Dark Matter Self-interactions and Small Scale Structure

Dark Matter Self-interactions and Small Scale Structure

Sean Tulin stulin@yorku.ca Department of Physics and Astronomy, York University, Toronto, Ontario M3J 1P3, Canada    Hai-Bo Yu haiboyu@ucr.edu Department of Physics and Astronomy, University of California, Riverside, California 92521, USA
July 17, 2019

We review theories of dark matter (DM) beyond the collisionless paradigm, known as self-interacting dark matter (SIDM), and their observable implications for astrophysical structure in the Universe. Self-interactions are motivated, in part, due to the potential to explain long-standing (and more recent) small scale structure observations that are in tension with collisionless cold DM (CDM) predictions. Simple particle physics models for SIDM can provide a universal explanation for these observations across a wide range of mass scales spanning dwarf galaxies, low and high surface brightness spiral galaxies, and clusters of galaxies. At the same time, SIDM leaves intact the success of CDM cosmology on large scales. This report covers the following topics: (1) small scale structure issues, including the core-cusp problem, the diversity problem for rotation curves, the missing satellites problem, and the too-big-to-fail problem, as well as recent progress in hydrodynamical simulations of galaxy formation; (2) N-body simulations for SIDM, including implications for density profiles, halo shapes, substructure, and the interplay between baryons and self-interactions; (3) semi-analytic Jeans-based methods that provide a complementary approach for connecting particle models with observations; (4) merging systems, such as cluster mergers (e.g., the Bullet Cluster) and minor infalls, along with recent simulation results for mergers; (5) particle physics models, including light mediator models and composite DM models; and (6) complementary probes for SIDM, including indirect and direct detection experiments, particle collider searches, and cosmological observations. We provide a summary and critical look for all current constraints on DM self-interactions and an outline for future directions.

I Introduction

i.1 The dark matter puzzle

It was long ago pointed out by Oort that the distributions of mass and light in galaxies have little resemblance to one another Oort (1940). At the time, observations of galaxy NGC 3115 found a rotation curve rising linearly with radius—indicating a constant mass density—despite its luminosity falling by over an order of magnitude. Oort derived a mass-to-light ratio that increased with distance up to in solar units111Modern distance estimates revise Oort’s value down to . at the outermost radius. He concluded that the luminous disk of NGC 3115 is “imbedded in a more or less homogeneous mass of great density” Oort (1940). The same phenomenon was observed in M31 by Babcock a year earlier Babcock (1939). These observations stood in contrast to the Milky Way (MW), where, by studying vertical dynamics of nearby stars, the mass-to-light ratio within the disk was known to be only at the solar radius Öpik (1915); Kapteyn (1922); Jeans (1922); Oort (1932). Earlier pioneering observations of the Coma cluster by Zwicky Zwicky (1933) and Virgo cluster by Smith Smith (1936) inferred similarly striking discrepencies between mass and light on much larger scales.

These issues were not fully appreciated for several decades, when astronomers were able to study galactic rotational velocities at much larger distances. While many rotation curves exhibited a linearly increasing velocity at small distances, they were expected to turn over eventually and fall as according to Kepler’s laws King (1963). Instead, optical observations by Rubin and Ford of M31 Rubin and Ford (1970)—soon after extended to larger radii using 21-cm radio observations Roberts and Whitehurst (1975); Roberts and Rots (1973)—revealed a circular velocity that did not fall off, but remained approximately constant. Many other spiral galaxies were found to exhibit the same behavior Freeman (1970); Roberts and Rots (1973); Rubin et al. (1980); Bosma (1981); Sofue and Rubin (2001), indicating that most of the mass in galaxies is found in massive, nonluminous halos extending far beyond the spatial extent of luminous stars and gas Ostriker et al. (1974); Einasto et al. (1974). Massive spherical dark halos could also explain the apparent stability of bulgeless spiral galaxies Ostriker and Peebles (1973), which by themselves are unstable to the formation of bars Toomre (1964); Hohl (1971).

The large amount of dark matter (DM) required by observations pointed toward its nonbaryonic nature.222In this context, the term “baryons” represents protons, neutrons, and electrons that constitute normal atomic matter. The total mass density found in halos was estimated to be around of the critical density Ostriker et al. (1974); Einasto et al. (1974), in remarkable agreement with present cosmological values Ade et al. (2016). If composed of baryons, the cosmological baryon density would be in tension with upper limits inferred from nucleosynthesis arguments Wagoner (1973), as well as being difficult to hide from astrophysical observations Hegyi and Olive (1983). Therefore, the “missing mass” puzzle in galaxies, as well as clusters, suggested the existence of a new dominant form of matter, such as an elementary particle leftover from the Big Bang.

At first, the neutrino appeared to be a promising DM candidate within the Standard Model (SM) of elementary particles Gershtein and Zeldovich (1966); Cowsik and McClelland (1972); Szalay and Marx (1976). A relic thermal bath of neutrinos, produced alongside the cosmic microwave background (CMB), could yield the required mass density if the neutrino mass was . However, because neutrinos are “hot” DM—they decouple from photons and electrons around the nucleosynthesis epoch while still relativistic—free-streaming erases density fluctuations below supercluster scales (see, e.g., Ref. Primack and Gross (2000)). Numerical simulations have shown that top-down structure formation, where superclusters form first in the neutrino-dominated Universe and subsequently fragment to produce galaxies, is incompatible with galaxy clustering constraints White et al. (1983, 1984); Hut and White (1984).

Figure 1: Matter power spectrum inferred through cosmological measurements. Red line shows the best fit for CDM cosmology for a simplified five-parameter model, assuming a flat spatial geometry and a scale-invariant primordial spectrum. Reprinted from Ref. Tegmark et al. (2004a). See therein and Ref. Tegmark et al. (2004b) for further information.

Cosmological data has converged upon the CDM paradigm as the standard model of cosmology (e.g., Ref. Bahcall et al. (1999)). Of the total mass-energy content of the Universe, approximately is cold dark matter (CDM) and is baryonic matter (while the remainder is consistent with a cosmological constant ), with a nearly scale-invariant spectrum of primordial fluctuations Ade et al. (2016). In this picture, structure in the Universe forms as primordial overdensities collapse under gravity. Since CDM, acting as a pressureless fluid, is more dominant and collapses more readily than baryonic matter, it provides the gravitational potential underlying the distribution of visible matter in the Universe. The observed matter power spectrum, as obtained from a variety of cosmological probes, is in remarkable agreement with CDM cosmology, shown in Fig. 1. In addition, the CDM model also explains many important aspects of galaxy formation Springel et al. (2006); Trujillo-Gomez et al. (2011).

Despite this success, all evidence to date for DM comes from its gravitational influence in the Universe. With no viable DM candidate within SM, the underlying theory for DM remains unknown. Many new particle physics theories proposed to address shortcomings of the SM simultaneously predict new particles that can be a DM candidate. Examples include weakly-interacting massive particles (WIMPs) motivated by the hierarchy problem, such as neutralinos in the supersymmetric models Goldberg (1983); Jungman et al. (1996) and Kaluza-Klein states in extra dimensional models Servant and Tait (2003); Cheng et al. (2002), as well as extremely light axion particles Preskill et al. (1983) associated with the solution to the strong CP problem in QCD Peccei and Quinn (1977). The comic abundance of these new particles can be naturally close to the DM abundance inferred from the cosmological observations (e.g., Ref. Lee and Weinberg (1977)). This coincidence has motivated decades of efforts to discover the particle physics realization of DM through experimental searches for new physics beyond the SM (e.g., see Bertone et al. (2005); Feng (2010); Arrenberg et al. (2013) and references therein).

On large scales, the structure of the Universe is consistent with DM particles that are cold, collisionless, and interact with each other and SM particles purely via gravity. WIMPs and axions have interactions with the SM that are potentially large enough to be detectable in the laboratory, while on astrophysical scales these interactions are negligible and these candidates behave as CDM. On the other hand, other particle physics candidates for DM may have interactions with the SM are too feeble to observe directly. Here, observational tests of structure and departures from the CDM paradigm play a complementary role in probing the particle physics of DM independently of its interactions with SM particles (as had been done for hot DM candidates).

i.2 Crisis on small scales

CDM is an extremely successful model for the large scale structure of the Universe, corresponding to distances greater than today (see Fig. 1). On smaller scales, structure formation becomes strongly nonlinear and N-body simulations have become the standard tool to explore this regime. Cosmological DM-only simulations have provided several predictions for the structure and abundance of CDM halos and their substructure. However, it remains unclear whether these predictions are borne out in nature.

Since the 1990s, four main discrepancies between CDM predictions and observations have come to light.

  • Core-cusp problem: High-resolution simulations show that the mass density profile for CDM halos increases toward the center, scaling approximately as in the central region Dubinski and Carlberg (1991); Navarro et al. (1996a); Navarro et al. (1997). However, many observed rotation curves of disk galaxies prefer a constant “cored” density profile  Flores and Primack (1994); Moore (1994); Moore et al. (1999a), indicated by linearly rising circular velocity in the inner regions. The issue is most prevalent for dwarf and low surface brightness (LSB) galaxies Burkert (1995); McGaugh and de Blok (1998); Côté et al. (2000); van den Bosch and Swaters (2001); Borriello and Salucci (2001); de Blok et al. (2001a, b); Marchesini et al. (2002); Gentile et al. (2005); Gentile et al. (2007a); Kuzio de Naray et al. (2006, 2008); Salucci et al. (2007), which, being highly DM-dominated, are appealing environments to test CDM predictions.

  • Diversity problem: Cosmological structure formation is predicted to be a self-similar process with a remarkably little scatter in density profiles for halos of a given mass Navarro et al. (1997); Bullock et al. (2001). However, disk galaxies with the same maximal circular velocity exhibit a much larger scatter in their interiors Oman et al. (2015) and inferred core densities vary by a factor of  Kuzio de Naray et al. (2010).

  • Missing satellites problem: CDM halos are rich with substructure, since they grow via hierarchical mergers of smaller halos that survive the merger process Kauffmann et al. (1993). Observationally, however, the number of small galaxies in the Local Group are far fewer than the number of predicted subhalos. In the MW, simulations predict subhalos large enough to host galaxies, while only 10 dwarf spheroidal galaxies had been discovered when this issue was first raised Moore et al. (1999b); Klypin et al. (1999). Nearby galaxies in the field exhibit a similar underabundance of small galaxies compared to the velocity function inferred through simulations Zavala et al. (2009); Zwaan et al. (2010); Trujillo-Gomez et al. (2011).

  • Too-big-to-fail problem (TBTF): In recent years, much attention has been paid to the most luminous satellites in the MW, which are expected to inhabit the most massive suhalos in CDM simulations. However, it has been shown that these subhalos are too dense in the central regions to be consistent with stellar dynamics of the brightest dwarf spheroidals Boylan-Kolchin et al. (2011, 2012). The origin of the name stems from the expectation that such massive subhalos are too big to fail in forming stars and should host observable galaxies. Studies of dwarf galaxies in Andromeda Tollerud et al. (2014) and the Local Group field Garrison-Kimmel et al. (2014) have found similar discrepancies.

It must be emphasized, however, that these issues originally gained prominence by comparing observations to theoretical predictions from DM-only simulations. Hence, there has been extensive debate in the literature whether these small scale issues can be alleviated or solved in the CDM framework once dissipative baryonic processes, such as gas cooling, star formation, and supernova feedback, are included in simulations Navarro et al. (1996b); Governato et al. (2010). We review these topics in detail in §II.

A more intriguing possibility is that the CDM paradigm may break down on galactic scales. One early attempt to solve these issues supposes that DM particles are warm, instead of cold, meaning that they were quasi-relativistic during kinetic decoupling from the thermal bath in the early Universe Colombi et al. (1996); Bode et al. (2001). Compared to CDM, warm DM predicts a damped linear power spectrum due to free-streaming, resulting in a suppression of the number of substructures. Warm DM halos are also typically less concentrated because they form later than CDM ones. Recent high-resolution simulations show that warm DM may provide a solution to the missing satellites and too-big-to-fail problems Lovell et al. (2012, 2014); Horiuchi et al. (2016). However, the favored mass range of thermal warm DM is in strong tension with Lyman- forest observations Iršič et al. (2017); Viel et al. (2013)333The Lyman- constraints may be weakened due to uncertainties in the evolution of intergalactic medium Kulkarni et al. (2015); Garzilli et al. (2017); Cherry and Horiuchi (2017). and the abundance of high redshift galaxies Menci et al. (2016). Also, while warm DM halos have constant density cores set by the phase space density limit, the core sizes are far too small to solve the core-cusp problem given Lyman- constraints Maccio et al. (2012).

i.3 Self-Interacting dark matter

Another promising alternative to collisionless CDM is self-interacting dark matter (SIDM), proposed by Spergel & Steinhardt to solve the core-cusp and missing satellites problems Spergel and Steinhardt (2000). In this scenario, DM particles scatter elastically with each other through interactions. Self-interactions lead to radical deviations from CDM predictions for the inner halo structure, shown in Fig. 2. We summarize the expectations for SIDM halos (blue) compared to CDM halos (black) as follows:

  • Isothermal velocity dispersion: Although a CDM halo is a virilized object, the DM velocity dispersion, indicating the “temperature” of DM particles, is not a constant and decreases towards the center in the inner halo. Self-interactions transport heat from the hotter outer to the cooler inner region of a DM halo, thermalizing the inner halo and driving the velocity dispersion to be uniform with radius (Fig. 2, left panel). The velocity distribution function for SIDM becomes more Maxwell-Boltzmann compared to CDM Vogelsberger and Zavala (2013).

