arXiv:1812.05204v1 [gr-qc] 13 Dec 2018 Tracker and scaling solutions in DHOST theories Noemi Frusciante 1 , Ryotaro Kase 2 , Kazuya Koyama 3 , Shinji Tsujikawa 2 and Daniele Vernieri 4 1 Instituto de Astrof´ ısica e Ciˆ encias do Espa¸ co, Faculdade de Ciˆ encias da Universidade de Lisboa, Campo Grande, PT1749-016 Lisboa, Portugal 2 Department of Physics, Faculty of Science, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan 3 Institute of Cosmology & Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, United Kingdom 4 Centro de Astrof´ ısica e Gravita¸ c˜ ao - CENTRA, Departamento de F´ ısica, Instituto Superior T´ ecnico - IST, Universidade de Lisboa - UL, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal In quartic-order degenerate higher-order scalar-tensor (DHOST) theories compatible with gravitational-wave constraints, we derive the most general Lagrangian allowing for tracker solu- tions characterized by ˙ φ/H p = 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, how- ever, there exists an approximate tracker solution in the early cosmological epoch with the nearly constant field equation of state w φ = −1 − 2p ˙ H/(3H 2 ). The scaling solution, which corresponds to p = 1, is the unique case in which all the terms in the field density ρ φ and the pressure P φ obey the scaling relation ρ φ ∝ P φ ∝ H 2 . 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. I. INTRODUCTION There have been numerous attempts to modify or ex- tend General Relativity (GR) at large distances [1–6]. One of such motivations is to explain the observational evidence of late-time cosmic acceleration by introducing a new ingredient beyond the scheme of standard model of particle physics. The simple candidate for such a new degree of freedom (DOF) is a scalar field φ [7–13], which has been widely exploited to describe the dynamics of dark energy. The theories in which the scalar field is directly coupled to gravity (with two tensor polarized DOFs) are gener- ally called scalar-tensor theories [14, 15]. It is known that Horndeski theories [16] are the most general scalar-tensor theories with second-order equations of motion [17–19]. The second-order property ensures the absence of an Os- trogradsky instability [20] associated with a linear depen- dence of the Hamiltonian arising from extra DOFs. Horndeski theories can be extended to more general theoretical schemes without increasing the number of propagating DOFs [21]. For example, Gleyzes-Langlois- Piazza-Vernizzi (GLPV) expressed the Horndeski La- grangian in terms of scalar quantities arising in the 3+1 decomposition of spacetime [22] and derived a beyond- Horndeski Lagrangian without imposing two conditions Horndeski theories obey [23]. The Hamiltonian analy- sis in the unitary gauge showed that the GLPV theories do not increase the number of DOFs relative to those in Horndeski gravity [24–26]. One can further perform a healthy extension of Horn- deski theories by keeping one scalar and two tensor DOFs. Even if Euler-Lagrange equations contain deriva- tives higher than second order in the scalar field and the metric, it is possible to maintain the same number of propagating DOFs by imposing the so-called degeneracy conditions of their Lagrangians [27–31]. They are dubbed degenerate higher-order scalar-tensor (DHOST) theories, which encompass GLPV theories as a special case. The absence of an extra DOF was confirmed by the Hamil- tonian analysis [26, 28] as well as by the field definition linking to Horndeski theories [29, 30, 32]. The DHOST theories contain the products of covari- ant derivatives of the field which are quadratic and cu- bic in ∇ µ ∇ ν φ, say, (φ) 2 and (φ) 3 , respectively. If we apply the DHOST theories to dark energy and adopt the bound of the speed c t of gravitational waves con- strained from the GW170817 event [33] together with the electromagnetic counterpart [34], the Lagrangians consis- tent with c t = 1 (in the unit where the speed of light is equivalent to 1) are up to quadratic in ∇ µ ∇ ν φ with one of the terms vanishing (A 1 = 0) [35] among six coeffi- cients of quartic derivative interactions. From the degen- eracy conditions there are three constraints among the other five coefficients [36–38], so we are left with two free quartic-order functions. If we take into account the de- cay of gravitational waves to dark energy [39], we have an