Boltzmann solvers are an important tool for the computation of cosmological observables in the linear regime. In the presence of massive neutrinos, they involve solving the Boltzmann equation followed by an integration in momentum space to arrive at the desired fluid properties, a procedure which is known to be computationally slow. In this work we introduce the so-called generalized Boltzmann hierarchy (GBH) for massive neutrinos in cosmology, an alternative to the usual Boltzmann hierarchy, where the momentum dependence is integrated out leaving us with a two-parameter infinite set of ordinary differential equations. Along with the usual expansion in multipoles, there is now also an expansion in higher velocity weight integrals of the distribution function. Using a toy code, we show that the GBH produces the density contrast neutrino transfer function to a $\lesssim 0.5\%$ accuracy at both large and intermediate scales compared to the neutrino free-streaming scale, thus providing a proof-of-principle for the GBH. We comment on the implementation of the GBH in a state of the art Boltzmann solver.

中文翻译：

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宇宙学中大量中微子的广义玻尔兹曼层次

玻尔兹曼求解器是计算线性区域中宇宙学可观测量的重要工具。在存在大量中微子的情况下，它们涉及求解 Boltzmann 方程，然后在动量空间中积分以获得所需的流体特性，这一过程已知计算速度缓慢。在这项工作中，我们为宇宙学中的大量中微子引入了所谓的广义玻尔兹曼层次（GBH），这是通常玻尔兹曼层次的替代方案，其中动量依赖性被积分，留下了一个双参数无限常微分方程组。除了通常的多极扩展之外，现在还有分布函数的更高速度权重积分的扩展。使用玩具代码，$\lesssim 0.5\%$与中微子自由流动尺度相比，在大尺度和中尺度上的准确性，从而为 GBH 提供了原理证明。我们评论了 GBH 在最先进的 Boltzmann 求解器中的实现。