I What is the nature of space and time in deep empty regions of the cosmos?

[Satyam]

Thank you all for clearing my doubt before. There is a question I want to ask on space and time and this time it is not about the absolute time as I have understood fairly that space and time are always relative.
So Here it is according to GR It is very beautifully explained that Very massive objects manipultes space and time , as they bends the space- time fabric and as a consequence give rise to gravity. This shows how gravity actually behaves.
I want to ask you then what we will call or how we will acknowledge that space and time which has not been manipulated or in other words describe something in domain of deep empty spaces in this ever expanding cosmos.?

 

[PeterDonis]

I want to ask you then what we will call or how we will acknowledge that space and time which has not been manipulated or in other words describe something in domain of deep empty spaces in this ever expanding cosmos.?

I don't understand the question.

 

[Satyam]

I don't understand the question.

How we will acknowledge that space and time which has not been manipulated i.e the space and time fabric which has not been bend as there is no planet , star or massive object to manipulate the space time fabric or in other words how to describe the domain of deep empty spaces in this ever expanding cosmos.?

 

[fresh_42]

How we will acknowledge that space and time which has not been manipulated i.e the space and time fabric which has not been bend as there is no planet , star or massive object to manipulate the space time fabric or in other words how to describe the domain of deep empty spaces in this ever expanding cosmos.?

There is no total emptiness. Energy counts, too, and gravitation acts infinitely. But what do you want to describe, if there is nothing to describe?

 

[Ibix]

How we will acknowledge that space and time which has not been manipulated i.e the space and time fabric which has not been bend as there is no planet , star or massive object to manipulate the space time fabric or in other words how to describe the domain of deep empty spaces in this ever expanding cosmos.?

It's all spacetime. The only difference is how curved it is and in what way it is curved.

 

[Dale]

I want to ask you then what we will call or how we will acknowledge that space and time which has not been manipulated or in other words describe something in domain of deep empty spaces in this ever expanding cosmos.?

I am not sure what you are asking since you seem to be asking about two different spacetimes.

A spacetime without any mass or energy is called a vacuum spacetime. The simplest vacuum spacetime is the Minkowski spacetime which is flat (no curvature) everywhere.

The spacetime which describes the cosmos as a whole is the FLRW spacetime also called LCDM for a specific equation of state.

 

[PeterDonis]

space and time which has not been manipulated i.e the space and time fabric which has not been bend as there is no planet , star or massive object to manipulate the space time fabric

Ok, then you're talking about the flat Minkowski spacetime of special relativity. Which does not actually exist; it's an idealization only, useful for certain purposes.

how to describe the domain of deep empty spaces in this ever expanding cosmos.?

As an appropriate curved spacetime geometry, since there is matter and energy in the universe. The fact that there might not be matter or energy in some particular region does not mean there is no spacetime curvature in that region; there can be spacetime curvature there due to the presence of matter or energy elsewhere in the universe.

 

[Satyam]

I am not sure what you are asking since you seem to be asking about two different spacetimes.

A spacetime without any mass or energy is called a vacuum spacetime. The simplest vacuum spacetime is the Minkowski spacetime which is flat (no curvature) everywhere.

The spacetime which describes the cosmos as a whole is the FLRW spacetime also called LCDM for a specific equation of state.

So are there actually two dimensions/ models of space-time. One is vacuum space-time and other FLRW. ?

 

[Satyam]

It's all spacetime. The only difference is how curved it is and in what way it is curved.

I'm asking about that space- time which is not curved. i.e what is the nature of that space-time which is not curved.

 

[Ibix]

So are there actually two dimensions/ models of space-time. One is vacuum space-time and other FLRW. ?

No. There's one model of spacetime, a 4d manifold. Depending on what's in that spacetime it has different curvature, and we give names to the different curvatures we get under circumstances. Spacetime around a non-rotating black hole is called Schwarzschild spacetime. Spacetime in a large enough empty region is flat and is called Minkowski spacetime. Spacetime on a large enough scale that galaxies are too small to see is FLRW spacetime.

I'm asking about that space- time which is not curved. i.e what is the nature of that space-time which is not curved.

It's spacetime with zero curvature. We do not have a more fundamental model, so there isn't any further layer of explanation.

