Title: Asymptotically Schwarzschild solutions in $f(R)$ extension of general relativity Show Chinese title

Title: 渐近施瓦西解在广义相对论的$f(R)$扩展中

Abstract: We address the question of how to build a class of $f(R)$ extensions of General Relativity which are compatible with solar system experiments, without making any preliminary assumption on the properties of $f$. The aim is reached by perturbatively solving the modified Einstein equations around a Schwarzschild background and retrieving a posteriori the corresponding $f(R)$. This turns out to be non analytical in $R=0$ and should be intended as the leading correction to the Einstein-Hilbert action in the low curvature limit. The parameters characterizing the $f(R)$ class are then set by constraining the corrections to four different local tests with the observations. The result is a class of $f(R)$ theories built up from a purely bottom-up approach and compatible with the local tests. At a more general level, this result can help constraining exact $f(R)$ models working in Cosmology, since it provides the correct local limit. Further developments and possible extensions of the approach to Cosmology are also discussed. Show Chinese abstract

Abstract: 我们探讨了如何构建一类与广义相对论（General Relativity）相兼容的$f(R)$扩展理论，且无需对$f$的性质作任何先验假设。目标是通过在史瓦西背景附近摄动求解修正后的爱因斯坦方程，并随后回溯得到对应的$f(R)$。结果表明该解对$R=0$并非解析，应被视为爱因斯坦-希尔伯特作用量在低曲率极限下的主要修正项。 随后，通过约束四种不同的局域实验观测结果来确定$f(R)$类参数。最终得到了一类从纯自下而上方法构建且符合局域实验的$f(R)$理论。 更广泛地说，这一结果有助于限制宇宙学中精确的$f(R)$模型，因为它提供了正确的局域极限。此外，还讨论了该方法在宇宙学中的进一步发展及其可能的扩展。