Friday, February 10, 2012

distance measures

which of the cosmological distance measures are additive, $D(z_1,z_2) + D(z_2,z_3) = D(z_1,z_3)$ and which ones are monotonic, $dD/dz > 0$?

and for bonus points, like always: what's $i^i$ with $i = \sqrt{-1}$?


  1. I hope I've got it right:

    1. Since the proper and comoving distances are integrals their linearity should allows to add them transitively.
    2. The proper and comoving distances is monotonically increasing, but I think that the angular diameter distance can violate this dependent on the cosmological model. If the universe has positive spatial curvature a maximum should occur, because two meridians on a spherical surface starting form the pole approach each other again after the equator. Since the luminosity distance is directly related to the angular diameter distance via the Etherington relation I think that it can have a maximum as well, but I'm not sure if the a^2 factor can correct for that in general.
    3. i^i = exp(i \pi/2)^i = exp(i^2 \pi/2) = exp(-\pi/2)

  2. i find it funny that i^i is a real number :)