 

[Satyam]

Ok, then you're talking about the flat Minkowski spacetime of special relativity. Which does not actually exist; it's an idealization only, useful for certain purposes.
As an appropriate curved spacetime geometry, since there is matter and energy in the universe. The fact that there might not be matter or energy in some particular region does not mean there is no spacetime curvature in that region; there can be spacetime curvature there due to the presence of matter or energy elsewhere in the universe.

The space - time is always curved is hard to digest. Because the % occupied by matter and energy I'd only 5% of the total cosmos. The respective is dark matter and energy . Can we say that flat or unmanipulated space- time anddark matter and energy are same?
Here is the link of the blog.
https://en.m.wikipedia.org/wiki/Dark_matter
Here some scientific theory have also proved that flat space-time exists. Have a look
https://en.m.wikipedia.org/wiki/Shape_of_the_universe

 

[jbriggs444]

The space - time is always curved is hard to digest. Because the % occupied by matter and energy I'd only 5% of the total cosmos.

Matter, energy, dark matter, dark energy. They all curve space time. The difference between the dark stuff and the non-dark stuff is that we can see the latter. Hence the names.

Can we say that flat or unmanipulated space- time anddark matter and energy are same?

No. Not even close.

Here is the link of the blog.
https://en.m.wikipedia.org/wiki/Dark_matter
Here some scientific theory have also proved that flat space-time exists. Have a look
https://en.m.wikipedia.org/wiki/Shape_of_the_universe

The latter reference does not say what you think it says. Space-time curvature is a tensor, not a scalar. The fact that we live in an expanding universe means that the tensor is not identically zero.

However, we can observe some symmetries. And if we adopt the "co-moving" coordinate system where the expansion is uniform in all directions then we can slice space-time up into spatial slices (each corresponding to a moment in time according to our coordinate system) and ask whether these spatial slices are each flat. The curvature of a 3-dimensional space is a scalar. It can be positive, negative or zero. It is this curvature of these spatial slices which the latter article is speaking of. Not space-time curvature.

 

[Dale]

So are there actually two dimensions/ models of space-time. One is vacuum space-time and other FLRW. ?

There are an infinite number of models of space-time corresponding to an infinite number of distributions of matter. Some of the more useful or easy to calculate models have names. You were asking about two of those (as far as I could tell).

I'm asking about that space- time which is not curved. i.e what is the nature of that space-time which is not curved.

That is the flat Minkowski solution. However you also asked:

domain of deep empty spaces in this ever expanding cosmos.?

Which is not flat and is the FLRW spacetime instead.

Because the % occupied by matter and energy I'd only 5% of the total cosmos.

This is not correct. In the FLRW spacetime one of the assumptions is homogeneity which means that 100% of the cosmos is occupied by matter at the largest scales. Remember, we are talking about cosmological scales, so at cosmological scales every part of the universe contains a mix of matter, dark matter, energy, and dark energy. The percentages you mention describe the ratio of this mixture, not the spatial distribution which is homogenous.

 

[Ibix]

The space - time is always curved is hard to digest.

Gravity has infinite range as far as we are aware. So spacetime will be (possibly very slightly) curved everywhere if it is curved anywhere.

Here some scientific theory have also proved that flat space-time exists. Have a look
https://en.m.wikipedia.org/wiki/Shape_of_the_universe

This says that there are spatially flat FLRW solutions, not that spacetime curvature is zero. These are not the same thing.

 

[Satyam]

There are an infinite number of models of space-time corresponding to an infinite number of distributions of matter. Some of the more useful or easy to calculate models have names. You were asking about two of those (as far as I could tell).

That is the flat Minkowski solution. However you also asked:
Which is not flat and is the FLRW spacetime instead.

This is not correct. In the FLRW spacetime one of the assumptions is homogeneity which means that 100% of the cosmos is occupied by matter at the largest scales. Remember, we are talking about cosmological scales, so at cosmological scales every part of the universe contains a mix of matter, dark matter, energy, and dark energy. The percentages you mention describe the ratio of this mixture, not the spatial distribution which is homogenous.

Can you throw more light on homogeneous spatial distribution and how would our universe look if it was heterogeneous..?

 

[Ibix]

Can you throw more light on homogeneous spatial distribution and how would our universe look if it was heterogeneous..?

We might see different numbers of galaxies in different parts of the sky, or more galaxies further away than nearby. On average at large scales, we don't see this - the universe is pretty much the same in every direction.

