# Is the total energy of the CMBR diminishing with time?

Is the total energy of the CMBR diminishing with time? Since it's existence the volume of the Universe (at least the visible part) has grown by a factor three, which is to say, the number of photons per unit volume is diminished by a factor 27. But the wavelength of these photons has grown by a much bigger factor. So you should think the total energy of all the CMBR photons has become less over time.

• The CMB was emitted at $z\simeq1100$, so the volume of the Universe has increased by a factor $1100^3\sim10^9$ since the CMB was emitted, i.e. much, much more that 27. – pela Jan 19 '17 at 10:09
• This question is ambiguous. Are you talking about total energy or energy density? They're not the same. You specifically say "total energy", but then mention photons per volume, indicating you're talking about energy density. – zephyr Jan 19 '17 at 14:36
• @pela-Since the emergence of the CMBR, about 300 000 years after the big bang, space has stretched with a factor 3 (that's why the radius of the visible Universe is now about 40 000 000 000 lightyears). So per unit volume the number of photons has decreased by a factor 27. I think you mean the radius of the visible Universe has increased by a factor $\ 10^9$ since 300 000 years after the big bang.This doesn't change the fact that every unit of volume has increased by a factor 27, and the number of photons décreased by the same factor. – descheleschilder Jan 19 '17 at 16:37
• @zephir-II mean the total energy of the CMBR, and mentioned that the number of photons per unit volume has decreased by a factor 27 because if the number of photons in one unit of volume has decreased, the number of photons for the whole volume of the Universe has decreased by the same factor. It's like the number of atoms in a gas with volume V. If you increaese each unit volume by a factor 3, then the whole volume is also increased by a factor three, and the number of atoms per unit volume dercreases by a factor 27, like is the case for the whole volume V. – descheleschilder Jan 19 '17 at 16:49

Yes. The energy density in the radiation field scales like $a^{-4}$, and the volume scales like $a^{3}$. Since the total energy is the density times the volume, the total energy scales like $a^{-1}$. Note that $a$ is called the "scale factor" and $a = \frac{1}{1+z}$, where $z$ is the redshift, assuming $z=0$ now.
• Time and redshift are related. $z=0$ now purely by convention/definition. In the future redshift will be negative, and if the universe doesn't recollapse, will go to $-1$ in the infinitely distant future. – Sean Lake Jan 19 '17 at 7:48