Dark matter concentrations in galactic nuclei according to polytropic models
Abstract
We calculate the radial profiles of galaxies where the nuclear region is selfgravitating, consisting of selfinteracting dark matter (SIDM) with degrees of freedom. For sufficiently high density this dark matter becomes collisional, regardless of its behaviour on galaxy scales. Our calculations show a spike in the central density profile, with properties determined by the dark matter microphysics, and the densities can reach the ‘mean density’ of a black hole (from dividing the blackhole mass by the volume enclosed by the Schwarzschild radius). For a galaxy halo of given compactness , certain values for the dark matter entropy yield a dense central object lacking an event horizon. For some soft equations of state of the SIDM (e.g. ), there are multiple horizonless solutions at given compactness. Although light propagates around and through a sphere composed of dark matter, it is gravitationally lensed and redshifted. While some calculations give nonsingular solutions, others yield solutions with a central singularity. In all cases the density transitions smoothly from the central body to the darkmatter envelope around it, and to the galaxy’s dark matter halo. We propose that pulsar timing observations will be able to distinguish between systems with a centrally dense dark matter sphere (for different equations of state) and conventional galactic nuclei that harbour a supermassive black hole.
keywords:
black hole physics — dark matter — galaxies: haloes — galaxies: nuclei — pulsars.1 Introduction
Invisibly compact, relativistic objects appear to reside in the central regions of most large galaxies. Their masses appear to correlate with certain host properties (e.g. Magorrian et al., 1998; Ferrarese & Merritt, 2000; Laor, 2001; Häring & Rix, 2004; Gültekin et al., 2009; Feoli & Mancini, 2009; Burkert & Tremaine, 2010; Graham et al., 2011; Xiao et al., 2011; Soker & Meiron, 2011; Rhode, 2012; Bogdán & Goulding, 2015; Ginat et al., 2016). If these objects are dense enough to possess an event horizon, then they are supermassive black holes (SMBH). More exotic alternatives may lack a horizon (e.g Müller Zum Hagen et al., 1974; Ori & Piran, 1987; Tkachev, 1991; Viollier et al., 1993; Tsiklauri & Viollier, 1998; Schunck & Torres, 2000; Kovács & Harko, 2010; Joshi et al., 2011; Diemer et al., 2013; Meliani et al., 2015). Whatever they are, some of these nuclei act as ‘quasars’ during episodes of bright, rapid gas accretion. Powerful quasars are found at high redshifts, (e.g. Fan et al., 2004; Mortlock et al., 2011; Venemans et al., 2013; Ghisellini et al., 2015; Wu et al., 2015), implying that their central objects were already present and grew on short timescales in the early Universe. The largest ultramassive black hole (UMBH) candidates are a few (McConnell et al., 2011; Postman et al., 2012; van den Bosch et al., 2012; Shields & Bonning, 2013; Fabian et al., 2013; Ghisellini et al., 2015; Yıldırım et al., 2015, 2016; Scharwächter et al., 2016; Thomas et al., 2016). These are difficult to reconcile with the conventional scenario in which SMBH grew via accretion of luminous gas and stars (Soltan, 1982; Yu & Tremaine, 2002; Shankar et al., 2009; Novak, 2013).
Galaxies possess another significant nonluminous component, in the form of invisible ‘dark matter’ (DM) that seems to reside in spheroidal haloes: more radially extended than the visible matter (Oort, 1932; Zwicky, 1937; Babcock, 1939; Ostriker & Peebles, 1973). The fundamental nature of DM is unknown, besides constraints on its electromagnetic traits (e.g. Sigurdson et al., 2004; McDermott et al., 2011; Cline et al., 2012; Khlopov, 2014; Kadota & Silk, 2014). Cosmic filaments and voids can form in collisionless cold dark matter (e.g. Frenk et al., 1983; Melott et al., 1983; Springel et al., 2006), selfinteracting dark fluid (Moore et al., 2000), or wavelike cosmic boson fields (Woo & Chiueh, 2009; Schive et al., 2014a, b). However, when simulations treat the DM like a collisionless gravitating dust, steep powerlaw density cusps emerge throughout the centres of selfbound systems (Gurevich & Zybin, 1988; Dubinski & Carlberg, 1991; Navarro et al., 1996). Observationally, at kiloparsec scales, dark matter in most types of galaxies exhibits nearly uniform central cores that attenuate at larger radii (e.g. Flores & Primack, 1994; Moore, 1994; Burkert, 1995; Salucci & Burkert, 2000; Gentile et al., 2004; Gilmore et al., 2007; Oh et al., 2008; Inoue, 2009; Herrmann & Ciardullo, 2009; de Blok, 2010; Saxton & Ferreras, 2010; Memola et al., 2011; Walker & Peñarrubia, 2011; Schuberth et al., 2012; Salucci et al., 2012; Lora et al., 2012, 2013; Agnello & Evans, 2012; Amorisco et al., 2013; Pota et al., 2015; Bottema & Pestaña, 2015). Among many interpretations, it has been suggested that the dark cores are supported by dark pressure due to DM selfinteractions via selfscattering, longer range dark forces, or more exotic mechanisms (e.g. Spergel & Steinhardt, 2000; Peebles, 2000; Ackerman et al., 2009; Hochberg et al., 2014; Boddy et al., 2014; Cline et al., 2014a; Heikinheimo et al., 2015). If selfinteracting dark matter (SIDM) is in this sense plasma or gaslike, then the manner of its interaction with the SMBH (or other exotic central object) could provide informative constraints on the physics of both these mysterious entities.
