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I was just told the CMBR energy diminishes with time. Can it be that the lost energy is somehow transferred to spacetime, causing the expansion to accelerate?

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No - the decreasing energy in the CMB is already well modeled in the Friedmann equations. The term in the density parameter that is proportional to $a^{-4}$ is the contribution of radiation energy density to the evolution of the universe, the term proportional to $a^{-3}$ is matter density (mostly dark, but includes ordinary matter), $a^{-2}$ is the contribution of the curvature of space-time itself, and the term without any factors of $a$ is the contribution of dark energy. The size of the radiation density, today, is already a small fraction of the matter density (about 0.03% of the matter density, 0.01% of the density of the universe overall - ordinary matter is about 5% overall).

The last time the energy density in the radiation fields was the same size as what's in the matter fields was around $z=3,300$.

I also disagree with @J.Chomel's answer - the energy stored in the radiation field is decreasing. Then energy density in the radiation field scales like $a^{-4}$, and the volume scales like $a^3$. Since the total energy is the energy density times the volume, the total energy scales like $a^{-1}$, just as you would expect with the number of photons being fixed, but the energy in each photon scaling as $a^{-1}$ as the wavelength increases.

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  • $\begingroup$ Then where does the diminished energy go? $\endgroup$ – descheleschilder Jan 19 '17 at 10:03
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    $\begingroup$ The question of "energy" in general relativity is more subtle than I can fully address in a comment, and care to address in an answer. Follow the link for a good analysis. $\endgroup$ – Sean Lake Jan 19 '17 at 10:19
  • $\begingroup$ The energy diminishes relatively. This is what I was trying to say; it diminishes because CMBR photons EM waves stretch. $\endgroup$ – J. Chomel Jan 19 '17 at 10:21
  • $\begingroup$ Okay, but if the total number of photons stays the same, the total CMBR-energy lowers. The cause of this lowering (the stretching of the photon wavelengths is due to the stretching of space) is not relevant. The energy diminishes absolutely. What do you mean if you say the energy diminishes relatively? Relative to the stretching space? Why doen't that diminish the energy absolutely? $\endgroup$ – descheleschilder Jan 19 '17 at 12:18
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    $\begingroup$ @J.Chomel: The energy does diminish. You cannot "look at the Universe from outside the Universe", not even theoretically. $\endgroup$ – pela Jan 20 '17 at 12:33

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