Signatures of Velocity-dependent Dark Matter Self-interactions in Milky Way-mass Halos

Self-interacting dark matter (SIDM) has long been an attractive alternative to purely cold, collisionless dark matter (CDM) owing to several potential "small-scale problems" attributed to CDM (see Bullock & Boylan-Kolchin 2017 for a review of these problems, and see Tulin & Yu 2018 for a review of their potential resolutions in SIDM). Historically, SIDM was motivated by the core–cusp problem, which concerns the apparent discrepancy between the steep, cuspy NFW profiles ubiquitous among DM halos in CDM simulations and the flatter, cored profiles inferred from the dynamics of various tracers in dwarf galaxies (Firmani et al. 2000; Spergel & Steinhardt 2000; Davé et al. 2001; Colín et al. 2002). The core–cusp problem is sensitive to the impact of baryonic feedback, and it remains an active area of study (e.g., Elbert et al. 2015). In particular, it has recently been cast in terms of the diversity of inner DM density profiles for field (Oman et al. 2015; Kamada et al. 2017; Kaplinghat et al. 2019a; Ren et al. 2019; Zavala et al. 2019; Santos-Santos et al. 2020) or satellite (Valli & Yu 2018; Kahlhoefer et al. 2019) galaxies at fixed halo properties. For Milky Way (MW) satellite galaxies, self-interactions in the presence of the tidal field of the Galactic disk may accelerate gravothermal core collapse, and this process has been proposed as a unique signature of SIDM (Kaplinghat et al. 2019b; Nishikawa et al. 2020; Sameie et al. 2020).

Self-interactions can also affect the abundance of DM substructure in a statistical fashion. In particular, many authors have studied the subhalo populations of MW-mass hosts in the context of SIDM (e.g., Davé et al. 2001; Colín et al. 2002; D'Onghia & Burkert 2003; Vogelsberger et al. 2012, 2019; Rocha et al. 2013; Zavala et al. 2013; Dooley et al. 2016; Robles et al. 2019), and a few cosmological SIDM-plus-hydrodynamic simulations of isolated dwarfs (Fry et al. 2015; Harvey et al. 2018; Fitts et al. 2019) and MW-mass systems (Robles et al. 2017) have been performed.

The consensus of these studies has been that SIDM models that do not drastically reduce the number counts of surviving subhalos have little effect on subhalo abundances in MW-mass systems (see, e.g., the discussion in Tulin & Yu 2018). However, this is a model-dependent statement that warrants scrutiny in light of increasingly precise constraints on the abundance of subhalos in MW-mass systems enabled by recent measurements of the luminosity function of satellite galaxies around the MW (e.g., Drlica-Wagner et al. 2020) and around MW analogs (e.g., Geha et al. 2017; Kondapally et al. 2018; Bennet et al. 2019). In addition, theoretical predictions for gaps and perturbations in nearby stellar streams (e.g., Banik et al. 2019; Bonaca et al. 2019), direct and indirect detection experiments (e.g., Ibarra et al. 2019; Ishiyama & Ando 2020), and flux ratios in strongly lensed systems (e.g., Gilman et al. 2020; Hsueh et al. 2020) are all sensitive to the abundance and internal properties of DM substructure in distinct ways.

Many of the aforementioned SIDM studies only consider isotropic, velocity-independent self-interactions, at least on the velocity scales relevant for the MW host halo and its subhalos. Nonetheless, there are theoretical and observational arguments in favor of a velocity-dependent SIDM cross section, which is a generic consequence of interactions in a nonminimal dark sector (see, e.g., Feng et al. 2009; Loeb & Weiner 2011; Tulin et al. 2013; Boddy et al. 2014; Kaplinghat et al. 2016; Tulin & Yu 2018). While velocity-dependent SIDM models have been explored on galaxy cluster scales (e.g., Robertson et al. 2017; Banerjee et al. 2020), a thorough study of the corresponding predictions on Galactic scales is now crucial. This is particularly relevant because the increasingly well-measured abundance and internal properties of MW satellites can be affected by self-interactions at both the host halo velocity scale (which depend on satellites' typical velocities relative to the host) and subhalo velocity scales (which depend on the internal velocity dispersion of individual subhalos).

Herein we investigate the effects of an SIDM model with a generic, velocity-dependent self-interaction cross section on the DM host halo and subhalos of MW-mass systems. We focus on the physical mechanisms that shape the abundance and properties of surviving and disrupted subhalos, setting the stage for future analyses to constrain the SIDM cross section at the velocity scale of MW-mass systems. This paper is organized as follows: We describe our SIDM model in Section 2 and our simulations in Section 3. We then present our results, focusing on the properties of our host halos in Section 4 and their subhalo populations in Section 5. We discuss challenges and caveats in Section 6, we compare to previous studies in Section 7, we consider prospects for SIDM constraints in Section 8, and we conclude in Section 9.

We consider an SIDM model in which a DM particle χ interacts under the exchange of a light mediator, which can be either a scalar particle ϕ or a vector particle ϕ^μ. These scenarios are respectively described by the interaction Lagrangians (Tulin & Yu 2018)

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{int}}=\left\{\begin{array}{l}{g}_{\chi }\bar{\chi }\chi \phi ,\mathrm{scalar}\,\mathrm{mediator}\\ {g}_{\chi }\bar{\chi }{\gamma }^{\mu }\chi {\phi }_{\mu },\mathrm{vector}\,\mathrm{mediator},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where g_χ is a coupling constant and γ^μ are Dirac matrices.

Self-interactions governed by Equation (1) can be described by a Yukawa potential

$$\begin{eqnarray}&&V(r)=\pm \displaystyle \frac{{\alpha }_{\chi }}{r}{e}^{-{m}_{\phi }/r},\end{eqnarray} \tag{ 2 }$$

where r is the separation between DM particles, ${\alpha }_{\chi }\equiv {g}_{\chi }^{2}/4\pi$ is the analog of the fine-structure constant in the dark sector, and m_ϕ is the mediator mass.

We focus on t-channel scattering, which leads to an effective differential scattering cross section (Ibe & Yu 2010; Kummer et al. 2018)

$$\begin{eqnarray}&&\displaystyle \frac{d\sigma }{d{\rm{\Omega }}}=\displaystyle \frac{{\sigma }_{0}}{2{\left[1+\tfrac{{v}^{2}}{{w}^{2}}{\sin }^{2}\left(\tfrac{\theta }{2}\right)\right]}^{2}},\end{eqnarray} \tag{ 3 }$$

where v is the relative velocity between interacting DM particles with mass m_χ, $w\equiv {m}_{\phi }/{m}_{\chi }$ is a characteristic velocity scale, ${\sigma }_{0}\equiv 4\pi {\alpha }_{\chi }^{2}{m}_{\chi }^{2}/{m}_{\phi }^{4}$ is the amplitude of the cross section, and θ is the scattering angle in the center-of-mass frame. For $w\ll v$ (in natural units), typically corresponding to an MeV-scale mediator for a GeV-scale DM particle mass, Equation (3) reduces to Rutherford-like scattering; for a heavy mediator, it reduces to velocity-independent isotropic scattering.

