The slope and scatter of the star-forming main sequence at z ∼ 5: Reconciling observations with simulations

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Volume 707 (March 2026)

A&A, 707 (2026) A129

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Table 2.

SFMS slope, normalization at log(M_★/M_⊙) = 10.5, and intrinsic scatter estimated using the Bayesian model with truncated normal distributions (Sect. 3.3).

ALT Hα observations
	ms_slope	ms_norm	σ_int at log(M_★/M_⊙) = 8	σ_int at log(M_★/M_⊙) = 9

Fiducial – Theios et al. (2019) (ref.)	$0 . 59_{- 0.09}^{+ 0.10}$ $Mathematical equation: $ 0.59^{+0.10}_{-0.09} $$	1.51_−0.16^+0.16	$0 . 32_{- 0.03}^{+ 0.04}$ $Mathematical equation: $ 0.32^{+0.04}_{-0.03} $$	$0 . 37_{- 0.04}^{+ 0.03}$ $Mathematical equation: $ 0.37^{+0.03}_{-0.04} $$
SFR = SFR(L_Hα) ¹	0.66_−0.08^+0.08	$1 . 79_{- 0.15}^{+ 0.14}$ $Mathematical equation: $ 1.79 ^{+0.14}_{-0.15} $$	$0 . 35_{- 0.03}^{+ 0.04}$ $Mathematical equation: $ 0.35 ^{+0.04}_{-0.03} $$	$0 . 37_{- 0.03}^{+ 0.04}$ $Mathematical equation: $ 0.37 ^{+0.04}_{-0.03} $$
SFR = SFR(L_Hα, EW(Hα))1	0.67_−0.05^+0.06	$1 . 95_{- 0.11}^{+ 0.11}$ $Mathematical equation: $ 1.95 ^{+0.11}_{-0.11} $$	$0 . 29_{- 0.02}^{+ 0.02}$ $Mathematical equation: $ 0.29 ^{+0.02}_{-0.02} $$	$0 . 30_{- 0.03}^{+ 0.02}$ $Mathematical equation: $ 0.30 ^{+0.02}_{-0.03} $$
Modified dust attenuation ²	0.73_−0.10^+0.11	$1 . 79_{- 0.18}^{+ 0.17}$ $Mathematical equation: $ 1.79 ^{+0.17}_{-0.18} $$	$0 . 35_{- 0.04}^{+ 0.04}$ $Mathematical equation: $ 0.35 ^{+0.04}_{-0.04} $$	$0 . 37_{- 0.03}^{+ 0.04}$ $Mathematical equation: $ 0.37 ^{+0.04}_{-0.03} $$

	ms_slope	ms_norm	σ_eff at log(M_★/M_⊙) = 8	σ_eff at log(M_★/M_⊙) = 9

Skewed distribution ³	0.63_−0.13^+0.14	$1 . 46_{- 0.26}^{+ 0.22}$ $Mathematical equation: $ 1.46 ^{+0.22}_{-0.26} $$	$0 . 46_{- 0.08}^{+ 0.11}$ $Mathematical equation: $ 0.46 ^{+0.11}_{-0.08} $$	$0 . 49_{- 0.07}^{+ 0.11}$ $Mathematical equation: $ 0.49 ^{+0.11}_{-0.07} $$

Hydrodynamical simulations

	ms_slope	ms_norm	σ_int at log(M_★/M_⊙) = 8	σ_int at log(M_★/M_⊙) = 9

DustyGadget ^a	$0 . 97_{- 0.02}^{+ 0.02}$ $Mathematical equation: $ 0.97 ^{+0.02}_{-0.02} $$	$1 . 86_{- 0.02}^{+ 0.02}$ $Mathematical equation: $ 1.86 ^{+0.02}_{-0.02} $$	$0 . 15_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.15 ^{+0.01}_{-0.01} $$	$0 . 14_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.14 ^{+0.01}_{-0.01} $$
EAGLE ^b	$1 . 00_{- 0.02}^{+ 0.02}$ $Mathematical equation: $ 1.00 ^{+0.02}_{-0.02} $$	$1 . 90_{- 0.03}^{+ 0.03}$ $Mathematical equation: $ 1.90 ^{+0.03}_{-0.03} $$	$0 . 19_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.19 ^{+0.01}_{-0.01} $$	$0 . 21_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.21 ^{+0.01}_{-0.01} $$
Illustris-TNG50 ^c	$1 . 09_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 1.09 ^{+0.01}_{-0.01} $$	$2 . 22_{- 0.02}^{+ 0.02}$ $Mathematical equation: $ 2.22 ^{+0.02}_{-0.02} $$	$0 . 13_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.13 ^{+0.01}_{-0.01} $$	$0 . 12_{- 0.01}^{+ 0.01}$ $Mathematical equation: $ 0.12 ^{+0.01}_{-0.01} $$
SPHINX ^d	$0 . 95_{- 0.07}^{+ 0.07}$ $Mathematical equation: $ 0.95 ^{+0.07}_{-0.07} $$	$1 . 71_{- 0.08}^{+ 0.08}$ $Mathematical equation: $ 1.71 ^{+0.08}_{-0.08} $$	$0 . 46_{- 0.04}^{+ 0.04}$ $Mathematical equation: $ 0.46 ^{+0.04}_{-0.04} $$	$0 . 35_{- 0.03}^{+ 0.03}$ $Mathematical equation: $ 0.35 ^{+0.03}_{-0.03} $$
THESAN-ZOOM ^e	$0 . 98_{- 0.06}^{+ 0.06}$ $Mathematical equation: $ 0.98^{+0.06}_{-0.06} $$	$1 . 70_{- 0.17}^{+ 0.17}$ $Mathematical equation: $ 1.70^{+0.17}_{-0.17} $$	$0 . 48_{- 0.03}^{+ 0.03}$ $Mathematical equation: $ 0.48^{+0.03}_{-0.03} $$	$0 . 51_{- 0.05}^{+ 0.05}$ $Mathematical equation: $ 0.51^{+0.05}_{-0.05} $$

Notes. The Bayesian model with truncated normal distributions was applied to observations (i.e., the ALT robust Hα sample), adopting a different conversion between the Hα luminosity and SFR (Theios et al. 2019 – fiducial – and Kramarenko et al. 2026) and modified dust attenuation corrections, and to data from hydrodynamical simulations (Graziani et al. 2020; Schaye et al. 2015; Pillepich et al. 2019; Rosdahl et al. 2018; McClymont et al. 2025). We also report the parameter estimates from the Bayesian model using truncated skewed distributions, applied to the observational robust sample.

Calibration from Kramarenko et al. (2026) using SPHINX simulations (see Sect. 5.2).

Modified dust attenuation values for high-mass galaxies (see Sect. 5.3).

Fit to the data using truncated skewed distributions in the Bayesian framework (see Sect. 5.3).

^(a)

Data at z = 4.495; priv. comm.

^(b)

Public dataset at z = 3.98 and z = 4.49 taken from https://eagle.strw.leidenuniv.nl/wordpress/.

^(c)

Public dataset at z = 4.66 taken from https://www.tng-project.org/data/.

^(d)

Public dataset at z = 4.64 taken from https://github.com/HarleyKatz/SPHINX-20-data.

^(e)

Data at z = 4.46; private communication.

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