While a realistic halo should be cored at kpc scales, dense concentrations of visible matter exert a gravitational influence that may steepen the innermost part of the DM profile: ‘adiabatic contraction’ of collisionless DM (Blumenthal et al., 1986; Gnedin et al., 2004), or SIDM (Saxton, 2013, figure 1). A central massive object could distort the innermost parts of the halo, forming a dark density ‘spike’ within the local sphere of influence (Huntley & Saslaw, 1975; Quinlan et al., 1995; Munyaneza & Biermann, 2005; Guzmán & LoraClavijo, 2011a, b). This DM substructure might continue to grow denser near a SMBH’s event horizon. Relaxation processes and star formation in galaxy nuclei can grow powerlaw stellar density cusps (e.g. Bahcall & Wolf, 1976, 1977; Freitag et al., 2006; Alexander & Hopman, 2009; Aharon & Perets, 2015), which could also help induce a dark spike. Scattering by stars would render the DM indirectly collisional, regardless of its collisionality in the rarefied outskirts of haloes (Ilyin et al., 2004; Merritt, 2004, 2010).
The most commonly predicted spike profile is . For dark matter with thermal degrees of freedom (and an adiabatic pressuredensity law ) the spike profile tends to in newtonian regions (far outside any event horizon). This is the maximum slope when the central mass dominates DM selfgravity. (In regions where DM selfgravity is more influential than the central mass, density gradients can be locally shallow; and concentric regions can alternate between steep and shallow, as we describe in Subsection 3.2). If SIDM consists of particles scattering with a velocitydependent crosssection () then the ratio of mean free path to radial position is in the spike^{1}^{1}1For any , and fixing a sign in Saxton et al. (2014) p.3427.. If the heat capacity is high (, a ‘soft’ equation of state) then shortens enough at small radii that the centre is maximally collisional, for microphysics ranging from hard spheres () to Coulomb scattering (). The possibility of centrally strengthening SIDM interactions has so far not been considered in papers that implicitly assumed , (e.g. Shapiro & Paschalidis, 2014). It is worth emphasising that collisional pressure is not the only conceivable type of interaction. For instance, a dark plasma might be mediated by a dark version of electromagnetism, and develop collisionless shocks like ionised plasmas do (e.g. Ackerman et al., 2009; Heikinheimo et al., 2015). Boson condensate and scalar field dark matter theories entail effective pressures due to quantum effects (e.g. Goodman, 2000; Peebles, 2000; Arbey et al., 2003; Böhmer & Harko, 2007; Harko, 2011a, b; Chavanis & Delfini, 2011; Robles & Matos, 2012; Meliani et al., 2015). Fermionic dark matter could exhibit degeneracy pressure (e.g. Munyaneza & Biermann, 2005; Destri et al., 2013; de Vega & Sanchez, 2014; Domcke & Urbano, 2015; Horiuchi et al., 2014; Kouvaris & Nielsen, 2015).
At galaxy scales, early simulations of weakly scattering, thermally conductive SIDM predicted unrealistic steeper cusps, forming via gravothermal catastrophe (e.g. Burkert, 2000; Kochanek & White, 2000). More detailed investigations defer this collapse to the far cosmological future, and show the existence of another plausible regime in which strong scattering (short mean free paths) inhibits conduction and enables adiabatic, fluidlike phenomena (Balberg et al., 2002; Ahn & Shapiro, 2005; Koda & Shapiro, 2011).
Much recent research concentrated on the conjecture that DM is a weakly interacting massive particle with cosmologically long selfscattering timescales (e.g. Buckley & Fox, 2010; Feng et al., 2010; Loeb & Weiner, 2011). These models raise hopes of detecting DM decay or annihilation byproducts from the central spike (e.g. Gondolo & Silk, 1999; Merritt, 2004, 2010). In most of these models, the DM particles are pointlike and lack substructure (possessing only translational degrees of freedom, ). This can be implemented in body simulations with infrequent Monte Carlo scattering. Some simulations predict overly large SIDM cores, which prompted suggestions that the scattering crosssection is small or velocity dependent (, e.g. Yoshida et al., 2000; Davé et al., 2001; Arabadjis et al., 2002; Katgert et al., 2004; Vogelsberger et al., 2012; Rocha et al., 2013; Peter et al., 2013; Elbert et al., 2015). Alternatively, SIDM may have a higher internal heat capacity (). Without restricting the scattering physics, analytic models show that the range of results in galaxy clusters with realistic –kpc cores (Saxton & Wu, 2008, 2014), while the range can fit elliptical galaxy kinematics (Saxton & Ferreras, 2010) and naturally provides the observed scaling relations between galaxies and their SMBH (Saxton et al., 2014). Isolated galaxies gain dynamical stability from a suitable concentration of collisionless stars permeating the SIDM halo (Saxton, 2013).
