General formulation of cosmological perturbations in scalar-tensor dark energy coupled to dark matter

TL;DR

This paper develops a comprehensive framework for analyzing cosmological perturbations in scalar-tensor dark energy models coupled to dark matter, deriving equations and effective gravitational couplings without fixing gauges.

Contribution

It introduces a general interacting Lagrangian for dark energy and dark matter, providing analytic formulas for perturbations and gravitational couplings in scalar-tensor theories.

Findings

Effective gravitational coupling can be smaller than G due to momentum transfer.

Derived formulas apply to various interacting theories.

CDM sound speed vanishes with specific interaction forms.

Abstract

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 $f_2$.