A scalar field model for explaining the anomalous acceleration and light deflection at galactic and cluster scales, without further dark matter, is presented. It is formulated in a scale covariant scalar tensor theory of gravity in the framework of integrable Weyl geometry and presupposes two different phases for the scalar field, like the superfluid approach of Berezhiani/Khoury. In low acceleration regimes of static gravitational fields (in the Einstein frame) with accordingly low values of the scalar field gradient, the scalar field Lagrangian combines a cubic kinetic term similar to the “a-quadratic” Lagrangian used in the first covariant generalization of MOND (RAQUAL) (Bekenstein and Milgrom in Astrophys J 286:7–14, 1984) and a second order derivative term introduced by Novello et al. in the context of a Weyl geometric approach to cosmology (Novello et al. in Int J Mod Phys D 1:641–677, 1993; de Oliveira et al. in Class Quantum Gravity 14(10):2833–2843, 1997). In varying with regard to $$\phi $$ ϕ the latter is variationally equivalent to a first order expression. The scalar field equation thus remains of order two. In the Einstein frame it assumes the form of a covariant generalization of the Milgrom equation known from the classical MOND approach in the deep MOND regime. It implies a corresponding “non-metrical” contribution to the acceleration of free fall trajectories. In contrast to pure RAQUAL, the second order derivative term of the Lagrangian leads to a non-negligible contribution to the energy momentum tensor and an add-on to the light deflection potential in beautiful agreement with the dynamics of low velocity trajectories. Although the model takes up important ingredients from the usual RAQUAL approach, it differs essentially from the latter.—In higher sectional curvature regions, respectively for higher accelerations in static fields, the scalar field Lagrangian consists of a Jordan–Brans–Dicke term with sufficiently high value of the JBD-constant to satisfy empirical constraints. Here the dynamics agrees effectively with the one of Einstein gravity.

中文翻译：

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标量场引起对重力加速度的非度量贡献和对光偏转的兼容附加

提出了一个标量场模型，用于解释星系和星团尺度的异常加速度和光偏转，无需进一步的暗物质。它是在可积 Weyl 几何框架内的尺度协变标量张量引力理论中制定的，并假设标量场有两个不同的阶段，如 Berezhiani/Khoury 的超流体方法。在具有相应低标量场梯度值的静态引力场（在爱因斯坦框架中）的低加速状态下，标量场拉格朗日结合了三次动力学项，类似于 MOND 的第一次协变推广中使用的“a-二次”拉格朗日(RAQUAL)（Bekenstein 和 Milgrom 在 Astrophys J 286:7-14, 1984）和 Novello 等人引入的二阶导数项。在外尔几何方法的宇宙学背景下（Novello 等人在 Int J Mod Phys D 1:641–677, 1993 中；de Oliveira 等人在 Class Quantum Gravity 14(10):2833–2843, 1997 中）。关于 $$\phi $$ ϕ 的变化，后者在变化上等价于一阶表达式。标量场方程因此保持二阶。在爱因斯坦框架中，它采用了从经典 MOND 方法在深度 MOND 机制中已知的 Milgrom 方程的协变推广的形式。它意味着对自由落体轨迹加速的相应“非度量”贡献。与纯粹的 RAQUAL 相比，拉格朗日的二阶导数项导致对能量动量张量的不可忽略的贡献和对光偏转势的附加，与低速轨迹的动力学非常吻合。尽管该模型从通常的 RAQUAL 方法中吸收了重要的成分，但它与后者有本质上的不同。 ——在更高的截面曲率区域，分别对于静态场中更高的加速度，标量场拉格朗日由一个 Jordan-Brans-Dicke 项组成，具有足够的满足经验约束的 JBD 常数的高值。这里的动力学与爱因斯坦引力有效地吻合。对于静态场中更高的加速度，标量场拉格朗日量分别由 Jordan-Brans-Dicke 项组成，其 JBD 常数值足够高以满足经验约束。这里的动力学与爱因斯坦引力有效地吻合。对于静态场中更高的加速度，标量场拉格朗日量分别由 Jordan-Brans-Dicke 项组成，其 JBD 常数值足够高以满足经验约束。这里的动力学与爱因斯坦引力有效地吻合。