Within this rich diversity of DM theories, it is interesting to investigate whether there might be any direct relationship between SIDM and SMBH, enabling falsifiable predictions about one or the other. Dark matter might contribute significantly to the origin and growth of SMBH. Ostriker (2000) and Hennawi & Ostriker (2002) assumed an initially cuspy profile with weakly interacting SIDM, and inferred that collisionality must be weak in order to prevent SMBH from growing larger than observed. Balberg & Shapiro (2002) began with a cored SIDM profile, and showed that some versions of SIDM (with ) could form realistic SMBH and halo cores, prior to gravothermal catastrophe in some future era. Other fluidlike accretion models (in various contexts, with or without selfgravity) affirm that DM could contribute significantly to BH growth (e.g. MacMillan & Henriksen, 2002; Munyaneza & Biermann, 2005; Richter et al., 2006; Hernandez & Lee, 2010; Guzmán & LoraClavijo, 2011a, b; Pepe et al., 2012; LoraClavijo et al., 2014). In galaxy cluster models combining DM with radiative gas (Saxton & Wu, 2008, 2014) the physically consistent solutions always have a compact central mass.
In the fully relativistic theory of selfgravitating spherical accretion, accretion rates are maximal when the surrounding fluid envelope is half the mass of the accretor (Karkowski et al., 2006; Mach, 2009). This condition assumes special cases with a sonic point in the flow. Alternative, entirely subsonic solutions might be longerlived, with relatively more more massive fluid envelopes. It is conceivable that hydrostatic pressure might support a nearstationary SIDM envelope around a black hole. This paper will focus on scenarios in which a quasistatic SIDM spike is itself relativistically dense and supermassive. For now, we set aside the complications of gaseous and stellar physics, and appraise the effects of a spike of SIDM at densities comparable to the black hole, in regions all the way down to the event horizon. We will also see that a SMBH (with an event horizon) can be entirely replaced by a SIDM condensate.
2 Model
2.1 Formulation
The interval between events within and around a spherical mass distribution is , with the proper time given by
(1) 
in spherical coordinates . Here, is the radius at a surface of circumference , and is a dimensionless gravitational potential. We abbreviate for the Schwarzschild radius of the enclosed gravitating mass, . We seek solutions of the TolmanOppenheimerVolkoff (‘TOV,’ Tolman, 1934, 1939; Oppenheimer & Volkoff, 1939) model for a hydrostatic selfgravitating sphere. Unlike those classic models of relativistic stars, we allow a singularity or event horizon to occur at some inner radius (which will be obtained numerically). At each radius , there is locally an isotropic pressure and energy density . These quantities are linked by coupled differential equations,
(2) 
(3) 
(4) 
We seek solutions with finite total mass () within an outer boundary () where the density vanishes (). At this boundary the potential matches that of the external Schwarzschild (1916) vacuum model:
(5) 
The total energy density includes restmass density () and internal energy components,
(6) 
where is the number of effective thermal degrees of freedom, which depends on the dark matter microphysics. In this paper, we assume that is spatially constant. If the dark matter behaves adiabatically then there is a polytropic^{2}^{2}2Many papers use a different ‘polytropic’ law, (e.g. Zurek & Page, 1984; de Felice et al., 1995). This leads to some simpler results, but is harder to interpret in terms of microphysical heat capacity. Our version describes truly adiabatic conditions, and prevents unphysical outcomes such as superluminal or subzero sound speeds. Mrázová et al. (2005) compare these assumptions further. equation of state,
(7) 
or equivalently
(8) 
From fundamental thermodynamics, the adiabatic index is
(9) 
The quantity is a pseudoentropy: it is spatially constant for a well mixed adiabatic system (as this paper assumes). The laxer constraint of convective stability would require that everywhere. The related value is a generalised phasespace density. The halo’s total mass and outer radius can be finite if . A SIDM phase or process with would ensure a flat, accelerating cosmology (obviating dark energy, e.g. Bento et al., 2002; Kleidis & Spyrou, 2015) but the selfbound haloes would be denser outside than in their centres.
