# Shape of the universe

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The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes general global properties of its shape as a continuous object.

The spatial curvature is described by general relativity, which describes how spacetime is curved due to the effect of gravity. The spatial topology cannot be determined from its curvature, due to the fact that there exist locally indistinguishable spaces that may be endowed with different topological invariants.[1]

Cosmologists distinguish between the observable universe and the entire universe, the former being a ball-shaped portion of the latter that can, in principle, be accessible by astronomical observations. Assuming the cosmological principle, the observable universe is similar from all contemporary vantage points, which allows cosmologists to discuss properties of the entire universe with only information from studying their observable universe. The main discussion in this context is whether the universe is finite, like the observable universe, or infinite.

Several potential topological and geometric properties of the universe need to be identified. Its topological characterization remains an open problem. Some of these properties are:[2]

1. Boundedness (whether the universe is finite or infinite)
2. Flatness (zero curvature), hyperbolic (negative curvature), or spherical (positive curvature)
3. Connectivity: how the universe is put together as a manifold, i.e., a simply connected space or a multiply connected space.

There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite.[3] Although it is usually assumed in the literature that a flat or negatively curved universe is infinite, this need not be the case if the topology is not the trivial one. For example, a multiply connected space may be flat and finite, as illustrated by the three-torus. Yet, in the case of simply connected spaces, flatness implies infinitude.[3]

The shape of the universe remains a matter of debate in physical cosmology. In this regard, experimental data from various independent sources (WMAP, BOOMERanG, and Planck for example) interpreted within the standard family of metrics imply that the universe is flat to within only a 0.4% margin of error of the curvature density parameter.[4][5][6] The issue of simple versus multiple connectivity is even more uncertain and as of 2023 has not yet been decided based on astronomical observation. On the other hand, any non-zero curvature is possible for a sufficiently large curved universe (analogous to how a small portion of a sphere can look flat). Theorists have been trying to construct a formal mathematical model of the shape of the (entire) universe relating connectivity, curvature and boundedness. In formal terms, this is a 3-manifold model corresponding to the spatial section (in comoving coordinates) of the four-dimensional spacetime of the universe. The family of models that most theorists use is the Friedmann–Lemaître–Robertson–Walker (FLRW) models. 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 is also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space,[8][9] the multiply connected three-torus, and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by a 2-dimensional lattice).[10]

Physical cosmology is based on the theory of General Relativity, a physical picture cast in terms of differential equations. Therefore, only the local geometric properties of the universe become theoretically accessible. Thus, Einstein's field equations determine only the local geometry but have absolutely no say on the topology of the universe. At present, the only possibility to elucidate such global properties relies on observational data, especially the fluctuations (anisotropies) of the temperature gradient field of the Cosmic Microwave Background (CMB).[11][12]