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Our cosmology course instructor said that

At a given cosmic time $t=\tau$, the comoving observers define a spacelike hypersurface $\Sigma$ in which the universe looks homogenous and isotropic. Another observer, who is moving with a uniform velocity w.r.t the class of comoving observers will find the universe to be anisotropic. An intergalactic spaceship moving at a high speed would see the galactic distribution quite differently.

I have several questions based on his statement.

  1. What does he mean by "the comoving observers define a spacelike hypersurface $\Sigma$."?

  2. Is it that the homogeneity and isotropy of the Universe true only for the comoving observers? If yes, why?

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  • $\begingroup$ re 1, tangent vectors to $\Sigma$ have space-like norm (ie positive or negative depending on sign convention of the metric tensor), re 2, length contraction will distort the matter distribution $\endgroup$ – Christoph Mar 17 '17 at 18:27
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The bit about the spacelike hypersurface is simply a complex way of saying those observers can define their own notion of simultaneity. They pick a slice of space and say "the time value is the same everywhere here. I HAVE SPOKEN!". Of course, they have to define it in a way that preserves homogeneity and isotropy, but they can always do this if they are truly comoving.

When he says the non-comoving observers will see the universe as anisotropic, this is because of special relativity. Any peculiar velocity blue-shifts the CMB in front of you and red-shifts it behind you. Furthermore, the distribution of structures in the universe changes when you have a peculiar velocity because of how length contraction alters how you perceive relative densities at various angle from your direction of motion. We can see these effects from Earth. Both the WMAP and Planck data sets had to have the dipole anisotropy removed from the CMB data because of our roughly $360\,{\rm km}\,{\rm s}^{-1}$ peculiar velocity.

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