Designing Horndeski and the effective fluid approach

Abstract

We present a family of designer Horndeski models, i.e. models that have a background exactly equal to that of the ΛCDM model but perturbations given by the Horndeski theory. Then, we extend the effective fluid approach to Horndeski theories, providing simple analytic formulae for the equivalent dark energy effective fluid pressure, density and velocity. We implement the dark energy effective fluid formulae in our code EFCLASS, a modified version of the widely used Boltzmann solver CLASS, and compare the solution of the perturbation equations with those of the code hi CLASS which already includes Horndeski models. We find that our simple modifications to the vanilla code are accurate to the level of ∼ 0.1% with respect to the more complicated hi CLASS code. Furthermore, we study the kinetic braiding model both on and off the attractor and we find that even though the full case has a proper ΛCDM limit for large n, it is not appropriately smooth, thus causing the quasistatic approximation to break down. Finally, we focus on our designer model (HDES), which has both a smooth ΛCDM limit and well-behaved perturbations, and we use it to perform Markov Chain Monte Carlo analyses to constrain its parameters with the latest cosmological data. We find that our HDES model can also alleviate the soft 2σ tension between the growth data and Planck 18 due to a degeneracy between σ8 and one of its model parameters that indicates the deviation from the ΛCDM model.

Measurements of $$H_0$$ and reconstruction of the dark energy properties from a model-independent joint analysis

“…The condition c 2 s > 0 is usually imposed to avoid gradient instability. However, the perturbations can still remain stable under c 2 s < 0 consideration [92][93][94][95]. Thus, if the effective sound speed is negative, this would be a smoking gun signature for the existence of an anisotropic stress and possible modifications of gravity.…”

Cosmological constraints on nonadiabatic dark energy perturbations

“…Since currently it is unknown if DE has a nonadiabatic component and, even if it does, the behavior of δ P nad ρ is unknown, we took advantage of the effective fluid approach of Refs. [29,30] in order to construct a realistic ansatz.…”

“…In particular, here we will consider a case which is motivated by the effective fluid approach of Refs. [29,30] and as an example we will consider the designer fðRÞ model (see Ref. [29]), which is constructed so that the background expansion corresponds exactly to ΛCDM but to linear order, it can have perturbations [43].…”

“…One way to do this is to map the MG models, to linear order, to some DE fluid via the effective fluid approach. Then, MG models can be interpreted as DE fluids described by an equation of state wðaÞ, a pressure perturbation δPðk; aÞ, and an anisotropic stress σðk; aÞ [26,[28][29][30][31][32][33]. Hence, the evolution of the background is determined by wðaÞ, while the evolution of the perturbations is governed by δPðk; aÞ and σðk; aÞ, both of which are time and scale dependent.…”

hi_class background evolution, initial conditions and approximation schemes

“…EFTCAMB [76], COOP [77]) and ones dealing with an approximation to it or specific models of gravity (e.g. DASh [78], GalCAMB [79], EoS_class [80] or EFCLASS [81]) reaching the level of accuracy needed by next-generation cosmological experiments [82]. Currently the publicly available version of the hi_class code incorporates Horndeski's scalar-tensor theory, a very general overarching framework that incorporates a large class of DE and beyond GR theories.…”