The corresponding momentum transfer cross section for identical particles is given by (Kahlhoefer et al. 2014)

$$\begin{eqnarray}&&{\sigma }_{T}=\int \displaystyle \frac{d\sigma }{d{\rm{\Omega }}}\left(1-\left|\cos \theta \right|\right)d{\rm{\Omega }}.\end{eqnarray} \tag{ 4 }$$

Note that σ_T is a function of v, w, and σ₀. Finally, the total cross section

$$\begin{eqnarray}&&\sigma =\int \displaystyle \frac{d\sigma }{d{\rm{\Omega }}}d{\rm{\Omega }}\end{eqnarray} \tag{ 5 }$$

determines the probability of DM self-interactions in our numerical implementation (see Section 3.2). For the case of isotropic, velocity-independent self-interactions, the total cross section is related to σ_T and σ₀ via σ = 2σ_T = 2πσ₀.

At velocity scales v > w, the momentum transfer cross section falls off as v⁻⁴; meanwhile, for $v\ll w$, it flattens toward its asymptotic value. Thus, for values of w that are small relative to the typical velocities in a virialized system, interactions at low relative velocities are more effective at transferring momentum than interactions at high relative velocities. In the context of MW-mass systems, this implies that interactions among host halo particles and between host halo and subhalo particles are less effective than interactions among subhalo particles if w is small relative to the characteristic velocity scale of the host ($\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$). On the other hand, as w increases toward the velocity scale of the host halo, subhalo–host halo and host halo–host halo interactions become more significant.

The characteristic velocity scale w is an easily interpretable quantity that captures much of the key physics that shapes MW-mass systems in our two-parameter SIDM model. Thus, we parameterize and refer to our simulations in terms of w, and we choose σ₀/m_χ for each model so that the momentum transfer cross section at the velocity scales of interest yields enough self-interactions to produce observable effects, but not so many that the models are likely ruled out already. In particular, we study the four following SIDM model variants:

• 1.  
  w₁₀: a model with w = 10 km s⁻¹, ${\sigma }_{0}/{m}_{\chi }=8/\pi \ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, for which self-interactions within subhalos are significant, but self-interactions at the host halo velocity scale are negligible;
• 2.  
  w₁₀₀: a model with w = 100 km s⁻¹, ${\sigma }_{0}/{m}_{\chi }=2/\pi \ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, for which self-interactions within low-mass subhalos are less significant than in w₁₀, but self-interactions at the host halo velocity scale are more significant;
• 3.  
  w₂₀₀: a model with w = 200 km s⁻¹, ${\sigma }_{0}/{m}_{\chi }=2/\pi \ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, for which self-interactions within low-mass subhalos are identical to w₁₀₀, but self-interactions at the host halo velocity scale are more significant;
• 4.  
  w₅₀₀: a model with w = 500 km s⁻¹, ${\sigma }_{0}/{m}_{\chi }=1/\pi \ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, for which self-interactions within low-mass subhalos are the least significant among our model variants, while interactions at the host halo velocity scale are similar to w₂₀₀ (though slightly more effective at high velocities). Scattering in this model is isotropic and velocity independent on the scales relevant for MW-mass systems, meaning that self-interactions with large scattering angles at high relative velocities are potentially significant.

Figure 1 shows the momentum transfer cross section corresponding to each model variant, and their main properties are summarized in Table 1.

Zoom In Zoom Out Reset image size

Table 1.  Summary of SIDM Model Variants and Simulation Results

Simulation	$w\ (\mathrm{km}\,{{\rm{s}}}^{-1})$	${\sigma }_{0}/{m}_{\chi }\ ({\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1})$	Cored Host Halo?	Cored Subhalos?	Ram Pressure Stripping?	${N}_{\mathrm{SIDM}}/{N}_{\mathrm{CDM}}$
CDM	⋯	⋯	X	X	X	1.0
w10	10	$8/\pi$	X	✓	X	0.82
w100	100	$2/\pi$	✓	✓	✓	0.64
w200	200	$2/\pi$	✓	✓	✓	0.65
w500	500	$1/\pi$	✓	✓	✓	0.44

Note. SIDM model variants considered in this work and the main qualitative results of our zoom-in simulation of an MW-mass system for each case. The first column lists the DM model, the second and third columns list the characteristic velocity scale and amplitude of the self-interaction cross section for our SIDM model variants, the fourth (fifth) column lists whether the host halo (subhalos) are cored by self-interactions, the sixth column lists whether subhalos are affected by ram pressure stripping due to self-interactions with the host, and the seventh column lists the fraction of subhalos among our matched subhalo populations that survive to z = 0 relative to the number of surviving subhalos with ${V}_{\mathrm{peak}}\gt 20\ \mathrm{km}\,{{\rm{s}}}^{-1}$ in our CDM simulation.

Download table as:  ASCIITypeset image

3.1. General Description

3.2. SIDM Implementation

3.3. Subhalo Definitions and Resolution Cuts

3.4. Subhalo Matching Procedure

We now present our results, focusing on the properties of the MW host halo and its subhalo population in each SIDM model variant described above. Table 1 lists the main qualitative results of each simulation.

The virial mass of the MW host halo is virtually identical in all of our simulations. However, the DM profile in the inner regions of the host varies significantly as a function of w. This is illustrated in Figure 2, which shows the density and velocity dispersion profiles of the host in CDM and in each of our SIDM model variants. As expected, SIDM models with larger self-interaction cross sections at the velocity scale set by the host's velocity dispersion ($\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$) exhibit cored density profiles, and the DM distribution in the inner regions of these hosts is roughly isothermal, consistent with previous findings (e.g., Davé et al. 2001; Colín et al. 2002; Vogelsberger et al. 2012, 2019; Rocha et al. 2013; Robles et al. 2019). In contrast, the host halo in w₁₀ is very similar to that in CDM because self-interactions at the host's velocity scale are negligible in this case.⁷

Zoom In Zoom Out Reset image size

As demonstrated by several authors (e.g., Kaplinghat et al. 2014; Sameie et al. 2018; Robles et al. 2019), these results are expected to change in the presence of a central baryonic component such as the Galactic disk. In particular, because DM dynamically responds to the total (i.e., DM-plus-baryonic) gravitational potential, we expect our host halos to exhibit much smaller cores or even cusps in the presence of baryons. Importantly, this implies that the findings discussed below on the disruption of SIDM subhalos are strictly lower limits, since tidal disruption and ram pressure stripping would be more severe for the denser, hotter host halo expected in the presence of baryons.

