The 21-cm hyperfine line of neutral hydrogen is a useful tool to probe the conditions of the Universe during the Dark Ages, Cosmic Dawn, and the Epoch of Reionisation. In most of the current calculations, the 21-cm line signals at given frequencies are computed, using an integrated line-of-sight line opacity, with the correction for cosmological expansion. These calculations have not fully captured the line and continuum interactions in the radiative transfer, in response to evolution of the radiation field and the variations of thermal and dynamic properties of the line-of-sight medium. We construct a covariant formulation for the radiative transfer of the 21-cm line and derive the cosmological 21-cm line radiative transfer (C21LRT) equation. The formulation properly accounts for local emission and absorption processes and the interaction between the line and continuum when the radiation propagates across the expanding Universe to the present observer. Our C21LRT calculations show that methods simply summing the line optical depth could lead to error of 5% in the 21-cm signals for redshift z ∼ 12 − 35 and of $>10~{{\%}}$ for redshift z ≲ 8. Proper covariant radiative transfer is therefore necessary for producing correct theoretical templates for extracting information of the structural evolution of the Universe through the Epoch of Reionisation from the 21-cm tomographic data.

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21 厘米线的宇宙学辐射传输的协变公式

中性氢的 21 厘米超细线是探测黑暗时代、宇宙黎明和再电离时代宇宙状况的有用工具。在当前的大多数计算中，使用积分视线线不透明度以及宇宙膨胀校正来计算给定频率下的 21 厘米线信号。这些计算尚未完全捕获辐射传输中的线和连续相互作用，以响应辐射场的演变以及视线介质的热和动态特性的变化。我们构建了 21 厘米线辐射传输的协变公式，并推导出宇宙学 21 厘米线辐射传输（C21LRT）方程。该公式正确地解释了当辐射穿过膨胀的宇宙传播到当前观察者时的局部发射和吸收过程以及线和连续体之间的相互作用。我们的 C21LRT 计算表明，简单地求和线光学深度的方法可能会导致红移 z ∼ 12 − 35 的 21 厘米信号误差为 5%，红移 z ≲ 的误差为 $>10~{{\%}}$ 8. 因此，适当的协变辐射传输对于生成正确的理论模板是必要的，以便从 21 厘米断层扫描数据中提取再电离时期宇宙结构演化的信息。