Peak star formation efficiency and no missing baryons in massive spirals

TL;DR

This study finds that massive spiral galaxies have very high star formation efficiencies, nearly converting all their baryons into stars, challenging the traditional view of low efficiency and missing baryons in such galaxies.

Contribution

It provides observational evidence that massive spirals can reach near 100% star formation efficiency, contradicting previous models of declining efficiency at high masses.

Findings

Massive spirals have star formation efficiencies of 30-100%.

No evidence of efficiency decline at high galaxy masses.

Massive spirals contain virtually no missing baryons.

Abstract

It is commonly believed that galaxies use, throughout the Hubble time, a very small fraction of the baryons associated to their dark matter halos to form stars. This so-called low "star formation efficiency" $f_\star\equiv M_\star/f_{\rm b} M_{\rm halo}$, where $f_{\rm b}\equiv\Omega_{\rm b}/\Omega_{\rm c}$ is the cosmological baryon fraction, is expected to reach its peak at nearly $L^\ast$ (at efficiency $\approx 20\%$) and decline steeply at lower and higher masses. We have tested this using a sample of nearby star-forming galaxies, from dwarfs ($M_\star\simeq 10^7 M_\odot$) to high-mass spirals ($M_\star\simeq 10^{11} M_\odot$) with HI rotation curves and 3.6$\mu$m photometry. We fit the observed rotation curves with a Bayesian approach by varying three parameters, stellar mass-to-light ratio, halo concentration and mass. We found two surprising results: 1) the star formation efficiency is a monotonically increasing function of $M_\star$ with no sign of a decline at high masses, and 2) the most massive spirals ($M_\star\simeq 1-3 \times 10^{11} M_\odot$) have $f_\star\approx 0.3-1$, i.e. they have turned nearly all the baryons associated to their haloes into stars. These results imply that the most efficient galaxies at forming stars are massive spirals (not $L^\ast$ galaxies), they reach nearly 100% efficiency and thus, once both their cold and hot gas is considered into the baryon budget, they have virtually no missing baryons. Moreover, there is no evidence of mass quenching of the star formation occurring in galaxies up to halo masses of $M_{\rm halo}\approx {\rm a\, few}\times 10^{12} M_\odot$.

Tables

Table 1. Table 1: Priors of our model. 𝒫 ​ ( M halo , c , Υ ⋆ ) 𝒫 subscript 𝑀 halo 𝑐 subscript Υ ⋆ \mathcal{P}(M_{\rm halo},c,\Upsilon_{\star}) in Eq.  3 is given by the product of the three terms.

Parameter	Type
${{\rm Y}}_{\star }$	uniform	$0.01\le {{\rm Y}}_{\star }\le 1.2$
$M_{\mathrm{halo}}$	uniform	$6\le \log M_{\mathrm{halo}}/M_{\odot }\le 15$
$c$	lognormal	$c-M_{\mathrm{halo}}$ from DM14

Table 2. Table 2: Results of the fits for individual galaxies. The near-infrared luminosity L [ 3.6 ] subscript 𝐿 delimited-[] 3.6 L_{[3.6]} is given in solar luminosities; the posteriors of the three model parameters, disc mass-to-light ratio Υ disc subscript Υ disc \Upsilon_{\rm disc} , halo mass M halo subscript 𝑀 halo M_{\rm halo} and concentration c 𝑐 c are represented with their 50th-16th-84th percentiles; χ red 2 subscript superscript 𝜒 2 red \chi^{2}_{\rm red} is the reduced χ 2 superscript 𝜒 2 \chi^{2} (Eq.  2 ) for the best fit model; the posterior on the derived parameter f ⋆ = M ⋆ / f b ​ M halo subscript 𝑓 ⋆ subscript 𝑀 ⋆ subscript 𝑓 b subscript 𝑀 halo f_{\star}=M_{\star}/f_{\rm b}M_{\rm halo} is represented with its 50th-16th-84th percentiles.

