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These classical energy conditions are violated by quantum effects of order ℏ \hbar , with typical quantum violations being approximately ⟨ T μ ​ ν ⟩ v ​ i ​ o ​ l ​ a ​ t ​ i ​ o ​ n ≈ ℏ ​ c 9 / ( G ​ M ) 4 \langle T_{\mu\nu}\rangle_{violation}\approx\hbar c^{9}/(GM)^{4} . See 24 pp. 128–129. Here ℏ \hbar is Planck’s constant, c c is the speed of light, G G is Newton’s constant, and M M is the mass of the body under consideration. These quantum effects are not expected to be significant for large classical systems—particularly in cosmological settings. For a general discussion of quantum effects in semiclassical gravity see N. D. Birrell and P. C. W. Davies, Quantum fields in curved spacetime

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