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The cosmic microwave background (CMB) radiation comprises about 98% of all electromagnetic radiation in the universe. And, from the creation of the CMB to today that electromagnetic radiation has red shifted to about 1100 times its original wavelength. And, the energy content of electromagnetic radiation is inversely proportionally to its wavelength. Therefore, the CMB has shed an immense amount of energy in the last 13.8 billion years because of the redshift due to the expanding universe.

How does this amount of energy compare to the estimate of Dark Energy in the universe?

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In terms of direct numbers at present, dark energy comprises about 70% of all of the energy in the universe. Radiation, on the other hand, makes up less than 0.005% of the energy in the universe. It's such a small fraction that it's less than the error associated with the values for matter and dark energy.

A good way to approximate how the two energies compare over time (with expansion included) is through the scale factor of the metric, $a$. The scale factor represents the ratio of the distance between two points at any given moment of time to the distance between those two points now. Naturally, as the universe expands, the amount of volume in a given region of the universe increases like $a^3$. With that said, let's take a look at the volume density of both your types of energy.

As you quite correctly pointed out, the expansion of the universe redshifts radiation, which means the universe loses that energy entirely. Add to that the fact that the number density of a fixed number of photons is proportional to $\frac{1}{a^3}$, and it's not hard to understand why relativity says the total energy density of radiation decreases like $\frac{1}{a^4}$. In other words the total amount of energy contained in radiation decreases approximately like $\frac{1}{a}$.

As for dark energy, the current accepted model, $\Lambda$-CDM, treats dark energy like a constant energy density. That means as the universe expands, the amount of dark energy per unit volume remains constant. Yikes. This means the total amount of dark energy increases like $a^3$.

Add them together and you see there is a net increase in total energy of the universe (the total energy of matter remains more or less constant). Clearly it isn't the case that the energy lost from radiation is picked up as dark energy. But, of course, you already knew that. You had already gone as far as realizing that the radiation energy fell off like $\frac{1}{a}$ and there would need to be a seriously funky (<-- technical term) relationship between $a$ and the energy density of dark energy for the two totals to sum to a constant. Kudos to you for having figured this out by yourself and asking an excellent follow-up question.

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  • $\begingroup$ +1, but since the OP asks "how does $E_\mathrm{CMB}$ compare to $E_\Lambda$", you could provide an actual number here. $\endgroup$ – pela Aug 5 '16 at 14:16
  • $\begingroup$ @Jim, do you have an estimate for the integral of the 1/a loss over 13.8 billion years as compared to DE? Was the CMB energy loss ever greater than the DE? $\endgroup$ – Allyn Shell Aug 5 '16 at 15:42
  • $\begingroup$ @AllynShell I don't have the exact numbers. But they aren't too hard to estimate. Blueshift the CMB in reverse by 1100 and that should give you the approximate order of magnitude of radiation energy. I do know that the total amount of dark energy gained is greater than the total amount of radiation energy lost. That's easy to see. No matter when you start counting, the amount of dark energy gained from one moment to the other is always greater than the radiation energy lost. $\frac{a_f^3}{a_i^3}>\frac{a_f}{a_i}>1$. That's simple math. $\endgroup$ – Jim Aug 5 '16 at 18:32
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The dynamics of FLRW cosmology is really not that different from elementary Newtonian mechanics. One can derive some of the salient aspects of this using Newtonian mechanics. The energy $E~=~K~+~V$ has the total energy $E$ constant, and the kinetic $K$ and potential $V$ energies add and subtract from each other to maintain a constant total energy. The scale factor for the evolution of the spacetime, which I work out in a Newtonian framework here How did the universe shift from "dark matter dominated" to "dark energy dominated"? within a Newtonian context.

The more complete general relativistic equation is $$ H^2~=~\left(\frac{\dot a}{a}\right)^2~=~H_0\left[\frac{\Omega_m}{a^3}~+~\frac{\Omega_r}{a^4}~+~(1~-~\Omega_r~-~\Omega_m)a^{-3(1+w)}\right] $$ Here $\Omega_m$ holds for matter, $\Omega_r$ for radiation. $\Omega_m$ is $.26$ for dark matter and $.04$ for luminous matter, so $\Omega_m~=~.3$, and currently $\Omega_r$ is very small. This equation describes the dynamics according to dark energy or the vacuum. In a Newtonian framework the left hand side is kinetic energy and the right potential. The loss of energy in photons means the photon by being stretched has its energy taken in by spacetime.

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