Study of gravitational fields and globular cluster systems of early-type galaxies

TL;DR

This study compares gravitational models LCDM and MOND using globular cluster velocities in early-type galaxies, finding LCDM fits are generally successful but with some discrepancies, while MOND models often predict better velocity profiles but face challenges in cluster centers.

Contribution

It provides a comprehensive analysis of ETGs with globular cluster data to evaluate the performance of LCDM and MOND theories, highlighting strengths and limitations of each.

Findings

LCDM fits are successful but show low halo concentrations and high stellar masses.

MOND models fit many galaxies well but struggle in galaxy cluster centers.

Statistical criteria favor LCDM, but MOND predicts velocity profiles more accurately.

Abstract

(abridged) Context. Gravitational fields at the outskirts of early-type galaxies (ETGs) are difficult to constrain observationally. It thus remains poorly explored how well the LCDM and MOND hypotheses agree with ETGs. Aims. This led us to gather a large sample of ETGs and examine homogeneously which dark matter halos they occupy, whether the halos follow the theoretically predicted stellar-to-halo mass relation (SHMR) and the halo mass-concentration relation (HMCR), whether ETGs obey MOND and the radial acceleration relation (RAR) observed for late-type galaxies (LTGs), and finally whether LCDM or MOND perform better in ETGs. Methods. We employed Jeans analysis of radial velocities of globular clusters (GCs). We analysed nearly all ETGs having more than about 100 archival GC radial velocity measurements. The GC systems of our 17 ETGs extend mostly over ten effective radii. A LCDM simulation of GC formation helped us to interpret the results. Results. Successful LCDM fits are found for all galaxies, but compared to the theoretical HMCR and SHMR, the best-fit halos usually have concentrations that are too low and stellar masses that are too high for their masses. This might be because of tidal stripping of the halos or because ETGs and LTGs occupy different halos. Most galaxies can be fitted by the MOND models successfully as well, but for some of the galaxies, especially those in centers of galaxy clusters, the observed GC velocity dispersions are too high. This might be a manifestation of the additional dark matter that MOND requires in galaxy clusters. Additionally, we find many signs that the GC systems were perturbed by galaxy interactions. Formal statistical criteria prefer the best-fit LCDM models over the MOND models, but this might be due to the higher flexibility of the LCDM models. The MOND approach can predict the GC velocity dispersion profiles better.

