We present a same-level comparison of the most prominent inversion methods for the reconstruction of the matter density field in the quasi-linear regime from the Ly$\alpha$ forest flux. Moreover, we present a pathway for refining the reconstruction in the framework of numerical optimization. We apply this approach to construct a novel hybrid method. The methods which are used so far for matter reconstructions are the Richardson-Lucy algorithm, an iterative Gauss-Newton method and a statistical approach assuming a one-to-one correspondence between matter and flux. We study these methods for high spectral resolutions such that thermal broadening becomes relevant. The inversion methods are compared on synthetic data (generated with the lognormal approach) with respect to their performance, accuracy, their stability against noise, and their robustness against systematic uncertainties. We conclude that the iterative Gauss-Newton method offers the most accurate reconstruction, in particular at small S/N, but has also the largest numerical complexity and requires the strongest assumptions. The other two algorithms are faster, comparably precise at small noise-levels, and, in the case of the statistical approach, more robust against inaccurate assumptions on the thermal history of the intergalactic medium (IGM). We use these results to refine the statistical approach using regularization. Our new approach has low numerical complexity and makes few assumptions about the history of the IGM, and is shown to be the most accurate reconstruction at small S/N, even if the thermal history of the IGM is not known. Our code will be made publicly available under this https URL.

中文翻译：

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基于现有重建方法定量比较的优化Lyα森林反演工具

我们对用于从 Ly$\alpha$ 森林通量重建准线性状态下的物质密度场的最突出反演方法进行了同级比较。此外，我们提出了一种在数值优化框架中改进重建的途径。我们应用这种方法来构建一种新的混合方法。迄今为止用于物质重建的方法是理查森-露西算法、迭代高斯-牛顿方法和假设物质和通量之间一一对应的统计方法。我们研究这些方法以获得高光谱分辨率，从而使热展宽变得相关。反演方法在合成数据（使用对数正态方法生成）上的性能、准确性、对噪声的稳定性、以及它们对系统不确定性的鲁棒性。我们得出结论，迭代高斯-牛顿法提供了最准确的重建，特别是在小 S/N 时，但也具有最大的数值复杂性并且需要最强的假设。其他两种算法速度更快，在小噪声水平下相对精确，并且在统计方法的情况下，对于星系间介质 (IGM) 的热历史的不准确假设更具有鲁棒性。我们使用这些结果来改进使用正则化的统计方法。我们的新方法具有较低的数值复杂性，并且对 IGM 的历史几乎没有假设，并且即使不知道 IGM 的热历史，也可以在小 S/N 下进行最准确的重建。我们的代码将在此 https URL 下公开提供。