Spatial Curvature in Cosmological Models

Standard cosmological models are classified by spatial curvature into three broad types: open, flat, and closed universes. These correspond to negative, zero, and positive spatial curvature respectively, and each case has distinct implications for the geometry and global volume of space.

Open, Flat, and Closed Universes

In the simplest Robertson–Walker models the three cases can be described as:

• Open universe: Negative spatial curvature; typically an infinite spatial volume.
• Flat universe: Zero curvature; usually infinite in spatial extent in the standard models.
• Closed universe: Positive spatial curvature with a finite spatial volume; it can be visualized as the three-dimensional analogue of a sphere.

All three types are valid solutions of Einstein’s field equations. This means the question of whether the universe is finite or infinite cannot be decided by theory alone; it must be determined by observations of curvature and topology.

Observational Constraints on Curvature

Measurements of the cosmic microwave background (CMB) anisotropies provide the strongest direct constraints on spatial curvature. Current CMB data indicate that the universe is very close to flat. Exact zero curvature is the mathematical boundary between the open and closed families of solutions.

Observational uncertainties, however, are not zero. Small positive or negative curvature remains observationally allowed. To tighten constraints, cosmologists combine CMB measurements with other probes such as baryon acoustic oscillations (BAO) and Type Ia supernova observations, which help constrain both curvature and the expansion history.

Universe vs. Observable Universe

The observable universe is the region from which light has had time to reach us since the Big Bang; it therefore has a finite radius and finite volume. By contrast, the term universe in its broadest sense denotes all of physical space and may be much larger than the observable portion.

Regions beyond our observable horizon could either extend indefinitely or wrap back on themselves depending on the universe’s global geometry and topology. Observational access to those regions is limited by the finite age of the universe and the finite speed of light.

Key Takeaways

• Spatial curvature determines whether a cosmological model is open, flat, or closed.
• Current observations favor a universe very near to flat, but small curvature is still allowed within uncertainties.
• Whether the entire universe is finite or infinite depends on both curvature and topology and cannot be settled by theory alone.

References

• Sean Carroll, “Lecture Notes on General Relativity,” http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll8.html
• Komatsu et al., Astrophys. J. Suppl. 192 (2011) 18, http://arxiv.org/abs/1001.4538

Contributors

• bcrowell
• George Jones
• jim mcnamara
• marcus
• PAllen
• tiny-tim
• vela

Questions and Answers

1. What are the three types of cosmological models?
   • Open universe: Negative spatial curvature with typically infinite spatial volume.
   • Flat universe: Zero spatial curvature; usually infinite in standard Robertson–Walker models.
   • Closed universe: Positive spatial curvature with finite spatial volume, like a three-dimensional sphere.
2. Can we determine whether the universe is finite or infinite?
   • No. All three curvature cases are solutions to Einstein’s field equations, so observations are required to determine which case describes our universe.
3. What do current observations say about the universe’s curvature?
   • Current CMB measurements suggest the universe is very nearly flat, but small positive or negative curvature is still allowed within observational uncertainties.
4. What is the difference between the “universe” and the “observable universe”?
   • The observable universe is the region we can, in principle, observe, limited by the finite time light has traveled since the Big Bang and thus has finite volume. The universe as a whole may be finite or infinite depending on its curvature and topology.
5. Why is a perfectly flat universe considered a special case?
   • Exact flatness is the boundary between open and closed geometries. Mathematically it is a limiting, finely tuned condition compared with slightly open or slightly closed scenarios.