We write down the Lagrangian bias expansion in general relativity up to 4th order in terms of operators describing the curvature of an early-time hypersurface for comoving observers. They can be easily expanded in synchronous or comoving gauges. This is necessary for the computation of the one-loop halo bispectrum, where relativistic effects can be degenerate with a primordial non-Gaussian signal. Since the bispectrum couples scales, an accurate prediction of the squeezed limit behavior needs to be both non-linear and relativistic. We then evolve the Lagrangian bias operators in time in comoving gauge, obtaining non-local operators analogous to what is known in the Newtonian limit. Finally, we show how to renormalize the bias expansion at an arbitrary time and find that this is crucial in order to cancel unphysical $1/k^2$ divergences in the large-scale power spectrum and bispectrum that could be mistaken for a contamination to the non-Gaussian signal.

中文翻译：

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从物质到星系：单环双谱的一般相对论偏差

我们根据描述共同移动观察者的早期超曲面曲率的算子将广义相对论中的拉格朗日偏差展开写到四阶。它们可以轻松扩展为同步或联动仪表。这对于计算单环晕双谱是必要的，其中相对论效应可以用原始非高斯信号退化。由于双谱偶标度，压缩极限行为的准确预测需要是非线性和相对论的。然后，我们在联动规范中及时演化拉格朗日偏差算子，获得类似于牛顿极限中已知的非局部算子。最后，