KiDS-Legacy: Constraints on Horndeski gravity from weak lensing combined with galaxy clustering and cosmic microwave background anisotropies

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Table 1.

Model parameters and their priors.

Type	Parameter	Prior	Description
Cosmological	ω_cdm	𝒰(0.051,0.255)	Reduced CDM density
	ω_b	𝒰(0.019,0.026)	Reduced baryon density
	h	𝒰(0.64,0.82)	Dimensionless Hubble parameter
	10⁻⁹A_s	𝒰(0.05,5.00)	Amplitude of the primordial power spectrum
	n_s	𝒰(0.84,1.1)	Spectral index of the primordial power spectrum
	τ	𝒰(0.01,0.20)	Optical depth to reionization
	Ω_K	0.0	Energy density of spatial curvature
	w₀	−1.0a	Dark energy equation of state today
	w_a	0.0^a	Linear slope of the dark energy equation of state
	∑m_ν[eV/c²]	0.06	Total neutrino mass

Nuisance	log₁₀T_AGN	𝒰(7.3,8.3)	Baryon feedback parameter
	A_IA	$N (μ_{A_{IA}, β_{IA}}, C_{A_{IA}, β_{IA}})$ $Mathematical equation: $ {-}\mathcal{N}(\boldsymbol{\mu}_{A_{\mathrm{IA}},\beta_{\mathrm{IA}}}, \mathbf{C}_{A_{\mathrm{IA}},\beta_{\mathrm{IA}}}) $$	Amplitude of intrinsic galaxy alignments for red galaxies
	β_IA	$N (μ_{A_{IA}, β_{IA}}, C_{A_{IA}, β_{IA}})$ $Mathematical equation: $ {-}\mathcal{N}(\boldsymbol{\mu}_{A_{\mathrm{IA}},\beta_{\mathrm{IA}}}, \mathbf{C}_{A_{\mathrm{IA}},\beta_{\mathrm{IA}}}) $$	Slope of the mass scaling of intrinsic galaxy alignments
	log₁₀M_i[h⁻¹ M_⊙]	$N (μ_{M}, C_{M})$ $Mathematical equation: $ {-}\mathcal{N}(\boldsymbol{\mu}_M, \mathbf{C}_M) $$	Mean halo mass of early-type galaxies per tomographic bin
	δz_i	𝒩(μ_δz,C_δz)	Shift of the mean of the redshift distribution per tomographic bin
	log₁₀k_V	𝒩(μ_M, C_M	Screening scale
	A_L	𝒰(#x2212;1.0,1.0)	Planck lensing amplitude

Stable 1	${\hat{D}}_{kin}$ $Mathematical equation: $ \hat{D}_{\mathrm{kin}} $$	𝒰(0.8,1.5)	De-mixed kinetic term of the scalar field perturbation
	${\hat{c}}_{s}^{2}$ $Mathematical equation: $ \hat{c}_{\mathrm{s}}^2 $$	𝒰(0.0,10.0)	Effective sound speed of the scalar field perturbation
	$Δ {\hat{M}}_{}^{2}$ $Mathematical equation: $ \Delta \hat{M}_^2 $$	1.0	Deviation of the effective Planck mass from its fiducial value

Stable 2	${\hat{D}}_{kin}$ $Mathematical equation: $ \hat{D}_{\mathrm{kin}} $$	𝒰(#x2212;1.0,10.0)	De-mixed kinetic term of the scalar field perturbation
	${\hat{c}}_{s}^{2}$ $Mathematical equation: $ \hat{c}_{\mathrm{s}}^2 $$	𝒰(0.0,10.0)	Effective sound speed of the scalar field perturbation
	$Δ {\hat{M}}_{}^{2}$ $Mathematical equation: $ \Delta \hat{M}_^2 $$	𝒰(0.0,2.0)	Deviation of the effective Planck mass from its fiducial value

Alpha	${\hat{α}}_{B}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{B}} $$	0.0	Braiding
	${\hat{α}}_{M}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{M}} $$	𝒰(#x2212;2.0,2.0)	Run rate of the effective Planck mass
	${\hat{α}}_{K}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{K}} $$	𝒰(#x2212;1.0,3.0)	Kineticity
	${\hat{α}}_{T}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{T}} $$	1.0	Tensor speed excess

Notes. The first two columns provide parameter names and types of the sampling parameters. The third column lists the adopted prior with uniform priors denoted by 𝒰 and Gaussian priors denoted by 𝒩 and the fourth column provides a brief parameter description. The parameters are grouped into cosmological parameters, nuisance parameters, and three parametrisations of Horndeski gravity. For the Horndeski model, we consider two parametrisations based on the stable basis functions, keeping either the sound speed or the Planck mass fixed to its fiducial value, as well as a parametrisation in terms of the α functions.

^(a)

In Sect. 4.4, we derive constraints on dynamical dark energy with a prior on w₀ ∈ 𝒰(−3.0, 0.0) and w_a ∈ 𝒰(−5.0, 5.0).

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