To visualize the present-day DM structure in our simulations, Figure 3 shows the DM particle density in the phase space of radial distance versus radial velocity with respect to the center of the host. We observe that the DM profiles of both the host halo and its subhalos are less concentrated in our SIDM simulations. This effect is more significant for larger values of w; for example, visual inspection of Figure 3 suggests that many prominent substructures that survive in CDM are completely disrupted in w₅₀₀. We also find that the phase-space density at small radii and low radial velocities with respect to the host center increases with w, implying that particles on radial orbits (i.e., those with high radial velocities in our CDM simulation) are scattered onto tangential orbits owing to self-interactions. Finally, we note that specific substructures in w₁₀ appear more diffuse than their counterparts in CDM, which arises as a result of the large momentum transfer cross section at low relative velocities for w₁₀ (see Figure 1).

Zoom In Zoom Out Reset image size

The differences in the host halo's density and velocity dispersion profiles discussed above impact the post-infall evolution of subhalos. In particular, it is expected that the present-day (z = 0) abundance and properties of subhalos are affected by ram-pressure-like stripping due to self-interactions between subhalo and host halo particles, as well as by tidal effects in the host halo's gravitational field. We investigate these effects in Section 5.2.

Thus, the host halo is cored and thermalized in the absence of baryons if self-interactions are significant at the velocity scale set by the host's velocity dispersion. These effects will be weakened in the presence of baryons.

Next, we examine the pre- and post-infall evolution of all subhalos that fall into the host in each of our simulations. We then compare the corresponding surviving subhalo populations at z = 0.

5.1. Pre-infall Subhalo Evolution

In this subsection, we explore the formation and evolution of subhalos before they fall into the host.

5.1.1. Subhalo Assembly

First, we investigate the formation and initial properties of subhalos in each of our simulations. To do so, we plot the distribution of V_peak for each SIDM model variant in the left panel of Figure 4, and we compare these to CDM. The V_peak distributions are calculated using the matched subhalo populations defined in Section 3.4, and they include both subhalos that survive to z = 0 and those that fall into the host halo before disrupting. We find that the V_peak distributions are unchanged relative to CDM. In addition, the distributions of the time at which V_peak is achieved are nearly identical among the simulations, although there is a small amount of scatter on a subhalo-by-subhalo basis.

Zoom In Zoom Out Reset image size

Thus, the formation times and initial properties of subhalos in all of our SIDM simulations—defined in terms of ${V}_{\mathrm{peak}}$ and the time at which V_peak occurs—are statistically identical to the corresponding quantities in our CDM simulation.

5.1.2. Effects of Early Self-interactions

Next, we assess the effects of self-interactions on subhalos before infall into the host. In particular, the right panel of Figure 4 shows the cumulative distribution of maximum circular velocity evaluated at the time of accretion onto the host, V_acc, divided by V_peak.⁸ This quantity captures the impact of self-interactions between the early time at which V_peak is usually achieved (z_peak ∼ 3 for surviving subhalos, on average) and the time of accretion (${z}_{\mathrm{acc}}\sim 1$ for surviving subhalos, on average). These characteristic peak and infall times are very similar among our CDM and SIDM simulations.

We observe a subtle but systematic trend in V_acc/V_peak as a function of the characteristic self-interaction velocity scale w. In particular, the low-V_acc/V_peak tail of this distribution, which is present in CDM and results from pre-infall tidal stripping (e.g., Behroozi et al. 2014; Wetzel et al. 2015), is more prominent in SIDM. This difference becomes increasingly pronounced for SIDM model variants with larger values of w, indicating that self-interactions at the typical relative velocity scale affect subhalos during pre-infall tidal stripping.

This finding is consistent with the fact that regions outside the conventional virial radius of the host halo can be quite dense (note that the self-interaction rate depends on the physical density, rather than the comoving density). In particular, typical splashback boundaries for MW-mass halos extend to ∼1.5 times the conventional virial radius (e.g., Adhikari et al. 2014; Diemer & Kravtsov 2014; More et al. 2015). Accordingly, we find that there is a noticeably smaller difference in the low-V_acc/V_peak tail relative to CDM if V_acc is evaluated when subhalos cross within twice the virial radius of the host.

In Section 5.3.3, we show that the density profiles of subhalos in our SIDM model variants that exhibit appreciable changes to their V_acc/V_peak distribution relative to CDM (i.e., all model variants other than w₁₀) already exhibit cored density profiles at infall. We provide a physical interpretation of the scaling with w in Section 5.2.1.

Thus, subhalos in our SIDM simulations are affected by self-interactions before accretion onto the host, leading to lower values of V_acc/V_peak relative to subhalos in CDM.

5.2. Post-infall Subhalo Evolution

We now explore the evolution of subhalos in our SIDM model variants after infall into the host. In particular, we describe the physical effects that influence subhalos after infall, and we investigate subhalo disruption in our simulations.

5.2.1. Physical Effects

After infall, two main effects shape the subhalo populations in our SIDM simulations:

• 1.  
  Tidal stripping. The gravitational field of the host halo tidally strips material from subhalos. Note that this effect is velocity independent; in particular, assuming spherical symmetry and considering tidal interactions with the host halo only, the mass-loss rate due to tidal stripping can be written as (van den Bosch et al. 2018)

  $$\begin{eqnarray}&&{\left(\displaystyle \frac{{\dot{M}}_{\mathrm{sub}}}{{M}_{\mathrm{sub}}}\right)}_{\mathrm{tidal}}=-\displaystyle \frac{1}{\alpha }\displaystyle \frac{1}{{\tau }_{\mathrm{orb}}}\displaystyle \frac{{M}_{\mathrm{sub}}(\gt {r}_{t})}{{M}_{\mathrm{sub}}},\end{eqnarray} \tag{ 6 }$$

  where M(>r_t) is the mass of a subhalo contained outside of its tidal radius, τ_orb is the orbital timescale, and α is an order-unity constant. Note that the tidal radius (i.e., the distance from the center of the subhalo at which the host's tidal force is balanced by the subhalo's own gravity) depends on both the host halo and subhalo density profiles and the distance of the subhalo from the center of the host. Although tidal stripping occurs in CDM-only simulations, differences in the strength of this effect may arise in SIDM owing to changes in the density profiles of both the host halo and its subhalos caused by self-interactions. In particular, we expect that the cored density profile of the host halo in SIDM reduces the efficacy of tidal stripping relative to CDM for a fixed subhalo profile. However, as discussed in, e.g., Kahlhoefer et al. (2019), the cores induced in SIDM subhalos make them more susceptible to disruption in a fixed tidal field.
• 2.  
  Ram pressure stripping. Self-interactions with host halo particles drive material out of subhalos. Taken to the extreme, this process can completely dissociate subhalos in a phenomenon known as subhalo evaporation (see, e.g., Vogelsberger et al. 2019). Unlike tidal stripping, which strips mass from the outskirts of subhalos, ram pressure interactions remove material throughout their extent. In particular, subhalos experience an effective pressure owing to interactions with the host; the mass-loss rate due to these ram-pressure-like interactions can be written as (Kummer et al. 2018)