The physical meanings of in various contexts were discussed in Saxton & Wu (2008); Saxton & Ferreras (2010); Saxton (2013); Saxton et al. (2014). The equations (7) and (8) might describe a SIDM fluid in adiabatic conditions (which is appropriate for a nonreactive, nonradiative, pressured entity). For example, if DM has composite bound states (e.g Kaplan et al., 2010; Boddy et al., 2014; Cline et al., 2014a; Wise & Zhang, 2014; Hardy et al., 2015; Choquette & Cline, 2015) that include dark molecules, then . Alternatively, might just as well describe the scalar field of Peebles (2000), where derives from a selfcoupling term in the particle lagrangian. Polytropic conditions also occur if the Tsallis thermostatistics apply to collisionless selfgravitating systems (Tsallis, 1988; Plastino & Plastino, 1993; Nunez et al., 2006; Zavala et al., 2006; Vignat et al., 2011; Frigerio Martins et al., 2015). It is conceivable that varies between astrophysical environments: e.g. due to phase changes; dark molecule formation / dissociation; or the transition to the relativistic regime of a dark fermion gas (Arbey, 2006; Slepian & Goodman, 2012; Domcke & Urbano, 2015; Cline et al., 2014b). These complications depend on specific detailed microphysical models, so for the present paper we prefer to focus on the ideal of uniform , and explore the generic consequences of low and high heat capacities.
The quantity is analogous to the newtonian 1D velocity dispersion (assumed to be isotropic). It is however possible that in sufficiently hot regions. The adiabatic sound speed is given by (Tooper, 1965)
(10) 
This is always subluminal if . The maximal sound speed is less if the heat capacity is greater^{3}^{3}3In adiabatic ultrarelativistic media, acoustic waves propagate slower than light or gravity waves. When there are two coterminous relativistic fluids, the lower medium (e.g. radiationdominated plasma) counducts sound faster than the high fluid (e.g. forms of dark matter). This may have consequences in the early Universe. CyrRacine et al. (2014) modelled some cosmic dark acoustic oscillations (DAO) for . (). The radial propagation time for sound waves and light is given by
(11) 
Pressure profiles in particular solutions obtained from (2) and (4) are steep and sensitive to , while the radial profiles of are more gently varying. For this practical reason, we solve gradient equations for ,
(12) 
and find and in postprocessing using equations (7) and (8). Relations (3), (5) and (12) imply
(13) 
The gravitational redshift relative to an observer at infinity () thus depends on local and surface boundary conditions. Evidently, at any point where . With locally infinite redshift, the term vanishes from the interval (1). Time is frozen at this surface, and the surrounding structure is longlasting (indeed eternal) to outside observers. This inner surface is a nonrotating naked singularity in a density spike, settled without ongoing inflow. In limiting cases where , it becomes an event horizon too. We describe these conditions further in Subsection 3.2.
2.2 Numerical integration
To obtain a solution for the radial profile, we may start at the outer boundary (), where we set the total mass () and vacuum conditions (, ). The degrees of freedom () and phasespace density () are chosen constants. The ODEs for each quantity are used in the forms , , or , depending on which gives the shallowest gradients. In the locally appropriate form, the set of ODEs is integrated radially inwards from the preceding reference point using RungeKutta methods (rkf45, rk4imp and rk8pd in the Gnu Scientific Library) until the inner boundary is found: or , whichever happens first. To initially launch the solver inwards from the outer boundary, the first partial integral is a small radial step using ODEs. Then there are tentative steps using ODEs, while . At medium radii, the integrator proceeds using ODEs, picking tentative target radii cautiously outside the local Schwarzschild value (). If this process becomes slow due to steep gradients when , the integrator swaps to another choice of independent variable, and proceeds in terms of or ODEs. Eventually the numerical integral halts at an impassable inner boundary. There are two possible types.
In many cases, the gradient of steepens at small , and the temperature and density blow up, inevitably to form a sharp inner boundary. Approaching that limit, it is informative to rewrite the differential equations as:
(14) 
(15) 
(16) 
As , the derivative (meaning that temperature and density rise sharply over a tiny radial step inwards). The mass derivative approaches a constant asymptotically. A thin dense inner shell, where rises by a ratio , can account for most of the remaining inner mass. These are singular profiles.
If, in other cases, the density gradient becomes shallow at small , then the inner mass , and the potential gradient flattens. This selfconsistently compels the gradients of , and to flatten towards the centre. Such solutions are nonsingular. In those cases, another integration method determines the radial profile more directly. We set nonsingular conditions at the origin: , , and positive values of and . Integration proceeds outwards adaptively in small steps, using the , and equations, until nearing the outer boundary . Iteration of trial steps in or direct integration to the limit in yields the outer boundary conditions (, , etc.). By construction, this method never finds any of the singular solutions.
Throughout the numerical integrals, our solver routines keep the relative error on each variable within . The code records all variable states at intermediate radial shells in an ordered data structure, which provides checkpoints for retrospective refinements. Finally the inner boundary conditions are recorded (, , , etc). With both boundaries identified, we can safely integrate the ODEs inwards or outwards from any checkpoint, to quickly find the conditions anywhere else. We refine the grid recursively around interesting features, e.g. the halfmass radius (); and any radii where the density index () is integer. Once the profile is recorded at satisfactory resolution, the solution can be rescaled (e.g. to unit radius ) using the innate homologies of the model (Appendix A).
