In this paper we show how the solar system, the galactic, and the cosmological scales, are accommodated in a single framework, namely, in the derivative expansion framework. We construct a locally inertial static metric, based on the Einstein equations and on the derivative expansion method, which describes a Schwarzschild black hole immersed in dark matter and dark energy. The leading order metric in the expansion corresponds to the solar system, the first order metric to the galaxy, and the second order metric to cosmology. It is shown how this metric captures the main observations at each scale: in the solar system it trivially gives the Keplerian physics, in the galaxy it gives the flat part of the rotation curve and the Baryonic Tully-Fisher relation, and in the cosmological scale it gives the cosmological redshift, the accelerating expansion, and it coincides with the Robertson–Walker spacetime in the appropriate limit and approximation.

在这篇论文中，我们展示了太阳系、银河系和宇宙尺度是如何容纳在一个单一的框架中的，即在导数膨胀框架中。我们基于爱因斯坦方程和导数展开方法构建了一个局部惯性静态度量，它描述了沉浸在暗物质和暗能量中的施瓦西黑洞。膨胀中的前序度量对应于太阳系，一阶度量对应于星系，二阶度量对应于宇宙学。显示了该度量如何在每个尺度上捕获主要观测结果：在太阳系中，它简单地给出了开普勒物理学，在星系中，它给出了旋转曲线的平坦部分和重子 Tully-Fisher 关系，以及在宇宙学尺度中它给出了宇宙学红移，加速膨胀，