Tracker and scaling solutions in DHOST theories

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

doi:10.1016/j.physletb.2019.01.009

Abstract

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.

Highlights

There have been numerous attempts to modify or extend General Relativity (GR) at large distances [1,2,3,4,5,6]
We extend the analysis to the case in which a field-dependent coupling Q (φ) between φ and matter is present and obtain the most general Lagrangian allowing for scaling solutions
We considered quadratic-order degenerate higher-order scalar–tensor (DHOST) theories satisfying degeneracy conditions to avoid the Ostrogradsky instability, the constraint on the speed of gravitational waves, and the bound on the decay of gravitational waves to dark energy perturbations

Summary

Introduction

There have been numerous attempts to modify or extend General Relativity (GR) at large distances [1,2,3,4,5,6]. Even if Euler– Lagrange equations contain derivatives 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 [27,28,29,30,31] They are dubbed degenerate higher-order scalar–tensor (DHOST) theories, which encompass GLPV theories as a special case. If the scalar field has a constant coupling Q with matter, the scaling solution satisfying the relation φ ∝ H exists for the cubic-order Horndeski Lagrangian L = X g2(Y ) − g3(Y ) φ, where g2, g3 are arbitrary functions of Y = Xeλφ (λ is a constant) [63] In this case, it is possible to construct viable dark energy models with a scaling φ-matter dominated epoch (φMDE).

Background equations in DHOST theories

Derivation of the scaling Lagrangian

Constant Q

Fixed points for constant Q

Conclusions