3 Results
3.1 Parameterspace domains
To standardise our description of the parameterspace, let us define some global properties of each solution, in terms that are invariant under the model’s natural scaling homologies. The halo’s mean density is and surface escape velocity is . As in Saxton et al. (2014), we quantify the gravitational compactness and phasespace density in dimensionless terms:
(17) 
(18) 
Characteristically, for galaxy clusters; for giant galaxies; for dwarf galaxies. These are upper limits since a small perturbation of the system can spread out a small mass element of the halo fringe, raising without greatly affecting the core structure. To lessen this sensitivity to the outskirts, we will sometimes specify compactness in terms of the equipotential containing the inner half of the mass, (i.e. and where ). In any case, the halo radius cannot exceed the separation between neighbouring galaxies. The known cosmic mean density gives a lower bound,
(19) 
which for Hinshaw et al. (2013) cosmic parameters gives .
Fig. 1 illustrates how the ratio of inner and outer radii () depends on , for fixed . The smallest values of give solutions where and most of the mass is concentrated near . At the opposite extreme (), the inner and outer radii are comparable (), which does not resemble any astronomical object. An intermediate domain contains nontrivial solutions where . If and is galaxylike, then has one special root where .
For models with , the landscape has more features. Across a finite domain of , there are conditions where . This interval is wider when is greater or is smaller. However, for many galaxylike choices, there exist multiple roots where . These states tend to be more abundant if is larger (implying high heat capacity in the matter) or is smaller (a less compact or less massive astronomical system). Solutions at lower values tend to appear at quasiregular logarithmic steps. The higher tend to bunch together. The medium are less regular, or show gaps (e.g. the interval when and ).
Taken at fixed , there is no obvious firstprinciples explanation for these patterns and irregularities; the values depend on nonlinearities of the TOV model. The topography of this parameterspace does however correspond to some features in a recent nonrelativistic model that successfully predicts the scaling relation between SMBH and galaxy haloes (Saxton et al., 2014). The higher values crowd around a maximum that is actually a limit where the halo becomes a nonsingular LaneEmden sphere (lacking a compact central mass). Lower values correspond to the ‘valley’ solutions of Saxton et al. (2014), where the envelope of dark matter immediately surrounding a SMBH attains densities comparable to the SMBH itself. The interval where corresponds to a ‘plateau’ where the nonrelativistic model predicted a maximum ratio of SMBH to halo core masses () for given halfmass compactness . In the newtonian halo model, was a continuum. The quantisation of models at discrete values is new to the relativistic version. In this fundamental picture, SMBH formation and growth is a simple and inexorable result of decreasing (rising entropy) through any unspecified dissipative processes in the DM halo.
3.2 Radial profiles and their classes
At large radii, where and , each density profile resembles a nonrelativistic LaneEmden sphere (e.g. Lane, 1870; Ritter, 1878; Emden, 1907). Fig. 2 depicts the radial density profiles differing in when and the halfmass compactness is fixed to . The plotted region spans the scales of galaxy haloes (kpc) to galaxy nuclei (a few au). In the outermost fringe, the density declines steeply with radius, .
The fringe surrounds a core of softer density gradients. The core is smaller (relative to ) if is greater or smaller (Saxton et al., 2014). In higher solutions (low entropy; darker curves in Fig. 2) the core is larger and sharperedged; the central density gradients flatten and may be nonsingular at the origin.
For lower (higher entropy), a powerlaw density spike occurs inside the core. As is lowered, the spike gains dominance and the core shrinks in relative radial terms. For very low , the core is indistinct (lightest curves in Fig. 2), as the spike and outer fringe merge. A strong spike occurs wherever a compact central mass dominates over the fluid’s local selfgravity, as in newtonian ‘loaded polytropes’ with a pointmass at the origin (e.g. Huntley & Saslaw, 1975). A newtonian spike has a powerlaw form () regardless of whether the fluid distribution is stationary (e.g. Kimura, 1981; Quinlan et al., 1995; Saxton et al., 2014) or an accretion flow (e.g. Bondi, 1952; Saxton & Wu, 2008; LoraClavijo et al., 2014). In relativistic regions () the spike profile becomes .
For , the spike’s locally steep density gradients can in some cases give way to more complicated structures. In spike conditions, (with ) and the local mass profile obeys . Wherever , which occurs easily when and , a small radial step inwards accounts for a large jump in mass. This leaves a weakergravity region inside the spike, and hydrostatic balance ensures locally shallow gradients (small , i.e. ‘core’ behaviour) until the steep spike behaviour resumes at much smaller radii. As undulates radially inwards, the profile is terraced: dense inner cores nest concentrically within outer cores. Density plots can resemble a ziggurat or wedding cake. Mathematically, terracing occurs because the coupling of the firstorder ODEs (2) and (12) is equivalent to an oscillatory secondorder ODE in . Such features emerged in the study of nonrelativistic polytropes: e.g. the nonsingular polytropes of Medvedev & Rybicki (2001), and the 610 galaxy halo models of Saxton et al. (2014).
