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

Ryotaro Kase, Shinji Tsujikawa

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


For a scalar field φ 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 nc and four-velocity ucμ. For a very general interacting Lagrangian f(nc, φ, X, Z), where f depends on nc, φ, X=-μ φ μ φ/2, and Z=ucμμ φ, 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=-f1(φ, X, Z)nc+f2(φ, 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 f2.

Original languageEnglish
Article number032
JournalJournal of Cosmology and Astroparticle Physics
Issue number11
Publication statusPublished - 2020 Nov


  • dark energy theory
  • modified gravity

ASJC Scopus subject areas

  • Astronomy and Astrophysics


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