For a scalar field $\phi$ coupled to cold dark matter (CDM), we provide a general framework for studying the background and perturbation dynamics on the isotropic cosmological background. The dark energy sector is described by a Horndeski Lagrangian with the speed of gravitational waves equivalent to that of light, whereas CDM is dealt as a perfect fluid characterized by the number density $n_c$ and four-velocity $u_c^\mu$. For a very general interacting Lagrangian $f(n_c, \phi, X, Z)$, where $f$ depends on $n_c$, $\phi$, $X=-\partial^{\mu} \phi \partial_{\mu} \phi/2$, and $Z=u_c^{\mu} \partial_{\mu} \phi$, we derive the full linear perturbation equations of motion without fixing any gauge conditions. To realize a vanishing CDM sound speed for the successful structure formation, the interacting function needs to be of the form $f=-f_1(\phi, X, Z)n_c+f_2(\phi, X, Z)$. Employing a quasi-static approximation for the modes deep inside the sound horizon, we obtain analytic formulas for the effective gravitational couplings of CDM and baryon density perturbations as well as gravitational and weak lensing potentials. We apply our general formulas to several interacting theories and show that, in many cases, the CDM gravitational coupling around the quasi de-Sitter background can be smaller than the Newton constant $G$ due to a momentum transfer induced by the $Z$-dependence in $f_2$.

中文翻译：

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与暗物质耦合的标量张量暗能量中宇宙学扰动的一般公式

对于与冷暗物质 (CDM) 耦合的标量场 $\phi$，我们提供了研究各向同性宇宙学背景上的背景和扰动动力学的一般框架。暗能量扇区由 Horndeski Lagrangian 描述，引力波的速度等于光速，而 CDM 被视为完美流体，其特征是数密度 $n_c$ 和四速度 $u_c^\mu$。对于非常一般的相互作用拉格朗日 $f(n_c, \phi, X, Z)$，其中 $f$ 取决于 $n_c$, $\phi$, $X=-\partial^{\mu} \phi \partial_ {\mu} \phi/2$ 和 $Z=u_c^{\mu} \partial_{\mu} \phi$，我们在不固定任何规范条件的情况下推导出完整的线性扰动运动方程。为了实现成功的结构形成的消失的 CDM 声速，交互函数需要是 $f=-f_1(\phi, X, Z)n_c+f_2(\phi, X, Z)$。对声音视界深处的模式采用准静态近似，我们获得了 CDM 和重子密度扰动的有效引力耦合以及引力和弱透镜势的解析公式。我们将我们的一般公式应用于几个相互作用的理论，并表明，在许多情况下，由于 $Z$-引起的动量转移，准去西特背景周围的 CDM 引力耦合可能小于牛顿常数 $G$依赖于 $f_2$。