In principle, terracing can continue inwards forever. However, once the temperature becomes relativistic, for all meaningful , which prevents any more undulations. When a relativistic core emerges, it is a unique and final central substructure. As long as the outer boundary is finite, the number of cores is finite. Fig. 3 shows some terraced profiles: their velocity dispersion; enclosed mass; and a score for the strength of relativistic effects. The example has two cores (left column); the example has four cores (right column).
Conditions at the inner boundary () complete the classification of radial solutions:

Sometimes at , with shallow density gradients and small there. This is a ‘vacant core’ case (Kimura, 1981). Its inner boundary lacks selfconsistent support and is unphysical. The implication is that the global mass within was badly estimated. We discard such profiles. In Fig. 1, vacant core solutions occur at high near the right border.

A density singularity can occur at , and possess a photonsphere shadow. If this happens at a place where and then we have a black hole. If however then we might call this object a ‘black bubble.’ The bubble surface is induced by pressure rather than mass concentration.^{4}^{4}4In a newtonian model we might expect the dense shell to fall radially inwards (e.g. cold gas shells in some cooling flow models, Saxton & Wu, 2008) but the timefrozen relativistic boundary need not evolve (from any external viewpoint). Black bubbles occur at above the roots; black holes arise in the limit .

In special configurations , the profile is continuous all the way to the origin () and the density gradients are shallow there (). There is no distinct massive central object (). This is a nonsingular polytropic sphere, resembling TOV toy models of stellar structure. This profile is the only solution when and is galaxylike. For the largest solution is this type.

As reported in §3.1, for discrete values , the density spike can appear at the origin ( and ). This is a nonrotating variety of naked singularity within a pressuresupported envelope.
At given , the highest state is nonsingular and singlecored. Lower solutions can be terraced or spikedominated, and are energetically extreme (Appendix B). For each root, there is a nonsingular solution and a singular solution, which are alike in their outer profiles; but differ by the presence or absence of a singularity at the origin. This means that a relativistic core is indifferent to whether or not it hosts a BH of much smaller mass.
3.3 Supermassive object & pseudohorizon
The dark matter core sizes in observed galaxies and clusters are consistent with (Saxton & Wu, 2008; Saxton & Ferreras, 2010; Saxton & Wu, 2014). With such equations of state, some halo solutions are terraced (at low enough and ). In the newtonian singlefluid context, Saxton et al. (2014) show that a galaxy halo can have a kpcsized outer core, surrounding a denser inner core or steep spike at subparsec scales. A particularly dense inner core or spike, with locally relativistic , might imitate the presence of a supermassive black hole. A true black hole (of much smaller mass) could reside at the centre of this invisible DM envelope. Alternatively, the envelope density can continue gradually rising into a central naked singularity, without any horizon.
The highest eigenvalue gives the simplest central structures. Collectively, we call them the bare solutions. In the nonsingular case, there is no distinct central mass, and the inner region is almost uniform. In the highest case containing a singularity, the density rises gradually at smaller radii, without any clear transition between this nuclear spike and the outer halo. Bare solutions represent either: (a) a young or undisturbed galaxy that has not yet formed a nuclear object; or else (b) the nucleus is a naked singularity in a continuous density spike.
Many other solutions feature a layer where is comparable to the Schwarzschild radius. (i.e. a local dip in the ratio , in the middle row of Fig. 3). We call this place a ‘pseudohorizon’ if the ratio is small (), and call the profile a loaded model. The object defined by pseudohorizon radius is a blurryedged relativistic SIDM ball, enclosing a mass . By these definitions, equation (2) implies a condition on the energy density, . Outside the pseudohorizon we find that , but not inside. Unlike a BH event horizon, the pseudohorizon does not censor the interior from sight.
For astrophysically relevant choices of the system parameters, the pseudohorizon typically occurs at . For a galaxysized halo (kpc), typical values of correspond to milliparsecs or less. This is compatible with the sizes of observed SMBH candidates (e.g. for Sgr A* in the Milky Way). Fig. 4 shows scatter plots of the central mass fractions () for and various compactness (). For fixed , the sequence of verses is ‘U’shaped: the lowest solution has the largest mass ; medium yields smaller ; and the mass fraction rises again with at the high end. Among the galaxylike models shown (e.g. with ), the central mass is and for small to larger respectively. For comparable models, the three lowest solutions have , and . The four lowest solutions when have ratios , , and . Generally for , the lowest solution represents a massive relativistic object under a tenuous and lightweight envelope extending to huge radii. The higher loaded solutions are more compatible with observed SMBH candidates’ values.
