Assessing the Fornax globular cluster timing problem in different models of dark matter

We investigate what the orbits of globular clusters (GCs) in the Fornax dwarf spheroidal (dSph) galaxy can teach us about dark matter (DM). This problem was recently studied for ultralight dark matter (ULDM). We consider two additional models: (i) fermionic degenerate dark matter (DDM), where Pauli blocking should be taken into account in the dynamical friction computation; and (ii) self-interacting dark matter (SIDM). We give a simple and direct Fokker-Planck derivation of dynamical friction, new in the case of DDM and reproducing previous results in the literature for ULDM and cold DM. ULDM, DDM and SIDM were considered in the past as leading to cores in dSphs, a feature that acts to suppress dynamical friction and prolong GC orbits. For DDM we derive a version of the cosmological free streaming limit that is independent of the DM production mechanism, finding that DDM cannot produce an appreciable core in Fornax without violating Ly-$\alpha$ limits. If the Ly-$\alpha$ limit is discounted for some reason, then stellar kinematics data does allow a DDM core which could prolong GC orbits. For SIDM we find that significant prolongation of GC orbits could be obtained for values of the self-interaction cross section considered in previous works. In addition to reassessing the inspiral time using updated observational data, we give a new perspective on the so-called GC timing problem, demonstrating that for a cuspy cold DM profile dynamical friction predicts a $z=0$ radial distribution for the innermost GCs that is independent of initial conditions. The observed orbits of Fornax GCs are consistent with this expectation with a mild apparent fine-tuning at the level of $\sim 25\%$.

Figure 1

Orbital radius vs time, calculated for the Fornax GC4 assuming a slightly eccentric orbit. $t=0$ represents today. The orbit calculation assumes the CDM Navarro-Frenk-White (NFW) [22] profile of Ref. [17].

Figure 2

Radius of an infalling GC with mass $m_{\star }=3\times {10}^5{M}_{\odot }$, based on the simulations of Ref. [17]. In dotted blue (thick dotted red) we plot the simulation result (Fig. 3 in Ref. [17]) for the NFW (ISO) halo. In solid lines, we plot our semianalytic integration. Horizontal dot-dashed lines show the radii $r_f$ where $M_{\mathrm{halo}}(r_f)=m_{\star }$, in which the semianalytic treatment should break down.

Figure 3

The ratio of the circular velocity $V_{\mathrm{circ}}$ to the radial velocity dispersion ${\sigma }_r$, reproduced from Ref. [17] for NFW (thick blue) and isothermal (ISO, red) halos.

Figure 4

Orbital decay time calculated using $\rho$ and $\sigma$ from Fig. 1 of Ref. [17] and assuming a test object on a circular orbit. NFW denotes a cuspy profile, and ISO denotes an isothermal cored profile. We use here the GC mass $m_{\star }=3\times {10}^5{M}_{\odot }$.

Figure 5

LOSVD compared with data [10] for DDM with $g=2$. The best-fit parameters of the profile are ${\mu }_0/T=3$ and ${\rho }_0=3.9\times {10}^7{M}_{\odot }/{\mathrm{kpc}}^3$.

Figure 6

The circular velocity $V_{\mathrm{circ}}$ and the Fermi velocity $v_F$ for the DDM halo with $m=135\mathrm{eV}$, ${\rho }_0=3.9\times {10}^7{M}_{\odot }/{\mathrm{kpc}}^3$, ${\mu }_0/T=3$ and $g=2$. The vertical lines show the estimated orbital radii ($r=r_{\perp }\times 2/\sqrt{3}$) of the three GCs closest to the dynamical center of Fornax, cf. Table 1.

Figure 7

A comparison of the velocity dispersions (in natural units) of WDM and DDM for various particle masses. Including CMB data, Ref. [74] put a bound of $m>2.96\mathrm{keV}$ on WDM (95% C.L.), for which we plot the velocity dispersion as the green solid line.

Figure 8

Contours of the inspiral time of GC3, defined in Appendix pp7, for the DDM (top) and cuspy CDM models (bottom). The “naive” estimates written on top are those given in Table 1 based on an evaluation of the instantaneous DF time at $r_{\mathrm{true}}/r_{\perp }=2/\sqrt{3}$ and $V_{\mathrm{true}}/V_{\mathrm{circ}}(r_{\mathrm{true}})=1$. The different line types correspond to different discrete choices in our scan of the initial conditions in phase space, explained in more detail in Appendix pp7.

Figure 9

LOSVD data of Fornax dSph modeled by different SIDM profiles. The central density is ${\rho }_c=2.6\times {10}^7{M}_{\odot }/{\mathrm{kpc}}^3$, the velocity dispersion is ${\sigma }_0=17\mathrm{km}/\sec$ and $r_1/r_{\mathrm{ic}}=6$.

