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arXiv 1503.08506astro-ph.CO
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从引力透镜看距离-红移关系的偏差

On the Bias of the Distance-Redshift Relation from Gravitational Lensing

Nick Kaiser, John A. Peacock

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32 pages, 5 figures, submitted to MNRAS
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摘要：

宇宙学中一个长期存在的问题是引力透镜是否改变了距离-红移关系$D(z)$或源的平均通量密度。最近的非线性相对论微扰理论的研究重新点燃了人们对这一点的兴趣，这些研究发现，在恒定红移的表面面积和到该表面的平均距离上都存在偏差，在这两种情况下都存在分数偏差，偏差的量级为均方收敛$\langle κ^2 \rangle$。任何这样的面积偏差都可能改变CMB宇宙学，而平均通量密度的相应偏差可能影响超新星宇宙学。在这里，我们表明，在一个合奏平均的意义上，扰动的表面面积的常数红移实际上是小得多，是在累积弯曲角平方的顺序，或大致的百万分之一的影响。这验证了Weinberg（1976）的论点，即源的平均放大率$μ$是1，以及Kibble \& Lieu（2005）的论点，即平均方向平均逆放大率是1。这也验证了在分析CMB各向异性时常规的透镜处理。但是，放大率的分散性的存在将导致这些守恒量的任何非线性函数在统计上有偏差。例如，距离$D$与$μ^{-1/2}$成比例，因此即使$\langle μ\rangle=1$，透镜效应也会使$\langle D\rangle$偏置。这些量的分数偏差通常为$\langle κ^2 \rangle$级，其数量级大于面积扰动。声称辐射源的面积或通量密度存在大偏差，似乎是由于对这种效应的误解：它们并不代表一种新的非牛顿效应，也不使标准宇宙学分析失效。

英文摘要：

A long standing question in cosmology is whether gravitational lensing changes the distance-redshift relation $D(z)$ or the mean flux density of sources. Interest in this has been rekindled by recent studies in non-linear relativistic perturbation theory that find biases in both the area of a surface of constant redshift and in the mean distance to this surface, with a fractional bias in both cases on the order of the mean squared convergence $\langle κ^2 \rangle$. Any such area bias could alter CMB cosmology, and the corresponding bias in mean flux density could affect supernova cosmology. Here we show that, in an ensemble averaged sense, the perturbation to the area of a surface of constant redshift is in reality much smaller, being on the order of the cumulative bending angle squared, or roughly a part-in-a-million effect. This validates the arguments of Weinberg (1976) that the mean magnification $μ$ of sources is unity and of Kibble \& Lieu (2005) that the mean direction-averaged inverse magnification is unity. It also validates the conventional treatment of lensing in analysis of CMB anisotropies. But the existence of a scatter in magnification will cause any non-linear function of these conserved quantities to be statistically biased. The distance $D$, for example, is proportional to $μ^{-1/2}$ so lensing will bias $\langle D\rangle$ even if $\langle μ\rangle=1$. The fractional bias in such quantities is generally of order $\langle κ^2 \rangle$, which is orders of magnitude larger than the area perturbation. Claims for large bias in area or flux density of sources appear to have resulted from misinterpretation of such effects: they do not represent a new non-Newtonian effect, nor do they invalidate standard cosmological analyses.

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