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Traditionally when deriving Friedmann–Lemaitre–Robertson–Walker metric, homogeneity and isotropy assumption is made. This assumption, however, holds only for large scales. To take into account some small-scale irregularities, cosmological perturbations are introduced. But as far as my knowledge goes, these do not explain how does the universe evolve inside those perturbations (e.g. galaxies). So my question is: what does the expansion of universe mean for regions where matter is concentrated - is there slower(or faster?) expansion, or no expansion at all? What would it look like for universe without cosmological constant (i.e. without dark energy)?

I-L-P
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Not every part of the universe expands at an equal rate. Your body is not expanding, and neither is the Earth, the Solar System, or the Milky Way. The local densities are greater than the critical density and that counteracts expansion.

Dark energy is not directly relevant to this. It's possible that, depending on the exact properties of dark energy, the universe's expansion will get faster and faster indefinitely. That could lead to the big rip scenario where everything, including individual atoms, is torn asunder. Without dark energy nothing much changes: the universe would still be expanding (albeit the rate of expansion would be slowing down), and your body, the Earth, etc would not be expanding.

Allure
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  • Your body is not expanding, and neither is the Earth, the Solar System, or the Milky Way. The local densities are greater than the critical density and that counteracts expansion. This is totally wrong. There is no critical density, and all of the systems you mentioned are expanding, but the effect is much too small to measure. –  Jan 18 '18 at 23:12
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    @BenCrowell Pretty confident the statement is correct. I heard it directly from one of the discoverers of Dark Energy, Prof. Brian Schmidt. A brief Google search concurs. If you still think it's wrong, I'd like to see sources. – Allure Jan 18 '18 at 23:28