  • Reduced central density: Hierarchical structure formation leads to a universal density profile for CDM halos Navarro et al. (1996a); Navarro et al. (1997). In the presence of collisions, the central density is reduced as low-entropy particles are heated within the dense inner halo, turning a cusp into a core (Fig. 2, center panel).

  • Spherical halo shape: While CDM halos are triaxial Dubinski and Carlberg (1991), collisions isotropize DM particle velocities and tend to erase ellipticity. The minor-to-major axis ratio is closer to unity toward the center of SIDM halos compared to CDM halos (Fig. 2, right panel).

Since the scattering rate is proportional to the DM density, SIDM halos have the same structure as CDM halos at sufficiently large radii where the collision rate is negligible.

Figure 2: Left: Density profiles (left), dispersion profiles (center), and median halo shapes (right) for SIDM with and its CDM counterpart. DM self-interactions cause heat transfer from the hot outer region to the cold inner region of a CDM halo and kinetically thermalize the inner halo, leading to a shallower density profile and a more spherical halo shape. Simulation data from Ref. Rocha et al. (2013); Peter et al. (2013).

The local collision rate is given by


where is the DM particle mass, while are the cross section and relative velocity, respectively, for scattering. Within the central region of a typical dwarf galaxy, we have and  Oh et al. (2011a). Therefore, the cross section per unit mass must be at least


to have an effect on the halo, corresponding to at least one scattering per particle over 10 Gyr galactic timescales. For , the mean free path of DM particles is larger than the core radii (Knudsen number larger than unity) and heat conduction is effective in the inner halo. Provided is not dramatically larger than this value, is negligible during the early Universe when structure forms.444Although self-interactions may be active in the very early Universe, long before matter-radiation equality, they rapidly fall out of equilibrium due to the Hubble dilution and redshifting of DM particles. Therefore SIDM retains the success of large-scale structure formation from CDM, affecting structure at late times and only on small scales in the dense inner regions of halos.

Cored density profiles lead to shallower rotation curves for dwarf and LSB spiral galaxies at small radii, in accord with observations, while cores in satellite galaxies ameliorate the too-big-to-fail problem by reducing the predicted stellar line-of-sight velocity dispersion for the largest subhalos. If is fixed as in Eq. (2), the effect of self-interactions on the halo scales approximately in a self-similar fashion, and larger halos, such as those for massive elliptical galaxies and clusters, may be impacted by self-interactions at proportionally larger radii.

In principle, self-interactions can also affect substructure, reducing the subhalo mass function to solve the missing satellites problem. SIDM subhalos are prone to tidal disruption, being less concentrated than their CDM counterparts, as well as evaporation due to ram pressure stripping from the host halo Spergel and Steinhardt (2000). However, this mechanism requires a too-large cross section that is excluded for reasons we discuss in §III.

The effect of self-interactions on DM halos has been borne out through N-body simulations. Soon after Spergel & Steinhardt’s proposal, a first wave of SIDM simulations was performed Moore et al. (2000); Yoshida et al. (2000a); Burkert (2000); Kochanek and White (2000); Yoshida et al. (2000b); Dave et al. (2001); Colin et al. (2002). In conjunction with these simulations, a number of constraints on SIDM emerged, the most stringent of which were strong lensing measurements of the ellipticity and central density of cluster MS2137-23 Miralda-Escude (2002); Meneghetti et al. (2001). These studies limited the cross section to be below , which excludes self-interactions at a level to explain small scale issues in galaxies. The Bullet Cluster provides additional evidence for the collisionless nature of DM, requiring  Randall et al. (2008). Although ad hoc velocity dependencies were put forth to evade these constraints Yoshida et al. (2000b); Firmani et al. (2000, 2001); Colin et al. (2002), SIDM largely fell into disfavor in light of these difficulties.

More recently, new N-body simulations for SIDM, with dramatically higher resolution and halo statistics, have revived DM self-interactions Vogelsberger et al. (2012); Rocha et al. (2013); Peter et al. (2013); Zavala et al. (2013); Elbert et al. (2015); Vogelsberger et al. (2014); Fry et al. (2015); Dooley et al. (2016). Many of the constraints from larger scales are much weaker than previously thought Rocha et al. (2013); Peter et al. (2013). In particular, constraints based on ellipticity have been overestimated: self-interactions do not erase triaxiality as effectively as previously supposed, and moreover, an observed ellipticity has contributions along the line-of-sight from regions outside the core where the halo remains triaxial Peter et al. (2013). The conclusion is that can solve the core-cusp and TBTF issues on small scales, while remaining approximately consistent with other astrophysical constraints on larger scales Rocha et al. (2013); Peter et al. (2013); Zavala et al. (2013); Elbert et al. (2015). However, more recent studies based on stacked merging clusters Harvey et al. (2015) and stellar kinematics within cluster cores Kaplinghat et al. (2016) suggest some tension with these values. Viable SIDM models are preferred to have a scattering cross section with a mild velocity-dependence from dwarf to cluster scales. We discuss these issues in further detail in §III.

On the theory side, a new semi-analytical SIDM halo model has been developed based on the Jeans equation Kaplinghat et al. (2014a); Kaplinghat et al. (2016); Kamada et al. (2016). It can reproduce the simulation results for SIDM profiles within 10–20% while being much cheaper computationally. Discussed in §IV, this approach provides insight for understanding the baryonic influence on SIDM halo properties Kaplinghat et al. (2014a), testing SIDM models from dwarf to cluster scales Kaplinghat et al. (2016), and addressing the diversity in rotation curves Kamada et al. (2016).

Furthermore, there has been important progress in particle physics models for SIDM. Both numerical and analytical methods have been developed to accurately calculate the cross section for SIDM models involving the Yukawa Feng et al. (2010a); Tulin et al. (2013a); Schutz and Slatyer (2015) or atomic interactions Cline et al. (2014a); Boddy et al. (2016). These studies make it possible to map astrophysical constraints on to the particle model parameters, such as the DM and mediator masses and coupling constant. In addition, an effective theory approach has been proposed in parametrizing SIDM models with a set of variables that are directly correlated with astrophysical observations Cyr-Racine et al. (2016); Vogelsberger et al. (2016).

Positive observations Observation Refs.
Cores in spiral galaxies Rotation curves Dave et al. (2001); Kaplinghat et al. (2016)
(dwarf/LSB galaxies)
Too-big-to-fail problem
Milky Way Stellar dispersion Zavala et al. (2013)
Local Group Stellar dispersion Elbert et al. (2015)
Cores in clusters Stellar dispersion, lensing Kaplinghat et al. (2016); Elbert et al. (2016)
Abell 3827 subhalo merger DM-galaxy offset Kahlhoefer et al. (2015)
Abell 520 cluster merger DM-galaxy offset Jee et al. (2014a); Kahlhoefer et al. (2014); Sepp et al. (2016)
Halo shapes/ellipticity Cluster lensing surveys Peter et al. (2013)
Substructure mergers DM-galaxy offset Harvey et al. (2015); Wittman et al. (2017)
Merging clusters Post-merger halo survival Table 2
(Scattering depth )
Bullet Cluster Mass-to-light ratio Randall et al. (2008)
Table 1: Summary of positive observations and constraints on self-interaction cross section per DM mass. Italicized observations are based on single individual systems, while the rest are derived from sets of multiple systems. Limits quoted, which assume constant , may be interpreted as a function of collisional velocity provided is not steeply velocity-dependent. References noted here are limited to those containing quoted self-interaction cross section values. Further references, including original studies of observations, are cited in the corresponding sections below.

In Table 1, we summarize the present status of astrophysical observations related to SIDM . The positive observations indicate discrepancies with CDM-only simulations and the required cross section assuming self-interactions are responsible for solving each issue. Dwarf and LSB galaxies favor cross sections of at least to produce large enough core radii in these systems, which is also consistent with alleviating the too-big-to-fail problem among MW satellites and field dwarfs of the Local Group. The cross section need not be particularly fine-tuned. Values as large as provide consistent density profiles on dwarf scales Elbert et al. (2015). (The upper limit on on dwarf scales due to the onset of gravothermal collapse, which we discuss in §III, is not well-known.)

Lastly, for completeness, let us mention that SIDM, in its original conception, was introduced much earlier by Carlson, Machacek & Hall Carlson et al. (1992) to modify large scale structure formation. In their scenario, DM particles (e.g., glueballs of a nonabelian dark sector) have (or ) number-changing interactions by which they annihilate and eventually freeze-out to set the relic density, as well as elastic self-interactions that maintain kinetic equilibrium. Unlike standard freeze-out Scherrer and Turner (1986), DM particles retain entropy as their number is depleted and therefore cool more slowly than the visible sector. When this theory was proposed, a mixture of both cold plus hot DM within a flat, matter-dominated Universe () was a viable and theoretically appealing alternative to CDM Davis et al. (1992); Klypin et al. (1993); Primack (1997). Although this original SIDM was proposed as a hybrid between hot plus cold DM, it provided too much small scale power suppression in Lyman- observations relative to larger scales and was found to be excluded de Laix et al. (1995).555We emphasize that this exclusion is for a matter-dominated cosmology. Recently, this SIDM scenario has been revived within an observationally consistent (i.e., -dominated) cosmology Hochberg et al. (2014), discussed in §VI.

i.4 From astrophysics to particle theory

The figure of merit for self-interactions, , depends on the underlying DM particle physics model. WIMPs have self-interactions mediated through the weak force, Higgs boson, or other heavy states. Since WIMP interactions and masses are set by the weak scale, yielding , they effectively behave as collisionless CDM. If self-interactions indeed explain the small scale issues, then DM cannot be a usual WIMP.

Figure 3: As an analogy to SIDM, we show for “nucleon self-interactions” (neutron-proton elastic scattering) as a function of scattering velocity . Nuclear interactions are similar in magnitude to what is required for DM self-interactions to explain small scale structure issues.

What sort of particle theory can have a large enough ? An analogy is provided by nuclear interactions, mediated by pion exchange. In Fig. 3, we show - elastic scattering data for kinetic energies 0.5 eV 10 MeV Obložinský et al. (2011), expressed in terms of and (here, is the nucleon mass). The required cross section for SIDM is comparable in magnitude to nuclear cross sections for visible matter. The lesson here is that or larger can be achieved if the interaction scale lies below GeV. However, unlike nuclear scattering, the theory of self-interactions need not be strongly-coupled, nor does the DM mass need to be below 1 GeV. For example, self-interactions can be a weakly-coupled dark force Ackerman et al. (2009); Feng et al. (2009, 2010a); Buckley and Fox (2010); Loeb and Weiner (2011); Tulin et al. (2013b, a), with the mediator particle denoted by . A perturbative calculation gives (in the limit )


where is the DM analog of the electromagnetic fine structure constant, . Self-interactions that are electomagnetic strength (or weaker) are sufficient, as are weak-scale DM masses, provided the mediator mass is light enough. Note that Eq. (3) is only valid in the weakly-coupled Born limit, , and there are important corrections outside of this regime Tulin et al. (2013a).

Another important point is that need not be constant in velocity. DM particles in larger mass halos have larger characteristic velocities compared to those in smaller halos. Therefore, observations from dwarf to cluster scales provide complementary handles for constraining the velocity-dependence of . Along these lines, Refs. Yoshida et al. (2000b); Firmani et al. (2000, 2001); Colin et al. (2002) advocated a cross section of the form in order to evade cluster constraints and fit a constant central density across all halo scales. We caution that such a dependence is not motivated by any theoretic framework for SIDM.

On the other hand, many well-motivated particle physics scenarios predict velocity-dependent cross sections that are not described by simple power laws. Scattering through a dark force can be a contact-type interaction at small velocity, with constant ; yet once the momentum transfer is larger than the mediator mass, scattering is described by a Rutherford-like cross section that falls with higher velocity. These models naturally predict a larger on dwarf scales compared to clusters, depending on the model parameters. The velocity-dependence may be qualitatively similar to - scattering, shown in Fig. 3, or it can have a more complicated form Tulin et al. (2013b, a). We discuss models for SIDM and their expected velocity-dependence in §VI.