Figure 10

DF time for an SIDM halo, analogously to Fig. 8.

Figure 11

Comparison of models. Top left: LOSVD data and fits. The M19 NFW and M19 ISO models refer to the halos of Ref. [17], for which we only fit the velocity anisotropy. The DDM and SIDM models are based on Sec. 4 and Sec. 5. The ${\chi }^2/\mathrm{d}.\mathrm{o}.\mathrm{f}.$ is $\approx 1.9$ for NFW, DDM and SIDM, and $\approx 2.5$ for ISO. The velocity anisotropy is taken to be constant in each fit. We find ${\beta }_{\mathrm{NFW}}=-0.4$, ${\beta }_{\mathrm{ISO}}=0.2$, ${\beta }_{\mathrm{DDM}}=-0.1$, ${\beta }_{\mathrm{SIDM}}=0.1$ and ${\beta }_{\text{coreNFW}}=-0.1$. Top right: Density profiles. In addition to DM, we also plot an estimate of the stellar density, assuming a Plummer profile with scale $r_p=851\mathrm{pc}$ [21] and mass $4\times {10}^7{M}_{\odot }$ [18]. Bottom left: Circular velocity. (Note how cored models require some tuning to explain the large radial velocity of GC4, $|\mathrm{\Delta }{v}_r|=8.26\pm 0.64\mathrm{km}/\mathrm{s}$, at its small projected radius $r_{\perp }\approx 0.154\mathrm{pc}$). Bottom right: Instantaneous DF time, evaluated for $m_{\star }=3\times {10}^5{M}_{\odot }$.

Figure 12

Example of a calculated CDF of projected radii using $\tau (r)$ from Fig. 11. Left: NFW halo. Right: Cored ISO halo. The solid blue line shows the CDF after projection effects are taken into account. The dashed green line shows the result before projection. For the NFW case, the small-$r$ prediction of Eq. (32) is shown by the red dashed line. Observed Fornax GCs are also shown. The initial radial GC PDFs used to make the plot are explained in the text.

Figure 13

The density profile found by solving Eq. (b12) and inserting into Eq. (b13) (blue line), compared with an analytical ansatz (orange) that demonstrates the asymptotic behavior of the solution. Evidently, for $x\lesssim 1$ the density profile is constant, whereas the density asymptotes to $1/x^2$ for large $x$.

Figure 14

The density profile found by solving Eq. (b12) and inserting into Eq. (b13) for different values of ${\mu }_0/T$, keeping ${\rho }_0$ constant. The dashed blue line (RSU) is based on the treatment of Ref. [45]. The dotted red line (LE) is based on solving the Lane-Emden equation, which is the ${\mu }_0/T\to \infty$ limit of the equations, as described in the text.

Figure 15

The circular velocities $\sqrt{GM(r)/r}$ for the density profiles that appear in Fig. 14.

Figure 16

Initial radial PDF $f_0$ (left) and resulting current radial CDF $F_{\mathrm{\Delta }t}$ (right) for a cuspy halo with $\tau (r)\propto r^{1.85}$ and $r_{\mathrm{cr}}=1.12$ (units on the $x$ axis are arbitrary). Two different examples for $f_0$ are shown, leading to nearly identical $F_{\mathrm{\Delta }t}$. The two versions of $f_0$ are normalized to yield $F_{\mathrm{\Delta }t}(\infty )=6$. In both panels, $r_{\mathrm{cr}}$ is marked with a vertical black line. On the left, the $r\ll r_{\mathrm{cr}}$ approximation $F_{\mathrm{\Delta }t}\approx A(\tau (r)/\mathrm{\Delta }t)$ is shown by the red dotted curve.

Figure 17

The effect of projection. Solid blue: unprojected radial CDF computed at the projected radius, $F_{\mathrm{\Delta }t}(r_{\perp })$. Dashed black: CDF of projected radii $F_{\mathrm{\Delta }t}^{\perp }(r_{\perp })$.

Figure 18

The coordinate system that we adopt to analyze a given GC. The galactic dynamical center is at the origin. The observer is located at a very large $X$. The GC is located somewhere on the line $Z=0$, $Y=r_{\perp }$. The true radius is therefore $r_{\mathrm{true}}=r_{\perp }/\sin \alpha$. We assume $\mathrm{\Delta }{v}_r$, the component of velocity in the $X$ direction, can be measured. We assume that the rest of the components cannot be measured for now. The dotted line is the quasistable orbit of the GC.