Name	$\log L_{\left[3.6\right]}$	${{\rm Y}}_{\mathrm{disc}}$	16th	84th	$\log M_{\mathrm{halo}}$	16th	84th	$\log c$	16th	84th	${\chi }_{\mathrm{red}}^2$	$f_{\star }$	16th	84th
D512-2	8.51	0.62	0.22	1.02	9.91	9.59	10.26	0.98	0.86	1.11	1.05	0.0852	0.0157	0.2845
DDO064	8.20	0.60	0.21	1.00	10.29	9.76	10.92	1.00	0.83	1.17	1.07	0.0237	0.0029	0.1213
DDO170	8.73	0.38	0.12	0.80	10.66	10.58	10.76	0.82	0.74	0.88	2.73	0.0215	0.0052	0.0431
ESO116-G012	9.63	0.44	0.20	0.69	11.72	11.49	12.05	0.89	0.74	1.01	2.52	0.0181	0.0064	0.0374
ESO444-G084	7.85	0.60	0.21	0.99	11.23	10.93	11.65	0.92	0.79	1.03	0.76	0.0011	0.0003	0.0030
F565-V2	8.75	0.58	0.19	1.00	11.14	10.88	11.51	0.89	0.73	1.02	1.10	0.0103	0.0026	0.0280
F568-V1	9.58	0.70	0.27	1.05	11.63	11.30	12.07	1.01	0.84	1.15	0.30	0.0314	0.0077	0.0919
F574-1	9.82	0.68	0.27	1.03	11.29	11.07	11.55	0.92	0.80	1.03	1.84	0.1097	0.0312	0.2466
F583-1	8.99	0.57	0.18	0.98	11.08	10.77	11.42	0.86	0.73	0.98	2.11	0.0255	0.0058	0.0731
F583-4	9.23	0.62	0.21	1.02	10.61	10.31	10.98	0.98	0.82	1.11	0.44	0.1377	0.0263	0.4602
NGC0024	9.59	1.02	0.77	1.15	11.27	11.06	11.56	1.06	0.91	1.20	0.66	0.1306	0.0670	0.2269
NGC0100	9.51	0.29	0.09	0.60	11.36	11.02	11.76	0.85	0.70	0.97	1.20	0.0208	0.0024	0.0820
NGC0247	9.87	0.64	0.25	1.01	11.35	11.09	11.62	0.82	0.72	0.90	2.14	0.1013	0.0280	0.2969
NGC0289	10.86	0.59	0.43	0.76	11.83	11.74	11.94	0.91	0.77	1.05	1.95	0.2833	0.1765	0.4227
NGC0300	9.47	0.46	0.17	0.79	11.37	11.18	11.63	0.89	0.75	1.01	0.72	0.0268	0.0102	0.0573
NGC0801	11.49	0.56	0.52	0.60	12.00	11.90	12.14	0.77	0.63	0.90	6.80	1.0564	0.7746	1.3789
NGC1003	9.83	0.46	0.24	0.66	11.49	11.39	11.62	0.78	0.67	0.88	3.09	0.0485	0.0240	0.0746
NGC1090	10.86	0.48	0.36	0.59	11.72	11.63	11.84	0.94	0.80	1.07	2.50	0.3931	0.2445	0.5734
NGC1705	8.73	0.99	0.72	1.15	10.86	10.57	11.26	1.16	0.99	1.31	0.66	0.0352	0.0133	0.0712
NGC2403	10.00	0.42	0.30	0.53	11.40	11.33	11.49	1.14	1.06	1.23	9.47	0.1012	0.0828	0.1164
NGC2683	10.91	0.66	0.58	0.73	11.63	11.46	11.82	0.96	0.82	1.11	1.31	0.4620	0.2927	0.7192
NGC2841	11.27	0.87	0.79	0.94	12.54	12.42	12.69	0.88	0.76	1.00	1.81	0.1796	0.1335	0.2283
NGC2903	10.91	0.37	0.31	0.41	11.75	11.67	11.85	1.24	1.14	1.34	7.61	0.3001	0.1718	0.4448
NGC2915	8.81	0.56	0.19	0.97	11.10	10.85	11.44	1.03	0.86	1.18	0.98	0.0106	0.0030	0.0270
NGC2955	11.50	0.47	0.44	0.51	12.13	11.80	12.48	0.88	0.71	1.03	4.81	0.6863	0.3052	1.4911
NGC2998	11.18	0.62	0.48	0.74	12.01	11.91	12.13	0.91	0.76	1.06	2.74	0.5532	0.3896	0.7568
NGC3198	10.58	0.51	0.38	0.61	11.67	11.60	11.75	0.98	0.87	1.09	1.43	0.2475	0.1981	0.2979
NGC3521	10.93	0.52	0.47	0.58	12.29	11.83	12.85	0.86	0.68	1.03	0.29	0.1212	0.0315	0.3787
NGC3726	10.85	0.39	0.28	0.47	11.76	11.59	11.98	0.87	0.73	1.02	2.96	0.1987	0.1058	0.3469
NGC3741	7.45	0.46	0.14	0.89	10.57	10.33	10.86	0.84	0.72	0.95	1.05	0.0013	0.0004	0.0031
NGC3769	10.27	0.35	0.21	0.51	11.40	11.25	11.57	1.01	0.88	1.14	0.68	0.0970	0.0495	0.1719
NGC3893	10.77	0.50	0.41	0.58	12.01	11.75	12.36	0.95	0.78	1.11	1.27	0.1227	0.0551	0.2310
NGC3972	10.16	0.40	0.14	0.73	12.03	11.57	12.52	0.86	0.70	0.98	1.19	0.0240	0.0038	0.1250
NGC3992	11.36	0.82	0.69	0.93	12.15	12.03	12.30	0.90	0.74	1.05	0.85	0.4339	0.3037	0.6160
NGC4010	10.24	0.25	0.09	0.45	11.96	11.62	12.36	0.81	0.68	0.95	2.44	0.0216	0.0045	0.0764
NGC4013	10.90	0.48	0.41	0.54	11.98	11.81	12.19	0.85	0.70	0.99	1.31	0.0776	0.0483	0.1161
NGC4088	11.03	0.31	0.24	0.37	11.77	11.54	12.05	0.91	0.74	1.06	0.57	0.2924	0.1393	0.5817
NGC4100	10.77	0.74	0.61	0.85	11.69	11.48	11.93	0.97	0.81	1.12	1.27	0.4199	0.2325	0.7423
NGC4138	10.64	0.68	0.58	0.80	11.46	11.09	11.82	0.99	0.82	1.16	1.68	0.2491	0.1029	0.6693
NGC4157	11.02	0.40	0.32	0.48	11.95	11.74	12.22	0.89	0.73	1.04	0.55	0.2388	0.1231	0.4311
NGC4183	10.03	0.75	0.38	1.04	11.16	10.97	11.35	1.01	0.87	1.13	0.18	0.3102	0.1236	0.6488
NGC4559	10.29	0.38	0.20	0.55	11.41	11.23	11.61	0.95	0.81	1.09	0.24	0.1513	0.0527	0.3176
NGC5033	11.04	0.40	0.31	0.48	11.91	11.86	11.96	1.23	1.14	1.31	3.81	0.3049	0.1732	0.4383
NGC5055	11.18	0.32	0.29	0.34	11.82	11.79	11.85	1.12	1.06	1.18	2.75	0.4220	0.3913	0.4514
NGC5371	11.53	0.44	0.34	0.53	11.64	11.53	11.74	1.21	1.02	1.34	6.59	1.9570	1.1181	3.1110
NGC5585	9.47	0.18	0.08	0.30	11.33	11.18	11.52	0.90	0.79	0.98	5.85	0.0142	0.0006	0.0294
NGC5907	11.24	0.68	0.56	0.78	12.02	11.93	12.16	0.89	0.71	1.07	6.38	0.5110	0.4049	0.6183
NGC5985	11.32	0.45	0.26	0.65	12.21	12.12	12.28	1.37	1.30	1.44	2.85	0.3156	0.1265	0.5595
NGC6015	10.51	0.78	0.65	0.87	11.67	11.52	11.88	0.94	0.77	1.10	8.45	0.3054	0.1972	0.4377
NGC6195	11.59	0.46	0.42	0.48	12.16	11.94	12.42	0.79	0.64	0.93	3.44	0.6961	0.3866	1.1779
NGC6503	10.11	0.45	0.36	0.53	11.28	11.21	11.36	1.11	1.02	1.19	1.61	0.1585	0.1316	0.1883
NGC6674	11.33	0.94	0.83	1.03	12.42	12.32	12.56	0.65	0.52	0.77	3.87	0.3996	0.2914	0.5274
NGC6946	10.82	0.44	0.38	0.48	11.83	11.62	12.12	0.95	0.79	1.09	1.88	0.2336	0.1103	0.4250
NGC7331	11.40	0.36	0.33	0.40	12.38	12.21	12.60	0.85	0.71	0.98	0.80	0.1527	0.0945	0.2232