The particle physics of SIDM is not just about self-interactions. While searching for their effect on structure is the only probe if DM is completely decoupled from the SM, this is unlikely from an effective field theory point of view. For example, a dark force can interact with the SM through hypercharge kinetic mixing Holdom (1986) if it is a vector or the Higgs portal operator Patt and Wilczek (2006) if it is a scalar. Such couplings allow mediator particles thermally produced in the early Universe to decay, avoiding overclosure Kaplinghat et al. (2014b). At the same time, these operators provide a window to probing SIDM in direct and indirect detection experiments, as well as collider searches. Similar to the WIMP paradigm, SIDM motivates a multifaceted program of study combining complementary data from both astrophysical and terrestrial measurements. Though the coupling between the dark and visible sectors must be much weaker than for WIMPs, this can be compensated by the fact that mediator states must be much lighter than the weak scale. We discuss in §VII these complementary searches for DM (and other dark sector states) within the context of the SIDM paradigm.

Ii Astrophysical observations

The kinematics of visible matter is a tracer of the gravitational potential in galaxies and clusters, allowing the underlying DM halo mass distribution to be inferred. Observations along these lines have pointed to the breakdown of the collisionless CDM paradigm on small scales. In this section, we review these astrophysical observations and discuss possible solutions from baryonic physics and other systematic effects. For other recent reviews of these issues, we direct the reader to Refs. de Blok (2010); Bullock (2010); Weinberg et al. (2014); Del Popolo and Le Delliou (2017).

The issues discussed here may share complementary solutions. For example, mechanisms that generate cored density profiles may help reconcile the too-big-to-fail problem by reducing central densities of MW subhalos, as well as accommodating the diversity of galactic rotation curves. On the other hand, the too-big-to-fail problem may share a common resolution with the missing satellites problem if the overall subhalo mass function is reduced.

ii.1 Core-Cusp Problem

Collisionless CDM-only simulations predict “cuspy” DM density profiles for which the logarithmic slope, defined by , tends to at small radii Dubinski and Carlberg (1991); Navarro et al. (1996a); Navarro et al. (1997); Moore et al. (1999a). Such halos are well-described by the Navarro-Frenk-White (NFW) profile Navarro et al. (1996a); Navarro et al. (1997),


where is the radial coordinate and and are characteristic density and scale radius of the halo, respectively.666High resolution simulations have found that collisionless CDM profiles become shallower than at small radii Navarro et al. (2010); Stadel et al. (2009), in better agreement with the Einasto profile Einasto (1965). However, the enclosed mass profile is slightly larger at small radii compared to NFW fits Navarro et al. (2010), further exacerbating the issues discussed here. On the other hand, as we discuss below, many observations do not find evidence for the steep inner density slope predicted for collisionless CDM, preferring “cored” profiles with inner slopes that are systematically shallower Flores and Primack (1994); Moore (1994). This discrepancy is known as the “core-cusp problem.”

A related issue, known as the “mass deficit problem,” emerges when observed halos are viewed within a cosmological context McGaugh and de Blok (1998); de Blok et al. (2001c); McGaugh et al. (2007); Oman et al. (2015). The mass-concentration relation predicted from cosmological CDM simulations implies a tight correlation between and (see, e.g., Refs. Ludlow et al. (2014); Dutton and Macci� (2014); Rodriguez-Puebla et al. (2016)). Since cosmological NFW halos are essentially a single-parameter profile (up to scatter), determination of the halo at large radius fixes the halo at small radius. However, many observed systems have less DM mass at small radii compared to these expectations. Alternatively, if NFW halos are fit to data at both large and small radii, the preferred profiles tend to be less concentrated than expected cosmologically.

Rotation curves: Late-type dwarf and LSB galaxies are ideal laboratories for halo structure. These systems are DM dominated down to small radii (or over all radii) and environmental disturbances are minimized. Flores & Primack Flores and Primack (1994) and Moore Moore (1994) first recognized the core-cusp issue based on HI rotation curves for several dwarfs, which, according to observations, are well described by cored profiles Carignan and Freeman (1988); Lake et al. (1990); Jobin and Carignan (1990).777In these early studies, the Hernquist profile Hernquist (1990) was used to model collisionless CDM halos. It has the same behavior as the NFW profile in the inner regions, , but falls as at large radii. On the other hand, a variety of cored profiles have been adopted in the literature, including the nonsingular isothermal profile Eddington (1926), Burkert profile Burkert (1995), and pseudo-isothermal profile , where and are the core density and radius, respectively van Albada et al. (1985).

Indeed, rotation curve measurements for dwarfs and LSBs have been a long-standing challenge to the CDM paradigm Burkert (1995); McGaugh and de Blok (1998); Côté et al. (2000); van den Bosch and Swaters (2001); Borriello and Salucci (2001); de Blok et al. (2001a, b); Marchesini et al. (2002); Gentile et al. (2005); Gentile et al. (2007a); Kuzio de Naray et al. (2006, 2008); Salucci et al. (2007). For axisymmetric disk galaxies, circular velocity can be decomposed into three terms


representing the contributions to the rotation curve from the DM halo, stars, and gas, respectively. The baryonic contributions to the rotation curve are modeled from the respective surface luminosities of stars and gas. However, the overall normalization between stellar mass and light remains notoriously uncertain: stellar mass is dominated by smaller and dimmer stars, while luminosity is dominated by more massive and brighter stars. Estimates for the stellar mass-to-light ratio—denoted by in Eq. (5)—rely on stellar population synthesis models and assumptions for the initial mass function, with uncertainties at the factor-of-two level Conroy (2013). Modulo this uncertainty, the DM profile can be fit to observations. For a spherical halo, the DM contribution to the rotation curve is , where is Newton’s constant and is the DM mass enclosed within .

Figure 4: Left: Observed rotation curve of dwarf galaxy DDO 154 (black data points) Oh et al. (2015) compared to models with an NFW profile (dotted blue) and cored profile (solid red). Stellar (gas) contributions indicated by pink (dot-)dashed lines. Right: Corresponding DM density profiles adopted in the fits. NFW halo parameters are and , while the cored density profile is generated using an analytical SIDM halo model developed in Kaplinghat et al. (2016); Kamada et al. (2016).

Fig. 4 illustrates these issues for dwarf galaxy DDO 154. The left panel shows the measured HI rotation curve Oh et al. (2015) compared to fits with cuspy (NFW) and cored profiles, which are shown in the right panel. The NFW halo has been chosen to fit the asymptotic velocity at large radii and match the median cosmological relation between and  Kamada et al. (2016). However, this profile overpredicts in the inner region. This discrepancy is a symptom of too much mass for , while the data favors a shallower cored profile with less enclosed mass. An NFW profile with alternative parameters can provide an equally good fit as the cored profile, but the required concentration is significantly smaller than preferred cosmologically Oh et al. (2015). We note not all dwarf galaxies require a density core, as we we will discuss later.

Recent high-resolution surveys of nearby dwarf galaxies have given further weight to this discrepancy. The HI Near Galaxy Survey (THINGS) presented rotation curves for seven nearby dwarfs, finding a mean inner slope  Oh et al. (2011a), while a similar analysis by LITTLE THINGS for 26 dwarfs found  Oh et al. (2015). These results stand in contrast to predicted for CDM.

However, this discrepancy may simply highlight the inadequacy of DM-only simulations to infer the properties of real galaxies containing both DM and baryons. One proposal along these lines is that supernova-driven outflows can potentially impact the DM halo gravitationally, softening cusps Navarro et al. (1996b); Oh et al. (2011b), which we discuss in further detail in §II E. Alternatively, the inner mass density in dwarf galaxies may be systematically underestimated if gas pressure—due to turbulence in the interstellar medium—provides radial support to the disk Dalcanton and Stilp (2010); Pineda et al. (2017). In this case, the observed circular velocity will be smaller than needed to balance the gravitational acceleration, as per Eq. (5), and purported cores may simply be an observational artifact.

In light of these uncertainties, LSB galaxies have become an attractive testing ground for DM halo structure. A variety of observables—low metallicities and star formation rates, high gas fractions and mass-to-light ratios, young stellar populations—all point to these galaxies being highly DM-dominated and having had a quiescent evolution de Blok and McGaugh (1997). Moreover, LSBs typically have larger circular velocities and therefore deeper potential wells compared to dwarfs. Hence, the effects of baryon feedback and pressure support are expected to be less pronounced.

Rotation curve studies find that cored DM profiles are a better fit for LSBs compared to cuspy profiles McGaugh and de Blok (1998); de Blok et al. (2001a, b); Kuzio de Naray et al. (2006, 2008). In some cases, NFW profiles can give reasonable fits, but the required halo concentrations are systematically lower than the mean value predicted cosmologically. Although early HI and long-slit H observations carried concerns that systematic effects—limited resolution (beam-smearing), slit misalignment, halo triaxiality and noncircular motions—may create cores artificially, these issues have largely been put to rest with the advent of high-resolution HI and optical velocity fields (see Ref. de Blok (2010) and references therein). Whether or not baryonic feedback can provide the solution remains actively debated Kuzio de Naray and Spekkens (2011); Oman et al. (2015); Katz et al. (2016); Pace (2016). Cored DM profiles have been further inferred for more luminous spiral galaxies as well Gentile et al. (2004); Salucci et al. (2007); Donato et al. (2009).

Although observational challenges remain in interpreting the very inner regions of galaxies, other studies have shown that small scale issues persist out to larger halo radii as well. McGaugh et al. McGaugh et al. (2007) examined rotation curves restricted to kpc for 60 disk galaxies spanning a large range of mass and types. Under plausible assumptions for , the inferred halo densities are systematically lower than predicted for CDM. Gentile et al. Gentile et al. (2007b) analyzed rotation curves of 37 spiral galaxies. For each galaxy, an NFW halo is identified by matching the total enclosed DM mass with the observed one at the last measured rotation curve point. The resulting NFW halos are overdense in the central regions, but less dense in the outer regions, compared to the DM distribution inferred observationally. These studies support the picture that cosmological NFW halos tend to be too concentrated to fit observed rotation curves.

Milky Way satellites: Stellar kinematics for dwarf spheroidal (dSph) galaxies around the MW support the existence of cored DM density profiles. These DM-dominated galaxies—unlike disk galaxies discussed above—are gas-poor, dispersion-supported systems in which the stellar kinematics are dominated by random motions. Since available observations only encode half of the full 6D phase space information (i.e., two spatial dimensions in the plane of the sky and one velocity dimension along the line of sight), there is a well-known degeneracy between halo mass and stellar velocity anisotropy, given by the parameter


where denotes the radial () and tangential () stellar velocity dispersions, which follows from the equilibrium Jeans equation. Fortunately, however, for a wide range of halo models and anisotropies, the halo mass can be robustly estimated at the half-light radius, the radius enclosing half of the luminosity of the galaxy, with little scatter Walker et al. (2009); Wolf et al. (2010).

Further studies have exploited the fact that Sculptor and Fornax dSphs have two chemo-dynamically distinct stellar subpopulations (metal-rich and metal-poor) Tolstoy et al. (2004). Both populations trace the same underlying gravitational potential—but at different radii—effectively constraining the halo mass at two distinct radii and allowing the DM halo profile to be determined. The analyses by Battaglia et al Battaglia et al. (2008), based on the Jeans equation, and by Amorisco & Evans Amorisco and Evans (2012), based on modeling stellar distribution functions, find that a cored DM profile provides a better fit to the velocity dispersion profiles of Sculptor compared to NFW. A third method by Walker & Peñarrubia Walker and Penarrubia (2011) reached a similar conclusion for both Sculptor and Fornax. Their method used a simple mass estimator to measure the slope of the mass profile, excluding a NFW profile at high significance.888Strigari et al. Strigari et al. (2014) performed a different analysis for the Sculptor galaxy and argued that an NFW profile is consistent with observations (see also Ref. Breddels and Helmi (2013)). The discrepancy could be due to different assumptions about the anisotropy of the stellar velocity dispersions. With the recent identification of three distinct stellar subpopulations in Fornax, Amorisco et al. Amorisco et al. (2013) further showed that the galaxy resides in a DM halo with a constant density core, while an NFW halo is incompatible with observations unless its concentration is unrealistically low.

Observations of kinematically cold stellar substructures in dSphs lend indirect support to the presence of DM cores. Kleyna et al. Kleyna et al. (2003) detected a stellar clump with low velocity dispersion in the Ursa Minor dSph and argued that its survival against tidal disruption is more likely in a cored—rather than cuspy—host halo. Walker et al. Walker et al. (2006) found a similar clump in the Sextans dSph, also implying a cored halo Lora et al. (2013). Lastly, the wide spatial distribution of five globular clusters in the Fornax dSph again favors a cored host halo Sanchez-Salcedo et al. (2006); Goerdt et al. (2006); Read et al. (2006); Cole et al. (2012), as dynamical friction within a cuspy halo would cause these clusters to sink to the center in less than a Hubble time.

