由WMAP天图的空间互相关联函数暗示的广义庞加莱十二面体空间假设的最佳相位
The optimal phase of the generalised Poincare dodecahedral space hypothesis implied by the spatial cross-correlation function of the WMAP sky maps
摘要：
一些研究提出，宇宙的形状可能是庞加莱十二面体空间（PDS），而不是一个无限的、简单连通的平坦空间。这两种模型都假设近似平坦的FLRW度量约为30%的物质密度。我们研究了PDS模型的两个预测。(i)对于正确的模型，覆盖空间中温度波动的空间两点交叉关联函数$\ximc$（其中任何对中的两个点位于最后散射表面（SLS）的不同副本上）应该与SLS单个副本上的自关联函数$\xisc$具有相似的数量级。(ii)通过在WMAP地图中最大化$\ximc$相对于$\xisc$，找到任意扭曲$ϕ$的“广义”PDS模型的最佳方向和确定的圆半径，应产生$ϕ\in \{\pm 36°\}$。我们使用马尔可夫链蒙特卡罗（MCMC）方法对方向、圆大小和$ϕ$进行了优化，得到了尺度小于4.0 h^-1 Gpc时的互相关。这两个预测都得到了满足：（i）对于WMAP三年全天空图的两个不同前景缩减版本（有和没有kp 2星系掩模），找到了一个最佳的“广义”PDS解，根据单连通假设，这些点之间有很强的互相关性，而这些点之间的相关性很弱：面中心是$(l,b)_{i=1,6}\approx (184d, 62d), (305d, 44d), (46d, 49d), (117d, 20d), (176d, -4d), (240d, 13d) to within ~2d, and their antipodes; (ii) this solution has twist ϕ= (+39 \pm 2.5)d, in agreement with the PDS model. The chance of this occurring in the simply connected model, assuming a uniform distribution $，面中心是\in [0，2 π]$，约为6- 9%。
英文摘要：
Several studies have proposed that the shape of the Universe may be a Poincare dodecahedral space (PDS) rather than an infinite, simply connected, flat space. Both models assume a close to flat FLRW metric of about 30% matter density. We study two predictions of the PDS model. (i) For the correct model, the spatial two-point cross-correlation function, $\ximc$, of temperature fluctuations in the covering space, where the two points in any pair are on different copies of the surface of last scattering (SLS), should be of a similar order of magnitude to the auto-correlation function, $\xisc$, on a single copy of the SLS. (ii) The optimal orientation and identified circle radius for a "generalised" PDS model of arbitrary twist $ϕ$, found by maximising $\ximc$ relative to $\xisc$ in the WMAP maps, should yield $ϕ\in \{\pm 36°\}$. We optimise the cross-correlation at scales < 4.0 h^-1 Gpc using a Markov chain Monte Carlo (MCMC) method over orientation, circle size and $ϕ$. Both predictions were satisfied: (i) an optimal "generalised" PDS solution was found, with a strong cross-correlation between points which would be distant and only weakly correlated according to the simply connected hypothesis, for two different foreground-reduced versions of the WMAP 3-year all-sky map, both with and without the kp2 Galaxy mask: the face centres are $(l,b)_{i=1,6}\approx (184d, 62d), (305d, 44d), (46d, 49d), (117d, 20d), (176d, -4d), (240d, 13d) to within ~2d, and their antipodes; (ii) this solution has twist ϕ= (+39 \pm 2.5)d, in agreement with the PDS model. The chance of this occurring in the simply connected model, assuming a uniform distribution $ϕ\in [0,2π]$, is about 6-9%.
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