 

[Satyam]

Matter, energy, dark matter, dark energy. They all curve space time. The difference between the dark stuff and the non-dark stuff is that we can see the latter. Hence the names.

No. Not even close.

The latter reference does not say what you think it says. Space-time curvature is a tensor, not a scalar. The fact that we live in an expanding universe means that the tensor is not identically zero.

However, we can observe some symmetries. And if we adopt the "co-moving" coordinate system where the expansion is uniform in all directions then we can slice space-time up into spatial slices (each corresponding to a moment in time according to our coordinate system) and ask whether these spatial slices are each flat. The curvature of a 3-dimensional space is a scalar. It can be positive, negative or zero. It is this curvature of these spatial slices which the latter article is speaking of. Not space-time curvature.

However if the observable universe is expanding which the results shows it is then it's volume must be increasing or the space in the universe is getting increasing.
For analogy we can say that if we put gas into a balloon it's volume get increased and as it volume increases it displaces the air around it. So it takes over air and displaces it as the volume increases.

So in the case of this cosmos as it is expanding then what are the characteristics of that domain in which it is expanding into..?

 

[Satyam]

We might see different numbers of galaxies in different parts of the sky, or more galaxies further away than nearby. On average at large scales, we don't see this - the universe is pretty much the same in every direction.

Okay the universe is same in every direction is an indication of it's isotropic behaviour. What about it's homogeneity?

 

[Ibix]

Okay the universe is same in every direction is an indication of it's isotropic behaviour. What about it's homogeneity?

Homogeneity means that it's the same everywhere, at all distances. That does imply isotropy too.

 

[Satyam]

Homogeneity means that it's the same everywhere, at all distances. That does imply isotropy too.

Your statement is incorrect.
Isotropy is about exhibiting same properties in every direction while homogeneity is about uniform composition. Here are two examples
1. An example of something that is homogeneous but not isotropic is a space that is filled with a uniform electric or magnetic field. Because the field is uniform it is homogeneous, but because the field has a direction, it is not isotropic.

Similarly, it's easy to see how something can be isotropic but not homogeneous. For example, a spherically symmetric distribution of mass is isotropic for an observer at the center of the sphere, but is not necessarily homogenous.

 

[PeterDonis]

Your statement is incorrect.

No, his statement was correct.

Isotropy is about exhibiting same properties in every direction while homogeneity is about uniform composition.

Homogeneity is not just about "uniform composition". It means uniformity everywhere in space, period. And that, as @Ibix said, implies isotropy as well; a universe which is the same at every point in space will also be the same in every direction from every point in space.

 

[Satyam]

No, his statement was correct.
Homogeneity is not just about "uniform composition". It means uniformity everywhere in space, period. And that, as @Ibix said, implies isotropy as well; a universe which is the same at every point in space will also be the same in every direction from every point in space.

Ok Sir. I'm not here to debate anyone. Can you tell me more about spatial distribution of matter as my purpose was to get a clear picture of this.

 

[PeterDonis]

An example of something that is homogeneous but not isotropic is a space that is filled with a uniform electric or magnetic field.

A region of space can be homogeneous in this sense, but the universe as a whole cannot; it's impossible to have a uniform EM field everywhere in the universe.

Can you tell me more about spatial distribution of matter

For what spacetime? So far we have discussed four possibilities in this thread: flat Minkowski spacetime, Schwarzschild spacetime, FRW spacetime, and our actual universe (which is not exactly described by any of those). Which one are you asking about?

 

[Satyam]

A region of space can be homogeneous in this sense, but the universe as a whole cannot; it's impossible to have a uniform EM field everywhere in the universe.
For what spacetime? So far we have discussed four possibilities in this thread: flat Minkowski spacetime, Schwarzschild spacetime, FRW spacetime, and our actual universe (which is not exactly described by any of those). Which one are you asking about?

How spatial distribution is homogeneous in our actual/ observable universe.

 

[PeterDonis]

About our actual/ observable universe.

Then, as I said, our actual/observable universe is not described exactly by any of the spacetimes we have discussed, but FRW spacetime is closest. Our actual/observable universe is not exactly isotropic or homogeneous, but it is to a good approximation on scales of about 100 million light-years and larger. So an FRW spacetime is a good approximation to the actual spacetime geometry on these scales.