Galaxy clusters: There is evidence that the core-cusp problem may be present for massive galaxy clusters as well. Sand et al. Sand et al. (2002, 2004) have advocated that the inner mass profile in relaxed clusters can be determined using stellar kinematics of the brightest cluster galaxy (BCG) that resides there. In combination with strong lensing data, they find a mean inner slope for the three best-resolved clusters in their sample, shallower than expected for CDM, albeit with considerable scatter between clusters Sand et al. (2004). Further studies by Newman et al. Newman et al. (2009, 2011, 2013a, 2013b) have supported these findings. A joint analysis of stellar BCG kinematics and lensing data for seven clusters with masses has found that, while the total mass density profiles are consistent with NFW, the density profiles for DM only become softer than NFW for 30 kpc Newman et al. (2013a, b). The halos are described equally well by a shallower inner slope or a cored profile with core radius kpc.

Baryon dynamics from active galactic nuclei (AGN) can be important, similar to the effect of supernovae in smaller halos. Hydrodynamical simulations of clusters by Martizzi et al. Martizzi et al. (2012, 2013) have found that repeated outflows from gas accretion and ejection by AGN can produce a cored cluster halo. On the other hand, weaker feedback prescriptions in simulations by Schaller et al. Schaller et al. (2015a)—argued to produce a more realistic stellar mass function for cluster galaxies—do not produce cored DM profiles and yield total mass profiles that are somewhat steeper than NFW, seemingly in tension with the Newman et al. results Newman et al. (2013a, b).

Other systematic effects may also be relevant for reconciling this descrepency. BCGs are dispersion-supported galaxies dominated by the stellar density in their interiors and there is a systematic degeneracy between and that is important when disentangling the DM and baryonic densities. Schaller et al. Schaller et al. (2015a) argue that radially-biased , as suggested by their simulations, can push their results into closer agreement with Refs. Newman et al. (2013a, b). On the other hand, Newman et al. Newman et al. (2009, 2013b) find that radially biased anisotropies only serve to make DM density profiles shallower in their fits. In any case, the comparison is not strictly apples-to-apples since the Newman clusters Newman et al. (2013a, b) are up to a factor of five larger than the largest simulated ones by Schaller et al. Schaller et al. (2015a).

ii.2 Diversity Problem

In CDM, hierarchical structure formation produces self-similar halos well-described by NFW profiles. Since the halo parameters (e.g.,  and ) are strongly correlated, there is only one parameter specifying a halo. For example, once the maximum circular velocity (or any other halo parameter) is fixed, the halo density profile is completely determined at all radii including the inner density cusp (up to the scatter). On the other hand, the inner rotation curves of observed galaxies exhibit considerable diversity. Galaxies of the same can have significant variation in their central densities. Any mechanism to explain the core-cusp issue must also accommodate this apparent diversity.

To illustrate this issue, Kuzio de Naray et al. Kuzio de Naray et al. (2010) fitted seven LSB galaxies with four different cored halo models, including, e.g., a pseudoisothermal profile . Fig. 5 (left) shows the central DM density —inferred by the inner slope of —versus for these galaxies. Within the sample, there is no clear correlation between the inner () and outer () parts of the halo. Moreover, the spread in can be large for galaxies with similar , up to a factor of when . The result is independent of the choice of halo model and mass-to-light ratio.

Figure 5: Left: Inferred central core density as a function of the maximum observed rotation velocity of seven low-surface-brightness galaxies. Each symbol represents a different model for the cored DM halo density profile. For a given model, is not a constant for a fixed . The small gray symbols indicate the results when a non-zero stellar mass-to-light ratio is assumed. Reprinted from Ref. Kuzio de Naray et al. (2010). Right: The total (mean) rotation speed measured at versus the maximum rotation speed for observed galaxies. Solid black line indicates CDM-only prediction expected for NFW haloes of average concentration. Thick red line shows the mean relation predicted in the cosmological hydrodynamical simulations Oman et al. (2015), and the shaded areas show the standard deviation. Data compiled in Oman et al. (2015).

Instead of fitting to a specific halo profile, Oman et al. Oman et al. (2015) parametrized the diversity of rotation curves more directly by comparing versus , which represent the inner and outer halos, respectively. Fig. 5 (right) shows the scatter in these velocities for observed galaxies (blue points) compared to the correlation expected from CDM-only halos (solid line) and CDM halos with baryons (red band). For in the range of , the spread in is a factor of for a given . For example, when , CDM (only) predicts (solid line), but observed galaxies span from to . Galaxies at the low end of this range suffer from the mass deficit problem discussed above. For these outliers, including the baryonic contribution will make the comparison worse. On the other hand, galaxies at the upper end of this range could be consistent with CDM predictions once the baryonic contribution is included Oman et al. (2015). However, the spread in the baryon distribution plays a less significant role in generating the scatter in for CDM halos since the enclosed DM mass in the cusp tends to dominate over the baryon mass.

ii.3 Missing Satellites Problem

The hierarchical nature of structure formation predicts that CDM halos should contain large numbers of subhalos Kauffmann et al. (1993). Numerical simulations predict that a MW-sized halo has a subhalo mass function that diverges at low masses as  Springel et al. (2008). Consequently, the MW should have several hundred subhalos with within its virial radius, each in principle hosting a galaxy Klypin et al. (1999); Moore et al. (1999b). However, only 11 dwarf satellite galaxies were known in the MW when the problem was originally raised in 1999, shown in Fig. 6 (left). The mismatch also exists between the abundance of observed satellites in the Local Group and that predicted in simulations. This conflict is referred as the “missing satellites problem.” We note that a similar descrepancy does not appear for galactic-scale substructure in galaxy clusters (shown in Fig. 6 (left) for the Virgo cluster).

Figure 6: Left: Abundance of subhalos within the MW (dashed) and Virgo cluster (solid) in CDM simulations, compared with the distribution of observed MW satellites (filled circles) and galaxies in the Virgo cluster (open circles). Reprinted from Ref. Moore et al. (1999b). Right: Circular velocity profiles for MW subhalos with predicted from CDM simulations (purple lines). Each data point corresponds to evaluated at the half-light radius for nine brightest MW dwarf spheroidal galaxies. Reprinted from Ref. Boylan-Kolchin et al. (2012).

One possibility is that these subhalos exist but are invisible because of the low baryon content. For low-mass subhalos, baryonic processes may play an important role for suppressing star formation. For instance, the ultraviolet photoionizing background can inhibit gas collapse into DM halos by heating the gas and reducing the gas cooling rate, which could suppress galaxy formation in halos with circular velocities less than  Thoul and Weinberg (1996); Bullock et al. (2000). In addition, after the initial star formation episode, supernova-driven winds could push the remaining gas out of the shallow potential wells of these low mass halos Dekel and Silk (1986).

The discovery of many faint new satellites in the Sloan Digital Sky Survey has suggested that as many as a factor of more dwarf galaxies could be still undiscovered due to faintness, luminosity bias, and limited sky coverage Tollerud et al. (2008); Walsh et al. (2009); Bullock et al. (2010). More recently, seventeen new candidate satellites have been found in the Dark Energy Survey Bechtol et al. (2015); Drlica-Wagner et al. (2015). Given these considerations, the dearth of MW subhalos may not be as severe as thought originally.

A similar abundance problem has arisen for dwarf galaxies in the field of the Local Volume. The velocity function—the number of galaxies as a function of their HI line widths—provides a useful metric for comparing to CDM predictions since HI gas typically extends out to large distances to probe for the halo Cole and Kaiser (1989); Shimasaku (1993). While in accord with observations for larger galaxies, the velocity function for CDM overpredicts the number smaller galaxies with  Zavala et al. (2009); Zwaan et al. (2010); Trujillo-Gomez et al. (2011). For example, Klypin et al. Klypin et al. (2015) find nearby galaxies within 10 Mpc with , while CDM predicts . Unlike the satellites, which are considerably smaller and fainter, these galaxies are relatively bright dwarf irregulars where observations are essentially complete within this volume.

One explanation for this missing dwarf problem is that HI line widths may be biased tracers for . HI measurements for many dwarf galaxies are limited to the rising part of the rotation curve and therefore do not sample the full gravitational potential of the DM halo. Whether this bias can reconcile the observed velocity function with CDM Brook and Shankar (2016); Macciò et al. (2016); Brooks et al. (2017), or if the discrepancy still persists Trujillo-Gomez et al. (2016), remains to settled.

ii.4 Too-Big-to-Fail Problem

Satellites within the Local Group: Boylan-Kolchin et al. Boylan-Kolchin et al. (2011, 2012) showed that the population of the MW’s brightest dSph galaxies, which are DM dominated at all radii, exhibit another type of discrepancy with CDM predictions. Since these satellites have the largest stellar velocities and luminosities, they are expected to live in the most massive MW subhalos. However, the most massive subhalos predicted by CDM-only simulations have central densities too large to host the observed satellites. As shown in Fig. 6 (right), simulations predict subhalos with , whereas the bright MW dSphs have stellar dispersions corresponding to CDM subhalos with . It is puzzling that the most massive subhalos should be missing luminous counterparts since their deep potential wells make it unlikely that photoionizing feedback can inhibit gas accretion and suppress galaxy formation. Hence, these substructures should be too big to fail to form stars.

Several proposed mechanisms may address the TBTF problem without invoking DM physics. First, the MW halo mass may be underestimated. The TBTF discrepancy is based on simulated MW-like halos with masses in the range  Boylan-Kolchin et al. (2011, 2012). However, since the number of subhalos scales with host halo mass, the apparent lack of massive subhalos might be accommodated if the MW halo mass is around , although Boylan-Kolchin et al. argue against this possibility (see also Ref. Wang et al. (2015) for a summary of different estimates.) Second, the MW may have less massive subhalos due to scatter from the stochastic nature of structure formation Purcell and Zentner (2012). However, a recent analysis finds that there is only chance that MW-sized host halos have a subhalo population in statistical agreement with that of the MW Jiang and Bosch (2015). In addition, similar discrepancies also exist for the brightest dwarf galaxies in Andromeda Tollerud et al. (2014) and the Local Group field Garrison-Kimmel et al. (2014), which further disfavor these explanations. Baryonic physics—including environmental effects from the MW disk—may also play a role, discussed in the next section.

Field galaxies: A similar TBTF problem also arises for dwarf galaxies in the field Ferrero et al. (2012); Papastergis et al. (2015); Garrison-Kimmel et al. (2014); Brook and Di Cintio (2015); Klypin et al. (2015). According to abundance matching, galaxies are expected to populate DM halos in a one-to-one relationship that is monotonic with mass (i.e., larger galaxies in larger halos). However, the galaxy stellar mass function, inferred by the Sloan Digital Sky Survey Baldry et al. (2008); Li and White (2009), is shallower at low mass than the halo mass function in CDM, which suggests that galaxy formation becomes inefficient for halos below  Guo et al. (2010). Hence, it is expected that most faint dwarf galaxies populate a narrow range of DM halos with , while halos with would have no galactic counterpart Ferrero et al. (2012).

Ferrero et al. Ferrero et al. (2012) find that rotation curves for faint dwarf galaxies do not support these conclusions. A large fraction of faint galaxies in their sample appear to inhabit CDM halos with masses below , which would imply that other more massive halos—which should be too big to fail—lack galaxies. On the other hand, optical galaxy counts may simply have missed a large fraction of low surface brightness objects.

Figure 7: The relation for galaxies. The colored points represent observed galaxies, and they are drawn as upper limits, as the baryonic contribution to the rotation curve is neglected. The blue line is inferred from abundance matching in CDM cosmology. Reprinted from Ref. Papastergis et al. (2015).

More recently, Papastergis et al. Papastergis et al. (2015) have confirmed the findings of Ref. Ferrero et al. (2012) by utilizing data from the Arecibo Legacy Fast ALFA 21-cm survey. Since the majority of faint galaxies are gas-rich late-type dwarfs, these observations provide a more complete census without a bias against low stellar luminosity objects. Fig. 7 shows their main results, illustrating the TBTF problem in the field.

  • The blue curve is the prediction of abundance matching between observed HI rotational velocities () and predicted maximum rotational velocities for CDM halos ().

  • The various symbols indicate the observed correlation for a sample of 194 nearby galaxies. For each galaxy, is the rotational velocity at the outermost measured radius and is determined by matching a CDM halo rotation curve at that point.

  • Since baryonic contributions to the rotation curve are neglected, each symbol is regarded as an upper limit on . Hence, the symbols are indicated by leftward arrows.

Galaxies to the right of the blue line can potentially be reconciled with abundance matching predictions since baryons can make up the difference. On the other hand, the lowest mass galaxies to the left of the blue line seem inconsistent with these arguments. Abundance matching predicts they should inhabit halos with , yet this is incompatible with their measured HI circular velocities.