  $$\begin{eqnarray}&&\begin{array}{l}{\left(\displaystyle \frac{{\dot{M}}_{\mathrm{sub}}}{{M}_{\mathrm{sub}}}\right)}_{\mathrm{ram}-\mathrm{pressure}}\\ =-{\chi }_{e}({v}_{\mathrm{sub}},{v}_{\mathrm{esc},\mathrm{sub}}){\rho }_{\mathrm{host}}({\boldsymbol{r}}){v}_{\mathrm{sub}}({\boldsymbol{r}})\displaystyle \frac{\sigma ({v}_{\mathrm{sub}})}{{m}_{\chi }},\end{array}\end{eqnarray} \tag{ 7 }$$

  where χ_e is an order-unity factor, ${v}_{\mathrm{sub}}({\boldsymbol{r}})$ is the velocity of the subhalo relative to the host evaluated along its orbit, v_esc,sub is the escape velocity from the subhalo (which depends on the subhalo's density profile), ${\rho }_{\mathrm{host}}({\boldsymbol{r}})$ is the density of the host halo evaluated along the subhalo's orbit, and $\sigma ({v}_{\mathrm{sub}})/{m}_{\chi }$ is the total SIDM cross section evaluated at v_sub.The mass-loss rate due to ram pressure stripping depends on the SIDM cross section at the typical relative velocity scale between the host halo and its subhalos, which is set by the gravitational potential of the host. Because SIDM models with larger values of w have larger self-interaction cross sections at this velocity scale, the strength of ram pressure stripping increases with w.

By integrating Equation (7) over the course of a typical orbit, we find that subhalos with close pericentric passages to the center of the host (${d}_{\mathrm{peri}}\lesssim 70\ \mathrm{kpc}$) can lose $\sim 10 \%$ of their infall mass owing to ram pressure stripping alone, for an isotropic cross section of $\sigma /{m}_{\chi }=2\,{\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. Combined with tidal effects, and particularly the fact that subhalos with cored density profiles are more susceptible to tidal disruption, our calculations suggest that self-interactions at the relevant velocity scales are sufficient to severely strip subhalos in SIDM model variants with large values of w. We have confirmed that this behavior—i.e., increased mass loss relative to CDM followed by disruption due to tidal forces at pericenter—is indeed the dominant disruption mechanism in our simulations by inspecting the histories of matched subhalos that survive in CDM but disrupt in our SIDM model variants.

Thus, the evolution of subhalos after infall in our SIDM simulations is driven by a combination of tidal forces and ram pressure stripping. Ram pressure stripping is more effective for models with larger values of w, which have larger self-interaction cross sections at the host halo velocity scale. This mass-loss mechanism makes subhalos more susceptible to tidal disruption, particularly during pericentric passages.

5.2.2. Subhalo Disruption

We now analyze our simulation results to quantitatively assess the impact of the effects described in the previous section. In our CDM simulation, roughly half of all subhalos with ${V}_{\mathrm{peak}}\gt 20\ \mathrm{km}\,{{\rm{s}}}^{-1}$ that fell into the host disrupt by z = 0. We find that even larger fractions of subhalos disrupt in our SIDM simulations, as indicated by the surviving subhalo fractions in Table 1. Based on the arguments in Section 5.2.1, we expect that extra subhalo disruption relative to CDM should be correlated with the SIDM cross section at the typical relative velocity scale set by the host's gravitational potential, such that subhalos are disrupted more efficiently in SIDM model variants with larger values of w. As demonstrated below, our results are consistent with this hypothesis.

To build intuition about subhalo disruption in our SIDM model, we first investigate the distribution of pericentric distance, d_peri, for disrupted subhalos in each simulation. We define d_peri as the distance of closest approach to the center of the host halo during each subhalo's first orbit around the host after infall.⁹ The top left and top right panels of Figure 5 show the distributions of d_peri for surviving and disrupted subhalos in each of our simulations. We observe that the amount of subhalo disruption at large pericentric distances increases strongly as a function of w, and we have verified that most of this disruption occurs during early pericentric passages. Subhalos in our SIDM simulations therefore disrupt at distances at which subhalos in our CDM simulation almost never disrupt, indicating that disruption is driven by tidal effects during early pericentric passages that are enhanced by ram-pressure-like interactions with the host.

Zoom In Zoom Out Reset image size

To explore the data further, the bottom left and bottom right panels of Figure 5 show the distributions of maximum relative velocity with respect to the host evaluated along the orbit of each surviving and disrupted subhalo. The maximum relative velocity distributions for disrupted subhalos in our SIDM model variants peak strongly relative to CDM above $\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$, and the strength of this effect is correlated with w. This demonstrates that extra subhalo disruption relative to CDM is mainly driven by self-interactions at relative velocities above $\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$, and that subhalos are more easily disrupted in SIDM model variants with larger momentum transfer cross sections at these velocity scales.¹⁰

This picture is consistent with the fact that w₅₀₀—which has the largest momentum transfer cross section at large relative velocity scales—exhibits the most subhalo disruption among our SIDM model variants (see Section 5.3.1). Thus, our findings support the hypothesis that a combination of ram pressure and tidal stripping increases the amount of subhalo disruption relative to CDM.

Thus, a significant fraction of the subhalos that survive in our CDM simulation are disrupted in our SIDM simulations owing to extra mass loss caused by ram pressure stripping from self-interactions with the host at large relative velocities, which makes these subhalos more susceptible to tidal disruption. This effect is more severe for models with larger values of w.

5.3. Surviving Subhalo Populations

We now examine the population statistics and properties of surviving subhalos in our SIDM simulations.

5.3.1. Population Statistics

In the left panel of Figure 6, we plot the cumulative number of surviving subhalos as a function of V_peak in each of our simulations. We find that the abundance of surviving subhalos monotonically decreases as a function of w. In particular, subhalo abundances are rescaled in an approximately V_peak-independent fashion, and the number of surviving subhalos in SIDM divided by the number of surviving subhalos in CDM ranges from ∼0.8 in w₁₀ to ∼0.4 in w₅₀₀. We have chosen to present these results in terms of V_peak because this quantity is expected to correlate more directly with satellite luminosity than, e.g., present-day virial mass or maximum circular velocity (e.g., Reddick et al. 2013; Lehmann et al. 2017; Nadler et al. 2019b, 2020).

