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宇宙学中的大质量中微子：解析解和流体近似

Massive Neutrinos in Cosmology: Analytic Solutions and Fluid Approximation

Masatoshi Shoji, Eiichiro Komatsu

备注：
18 pages, 7 figures. (v2) with up-dated reference, minor typos corrected. (v3) An error in the normalization on the scale factor has been corrected. We now find that the fluid approximation is accurate at a few~25% accuracy. This version supersedes the published version. The code for calculating exact solutions of Ψ_l is publicly available at http://gyudon.as.utexas.edu/~komatsu/CRL/index.html
发表信息：
Phys.Rev.D81:123516,2010; Erratum-ibid.D82:089901,2010
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摘要：

我们研究了z<1000红移下自由流态大质量中微子线性密度涨落的演化，并对流体近似的使用给出了明确的解释。我们求解了Einstein deSitter（EdS）宇宙中的无碰撞玻尔兹曼方程，在lmax=1和2处截断玻尔兹曼方程，并将由此得到的中微子密度对比δ ν^{fluid}与本文导出的玻尔兹曼方程的精确解进行了比较。粗略地说，如果当一个给定波数的中微子密度涨落进入视界时，中微子已经是非相对论性的，那么流体近似是准确的。我们发现，在k<0.4 hMpc^{-1}，红移z<10的情况下，对于0.05<mv <0.5eV的大质量中微子，流体近似的精度只有几%到25%.这一结果量化了流体近似对于m_v <0.5eV的大质量中微子的局限性。我们还发现，从流体方程（即，连续性和欧拉方程）在较低的红移处变得更好的近似，并且可以通过在欧拉方程中包括各向异性应力项来进一步提高精度。各向异性应力项有效地将压力项增加了9/5倍。

英文摘要：

We study the evolution of linear density fluctuations of free-streaming massive neutrinos at redshift of z<1000, with an explicit justification on the use of a fluid approximation. We solve the collisionless Boltzmann equation in an Einstein de-Sitter (EdS) universe, truncating the Boltzmann hierarchy at lmax=1 and 2, and compare the resulting density contrast of neutrinos, δ_ν^{fluid}, with that of the exact solutions of the Boltzmann equation that we derive in this paper. Roughly speaking, the fluid approximation is accurate if neutrinos were already non-relativistic when the neutrino density fluctuation of a given wavenumber entered the horizon. We find that the fluid approximation is accurate at few to 25% for massive neutrinos with 0.05<m_ν<0.5eV at the scale of k<0.4~hMpc^{-1} and redshift of z<10. This result quantifies the limitation of the fluid approximation, for the massive neutrinos with m_ν<0.5eV. We also find that the density contrast calculated from fluid equations (i.e., continuity and Euler equations) becomes a better approximation at a lower redshift, and the accuracy can be further improved by including an anisotropic stress term in the Euler equation. The anisotropic stress term effectively increases the pressure term by a factor of 9/5.

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