度量仿射宇宙学模型与变分法的反问题。第二部分：$\Lambda$CDM 模型的变分自举
Metric-affine cosmological models and the inverse problem of the calculus of variations. Part II: Variational bootstrapping of the $Λ$CDM model
AI总结：
本文将基于正则变分完备化的变分自举方法应用于度量仿射引力，从类爱因斯坦场方程的猜测出发，构建变分意义上最接近$\Lambda$CDM模型的度量仿射模型，得到涵盖多种模型的拉格朗日量及联络场方程。
摘要：
基于正则变分完备化的变分自举方法，能够从仅对一组场变量的场方程猜测出发，构建依赖两组场变量的物理理论的拉格朗日量。这种构建方式对修正引力理论的构建尤其具有吸引力，因为人们可以借鉴广义相对论（GR）来“有依据地猜测”度规场方程；而其他场的场方程则由得到的拉格朗日量确定（仅差完全独立于度规张量的项）。在本文中，我们应用变分自举方法来确定在变分意义上最接近宇宙学$\Lambda$CDM模型的度量仿射模型。从形式上类似于带宇宙学“常数”（实际上是由度规和联络构建的标量函数）和暗物质项的爱因斯坦场方程的“有依据猜测”出发，该方法能够找到“修正的”度规方程，并“自举”得到联络场方程。通过该方法得到的拉格朗日量尽管施加了一些限制性准则，仍涵盖了种类繁多的度量仿射模型。特别地，它们包含了一类限制为曲率张量线性项的二次度量仿射理论子类。
英文摘要：
The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only one such set. This setup is particularly appealing for the construction of modified theories of gravity, since one can take lessons from GR for an "educated guess" of the metric field equations; the field equations for the other fields are then fixed by the obtained Lagrangian (up to terms that are completely independent from the metric tensor). In the present paper, we apply variational bootstrapping to determine metric-affine models which are, in a variational sense, closest to the $Λ$CDM model of cosmology. Starting from an "educated guess" that formally resembles the Einstein field equations with a cosmological "constant" (actually, a scalar function built from the metric and the connection) and a dark matter term, the method then allows to find "corrected" metric equations and to "bootstrap" the connection field equations. Lagrangians obtained via this method, though imposing some restricting criteria, still encompass a wide variety of metric-affine models. In particular, they allow for a subclass of quadratic metric-affine theories restricted to linear terms in the curvature tensor.
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