Hubble, Hubble toil and trouble

(With apologies to Shakespeare)

"Double, double toil and trouble; Fire burn and cauldron bubble."

William Shakespeare, Macbeth (1606) act 4, scene 1, line 14

The 2003-02-15 issue of NewScientist has an article titled "Our Universe in glorious detail" on pages 12-13, about results from the Microwave Anisotropy Probe (MAP). On page 12 is a small sidebar "Our Universe: The Facts", which contains (among others) the following two items:

  • Age: 13.7 billion years
  • Hubble constant (expansion rate): 71 km/sec/megaparsec

Since I'm interested in pregeometry (a term coined in the 1977 paper "Is physics legislated by cosmogony?" by J. A. Wheeler and C. M. Patton, and meaning "...something deeper than geometry, that underlies both geometry and particles."), I wondered what the above would entail at the Planck scale.

First, I converted the Hubble Constant (Ho) from the given units of km/sec/parsec to mks units, and ended up with 2.300953293×10-18 m/s/m (a parsec is 3.0856775854×1016 m).

Then, I considered that dividing the Planck length (λp) by the Planck time (tp) yields the speed of light (c), which is 2.99792458×108 m/s (the speed of light entails moving the Planck length in each unit Planck time).

So, I decided I was interested in knowing at what distance the given value of Ho implies a relative velocity of c. The simple calculation (c/Ho) yields 1.302905448×1026 m.

And, since distances such as these are commonly expressed in light-years (1 ly = 9.46052973&times1015 m), we find that this critical distance is... drum roll, please... 1.377201×1010 (about 13.77 billion) light-years.

Curiously, the age of the universe given in the sidebar is 13.7 billion years. So, the given age of the universe is just about the number of years that would be required for the universe to reach the critical size if its net expansion rate over its history was c! Actually, its not quite so amazing, since the number probably came from the typical extrapolation based on Ho.

However, there is something interesting to be considered here. Estimates of the size of the visible universe put its diameter at somewhere around 28 Billion light years. For both this estimate and the Ho estimate from MAP to be true, two points at the edge of the visible universe, and diametrically opposite relative to our view point, have a relative velocity greater than c, according to Hubble's Law (a little over 2c for a diameter of 28 Billion light years)!

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