The central object lacks a truly concealing horizon, and the interior regions are significantly gravitationally redshifted. When light emits from the interior, the ratio of emitted and detected frequencies is , which for the SIDM model gives for an intergalactic observer. For the astronomical solutions we have shown, the internal redshift of the central mass ranges from up to . The higherredshift region around the singularity (if present) is only a tiny subvolume, orders of magnitude thinner than . If luminous matter traverses or resides within the supermassive SIDM ball, it will appear mildly to severely dimmed and reddened. The nucleus is less a black hole than a gloomy red pit. Comparable but milder gravitational redshifts were derived for nonsingular supermassive ‘boson star’ models (e.g. , Schunck & Liddle, 1997). For each there is a unique naked singularity solution, with infinite central redshift in a powerlaw density spike (see Appendix C).
Fig. 5 illustrates the radial profiles immediately surrounding the pseudohorizon, in families of models that have identical but different . These curves have been rescaled to pseudohorizon units ( and ). We omit the bare solutions, since they lack a pseudohorizon (). Many solutions come in pairs that have congruent profiles around the central object, but differing profiles in the galaxy fringe. Pairs include a low and high solution. In the Figure, many of the low profiles (faint shaded) overlap a high counterpart (dark dashed curves). In the rich family of solutions for , there are three pairs plus two unique solutions at medium . The velocity dispersion inside the pseudohorizon is almost identical for paired solutions, and unequal for unrelated solutions.
At fixed global compactness , the supermassive objects tend to have shallower internal potential if is larger. Within each family, the extreme (low and high) loaded solutions have:

the weakest pseudohorizon (larger at the dip);

shallower interior potential () and weaker redshift;

steeper decline in just outside .

The dark envelope within is less massive compared to the central object ().
Conversely, the medium models have:

the strongest pseudohorizon (smaller at the dip);

a deeper interior potential () and stronger redshift;

a fuzzier outer density profile, with less distinction between the central object and its envelope.

The dark envelope within is more massive compared to .
A proportionally more massive dark envelope will induce stronger deviations from Schwarzschild predictions for lightbending and circumnuclear orbital motions. A smaller value of and deeper potential imply a sharper transition between the interior and exterior, so that the object might be harder to distinguish from a black hole observationally.
The innermost individually observed stars in the Milky Way pass the centre no closer than during ‘perimelasma’ (e.g. Ghez et al., 2008; Gillessen et al., 2009; Meyer et al., 2012). In this region around most of the models in Fig. 5, especially those with shallow , the orbital velocity profiles are effectively Keplerian (, calculated as in Appendix D). For the deeper solutions, mpc and pcscale rotation curves are only subtly deviant from Keplerian (no flatter than ). For fitting imperfectly measured stellar orbits, the steep density profile of a spike could be intrinsically difficult to distinguish from a pointmass or SMBH. With enough precision, precession effects might reveal the dark envelope, though most papers to date apply only to spikes or Plummer cored profiles (e.g. Rubilar & Eckart, 2001; Schödel et al., 2002; Mouawad et al., 2005; Zakharov et al., 2007, 2010; Iorio, 2013; Dokuchaev & Eroshenko, 2015). At kpc radii, our model velocity profiles can rise as just expected within the DM core of a galaxy, then flatten and decline in the outer fringes of the halo. In order to distinguish a central SMBH from a compact SIDM object with a dark envelope, it would be preferable to rely on more direct probes of the interior.
4 An Observational Test
The propagation path of light in spacetime is bent under gravity and the wavelength is stretched when viewed by a distant observer. Thus, a massive black hole would distort the apparent background stellar surface density around it, casting multiple images of some background stars (Wardle & YusefZadeh, 1992; Jaroszynski, 1998; Alexander & Sternberg, 1999). A massive DM envelope is transparent to light, but it can cause gravitational redshifts and lensing. Its presence around a massive black hole would further complicate the gravitational lensing process. Its sole presence, with a highly dense concentration at the centre of a DM halo, is expected to show observable gravitational effects like those of a black hole, despite the absence of an event horizon. A dense and massive darkmatter sphere can trap light (Bilić et al., 2000; Dabrowski & Schunck, 2000; Nusser & Broadhurst, 2004; BinNun, 2013; Horvat et al., 2013). It can cause light rays to circulate around and also allow them to pass through it, forming an optically scrambled ‘photon sphere.’
When starlight is gravitationally lensed, the optical path length to the observer increases. The differing optical path lengths of the rays in multiplylensed variable pointsources behind a deep gravitational well results in differing timing lags in their variable emissions (e.g. Bozza & Mancini, 2004). Timing observations therefore provide a useful means to study the properties of spacetime around extreme gravity systems, such as black holes, or the dense DM envelopes and spheres described in the previous sections.