NGC7814	10.87	0.50	0.43	0.56	12.21	12.01	12.50	1.01	0.86	1.15	1.30	0.1245	0.0688	0.1869
UGC00128	10.08	0.53	0.18	0.92	11.56	11.53	11.59	0.93	0.86	0.99	3.19	0.1058	0.0370	0.1797
UGC00191	9.30	0.83	0.51	1.08	10.96	10.87	11.10	0.93	0.82	1.02	3.68	0.0947	0.0586	0.1368
UGC00731	8.51	0.59	0.19	1.01	10.77	10.64	10.91	0.99	0.91	1.08	0.36	0.0176	0.0051	0.0338
UGC02259	9.24	0.86	0.46	1.11	10.78	10.69	10.89	1.23	1.15	1.31	1.37	0.1220	0.0610	0.1851
UGC02487	11.69	0.98	0.85	1.08	12.58	12.52	12.67	0.94	0.81	1.06	5.28	0.3968	0.3302	0.4704
UGC02885	11.61	0.63	0.55	0.72	12.62	12.48	12.79	0.75	0.62	0.88	1.47	0.3448	0.2284	0.5073
UGC02916	11.09	0.34	0.31	0.36	12.10	11.93	12.31	1.05	0.95	1.15	10.88	0.2354	0.1404	0.3645
UGC02953	11.41	0.56	0.51	0.60	12.29	12.22	12.36	1.11	1.02	1.20	6.78	0.4796	0.3421	0.6312
UGC03205	11.06	0.72	0.64	0.79	12.12	11.95	12.33	0.85	0.70	1.01	3.51	0.4040	0.2531	0.5862
UGC03546	11.01	0.41	0.34	0.46	11.92	11.80	12.06	1.07	0.96	1.18	1.52	0.2236	0.1352	0.3344
UGC03580	10.12	0.18	0.13	0.22	11.52	11.42	11.64	0.95	0.87	1.04	3.52	0.0459	0.0121	0.0823
UGC04278	9.12	0.36	0.10	0.76	11.41	11.00	11.89	0.80	0.65	0.94	2.19	0.0095	0.0011	0.0430
UGC04483	7.11	0.52	0.17	0.93	9.30	8.97	9.74	1.11	0.95	1.26	0.74	0.0160	0.0038	0.0485
UGC04499	9.19	0.34	0.11	0.69	10.89	10.70	11.12	0.93	0.81	1.04	0.95	0.0322	0.0070	0.0839
UGC05005	9.61	0.36	0.10	0.78	11.10	10.84	11.36	0.85	0.71	0.97	1.11	0.0718	0.0151	0.2207
UGC05253	11.23	0.46	0.43	0.48	12.16	12.08	12.27	1.05	0.98	1.12	3.22	0.3759	0.2567	0.5165
UGC05414	9.05	0.20	0.06	0.46	11.17	10.82	11.57	0.77	0.64	0.89	1.68	0.0061	0.0002	0.0256
UGC05716	8.77	0.44	0.15	0.83	10.81	10.75	10.89	0.98	0.91	1.03	1.76	0.0186	0.0062	0.0312
UGC05721	8.73	0.93	0.60	1.12	10.91	10.68	11.23	1.17	1.01	1.30	1.90	0.0317	0.0142	0.0596
UGC05829	8.75	0.59	0.18	1.01	10.47	10.16	10.83	0.95	0.80	1.09	0.84	0.0539	0.0106	0.1593
UGC05918	8.37	0.63	0.21	1.02	10.07	9.81	10.43	1.04	0.89	1.17	0.35	0.0580	0.0124	0.1611
UGC06399	9.36	0.61	0.22	0.99	11.27	10.95	11.67	0.89	0.75	1.02	0.97	0.0362	0.0077	0.1135
UGC06446	8.99	0.75	0.32	1.08	10.96	10.75	11.23	1.06	0.92	1.18	0.22	0.0385	0.0133	0.0808
UGC06614	11.09	0.27	0.17	0.36	12.20	12.03	12.41	0.83	0.68	0.96	0.44	0.0828	0.0428	0.1474
UGC06667	9.15	0.63	0.21	1.03	11.41	11.18	11.72	0.88	0.76	0.98	1.57	0.0113	0.0029	0.0275
UGC06786	10.87	0.57	0.49	0.65	12.22	12.10	12.37	1.05	0.94	1.16	1.47	0.1669	0.1166	0.2240
UGC06787	10.99	0.43	0.38	0.47	12.17	12.10	12.24	1.19	1.12	1.26	27.20	0.2041	0.1410	0.2737
UGC06917	9.83	0.46	0.18	0.78	11.46	11.23	11.77	0.93	0.79	1.05	0.75	0.0438	0.0137	0.1163
UGC06923	9.46	0.30	0.11	0.59	11.20	10.83	11.68	0.94	0.78	1.08	0.85	0.0194	0.0035	0.0809
UGC06930	9.95	0.68	0.28	1.02	11.15	10.93	11.38	0.99	0.86	1.12	0.33	0.2057	0.0617	0.4919
UGC06973	10.73	0.18	0.16	0.20	12.83	12.24	13.53	0.86	0.65	1.06	1.11	0.0032	0.0006	0.0126
UGC06983	9.72	0.76	0.38	1.06	11.31	11.11	11.57	1.00	0.85	1.13	0.70	0.0767	0.0301	0.1557
UGC07089	9.55	0.44	0.13	1.05	10.68	9.71	11.15	0.91	0.75	1.13	1.01	0.1587	0.0203	3.9876
UGC07125	9.43	0.28	0.09	0.57	10.46	10.33	10.60	0.91	0.81	1.01	1.08	0.1392	0.0239	0.3678
UGC07151	9.36	0.84	0.58	1.06	10.77	10.45	11.14	0.95	0.80	1.07	2.64	0.1613	0.0586	0.4149
UGC07399	9.06	0.84	0.45	1.10	11.39	11.17	11.70	1.13	1.01	1.23	1.74	0.0163	0.0066	0.0315
UGC07524	9.39	0.50	0.17	0.94	11.00	10.77	11.27	0.87	0.75	0.97	0.94	0.0657	0.0155	0.1930
UGC07559	8.04	0.53	0.15	1.02	9.31	8.70	9.76	1.08	0.92	1.27	1.29	0.1263	0.0185	1.0510
UGC07603	8.58	0.53	0.20	0.88	11.01	10.70	11.44	0.97	0.82	1.11	1.62	0.0084	0.0021	0.0224
UGC07690	8.93	0.89	0.66	1.08	10.18	9.87	10.53	1.09	0.94	1.25	0.48	0.1986	0.0754	0.4751
UGC07866	8.09	0.66	0.22	1.06	9.31	8.78	9.80	1.14	0.97	1.30	0.23	0.1754	0.0266	0.9472
UGC08286	9.10	0.94	0.61	1.13	10.90	10.78	11.05	1.11	1.02	1.20	2.13	0.0801	0.0490	0.1160
UGC08490	9.01	0.92	0.58	1.12	10.79	10.64	10.99	1.15	1.01	1.27	0.29	0.0746	0.0425	0.1147
UGC08550	8.46	0.79	0.42	1.07	10.51	10.33	10.74	1.05	0.93	1.16	0.66	0.0314	0.0154	0.0546
UGC08699	10.70	0.56	0.51	0.60	11.95	11.75	12.21	0.99	0.85	1.11	1.13	0.1982	0.1076	0.3284
UGC09037	10.84	0.11	0.04	0.20	11.91	11.74	12.13	0.87	0.74	0.98	1.03	0.0381	0.0101	0.0852
UGC09133	11.45	0.47	0.44	0.50	12.22	12.18	12.25	0.99	0.92	1.05	8.84	0.5423	0.4231	0.6673
UGC10310	9.24	0.73	0.30	1.06	10.67	10.42	10.96	1.02	0.88	1.14	0.49	0.1258	0.0341	0.3281
UGC11820	8.99	0.52	0.17	0.90	11.15	11.04	11.28	0.74	0.65	0.81	2.20	0.0221	0.0079	0.0377
UGC11914	11.18	0.64	0.61	0.67	13.04	12.44	13.67	0.75	0.58	0.94	2.55	0.0492	0.0110	0.2009
UGC12506	11.14	0.97	0.66	1.14	12.14	11.96	12.33	0.99	0.84	1.13	0.67	0.5698	0.2753	0.9742
UGC12632	9.11	0.66	0.23	1.04	10.73	10.56	10.92	0.98	0.87	1.09	0.41	0.0878	0.0252	0.1817
UGC12732	9.22	0.54	0.18	0.95	11.11	10.96	11.30	0.92	0.80	1.02	0.29	0.0361	0.0109	0.0741
UGCA281	8.29	0.66	0.28	1.01	9.86	9.36	10.46	1.04	0.88	1.18	0.89	0.0382	0.0054	0.1712
UGCA444	7.08	0.61	0.21	1.02	9.62	9.19	10.14	1.08	0.91	1.25	0.55	0.0088	0.0018	0.0316

