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Symmetries correspond to specific properties of the space in question.

  • translation symmetry $\leftrightarrow$ homogeneity,
  • rotational symmetry $\leftrightarrow$ isotropy

What property is related to invariance under dilatations?

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It's

  • dilatation symmetry $\leftrightarrow$ scale invariant.
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  • $\begingroup$ The space expansion does not make the universe scale invariant. I think the question is unclear and so is your answer. $\endgroup$ – safesphere Oct 4 '18 at 15:29

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