Shape of the universe

General relativity explains how spatial curvature (local geometry) is constrained by gravity.

The global topology of the universe cannot be deduced from measurements of curvature inferred from observations within the family of homogeneous general relativistic models alone, due to the existence of locally indistinguishable spaces with varying global topological characteristics.

To date, no compelling evidence has been found suggesting the topology of the universe is not simply connected, though it has not been ruled out by astronomical observations.

Different mathematical models of the universe's global geometry can be constructed, all consistent with current observations and general relativity.

For instance, a small closed universe would produce multiple images of the same object in the sky, though not necessarily of the same age.

Data from the 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 (predominantly photons and neutrinos), and dark energy or the cosmological constant:[4][5] The actual value for critical density value is measured as ρcritical = 9.47×10−27 kg⋅m−3.

Using a method similar to this, the BOOMERanG experiment has determined that the sum of the angles to 180° within experimental error, corresponding to Ωtotal ≈ 1.00±0.12.

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 when analyzed at a sufficiently large spatial scale.

The universe is often taken to be a geodesic manifold, free of topological defects; relaxing either of these complicates the analysis considerably.

A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold.

In the 1990s and early 2000s, empirical methods for determining the global topology using measurements on scales that would show multiple imaging were proposed[8] and applied to cosmological observations.

[9][10] In the 2000s and 2010s, it was shown that, since the universe is inhomogeneous as shown in the cosmic web of large-scale structure, acceleration effects measured on local scales in the patterns of the movements of galaxies should, in principle, reveal the global topology of the universe.

The latest research shows that even the most powerful future experiments (like the SKA) will not be able to distinguish between a flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10−4.

If the true value of the cosmological curvature parameter is larger than 10−3 we will be able to distinguish between these three models even now.

[14] Final results of the Planck mission, released in 2018, show the cosmological curvature parameter, 1 − Ω = ΩK = −Kc2/a2H2, to be 0.0007±0.0019, consistent with a flat universe.

In the absence of dark energy, a flat universe expands forever but at a continually decelerating rate, with expansion asymptotically approaching zero.

With dark energy, the expansion rate of the universe initially slows down, due to the effect of gravity, but eventually increases.

[16] A positively curved universe is described by elliptic geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold (such as the Poincaré dodecahedral space), all of which are quotients of the 3-sphere.

This was proposed by Jean-Pierre Luminet and colleagues in 2003[9][17] and an optimal orientation on the sky for the model was estimated in 2008.

[18] When cosmologists speak of the universe as being "open" or "closed", they most commonly are referring to whether the curvature is negative or positive, respectively.