Equations

$$V_{\mathrm{c}}=\sqrt{V_{\mathrm{DM}}^2+V_{\mathrm{gas}}^2+V_{\star }^2}.$$

$$\begin{array}{rl}{\chi }^2& =-\ln \mathcal{P}(\theta |M_{\mathrm{halo}},c,{{\rm Y}}_{\mathrm{disc}})\\ & =\sum _{i=0}^N\frac{1}{2}{\left[\frac{V_{\mathrm{obs},i}-V_{\mathrm{c}}(R_i|M_{\mathrm{halo}},c,{{\rm Y}}_{\mathrm{disc}})}{{\sigma }_{V_{\mathrm{obs},i}}}\right]}^2\end{array}$$

$$\mathcal{P}(M_{\mathrm{halo}},c,{{\rm Y}}_{\mathrm{disc}}|\theta )\propto \mathcal{P}(\theta |M_{\mathrm{halo}},c,{{\rm Y}}_{\mathrm{disc}})\mathcal{P}(M_{\mathrm{halo}},c,{{\rm Y}}_{\mathrm{disc}})$$

Full text

1 Introduction

In our Universe, only about one-sixth of the total matter is baryonic, while the rest is widely thought to be in form of non-baryonic, collisionless, non-relativistic dark matter (e.g. Planck Collaboration et al. 2018). In the so-called standard $\Lambda$ Cold Dark Matter ($\Lambda$CDM) paradigm, galaxies form within extended haloes of dark matter that were able to grow enough to become gravitationally bound (e.g. White & Rees 1978). In this scenario it is then reasonable to expect that the amount baryons present in galaxies today is roughly a fraction $f_{\rm b}\equiv\Omega_{\rm b}/\Omega_{\rm c}=0.188$ (the “cosmological baryon fraction”, e.g. Planck Collaboration et al. 2018) of the mass in dark matter. However, it was realised that the total amount of baryons that we can directly observe in galaxies (stars, gas, dust etc.) is instead at most only about 20$\%$ of the cosmological value (e.g. Persic & Salucci 1992; Fukugita et al. 1998). This became known as the “missing baryons” problem and has prompted the search for large resevoirs of baryons within the diffuse, multi-phase circumgalactic medium of galaxies (Bregman 2007; Tumlinson et al. 2017).

Arguably the most important indicator of this issue is the so-called stellar-to-halo mass relation, which connects the stellar mass $M_{\star}$ of a galaxy to its dark matter halo of mass $M_{\rm halo}$ (see Wechsler & Tinker 2018, for a recent review). This relation can be probed observationally through many different techniques, e.g. galaxy abundance as a function of stellar mass (e.g. Vale & Ostriker 2004; Behroozi et al. 2010; Moster et al. 2013), galaxy clustering (e.g. Kravtsov et al. 2004; Zheng et al. 2007), group catalogues (e.g. Yang et al. 2008), weak galaxy-galaxy lensing (e.g. Mandelbaum et al. 2006; Leauthaud et al. 2012), satellite kinematics (e.g. van den Bosch et al. 2004; More et al. 2011; Wojtak & Mamon 2013) and internal galaxy dynamics (e.g. Persic et al. 1996; McConnachie 2012; Cappellari et al. 2013; Desmond & Wechsler 2015; Read et al. 2017; Katz et al. 2017, hereafter K17). Amongst all these determinations there is wide consensus on the overall shape of the relation and, in particular, on the fact that the ratio of stellar-to-halo mass $f_{\star}=M_{\star}/f_{\rm b}M_{\rm halo}$, sometimes called star-formation efficiency, is a non-monotonic function of mass with a peak ($f_{\star}\approx 0.2$) at $M_{\rm halo}\approx 10^{12}M_{\odot}$ (roughly the mass of the Milky Way). This can be interpreted as galaxies of these characteristic mass having been overall, during the course of their life, the most efficient at turning gas into stars. And yet, efficiencies of the order of $20\%$ are still relatively low, implying that most baryons are still undetected even in these systems111Since molecular, atomic and ionized gas is typically dynamically sub-dominant in $M_{\star}>10^{10}M_{\odot}$ galaxies..

