Shape of the universe


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. General relativity explains how spatial curvature 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.
Observational evidence indicates that the observable universe is spatially flat. It is unknown whether the universe is simply connected like euclidean space or multiply connected like a torus.

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 about 46 billion light-years in every direction. It appears older and more redshifted the deeper we look into space. In theory, we could look all the way back to the Big Bang, but in practice, we can only see up to the cosmic microwave background as anything beyond that is opaque. Studies show that the observable universe is isotropic and homogeneous 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 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 2024, observational evidence indicates that the observable universe is spatially flat with an unknown global structure.

Curvature of the universe

The curvature is a quantity describing how the geometry of a space differs locally from flat space. The curvature of any locally isotropic space falls into one of the three following cases:
  1. Zero curvature a drawn triangle's angles add up to 180° and the Pythagorean theorem holds; such 3-dimensional space is locally modeled by Euclidean space.
  2. Positive curvaturea drawn triangle's angles add up to more than 180°; such 3-dimensional space is locally modeled by a region of a 3-sphere.
  3. Negative curvaturea drawn triangle's angles add up to less than 180°; such 3-dimensional space is locally modeled by a region of a hyperbolic space.
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°.
File:End of universe.jpg|thumb|275px|The local geometry of the universe is determined by whether the density parameter is greater than, less than, or equal to 1. From top to bottom: a spherical universe with, a hyperbolic universe with, and a flat universe with. These depictions of two-dimensional surfaces are merely easily visualizable analogs to the 3-dimensional structure of space.File:Spacetime-diagram-flat-universe-proper-coordinates.png|thumb|275px|Proper distance spacetime diagram of our flat ΛCDM universe. Particle horizon: green, Hubble radius: blue, Event horizon: purple, Light cone: orange.
General relativity explains that mass and energy bend the curvature of spacetime and is used to determine what curvature the universe has by using a value called the density parameter, represented with Omega. 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, the universe is flat.
  • If, there is positive curvature.
  • If, there is negative curvature.
Scientists could experimentally calculate to determine the curvature two ways. One is to count all the mass–energy in the universe and take its average density, then divide that average by the critical energy density. Data from the Wilkinson Microwave Anisotropy Probe as well as the Planck spacecraft give values for the three constituents of all the mass–energy in the universe – normal mass, relativistic particles, and dark energy or the cosmological constant:
The actual value for critical density value is measured as ρcritical =. 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. This can be done by using the CMB and measuring the power spectrum and temperature anisotropy. For instance, one can imagine finding a gas cloud that is not in thermal equilibrium due to being so large that light speed cannot propagate the thermal information. Knowing this propagation speed, we then know the size of the gas cloud as well as the distance to the gas cloud, we then have two sides of a triangle and can then determine the angles. Using a method similar to this, the BOOMERanG experiment has determined that the sum of the angles to 180° within experimental error, corresponding to.
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 spacetime intervals, we can approximate 3-space by the familiar Euclidean geometry.
The Friedmann–Lemaître–Robertson–Walker model using Friedmann equations is commonly used to model the universe. The FLRW model provides a curvature of the universe based on the mathematics of fluid dynamics, that is, modeling the matter within the universe as a perfect fluid. Although stars and structures of mass can be introduced into an "almost FLRW" model, a strictly FLRW model is used to approximate the local geometry of the observable universe. Another way of saying this is that, if all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed. This assumption is justified by the observations that, while the universe is "weakly" inhomogeneous and anisotropic, it is on average homogeneous and isotropic when analyzed at a sufficiently large spatial scale.

Global universal 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 or else multiply connected.

    Infinite or finite

One of the unresolved questions about the universe is whether it is infinite or finite in extent. Answers within the 21st century depend on the current standard cosmological model.
Ancient mythologies variously described the universe as finite.
By way of the account of Diogenes Laërtius, for Leucippus the universe is spatially infinite. Eudoxus in thought of motion considered the stars integral to a sphere. The concept of Aristotle, concentric spheres existed outgoing from Earth, the furthest contained the stars and was sometimes termed the kosmos, outside of which there was nothing; neither any place, time, or void extracosmic.
From the concepts of Aristotle which became the mode for Ptolemy the preferred general cosmology into the Middle Ages was the cosmos was finite because of Aristotelian cosmology. Dante Alighieri, Paradiso, conceived of a Ptolemaic understanding universe which explained the Earth was central to spheres the outer of which was the realm of God, the perception of all prominent medieval era thinkers. Bradwardine and Oresme during the 14th century contested the Aristotlian view on the basis of infinite God.
The advent of the heliocentric model produced in scientific thought the possibility of an infinite universe. A Universe infinite in size, using Copernicus, explained by Thomas Digges in: A perfit description of the caelestiall orbs, published 1576, was a conceptual break from the tradition of the reality of a celestial outer realm known as Paradise.
Einstein in consideration of his general theory of relativity demonstrated in Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie a finite universe. The de Sitter infinite universe was caused by incompatibility of Relativity and Euclidean space. Hilbert thought the universe was determined finite by elliptical geometry or infinite by Euclidean geometry.
The factor which could determine from our position in the universe a scientific answer of which version of the universe is thought reality with regards to the geometry of the universe is: if positively curved is finite, if flat or negatively curved is infinite. A finite universe is volumetrical, an infinite universe could encompass an infinity of space with a finite amount of matter. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe means that there are points arbitrarily far apart: for any distance, there are points that are of a distance at least apart. A finite universe is a bounded metric space, where there is some distance such that all points are within distance of each other. The smallest such is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale".