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 (WMAP, BOOMERanG, and Planck for example) imply that the observable universe is flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology. 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: Local geometry: This relates to the curvature of the universe, primarily concerning what we can observe. Global geometry: This pertains to the universe's overall shape and structure. The observable universe is like a sphere extending 46.5 billion light-years in all directions from any observer. 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 (CMB) 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 current observations. Hence, it is unclear whether the observable universe matches the entire universe or is significantly smaller.
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 topology.
Curvature of the universe
The curvature is a quantity describing how the geometry of a space differs locally from the one of the flat space. The curvature of any locally isotropic space (and hence of a locally isotropic universe) falls into one of the three following cases: 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 . 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 . 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 .
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°.
