page_title: Shape of the universe
section_title: Curvature of the universe
hierarchical_section_title: Shape of the universe / Curvature of the universe
context_page_description: 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, while the spatial topology cannot be determined from its curvature; locally indistinguishable spaces with different topologies exist mathematically.
  Cosmologists distinguish between the observable universe and the entire universe, the former being a spherical portion of the latter that can, in principle, be accessible by astronomical observations. Assuming the cosmological principle, the observable universe is similar for all contemporary vantage points, which allows cosmologists to discuss properties of the entire universe with only information inside their observable universe.
  The shape of the entire universe can be described with three attributes:
  Finite or infinite
  Flat, open, or closed
context_section_description: 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 E³.
  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.
  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°.
  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 (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 mass-energy in the universe and take its average density then divide that average by the critical energy density. Data from Wilkinson Microwave Anisotropy Probe (WMAP) as well as the Planck spacecraft give values for the three constituents of all the mass-energy in the universe – normal mass (baryonic matter and dark matter), relativistic particles (photons and neutrinos), and dark energy or the cosmological constant:
  Ωₘₐₛₛ ≈ 0.315±0.018
  Ωrelativistic ≈ 9.24×10⁻⁵
  ΩΛ ≈ 0.6817±0.0018
  Ωₜₒₜₐₗ=  Ωₘₐₛₛ + Ωrelativistic + ΩΛ=  1.00±0.02
  The actual value for critical density value is measured as ρcritical=  9.47×10⁻²⁷ kg m⁻³. From these values, within experimental error, the universe seems to be 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 CMB and measuring the power spectrum and temperature anisotropy. For an intuition, 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 an Ωₜₒₜₐₗ ≈ 1.00±0.12.
  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 (FLRW) 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 ignor
caption: The local geometry of the universe is determined by whether the 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.
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