Several works have suggested that the exact shape of the stellar-to-halo mass relation depends on galaxy morphology (e.g. Mandelbaum et al. 2006; Conroy et al. 2007; Dutton et al. 2010; More et al. 2011; Rodríguez-Puebla et al. 2015; Lange et al. 2018), especially on the high-mass side ($\log\,M_{\star}/M_{\odot}\gtrsim 10$) where red, passive early-type systems appear to reside in more massive halos with respect to blue, star-forming late-type galaxies. This is intriguing, since it is suggesting that galaxies with different morphologies likely followed different evolutionary pathways that led the late-type ones, at a given M⋆, to live in relatively lighter halos and to have a somewhat smaller fraction of missing baryons with respect to early-type systems222 Blue galaxies also have typically larger reservoirs of cold gas with respect to red ones. However, on average, the amount cold gas is sub-dominant with respect to stars for $M_{\star}>10^{10}M_{\odot}$. (e.g. Papastergis et al. 2012). . However, one of the main difficulties associated to these measurements is the paucity of high-mass galaxies in the nearby Universe (e.g. Kelvin et al. 2014), given that most of the aforementioned observational probes use statistical estimates based on on large galaxy samples.

In this paper we use another, complementary approach to estimate the stellar-to-halo mass relation through accurate modelling of the gas dynamics within spiral galaxies. We use the observed Hi rotation curves of a sample of regularly rotating, nearby disc galaxies to fit mass models comprising of a baryonic plus a dark matter component. We then extrapolate the dark matter profile to the virial radius, with cosmologically motivated assumptions, to yield the halo mass. A considerable advantage of this method is that each system can be studied individually and halo masses, along with their associated uncertainties, can be determined in great detail for each object. We show that this approach leads to a coherent picture of the relation between stellar and halo mass in late-type galaxies, which in turns profoundly affects our perspective on the star-formation efficiency in the high-mass regime.

The paper is organised as follows: we present our sample and methodology to derive stellar and halo masses in Section 2; we describe our results in Section 3 and we discuss them in detail in Section 4.

2 Method

Here we describe the data and methodology of our analysis. We adopt a standard $\Lambda$CDM cosmology, with parameters estimated by the Planck Collaboration et al. (2018). In particular, we use a Hubble constant of $H_{0}=67.66$ km s*-1* Mpc*-1* and a cosmological baryon fraction of $f_{\rm b}\equiv\Omega_{\rm b}/\Omega_{\rm c}=0.188$.

2.1 Data

We use the sample of 175 disc galaxies with near-infrared photometry and Hi rotation curves (SPARC) collected by Lelli et al. (2016a, hereafter LMS16). This sample of spirals in the nearby Universe spans more than 4 orders of magnitude in luminosity at 3.6$\mu$m and all morphological types, from irregulars to lenticulars. The galaxies have been selected to have extended, regular, high-quality Hi rotation curves and measured near-infrared photometry; thus it is not volume limited. Nevertheless, it still provides a fair representation of the population of (regularly rotating) spirals at $z=0$ and most importantly is best suited for our dynamical study.

The Hi rotation curves are used as tracers of the circular velocity of the galaxies, while the individual contributions of the atomic gas ($V_{\rm gas}$) and stars ($V_{\star}$) to the circular velocity are derived from the Hi and 3.6$\mu$m total intensity maps respectively (see LMS16, for further details). $V_{\rm gas}$ traces the distribution of atomic hydrogen, corrected for the presence of helium, while the near-infrared surface brightness is decomposed into and exponential disc ($V_{\rm disc}$) and a spherical bulge ($V_{\rm bulge}$). The contribution of the stars to the circular velocity is then $V_{\star}^{2}=\Upsilon_{\rm disc}V_{\rm disc}^{2}+\Upsilon_{\rm bulge}V_{\rm bulge}^{2}$, given stellar mass-to-light ratios of the disc ($\Upsilon_{\rm disc}$) and bulge populations ($\Upsilon_{\rm bulge}$) respectively.

2.2 Model

We model the observed rotation curve as

[TABLE]

Here $V_{\rm DM}$ is the dark matter contribution to the circular velocity and, for simplicity, we have assumed that $\Upsilon_{\rm bulge}=1.4\Upsilon_{\rm disc}$, as suggested by stellar population synthesis models (e.g. Schombert & McGaugh 2014), thus $V_{\star}^{2}=\Upsilon_{\rm disc}\left(V_{\rm disc}^{2}+1.4V_{\rm bulge}^{2}\right)$. In Appendix A we explore the effect of fixing different mass-to-light ratios $\Upsilon_{\rm disc}$ and $\Upsilon_{\rm bulge}$ for disc and bulge respectively: our findings on the stellar-to-halo mass relation do not change significantly if we assume $\Upsilon_{\rm disc}=0.5$ and $\Upsilon_{\rm bulge}=0.7$, for which the scatter of the baryonic Tully-Fisher relation is minimised (Lelli et al. 2016b).

The dark matter distribution is modelled as a Navarro et al. (1996, hereafter NFW) spherical halo, which is characterised by a dimensionless concentration parameter ($c$) and the halo mass ($M_{\rm halo}$), which we take as that within a radius enclosing 200 times the critical density of the Universe. Thus our rotation curve model has three free parameters: $M_{\rm halo}$, $c$ and $\Upsilon_{\star}$.

We compute the posterior distributions of these parameters with a Bayesian approach. We define a standard $\chi^{2}$ likelihood $\mathcal{P}$, given the data $\theta$, as

[TABLE]

where $V_{{\rm obs},i}$ is the $i$-th point of the observed rotation curve at radius $R_{i}$ and $\sigma_{V_{{\rm obs},i}}$ is its observed uncertainty. The posterior distribution of the three parameters is then given by Bayes’ theorem

[TABLE]

where $\mathcal{P}(M_{\rm halo},c,\Upsilon_{\rm disc})$ is the prior. We sample the posterior with an affine-invariant Markov Chain Monte Carlo method (MCMC, in particular, we use the python implementation by Foreman-Mackey et al. 2013).