Zoom In Zoom Out Reset image size

We have verified that similar trends hold for the present-day M_vir and V_max functions; however, because these distributions mix the pre-infall properties and post-infall evolution of surviving subhalos, their shapes differ in detail from the corresponding peak functions. For example, we observe an enhancement in subhalo abundance at large values of V_max for all of our SIDM model variants relative to CDM. We speculate that this is a consequence of less effective tidal stripping for surviving subhalos in our SIDM simulations, which occupy tangential orbits around a cored host halo. Previous authors have also found hints of this trend (e.g., Rocha et al. 2013; also see Section 7).

In the right panel of Figure 6, we plot the radial distribution of surviving subhalos in each simulation. Similar to the V_peak functions, we find that the radial distributions are approximately rescaled in the outer regions. However, disruption becomes increasingly severe near the center of the host (i.e., $r/{R}_{\mathrm{vir}}\lesssim 0.5$), causing ${N}_{\mathrm{SIDM}}/{N}_{\mathrm{CDM}}$ to decrease sharply in that region, except for the "spikes" at small radii observed for w₁₀ and w₂₀₀. These correspond to either one or two additional subhalos near the center of these hosts; we investigate these subhalos further in Appendix C.

Thus, subhalo disruption due to self-interactions approximately rescales the number of surviving subhalos relative to CDM as a function of V_peak. Subhalo disruption is particularly effective in the inner regions of the host halo, and it is more severe for models with larger values of w.

5.3.2. Orbital Anisotropy Profile

An immediate consequence of the ram-pressure-plus-tidal-stripping disruption mechanism described above is that subhalos on radial orbits are preferentially disrupted. We therefore expect surviving subhalos in SIDM to preferentially occupy tangential orbits relative to surviving subhalos in CDM. We test this by measuring the orbital anisotropy profile

$$\begin{eqnarray}&&\beta (r)\equiv 1-\displaystyle \frac{{\sigma }_{t}{\left(r\right)}^{2}}{2{\sigma }_{r}{\left(r\right)}^{2}},\end{eqnarray} \tag{ 8 }$$

where σ_r and σ_t denote the radial and tangential velocity dispersion of subhalos, respectively. Note that β < 0 corresponds to a tangentially biased orbital distribution, β = 0 corresponds to an isotropic orbital distribution, and the maximum allowed value of β = 1 corresponds to a purely radial orbital distribution.

We measure β(r) by calculating the radial and tangential velocity distributions of subhalos in the frame of the host halo, binning subhalos in distance from the center of the host. The result is shown in Figure 7. We find that the orbital anisotropy profile rises toward the outskirts of the host halo in our SIDM simulations, while it is roughly flat in CDM. As expected based on the results in Section 5.2.2, surviving subhalos preferentially occupy tangential orbits in our SIDM simulations, and this effect is more significant for model variants with larger values of w, for which subhalo disruption via ram pressure plus tidal stripping is more severe.¹¹ We observe a similar trend for anisotropy profiles computed using the DM particles belonging to the host halo in each simulation, although the magnitude of the effect is less severe in this case. An interesting consequence of this effect is that, at fixed V_peak, surviving subhalos in our SIDM simulations tend to be more massive on average than their counterparts in CDM, since they are less susceptible to tidal stripping on tangential orbits; however, this difference largely vanishes if subhalos are matched based on their orbital properties.

Zoom In Zoom Out Reset image size

These results are interesting in light of recent high-precision measurements of the orbital properties of MW satellites enabled by Gaia (e.g., Fritz et al. 2018; Gaia Collaboration et al. 2018; Simon 2018). In particular, recent studies of the orbital anisotropy profile find that the typically flat $\beta (r)$ profiles found in DM–only simulations are in tension with the observed velocity anisotropy profile of MW satellites (Riley et al. 2019). Baryonic effects, and particularly the tidal influence of the Galactic disk, affect the β(r) profile in a similar manner to ram pressure stripping in our SIDM simulations by disrupting subhalos on radial orbits near the center of the host. However, as noted by Riley et al. (2019), the changes to β(r) due to baryonic physics only resolve the discrepancy with the observed profile if the Galactic disk is sufficiently massive, which in turn depends on the prescription for baryonic feedback and the mass accretion history of the MW system. Adding baryonic subhalo disruption to an orbital distribution that is already tangentially biased owing to self-interactions may result in even more tangential bias than observed in the inner regions of the MW, potentially yielding an upper limit on the SIDM cross section at ∼200 km s⁻¹.

Thus, surviving subhalos in SIDM occupy tangentially biased orbits relative to subhalos in CDM, leaving a systematic imprint on the orbital anisotropy profile.

5.3.3. Subhalo Profiles

Because the momentum transfer cross section in our SIDM model scales as ${v}^{-4}$ above the characteristic velocity scale w, self-interactions at low relative velocities are strictly more efficient at transferring momentum than interactions at high relative velocities. Thus, we expect the density profiles of low-mass subhalos in our SIDM simulations to be impacted by self-interactions. In contrast to subhalo disruption, it is not clear a priori which SIDM model variants will most significantly affect subhalo profiles, since both self-interactions within subhalos and self-interactions with the host halo potentially impact subhalo density profiles.

To investigate the impact of self-interactions on subhalo density profiles, we select representative subhalos in our CDM simulation, and we find the corresponding subhalos in our SIDM simulations by matching on V_peak, accretion time, and pericentric distance. We then select the particles in the initial phase-space region associated with these matched subhalos and track the evolution of their density profiles until z = 0. Figure 8 shows the results of this procedure for a particular set of subhalos matched to a CDM subhalo with ${V}_{\mathrm{peak}}\approx 40\ \mathrm{km}\,{{\rm{s}}}^{-1}$ and accretion time z_acc ≈ 1 that passes close to the center of the host (d_peri ≈ 70 kpc). We show the density profiles of these matched subhalos at the time of accretion onto the host and at z = 0; we also show the density profile of a subhalo in w₅₀₀ with similar V_peak and z_acc that does not pass close to the center of its host (d_peri ≈ 160 kpc).

Zoom In Zoom Out Reset image size

We find that subhalos in SIDM model variants with larger values of w have lower inner densities and are more cored, i.e., their inner density profiles are flatter as a function of distance from the center of the subhalo than in CDM. We interpret this as a consequence of ram pressure stripping; even though self-interactions within subhalos are more effective for SIDM models with smaller values of w, interactions with the host at large relative velocities significantly alter the inner profiles of subhalos in model variants with larger values of w. As indicated by Figure 8, we find that—at fixed V_peak and z_acc—subhalos with closer pericentric passages to the center of the host are significantly more cored. Detailed orbital modeling of satellites in MW-mass systems is therefore crucial in order to interpret the inferred DM density profiles of their subhalos.