 

[Satyam]

Sir on what reference we are saying that it is homogeneous. In order to understand FLRW Model. It is important to grasp why it is considered homogeneous?

 

[Ibix]

It is important to grasp why it is considered homogeneous?

Because that's a pretty good approximation to what we see through telescopes.

 

[jbriggs444]

However if the observable universe is expanding which the results shows it is then it's volume must be increasing or the space in the universe is getting increasing.

When we talk about the universe expanding, we are not normally talking about an increase in the radius of the observable universe. Rather we are talking about the fact that all of the [large scale] things in the universe are getting farther apart.

If the universe as a whole is infinite, then it does not have a defined volume. So it is not correct to talk about its volume increasing or decreasing.

So in the case of this cosmos as it is expanding then what are the characteristics of that domain in which it is expanding into..?

There is no such domain. It is not that sort of expansion. It is not expanding to fill some existing empty space. It is simply expanding. Distances are getting greater. There is no need consider our universe as somehow embedded within a higher dimensional space in order to describe its expansion.

Edit...

You had invoked the balloon analogy, pointing out that when the balloon expends, it displaces air. This is a good example of what it means to have a lower-dimensional space embedded in a higher dimensional space. The surface of the balloon is a two dimensional space. We picture it embedded in a pre-existing three dimensional space. We do that because it is easy to imagine.

But there is no requirement for the three dimensional space to actually exist. One can describe all of the relevant properties of a two dimensional surface with a spherical topology without ever considering it to exist within a three dimensional space. One can do it with a two dimensional coordinate system (like latitude and longitude). The trick is to use a distance metric that is different from the euclidean $\sqrt{x^2 + y^2}$ one. [One also has to split it up into multiple patches -- that's what we call a manifold].

Same for our four-dimensional space-time. We can describe the relevant properties in terms of a metric rather than in terms of some euclidean hyper-space within which it is hypothetically embedded.

 

[PeterDonis]

In order to understand FLRW Model. It is important to grasp why it is considered homogeneous?

The FRW model is homogeneous by construction; it is a model explicitly constructed to be homogeneous.

Our actual universe, as I said, is not exactly homogeneous, but is homogeneous to a good approximation on scales of about 100 million light years and larger. We know that from astronomical observations.

 

[Dale]

Can you throw more light on homogeneous spatial distribution and how would our universe look if it was heterogeneous..?

If the universe were not spatially homogenous and isotropic then it would look different at different directions and distances (after accounting for light travels time). For instance, if the universe had an edge and if we were close to it then we would not see distant objects in that direction.

So in the case of this cosmos as it is expanding then what are the characteristics of that domain in which it is expanding into..?

We have no evidence that such a domain exists. It certainly is not necessary for general relativity.

 

[Satyam]

When we talk about the universe expanding, we are not normally talking about an increase in the radius of the observable universe. Rather we are talking about the fact that all of the [large scale] things in the universe are getting farther apart.

If the universe as a whole is infinite, then it does not have a defined volume. So it is not correct to talk about its volume increasing or decreasing.

There is no such domain. It is not that sort of expansion. It is not expanding to fill some existing empty space. It is simply expanding. Distances are getting greater. There is no need consider our universe as somehow embedded within a higher dimensional space in order to describe its expansion.

Edit...

You had invoked the balloon analogy, pointing out that when the balloon expends, it displaces air. This is a good example of what it means to have a lower-dimensional space embedded in a higher dimensional space. The surface of the balloon is a two dimensional space. We picture it embedded in a pre-existing three dimensional space. We do that because it is easy to imagine.

But there is no requirement for the three dimensional space to actually exist. One can describe all of the relevant properties of a two dimensional surface with a spherical topology without ever considering it to exist within a three dimensional space. One can do it with a two dimensional coordinate system (like latitude and longitude). The trick is to use a distance metric that is different from the euclidean $\sqrt{x^2 + y^2}$ one. [One also has to split it up into multiple patches -- that's what we call a manifold].

Same for our four-dimensional space-time. We can describe the relevant properties in terms of a metric rather than in terms of some euclidean hyper-space within which it is hypothetically embedded.

