Flat universe

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 of a continuous object.
The spatial curvature is related to general relativity, which describes how spacetime is curved and bent by mass and energy. The spatial topology cannot be determined from its curvature, due to the fact that there exist (mathematically) locally indistinguishable spaces with different topologies.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.
Several potential topological or geometric attributes of the universe interest may be discussed. Some of these are:
Boundedness (whether the universe is finite or infinite)
Flat (zero curvature), hyperbolic (negative curvature), or spherical (positive curvature)
Connectivity: how the universe is put together, i.e., simply connected space or multiply connected space.There are certain logical connections among these properties. For example, a universe with positive curvature is necessarily finite. 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 three-torus is flat but finite.The exact shape is still a matter of debate in physical cosmology, but experimental data from various independent sources (WMAP, BOOMERanG, and Planck for example) confirm that the universe is flat with only a 0.4% margin of error. On the other hand, any non-zero curvature is possible for a sufficiently large curved universe (analogously 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 universe. 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 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, but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space and the Sokolov–Starobinskii space (quotient of the upper half-space model of hyperbolic space by a 2-dimensional lattice).

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