We use a flat prior on the stellar mass-to-light ratio $\Upsilon_{\rm disc}$ limited to a reasonable range, $0.01\lesssim\Upsilon_{\rm disc}\lesssim 1.2$, which encompasses estimates obtained with stellar population models (Meidt et al. 2014; McGaugh & Schombert 2014). In a $\Lambda$CDM Universe the halo mass and concentration are well known to be anti-correlated. Thus, in order to test whether standard $\Lambda$CDM haloes can be used to fit galaxy rotation curves and then yield a stellar-to-halo mass relation, for the halo concentration we assume a prior which follows the $c-M_{\rm halo}$ relation as estimated in N-body cosmological simulations (e.g. Dutton & Macciò 2014, hereafter DM14): for each $M_{\rm halo}$, the prior on $c$ is lognormal with mean and uncertainty given by the $c=c(M_{\rm halo})$ of DM14 (their Eq. 8). The prior on the dark matter halo mass $M_{\rm halo}$ is, instead, flat over a wide range: $6\leq\log\,M_{\rm halo}/M_{\odot}\leq 15$.

A non-uniform prior on the halo concentration is needed to infer reasonable constraints on the halo parameters (see e.g. K17). The reason for this is that the Hi rotation curves do not typically extend out enough to probe the region where the NFW density profile steepens, thus yielding only a weak inference on $c$. The $\Lambda$CDM-motivated prior on the $c-M_{\rm halo}$ relation proves to be enough to well constrain all the model parameters. Furthermore, we notice that the DM14 $c-M_{\rm halo}$ relation does not distinguish between haloes hosting late-type or early-type galaxies, so we use it under the assumption that it provides a reasonable description of the correlation for the haloes where late-type galaxies form. We summarise in Table 1 our choice of priors.

3 Results

We modelled the rotation curves and we have measured the posterior distributions of $\Upsilon_{\rm disc},M_{\rm halo}$ and $c$ for all the 158 SPARC galaxies with inclination on the sky larger than 30 degrees – since for nearly face-on systems the rotation curves are very uncertain. For each parameter, we define the “best-value” to be the median of the posterior distribution and its uncertainty as the 16th - 84th percentiles. In Appendix A we provide in tabular form all the measurements and uncertainties, together with the value of the likelihood associated to the best model (Table LABEL:tab:results). We also present the full rotation curve decomposition for one case as an example (NGC 3992, Figure 3), while we make available the plots of all the other galaxies online at http://astro.u-strasbg.fr/~posti/PFM19_fiducial_fits/.

Unsurprisingly, we find that our model typically does not give very stringent constraints on the stellar mass-to-light ratio, with only 84 (68) galaxies having an uncertainty on $\Upsilon_{\star}$ smaller than 50$\%$ (30$\%$). In these cases, which are mostly for $M_{\star}>10^{10}M_{\odot}$ where the signal-to-noise is large, the $V_{\rm obs}$ and $V_{\star}$ profiles are similar enough to yield good constraints on $\Upsilon_{\rm disc}$. We find that these galaxies are not all maximal discs, as their $\Upsilon_{\rm disc}$ is homogeneously distributed in the range allowed by our prior. We find the highest mass spirals ($M_{\star}\gtrsim 10^{11}M_{\odot}$) to have much better fits with a slightly larger mass-to-light ratio ($\Upsilon_{\rm disc}\sim 0.7$) than the mean of our prior ($\Upsilon_{\rm disc}=0.6$), consistently with previous works who found that high mass discs are close to maximal (e.g. Lapi et al. 2018; Starkman et al. 2018; Li et al. 2018). Smaller systems, instead, have typically a poorer inference on the mass-to-light ratio, with about $\sim 50$ cases in which the posterior on $\Upsilon_{\rm disc}$ is quite flat. Even in these extreme cases it is nevertheless useful to let the MCMC explore the full range of possible mass-to-light ratios ($0.01\leq\Upsilon_{\rm disc}\leq 1.2$) as opposed to just fixing a value for $\Upsilon_{\rm disc}$, because this provides a more realistic estimate of the uncertainty on the other parameters of the dark matter halo. In other words, when the inference on $\Upsilon_{\rm disc}$ is poor, it may be thought as a nuisance parameter over which the posterior distributions of the other two more interesting halo parameters is marginalised.

For 137 (out of 158) galaxies we obtain a unimodal posterior distribution for the halo mass, thus we can associate a measurement and an uncertainty to $M_{\rm halo}$; the remaining 21 galaxies have, instead, either a multi-modal or a flat posterior on the halo mass and thus we discard them. These 21 are mostly low-mass systems ($M_{\star}\lesssim 2\times 10^{9}M_{\odot}$) and their removal does not alter in any way the high-mass end of the population, which is the main focus of our work. For some of the remaining 137 galaxies, we find that the NFW halo model provides a poor fit to the observed rotation curve, as their best-fit $\chi^{2}$ is large. This is not surprising, since it is well known that especially low-mass discs tend to have slowly rising rotation curves, which makes them more compatible with having centrally cored haloes (e.g. de Blok et al. 2001, K17). Indeed, by re-fitting all rotation curves with a cored halo model from Burkert (1995), we have found 27, mostly low-mass ($M_{\star}\lesssim 10^{10}M_{\odot}$), systems for which such cored profile is preferred to the NFW at a 3-$\sigma$ confidence level. For consistency we have decided to remove these 27 systems from our sample, but in Appendix A we demonstrate that their stellar and halo masses, derived by extrapolating the Burkert profile to the virial radius, are perfectly consistent with the picture that we present below.

