Threshold of primordial black hole formation

@article{Harada2013ThresholdOP,
  title={Threshold of primordial black hole formation},
  author={Tomohiro Harada and Chul-Moon Yoo and Kazunori Kohri},
  journal={Physical Review D},
  year={2013},
  volume={88},
  pages={029903}
}
Based on a physical argument, we derive a new analytic formula for the amplitude of density perturbation at the threshold of primordial black hole formation in the universe dominated by a perfect fluid with the equation of state $p=w\rho c^{2}$ for $w\ge 0$. The formula gives $\delta^{\rm UH}_{H c}=\sin^{2}[\pi \sqrt{w}/(1+3w)]$ and $\tilde{\delta}_{c}=[3(1+w)/(5+3w)]\sin^{2}[\pi\sqrt{w}/(1+3w)]$, where $\delta^{\rm UH}_{H c}$ and $\tilde{\delta}_{c}$ are the amplitude of the density… 

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