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How big is the universe?

Two questions hide in that one. How far can we see? And how big is all of space? The first has an answer for any model. The second has only a lower limit.

How far can we see?

Light has had only the age of the universe to travel. The farthest matter whose light could have reached us by now lies at the particle horizon, and everything inside it is the observable universe. For the Planck 2018 model its radius today is 14.2 Gpc, or 46.2 billion light years. That is more than three times the 13.8 billion light years that light covers in the age of the universe, because space went on expanding behind the light while it travelled. The CMB comes from almost that far away. The number is calculated with the standard radiation content included, since radiation controls the earliest part of the journey. The lesson computes the same quantity for your own model.

How big is all of space?

That depends on the curvature of space. If space is exactly flat or negatively curved (open), and has the simplest possible shape, it is infinite. If it is positively curved (closed) it is finite, like the surface of a sphere in one more dimension: it has a radius of curvature Rc = c/(H0√|Ωk|) and the far side of the universe is a distance πRc away. Measured curvature is very small, so Rc is very large, and what observations give is a lower limit: space is at least this big.

For example, giving the Planck model a slight positive curvature, Ωk = −0.01, makes Rc = 44.3 Gpc: the observable universe would reach only 10 % of the way to the far side.

A published lower limit

Bayesian model averaging gives parameter constraints that account for the uncertainty about which cosmological model is correct. Applied to curvature while allowing for evolving dark energy, it gave a curvature radius Rc > 42 Gpc (99%), corresponding to a lower limit of NU > 251 Hubble spheres (99%) on the size of the Universe.

For H0 = 67.66 km s−1 Mpc−1, a curvature radius of 42 Gpc corresponds to |Ωk| = 0.011. The limit is a statistical statement: it depends on the data used, which were those available when the paper was written, and on the prior assumptions of the analysis. For comparison, Planck 2018 combined with BAO gives Ωk = 0.001 ± 0.002, consistent with a flat universe.

What is computed and what is not

The radius of the observable universe is computed from the model, as exactly as any distance on this site. The size of the whole universe is not computed and cannot be: for the simplest shape it is bounded from below by how flat space is measured to be. An infinite universe and a very large finite one look the same from inside the horizon.

M. Vardanyan, R. Trotta & J. Silk (2011), Applications of Bayesian model averaging to the curvature and size of the Universe, MNRAS Letters 413, L91. More on the author page.

Questions for a class

  1. For the preset “Planck 2018 + BAO”, what is the radius of the observable universe in gigaparsecs and in billions of light years? Why is it larger than the distance light travels in the age of the universe?
  2. Raise ΩΛ by 0.01 so that Ωk = −0.01. What is the radius of curvature, how far away is the far side of this universe, and what fraction of that distance does the observable universe cover?
  3. For H0 = 67.66, what value of |Ωk| gives a radius of curvature of 42 Gpc, the lower limit quoted above?

Answers

  1. 14.2 Gpc, or 46.2 billion light years, against 13.8 billion light years for c × age. The space the light has already crossed keeps expanding behind it, so the matter that emitted it is now much farther away.
  2. Rc = 44.3 Gpc; the far side is πRc = 139 Gpc away; the observable universe (radius 14.2 Gpc) covers 10 % of it.
  3. |Ωk| = (c/H0 ÷ 42 Gpc)² = (4.43 ÷ 42)² = 0.0111.

Related: the curvature of the universe, the standard ruler, comoving distance.

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