I thought we do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper like an 'infinite' sheet of paper and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus i.e like a doughnut. The surface of the torus is also spatially flat, but it is finite. So we have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

 

[jbriggs444]

I thought we do not know whether the Universe is finite or not. To give you an example, imagine the geometry of the Universe in two dimensions as a plane. It is flat, and a plane is normally infinite. But you can take a sheet of paper like an 'infinite' sheet of paper and you can roll it up and make a cylinder, and you can roll the cylinder again and make a torus i.e like a doughnut. The surface of the torus is also spatially flat, but it is finite. So we have two possibilities for a flat Universe: one infinite, like a plane, and one finite, like a torus, which is also flat.

Again, that is not a flat Universe. That is a flat spatial slice of a universe. I did not make a claim one way or the other about the size of such a slice.

 

[Dale]

Sir on what reference we are saying that it is homogeneous. In order to understand FLRW Model. It is important to grasp why it is considered homogeneous?

The frame where it is homogenous and isotropic is called the comoving frame or comoving coordinates. Those are the standard coordinates for cosmology.

Regarding if it is important: I would say yes. The assumption of homogeneity and isotropy greatly constrains the form of the possible solutions. That is what makes it one of the few analytically tractable spacetime models.

 

[PAllen]

Isotropy everywhere implies homogeneity everywhere. However homogeneity (everywhere) does not imply isotropy. Consider a geometrically flat cylinder. It is homogeneous but not isotropic.

 

[Satyam]

Again, that is not a flat Universe. That is a flat spatial slice of a universe. I did not make a claim one way or the other about the size of such a slice.

The surface of the torus is termed as flat and surface is a slice of 3d as it is 2d. The point on which I want to drag your attention is that the universe can be in the form of torus and as there are no ends to it , it can be termed as endless universe. But being endless doesn't mean it is infinite.
We don't know yet the universe is finite or infinite.

If the universe as a whole is infinite, then it does not have a defined volume. So it is not correct to talk about its volume increasing or decreasing.

Is this correct to term universe as infinite and then making a statement based on that..?

 

[Dale]

The point on which I want to drag your attention is that the universe can be in the form of torus

Do you have a reference for this? I am skeptical that a toroidal universe is compatible with observation.

 

[PAllen]

A universe with toroidal spatial slices would not be isotropic. This would have observational consequences (and could not be described by an FLRW metric). (I see @Dale made a similar point).

 

[Satyam]

Do you have a reference for this? I am skeptical that a toroidal universe is compatible with observation.

Yes Sir
https://en.m.wikipedia.org/wiki/Shape_of_the_universe
Here I will tell you the headings in which you can look into this matter more quickly.
•Shape of the observable universe.
•Global universe structure

 

[Dale]

Yes Sir
https://en.m.wikipedia.org/wiki/Shape_of_the_universe
Here I will tell you the headings in which you can look into this matter more quickly.
•Shape of the observable universe.
•Global universe structure

Nothing on that page says that a toroidal universe is consistent with observation. In fact, it says:

"The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat,[7] but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space[8][9] and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by 2-dimensional lattice).[10] "

A torus is not listed as one of the other possible shapes consistent with the data, despite its various mathematical properties being mentioned several times in the article. I think that your point in post 35 is incorrect. A torus-shaped universe is not consistent with the data.

 

[PeterDonis]

A torus is not listed as one of the other possible shapes consistent with the data

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes. For one thing, I have seen references given in previous PF threads to papers stating that a finite, closed 3-sphere universe is not completely ruled out by the data, just highly unlikely; but a finite closed 3-sphere is not one of the possibilities listed in that paragraph.

(Quite frankly, to me the two possibilities other than "infinite and flat" that are listed seem considerably more esoteric to me than a 3-sphere, so the fact that those esoteric possibilities were listed and the 3-sphere was not makes me give less credibility to the article as a whole.)

A flat 3-torus with a large enough finite volume would be indistinguishable from a flat infinite Euclidean 3-space given the finite age of the universe, so I don't see how it could be ruled out completely; the most we could do from the data would be to set a lower bound on the finite volume of the 3-torus.

 

[Dale]

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes.

Sure, but it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe. I had requested such a reference to support the claim made and this is not one.

 

[PeterDonis]

it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe.

Yes, agreed.

 

[Satyam]

Nothing on that page says that a toroidal universe is consistent with observation. In fact, it says:

"The model most theorists currently use is the Friedmann–Lemaître–Robertson–Walker (FLRW) model. Arguments have been put forward that the observational data best fit with the conclusion that the shape of the global universe is infinite and flat,[7] but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space[8][9] and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by 2-dimensional lattice).[10] "

A torus is not listed as one of the other possible shapes consistent with the data, despite its various mathematical properties being mentioned several times in the article. I think that your point in post 35 is incorrect. A torus-shaped universe is not consistent with the data.

