Shape of the universe

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The shape of the universe is the local and global geometry of the universe, in terms of both curvature and topology (though, strictly speaking, it goes beyond both). When physicists describe the universe as being flat or nearly flat, they're talking geometry: how space and time are warped according to general relativity. When they talk about whether it is open or closed, they're referring to its topology.[1] Although the shape of the universe is still a matter of debate in physical cosmology, based on the recent Wilkinson Microwave Anisotropy Probe (WMAP) measurements "We now know that the universe is flat with only a 0.4% margin of error", according to NASA scientists.[2] 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 4-dimensional space-time of the universe. The model most theorists currently use is the so-called Friedmann–Lemaître–Robertson–Walker (FLRW) model. According to cosmologists, on this model the observational data best fit with the conclusion that the shape of the universe is infinite and flat,[3] but the data are also consistent with other possible shapes, such as the so-called Poincaré dodecahedral space[4][5] and the Picard horn.[6]

Two aspects of shape[edit]


Describing the shape of the universe requires a consideration of two aspects:

  1. its local geometry, which mostly concerns the curvature of the universe, particularly the observable universe, and

If the observable universe encompasses the entire universe, we may be able to determine the global structure of the entire universe by observation. However, if the observable universe is smaller than the entire universe, our observations will be limited to only a part of the whole, and we may not be able to determine its global geometry through measurement. It is possible to construct different mathematical models of the global geometry of the entire universe all of which are consistent with current observational data. For example, the observable universe may be many orders of magnitude smaller than the entire universe. The universe may be small in some dimensions and not in others (analogous to the way a cylinder is longer in the dimension of length than it is in the dimensions of width and depth). To test whether a given mathematical model describes the universe accurately, scientists look for the model's novel implications—what are some phenomena in the universe that we have not yet observed, but that must exist if the model is correct—and they devise experiments to test whether those phenomena occur or not. For example, if the universe is a small closed loop, one would expect to see multiple images of an object in the sky, although not necessarily images of the same age.


Local geometry (spatial curvature)[edit]


FLRW model of the universe[edit]

In General Relativity, this is modelled by the Friedmann–Lemaître–Robertson–Walker (FLRW) model. This model, which can be represented by the Friedmann equations, provides a curvature (often referred to as geometry) of the universe based on the mathematics of fluid dynamics, i.e. it models 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.



The homogeneous and isotropic universe allows for a spatial geometry with a constant curvature. One aspect of local geometry to emerge from General Relativity and the FLRW model is that the density parameter, Omega (Ω), is related to the curvature of space. Omega is the average density of the universe divided by the critical energy density, i.e. that required for the universe to be flat (zero curvature).

The curvature of space is a mathematical description of whether or not the Pythagorean theorem is valid for spatial coordinates. In the latter case, it provides an alternative formula for expressing local relationships between distances:

  • If the curvature is zero, then Ω = 1, and the Pythagorean theorem is correct;
  • If Ω > 1, there is positive curvature; and
  • if Ω < 1 there is negative curvature.

In the last two cases, the Pythagorean theorem is invalid (but discrepancies are only detectable in triangles whose sides' lengths are of cosmological scale).

If you measure the circumferences of circles of steadily larger diameters and divide the former by the latter, all three geometries give a value very close to π for small enough diameters but the ratio departs from π for larger diameters unless Ω = 1:

  • For Ω > 1 (the sphere, see diagram) the ratio falls below π: indeed, a great circle on a sphere has circumference only twice its diameter.
  • For Ω < 1 the ratio rises above π.

Astronomical measurements of both matter-energy density of the universe and spacetime intervals using supernova events 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.

Possible local geometries[edit]





Global structure: geometry and topology[edit]



Investigations within the study of global structure of include

  • Whether the universe is infinite or finite in extent
  • The scale or size of the entire universe (if it is finite)
  • Whether the geometry is flat, positively curved, or negatively curved
  • Whether the topology is simply connected like a sphere or multiply connected like a torus

Infinite or finite[edit]


Closed manifolds[edit]






Analysis of data from WMAP implies that on the scale to the surface of last scattering, the density parameter of the Universe is within about 0.5% of the value representing spatial flatness.[8]


The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e. possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite.[9] Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe.[9] For example, Euclidean space is flat, simply connected and infinite, but the torus is flat, multiply connected, finite and compact.



Universe with zero curvature[edit]



A flat universe can have zero total energy.

Universe with positive curvature[edit]


Poincaré dodecahedral space, a positively curved space, colloquially described as "soccerball-shaped", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball. This was proposed by Jean-Pierre Luminet and colleagues in 2003[4][11] and an optimal orientation on the sky for the model was estimated in 2008.[5]

Universe with negative curvature[edit]


Curvature: Open or closed[edit]

When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive. These meanings of open and closed are different from the mathematical meaning of open and closed used for sets in metric spaces and for the mathematical meaning of open and closed manifolds, which gives rise to ambiguity and confusion. In mathematics, there are definitions for a closed manifold (i.e. compact without boundary) and open manifold (i.e. one that is not compact and without boundary,[12]). A "closed universe" is necessarily a closed manifold. An "open universe" can be either a closed or open manifold. For example, the Friedmann–Lemaître–Robertson–Walker (FLRW) model the universe is considered to be without boundaries, in which case "compact universe" could describe a universe that is a closed manifold.

Milne model ("spherical" expanding)[edit]

Main article: Milne model



See also[edit]


  1. ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science (Caltech). LXXIV1: 30. 
  2. ^
  3. ^ Demianski, Marek; Sánchez, Norma; Parijskij, Yuri N. (2003). "Topology of the universe and the cosmic microwave background radiation". The Early Universe and the Cosmic Microwave Background: Theory and Observations. Proceedings of the NATO Advanced Study Institute. The early universe and the cosmic microwave background: theory and observations (Springer) 130: 161. ISBN 1-4020-1800-2. , Extract of page 161
  7. ^ Carl F. Gauss, Werke 8, 175–239, cited and translated in John W. Milnor (1982) Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. (N.S.) 6(1), p. 10. Milnor's translation reads:
    "The assumption that the sum of the three angles [of a triangle] is smaller than 180° leads to a geometry which is quite different from our (euclidean) geometry, but which is in itself completely consistent. I have satisfactorily constructed this geometry for myself so that I can solve every problem, except for the determination of one constant, which cannot be ascertained a priori. The larger one chooses this constant, the closer one approximates euclidean geometry. . . . If non-euclidean geometry were the true geometry, and if this constant were comparable to distances which we can measure on earth or in the heavens, then it could be determined a posteriori. Hence I have sometimes in jest expressed the wish that euclidean geometry is not true. For then we would have an absolute a priori unit of measurement."
  8. ^ a b Luminet, Jean-Pierre; Lachi`eze-Rey, Marc (1995). "Cosmic Topology". Physics Reports 254 (3): 135–214. arXiv:gr-qc/9605010. doi:10.1016/0370-1573(94)00085-h. 
  9. ^ Mihran Vardanyan et al. How flat can you get?, A model comparison perspective on the curvature of the Universe
  10. ^ "Is the universe a dodecahedron?", article at PhysicsWeb.
  11. ^ Since the universe is assumed connected, we do not need to specify the more technical "an open manifold is one without compact component".

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