# How small could the universe be?

In the article Will the Universe expand forever?, NASA states:

We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in extent

However, it seems possible that the universe is not infinite; merely very large. What is the smallest possible universe consistent with this observation, assuming our observable universe is representative of the whole?

• My up-vote was motivated mainly by the good answers you brought out, such as Bapowell's relating the observed flatness to shape. The Planck data he cites favors a toroidal shape (in spacetime), which is particularly consistent with "universes" (regions that may become accessible) that are temporally or spatially local, and are described in such cosmologies as Nikodem J. Poplawski's "cosmology with torsion" (in a 2010 paper of that name and several more recent ones). They rely heavily on the Kerr metric for rotating black holes that are, for now, causally separated from each other. – Edouard Feb 23 '19 at 17:05

From Friedmann equations:

$$K=\frac{H_0^2 R_0^2}{c^2}(\Omega - 1)$$

For curvature K=-1 the universe is hyperbolic (infinite)

For K=0 the universe is flat (infinite)

For K=1 the universe is spheric (finite), then:

$$R_0=\frac c{H_0 \sqrt{\Omega-1}}$$

The best value that we have for the parameters, come from Planck Collaboration 2015:

Hubble Constant $$H_0=67.74 \ (km/s)/Mpc$$

Density ratio

$$\Omega=1.000 \pm 0.005$$ Of course, c is the speed oh light. The minimum size of the universe's radius will be for the maximum value of the density ratio $$\Omega=1.005$$ then $$R_0=204 \ billion \ light \ years$$ The radius of the observable universe is 46 billion light year. So, the radius of the universe is at least 4.4 times the radius of the observable universe.

It depends on the topology, but to summarize the results of the Planck collaboration (http://xxx.lanl.gov/abs/1303.5086): for a flat universe with toroidal topology, its radius is at least $0.9 \chi_r$, where $\chi_r$ is the comoving distance to the last scattering surface. Positively curved spaces are of similar size. The general idea is that if it was any smaller, we'd see 'matched' circles in the CMB arising from self-intersections of the last scattering surface.

• Do we know a lower bound on 𝜒𝑟? – Doradus Mar 27 '19 at 13:37