Deeper multi-redshift upper limits on the epoch of reionisation 21 cm signal power spectrum from LOFAR between z = 8.3 and z = 10.1

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Volume 698 (June 2025)

A&A, 698 (2025) A186

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Table 2.

GP model: summary of the model parameters, priors, and posteriors (median with 68% intervals).

Component Covariance function^a	Parameter	z≈8.3		z≈9.1		z≈10.1

		Prior	Posterior	Prior	Posterior	Prior	Posterior
Sky RBF	σ²	−	$393.5 \binom{+ 16.4}{- 16.4}$ $393.5{{+16.4}\atop { -16.4}}$	−	$680.3 \binom{+ 28.6}{- 27.9}$ $680.3{{+28.6}\atop { -27.9}}$	−	$1059.5 \binom{+ 54.4}{- 52.1}$ $1059.5{{+54.4}\atop { -52.1}}$
Sky RBF	l [MHz]	=80	=80	$U (30, 60)$ ${\cal {{U}}}(30, 60)$	$41.8 \binom{+ 6.5}{- 4.8}$ $41.8{{+6.5}\atop { -4.8}}$	$U (10, 50)$ ${\cal {{U}}}(10, 50)$	$31.8 \binom{+ 5.2}{- 3.4}$ $31.8{{+5.2}\atop { -3.4}}$

Mode mixing Matérn 3/2	σ²	−	$81.5 \binom{+ 2.7}{- 2.7}$ $81.5{{+2.7}\atop { -2.7}}$	−	$114.8 \binom{+ 4.6}{- 4.4}$ $114.8{{+4.6}\atop { -4.4}}$	−	$290.5 \binom{+ 25.7}{- 23.5}$ $290.5{{+25.7}\atop { -23.5}}$
	θ [radians]^b	$U (0.15, 0.25)$ ${\cal {{U}}}(0.15, 0.25)$	$0.210 \binom{+ 0.004}{- 0.004}$ $0.210{{+0.004}\atop { -0.004}}$	$U (0.2, 0.3)$ ${\cal {{U}}}(0.2, 0.3)$	$0.238 \binom{+ 0.006}{- 0.006}$ $0.238{{+0.006}\atop { -0.006}}$	$U (0.05, 0.15)$ ${\cal {{U}}}(0.05, 0.15)$	$0.106 \binom{+ 0.008}{- 0.008}$ $0.106{{+0.008}\atop { -0.008}}$
	δ_buffer [μ s]	=0.02	=0.02	=0.01	=0.01	=0.04	=0.04

Mode mixing 2 Matérn 3/2	σ²	−	−	−	−	−	$18.8 \binom{+ 3.0}{- 2.9}$ $18.8{{+3.0}\atop { -2.9}}$
Mode mixing 2 Matérn 3/2	θ [radians]^b	−	−	−	−	$U (0.4, 0.8)$ ${\cal {{U}}}(0.4, 0.8)$	$0.57 \binom{+ 0.04}{- 0.04}$ $0.57{{+0.04}\atop { -0.04}}$

Excess RBF	σ²	−	$1.08 \binom{+ 0.06}{- 0.06}$ $1.08{{+0.06}\atop { -0.06}}$	−	$1.4 \binom{+ 0.1}{- 0.1}$ $1.4{{+0.1}\atop { -0.1}}$	−	$1.3 \binom{+ 0.2}{- 0.1}$ $1.3{{+0.2}\atop { -0.1}}$
Excess RBF	l [MHz]	$U (0.3, 0.6)$ ${\cal {{U}}}(0.3, 0.6)$	$0.41 \binom{+ 0.01}{- 0.01}$ $0.41{{+0.01}\atop { -0.01}}$	$U (0.2, 0.3)$ ${\cal {{U}}}(0.2, 0.3)$	$0.40 \binom{+ 0.02}{- 0.02}$ $0.40{{+0.02}\atop { -0.02}}$	$U (0.4, 0.8)$ ${\cal {{U}}}(0.4, 0.8)$	$0.36 \binom{+ 0.02}{- 0.02}$ $0.36{{+0.02}\atop { -0.02}}$

21 cm Learned kernel^c	σ²	−	<0.02	−	<0.05	−	<0.02
	x₁	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$0.0 \binom{+ 2.0}{- 1.9}$ $0.0{{+2.0}\atop { -1.9}}$	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$- 0.2 \binom{+ 2.1}{- 1.9}$ $-0.2{{+2.1}\atop { -1.9}}$	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$0.2 \binom{+ 1.9}{- 2.1}$ $0.2{{+1.9}\atop { -2.1}}$
	x₂	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$0.9 \binom{+ 1.5}{- 2.3}$ $0.9{{+1.5}\atop { -2.3}}$	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$- 0.6 \binom{+ 2.2}{- 1.6}$ $-0.6{{+2.2}\atop { -1.6}}$	$U (- 3, 3)$ ${\cal {{U}}}(-3, 3)$	$- 0.1 \binom{+ 2.2}{- 1.9}$ $-0.1{{+2.2}\atop { -1.9}}$

Noise Identity	α	$U (1.2, 1.9)$ ${\cal {{U}}}(1.2, 1.9)$	$1.74 \binom{+ 0.01}{- 0.01}$ $1.74{{+0.01}\atop { -0.01}}$	$U (1.4, 1.7)$ ${\cal {{U}}}(1.4, 1.7)$	$1.59 \binom{+ 0.02}{- 0.02}$ $1.59{{+0.02}\atop { -0.02}}$	$U (1.1, 1.5)$ ${\cal {{U}}}(1.1, 1.5)$	$1.31 \binom{+ 0.02}{- 0.02}$ $1.31{{+0.02}\atop { -0.02}}$

Notes. All variance parameters are expressed relative to the noise variance estimated from the time-differenced data. The prior ranges for the variance parameters of the different components are chosen to be very broad, having no impact on this analysis.

‘RBF’ is the radial basis function; ‘Matérn 3/2’ and ‘Matérn 5/2’ are Matérn class functions (Rasmussen & Williams 2006) with ν=3/2 and ν=5/2, respectively.

The coherence-scale of the mode mixing components is baseline dependent, as defined by the foregrounds wedge equation. The corresponding coherence-scales for each redshift bins can be seen in Figure 8.

The 21 cm component is learned from Grizzly simulations.

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