We find that the circular velocity profiles of subhalos with close pericentric passages are significantly altered relative to CDM, as expected based on the changes to their density profiles and consistent with previous findings (Vogelsberger et al. 2012; Zavala et al. 2013; Robles et al. 2017, 2019; Fitts et al. 2019; Sameie et al. 2020). This has implications for SIDM solutions to the diversity and "too big to fail" problems concerning MW satellites (Kahlhoefer et al. 2019; Zavala et al. 2019), although baryonic feedback mechanisms, including heating from supernova feedback, can also reduce and flatten subhalos' central density profiles (Pontzen & Governato 2012; Brooks et al. 2013; Creasey et al. 2017; Santos-Santos et al. 2018; Read et al. 2019). However, if cores are created by stellar feedback, then subhalos' inner density profiles are expected to be correlated with their galaxies' star formation histories (e.g., Read et al. 2019). Meanwhile, in SIDM, we expect the inner amplitude and flatness of subhalos' density profiles to correlate most directly with their orbital histories, which can be encapsulated by their infall times and orbital eccentricities.

Thus, surviving subhalos in SIDM models with larger values of w have lower-amplitude, flatter density profiles relative to their CDM counterparts. The magnitude of this effect depends on subhalos' orbital properties such as their infall time and distance of closest approach to the center of the host, and this effect is more significant for larger values of w.

We now discuss several caveats and systematics associated with our analysis and main results.

6.1. Numerical Effects

6.2. Sample Variance

6.3. Baryonic Effects

Comparing our results to previous studies of SIDM effects on MW-mass systems is not straightforward. For clarity, we limit this discussion to a comparison of results for subhalo abundances. The following theoretical systematics should be kept in mind throughout the discussion in this section:

• 1.  
  SIDM cross section (amplitude and velocity dependence). Authors have studied a variety of velocity-dependent SIDM models with different cross section amplitudes. We comment on case-by-case comparisons with our model below.
• 2.  
  SIDM implementation. Most SIDM implementations in the literature are functionally identical to ours, with the exception of Vogelsberger et al. (2012) (and, by extension, Zavala et al. 2013 and Dooley et al. 2016). In particular, we determine whether a given DM particle scatters by calculating the interaction probability with all neighboring particles within a sphere of radius equal to the softening length, while Vogelsberger et al. (2012) calculate interaction probabilities using the k = 38 nearest neighbors of each particle.
• 3.  
  Halo finding. Various halo-finding algorithms have been used to analyze SIDM simulations of MW-mass systems.

Our findings are in reasonable agreement with Vogelsberger et al. (2012). These authors claim that a velocity-independent SIDM model with ${\sigma }_{T}/{m}_{\chi }=10\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ (i.e., our w₅₀₀ model with a 10 times higher cross section) is ruled out by the abundance and density profiles of bright MW satellites. These authors find that subhalo disruption in this velocity-independent model is driven by evaporation due to subhalo–host halo interactions, similar to our conclusion that ram pressure stripping plus tidal effects drive subhalo disruption in w₅₀₀. Vogelsberger et al. (2012) also find that two velocity-dependent SIDM models, both of which are similar to our w₁₀ model but with slightly larger momentum transfer cross sections at the host halo velocity scale, are allowed by the MW satellite data. By visual inspection, it appears that subhalo abundances are suppressed by ∼20% relative to CDM in these models, with the largest differences at low subhalo masses. The amount of subhalo disruption in our w₁₀ model is consistent with this result (see, e.g., Figure 6), although we find that the suppression of subhalo abundance relative to CDM is a much weaker function of subhalo mass, possibly due to our use of a matched subhalo sample.

Rocha et al. (2013) find that the abundance of MW subhalos in SIDM models with velocity-independent cross sections of σ/m_χ = 0.1 and $1\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ is very similar to that in CDM. However, they find that $\sim 20 \%$ fewer subhalos survive near the inner regions of their host halo ($r\lesssim 0.5{R}_{\mathrm{vir}}$) for $\sigma /{m}_{\chi }=1\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, corresponding to ${\sigma }_{T}/{m}_{\chi }=0.5\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$. Our w₁₀₀ model, in which $\sim 35 \%$ of subhalos are disrupted relative to CDM, has a similar momentum transfer cross section at the MW-mass host halo velocity scale. Thus, our results are roughly consistent with Rocha et al. (2013), although further investigation into halo finder differences is needed to assess whether the remaining $\sim 15 \%$ discrepancy is due to the velocity dependence of our w₁₀₀ model. Interestingly, Rocha et al. (2013) find that subhalo abundances for $\sigma /{m}_{\chi }=0.1\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ are identical to CDM at low ${V}_{\max }$, and that there are actually more high-${V}_{\max }$ subhalos than CDM in this case. Although we cannot compare our results to their $0.1\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ model directly, we similarly find that the high-${V}_{\max }$ tail of the surviving subhalo distribution is enhanced in our SIDM simulations relative to CDM, which might be due to the fact that surviving subhalos occupy tangential orbits on which tidal stripping is less severe.

Zavala et al. (2013) simulate velocity-independent SIDM models with ${\sigma }_{T}/{m}_{\chi }=0.1$, 1, and $10\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, as well as two velocity-dependent models that lie between our w₁₀ and w₁₀₀ models at the MW-mass host halo velocity scale, but which rise more steeply at low relative velocities. They find that the only model that leads to a difference in substructure abundance relative to CDM is ${\sigma }_{T}/{m}_{\chi }=10\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$, in stark contrast to our finding that subhalo disruption is significant in all of our model variants. This may be explained by the difference in SIDM implementation, since interactions between subhalo and host halo particles are less likely in the Zavala et al. (2013) implementation than in ours. In particular, our implementation allows interactions between any DM particles within a softening-length-sized sphere, while the k-nearest neighbors technique used in Zavala et al. (2013) does not. Although Zavala et al. (2013) use the same SIDM implementation as Vogelsberger et al. (2012), we speculate that there is less of a discrepancy between our results and those of Vogelsberger et al. (2012) because these authors only study models in which the cross section is either negligible or extremely large at the host halo velocity scale.

We note that Dooley et al. (2016) study stellar stripping using the same set of SIDM models and implementation as Zavala et al. (2013). These authors reiterate that substructure is only affected in the most extreme SIDM model, although they remark that subhalos with close pericentric passages in any of the models typically evaporate within $\sim 6\ \mathrm{Gyr}$, which is qualitatively consistent with our findings.

