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

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Table A.1.

Marginal parameter constraints for KiDS-Legacy in combination with DESI DR2 BAO, eBOSS DR16 RSD, and Planck 2018 TTTEEE, low-ℓ TT, and low-ℓ EE datasets.

		ΛCDM	stable: ${\hat{D}}_{kin}$ $Mathematical equation: $ \hat{D}_{\mathrm{kin}} $$ , $Δ {\hat{M}}_{}^{2}$ $Mathematical equation: $ \Delta \hat{M}_^2 $$	stable: ${\hat{D}}_{kin}$ $Mathematical equation: $ \hat{D}_{\mathrm{kin}} $$ , ${\hat{c}}_{s}^{2}$ $Mathematical equation: $ \hat{c}_{\mathrm{s}}^2 $$	alpha: ${\hat{α}}_{B}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{B}} $$ , ${\hat{α}}_{M}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{M}} $$
Cosmological	ω_cdm	$0 . 117_{- 0.001}^{+ 0.001}$ $Mathematical equation: $ 0.117^{+0.001}_{-0.001} $$	0.0	$0 . 117_{- 0.001}^{+ 0.001}$ $Mathematical equation: $ 0.117^{+0.001}_{-0.001} $$	$0 . 117_{- 0.001}^{+ 0.001}$ $Mathematical equation: $ {-}0.117^{+0.001}_{-0.001} $$
	100ω_b	$2 . 262_{- 0.015}^{+ 0.012}$ $Mathematical equation: $ 2.262^{+0.012}_{-0.015} $$	$2 . 263_{- 0.013}^{+ 0.013}$ $Mathematical equation: $ {-}2.263^{+0.013}_{-0.013} $$	$2 . 265_{- 0.015}^{+ 0.012}$ $Mathematical equation: $ 2.265^{+0.012}_{-0.015} $$	$2 . 265_{- 0.013}^{+ 0.013}$ $Mathematical equation: $ {-}2.265^{+0.013}_{-0.013} $$
	h	$0 . 685_{- 0.003}^{+ 0.002}$ $Mathematical equation: $ 0.685^{+0.002}_{-0.003} $$	$0 . 686_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ {-}0.686^{+0.003}_{-0.003} $$	$0 . 686_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ 0.686^{+0.003}_{-0.003} $$	$0 . 687_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ {-}0.687^{+0.003}_{-0.003} $$
	n_s	$0 . 972_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ 0.972^{+0.003}_{-0.003} $$	$0 . 972_{- 0.003}^{+ 0.004}$ $Mathematical equation: $ {-}0.972^{+0.004}_{-0.003} $$	$0 . 973_{- 0.004}^{+ 0.003}$ $Mathematical equation: $ 0.973^{+0.003}_{-0.004} $$	$0 . 973_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ {-}0.973^{+0.003}_{-0.003} $$
	τ	$0 . 056_{- 0.007}^{+ 0.008}$ $Mathematical equation: $ 0.056^{+0.008}_{-0.007} $$	$0 . 053_{- 0.007}^{+ 0.008}$ $Mathematical equation: $ {-}0.053^{+0.008}_{-0.007} $$	$0 . 054_{- 0.008}^{+ 0.008}$ $Mathematical equation: $ 0.054^{+0.008}_{-0.008} $$	$0 . 052_{- 0.008}^{+ 0.007}$ $Mathematical equation: $ {-}0.052^{+0.007}_{-0.008} $$
	10⁹A_s	$2 . 097_{- 0.031}^{+ 0.032}$ $Mathematical equation: $ 2.097^{+0.032}_{-0.031} $$	$2 . 083_{- 0.033}^{+ 0.032}$ $Mathematical equation: $ {-}2.083^{+0.032}_{-0.033} $$	$2 . 085_{- 0.033}^{+ 0.033}$ $Mathematical equation: $ 2.085^{+0.033}_{-0.033} $$	$2 . 077_{- 0.040}^{+ 0.028}$ $Mathematical equation: $ {-}2.077^{+0.028}_{-0.040} $$

Derived	Ω_m	$0 . 299_{- 0.003}^{+ 0.003}$ $Mathematical equation: $ 0.299^{+0.003}_{-0.003} $$	$0 . 298_{- 0.004}^{+ 0.003}$ $Mathematical equation: $ {-}0.298^{+0.003}_{-0.004} $$	$0 . 297_{- 0.004}^{+ 0.003}$ $Mathematical equation: $ 0.297^{+0.003}_{-0.004} $$	$0 . 296_{- 0.004}^{+ 0.004}$ $Mathematical equation: $ {-}0.296^{+0.004}_{-0.004} $$
	S₈	$0 . 804_{- 0.008}^{+ 0.008}$ $Mathematical equation: $ 0.804^{+0.008}_{-0.008} $$	$0 . 813_{- 0.011}^{+ 0.008}$ $Mathematical equation: $ {-}0.813^{+0.008}_{-0.011} $$	$0 . 803_{- 0.009}^{+ 0.008}$ $Mathematical equation: $ 0.803^{+0.008}_{-0.009} $$	$0 . 822_{- 0.015}^{+ 0.011}$ $Mathematical equation: $ {-}0.822^{+0.011}_{-0.015} $$
	σ₈	$0 . 803_{- 0.005}^{+ 0.007}$ $Mathematical equation: $ 0.803^{+0.007}_{-0.005} $$	$0 . 813_{- 0.007}^{+ 0.008}$ $Mathematical equation: $ {-}0.813^{+0.008}_{-0.007} $$	$0 . 804_{- 0.006}^{+ 0.007}$ $Mathematical equation: $ 0.804^{+0.007}_{-0.006} $$	$0 . 823_{- 0.011}^{+ 0.015}$ $Mathematical equation: $ {-}0.823^{+0.015}_{-0.011} $$
	α_B, 0	—	$1 . 439_{- 0.763}^{+ 0.390}$ $Mathematical equation: $ {-}1.439^{+0.390}_{-0.763} $$	$0 . 180_{- 0.153}^{+ 0.162}$ $Mathematical equation: $ 0.180^{+0.162}_{-0.153} $$	—