Common ground with the core-cusp problem: The essence of the TBTF problem is that low-mass galaxies have gas or stellar velocities that are too small to be consistent with the CDM halos they are predicted to inhabit. At face value, the issue is reminiscent of the core-cusp/mass deficit problems for rotation curves and other observations. Thus, one way to resolve the TBTF problem is if these galaxies have reduced central densities compared to CDM halos. By generating cored profiles in low-mass halos, self-interactions may resolve this issue for the dwarf galaxies in the MW Vogelsberger et al. (2012); Zavala et al. (2013), Local Group Elbert et al. (2015), and the field Schneider et al. (2016).

ii.5 Baryon feedback

It has been more than 20 years, since the small scale “crisis” was first posed in the 1990s. Since then, there has been extensive discussion on whether all issues can be resolved within the CDM paradigm once baryonic feedback processes—gas cooling, star formation, supernovae, and active galactic nuclei—are accounted for. Here we give a brief review of feedback on galactic scales.

Cores in galaxies: As galaxies form, gas sinks into the inner halo to produce stars. The deepening gravitational potential of baryons causes the central density and velocity dispersion of DM to increase through adiabatic contraction Blumenthal et al. (1986). Although at first sight this makes the core-cusp problem worse, the situation can be very different due to non-adiabatic feedback from supernova-driven gas outflows Navarro et al. (1996b); Governato et al. (2010). If a sizable fraction of baryons is suddenly removed from the inner halo, DM particles migrate out to larger orbits. In this way, repeated bursts of star formation and outflows (followed by reaccretion) can reduce the central halo density through a purely gravitational interaction between DM and baryons. Non-adiabaticity is essential for the mechanism to work. This is achieved by assuming a “bursty” star formation history with variation in the star formation rate over time scales comparable to the dynamical time scale of the galaxy Teyssier et al. (2013). (See Refs. Pontzen and Governato (2014, 2012) for a pedagogical explanation.)

Governato et al. Governato et al. (2010) argued that bursty star formation in dwarf galaxies could be connected to another long-standing puzzle in galaxy formation: bulgeless disk galaxies. If strong outflows are necessary to remove low angular momentum gas to prevent bulge formation in certain galaxies, they may also induce DM cores. High-resolution hydrodynamical simulations show that supernova feedback can generate cores in CDM halos for dwarf galaxies Governato et al. (2010). Notably, the shallow inner slope of the DM distribution inferred from THINGS and LITTLE THINGS dwarf galaxies could be consistent with CDM halos once feedback is included (except DDO 210) Oh et al. (2011b); Governato et al. (2012); Oh et al. (2015) (see, e.g., Fig. 6 of Oh et al. (2015)).

Further studies by Di Cintio et al. Di Cintio et al. (2014a, b) investigated the effect of feedback as a function of halo mass. Their sample includes 31 galaxies from hydrodynamical simulations spanning halo mass . The corresponding stellar masses are fixed by abundance matching, with stellar-to-halo mass ratio in the range . Fig. 8 (left) shows how the DM inner density slope (measured at of the virial radius) depends strongly on . Di Cintio et al. Di Cintio et al. (2014a) conclude the following points:

  • Maximal flattening of the inner DM halo occurs for . According to abundance matching, this corresponds to stellar mass , halo mass , and asymptotic circular velocity . The THINGS dwarfs are right in the “sweet spot” for supernova feedback to be maximally effective.

  • Supernova feedback is less effective in larger halos, which (by abundance matching) correspond to larger values of . Despite more star formation available to drive feedback, the deeper potential well for the halo suppresses the effect. For these galaxies, adiabatic contraction dominates and halos are cuspy.

  • When the ratio is less than , there is too little star formation for feedback to affect the inner DM density. This implies that halos hosting galaxies with remain cuspy, consistent with earlier findings Governato et al. (2012)

These results have been further confirmed by the Numerical Investigation of Hundred Astrophysical Objects (NIHAO) simulations Tollet et al. (2016), where the same stellar feedback model was used for a larger halo sample. Other simulations with similar feedback prescriptions have also confirmed core formation in halos Teyssier et al. (2013); Madau et al. (2014).

Figure 8: Left: Slope of DM density profile vs the ratio of the stellar mass to the halo mass, predicted in CDM hydrodynamical simulations with strong stellar feedback. Different colors denote the feedback schemes, and different simulated galaxies are represented with symbols. The best fitted function is indicated with the dashed curve. Reprinted from Ref. Di Cintio et al. (2014a). Right: Observed rotation curves of dwarf galaxy IC 2574 (blue), compared to simulated ones from Oman et al. Oman et al. (2015) (green band) and Oh et al. Oh et al. (2011b) (open circles and triangles). Reprinted from Ref. Oman et al. (2015).

Cosmological simulations with Feedback In Realistic Environments (FIRE) Hopkins et al. (2014) have emphasized that core formation in DM halos is tightly linked with star formation history O�orbe et al. (2015); Chan et al. (2015). Oñorbe et al. O�orbe et al. (2015) targeted low mass galaxies relevant for the TBTF problem and showed that kpc-sized cores may arise through feedback. However, cores form only when star formation remains active after most of the halo growth has occurred (). Otherwise, even if a core forms at early times, subsequent halo mergers may erase it if feedback has ceased. Lower mass galaxies, such as ultra-faint dwarfs, have too little star formation to affect the DM halo.

Chan et al. Chan et al. (2015) explored a wider mass range using the FIRE simulations, and . The DM density profiles become shallow for due to strong feedback, with the maximal effect at . The result is broadly consistent with Governato et al. Governato et al. (2012) and Di Cintio et al. Di Cintio et al. (2014a). Chan et al. also found that large cores can form only if bursty star formation occurs at a late epoch when cusp-building mergers have stopped, as pointed out in O�orbe et al. (2015).

Although baryonic feedback may reconcile galactic rotation curves with the CDM paradigm (see, e.g., Refs. Brook (2015); Katz et al. (2016)), questions yet remain:

  • Can feedback generate ultra-low density cores? Feedback-created cores are limited to the inner kpc where the star formation rate is high. However, some galaxies, such as dwarf IC 2574, are extreme outliers with huge core sizes and mass deficits that are far beyond what feedback can do. Fig. 8 (right) shows the rotation curve for IC 2574, together with simulated ones with similar values of . The observed core size is kpc Oh et al. (2011a)—set by the region over where rises linearly with radius—which is too large to be created with bursty feedback (open symbols) Oh et al. (2011b) or smoother implementations (green band) Oman et al. (2015). Fig. 5 (right) shows that many galaxies have inner mass deficits out to 2 kpc and it is challenging for feedback to remove enough DM from their central regions to be consistent with observations.

  • Does feedback explain the full spectrum of observed rotation curves? Galaxies exhibit a broad spread in rotation curves, encompassing both cored and cuspy profiles Oman et al. (2015). While some systems have large cores, there are galaxies with the same which are consistent with a cuspy CDM halo. For fixed , the spread in the velocity at 2 kpc is a factor of , as shown in Fig. 5 (right). It is unclear whether feedback prescriptions can account for such a scatter.

  • Is bursty feedback required? Since star formation is far below the resolution of simulations, baryon dynamics depend on how feedback is modeled, especially the density threshold for star formation.999In various simulations, the gas density thresholds for star formation are taken to be (Brooks et al. Brooks et al. (2013)), (FIRE, Chan et al. Chan et al. (2015)), (FIRE-2, Wetzel et al. Wetzel et al. (2016)), and (Oman et al. Oman et al. (2015)). Large thresholds lead to bursty star formation since energy injection to the dense medium causes strong outflows, which disturb the potential violently to generate DM cores Governato et al. (2010, 2012). However, Oman et al. Oman et al. (2015) have analyzed galaxies from the EAGLE and LOCAL GROUP simulation projects—which adopt a far smaller threshold—and find that feedback is negligible. These simulations have star formation occuring throughout the gaseous disk—without sudden fluctuations in the gravitational potential needed to produce DM cores—and yet produce galaxies that are consistent with other observational constraints.

  • What is the epoch of core formation? Detailed comparisons between hydrodynamical simulations with bursty feedback would be useful to understand what systematic differences may be present (if any). One potential point of divergence is the epoch when star formation bursts need to occur to yield cores. Pontzen & Governato Pontzen and Governato (2012) argue that DM cores may form due to feedback at an early time . However, Refs. O�orbe et al. (2015); Chan et al. (2015) find that cores remain stable only if outflows remain active at later times, , once halo growth has slowed. This question is important for connecting observational tracers of starbursts in galaxies to feedback-driven core formation (e.g., Ref. Weisz et al. (2012); Kauffmann (2014)).

Substructure: A number of studies have found that baryon dynamics may solve the missing satellites and TBTF problem within CDM, independently of whether star formation is bursty or not. Sawala et al. Sawala et al. (2016) performed a suite of cosmological hydrodynamical simulations of 12 volumes selected to match the Local Group. These simulations adopt a smooth star formation history, as in Ref. Oman et al. (2015), and do not yield cored profiles. Regardless, the number of satellite galaxies is reduced significantly in MW and M31-like halos (within 300 kpc), as well as in the broader Local Group (within 2 Mpc), in better accord with observations compared to the expectation from DM-only simulations. Supernova feedback and reionization deplete baryons in low mass halos and only a subset of them can form galaxies. In addition, the most massive subhalos in MW-like galaxies have an inner mass deficit due to ram pressure stripping and a suppressed halo growth rate due to the baryon loss. These massive subhalos could be consistent with kinematical observations of the MW dSphs, solving the TBTF problem in the MW, within measurement uncertainties Fattahi et al. (2016).

Earlier studies by Zolotov et al. Zolotov et al. (2012) and Brooks et al. Brooks and Zolotov (2014); Brooks et al. (2013) reached a similar conclusion in simulations with a bursty star formation scheme, as in Ref. Governato et al. (2012) (see Ref. Stinson et al. (2006) for details). Strong feedback produces cored profiles in subhalos, which in turn enhances tidal effects from the stellar disk of host galaxies (see also Penarrubia et al. (2010)). More recently, FIRE simulations in the Local Group environment by Wetzel et al. Wetzel et al. (2016) also showed that the population of satellite galaxies with does not suffer from the missing satellites and TBTF problems.

Too-big-to-fail in the field: Papastergis & Shankar Papastergis and Shankar (2016) argued that the TBTF problem for field dwarf galaxies cannot be solved in CDM even with baryonic feedback effects. They adapted the halo velocity function proposed in Sawala et al. Sawala et al. (2016), which includes the effects of baryon depletion and reionization feedback, and the modified CDM halo density profile due to supernova feedback from Di Cintio et al. Di Cintio et al. (2014a). While reionization effectively suppresses star formation for halos with , the TBTF problem here concerns halos with , which are too massive to be affected significantly. In addition, cored profiles may help alleviate the tension for galaxies whose stellar kinematics are measured only in the very inner region, but not those with observed at large radii.

More recently, Verbeke et al. Verbeke et al. (2017) argued that the TBTF problem in the field may be resolved by noncircular motions in HI gas. According to their simulations, HI kinematics is an imperfect tracer for mass in small galaxies due to turbulence (as pointed out in Refs. Dalcanton and Stilp (2010); Pineda et al. (2017)). In particular, galaxies with km/s (see Fig. 7) may be consistent with living in larger halos than would otherwise be inferred without properly accounting for this systematic effect.

Summary: The predictions of CDM cosmology can be modified on small scales when dissipative baryon physics is included in simulations. Strong baryonic feedback from supernova explosions and stellar winds disturb the gravitational potential violently, resulting in a shallow halo density profile. Reionization can suppress galaxy formation in low mass halos and strong tides from host’s stellar disk can destroy satellite halos. These effects can also be combined in shaping galaxies. For example, if feedback induces cores in low mass halos, galaxy formation is more effectively suppressed by reionization and the halo is more vulnerable to tidal disruption. In simulations, the significance of these effects depends on the specific feedback models, in particular, the gas density threshold for star formation. It seems that strong feedback—leading to the core formation and significant suppression of star formation in low mass halos—is not required for reproducing general properties of observed galaxies in simulations (see, e.g., Refs. Oman et al. (2015); Sawala et al. (2016); Sales et al. (2017)). Thus, it remains unclear to what degree baryon dynamics affect halo properties in reality. Moreover, cored profiles generated by feedback, as proposed in Refs. Di Cintio et al. (2014a, b), may be inconsistent with the correlations predicted in CDM cosmology, namely the mass-concentration and abundance matching relations Pace (2016) (however, see Ref. Katz et al. (2016)). Thus, it remains an intriguing possibility that small scale issues may imply the breakdown of the CDM paradigm on galactic scales, as we will focus on in the rest of this article.

