The shape of the Universe is an informal name for a subject of investigation within physical cosmology which describes the geometry of the universe including both local geometry and global geometry. It is loosely divided into curvature and topology, even though strictly speaking, it goes beyond both. More formally, the subject in practice investigates which 3-manifold corresponds to the spatial section in comoving coordinates of the 4-dimensional space-time of the Universe.
Cosmologists normally work with a given space-like slice of spacetime called the comoving coordinate system. In terms of observation, the section of spacetime that can be observed is the backward light cone (points within the cosmic light horizon, given time to reach a given observer). For related issues, see distance measures (cosmology). The related term Hubble volume can be used to describe either the past light cone or comoving space up to the surface of last scattering. From the point of view of special relativity alone, speaking of "the shape of the universe (at a point in time)" is ontologically naive because of the issue of relativity of simultaneity: you cannot speak of different points in space being "at the same point in time", thus you cannot speak of "the shape of the universe at some point in time". However, the existence of a preferred set of comoving is possible and widely accepted in present-day physical cosmology.
If the observable universe is smaller than the entire universe (in some models it is many orders of magnitude smaller), one cannot determine the global structure by observation: one is limited to a small patch. Conversely, if the observable universe encompasses the entire universe, one can determine the global structure by observation. Further, the universe could be small in some dimension and not in others (like a cylinder): if a small closed loop exists, one would see multiple images of objects in the sky.
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:
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:
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 Space-time intervals, we can approximate 3-space by the familiar Euclidean geometry.
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 then 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 (in mathematics these are called the model geometries):
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.
In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.
A global geometry is also called a topology, as a global geometry is a local geometry plus a topology, but this terminology is misleading because a topology does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.
Two strongly overlapping investigations within the study of global geometry are whether the universe:
For spherical and hyperbolic spatial geometries, the curvature gives a scale (either by using the radius of curvature or its inverse), a fact noted by Carl Friedrich Gauss in an 1824 letter to Franz Taurinus.
The probability of detection of the topology by direct observation depends on the spatial curvature: 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.
A compact space is a stronger condition: in the context of Riemannian manifolds, it is equivalent to bounded and geodesically complete. If we assume that the universe is geodesically complete, then boundedness and compactness are equivalent (by the Hopf–Rinow theorem), and they are thus used interchangeably, if completeness is understood.
If the spatial geometry is spherical, the topology is compact. For a flat or a hyperbolic spatial geometry, the topology can be either compact or infinite: for example, Euclidean space is flat and infinite, but the torus is flat and compact.
In cosmological models (geometric 3-manifolds), a compact space is either a spherical geometry, or has infinite fundamental group (and thus is called "multiply connected", or more strictly non-simply connected), by general results on geometric 3-manifolds.
Compact geometries can be visualized by means of closed geodesics: on a sphere, a straight line, when extended far enough in the same direction, will reach the starting point.
Note that on a compact geometry, not every straight line comes back to its starting point. For instance, a line of irrational slope on a torus never returns to its origin. Conversely, a non-compact geometry can have closed geodesics: on a cylinder, which is a non-compact flat geometry, a loop around the cylinder is a closed geodesic.
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.
The alternative two-dimensional spaces with a Euclidean metric are the cylinder and the Möbius strip, which are bounded in one direction but not the other, and the torus and Klein bottle, which are compact.
In three dimensions, there are 10 finite closed flat 3-manifolds, of which 6 are orientable and 4 are non-orientable. The most familiar is the 3-Torus.
Absent dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching some fixed rate. With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases. The ultimate fate of the universe is the same as that of an open universe.
Analysis of data from the Wilkinson Microwave Anisotropy Probe (WMAP) looks for multiple "back-to-back" images of the distant universe in the cosmic microwave background radiation. It may be possible to observe multiple images of a given object, if the light it emits has had sufficient time to make one or more complete circuits of a bounded universe. Current results and analysis do not rule out a bounded global geometry (i.e. a closed universe), but they do confirm that 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. If the universe is bounded, this does not imply anything about the sign or zeroness of its curvature.
In a closed universe lacking the repulsive effect of dark energy, gravity eventually stops the expansion of the universe, after which it starts to contract until all matter in the observable universe collapses to a point, a final singularity termed the Big Crunch, by analogy with Big Bang. However, if the universe has a large amount of dark energy (as suggested by recent findings), then the expansion of the universe could continue forever.
Based on analyses of the WMAP data, cosmologists during 2004-2006 focused on the Poincaré dodecahedral space (PDS), but horn topologies (which are hyperbolic) were also deemed compatible with the data.
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