Threshold of primordial black hole formation (1309.4201v4)

Abstract: 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 perturbation at the horizon crossing time in the uniform Hubble slice and the amplitude measure used in numerical simulations, respectively, while the conventional one gives $\delta^{\rm UH}{H c}=w$ and $\tilde{\delta}{c}=3w(1+w)/(5+3w)$. Our formula shows a much better agreement with the result of recent numerical simulations both qualitatively and quantitatively than the conventional formula. For a radiation fluid, our formula gives $\delta^{\rm UH}{H c}=\sin^{2}(\sqrt{3}\pi/6)\simeq 0.6203$ and $\tilde{\delta}{c}=(2/3)\sin^{2}(\sqrt{3}\pi/6)\simeq 0.4135$. We also discuss the maximum amplitude and the cosmological implications of the present result.

• The paper introduces a new analytic threshold formula for PBH formation, offering improved predictability over traditional estimates.
• Methodology involves deriving density perturbation amplitudes using uniform Hubble slices and simulation measures that match numerical data.
• The refined model deepens our understanding of early universe fluctuations and informs future predictions of primordial black hole abundance.

Threshold of Primordial Black Hole Formation: A Summary

This paper presents a thorough analysis of the conditions under which primordial black holes (PBHs) can form in an early universe dominated by a perfect fluid with the equation of state $p=w\rho c^2$. The authors propose a new analytic formula aimed at improving the predictability and reliability of this threshold when compared to the conventional estimates.

Derivation and Implications of the New Threshold Formula

The paper derives a new formula for the amplitude of density perturbation at the threshold of PBH formation. This formula applies in a universe where the equation of state $p=w\rho c^2$ holds true for $w \ge 0$. The authors present $$\delta_{Hc}^{\rm UH} = \sin^{2}[\pi \sqrt{w}/(1+3w)]$$ for the uniform Hubble slice and $$\tilde{\delta_c} = [3(1+w)/(5+3w)]\sin^{2}[\pi\sqrt{w}/(1+3w)]$$ for the amplitude measure used in numerical simulations. This stands in contrast to the conventional estimates, which suggest $\delta_{Hc}^{\rm UH} = w$ and $\tilde{\delta_c} = 3w(1+w)/(5+3w)$. The new formulaoffers a more precise agreement with recent numerical simulations both qualitatively and quantitatively.

For a radiation fluid (where $w=1/3$), the new formula predicts $\delta_{Hc}^{\rm UH} \approx 0.6203$ and $\tilde{\delta_c} \approx 0.4135$. These values are notably different and suggest modifications in the interpretation of PBH formation dynamics.

Numerical Simulations and Comparisons

Addressing the discrepancies observed in past formulations, this paper's threshold formula aligns more closely with the outcomes observed in numerical simulations, displaying a strong agreement — within approximately 20% deviation — over a wide range of $w$. Previous methods, particularly those employing Carr's condition, provide less precise insight into the dynamics at play during PBH formation. The comparison illustrates how the maximum amplitude of density perturbation derived here adheres more closely to numerical data, therefore suggesting potential refinements in the modeling of the early universe's conditions.

Future Directions and Theoretical Implications

On a practical level, this research holds implications for theoretical modeling of the early universe through the lens of primordial black holes. Understanding the conditions under which these black holes form can offer a window into the nature of early cosmological fluctuations.

Theoretically, the refined model underscores the complex interplay between matter properties encapsulated in the parameter $w$ and gravitational instability required for PBH formation. Future work might extend these insights to paper how varying initial conditions influence the cosmological landscape, possibly contributing to new interpretations of universe inflationary models and the distribution of cosmic structures.

Conclusion

The paper provides an analytically derived threshold for PBH formation that improves upon conventional models by incorporating recent numerical findings. It bridges the gap between theoretical astrophysics and practical simulation, offering a more consistent understanding of early universe conditions conducive to black hole formation. This research invites further investigation into the intersection of cosmology, gravitational theory, and high-energy physics, potentially translating these results into predictive tools for PBH abundance and distribution.