Iii N-body simulations and SIDM halo properties

N-body simulations have been the primary tools for understanding the effect of self-interactions on structure Moore et al. (2000); Yoshida et al. (2000a); Burkert (2000); Kochanek and White (2000); Yoshida et al. (2000b); Dave et al. (2001); Colin et al. (2002); Vogelsberger et al. (2012); Rocha et al. (2013); Peter et al. (2013); Zavala et al. (2013); Elbert et al. (2015); Vogelsberger et al. (2014); Fry et al. (2015); Dooley et al. (2016). Early SIDM simulations used smoothed particle hydrodynamics to model DM collisions Moore et al. (2000); Yoshida et al. (2000a). This approach treats SIDM as an ideal gas described by fluid equations, which are valid in the optically-thick regime where the mean free path is much smaller than typical galactic length scales. Due to efficient thermalization, SIDM halos in this context form a singular isothermal profile, , which is steeper than collisionless CDM halos and exacerbates the core-cusp issue rather than solving it Moore et al. (2000); Yoshida et al. (2000a).101010The fluid equations approach has been used as a semi-analytical framework for modeling SIDM halos under simplifying assumptions (spherical symmetry and self-similar evolution) Balberg et al. (2002); Ahn and Shapiro (2005); Koda and Shapiro (2011). In fact, this model agrees well with N-body simulations for isolated halos, even for the optically-thin regime that is beyond the validity of the fluid approximation; however, the model is not yet able to reproduce N-body results for cosmological halos due to departure from self-similar evolution Koda and Shapiro (2011).

Consequently, most SIDM simulations have focused the more promising case of self-interactions in the optically-thin regime, with cross sections spanning  Burkert (2000); Kochanek and White (2000); Yoshida et al. (2000b); Dave et al. (2001); Colin et al. (2002); Vogelsberger et al. (2012); Rocha et al. (2013); Peter et al. (2013); Zavala et al. (2013); Elbert et al. (2015); Vogelsberger et al. (2014); Fry et al. (2015); Dooley et al. (2016). In this case, is larger than the typical core radius over which self-interactions are active. Here we discuss these simulations and their implications for astrophysical observables.

The majority of simulations assume a contact-type interaction where scattering is isotropic and velocity-independent, described by a fixed . These studies have converged on to solve the core-cusp and TBTF issues on small scales, while remaining approximately consistent with other astrophysical constraints on larger scales, such as ellipticity measurements.111111This value is consistent with Spergel & Steinhardt’s original estimate, , based on having in the range with DM density as in the local solar neighborhood Spergel and Steinhardt (2000). However, more recent studies based on massive clusters disfavor the constant cross section solution, prefering somewhat smaller values on these scales Kaplinghat et al. (2016); Elbert et al. (2016).

Before we discuss these topics in greater detail, we emphasize a few key points:

  • Self-interaction cross sections are generically velocity-dependent, as predicted in many particle models for SIDM (see §VI). Therefore, constraints on must be interpreted as a function of halo mass, since DM particles in more massive halos will have larger typical velocities for scattering. (“Halo mass” refers to the virial mass, for which we note that different studies have adopted somewhat different conventions in defining.)

  • The effect from self-interactions on a halo is not a monotonic function of , which can be understood as follows. If DM is collisionless, its velocity dispersion in the halo is peaked near the scale radius , while particles in both the center and outskirts of the halo are colder. Once collisions begin to occur, only the inner region of the halo is in thermal contact. The center of the halo forms a core that grows larger as more heat flows inward due to the positive temperature gradient. Eventually, once thermal contact is reached with the outskirts of the halo where the temperature gradient is negative, heat is lost outward and gravothermal collapse of the core ensues Lynden-Bell and Wood (1968); Colin et al. (2002).

  • Our discussion mostly focuses on DM-only simulations and is subject to the usual caveat that baryons have been ignored. While baryonic feedback may explain away the issues that SIDM aims to address, this depends strongly on the prescription for feedback adopted. Without a more definitive understanding, making quantitative statements about the evidence for SIDM remains limited by these systematics. At the end of this section, we discuss recent simulations for SIDM including baryonic dynamics Vogelsberger et al. (2014); Fry et al. (2015).

  • The present section focuses on the quasi-equilibrium structure of SIDM halos. We discuss simulations of merging halos, such as Bullet Cluster-like systems, in a later section (§V).

iii.1 Implementing self-interactions in simulations

In N-body simulations, DM particles are represented by “macroparticles” of mass , each representing a phase space patch covering a vast number of individual DM particles. Self-interactions, assumed to be a short-range force on galactic scales, are treated using a Monte Carlo approach Burkert (2000). Two macroparticles scatter if a random number between is less than the local scattering probability within a given simulation time step , which must be small enough to avoid multiple scatterings. Outgoing trajectories preserve energy and linear momentum121212Angular momentum is not conserved for individual scatterings, due to finite separations between macroparticles, but any nonconservation is expected to average to zero over the halo since separation vectors are randomly oriented Kochanek and White (2000)., with a scattering angle chosen randomly, under the assumption that scattering is isotropic in the center-of-mass frame.

Different simulations have adopted different prescriptions for determining the scattering probability. Many earlier studies used a background density method Burkert (2000); Kochanek and White (2000); Yoshida et al. (2000b); Colin et al. (2002). In this approach, the probability for an individual macroparticle to scatter is


where is the mean local background density, spatially averaged over its nearest neighbors, and is the relative velocity between and one or more of its neighbors. If a scattering occurs, particle is paired up with a nearby recoiling partner .

An alternative approach treats scattering as a pair-wise process between macroparticles with velocities  Dave et al. (2001); Rocha et al. (2013). The probability is


where represents the target density from to be scattered by . Rocha et al. Rocha et al. (2013) provide an insightful derivation of Eq. (8) starting from the Boltzmann collision term. Each macroparticle , centered at , is coarse-grained over a finite spatial patch using a cubic spline kernel with smoothing length  Monaghan and Lattanzio (1985).131313The smoothing kernel is defined as , where (9) is normalized as and has dimensions of number density. The resulting collision rate between patches is determined by the overlap integral


Numerical convergence studies show that must not be too small compared to the mean particle spacing , requiring  Rocha et al. (2013). Simulations in Ref. Rocha et al. (2013) have fixed such that throughout the inner halo where the self-interaction rate is large. However, self-interactions are artificially quenched in the outer halo where , although scattering is not expected to be relevant here due to simple rate estimates.

Alternatively, Vogelsberger et al. Vogelsberger et al. (2012) follow a hybrid between these approaches. First, is computed in Eq. (8) with . Here, can be interpretted as the “background” density from at the position of (which follows from Eq. (10) by setting the smoothing kernel for to be a delta function). The smoothing length is also taken to be much larger than the Ref. Rocha et al. (2013) approach, with adjusted dynamically such that particles are in range of . Since , scatterings are less localized. The total probability for to scatter is determined by , and if a scattering occurs, the scattering partner is chosen with probabilty weighted by . It is unknown what differences, if any, may arise between these various methods (although both Refs. Rocha et al. (2013); Vogelsberger et al. (2012) are in mutual agreement with the Jeans modeling approach Kaplinghat et al. (2016) within 10–20%).

Next, we turn to convergence issues. Since self-interactions affect mainly the innermost radii in the halo, simulations must have sufficient resolution to robustly model these dynamics. It is well-known that coarse-graining introduces an artificial relaxation process due to two-body gravitational scattering that can alter the inner halo profile, with a timescale that scales with particle number,  Binney and Tremaine (1987). Power et al. Power et al. (2003) showed that this process is the main limit for resolving the innermost halo structure for collisionless CDM. The innermost radius of convergence, termed the Power radius, occurs where there are too few enclosed particles, , such that becomes shorter than the Hubble time. This radius is determined by the condition , where is the enclosed mean DM density contrast relative to the critical density Power et al. (2003).

In fact, SIDM halos are more robust and better converged below the Power radius compared to their CDM counterparts. This is sensible since the rate for self-interactions is larger—by construction, the goal is to have more than one self-interaction per Hubble time—than the rate for gravitational scattering Vogelsberger et al. (2012); Elbert et al. (2015).

At the same time, the cosmological environment at the outermost radii is also important for the SIDM halo. Its thermal evolution is affected by heating from mergers and infall, which slows the gravothermal collapse of cores. Hence, simulations of isolated halos are not sufficient for SIDM, as we discuss below.

iii.2 Halo density profiles

Cosmological simulations of SIDM halos show a mass deficit at small radii compared to collisionless CDM halos, provided self-interactions are in the optically-thin regime Dave et al. (2001); Yoshida et al. (2000b). Here we discuss these results and the implications for .

Galactic scales: Davé et al. Dave et al. (2001) performed the first cosmological SIDM simulations targeting dwarf galaxies (). Their simulation volume contained several dwarf galaxies, resolved down to scales, simulated with and . Their results preferred in order to produce galaxies with core densities , broadly consistent with observed galaxies Firmani et al. (2000), while yielded densities a few times larger but still viable. Moreover, for their largest simulated halo (), no evidence of core collapse was seen with cross sections as large as .

Figure 9: Left: Density profiles for halo with mass (dubbed “Pippin”) from DM-only simulations with varying values of . Right: Rotation curves for Pippin halo with are broadly consistent with measured stellar velocities (evaluated at their half-light radii) for field dwarf galaxies of the Local Group. Reprinted from Ref. Elbert et al. (2015).

More recently, Elbert et al. Elbert et al. (2015) simulated two (cosmological) dwarf galaxies, resolved down to scales and for many values of . Fig. 9 (left) shows their results for one dwarf (“Pippin”) and demonstrates that central density is not a monotonic function of . For , self-interactions are predominantly in the core-growth regime (heat flowing in), and the central density decreases with increasing . However, a larger self-interaction rate, , leads to an increasing central density, indicating this halo has entered core collapse. Nevertheless, core collapse is mild. Density profiles with , spanning two orders of magnitude, vary in their central densities by only a factor of . Comparing with data for field dwarfs in the Local Group, Fig. 9 (right) shows that predicted SIDM rotation curves for are consistent with the velocities and half-light radii inferred from several observed galaxies. This illustrates not only how SIDM affects both the core-cusp and TBTF problems simultaneously, but that need not be fine-tuned to address these issues.

The conclusion from these studies is that can produce cores needed to resolve dwarf-scale anomalies Elbert et al. (2015). However, the upper limit on at these scales—due to core collapse producing a too-cuspy profile—remains unknown.

Cluster scales: Next, we turn to clusters (. The first cosmological simulations at these scales were performed by Yoshida et al. Yoshida et al. (2000b), which studied a single halo for , , and . More recently, Rocha et al. Rocha et al. (2013) performed simulations targeting similar scales, but over much larger cosmological volume, for and . The best-resolved halos in their volume span . For , the central density profiles are clearly resolved for the Yoshida halo and for Rocha halos. On cluster scales, SIDM halos have radius cores and central densities . For , the simulations lack sufficient resolution to fully resolve the cored inner halo, though radius cores seem a reasonable estimate. For , the Yoshida halo has a similar density profile compared to , although the former is considerably more spherical (ellipticity is discussed below).

It is important to note that SIDM halos exhibit variability in their structure. Within the Rocha et al. Rocha et al. (2013) halo sample, SIDM halos, with fixed and fixed , show an order-of-magnitude scatter in their central densities. The dwarf halo samples from Davé et al. Dave et al. (2001) show a similar scatter in central density, albeit with lower resolution. This variation reflects the different mass assembly histories for different halos.

Strong gravitational lensing data has been used to constrain the core size and density in clusters relevant for SIDM Firmani et al. (2000, 2001); Wyithe et al. (2001); Meneghetti et al. (2001). Mass modeling of cluster CL0024+1654 found a cored DM profile with radius kpc Tyson et al. (1998)—quite similar to the Yoshida halo—which was interpretted as evidence for self-interactions at these scales Firmani et al. (2000, 2001). However, interpretation for this particular cluster is complicated by the fact that it has undergone a recent merger along the line of sight (see Ref. Umetsu et al. (2010) and references therein).

Meneghetti et al. Meneghetti et al. (2001) placed the strongest constraint on cluster cores by examining the ability of SIDM halos (specifically, the halo from Yoshida et al. Yoshida et al. (2000b, a)) to produce “extreme” strong lensing arcs, i.e., radial arcs or giant tangential arcs. Giant tangential arcs, with length-to-width ratio , are present in many lensing observations Wu and Hammer (1993), but simulated SIDM halos with or lack sufficient surface mass density, limiting arcs to . Observations of radial arcs pose a more severe constraint. Since they do not occur even for the SIDM halo with (as well as for larger ), Meneghetti et al. conclude on cluster scales. However, there are several caveats to keep in mind (as acknowledged in Meneghetti et al. (2001)). First, due to the aforementioned variation in SIDM density profiles, constraints based on a single simulated halo require caution. Second, and more importantly, the simulated SIDM halo does not include the baryonic density from a central galaxy. Recent studies of massive clusters by Newman et al. Newman et al. (2013b, a)—two of which exhibit radial arcs (MS2137-23 and A383)—demonstrate that an radius DM core is consistent with lensing data provided the baryonic mass is included. The important point is that the total density profile is well-described by an NFW profile, which is well-known to permit radial arcs Bartelmann (1996), even though the DM density by itself may have a cored profile. Thus, we conclude that cluster cores are consistent with , but is excluded. (We make these statements more precise in §IV).