Figures

Tables

Table 1. Table 1: Galaxy sample

Name	$d$	1′	$\log L$	$R_e$	$n$	$B-V$	Type	Env	Prof	Iso	${\lambda }_{R_e}$	$ϵ$	Rot
	[Mpc]	[kpc]	[$L_{☉}$]	[kpc]		[mag]
N 821	24${}_{-2}{}^{}{}_{}{}^{+2}$	7.0	10.5	4.714	4.714	0.87	E619	F19	$\mid$1	D7,9	0.278	0.358	f
N 1023	11.4${}_{-0.8}{}^{}{}_{}{}^{+0.9}$	3.3	10.5	2.728	4.228	0.91	S08	G28	\1	D5	0.398	0.638	f
N 1399	20${}_{-1}{}^{}{}_{}{}^{+2}$	5.8	10.7	3.515	5.615	0.93	E1pec24	C24	$\cap$1	D10	0.0811	0.0911	s
N 1400	26${}_{-4}{}^{}{}_{}{}^{+4}$	7.7	10.4	3.414	4.014	0.89	S0/E019	G19	$\cap$1	02	0.278	0.118	f
N 1407	29${}_{-3}{}^{}{}_{}{}^{+4}$	8.4	11.0	9.414	8.314	0.95	E019	G19	$\cap$2	02	0.088	0.058	f
N 2768	22${}_{-2}{}^{}{}_{}{}^{+3}$	6.5	10.7	8.916	3.316	0.91	E6/S01/219	G19	?3	D5	0.258	0.578	f
N 3115	9.7${}_{-0.4}{}^{}{}_{}{}^{+0.4}$	2.8	10.2	4.817	4.417	0.90	S019	F19	\1	D6	0.588	0.498	f
N 3377	11.2${}_{-0.5}{}^{}{}_{}{}^{+0.5}$	3.3	9.9	2.916	5.016	0.82	E619	G19	\1	D5	0.528	0.338	f
N 4278	16${}_{-1}{}^{}{}_{}{}^{+2}$	4.7	10.2	2.516	4.816	0.90	E1-219	G19	$\cap$1	B5	0.188	0.098	f
N 4365	20${}_{-2}{}^{}{}_{}{}^{+2}$	5.9	10.7	8.518	5.218	0.93	E319	G19	$\cap$1	B7	0.098	0.248	s
N 4472	16.3${}_{-0.7}{}^{}{}_{}{}^{+0.8}$	4.7	10.9	3.915	3.015	0.93	E224	C24	$\cap$1	B7,9	0.0779	0.1729	s
N 4486	16${}_{-1}{}^{}{}_{}{}^{+1}$	4.7	10.8	5.818	2.918	0.92	E019	C19	$\cap$1	B5	0.028	0.168	s
N 4494	17.1${}_{-0.8}{}^{}{}_{}{}^{+0.9}$	5.0	10.4	3.716	3.416	0.85	E1-E219	G19	\1	D7	0.218	0.148	f
N 4526	17${}_{-1}{}^{}{}_{}{}^{+2}$	4.9	10.4	2.728	3.628	0.89	S08	C28	?	B5	0.458	0.768	f
N 4649	17${}_{-1}{}^{}{}_{}{}^{+1}$	4.9	10.8	5.115	3.615	0.93	E2/S08	C24	$\cap$1	B7,9	0.1279	0.1569	f
N 5128	4.2${}_{-0.3}{}^{}{}_{}{}^{+0.3}$	1.2	10.5	6.214	4.014	0.87	S0pec/Epec25	G26	?12	?13	0.1513	0.0527	f
N 5846	25${}_{-2}{}^{}{}_{}{}^{+2}$	7.2	10.7	8.116	3.916	0.94	E019	G19	?4	B5	0.038	0.088	s
R17x	19${}_{-2}{}^{}{}_{}{}^{+2}$	5.5	10.2	1.5	2.6	0.90	-	-		-	-	-	-
R17y	19${}_{-2}{}^{}{}_{}{}^{+2}$	5.5	10.2	1.7	2.0	0.90	-	-		-	-	-	-
R17z	19${}_{-2}{}^{}{}_{}{}^{+2}$	5.5	10.2	2.0	2.0	0.90	-	-		-	-	-	-