Finally, although they focus on the impact of adding a Galactic disk to SIDM simulations, Robles et al. (2019) simulate MW host halos without disks in CDM and SIDM. The subhalo ${V}_{\max }$ functions and radial distributions shown in Robles et al. (2019) suggest that $\sim 60 \%$ of subhalos survive in their ${\sigma }_{T}/{m}_{\chi }=1\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ model relative to CDM, which is in reasonable agreement with our findings. Again, this effect is only noticeable at low ${V}_{\max }$. Like the results in Rocha et al. (2013), and consistent with our findings, the SIDM-only model in Robles et al. (2019) shows an enhancement in the abundance of high-${V}_{\max }$ subhalos relative to CDM.

The effects reported in this work can be incorporated in models of DM substructure in MW-mass systems in order to constrain the SIDM cross section at relatively low velocity scales. For example, several authors have recently proposed forward-modeling frameworks in which various DM properties can be constrained based on the abundance of observed MW satellites (e.g., Jethwa et al. 2018; Nadler et al. 2019a, 2019b). These constraints are set by the fact that deviations from CDM yield a suppression in the abundance of the low-mass subhalos that are inferred to host faint satellite galaxies. Incorporating our finding that subhalo disruption is driven by the self-interaction cross section at the host halo velocity scale will therefore yield an upper limit on the SIDM cross section at $\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$. The precise value of this limit is difficult to forecast given the theoretical uncertainties discussed above; thus, we plan to carry out this study comprehensively in future work. Meanwhile, gaps and perturbations in stellar streams—which are also sensitive to the abundance, radial distribution, and density profiles of both surviving and disrupted subhalos in the MW—can be used to place complementary constraints (e.g., Bovy 2016).

Our results suggest that placing an upper limit on the SIDM cross section at velocity scales below $\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$ by probing subhalo abundances is challenging, or at least highly model dependent, since the abundance of surviving subhalos is not very sensitive to the SIDM cross section below the velocity scale of the host. However, if observations of the inner density profiles of the faintest MW satellites unambiguously favor cored or cuspy inner DM density profiles, it may be possible to place a stringent limit on ${\sigma }_{T}/{m}_{\chi }$ at low relative velocities (e.g., $\sim 10\ \mathrm{km}\,{{\rm{s}}}^{-1}$). Read et al. (2018) claim that the cuspy halo profile inferred for the MW satellite Draco places an upper bound on the velocity-independent SIDM cross section of $\sigma /{m}_{\chi }\lesssim 0.6\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$ (${\sigma }_{T}/{m}_{\chi }\lesssim 0.3\ {\mathrm{cm}}^{2}\,{{\rm{g}}}^{-1}$) at 99% confidence. However, as we have argued, modulations to subhalos' density profiles in the presence of self-interactions are highly dependent on their orbital histories; in addition, it is not clear how the constraint in Read et al. (2018) translates to the velocity-dependent SIDM model considered here. We also note that the stellar profiles of surviving satellites can evolve differently depending on whether they occupy a cored or cuspy halo (e.g., Errani et al. 2015).

High-resolution spectroscopy on giant segmented mirror telescopes (e.g., Simon et al. 2019), along with improvements in mass profile modeling techniques (e.g., Genina et al. 2019; Read et al. 2019; Lazar & Bullock 2020), will increase the precision of density profile measurements and constraints. However, observational systematics associated with inferring the underlying DM density profiles (e.g., Pineda et al. 2017; Oman 2018) and degeneracies with baryonic coring mechanisms (e.g., Read et al. 2019) will likely make a detection of DM self-interactions at low relative velocities challenging. On a positive note, our results demonstrate that observables that cover a range of velocity scales related to MW-mass systems—such as the inferred abundances and density profiles of low-mass subhalos—are highly complementary. In particular, combining such observables informs the velocity dependence of the SIDM cross section, which is a key facet of many well-motivated SIDM models.

Given increasingly precise constraints on the abundance and internal properties of DM substructure enabled by observations of satellite galaxies, strongly lensed systems, and stellar streams, a detailed examination of DM models at the edge of allowed parameter space is crucial. SIDM is particularly interesting in this context, since constraints on the velocity-dependent self-interaction cross section at relative velocities that are small compared to those typical for galaxy clusters provide an important handle on particle physics in the dark sector. In this paper, we studied the phenomenology of a generic, velocity-dependent SIDM model in the context of the DM host halo and subhalos of MW-mass systems. Due to its velocity dependence, the effects of our SIDM model are inherently scale dependent. Thus, observations of MW-mass halos and their subhalos provide a range of velocity scales with which to probe DM self-interactions.

We have demonstrated that the characteristic velocity above which momentum transfer due to self-interactions becomes inefficient, which is related to the DM and mediator masses in our SIDM model, has a variety of phenomenological consequences for MW-mass host halos and their subhalos. Our main findings are as follows:

• 1.  
  In the absence of baryons, the DM distribution in the inner regions of the host is cored and thermalized owing to self-interactions (Figure 2). This effect is stronger for larger values of the cross section at the host halo velocity scale, corresponding to our model variants with larger values of w.
• 2.  
  The initial assembly of subhalos in our SIDM simulations, quantified by ${V}_{\mathrm{peak}}$ and the time at which ${V}_{\mathrm{peak}}$ is achieved, is statistically similar to that in CDM (Figure 4).
• 3.  
  Subhalos are mildly affected by self-interactions before infall into the host, leading to lower maximum circular velocity values and moderately cored density profiles at infall relative to subhalos in CDM (Figures 4 and 8).
• 4.  
  A significant fraction of the subhalos that survive in CDM are not found in SIDM owing to extra mass loss from ram pressure stripping caused by self-interactions with the host halo, which makes subhalos more susceptible to tidal disruption. This effect is more severe for models with larger values of w (Table 1, Figures 5 and 6).
• 5.  
  Surviving subhalos in SIDM occupy tangentially biased orbits relative to those in CDM, causing a systematic trend in the orbital anisotropy profile that is more significant for models with larger values of w (Figure 7).
• 6.  
  Surviving subhalos in SIDM models with larger values of w are less dense and more cored than surviving subhalos in CDM, and the magnitude of this effect depends sensitively on orbital properties such as infall time and pericentric distance, with more severe coring for subhalos that pass closer to the center of their host (Figure 8).

Given these findings, we plan to carry out a comprehensive study of these effects for a simulated MW-like host halo that includes a realistic Large Magellanic Cloud analog system, which has recently been used to fit the full-sky MW satellite luminosity function (Nadler et al. 2020). This analysis will allow us to constrain the SIDM cross section at the MW host halo velocity scale ($\sim 200\ \mathrm{km}\,{{\rm{s}}}^{-1}$) using the abundance, surface brightness distribution, and radial distribution of observed MW satellites.

