Critical density

The galaxies in the universe are all flying away from each other. The speeds of nearby galaxies are proportional to the distance the galaxy is away from us. This relation, v=Hdv=Hd is known as Hubble's law and the constant HH is known as Hubble's constant. The evolution of our universe is determined by general relativity and the amount of matter, dark matter, and dark energy in our universe. If we ignore dark energy (pretend there is none) one can determine the "critical density" of the universe. If the universe is more dense than the critical density of the universe, the universe will eventually crash back together, whereas if the density is less than the critical density, the universe will fly apart forever. This was a big question up until the discovery of dark energy.

Interestingly enough, one can determine the critical density through Newtonian physics. Consider a galaxy a distance dd away from us, moving radially away with a velocity given by Hubble's law. If the galaxy is not to escape to infinity, what is the critical density in # atoms of hydrogen per cubic meter?

Note: enter your answer to the nearest hundredth.

Details and assumptions

  • The Hubble constant is 68 km/s/Megaparsec.
  • Assume that all the matter is uniformly distributed throughout space.
  • Newton's gravitational constant is 6.67×1011 Nm2/kg26.67 \times 10^{-11}~\mbox{Nm}^2/\mbox{kg}^2.

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