Shape of the universe

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In physical cosmology, the shape of the universe refers to both its local and global geometry. Local geometry is defined primarily by its curvature, while the global geometry is characterised by its topology (which itself is constrained by curvature). General relativity explains how spatial curvature (local geometry) is constrained by gravity. The global topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics. For example; a multiply connected space like a 3 torus has everywhere zero curvature but is finite in extent, whereas a flat simply connected space is infinite in extent (euclidean space).

Current observational evidence (

It is currently unknown if the universe is simply connected like euclidean space or multiply connected like a torus. To date, no compelling evidence has been found suggesting the universe has a non-trivial (i.e.; not simply connected) topology, though it has not been ruled out by astronomical observations.

Shape of the observable universe

The universe's structure can be examined from two angles:

1. Local geometry: This relates to the curvature of the universe, primarily concerning what we can observe.
2. Global geometry: This pertains to the universe's overall shape and structure.

The observable universe is a roughly spherical region extending 46.5 billion light-years in all directions from any observer. It appears older and more

on the largest scales.

If the observable universe encompasses the entire universe, we might determine its structure through observation. However, if the observable universe is smaller, we can only grasp a portion of it, making it impossible to deduce the global geometry through observation. Different mathematical models of the universe's global geometry can be constructed, all consistent with current observations and general relativity. Hence, it is unclear whether the observable universe matches the entire universe or is significantly smaller, though it is generally accepted that the universe is larger than the observable universe.

The universe may be compact in some dimensions and not in others, similar to how a cuboid is longer in one dimension than the others. Scientists test these models by looking for novel implications – phenomena not yet observed but necessary if the model is accurate. For instance, a small closed universe would produce multiple images of the same object in the sky, though not necessarily of the same age. As of 2023 current observational evidence suggests that the observable universe is spatially flat with an unknown global structure.

Curvature of the universe

The

(and hence of a locally isotropic universe) falls into one of the three following cases:

1. Zero curvature (flat) — a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space E³.
2. Positive curvature — a drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere S³.
3. Negative curvature — a drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space H³.

Curved geometries are in the domain of Non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°.

density parameter Ω is greater than, less than, or equal to 1. From top to bottom: a spherical universe with Ω > 1, a hyperbolic universe with Ω < 1, and a flat universe

with Ω = 1. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of (local) space.

density parameter

, represented with Omega (Ω). The density parameter is the average density of the universe divided by the critical energy density, that is, the mass energy needed for a universe to be flat. Put another way,

• If Ω = 1, the universe is flat.
• If Ω > 1, there is positive curvature.
• If Ω < 1, there is negative curvature.

One can experimentally calculate this Ω to determine the curvature two ways. One is to count up all the

Ω_mass ≈ 0.315±0.018

Ω_relativistic ≈ 9.24×10⁻⁵

Ω_Λ ≈ 0.6817±0.0018

Ω_total = Ω_mass + Ω_relativistic + Ω_Λ = 1.00±0.02

The actual value for critical density value is measured as ρ_critical = 9.47×10⁻²⁷ kg m⁻³. From these values, within experimental error, the universe seems to be spatially flat.

Another way to measure Ω is to do so geometrically by measuring an angle across the observable universe. We can do this by using the

These and other astronomical measurements constrain the spatial curvature to be very close to zero, although they do not constrain its sign. This means that although the local geometries of spacetime are generated by the theory of relativity based on

.

The

isotropic

when analyzed at a sufficiently large spatial scale.

Global universe structure

Global structure covers the geometry and the topology of the whole universe—both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. The universe is often taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably. A global geometry is a local geometry plus a topology. It follows that a topology alone does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

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, and,
• whether the topology is simply connected (for example, like a sphere) or else multiply connected (for example, like a torus).^[6]

Infinite or finite

One of the unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as

. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".

With or without boundary

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

. The 3-sphere and 3-torus are both closed manifolds.

Observational methods

In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed[7] and applied to cosmological observations.^[8][9]

In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the cosmic web of large-scale structure, acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.^[10][11][12]

Curvature

The curvature of the universe places constraints on the topology. If the spatial geometry is

).

In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.

The latest research shows that even the most powerful future experiments (like the SKA) will not be able to distinguish between flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10⁻⁴. If the true value of the cosmological curvature parameter is larger than 10⁻³ we will be able to distinguish between these three models even now.^[13]

Final results of the Planck mission, released in 2018 show the cosmological curvature parameter, 1 − Ω = Ω_K = −Kc²/a²H², to be 0.0007±0.0019, consistent with a flat universe.^[14] (i.e. positive curvature: K = +1, Ω_K < 0, Ω > 1, negative curvature: K = −1, Ω_K > 0, Ω < 1, zero curvature: K = 0, Ω_K = 0, Ω = 1).

Universe with zero curvature

In a universe with zero curvature, the local geometry is

.

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe in the sense that space will continue expanding forever.

A flat universe can have zero total energy.^[15]

Universe with positive curvature

A positively curved universe is described by

of the 3-sphere.

Poincaré dodecahedral space is a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003^[8][16] and an optimal orientation on the sky for the model was estimated in 2008.^[9]

Universe with negative curvature

A hyperbolic universe, one of a negative spatial curvature, is described by hyperbolic geometry, and can be thought of locally as a three-dimensional analog of an infinitely extended saddle shape. There are a great variety of hyperbolic 3-manifolds, and their classification is not completely understood. Those of finite volume can be understood via the Mostow rigidity theorem. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called "horn topologies", so called because of the shape of the pseudosphere, a canonical model of hyperbolic geometry. An example is the Picard horn, a negatively curved space, colloquially described as "funnel-shaped".^[17]

Curvature: open or closed

When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively. These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in topological spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a

(FLRW) model, the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.

See also

References

External links

• Geometry of the Universe at icosmos.co.uk
• Janna Levin, Evan Scannapieco & Joseph Silk (1998). "The topology of the universe: the biggest manifold of them all". Classical and Quantum Gravity. 15 (9): 2689–2697.
  S2CID 119080782
  .
• Lachièze-Rey, M., Luminet, J.P. (1995). "Cosmic Topology". Physics Reports. 254 (3): 135–214.
  S2CID 119500217.{{cite journal}}: CS1 maint: multiple names: authors list (link
  )
• Luminet, J.P. (2016). "The Status of Cosmic Topology after Planck Data". Universe. 2 (1): 1–9.
  S2CID 7331164
  .
• Universe is Finite, "Soccer Ball"-Shaped, Study Hints. Possible wrap-around dodecahedral shape of the universe
• Classification of possible universes in the
  Lambda-CDM
  model.
• Fagundes, Helio V. (2002). "Exploring the global topology of the universe". Brazilian Journal of Physics. 32 (4): 891–894.
  S2CID 119495347
  .
• Grime, James. "π₃₉ (Pi and the size of the Universe)". Numberphile. Brady Haran. Archived from the original on 2015-04-30. Retrieved 2013-04-07.
• What do you mean the universe is flat? Scientific American Blog explanation of a flat universe and the curved spacetime in the universe.