We emphasize that self-interactions also affect the orbital distribution and density profiles of subhalos in MW-mass systems. Comparing these quantities to data in a forward-modeling approach will help break degeneracies among SIDM models that are not ruled out by subhalo abundances alone, and will therefore inform the velocity dependence of the SIDM cross section. For example, surviving subhalos in our SIDM simulations preferentially occupy tangential orbits, and this prediction can be compared to the measured orbital distribution of MW satellites using proper-motion measurements from Gaia and its future data releases. Meanwhile, surviving subhalos in SIDM exhibit cored density profiles, and the strength of this effect is a function of both microphysical SIDM parameters and the orbital history of each subhalo.

We expect that combining high-precision spectroscopic and proper-motion measurements of satellite galaxies with analyses of strongly lensed systems and perturbations in stellar streams will test these predictions, providing a new window into velocity-dependent DM self-interactions.

We thank Manoj Kaplinghat, Annika Peter, and the anonymous referee for comments on the manuscript, and we thank Neal Dalal, Xiaolong Du, and Carton Zeng for useful discussions. This research received support from the National Science Foundation (NSF) under grant No. NSF DGE-1656518 through the NSF Graduate Research Fellowship received by E.O.N., by the U.S. Department of Energy contract to SLAC No. DE-AC02-76SF00515, and by Stanford University. Y.-Y.M. is supported by NASA through the NASA Hubble Fellowship grant No. HST-HF2-51441.001 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. We thank the LSST Dark Matter Group (https://lsstdarkmatter.github.io/) for feedback at workshops supported by the LSSTC Enabling Science program (grant No. 2017-11).

This research made use of computational resources at SLAC National Accelerator Laboratory, a U.S. Department of Energy Office, and the Sherlock cluster at the Stanford Research Computing Center (SRCC); the authors are thankful for the support of the SLAC and SRCC computing teams. This research made extensive use of arXiv.org and NASA's Astrophysics Data System for bibliographic information.

To test for convergence, we rerun our CDM and w₅₀₀ simulations at higher resolution. In particular, these resimulations are run with a $4.0\times {10}^{4}\ {M}_{\odot }\ {h}^{-1}$ high-resolution particle mass and an $85\ \mathrm{pc}\ {h}^{-1}$ minimum softening length, corresponding to a factor of eight increase in mass resolution and a factor of two decrease in softening length relative to our fiducial simulations.

In Figure A1, we compare the subhalo ${V}_{\mathrm{peak}}$ functions and radial distributions from these high-resolution resimulations to our fiducial results. We find fairly good agreement, at the $\sim 15 \%$ level, between the standard and high-resolution results for subhalos above our fiducial ${V}_{\mathrm{peak}}$ thresholds. Interestingly, we find that there are more resolved subhalos above the relevant ${V}_{\mathrm{peak}}$ threshold in the lower-resolution simulations, which might be due to the fact that early interactions within subhalos are better resolved in the higher-resolution simulations, leading to more efficient coring. However, we note that there are also more resolved subhalos in the lower-resolution CDM simulation, so this also might indicate that the ${V}_{\mathrm{peak}}$ distributions are slightly different in the fiducial and high-resolution simulations; we have not attempted to match subhalo populations precisely for this comparison, since the level of agreement is reasonable. These findings suggest that artificial subhalo disruption is not a large effect relative to the physical disruption (or stripping below the resolution limit) of subhalos in our SIDM simulations. Thus, we conclude that our main results, including the amount of subhalo disruption in our SIDM simulations, are not highly sensitive to the resolution threshold of our fiducial simulations.

Zoom In Zoom Out Reset image size

In our fiducial analysis, we employed a conservative ${V}_{\mathrm{peak}}$ resolution cut of $20\ \mathrm{km}\,{{\rm{s}}}^{-1}$ for our CDM simulation, and we matched the number of surviving-plus-disrupted subhalos in CDM and each SIDM model variant using variable ${V}_{\mathrm{peak}}$ thresholds, denoted by ${V}_{\mathrm{thresh}}$. In this appendix, we show that this statistical subhalo matching method is necessary given the scatter in the ${V}_{\mathrm{peak}}$ distributions measured by our halo finder. We then demonstrate that our matched subhalo populations are well converged given our fiducial choice of ${V}_{\mathrm{thresh}}=20\ \mathrm{km}\,{{\rm{s}}}^{-1}$ in CDM. Finally, we summarize our main results for a lower ${V}_{\mathrm{peak}}$ threshold.

B.1. Subhalo Abundance for a Fixed ${V}_{\mathrm{peak}}$ Threshold

Figure B1 shows the number of surviving, disrupted, and surviving-plus-disrupted subhalos in SIDM relative to that in CDM if the same value of ${V}_{\mathrm{thresh}}$ is used for all of the simulations. The scatter among the SIDM model variants in all three panels of Figure B1 strongly suggests that a variable ${V}_{\mathrm{peak}}$ threshold must be adopted in order to match subhalo populations in our CDM and SIDM simulations. Physically, this is reasonable because of the differences in subhalo assembly on an object-by-object basis caused by early self-interactions described in Section 5.1.2; however, the differences could also be driven by halo finder issues. Note that the scatter among our SIDM model variants increases for lower values of ${V}_{\mathrm{thresh}}$, making precise subhalo population matching more difficult near the CDM resolution limit of ${V}_{\mathrm{peak}}\approx 10\ \mathrm{km}\,{{\rm{s}}}^{-1}$.

Zoom In Zoom Out Reset image size

B.2. Choice of Fiducial ${V}_{\mathrm{peak}}$ Threshold

B.3. Severity of Subhalo Disruption for a Less Conservative ${V}_{\mathrm{peak}}$ Threshold

As noted in Section 5.3.1, there is a curious "spike" in the inner radial distribution of surviving subhalos for w₁₀ and w₂₀₀, shown in the right panel of Figure 6. The spike is particularly significant for w₁₀, where the number of subhalos in one of the inner radial bins is enhanced by a factor of two with respect to CDM. We emphasize that these spikes are not statistically significant: they only correspond to one or two additional subhalos near the center of the host in the SIDM simulations. Given the increased efficiency of tidal disruption in SIDM, it might seem surprising that these subhalos survive. While the survival (relative to our other SIDM model variants) and sinking (relative to CDM) of these objects are plausible consequences of self-interactions, we cannot distinguish the presence of these subhalos at small radii from statistical fluctuations. In particular, it is possible that pericentric passages at $z\approx 0$ determined by the orbital phases of the subhalos in w₁₀ and w₂₀₀ happen to align more closely with the z = 0 snapshot than for subhalos in CDM or our other SIDM model variants. On the other hand, if this behavior persists in a larger suite of SIDM simulations, it may be a physical consequence of increased drag due to self-interactions.