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arXiv 1503.03934gr-qcastro-ph.COhep-th
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宇宙学长波解与原初黑洞的形成

Cosmological long-wavelength solutions and primordial black hole formation

Tomohiro Harada, Chul-Moon Yoo, Tomohiro Nakama, Yasutaka Koga

备注：
33 pages, 4 figures, accepted for publication in Physical Review D
发表信息：
Phys. Rev. D 91, 084057 (2015)
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摘要：

在与长波方案相容的一般规范条件下，我们构造了无对称性的宇宙学长波解。然后，我们指定在不同的时间切片的解决方案之间的关系。将这个一般框架应用于球对称，我们导出了共形平坦空间坐标下常平均曲率切片中的长波解与共动规范下的渐近拟齐次解之间的对应关系，并比较了这两种方法下PBH形成的数值结果.为了讨论PBH的形成，使用$\tildeδ_{c}$是方便和常规的，该值是在最低阶长波长展开中，在层位入口处，在同动切片中在阈值处的平均密度扰动所取的值。数值计算发现，在补偿模型中，从过密区到FRW宇宙的跃迁越尖锐，$\tildeδ_{c}$就越大。我们建议，对于状态方程$p=(Γ-1)ρ$，我们可以应用最小值$\tildeδ_{c, {\rm min}}\simeq [3Γ/(3Γ+2)]\sin^{2}\left[π\sqrt{Γ-1}/(3Γ-2)\right]$和最大值$\tildeδ_{c, {\rm max}}\simeq 3Γ/(3Γ+2)$的解析公式。对于曲率变量$ψ_{0,c}$的阈值峰值，我们发现转变越尖锐，$ψ_{0,c}$变得越小。我们对这一特点进行了分析。使用简化的模型，我们还分析推导出的环境效应，$ψ_{0,c}$可以显着更大（更小），如果潜在的更长的波长的密度扰动是积极的（消极的）。

英文摘要：

We construct cosmological long-wavelength solutions without symmetry in general gauge conditions compatible with the long-wavelength scheme. We then specify the relationship among the solutions in different time slicings. Applying this general framework to spherical symmetry, we derive the correspondence relation between long-wavelength solutions in the constant mean curvature slicing with conformally flat spatial coordinates and asymptotic quasihomogeneous solutions in the comoving gauge and compare the numerical results of PBH formation in these two different approaches. To discuss the PBH formation, it is convenient and conventional to use $\tildeδ_{c}$, the value which the averaged density perturbation at threshold in the comoving slicing would take at horizon entry in the lowest-order long-wavelength expansion. We numerically find that within compensated models, the sharper the transition from the overdense region to the FRW universe is, the larger the $\tildeδ_{c}$ becomes. We suggest that, for the equation of state $p=(Γ-1)ρ$, we can apply the analytic formulas for the minimum $\tildeδ_{c, {\rm min}}\simeq [3Γ/(3Γ+2)]\sin^{2}\left[π\sqrt{Γ-1}/(3Γ-2)\right]$ and the maximum $\tildeδ_{c, {\rm max}}\simeq 3Γ/(3Γ+2)$. As for the threshold peak value of the curvature variable $ψ_{0,c}$, we find that the sharper the transition is, the smaller the $ψ_{0,c}$ becomes. We analytically explain this feature. Using simplified models, we also analytically deduce an environmental effect that $ψ_{0,c}$ can be significantly larger (smaller) if the underlying density perturbation of much longer wavelength is positive (negative).

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