On (minuscule) dark matter production in supernovae

It is believed that dark matter is made of particles, which interact with matter only weakly and gravitationally. One common candidate for dark matter are so called WIMPs. WIMPs, specifically, are heavy and may be their own antiparticles.

And as any other particles dark matter particles can be produced at sufficiently high energies. The mass of dark matter particles is unknown, but is estimated to be of order $1$-$100 \textrm{GeV}$, which corresponds to temperatures of $T_{DM}\approx 10^{13}$-$10^{15}\textrm{K}$, at which these particles may be expected to be produced.

Such enormous temperatures are barely attainable in any reasonable astrophysical processes, but say in core-collapse supernovae newly formed core has temperatures of $T_{SN,after}\approx 10^{11}\textrm{K}$, and probably more during the collapse phase. Then a crude estimate would suggest that the amount of dark matter produced is $M_{DM}\approx e^{-T_{DM}/T_{SN,max}}M_\odot$. Or, in number form $\log_{10}(M_{DM}/\textrm{kg})=30.3-0.43(T_{DM}/T_{SN})$. This means that at $T_{SN}=1.4\cdot 10^{-2}T_{DM}$ the amount of dark matter produced during a supernova will be around one kilogram. Such temperatures are fairly reachable for $1 \textrm{GeV}$ DM particles. So one can optimistically expect few kilograms of dark matter produced per supernova.

Now the question. What is a typical dark matter production in core-collapse supernovae? A good answer, I imagine, would be a more robust expansion on the existing estimate. Any constructive comments are welcome.

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