Tracker and scaling solutions in DHOST theories

Noemi Frusciante, Ryotaro Kase, Kazuya Koyama, Shinji Tsujikawa, Daniele Vernieri

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In quartic-order degenerate higher-order scalar-tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solutions characterized by φ˙/Hp=constant, where φ˙ is the time derivative of a scalar field φ, H is the Hubble expansion rate, and p is a constant. While the tracker is present up to the cubic-order Horndeski Lagrangian L=c2X−c3X(p−1)/(2p)□φ, where c2,c3 are constants and X is the kinetic energy of φ, the DHOST interaction breaks this structure for p≠1. Even in the latter case, however, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state =−1−2pH˙/(3H2). The scaling solution, which corresponds to p=1, is the unique case in which all the terms in the field density ρφ and the pressure obey the scaling relation ρφH2. Extending the analysis to the coupled DHOST theories with the field-dependent coupling Q(φ) between the scalar field and matter, we show that the scaling solution exists for Q(φ)=1/(μ1φ+μ2), where μ1 and μ2 are constants. For the constant Q, i.e., μ1=0, we derive fixed points of the dynamical system by using the general Lagrangian with scaling solutions. This result can be applied to the model construction of late-time cosmic acceleration preceded by the scaling φ-matter-dominated epoch.
Original languageEnglish
Pages (from-to)167-175
Number of pages9
JournalPhysical Letters B
Early online date17 Jan 2019
Publication statusPublished - 10 Mar 2019


  • gr-qc
  • astro-ph.CO
  • hep-ph
  • hep-th
  • RCUK
  • STFC
  • ST/N000668/1


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