Comparison of different approaches to the quasi-static approximation in Horndeski models

Abstract

A quasi-static approximation (QSA) for modified gravity can be applied in a number of ways. We consider three different analytical formulations based on applying this approximation to: (1) the field equations; (2) the equations for the two metric potentials; (3) the use of the attractor solution derived within the Equation of State (EoS) approach. We assess the veracity of these implementations on the effective gravitational constant ($\mu$) and the slip parameter ($\eta$), within the framework of Horndeski models. In particular, for a set of models we compare cosmological observables, i.e., the matter power spectrum and the CMB temperature and lensing angular power spectra, computed using the QSA, with exact numerical solutions. To do that, we use a newly developed branch of the CLASS code: QSA_class. All three approaches agree exactly on very small scales. Typically, we find that, except for $f(R)$ models where all the three approaches lead to the same result, the quasi-static approximations differ from the numerical calculations on large scales ($k\lesssim 3-4\times 10^{-3}h{\mathrm{Mpc}}^{-1}$). Cosmological observables are reproduced to within 1% up to scales $\mathrm{K}=k/H_0$ of the order of a few and $\ell >5$ for the approaches based on the field equations and on the EoS, and we also do not find any appreciable difference if we use the scale-dependent expressions for $\mu$ and $\eta$ with respect to the value on small scales, showing that the formalism and the conclusions are reliable and robust, fixing the range of applicability of the formalism. We discuss why the expressions derived from the equations for the potentials have limited applicability. Our results are in agreement with previous analytical estimates and show that the QSA is a reliable tool and can be used for comparison with current and future observations to constrain models beyond $\Lambda$CDM.

arXiv commentThe manuscript is divided into two parts: the main paper (37 pages, 3 figures, 1 table) and the Supplementary Materials (24 pages, 1 table). Improved discussion on the validity of the quasi-static approximation. Matches the published version on JCAP

References in corpus (106)

• Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
• Measurements of Omega and Lambda from 42 High-Redshift Supernovae
• GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral
• Efficient Computation of CMB anisotropies in closed FRW models
• f(R) Theories Of Gravity
• The Confrontation between General Relativity and Experiment
• f(R) theories
• Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A
• Planck 2018 results. X. Constraints on inflation
• The Cosmic Linear Anisotropy Solving System (CLASS) II: Approximation schemes
• A Dynamical Solution to the Problem of a Small Cosmological Constant and Late-time Cosmic Acceleration
• New Hubble Space Telescope Discoveries of Type Ia Supernovae at z > 1: Narrowing Constraints on the Early Behavior of Dark Energy
• Models of f(R) Cosmic Acceleration that Evade Solar-System Tests
• A gravitational-wave standard siren measurement of the Hubble constant
• Dark Energy Survey Year 1 Results: Cosmological Constraints from Galaxy Clustering and Weak Lensing
• Beyond the Cosmological Standard Model
• Generalized G-inflation: Inflation with the most general second-order field equations
• From k-essence to generalised Galileons
• Causality, Analyticity and an IR Obstruction to UV Completion
• Cosmology and Fundamental Physics with the Euclid Satellite
• Dark Energy after GW170817: dead ends and the road ahead
• Strong constraints on cosmological gravity from GW170817 and GRB 170817A
• Dark Energy after GW170817 and GRB170817A
• Planck 2018 results. VIII. Gravitational lensing
• Cosmological Tests of Modified Gravity
• Imperfect Dark Energy from Kinetic Gravity Braiding
• The Cosmic Linear Anisotropy Solving System (CLASS) I: Overview
• G-inflation: inflation driven by the Galileon field
• The Effective Field Theory of Dark Energy
• Testing General Relativity in Cosmology
• Dark Energy vs. Modified Gravity
• k-Essence, superluminal propagation, causality and emergent geometry
• Maximal freedom at minimum cost: linear large-scale structure in general modifications of gravity
• A Parameterized Post-Friedmann Framework for Modified Gravity
• Measuring the dark side (with weak lensing)
• Essential Building Blocks of Dark Energy
• A discriminating probe of gravity at cosmological scales
• Approaches to Understanding Cosmic Acceleration
• A systematic approach to generalisations of General Relativity and their cosmological implications
• Breaking a Dark Degeneracy with Gravitational Waves
• Dark Energy or Modified Gravity? An Effective Field Theory Approach
• Distinguishing Modified Gravity from Dark Energy
• Born-Infeld inspired modifications of gravity
• Searching for modified growth patterns with tomographic surveys
• Challenges to Self-Acceleration in Modified Gravity from Gravitational Waves and Large-Scale Structure
• Cuscuton: A Causal Field Theory with an Infinite Speed of Sound
• Conditions for the cosmological viability of the most general scalar-tensor theories and their applications to extended Galileon dark energy models
• Effective gravitational couplings for cosmological perturbations in the most general scalar-tensor theories with second-order field equations
• Testing gravity with CAMB and CosmoMC
• Dark Energy versus Modified Gravity
• A unifying description of dark energy
• Effective Field Theory of Dark Energy: a Review
• The Imperfect Fluid behind Kinetic Gravity Braiding
• Current constraints on the cosmic growth history
• Effective Field Theory of Cosmic Acceleration: constraining dark energy with CMB data
• hi_class: Horndeski in the Cosmic Linear Anisotropy Solving System
• How to optimally parametrize deviations from General Relativity in the evolution of cosmological perturbations
• Observables and unobservables in dark energy cosmologies
• A practical approach to cosmological perturbations in modified gravity
• Effective Field Theory of Cosmological Perturbations
• Limits of Quasi-Static Approximation in Modified-Gravity Cosmologies
• On the evolution of density perturbations in f(R) theories of gravity
• Constraints on a New Post-General Relativity Cosmological Parameter
• The Observational Future of Cosmological Scalar-Tensor Theories
• Cuscuton Cosmology: Dark Energy meets Modified Gravity
• Lectures on the Cosmological Constant Problem
• A Simplified Approach to General Scalar-Tensor Theories
• Density split statistics: Cosmological constraints from counts and lensing in cells in DES Y1 and SDSS data
• What can Cosmology tell us about Gravity? Constraining Horndeski with Sigma and Mu
• No-go theorem for k-essence dark energy
• Effective action approach to cosmological perturbations in dark energy and modified gravity
• Weakly Broken Galileon Symmetry
• Cosmological parameter constraints for Horndeski scalar-tensor gravity
• Relativistic scalar fields and the quasi-static approximation in theories of modified gravity
• Effective Theory of Dark Energy at Redshift Survey Scales
• No Slip Gravity
• Vainshtein regime in Scalar-Tensor gravity: constraints on DHOST theories
• MGCAMB with massive neutrinos and dynamical dark energy
• Large Scale Structures in Kinetic Gravity Braiding Model That Can Be Unbraided
• Euclid preparation: VII. Forecast validation for Euclid cosmological probes
• Phenomenology of dark energy: exploring the space of theories with future redshift surveys
• hi_class: Background Evolution, Initial Conditions and Approximation Schemes
• A comparison of Einstein-Boltzmann solvers for testing General Relativity
• Phenomenology of dark energy: general features of large-scale perturbations
• The effective field theory of inflation/dark energy and the Horndeski theory
• Cosmological data favor Galileon ghost condensate over $\Lambda$CDM
• Designing Horndeski and the effective fluid approach
• Reconstructing Horndeski theories from phenomenological modified gravity and dark energy models on cosmological scales
• Cosmological Perturbations in Elastic Dark Energy Models
• Measuring gravity at cosmological scales
• Parameterizing dark sector perturbations via equations of state
• Reconstructing Horndeski models from the effective field theory of dark energy
• Semi-dynamical perturbations of unified dark energy
• Classifying Linearly Shielded Modified Gravity Models in Effective Field Theory
• f(R) as a dark energy fluid
• Cosmological perturbation theory in Generalized Einstein-Aether models
• Diagnostic of Horndeski Theories
• The Cosmological Constant Problem: Why it's hard to get Dark Energy from Micro-physics
• Computing model independent perturbations in dark energy and modified gravity
• Massive gravity, the elasticity of space-time and perturbations in the dark sector
• Do cosmological data rule out $f(\mathcal{R})$ with $w\ne -1$?
• No Slip CMB
• Modified Gravity Away from a $\Lambda$CDM Background
• Cosmologically viable generalized Einstein-Aether theories
• Dark sector evolution in Horndeski models
• Late time cosmic acceleration: ABCD of dark energy and modified theories of gravity

Cited by in corpus (19)

• Testing Screened Modified Gravity
• Fast and accurate predictions of the nonlinear matter power spectrum for general models of Dark Energy and Modified Gravity
• Cosmological Constraints on sub-horizon scales modified gravity theories with MGCLASS II
• New cosmological constraints on $f(T)$ gravity in light of full Planck-CMB and type Ia supernovae data
• New constraints on cosmological modified gravity theories from anisotropic three-point correlation functions of BOSS DR12 galaxies
• $\text{Hi-COLA}$: Fast, approximate simulations of structure formation in Horndeski gravity
• Euclid: Constraining linearly scale-independent modifications of gravity with the spectroscopic and photometric primary probes
• Revisiting Vainshtein Screening for fast N-body simulations
• Cosmological Tests of Gravity: A Future Perspective
• Cosmological constraints with the Effective Fluid approach for Modified Gravity
• The Effective Fluid approach for Modified Gravity and its applications
• Fully relativistic predictions in Horndeski gravity from standard Newtonian N-body simulations
• Tracking the validity of the quasi-static and sub-horizon approximations in modified gravity
• Testing $f(T)$ Gravity with Cosmological Observations: Confronting the Hubble Tension and Implications for the Late-Time Universe
• Non-linear density-velocity dynamics in $f(R)$ gravity from spherical collapse
• K-mouflage at high $k$: extending the reach of $\text{Hi-COLA}$
• Probing gravity beyond general relativity with bispectrum multipoles of cosmological tracers: I. Theoretical Foundations
• An effective fluid description of scalar-vector-tensor theories under the sub-horizon and quasi-static approximations
• Halos of dark energy