# Implications of SMBH Entropy on DM

1. Aug 25, 2015

### michael879

I stumbled upon this article (among many other similar ones) http://arxiv.org/abs/0909.3983, that seems to correct the outdated assumption that the CMB dominates the entropy density of the universe. They find that super massive black holes are actually the primary contribution by many orders of magnitude.

The thing is, I was looking for the entropy density of the universe today in order to calculate the predicted relic density of a generic WIMP. The standard method of estimating this gives,
$\Omega \approx 8\sqrt{\dfrac{5\pi}{g_*}}\dfrac{x_fs_0}{H_0^2}\left(a+\dfrac{3b}{2x_f}\right)^{-1}$
where $g_*$ are the relativistic degrees of freedom at the freeze-out point xf=m/Tf, s0 is the current entropy density, H0 is the current Hubble parameter, and a and b represent the s/p-wave annihilation cross sections of the WIMP. You can see that the ratio of the current entropy density to the cross section of the WIMP is directly proportional to the relic density. Every source I can find assumes that the CMB determines the total entropy density, meaning they are underestimating the "WIMP" cross-sections by a factor of 1026! This appears to completely destroy the "WIMP miracle", forcing thermally produced dark matter to interact much more strongly..

This is crazy, so I'm assuming I made a mistake somewhere. However, the above formula follows pretty simply from the Boltzmann equation, H0 and Ω are measured quantities, and although you could adjust xf a little bit, any major change would violate the initial assumptions about WIMP freezeout (non-relativistic at the time of freezeout, which occurs during the radiation-dominated period)...

*edit* I think I may have found where my problem was, but it would be nice to get some confirmation. The above equation was derived under the assumption that the WIMP froze out during a radiation-dominated period, meaning that only the radiation entropy of the universe was considered. When I solved for the comoving density y=n/s in the radiation dominated period, I implicitly defined it as using the radiation entropy rather than the total entropy of the universe. This would explain why only the CMB entropy matters in the calculation, but it's still a little strange...

Last edited: Aug 25, 2015
2. Aug 25, 2015

### Staff: Mentor

I would be surprised if the entropy of black holes influences WIMP densities in any way. How should it?
I agree with your edit: the formula probably has been derived using the CMB entropy and labeling it as "total entropy". Might be a bit misleading, but the WIMP density is right if you plug in the entropy without black holes.

3. Aug 25, 2015

### michael879

Well I was thinking of it more as the total entropy of the universe "influencing" its thermal history. It wouldn't be that surprising if miscalculating the current entropy by a factor 1026 significantly altered our reconstructed past. Although now that I think about it, what matters is only the state of the universe at recombination when the CMB froze out, since it contained all the entropy during WIMP freezeout. Any entropy losses to SMBHs after that would surely be negligible, and since I don't think there even were any before hand, you're right that the CMB entropy and therefore the WIMP abundance should be completely independent of SMBH formation