Beyond the Hellings–Downs curve: Non-Einsteinian gravitational waves in pulsar timing array correlations

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Volume 691 (November 2024)

A&A, 691 (2024) A126

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Table 1

Constraints on non-Einsteinian GW polarizations using NANOGrav (Agazie et al. 2023b) and Chinese PTA (Xu et al. 2023) data.

GW Mode/s	Data	CV	Speed v/c	Mass	r²	${\bar{χ}}^{2}$ ${\bar \chi ^2}$	log 𝓑_GRN	log 𝓑_HD
	NG15		1	0	0	0.64	5.8 ± 0.1	0
HD	CPTA		1	0	0	1.29	5.6 ± 0.2	0
	NG15+CPTA		1	0	0	0.94	9.9 ± 0.3	0

	NG15		>0.38	≲1.21	0	0.54	5.7 ± 0.1	−0.10 ± 0.01
	NG15	✓	>0.42	≲1.19	0	0.56	6.0 ± 0.1	0.23 ± 0.01
Tensor	CPTA		>0.46	≲1.16	0	1.03	6.1 ± 0.2	0.44 ± 0.01
	CPTA	✓	>0.52	≲1.12	0	0.70	8.8 ± 0.2	3.16 ± 0.01
	NG15+CPTA		>0.57	≲1.08	0	0.86	10.3 ± 0.3	0.41 ± 0.01
	NG15+CPTA	✓	>0.61	≲1.04	0	0.62	13.4 ± 0.3	3.48 ± 0.02

	NG15		${0.53}_{- 0.13}^{+ 0.29}$ $0.53_{ - 0.13}^{ + 0.29}$	≃(0.75, 1.20)	0	0.50	6.0 ± 0.1	0.20 ± 0.02
	NG15	✓	${0.58}_{- 0.12}^{+ 0.29}$ $0.58_{ - 0.12}^{ + 0.29}$	≃(0.66, 1.17)	0	0.44	6.5 ± 0.1	0.71 ± 0.02
Vector	CPTA		${0.49}_{- 0.10}^{+ 0.23}$ $0.49_{ - 0.10}^{ + 0.23}$	≃(0.92, 1.21)	0	1.04	6.3 ± 0.2	0.65 ± 0.02
	CPTA	✓	${0.60}_{- 0.10}^{+ 0.25}$ $0.60_{ - 0.10}^{ + 0.25}$	≃(0.70, 1.14)	0	0.56	9.4 ± 0.2	3.78 ± 0.02
	NG15+CPTA		${0.55}_{- 0.06}^{+ 0.17}$ $0.55_{ - 0.06}^{ + 0.17}$	≃(0.91, 1.14)	0	0.76	11.1 ± 0.3	1.15 ± 0.03
	NG15+CPTA	✓	${0.66}_{- 0.07}^{+ 0.16}$ $0.66_{ - 0.07}^{ + 0.16}$	≃(0.74, 1.05)	0	0.50	14.8 ± 0.3	4.86 ± 0.03

	NG15		<0.53	–	<0.56	0.54	5.6 ± 0.1	−0.16 ± 0.01
	NG15	✓	${0.43}_{- 0.39}^{+ 0.18}$ $0.43_{ - 0.39}^{ + 0.18}$	–	<0.61	0.54	6.1 ± 0.1	0.34 ± 0.01
HD-V	CPTA		<0.54	–	<0.14	1.29	3.5 ± 0.2	−2.12 ± 0.03
	CPTA	✓	<0.57	–	<0.29	0.72	7.9 ± 0.2	2.32 ± 0.02
	NG15+CPTA		<0.52	–	<0.14	0.94	7.9 ± 0.3	−1.99 ± 0.03
	NG15+CPTA	✓	<0.56	–	<0.28	0.64	12.8 ± 0.3	2.86 ± 0.02

	NG15		✗	–	0.33 ± 0.14	0.26	7.5 ± 0.1	1.68 ± 0.02
	NG15	✓	✗	–	${0.55}_{- 0.29}^{+ 0.23}$ $0.55_{ - 0.29}^{ + 0.23}$	0.14	9.1 ± 0.1	3.29 ± 0.01
HD-ϕ	CPTA		✗	–	<0.05	1.29	2.7 ± 0.2	−2.96 ± 0.05
	CPTA	✓	✗	–	✗	0.68	8.9 ± 0.2	3.27 ± 0.01
	NG15+CPTA		✗	–	<0.08	0.93	7.6 ± 0.3	−2.32 ± 0.04
	NG15+CPTA	✓	✗	–	>0.49	0.44	16.4 ± 0.3	6.51 ± 0.01

	NG15		✗	✗	0.36 ± 0.14	0.21	7.7 ± 0.1	1.94 ± 0.02
	NG15	✓	✗	✗	${0.55}_{- 0.28}^{+ 0.23}$ $0.55_{ - 0.28}^{ + 0.23}$	0.11	9.2 ± 0.1	3.40 ± 0.01
	CPTA		>0.46	<1.16	<0.07	1.14	3.5 ± 0.2	−2.14 ± 0.04
T-ϕ	CPTA	✓	✗	✗	>0.54	0.58	9.1 ± 0.2	3.49 ± 0.01
	NG15+CPTA		>0.56	<1.08	<0.11	0.83	8.8 ± 0.3	−1.14 ± 0.04
	NG15+CPTA	✓	✗	✗	>0.53	0.40	16.9 ± 0.3	6.93 ± 0.01

Notes. HD: Hellings-Downs curve; ‘Tensor’/’Vector’: subluminal GWs; HD-ϕ/V: luminal tensor GWs + scalar/vector GWs with speed v and fraction r²; T-ϕ: tensor and scalar GWs with the same speed. ‘✓’ in CV cells: cosmic variance included (Allen 2023; Bernardo & Ng 2022b). ‘✗’/‘−’: unconstrained/inapplicable parameters. ${\bar{χ}}^{2}$ ${\bar \chi ^2}$ : reduced chi-squared; log 𝓑_𝓗: Bayes factor vs. null hypothesis 𝓗 (Gaussian random noise/GRN or HD); log 𝓑_𝓗 > (<) 0: evidence for (against) correlations model. Masses in units of 10⁻²² eV/c². Data: NANOGrav (NG15), Chinese PTA (CPTA).

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