Table 2. Table 6: MOND fits

Name	$\beta$	$M/L$	$d$	$\log M_{\ast }$	${\chi }^2$	conf.	AICc	LBF
		[$M_{☉}/L_{☉}$]	[kpc]	[$M_{☉}$]		[%]
N 821	iso	$6\pm 1$	$25\pm 2$	11.3	75	52	-56	11.4
	neg	$6\pm 1$	$25\pm 2$	11.3	74	53	-56	11.4
	lit	$6\pm 1$	$25\pm 2$	11.3	75	52	-56	11.4
N 1023	iso	$3.2\pm 0.7$	$11.0\pm 0.8$	11.0	90	13	-110	22.9
	neg	$3.2\pm 0.7$	$10.9\pm 0.8$	10.9	90	12	-110	22.8
	lit	$3.3\pm 0.7$	$11.0\pm 0.8$	11.0	91	13	-110	23.0
N 1399	iso	$42\pm 5$	$25\pm 2$	12.5	922	0.15	590	-226
	neg	$37\pm 4$	$25\pm 2$	12.4	909	0.39	560	-195
	lit	$53\pm 6$	$26\pm 2$	12.6	945	0.021	660	-294
N 1400	iso	$5\pm 1$	$27\pm 3$	11.1	70	80	-79	16.6
	neg	$5\pm 1$	$27\pm 3$	11.1	68	91	-80	16.8
	lit	$6\pm 1$	$28\pm 3$	11.2	72	64	-77	16.1
N 1407	iso	$11\pm 2$	$35\pm 3$	12.2	407	23	42	-10.3
	neg	$10\pm 2$	$34\pm 3$	12.1	402	30	30	-7.54
	lit	$12\pm 2$	$36\pm 4$	12.2	413	15	61	-14.5
N 2768	iso	$6\pm 1$	$22\pm 2$	11.4	106	97	-91	19.0
	neg	$6\pm 1$	$22\pm 2$	11.4	105	95	-90	18.8
	lit	$6\pm 1$	$23\pm 2$	11.4	108	89	-91	19.1
N 3115	iso	$8\pm 1$	$9.8\pm 0.4$	11.1	164	38	-110	22.5
	neg	$8\pm 1$	$9.8\pm 0.4$	11.1	162	46	-110	22.6
	lit	$8\pm 1$	$9.8\pm 0.4$	11.1	167	32	-110	22.3
N 3377	iso	$4.7\pm 0.8$	$11.3\pm 0.5$	10.6	127	68	-220	46.3
	neg	$4.7\pm 0.7$	$11.3\pm 0.5$	10.6	126	74	-220	46.1
	lit	$4.7\pm 0.8$	$11.3\pm 0.5$	10.6	128	64	-220	46.3
N 4278	iso	$12\pm 2$	$18\pm 1$	11.4	301	17	-140	29.3
	neg	$12\pm 2$	$18\pm 1$	11.4	297	23	-140	30.4
	lit	$13\pm 2$	$18\pm 2$	11.5	308	9.7	-130	27.1
N 4365	iso	$18\pm 2$	$23\pm 2$	12.0	294	3.0	44	-10.6
	neg	$17\pm 2$	$23\pm 2$	12.0	288	5.5	36	-8.85
	lit	$19\pm 3$	$23\pm 2$	12.1	302	1.2	59	-14.0
N 4472	iso	$20\pm 2$	$16.9\pm 0.8$	12.3	313	3.5	220	-50.9
	neg	$18\pm 2$	$16.8\pm 0.8$	12.2	305	7.3	200	-45.7
	lit	$25\pm 3$	$17.0\pm 0.7$	12.4	326	0.96	240	-61.1
N 4486	iso	$31\pm 3$	$20\pm 1$	12.5	738	0.50	610	-149
	neg	$29\pm 3$	$19\pm 1$	12.4	732	0.83	590	-142
	lit	$32\pm 4$	$20\pm 1$	12.5	748	0.23	630	-157
N 4494	iso	$2\pm 1$	$16.6\pm 0.8$	10.7	83	13	-180	38.0
	neg	$2\pm 1$	$16.6\pm 0.8$	10.7	82	11	-180	37.9
	lit	$2.0\pm 0.4$	$16.6\pm 0.8$	10.7	84	15	-180	38.0
N 4526	iso	$5\pm 1$	$17\pm 1$	11.1	105	96	-57	11.7
	neg	$5\pm 1$	$17\pm 1$	11.1	104	92	-57	11.7
	lit	$5\pm 1$	$17\pm 1$	11.1	106	100	-57	11.6
N 4649	iso	$9\pm 1$	$18\pm 1$	11.8	467	14	50	-12.0
	neg	$9\pm 1$	$17\pm 1$	11.8	454	27	44	-10.6
	lit	$9\pm 1$	$18\pm 1$	11.8	487	3.2	61	-14.4
N 5128	iso	$6.9\pm 0.8$	$4.5\pm 0.3$	11.4	571	20	-360	76.6
	neg	$6.3\pm 0.7$	$4.5\pm 0.3$	11.4	563	30	-380	80.7
	lit	$7.6\pm 0.9$	$4.6\pm 0.3$	11.5	581	12	-330	71.1
N 5846	iso	$17\pm 3$	$29\pm 2$	12.0	246	4.8	39	-9.28
	neg	$16\pm 3$	$28\pm 2$	12.0	243	6.9	31	-7.63
	lit	$18\pm 3$	$29\pm 2$	12.0	252	2.8	50	-11.7
R17x	iso	$3.5\pm 0.5$	$18\pm 2$	10.7	185	49	-270	57.2
	neg	$3.4\pm 0.5$	$17\pm 2$	10.7	184	45	-270	57.2
	lit	$3.6\pm 0.5$	$18\pm 2$	10.7	187	54	-270	57.3
R17y	iso	$3.6\pm 0.5$	$18\pm 2$	10.7	185	50	-260	54.7
	neg	$3.6\pm 0.5$	$18\pm 2$	10.7	183	46	-250	53.4
	lit	$3.5\pm 0.5$	$18\pm 2$	10.7	186	54	-260	55.4
R17z	iso	$3.3\pm 0.5$	$18\pm 2$	10.7	185	48	-280	59.6
	neg	$3.3\pm 0.5$	$17\pm 2$	10.7	184	44	-280	59.0
	lit	$3.3\pm 0.5$	$18\pm 2$	10.7	187	53	-280	59.9

