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Does our "energy scale" inflate as the universe expands? In other words, is the amount of energy we measure as 1 joule today slightly higher than the amount of energy we measured as 1 joule yesterday?

So for example, if you take a cube of length $l$ and measure how much mass there is in it, call it $m$, you can calculate the energy density as

$$ \rho_{m,1} = \frac{mc^2}{l^3} $$

Now imagine that the universe expands, such that the size of the box is now $al$. The amount of matter inside the box remains unchanged. Does the energy density inside it then become

$$ \rho_{m,2} = \frac{amc^2}{(al)^3} = \frac{1}{a^2}\frac{mc^2}{l^3} = a^{-2}\rho_{m,1} $$

despite the fact that we observe the new energy density to be

$$ \rho_{m,2} = \frac{mc^2}{(al)^3} = \frac{1}{a^3}\frac{mc^2}{l^3} = a^{-3}\rho_{m,1} $$

? That is, does the energy density associated with the component $m$ change with the scale of the universe according to

$$ \rho_M \sim a^{-2} \tag{1} $$

?

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There are roughly three types of components in our universe: Matter ($M$), Radiation ($R$), and some sort of fluid ($\Lambda$). The first two components get diluted when the universe expands, so does the energy associated with them.

So for example, if you take a cube of length $l$ and measure how much mass there is in it, call it $m$, you can calculate the energy density as

$$ \rho_{m,1} = \frac{mc^2}{l^3} $$

Now imagine that the universe expands, such that the size of the box is now $al$. The amount of matter inside the box remains unchanged. The energy density inside it then becomes

$$ \rho_{m,2} = \frac{mc^2}{(al)^3} = \frac{1}{a^3}\frac{mc^2}{l^3} = a^{-3}\rho_{m,1} $$

That is, the energy density associated with the component $m$ changes with the scale of the universe according to

$$ \rho_M \sim a^{-3} \tag{1} $$

You can repeat the experiment for radiation, and what you get is

$$ \rho_R\sim a^{-4} \tag{2} $$

Now comes the interesting part: the energy density associated with the third component does not change

$$ \rho_\Lambda \sim a^0 \tag{3} $$

That is, the more meters of mater you create, the more joules of the $\Lambda$ component are created. You can think about it for a bit, and probably will reach the conclusion that the only substance that does that is vacuum! But we are far from understanding this for sure

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