Einstein’s General Relativity (GR) is possibly one of the greatest intellectual achievements ever conceived by the human mind. In fact, over the last century, GR has proven to be an extremely successful theory, with a well established experimental footing, at least for weak gravitational fields. Its predictions range from the existence of black holes and gravitational radiation (now confirmed) to the cosmological models. Indeed, a central theme in modern Cosmology is the perplexing fact that the Universe is undergoing an accelerating expansion, which represents a new imbalance in the governing gravitational equations. The cause of the late-time cosmic acceleration remains an open and tantalizing question, and has forced theorists and experimentalists to question whether GR is the correct relativistic theory of gravitation. This has spurred much research in modified theories of gravity, where extensions of the Hilbert–Einstein action describe the gravitational field, in particular, [Formula: see text] gravity, where [Formula: see text] is the curvature scalar. In this review, we perform a detailed theoretical and phenomenological analysis of specific modified theories of gravity and investigate their astrophysical and cosmological applications. We present essentially two largely explored extensions of [Formula: see text] gravity, namely: (i) the hybrid metric-Palatini theory; (ii) and modified gravity with curvature-matter couplings. Relative to the former, it has been established that both metric and Palatini versions of [Formula: see text] gravity possess interesting features but also manifest severe drawbacks. A hybrid combination, containing elements from both of these formalisms, turns out to be very successful in accounting for the observed phenomenology and avoids some drawbacks of the original approaches. Relative to the curvature-matter coupling theories, these offer interesting extensions of [Formula: see text] gravity, where the explicit nonminimal couplings between an arbitrary function of the scalar curvature [Formula: see text] and the Lagrangian density of matter, induces a nonvanishing covariant derivative of the energy-momentum tensor, which implies nongeodesic motion and consequently leads to the appearance of an extra force. We extensively explore both theories in a plethora of applications, namely, the weak-field limit, galactic and extragalactic dynamics, cosmology, stellar-type compact objects, irreversible matter creation processes and the quantum cosmology of a specific curvature-matter coupling theory.

中文翻译：

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超越爱因斯坦的广义相对论：混合度量-帕拉蒂尼引力和曲率-物质耦合

爱因斯坦的广义相对论（GR）可能是人类思想有史以来最伟大的智力成就之一。事实上，在上个世纪，GR 已被证明是一个非常成功的理论，具有良好的实验基础，至少对于弱引力场而言是这样。它的预测范围从黑洞的存在和引力辐射（现已确认）到宇宙模型。事实上，现代宇宙学的一个中心主题是一个令人困惑的事实，即宇宙正在加速膨胀，这代表了主导引力方程中的一种新的不平衡。后期宇宙加速的原因仍然是一个开放而诱人的问题，并迫使理论家和实验家质疑 GR 是否是正确的相对论引力理论。这激发了对修正的引力理论的大量研究，其中希尔伯特-爱因斯坦作用的扩展描述了引力场，特别是 [公式：见文本] 重力，其中 [公式：见文本] 是曲率标量。在这篇综述中，我们对特定修正的引力理论进行了详细的理论和现象学分析，并研究了它们的天体物理学和宇宙学应用。我们基本上提出了[公式：见文本]重力的两个主要探索的扩展，即：（i）混合度量-帕拉蒂尼理论；(ii) 并通过曲率物质耦合修改重力。相对于前者，已经确定[公式：见文本]重力的公制和帕拉蒂尼版本都具有有趣的特征，但也表现出严重的缺陷。混合组合，包含来自这两种形式的元素，结果证明在解释观察到的现象学方面非常成功，并避免了原始方法的一些缺点。相对于曲率-物质耦合理论，这些提供了[公式：见文本]引力的有趣扩展，其中标量曲率[公式：见文本]的任意函数与物质的拉格朗日密度之间的显式非最小耦合，导致了能量-动量张量的非零协变导数，这意味着非测地线运动并因此导致出现额外力。我们在大量应用中广泛探索这两种理论，即弱场极限、星系和河外动力学、宇宙学、恒星型致密物体、