Equations

$$\text{@vec}{a}_{\mathrm{N}}=\text{@vec}a\mu \left(\frac{a}{a_0}\right),$$

$$\rho (r)$$

$${\sigma }_{\mathrm{los},\mathrm{obs}}=\sqrt{\frac{1}{N-1}\sum _{i=1}^N(v_i-v_{\mathrm{sys}}{)}^2};\Delta {\sigma }_{\mathrm{obs}}=\frac{{\sigma }_{\mathrm{los},\mathrm{obs}}}{\sqrt{2(N-1)}},$$

$$s_3=\frac{N}{(N-1)(N-2)}\sum _{i=1}^N{\left(\frac{v_i-v_{\mathrm{sys}}}{{\sigma }_{\mathrm{los},\mathrm{obs}}}\right)}^3\pm \sqrt{\frac{6}{N}}$$

$$\begin{array}{rl}{s}_4=& \frac{N(N+1)}{(N-1)(N-2)(N-3)}\sum _{i=1}^N{\left(\frac{v_i-v_{\mathrm{sys}}}{{\sigma }_{\mathrm{los},\mathrm{obs}}}\right)}^4\\ & -\frac{3(N-1{)}^2}{(N-2)(N-3)}\pm \sqrt{\frac{24}{N}}.\end{array}$$

$$\frac{1}{\rho }\frac{\mathrm{d}(\rho {\sigma }_r^2)}{\mathrm{d}r}+2\frac{\beta (r)}{r}{\sigma }_r^2\text{leavevmode}=\text{leavevmode}a(r)+\frac{v_{\mathrm{rot}}^2}{r}.$$

$${\sigma }_{\mathrm{los}}^2(R)=\frac{{\int }_R^{\infty }{\sigma }_r^2(r)\left[1-(R/r{)}^2\beta \right]\rho (r){\left(r^2-R^2\right)}^{-1/2}r\mathrm{d}r}{{\int }_R^{\infty }\rho (r){\left(r^2-R^2\right)}^{-1/2}r\mathrm{d}r}.$$

$${\sigma }_{\mathrm{los},\mathrm{mod}}(R)=\sqrt{{\sigma }_{\mathrm{los}}^2+v_{\mathrm{rot}}^2},$$

$${\beta }_{\mathrm{lit}}(r)=\frac{0.5r}{r+1.4R_{\mathrm{e}}},$$

$$\mu (x)=x/(1+x).$$

$$a_{\mathrm{N}}=-G{M}_{\ast }(<r)/r^2,$$

$$M_{\mathrm{DM}}(<r)=4\pi {\rho }_s{r}_s^3\left\{\ln \left[\left(r_s+r\right)/r_s\right]-r/\left(r+r_s\right)\right\},$$

$$a=-G\left[M_{\ast }(<r)+M_{\mathrm{DM}}(<r)\right]/r^2.$$

$$\mathcal{F}(\text{@vec}p)=\prod _{i=1}^{N_{\mathrm{data}}}\frac{1}{\sqrt{2\pi {\sigma }_{\mathrm{mod},i}^2}}{e}^{\frac{-v_i^2}{2{\sigma }_{\mathrm{mod},i}^2}}\prod _{j=1}^{N_{\mathrm{par}}}\frac{1}{\sqrt{2\pi \Delta p_j^2}}{e}^{\frac{-(p_{j,\exp }-p_j{)}^2}{2\Delta p_j^2}}.$$

$$p_d=\log \frac{d}{d_0}.$$

$$p_{M/L}=\log \frac{M/L}{M/L_0}$$

$$p_{SH}=\frac{1}{\Delta \log M_h(M_{\ast })}\log \left[\frac{M_h}{M_h(M_{\ast })}\right].$$

$$p_c=\log \left(\frac{c}{c(M_h)}\right).$$

$$\ln \mathcal{F}(\text{@vec}\hat{p})-\ln \mathcal{F}(\text{@vec}p)\le 1/2.$$

$$X^2=\sum _{i=1}^{N_{\mathrm{data}}}\frac{v_i^2}{{\sigma }_{\mathrm{mod},i}^2}+\sum _{j=1}^{N_{\mathrm{par}}}\frac{(p_{j,\exp }-p_j{)}^2}{\Delta p_j^2}$$

$$\mathrm{AICc}=2X^2+2k+\frac{2k^2+2k}{n-k-1},$$

$$\mathrm{LBF}=\log \int \mathcal{L}\left[\mathrm{data}|M(\text{@vec}p)\right]f(\text{@vec}p)\mathrm{d}\text{@vec}p.$$

