Trying to plot a graph of energy densities for radiation and PBHs

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di_a_tonic17
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So a bit of context, I've been trying to model a plot of energy density relation for radiation and Primordial BH, but it is not right. Would someone be able to help me out? This is what I have so far. I am assuming the PBHs decay into a Planck relic, and the decay is instantaneous. Does the conservation law not follow through for the 2nd, 3rd, and 4th plots? Also, it completely breaks down for smaller values of beta. I'm attaching my calculation breakdown in the images below.


Any help would be appreciated.


If any more details are required, I'd be happy to provide them
 
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di_a_tonic17 said:
I'm attaching my calculation breakdown in the images below.
Images for equations are not acceptable here. Please use the PF LaTeX feature to post equations directly. There is a "LaTeX Guide" link at the bottom left of each post window.