In Figure 1 we plot the $M_{\star}-M_{\rm halo}$ relation for the 110 SPARC galaxies in our final sample. Points are the median of the posterior distributions of $M_{\rm halo}$ and $M_{\star}$; the 16th-84th percentiles of the $M_{\rm halo}$ distribution define the errorbar, while the uncertainty on the stellar mass is calculated as in Lelli et al. (2016b, their Eq. 5) where the uncertainty on $\Upsilon_{\rm disc}$ is given by the 16th-84th percentiles of its posterior. For comparison we also plot the $M_{\star}-M_{\rm halo}$ relation estimated by Moster et al. (2013) using abundance matching. In general we find that the abundance matching model is in good agreement with our measurements for $M_{\star}\lesssim 5\times 10^{10}M_{\odot}$, albeit our points have a large scatter especially at the lowest masses. The agreement is instead much poorer at high stellar masses, where the Moster et al. (2013) model predicts significantly larger halo masses with respect to our estimates. Our measurements indicate that there is no sign of a break in the stellar-to-halo mass relation of spirals and that it is consistent with being an increasing function of mass with roughly the same slope at all masses.

The tension at the high-mass end between our measurements and the abundance matching model is much clearer if one plots the stellar fraction, i.e. $f_{\star}\equiv M_{\star}/f_{\rm b}M_{\rm halo}$, also sometimes called star-formation efficiency, as a function of the stellar mass: we show this in Figure 2. This plot highlights the two main findings of our work, the first being that $f_{\star}$ appears to increase monotonically with galaxy stellar mass with no indication of a peak in the range $10\leq\log\,M_{\star}/M_{\odot}\leq 11$, where most abundance matching models find a maximum star-formation efficiency. For instance, a galaxy with $M_{\star}=2\times 10^{11}M_{\odot}$ has $f_{\star}\simeq 0.04$ in the Moster et al. (2013) model, while we find $f_{\star}\simeq 0.5$. By computing the difference between the measured $f_{\star}$ and that expected in the Moster et al. (2013) model, normalised by the sum in quadrature of the measured uncertainty on $f_{\star}$ and of the intrinsic scatter of the model, we find the measurement for the high-mass systems to be inconsistent at $2-3\sigma$ with the model (see the colours of the points in Fig. 2). Such a discrepancy is very robust and holds for all the tests we have run (we show in Appendix A, Figure 4, the $f_{\star}-M_{\star}$ diagram in all these cases):

• •

we have fitted the rotation curves assuming a cored (Burkert 1995) instead of a cuspy (NFW) profile. In general, this yields better fits for many low-mass systems, slightly larger stellar masses and smaller halo virial masses for all galaxies;

• •

we have used the fits recently obtained by Ghari et al. (2018), who used Einasto (1965) halo profiles (and distances and mass-to-light ratios from Li et al. 2018). In general, we typically find slightly smaller halo virial masses, but broadly consistent with our estimates with NFW profiles;

• •

we have fixed the mass-to-light ratio of the bulge and disc components to reasonable values suggested by stellar population synthesis models ($\Upsilon_{\rm disc}=0.5,\Upsilon_{\rm bulge}=0.7$, see e.g. Meidt et al. 2014; Schombert & McGaugh 2014);

• •

we tried allowing both $\Upsilon_{\rm disc}$ and $\Upsilon_{\rm bulge}$ to vary in our fits, with the additional constraint ($\Upsilon_{\rm disc}\leq\Upsilon_{\rm bulge}$). This has an effect only on the 28 (out of 110) galaxies in our final sample that have non-negligible bulges. We find the resulting uncertainties on $\Upsilon_{\rm disc}$ to be typically significantly larger in this case, but never dramatic.

In all these cases the final result is that the $f_{\star}-M_{\star}$ diagram is not significantly different from the one presented in Fig. 2.

Additionally, as shown by Katz et al. (2014, see their Fig. 20 and 23), the effect of adiabatic contraction of the dark matter halos due to the formation of stellar discs has a negligible impact on $f_{\star}$ for galaxies in the interested mass regime.

The other main finding highlighted by Fig. 2 is even more surprising: we find that all spirals with $M_{\star}\gtrsim 10^{11}M_{\odot}$ have stellar fraction very close to unity, in the range $f_{\star}\approx 0.3-1$, with a handful of them being consistent with $f_{\star}=1$ within the uncertainties. This implies that these galaxies were extremely efficient at turning gas into stars and that the amount of mass collapsed in stars is a considerable portion of the total amount of baryons expected to be associated with their haloes. In fact, if we include also the contribution of atomic and molecular hydrogen (the latter estimated through the $M_{\rm HI}-M_{H_{2}}$ relation given by Catinella et al. 2018), spirals with $M_{\star}\geq 10^{11}M_{\odot}$ are found to be consistent with a cold baryon budget of $f_{\rm baryons}=f_{\star}+1.4f_{\rm HI}+f_{{\rm H}_{2}}\approx 1$ within the uncertainties (where the factor 1.4 accounts for helium, e.g. Lelli et al. 2016a), with a mean value of $\sim 0.6$ and uncertainties of $[-0.3,+0.5]$. Moreover, considering that galaxies are known to be surrounded by massive, hot coronae, which are detected both in X-ray and with the Sunyaev-Zeldovich effect and account for about $0.1-0.3f_{\rm b}M_{\rm halo}$ (typically estimated statistically by stacking over many galaxies with a given stellar mass, e.g. Planck Collaboration et al. 2013; Bregman et al. 2018, and references therein), the total (cold+hot) baryon budget is easily compatible with unity at the high-mass end, with very little room for other baryonic components. In other words, we have found that the most massive, regularly rotating spirals in the nearby Universe have virtually no missing baryons.

4 Discussion

Our analysis allowed us to have a robust and unbiased estimate of the halo virial mass for a sample of 108 spiral galaxies in the nearby Universe using their high-quality Hi rotation curves. While we find good agreement with previous determinations of the stellar-to-halo mass relation for galaxies roughly up to the mass of Milky Way ($M_{\star}=5\times 10^{10}M_{\odot}$), we also find systematically smaller halo masses (factor $\sim 10$), corresponding to higher stellar-to-halo mass ratios, for the most massive spirals with respect to expectations from most up-to-date abundance matching models (e.g. Wechsler & Tinker 2018).

A possible explanation for this discrepancy is that, while the high-mass end ($M_{\star}\gtrsim 10^{11}M_{\odot}$) of the galaxy stellar mass function is vastly dominated by passive early-type galaxies which occupy massive ($M_{\rm halo}\gtrsim 5\times 10^{12}M_{\odot}$) dark matter halos, there still exists a population of star-forming spirals that inhabit halos of lower masses. The presence of this second population – which is not well represented by current abundance matching models – implies the existence of different evolutionary pathways for building galaxies of a given stellar mass. This suggests that e.g. a massive system that has evolved in isolation may have had the chance to sustain star-formation unimpeded for its entire life, potentially converting most of its available baryons into stars. While this is certainly not the case for high-mass early-types galaxies, which tend to live in high-density environments, it may well be the pathway taken by the high-mass population of spirals studied in this work. In fact, also McGaugh et al. (2010) by simply analysing the Tully-Fisher relation of a similar sample of spirals concluded that $f_{\star}$ does not turn over at the highest masses.