Isolated vs cosmological simulations: Lastly, we note that simulations of isolated SIDM halos can evolve much differently than cosmological halos. Kochanek & White Kochanek and White (2000) found that cores in isolated halos can be short-lived, collapsing very soon after formation and evolving toward a steeper profile. For example, an isolated halo similar to the Pippen halo with is expected to form a core and recollapse all within , in contrast to results from Elbert et al. Elbert et al. (2015). On the other hand, isolated halos with smaller cross sections can have cores that persist over a Hubble time. This points toward the importance of cosmological infall for mitigating gravothermal collapse—especially for larger values of —since the influx of high entropy DM particles slows energy loss from the core. In fact, it is the occasional violent major merger, rather than the smoother continuous infall of smaller clumps, which is predominantly responsible for “resetting the clock” for an otherwise collapsing core Yoshida et al. (2000b); Colin et al. (2002). This further supports the fact that SIDM halos are expected to have scatter in their structure that reflects the stochastic halo assembly process Colin et al. (2002); Brinckmann et al. (2017).

In addition, core collapse for isolated halos is dependent on the initial density profile, which must be set by hand as an initial condition for non-cosmological simulations. For example, SIDM cores evolving from an initial Hernquist profile, as assumed in Ref. Kochanek and White (2000), collapse twice as quickly relative to an initial NFW profile Koda and Shapiro (2011).

iii.3 Halo shapes: ellipticity

Self-interactions are expected to make DM halos more spherical compared to triaxial collisionless CDM halos, at least in the inner regions where the scattering rate is largest. In fact, halo shape observations of elliptical galaxies and clusters have provided some of the most stringent constraints on self-interactions that exist in the literature. These limits are based on the assumption that one scattering per particle is sufficient to substantially affect the ellipticity of a halo. Early simulations by Davé et al. Dave et al. (2001) supported this conclusion. For their largest and best-resolved halos (), SIDM halos yield a minor-to-major axis ratio at radii where at least one scattering has occurred, while CDM halos have at these radii. However, these conclusions have been revisited in light of recent simulations with larger halo masses and statistics, as we now discuss.

Cluster ellipticity: One of the strongest quoted limits on SIDM is due strong lensing measurements of cluster MS2137-23, based on its apparent ellipticity at distances down to kpc Mellier et al. (1993). Using these observations and assuming the halo should be spherical where Gyr, Miralda-Escudé obtained a limit on cluster scales Miralda-Escude (2002), which utterly rules out a contact-type interaction for SIDM. However, it is crucial to clarify the extent to which these arguments are borne out in N-body simulations.

More recently, Peter et al. Peter et al. (2013) revisted these issues in detail. Their study (companion to Rocha et al. Rocha et al. (2013), discussed above) involves a sample of simulated SIDM halos in the range with , , and . They conclude that the Miralda-Escudé limit at is far overestimated, due to several reasons:

  • For (close to the Miralda-Escudé limit), ellipticities for SIDM and CDM halos were found to be virtually identical, at least down the inner as limited by resolution. Although MS2137-23 is a factor of four larger than the largest Peter et al. halo, large deviations from CDM should be visible in this halo at kpc, which matches the same scattering rate in MS2137-23 at kpc, if Miralda-Escudé’s argument is correct.

  • Even for larger cross sections, SIDM halos are not spherical at the radius where one or two scatterings have occured, but rather remain somewhat elliptical with a median value of for halos in the range . Brinckmann et al. Brinckmann et al. (2017) have largely corroborated these results for larger halo masses comparable to MS2137-23.

  • Differing mass assembly histories lead to scatter in at the for individual halos, which complicates drawing a limit based on a single system.

  • Lensing observables are sensitive to the projected mass density along the line of sight. Thus, the surface density at small projected radius includes contributions from the outer halo, which may remain elliptical since self-interactions are not efficient.

Peter et al. conclude that even is not excluded by strong lensing for MS2137-23.141414This conclusion is further supported by the Jeans analysis in Sec. IV below, which combines lensing and stellar kinematics data to show MS2137-23 is consistent with . See Fig. 12 (right).

Statistical studies of ensembles of lensing observations, as opposed to single measurements, offer a more powerful probe of self-interactions. To illustrate the potential of this approach, Peter et al. Peter et al. (2013) consider a subset of five clusters from the Local Cluster Substructure Survey (LoCuSS) Richard et al. (2009), which are chosen based on having a parametrically similar surface density profile compared to their five most massive simulated SIDM halos. The distribution of ellipticities for SIDM halos with is consistent with the LoCuSS observations, while those with are not. Peter et al. conclude that these data tentatively suggest , albeit with caution due to the limited statistics, unknown selection bias, and lack of baryons included in the SIDM lens modeling. Indeed, for , noticible differences in ellipticity between SIDM and CDM halos become apparent at only small radii where the stellar density becomes important Peter et al. (2013); Brinckmann et al. (2017).

Elliptical galaxies: These massive systems offer the opportunity to constrain self-interactions in halos on scales. Constraints on SIDM have focused on the isolated elliptical galaxy NGC 720 Feng et al. (2010a, 2009); Lin et al. (2012), which has  Humphrey et al. (2006), based on X-ray shape measurements from the Chandra telescope Buote et al. (2002). Buote et al. Buote et al. (2002) have shown that the X-ray isophotes remain elliptical at least down to projected radii kpc. (At smaller radii, shape measurements suffer from systematic uncertainties due to point source subtraction.) Since the X-ray emissivity scales as , where the gas density has a fairly steep radial dependence, emission along the line of sight is strongly weighted toward physical radii near the projected radius (in contrast with lensing measurements, as described above). To determine the shape of the DM density, Buote et al. treat the gas as a single isothermal component in hydrostatic equilibrium and model the total mass density with a spheroidal profile that is either NFW, Hernquist Hernquist (1990), or isothermal (). While the isothermal model provides the best fit, all models (whether prolate or oblate) produce a similar ellipticity .

Constraints from NGC 720 have been based on the assumption that one self-interaction per particle within the inner 5 kpc over 10 Gyr is enough sphericalize the inner halo Feng et al. (2010a, 2009); Lin et al. (2012). This yields a stringent constraint, , assuming a mean DM density and relative velocity km/s Humphrey et al. (2006); Feng et al. (2010a).

However, the simulations by Peter et al. Peter et al. (2013) show such constraints to be substantially overestimated as well. As above, SIDM halos retain ellipticity even where . Based on their sample of simulated SIDM halos with mass , the distribution of ellipticities for is perfectly consistent with , and even halos are marginally allowed. On the other hand, the mean central densities for SIDM halos with are typically too small compared to NGC 720 (by a factor of a few), while provides better agreement. But before any robust limit can be made, it is essential to consider the gravitational influence of baryons on the DM density Kaplinghat et al. (2014a). If baryons dominate the total mass density within the inner kpc (as in the case for several elliptical galaxies that have received detailed mass modeling Humphrey et al. (2006)), the SIDM density profile can be steeper and more elliptical than expected without baryons.

iii.4 Substructure

In the collisionless CDM paradigm, hierarchical structure formation leads to an abundance of substructure within halos Kauffmann et al. (1993). Observationally, these “halos within halos” manifest as dwarf galaxies around larger galaxies, or galaxies within groups or clusters. Self-interactions tend to erase substructure, provided the scattering rate is sufficiently large, through two effects Spergel and Steinhardt (2000). First, self-interactions within subhalos lead to density profiles that are less concentrated and more prone to tidal disruption. Second, self-interactions lead to evaporation of subhalos via ram pressure stripping as they pass through their host halo, since the former have a much lower velocity dispersion compared to the latter. Suppressing the subhalo mass function on MW scales is relevant for addressing the missing satellites problem Moore et al. (1999b); Klypin et al. (1999), while reducing the central density profiles of subhalos is relevant for addressing the TBTF problem Boylan-Kolchin et al. (2011, 2012). However, substructure on cluster scales must be preserved Moore et al. (1999b); Gnedin and Ostriker (2001).

Figure 10: MW-like DM halo and its substructure for three scenarios: collisionless CDM (left); SIDM with a large, constant cross section (center); and SIDM with velocity-dependent scattering (right). Velocity-dependent SIDM model has at (dwarf scales) and for (MW and larger scales). Each panel shows projected densities for a cube. Reprinted from Ref. Vogelsberger et al. (2012).

Milky Way substructure: Early simulations by Davé et al. Dave et al. (2001) reported modest reductions () in the subhalo mass function below for and , which could help, but not alleviate, the missing satellites problem. However, more recent, higher resolution SIDM simulations have reached a more pessimistic conclusion Colin et al. (2002); Vogelsberger et al. (2012); Rocha et al. (2013); Zavala et al. (2013); Dooley et al. (2016). Fig. 10 shows the projected densities for a MW-like halo from Ref. Vogelsberger et al. (2012) for different DM models. For constant cross sections of or less, the subhalo mass function remains unchanged compared to collisionless CDM Zavala et al. (2013), which is illustrated in Fig. 10 (left). For a larger cross section of , the subhalo mass function could be reduced by for subhalos below , while the number of larger subhalos remains unaffected. This scenario corresponds to Fig. 10 (center). While it is evident that substructure is suppressed—particularly for smaller halos located nearer to the center of the host halo—even this modest reduction in substructure comes at a price of making the host halo spherical out to distances . This scenario is excluded according to the ellipticity constraints discussed above (albeit for halos a factor of a few more massive than the MW).

Alternatively, self-interactions may be velocity-dependent, which effectively allows the cross section to vary across halo mass scales. Fig. 10 (right) shows a MW-like SIDM halo with velocity-dependent self-interactions Vogelsberger et al. (2012). The cross section chosen, motivated by classical scattering from a Yukawa potential Feng et al. (2010a); Loeb and Weiner (2011), is sizable on dwarf scales but suppressed on MW and larger scales. It is clear from Fig. 10 (right) that the subhalo mass function is indistinguishable from collisionless CDM. Even though the scattering rate is large on dwarf scales, this does not translate into evaporation of substructure since the relative velocity between the subhalo and its host halo is set by the velocity dispersion of the latter, for which the cross section is suppressed.

Despite being one of its original motivations Spergel and Steinhardt (2000), the conclusion is that SIDM cannot solve the missing satellites problem.151515This conclusion only applies to the minimal SIDM scenario where the only interaction is elastic scattering between DM particles. Nonminimal variations of SIDM can have a substantial impact on substructure (see §VII). Having a constant cross section around is insufficient to erase MW substructure Colin et al. (2002); D’Onghia and Burkert (2003). Reducing substructure requires a much larger cross section on MW scales, which is excluded based on halo shape constraints, while merely having a large cross section on dwarf scales is insufficient to erase substructure Vogelsberger et al. (2012); Zavala et al. (2013). However, allowed SIDM models can still impact the stellar mass function for satellite galaxies Dooley et al. (2016). SIDM subhalos, while negligibly impacted by DM evaporation, have cored density profiles, resulting in a stellar density that is less tightly bound and more prone to tidal stripping.

Figure 11: Circular velocity profiles for 15 most massive subhalos for three scenarios shown in Fig. 10: collisionless CDM (left); SIDM with (center); and SIDM with velocity-dependent scattering (right). Data points show the inferred circular velocities at the half-light radii for MW dwarf spheroidal galaxies. Reprinted from Ref. Vogelsberger et al. (2012).

Next, we turn to the TBTF problem. For collisionless CDM, the most massive subhalos within a simulated MW-like halo are too dense, with too-large predicted stellar velocity profiles, to match observed velocity dispersions for the MW dSphs Boylan-Kolchin et al. (2012). Despite having little effect on the abundance or total mass of subhalos, self-interactions can reduce their central densities and thereby reduce their velocity profiles in accord with observations, as shown in Fig. 11. Self-interactions at the level of (or the velocity-dependent models considered therein) yield a reduced central density of  Vogelsberger et al. (2012); Zavala et al. (2013). These SIDM scenarios are consistent with the observed stellar kinematics for the MW dSphs Strigari et al. (2008); Wolf et al. (2010), while for self-interactions are insufficient Zavala et al. (2013).

Cluster-scale substructure: In contrast to MW scales, there is no “missing galaxies” problem for clusters. Moore et al. Moore et al. (1999b) showed that the abundance of substructure within the Virgo cluster is well-described by simulations of a collisionless CDM halo with mass . However, this effect does not provide a strong constraint on self-interactions. SIDM simulations with find only a modest effect from subhalo evaporation on cluster scales, while observables related to the central cluster densities and core sizes are in principle much more stringent Rocha et al. (2013).