Sure, but it is also not a reference supporting the claim that the evidence is compatible with a toroidal universe. I had requested such a reference to support the claim made and this is not one.

Sir I'm not making a claim but just asking that could it be possible. As I have read it in some articles.
Here I will post the lines from the article of Wikipedia which makes me think this way.
If we assume a finite universe then possible considerations Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.
In mathematics, there are definitions for a closed manifold (i.e., compact without boundary) and open manifold (i.e., one that is not compact and without boundary). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, in the Friedmann–Lemaître–Robertson–Walker (FLRW) model the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.
Here it is being said as in FLRW model the universe is considered to be without boundries in which case compact universe could describe a universe that is a closed manifold..

 

[Satyam]

I don't think we can take this Wikipedia article's listing in one particular paragraph as a definitive limit on the possible shapes. For one thing, I have seen references given in previous PF threads to papers stating that a finite, closed 3-sphere universe is not completely ruled out by the data, just highly unlikely; but a finite closed 3-sphere is not one of the possibilities listed in that paragraph.

As stated in the introduction, investigations within the study of the global structure of the universe include:

•Whether the universe is infinite or finite in extent
Whether the geometry of the global universe is flat, positively curved, or negatively curved
Whether the topology is simply connected like a sphere or multiply connected, like a torus[14]
A finite closed 3-sphere is one of the possibilities listed in that paragraph.
However there is also an interview given by Joseph silk who was Head of Astrophysics, Department of Physics, University of Oxford, United Kingdom http://www.esa.int/Science_Explorat...ite_or_infinite_An_interview_with_Joseph_Silk

 

[Dale]

Sir I'm not making a claim but just asking that could it be possible.

Ah ok. It may just be a language issue. You said “the universe can be in the form of torus” which in English is a statement of fact rather than “can the universe be in the form of torus” which would be the corresponding question.

So as a question I don’t know of any analysis of current cosmological evidence in terms of a torus. There may be some literature on the topic but I am not aware of it.

However, as a moderate Bayesian I have a preference for parsimonious models. If the evidence equally supports a simply-connected geometry and a toroidal geometry then I would use the simply-connected model and would consider the non-simply connected geometry to not be supported by the data.

 

[PeterDonis]

As stated in the introduction

That Wikipedia article is not a good reference, even more so than the average Wikipedia article. If you look at the non-mobile version, you will see that a number of statements in the article are disputed.

there is also an interview

He says basically what I said in post #40:

A flat 3-torus with a large enough finite volume would be indistinguishable from a flat infinite Euclidean 3-space given the finite age of the universe

Note, however, that a flat 3-torus as a model requires more assumptions than the standard infinite flat FRW universe, so if the data is equally consistent with both, the flat 3-torus gets ruled out by Occam's razor. If we ever find evidence in favor of a flat 3-torus, that would be different, but we haven't.

 

[Satyam]

Ok sir I have understood it as there are no evidence till yet in support of the flat 3 torus and we have to make more assumptions in order to consider this shape. Data is consistent for the one which requires minimum assumptions.
So do we know the shape of the universe yet , if we do then of what shape it is?

 

[Dale]

So do we know the shape of the universe yet , if we do then of what shape it is?

The shape is the FLRW shape. It doesn’t have an English name, but the shape is specified by the FLRW metric. The closest English word would be trumpet-shaped, but that isn’t exactly right. Hence the need for the math.

 

[PAllen]

A flat 3 torus would have the same FLRW metric as for infinite flat spatial slices, but this metric would be over one chart of several that are joined by transition functions to specify the topology. So the difference is not the metric over some region, but that by assumption of homogeneity and isotropy, the metric is global, and one chart covers the whole manifold. Whereas, for the 3 torus, you have several charts with transition functions, but the metric in each chart can be arranged to be same as the FLRW metric. This atlas of charts for the flat 3-torus violates isotropy (which is always assumed to mean isotropy everywhere, in cosmology), but as @PeterDonis noted, for a big enough torus up to the current age of the universe, it could be that this anisotropy is not yet observable (i.e. the causal past from present day Earth can be contained in one chart).