asking this in context of the calculation of the observable universe radius, where the time integral of the inverse of the scale factor is multiplied by the constant speed of light ##c##.
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Thread 'Rutgers finds a transparent Einstein Cross - apparently a dark matter "halo".'

The formal paper is here. The Rutgers University news has published a story about an image being closely examined at their New Brunswick campus. Here is an excerpt: Computer modeling of the gravitational lens by Keeton and Eid showed that the four visible foreground galaxies causing the gravitational bending couldn’t explain the details of the five-image pattern. Only with the addition of a large, invisible mass, in this case, a dark matter halo, could the model match the observations...
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Thread 'Big Bang and Infinite Universe'

Hi, I’m pretty new to cosmology and I’m trying to get my head around the Big Bang and the potential infinite extent of the universe as a whole. There’s lots of misleading info out there but this forum and a few others have helped me and I just wanted to check I have the right idea. The Big Bang was the creation of space and time. At this instant t=0 space was infinite in size but the scale factor was zero. I’m picturing it (hopefully correctly) like an excel spreadsheet with infinite...
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