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If I lift a book up, I applied a force opposite its weight so there's positive work done. Dark energy is said to be pushing galaxies apart and the energy is coming from this negative work done.

  1. first question: is expansion of universe negative work done?

  2. Second question: how does negative work done explain dark energy must exert a negative pressure?

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To the contrary, dark energy has positive energy density. It does have negative pressure though, but it has to do work against the gravitational pull of the matter. Like the comments mentioned, energy is not conserved; photons are losing energy due to the cosmic redshift, and on the other hand the dark energy density is staying constant so the total energy is increasing with volume. Also the potential energy is going to zero the farther the galaxies recede from each other (it is negative when objects are close to each other and zero if they are infinitely far apart).

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