$$\Delta \log a_{\mathrm{N}}=\sqrt{(\Delta \log M/L{)}^2+(2\Delta \log d{)}^2}.$$

$${\lambda }_{R_{\mathrm{e}}}=\frac{{\sum }_i{F}_i{R}_i|V_i|}{{\sum }_i{F}_i{R}_i\sqrt{V_i^2+{\sigma }_i^2}},$$

$$\Delta v=\frac{2G{M}_{\mathrm{p}}x}{v_{\mathrm{p}}{b}^2}.$$

Full text

*Aims. * The dearth of studies on this topic motivated us to gather a large sample of ETGs and examine homogeneously which dark matter halos they occupy, whether the halos follow the theoretically predicted stellar-to-halo mass relation (SHMR) and the halo mass-concentration relation (HMCR), whether ETGs obey MOND and the radial acceleration relation (RAR) observed for late-type galaxies (LTGs), and finally whether $\Lambda$CDM or MOND perform better in ETGs.

*Methods. *We employed Jeans analysis of radial velocities of globular clusters (GCs). We analysed nearly all ETGs having more than about 100 archival GC radial velocity measurements available. The GC systems of our 17 ETGs extend mostly over ten effective radii. A $\Lambda$CDM simulation of GC formation helped us to interpret the results.

*Results. *Successful $\Lambda$CDM fits are found for all galaxies, but compared to the theoretical HMCR and SHMR, the best-fit halos usually have concentrations that are too low and stellar masses that are too high for their masses. This might be because of tidal stripping of the halos or because ETGs and LTGs occupy different halos. Most galaxies can be fitted by the MOND models successfully as well, but for some of the galaxies, especially those in centers of galaxy clusters, the observed GCs velocity dispersions are too high. This might be a manifestation of the additional dark matter that MOND requires in galaxy clusters. Additionally, we find many signs that the GC systems were perturbed by galaxy interactions. Formal statistical criteria prefer the best-fit $\Lambda$CDM models over the MOND models, but this might be due to the higher flexibility of the $\Lambda$CDM models. The MOND approach can predict the GC velocity dispersion profiles better.

1 Introduction

The missing mass problem has not been solved decisively yet. The two most discussed solutions are the standard cosmological $\Lambda$CDM model (e.g., Mo et al., 2010) and the MOND paradigm of modified dynamics (Milgrom, 1983; Famaey & McGaugh, 2012; Milgrom, 2015). In this paper we aim to test their predictions in early-type galaxies (ETGs).

Assuming the $\Lambda$CDM paradigm, galaxies are surrounded by dark matter halos whose density can be described well by the Navarro-Frenk-White (NFW) profile (Navarro et al., 1996) according to dark-matter-only cosmological simulations. These halos are expected to meet the stellar-to-halo mass relation (SHMR) between the stellar mass of the galaxy and the mass of its dark halo (e.g., Behroozi et al., 2013). This relation is supposed to exist because the transport of baryons inside or outside the halo depends on the mass of the halo; the more massive the halo is, the more it accretes satellites and intergalactic gas. On the other side, the processes expelling the baryons from the halos such as active galactic nuclei outflows, supernova explosions, and stellar winds are less effective if the baryons reside in a deeper potential well. This relation can be recovered, for example, by the abundance matching technique based on comparing the halo mass function deduced from cosmological simulations to the observed galaxy stellar mass function while assuming that the most massive galaxies lie within the most massive halos; i.e., the recovered SHMR is a combination of results from observations and simulations. We must remember that the recovered SHMR is not necessarily the SHMR that $\Lambda$CDM would predict in a perfect simulation, which would include, for example, all baryonic processes correctly. In the current hydrodynamical cosmological simulations the parameters regulating the baryonic processes have to be tuned so that the galaxy stellar mass function matches the observations (Genel et al., 2014; Crain et al., 2015; Pillepich et al., 2018). The simulations that are not explicitly tuned to reproduce the stellar mass function do not do so properly (Khandai et al., 2015; Kaviraj et al., 2017).

Another correlation that the dark matter halos are expected to follow is that between the halo mass and its concentration referred to as the halo mass-concentration relation (HMCR). The concentration, $c$, of a NFW halo is the virial radius of the halo divided by its scale radius. The HMCR is a result deduced from $\Lambda$CDM cosmological simulations.