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That is, does the vacuum energy density change with the scale $a$ of the universe according to

$$ \rho_Λ \sim a $$

?

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    $\begingroup$ Why would it obey that particular scaling law? Don't make us guess about the motivation for the question. $\endgroup$ – Ben Crowell Mar 17 '18 at 1:52
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If the vacuum energy behaves like a cosmological constant then it does not change as the universe expands. That is, the vacuum energy per cubic metre is the same now as it was billions of years ago and the same as it will be billions of years from now. It is this constant value of the vacuum energy density that causes vacuum energy to behave as if it had a negative pressure. It also means vacuum energy appears from nowhere as space expands i.e. it violates conservation of energy. Although this appears odd it is entirely reasonable because the law of conservation of energy does not apply to the expanding universe.

But we don't know the origin of dark energy or how it behaves and there are alternative theories such as quintessence in which the energy density does change with time. So for now we cannot make any definitive statement.

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No. $\rho_{\Lambda}\propto a^3$ (because we have 3 spacial dimensions).

In a sense, $a$ is a 1D measure of a characteristic volume of the universe (e.g. think the side length if your characteristic volume of universe is a box, or the radius if your characteristic volume of universe is a sphere).

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  • $\begingroup$ oh wow........I have no clue what I was thinking when I wrote that. (I must have been thinking $E_{total}$ or something, not $\rho$.) Correction: $\rho_{\Lambda}=$ exactly constant (for cosmological constant only). Energy will scale $E\propto a^3$. $\endgroup$ – Bob Mar 19 '18 at 20:21
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Is the vacuum energy density proportional to the scale of the universe?

I think it does. I do not buy the assertions that claim it's constant, and that energy is being created as the universe expands. In ordinary life, or real life, I know of no circumstances in which energy is not conserved. So I am suspicious when people make claims such as "energy is not conserved in GR". When you drop your pencil energy is conserved. When you lift a bag of cement energy is conserved. Conservation of energy is one of the most important tenets of physics. So I treat claims which say energy is magicked up out of nothing with suspicion.

That is, does the vacuum energy density change with the scale a a of the universe according to $\rho_Λ \sim a$.

No, see what Bob said. But I think yes, the vacuum energy reduces as the universe expands, and I think we have hard scientific evidence of this. See the NASA article galaxy clusters reveal new dark matter insights. It concerns a study of 8,648 galactic clusters by Hironao Miyatake and 6 other authors. The article says the internal structure of a cluster is linked to the dark matter environment it’s in. It quotes co-author David Spergel saying this: “our work has shown that ‘age matters’: Younger clusters live in different large-scale dark-matter environments than older clusters”.

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