A discrepancy between the expected halo mass for a typical passive (red) $10^{11}M_{\odot}$ galaxy and an active (blue) one of the same $M_{\star}$, was also noted by other authors using various probes, such as satellite kinematics (e.g. Conroy et al. 2007; More et al. 2011; Wojtak & Mamon 2013), galaxy-galaxy weak lensing (e.g. Mandelbaum et al. 2006, 2016; Reyes et al. 2012), abundance matching (e.g. Rodríguez-Puebla et al. 2015) or combinations (e.g. Dutton et al. 2010). The works most similar to ours are those of K17 and Lapi et al. (2018). We use the same galaxy sample as in K17 (SPARC) and we perform a similar analysis as them, but with the crucial difference that we do not impose a prior on halo mass that follows an $M_{\star}-M_{\rm halo}$ relation from abundance matching, which slightly biases towards higher halo masses some of the high-mass galaxies333 Taking into account this difference in the priors used, our results are very well compatible with theirs: our conclusions sit in the middle between their case with uniform priors (their Fig. 3) and that in which they impose a prior following the Moster et al. (2013) $M_{\star}-M_{\rm halo}$ relation (their Fig. 5). Lapi et al. (2018), on the other hand, have a much larger sample of spirals than ours, but they rely on “stacked” rotation curves for their mass decompositions – i.e. they stack individual curves of galaxies in bins of absolute magnitude – whereas we focus on individual, well studied systems. Finally, we notice that, amongst the detailed studies of individual systems, i) Corbelli et al. (2010) measured the dynamical mass of M31 by decomposing its Hi rotation cureve, to find a surprisingly high $f_{\star}\simeq 0.6$, and ii) Martinsson et al. (2013) decomposed the Hi rotation curves of a small sample of 30 spirals from the DiskMass Survey, to find the highest star-formation efficiencies $f_{\star}\gtrsim 0.3$ for their three most massive galaxies ($\log M_{\star}/M_{\odot}\gtrsim 10.9$). While our results align with these previous works, to our knowledge we are the first to focus specifically on the $f_{\star}-M_{\star}$ relation and to highlight the fact that i) the highest-mass spirals are the most efficient galaxies at turning gas into stars, ii) that $f_{\star}$ increases monotonically with stellar mass for regularly rotating nearby discs and that iii) virtually all high-mass discs have $\gtrsim 30\%$ of the total baryons within their haloes in stars.

Our analysis establishes that the most efficient galaxies at forming stars are not $L^{\ast}$ galaxies, as previously thought (e.g. Wechsler & Tinker 2018), but much more massive systems, some of the most massive spiral galaxies in the nearby Universe ($M_{\star}\gtrsim 10^{11}M_{\odot}$). Not only the galactic star-formation efficiency peaks at much larger masses than we knew before, but we also showed that several massive discs have efficiencies $f_{\star}$ of the order unity. This result alone is of key importance since it demonstrates that there is no universal physical mechanism that sets the maximum star-formation efficiency to $20-30\%$.

Furthermore, the fact that some massive galaxies with high $f_{\star}$ exist has fundamental implications for star-formation quenching. Since these galaxies live in haloes with $M_{\rm halo}\sim 2-5\times 10^{12}M_{\odot}$, if mass is the main driver of quenching and if a critical mass for quenching exists (e.g. as expected in scenarios where virial shock heating of the circumgalactic medium is the key process, see Birnboim & Dekel 2003; Dekel & Birnboim 2008), then it follows that this critical mass can not be smaller than $\sim 5\times 10^{12}M_{\odot}$, which is almost an order of magnitude higher than previously thought (e.g. Dekel & Birnboim 2006). Interestingly, such a high threshold is instead expected in scenarios where the accretion of cool gas is hampered (“starvation”), e.g. by the high virial temperature of the circumgalactic gas in a galactic fountain cycle (e.g. Armillotta et al. 2016) or by the complex interplay of radiative cooling and feedback in the smooth gas accretion from cold filaments (e.g. van de Voort et al. 2011).

Even if we have measured high $f_{\star}$ for some massive spirals, still the vast majority of galaxies living in $M_{\rm halo}>10^{12}M_{\odot}$ haloes has $f_{\star}\ll 1$, which means that they managed to efficiently quench their star-formation. Our results imply that since mass can not be the major player in quenching galaxies, at least for $M_{\rm halo}\lesssim 5\times 10^{12}M_{\odot}$, and some other mechanism must play a fundamental role in the transition from actively to passively star-forming. One of the main suspects is clearly environment, since gas removal happens more frequently and also gas accretion is more difficult in high-density environments (e.g. Peng et al. 2010; van de Voort et al. 2017). Another is the powerful feedback from the active galactic nucleus (AGN), which can episodically suppress any gas condensation throughout the galaxy (e.g. Croton et al. 2006; Fabian 2012). Finally, another key process is the interaction with other galaxies, with passive galaxies being hosted in haloes with an active merger history, which can result in bursty star-formation histories and subsequent suppressive stellar/AGN feedback (e.g. Cox et al. 2006b; Gabor et al. 2010). This latter scenario also naturally accounts for the morphological transformation of disc galaxies, living in haloes with quiet merger histories, to spheroids, which are the dominant galaxy population at the high-mass end, where also mergers are more frequent (e.g. Cox et al. 2006a). This scenario is, in principle, testable both with current cosmological simulations and with a new abundance matching model which depends also on secondary halo parameters, such as merger history or formation time, and it is able to predict not only stellar masses but also other galaxy properties, such as morphology or colour.

Acknowledgements

We thank E. Corbelli, B. Famaey, A. Lapi, F. Lelli, A. Robertson, J. Sellwood and F. van den Bosch for useful discussions and A. Ghari for making their Einasto fits available to us. LP acknowledges financial support from a VICI grant from the Netherlands Organisation for Scientific Research (NWO) and from the Centre National d’Etudes Spatiales (CNES).

Appendix A Supplementary material