Using analytic estimates, Gnedin & Ostriker Gnedin and Ostriker (2001) excluded between based on the effect of DM evaporation on elliptical galaxies in cluster halos. As an elliptical galaxy passes through its host cluster halo, stars in the elliptical galaxy expand adiabatically as DM mass is lost through self-interactions, while the luminosity remains unchanged. This causes a shift in the fundamental plane for elliptical galaxies found in clusters and in the field, since the latter do not experience evaporation, and no significant environmental dependence is observed Kochanek et al. (2000).161616More recent studies have found a weak dependence to the fundamental plane intercept dependent on the local galaxy density. However, this dependence correlates with the stellar mass-to-light ratio, while the total dynamical mass (which would be affected by evaporation in denser environments) shows no such correlation La Barbera et al. (2010). However, based on simulation results for , Gnedin & Ostriker’s constraint may be overestimated, although detailed comparisons between N-body simulations and the fundamental plane relation are lacking Rocha et al. (2013).

iii.5 SIDM simulations with baryons

While DM-only simulations have played a crucial role toward understanding the impact of self-interactions on DM halos, the next step is to incorporate self-interactions within hydrodynamic simulations that include the dynamics of baryons. Feedback processes remain the leading “vanilla” explanation for small scale structure issues apparent in CDM-only simulations, despite considerable debate over precisely how they are implemented. Hydrodynamic simulations with SIDM may provide guidance for how to disentangle and distinguish the effects of self-interactions and feedback. Additionally, it is important to understand how the predictions of SIDM are influenced by the presence of baryons, as well as conversely how baryonic tracers for DM are influenced by self-interactions.

Recently, Vogelsberger et al. Vogelsberger et al. (2014) and Fry et al. Fry et al. (2015) have performed the first N-body simulations including both self-interactions and baryons, both targeting dwarf scales. However, both sets of simulations have implemented star formation and feedback to opposite effect. In Ref. Fry et al. (2015), galaxies have bursty star formation histories (provided enough baryons are present) marked by episodic supernovae-driven gas outflows over times much shorter than the dynamical time scale of the galaxy. This process—motivated in part by explaining the formation of bulgeless dwarf galaxies by expelling low angular momentum gas that would otherwise form a stellar bulge—leads to nonadiabatic changes in the gravitational potential that can significantly impact the central DM distribution Governato et al. (2010); Pontzen and Governato (2012); Governato et al. (2012). On the other hand, Ref. Vogelsberger et al. (2014) has implemented a smoother prescription for star formation and feedback, that, while successfully suppressing bulge formation, provides a much smaller influence on the DM halo.

Vogelsberger et al. Vogelsberger et al. (2014) present results for two dwarf galaxies, simulated for a range of self-interaction cross sections. These galaxies are DM-dominated at all radii and are comparable to observed THINGS dwarfs Oh et al. (2011a) in terms of their stellar mass and maximum circular velocity, . Whether DM is self-interacting or collisionless, baryonic feedback provides a negligible impact on the DM halo, and the DM central density is primarily influenced by the effect of self-interactions as in SIDM-only simulations. However, the presence of an core in SIDM halos can affect the baryon component compared to a more concentrated DM profile, resulting in an reduction in star formation and reducing the central densities of stars and gas. In particular, the stellar distribution inherits a core radius that correlates with the core radius for DM, which may provide a useful observational handle for SIDM.

Fry et al. Fry et al. (2015) have simulated somewhat smaller systems, including three dwarf galaxies with —comparable to field dwarfs of the Local Group Wolf et al. (2010)—as well as making predictions for even smaller dwarfs that are below current observational sensitivities. For the larger dwarfs, baryonic feedback by itself produces reduced central densities and cores for collisionless CDM, upon which the further inclusion of self-interactions with has only marginal effect. For smaller dwarfs with , baryonic feedback is insufficient to form cores due to a lack of star formation. At the same time, self-interactions with are unable to form cores larger than 500 pc in these smaller halos (consistent with simple rate arguments).

It is evident that the conclusions of Refs. Vogelsberger et al. (2014); Fry et al. (2015) are quite different due to the feedback prescriptions adopted therein. This underscores the role of baryonic physics as a systematic uncertainty in the small scale structure puzzle, as well the importance of taking new approaches to test the consistency of feedback models with collisionless CDM (e.g., Refs. Katz et al. (2016); Pace (2016); El-Badry et al. (2016, 2017)).

Iv Jeans approach to relaxed SIDM halos

iv.1 Isothermal solutions to the Jeans equations

Despite the importance of N-body simulations, these methods are limited by their intensive computational nature. Even minimal particle models for SIDM exhibit rich dynamics for elastic scattering Tulin et al. (2013b, a), and it is not feasible to explore the full range of possibilities with simulations. To complement these studies, there is a useful semi-analytic method based on the Jeans equation for understanding SIDM halo profiles in relaxed systems Rocha et al. (2013); Kaplinghat et al. (2014a); Kaplinghat et al. (2016). This approach is well-suited to the intermediate cross section regime, , where DM is neither fully collisionless nor collisional throughout an entire halo. The Jeans method can be fit directly to observations for individual systems, from dwarf galaxies to massive clusters, including the gravitational effect on the halo from the observed baryonic mass distribution. Therefore, this method provides a bridge between simulations, astrophysical observations, and SIDM particle models.

The Jeans equation may be derived from the collisional Boltzmann equation (see, e.g., Binney and Tremaine (1987))


where is the DM distribution function, is the total gravitational potential for both DM and baryons, and is the collision term from self-interactions. In the inner halo, the collision term drives the distribution function toward kinetic equilibrium, , where is the isotropic one-dimensional velocity dispersion. Taking the first moment of Eq. (11) and searching for quasi-equilibrium solutions in which time derivatives can be neglected, we have the (time-independent) Jeans equation


This is simply the condition for hydrostatic equilibrium for an ideal gas of pressure . The velocity anisotropy is assumed to vanish since collisions isotropize trajectories. Moreover, N-body simulations have shown that DM particles are approximately isothermal within the inner halo, such that can be taken as a constant.171717The velocity dispersion is expected to increase (decrease) with halo radius for heat transfer flowing in (out), corresponding the core growth (collapse) phase of the SIDM halo, depending on the value of . However, for a range of , N-body simulations have shown that the spatial variation in is only within the inner halo Elbert et al. (2015). Combining with Poisson’s equation, we have


where is the baryon mass density and is Newton’s constant.

On the other hand, DM is effectively collisionless in the outer halo due to the reduced particle number density. The delineation between the inner and outer halo occurs at radius where, on average, one collision per particle has occurred over the age of the halo, . This condition is , where the scattering rate is given in Eq. (1). The full density profile is taken to be a hybrid of collisional and collisionless (NFW) profiles, matched together at :


Here, is the isothermal density profile defined as the solution to Eq. (13). To match the two regions, the density profiles and enclosed mass are assumed to be equal at . The physical picture is that self-interactions simply rearrange and thermalize the DM particle distribution in the inner halo, while leaving the outer halo as it would be in the absence of collisions.

With this simple halo model in mind, we turn to several important questions: Do self-interactions provide a consistent solution to the core-cusp problem in all astrophysical systems? What are the particle physics implications for observations spanning widely different halo mass scales, from dwarf galaxies to clusters? And lastly, does the Jeans approach for SIDM halos agree with N-body simulations?

Figure 12: DM density profiles obtained for clusters A2537 (left) and MS2137 (right) using the Jeans method as per Eq. (13). Orange band shows the full SIDM profile from fitting stellar velocity kinematics at small radii (red data points in inset) and lensing data at large radii (not shown). Red band shows the stellar density profile. Cyan band shows the collisionless profile obtained by fitting only lensing data, which overpredicts stellar velocities in the central halo (see inset). Both clusters are consistent with . Reprinted from Ref. Kaplinghat et al. (2016).

To address these questions, Kaplinghat, Tulin & Yu Kaplinghat et al. (2016) applied the Jeans method to an astrophysical data set spanning DM halo masses in the range . These data include rotation curves from twelve DM-dominated galaxies, including those from THINGS Oh et al. (2011) and LSB galaxies from Kuzio de Naray et al. Kuzio de Naray et al. (2008), combined with kinematic and lensing studies from six massive clusters by Newman et al. Newman et al. (2013b, a), all of which exhibit cored profiles. For each system, the cross section is obtained by


following Eq. (1). Here, the quantity on the left-hand side represents the velocity-weighted cross section per unit mass, statistically averaged over velocity, while the on right-hand side is obtained by fitting Eq. (14) to astrophysical data for each system. Since more massive halos correlate with higher average relative velocities for DM particles, the range of halos provides an important probe of the velocity-dependence of self-interactions. Analogous to tuning the beam energy in a particle collider, the energy-dependence of scattering is crucially important for probing the underlying particle physics of SIDM.

To illustrate the Jeans method, Fig. 12 shows results for two clusters from Ref. Kaplinghat et al. (2016). The full SIDM profile has been fit to the stellar velocity dispersions for the brightest central galaxy at small radii ( kpc) and strong and weak lensing data at larger radii ( kpc). These data prefer cluster profiles with cores. A cuspy (NFW) profile fit only from lensing data does not agree with stellar data in the central halo (see inset). By matching the collisional and collisionless regions of the halo together to determine , the preferred cross section for these clusters is . These conclusions may be weakened if stellar anisotropies are far more significant that assumed Schaller et al. (2015a) or if AGN feedback is relevant Martizzi et al. (2012), in which case .

Figure 13: Left: Velocity-weighted self-interaction cross section per unit mass as a function of average relative particle velocity in a halo. Data points from astrophysical observations correspond to THINGS dwarf galaxies (red), LSB galaxies (blue), and clusters (green). Diagonal lines show constant values of . Gray points are fits to mock data from SIDM simulations, with fixed , as a test of the Jeans method to reproduce the input cross section. Reprinted from Ref. Kaplinghat et al. (2016); see therein for further details. Right: Comparison of DM density profiles for simulated SIDM-only halo (green dots) to SIDM halo with baryons (dashed curves), either with (black) or without (red) adiabatic contraction from stellar disk, where . The SIDM profile with baryons is virtually identical to the collisionless DM profile (NFW) except for the innermost kpc. Reprinted from Ref. Kaplinghat et al. (2014a).

Jointly analyzing both galaxies and clusters, Fig. 13 (left) illustrates how depends on the average collision velocity , assuming self-interactions are responsible for the observed cores in these systems. While galaxy-scale observations favor , data from clusters prefers a much smaller cross section,  Kaplinghat et al. (2016). Taken at face value, these data imply that SIDM can provide a consistent solution to the core-cusp problem, provided self-interactions are relatively suppressed in clusters compared to dwarf galaxies. Such a behavior is well-motivated from a particle physics perspective, as discussed below. The data given here may be fit by a massive dark photon model (dashed orange curve in Fig. 13). Lastly, to verify the validity of the Jeans approach, Ref. Kaplinghat et al. (2016) analyzed mock rotation curves produced from eight SIDM halos in a similar mass range from N-body simulations Rocha et al. (2013); Elbert et al. (2015), reproducing the input cross section value in those simulations.

Next, we turn to another important question: what is the interplay between self-interactions and baryons? The first simulations including both baryons with feedback processes and self-interactions for DM have only recently been performed (see §III.5), mainly targeting dwarf halos Vogelsberger et al. (2014); Fry et al. (2015). The Jeans method provides complementary insights into the effect of baryons on SIDM, especially in systems like the MW or larger that have a significant baryon fraction Kaplinghat et al. (2014a).

Although the Jeans approach is limited to quasi-equilibrium solutions (hence the dynamics of feedback is ignored), the static gravitational potential from baryons can dramatically change the predictions for observations compared to SIDM-only simulations. The baryon density enters through in Eq. (13), modifying the solution for from the usual cored isothermal profile. If dominates over in the inner halo, the core radius shrinks substantially compared to the SIDM-only halo without baryons. This effect is shown in Fig. 13 (right) for a MW-like halo and a self-interaction cross section of . While the SIDM-only halo has a core of size kpc, the core size is reduced by an order of magnitude due to the baryonic potential from stars. Except for the innermost kpc, the density profiles for both collisionless and collisional DM are virtually identical. Since in this case, where and are comparable, adiabatic contraction Blumenthal et al. (1986) may modify the initial NFW profile matched to the inner isothermal profile, as indicated in Fig. 13 (right). However, it is negligible for the SIDM fits of the clusters shown in Fig. 12, because the contraction effect is very mild and it occurs  Newman et al. (2013b).

The baryon density may affect the shape of the DM halo as well. While SIDM-only simulations predict halos that are spherical within the core radius, this conclusion changes once baryons are present. For a halo like the MW, the gravitational potential of the baryonic disk causes the SIDM halo to become oblate, with aspect ratio , out to kpc, the would-be core radius for SIDM without baryons Kaplinghat et al. (2014a).

To summarize, recent studies using the Jeans method have challenged the conventional SIDM paradigm in several important ways.

  • Astrophysical data from galaxies to clusters disfavor a constant self-interaction cross section. The velocity-dependence of scattering is important for constraining the particle physics underlying SIDM.

  • Expectations for SIDM halo profiles can change dramatically from SIDM-only halos if there is a sizable baryonic component. The central regions of halos need not have large spherical cores with slope