Shape of the Universe
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The shape of the Universe is an informal name for a subject of investigation within physical cosmology. Cosmologists and astronomers describe the geometry of the universe which includes both local geometry and global geometry. It is loosely divided into curvature and topology, even though strictly speaking, it goes beyond both.
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[edit] Introduction
Considerations of the shape of the universe can be split into two parts; the local geometry relates especially to the curvature of the observable universe, while the global geometry relates especially to the topology of the universe as a whole—which may or may not be within our ability to measure.
The extrapolation of the local geometry of space to the geometry of the whole universe is not without a specific ontological stance regarding how space and time coexist. Current thinking demands that space and time be considered as two aspects of a single entity 'spacetime'.
Nevertheless it still makes sense to speak about three-dimensional concepts referring to the universe, like the Hubble volume
[edit] Local geometry (spatial curvature)
The local geometry is the curvature describing any arbitrary point in the observable universe (averaged on a sufficiently large scale). Many astronomical observations, such as those from supernovae and the Cosmic Microwave Background (CMB) radiation, show the observable universe to be very close to homogeneous and isotropic and infer it to be accelerating. 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.
Another way of saying this is that if all forms of dark energy are ignored, then the curvature of the universe can be determined by measuring the average density of matter within it, assuming that all matter is evenly distributed (rather than the distortions caused by 'dense' objects such as galaxies). This assumption is justified by the observations that, while the universe is "weakly" inhomogeneous and anisotropic (see the large-scale structure of the cosmos), it is on average homogeneous and isotropic. 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 either of these 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 the value 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 are generated by the theory of relativity based on spacetime intervals, we can approximate it to the familiar Euclidean geometry.
[edit] Local geometries
There are three categories for the possible spatial geometries of constant curvature, depending on the sign of the curvature. If the curvature is exactly zero, then the local geometry is flat; if it is positive, then the local geometry is spherical, and if it is negative than the local geometry is hyperbolic.
The geometry of the universe is usually represented in the system of comoving coordinates, according to which the expansion of the universe can be ignored. Comoving coordinates form a single frame of reference according to which the universe has a static geometry of three spatial dimensions.
Under the assumption that the universe is homogeneous and isotropic, the curvature of the observable universe, or the local geometry, is described by one of the three "primitive" geometries:
- 3-dimensional Euclidean geometry, generally annotated as E3
- 3-dimensional spherical geometry with a small curvature, often annotated as S3
- 3-dimensional hyperbolic geometry with a small curvature, often annotated as H3
Even if the universe is not exactly spatially flat, the spatial curvature is close enough to zero to place the radius at approximately the horizon of the observable universe or beyond.
[edit] Global geometry
Global geometry covers the geometry, in particular the topology, of the whole universe—both the observable universe and beyond. While the local geometry does not determine the global geometry completely, it does limit the possibilities, particularly a geometry of a constant curvature. For a flat spatial geometry, it used to be thought that the scale of any properties of the topology is arbitrary, though recent research suggests that the three spatial dimensions may tend to equalise in length.[1]. The length scale of a flat geometry may or may not be directly detectable. For spherical and hyperbolic spatial geometries, the probability of detection of the topology by direct observation depends on the spatial curvature. Using the radius of curvature or its inverse as a scale, a small curvature of the local geometry, with a corresponding radius of curvature greater than the observable horizon, makes the topology difficult or impossible to detect if the curvature is hyperbolic. A spherical geometry with a small curvature (large radius of curvature) does not make detection difficult.
Two strongly overlapping investigations within the study of global geometry are:
- whether the universe is infinite in extent or is a compact space
- whether the universe has a simply or non-simply connected topology
[edit] Compactness of the global shape
A compact space is a general topological definition that encompasses the more applicable notion of a bounded metric space. In cosmological models, it requires either one or both of: the space has positive curvature (like a sphere), and/or it is "multiply connected", or more strictly non-simply connected.
If the 3-manifold of a spatial section of the universe is compact then, as on a sphere, straight lines pointing in certain directions, when extended far enough in the same direction will reach the starting point and the space will have a definable "volume" or "scale". If the geometry of the universe is not compact, then it is infinite in extent with infinite paths of constant direction that, generally do not return and the space has no definable volume, such as the Euclidean plane.
If the spatial geometry is spherical, the topology is compact. Otherwise, for a flat or a hyperbolic spatial geometry, the topology can be either compact or infinite.
[edit] Flat universe
In a flat universe, all of the local curvature and local geometry is flat. In general it can be described by Euclidean space, however there are some spatial geometries which are flat and bounded in one or more directions. These include, in two dimensions, the cylinder and the torus. Similar spaces in three dimensions also exist.
[edit] Spherical universe
A positively curved universe is described by spherical geometry, and can be thought of as a three-dimensional hypersphere.
One of the endeavors in the analysis of data from the Wilkinson Microwave Anisotropy Probe (WMAP) is to detect multiple "back-to-back" images of the distant universe in the cosmic microwave background radiation. Assuming the light has enough time since its origin to travel around a bounded universe, multiple images may be observed. While current results and analysis do not rule out a bounded topology, if the universe is bounded then the spatial curvature is small, just as the spatial curvature of the surface of the Earth is small compared to a horizon of a thousand kilometers or so.
Based on analyses of the WMAP data, cosmologists during 2004-2006 focused on the Poincaré dodecahedral space (PDS), but also considered horn topologies to be compatible with the data.
[edit] Hyperbolic universe
A hyperbolic universe (frequently but confusingly called "open") is described by hyperbolic geometry, and can be thought of as something like a three-dimensional equivalent of an infinitely extended saddle shape. For hyperbolic local geometry, many of the possible three-dimensional spaces are informally called horn topologies.
The ultimate fate of an open universe is that it will continue to expand forever, ending in a Heat Death, a Big Freeze or a Big Rip.
[edit] See also
- Theorema Egregium − The "remarkable theorem" discovered by Gauss which showed there is an intrinsic notion of curvature for surfaces. This is used by Riemann to generalize the (intrinsic) notion of curvature to higher dimensional spaces.
- Extra dimensions in String Theory for 6 or 7 extra space-like dimensions all with a compact topology.
[edit] External links
- Oct 2003 − Poincaré dodecahedral model − WMAP statistics model predictions for a spherical local geometry
- Feb 2004 − Poincaré dodecahedral model − 11 degree matched circles
- Universe is Finite, "Soccer Ball"-Shaped, Study Hints. Possible wrap-around dodecahedral shape of the universe
- Hyperbolic universes with a Horned Topology and the CMB Anisotropy
- ^ Roukema, Boudewijn F.; Stanislaw Bajtlik, Marek Biesiada, Agnieszka Szaniewska, Helena Jurkiewicz (8 Dec 2006). "A weak acceleration effect due to residual gravity in a multiply connected universe". Astronomy and Astrophysics. Retrieved on 2006-12-08.
