Tracker and scaling solutions in DHOST theories

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Citations (Scopus)


In quadratic-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 wϕ=−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 Pϕ obey the scaling relation ρϕ∝Pϕ∝H2. 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
JournalPhysics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics
Publication statusPublished - 2019 Mar 10
Externally publishedYes

ASJC Scopus subject areas

  • Nuclear and High Energy Physics


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