Nuisance	A_L	$1 . 181_{- 0.054}^{+ 0.060}$ $Mathematical equation: $ 1.181^{+0.060}_{-0.054} $$	$1 . 181_{- 0.057}^{+ 0.061}$ $Mathematical equation: $ {-}1.181^{+0.061}_{-0.057} $$	$1 . 174_{- 0.055}^{+ 0.059}$ $Mathematical equation: $ 1.174^{+0.059}_{-0.055} $$	$1 . 175_{- 0.056}^{+ 0.061}$ $Mathematical equation: $ {-}1.175^{+0.061}_{-0.056} $$
	log T_AGN	$7 . 350_{- 0.048}^{+ 0.208}$ $Mathematical equation: $ 7.350^{+0.208}_{-0.048} $$	$7 . 355_{- 0.053}^{+ 0.227}$ $Mathematical equation: $ {-}7.355^{+0.227}_{-0.053} $$	$7 . 356_{- 0.054}^{+ 0.213}$ $Mathematical equation: $ 7.356^{+0.213}_{-0.054} $$	$7 . 369_{- 0.067}^{+ 0.265}$ $Mathematical equation: $ {-}7.369^{+0.265}_{-0.067} $$
	A_Planck	$1 . 001_{- 0.003}^{+ 0.002}$ $Mathematical equation: $ 1.001^{+0.002}_{-0.003} $$	$1 . 000_{- 0.002}^{+ 0.003}$ $Mathematical equation: $ {-}1.000^{+0.003}_{-0.002} $$	$1 . 000_{- 0.002}^{+ 0.003}$ $Mathematical equation: $ 1.000^{+0.003}_{-0.002} $$	$1 . 001_{- 0.003}^{+ 0.002}$ $Mathematical equation: $ {-}1.001^{+0.002}_{-0.003} $$

Horndeski	${\hat{D}}_{kin}$ $Mathematical equation: $ \hat{D}_{\mathrm{kin}} $$	—	$3 . 741_{- 1.920}^{+ 0.692}$ $Mathematical equation: $ {-}3.741^{+0.692}_{-1.920} $$	$0 . 520_{- 0.520}^{+ 3.233}$ $Mathematical equation: $ 0.520^{+3.233}_{-0.520} $$	—
	$Δ {\hat{M}}_{}^{2}$ $Mathematical equation: $ \Delta \hat{M}^2_ $$	—	$0 . 317_{- 0.212}^{+ 0.069}$ $Mathematical equation: $ {-}0.317^{+0.069}_{-0.212} $$	—	—
	${\hat{c}}_{s}^{2}$ $Mathematical equation: $ \hat{c}_{\mathrm{s}}^2 $$	—	—	$0 . 105_{- 0.105}^{+ 0.700}$ $Mathematical equation: $ 0.105^{+0.700}_{-0.105} $$	—
	log₁₀k_V	—	$- 0 . 805_{- 0.186}^{+ 0.767}$ $Mathematical equation: $ -0.805^{+0.767}_{-0.186} $$	$0 . 817_{- 0.749}^{+ 0.183}$ $Mathematical equation: $ 0.817^{+0.183}_{-0.749} $$	$- 0 . 767_{- 0.198}^{+ 1.044}$ $Mathematical equation: $ -0.767^{+1.044}_{-0.198} $$
	${\hat{α}}_{B}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{B}} $$	—	—	—	$0 . 431_{- 0.239}^{+ 0.275}$ $Mathematical equation: $ {-}0.431^{+0.275}_{-0.239} $$
	${\hat{α}}_{M}$ $Mathematical equation: $ \hat{\alpha}_{\mathrm{M}} $$	—	—	—	$1 . 179_{- 0.747}^{+ 0.611}$ $Mathematical equation: $ {-}1.179^{+0.611}_{-0.747} $$

Notes. We report the marginal mode + 68 % HPDI of all sampling parameters in ΛCDM and three modified gravity parametrisations. We do not show redshift and IA nuisance parameters since their posteriors are driven by their Gaussian prior, as listed in Table 1. Additionally, we provide constraints on four derived parameters.

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