标题： 从Weyl-Schrödinger联系到加速的宇宙——通过长度保持的非度规性扩展爱因斯坦引力 显示英文标题

标题： From the Weyl-Schrödinger connection to the accelerating Universe -- extending Einstein's gravity via a length preserving nonmetricity

摘要： 黎曼几何的一个重要扩展是外尔几何，它本质上基于共形不变性和非度规性思想。 埃文·薛定谔在1940年代末提出了一种类似的非黎曼几何，在这种几何中，它更简单，并且（可能）更优雅 than 外尔几何。 即使它包含非度规性，薛定谔联络在平行传输下保持向量长度不变，因此似乎比外尔联络更物理。 有趣的是， 薛定谔的方法在引力物理领域并未引起太多关注。 本文的目标是重新考虑薛定谔几何作为扩展标准广义相对论的引力理论的潜在候选者。 我们考虑一个在无扭力情况下由保持长度的非度规性构建的引力作用量，并在Palatini和度规形式主义下研究其变分。 虽然Palatini变分导致标准广义相对论，但该理论的度规版本在引力爱因斯坦方程中添加了一些依赖于非度规性的额外项，这些项可以解释为代表一种几何类型的暗能量。 在获得广义弗里德曼方程后，我们通过考虑两种不同的模型，分别对应于满足线性状态方程的暗能量和守恒物质能量，详细分析了该理论的宇宙学含义。 在两种情况下，我们将Weyl-Schrödinger宇宙学的预测与一组观测数据中的哈勃函数进行了比较，并与$\Lambda$CDM标准范式的结果进行了比较。 显示英文摘要

摘要： One of the important extensions of Riemann geometry is Weyl geometry, which is essentially based on the ideas of conformal invariance and nonmetricity. A similar non-Riemannian geometry was proposed by Erwin Schr\"{o}dinger in the late 1940s, in a geometry which is simpler, and (probably) more elegant than the Weyl geometry. Even it contains nonmetricity, the Schr\"{o}dinger connection preserves the length of vectors under parallel transport, and thus seems to be more physical than the Weyl connection. Interestingly enough, Schr\"{o}dinger's approach did not attract much interest in the field of gravitational physics. It is the goal of the present paper to reconsider the Schr\"{o}dinger geometry as a potential candidate for a gravitational theory extending standard general relativity. We consider a gravitational action constructed from a length preserving non-metricity, in the absence of torsion, and investigate its variation in both Palatini and metric formalisms. While the Palatini variation leads to standard general relativity, the metric version of the theory adds some non-metricity dependent extra terms in the gravitational Einstein equations, which can be interpreted as representing a geometric type dark energy. After obtaining the generalized Friedmann equations, we analyze in detail the cosmological implications of the theory, by considering two distinct models, corresponding to a dark energy satisfying a linear equation of state, and to conserved matter energy, respectively. In both cases we compare the predictions of the Weyl-Schr\"{o}dinger cosmology with a set of observational data for the Hubble function, and with the results of the $\Lambda$CDM standard paradigm.