Pulsar timing has been identified as a spacetime probe because of the high precision achievable in the timing measurements. (e.g. Manchester, 2013). It is also because of the unique nature of pulsars (neutron stars) – highly compact (practically a point mass with respect to a massive black hole) and thus uneasily disrupted; narrow mass range; and for millisecond pulsars, high stability in the rotation rate (a stable, reliable clock). Moreover, rotating neutron stars will exhibit various relativistic couplings (see Wex & Kopeikin, 1999; Pfahl & Loeb, 2004; Kramer et al., 2004; Liu et al., 2012; Kocsis et al., 2012; Nampalliwar et al., 2013; Remmen & Wu, 2013; Singh et al., 2014; Angélil & Saha, 2014; Psaltis et al., 2016) that would otherwise be unobservable in the less compact stellar objects. These couplings provide additional handles in the analysis of spacetime structures around gravitating objects. Also, there are plausible theoretical reasons to expect swarms of pulsars (and other compact stars) to concentrate in galaxy nuclei (MiraldaEscudé & Gould, 2000; Pfahl & Loeb, 2004; Freitag et al., 2006). So far, one magnetar is known near Sgr A*, and there is debate about how many pulsars might also be discoverable (Macquart et al., 2010; Wharton et al., 2012; Rea et al., 2013; Dexter & O’Leary, 2014; Bramante & Linden, 2014; Macquart & Kanekar, 2015).
Here we illustrate how the dynamics of a pulsar (a test particle) responds to the different gravitational fields of polytropic SIDM spheres, and how the radio pulsation properties (i.e. ticks of the clock carried by an orbiting test particle) are affected. Fig. 6 shows the potentials and the gradients of potential of systems with , and various values. The potential and the potential gradient of a Schwarzschild black hole are also shown as a reference. The different potentials give rise to different pulsar orbital dynamics. For a pulsar orbiting around a Schwarzschild black hole, there is a limiting radius within which a stable circular orbit is impossible, i.e. the presence of an innermost stable circular orbit (ISCO). A pulsar would encounter a potential barrier for a central dense polytropic sphere (see Fig. 7, top panel), and hence it can have orbits for all nonzero radii, i.e. an ISCO does not exist. The Keplerian orbital velocity () profiles for the cases of polytropic DM spheres and for the case of a Schwarzschild black hole are different (Fig. 7, bottom panel). In each of these polytropic DM spheres, approaches a constant value as the orbital radius decreases.
The differences in gravitational potentials among these cases implies that radiation from an orbiting pulsar is subject to different gravitational redshifts. This frequency shift is a manifestation of time dilation induced by gravity, and the time dilation factors are thus always larger than one. The radiation from the pulsar is also affected by the pulsar’s orbital motion. This is due to the relativistic Doppler effect, not a direct consequence of gravitational effects, and can result in frequency blueshift or redshift, depending on the projected orbital velocity of the pulsar along the lineofsight. The pulsar’s orbital motion is however determined by the gravitational force that confines the pulsar in its orbit, and different gravitational fields will result in different orbital motions. The frequency shift from the pulsar radiation, and hence the apparent modulation of the pulsar’s pulse periods as measured by a distant observer, are a combination of the relativistic Doppler shift caused by the pulsar’s motion and the time dilation of the radiation that is climbing up a gravitational well (Appendix E). Fig. 8 shows the time dilation factor of radiation from the pulsar at (i) different azimuthal locations in the orbit and (ii) as a function of time. These calculations are performed using a generalrelativistic radiative transfer code (see Younsi et al., 2012; Younsi & Wu, 2015). This factor gives the fractional period variations of the pulses from the pulsar as measured by a distant observer. As shown, the polytropic DM models and the Schwarzschild black hole are distinguishable by measuring the pulsar’s orbital period and the variations in the pulse periods across the orbital phases.
Fig. 9 further elaborates the differences between pulse period variations among DM polytropic spheres, by showing the distinctive differences between the pulse period modulations of a pulsar in Keplerian orbits at various radii. In an orbital plane inclined at , each panel illustrates the timing factor at points around circular orbits, for each possible orbit in the radial range . The set of concentric pulsar orbits is rendered as if it were a disc, including the gravitational lensing effects. Most noticeably, the shortening of the pulse period (corresponding to frequency blueshift) always occurs when the pulsar orbiting a Schwarzschild black hole is approaching the observer. However, this pulse period shortening is not guaranteed for a DM polytropic sphere when is sufficiently large. In these cases, the pulse period shortening occurs only when the orbit is wide enough that orbital Doppler blueshift dominates the gravitational time dilation. In summary, DM polytropic spheres are distinguishable both amongst themselves and from a Schwarschild black hole via timing observations of the pulsar’s pulse period variations and the orbital period.
5 Astrophysical Implications
5.1 Accretion of visible matter
Any luminous matter which settles inside the pseudohorizon appears dimmed, reddened and timeretarded. Orbiting stars (and pulsars) can enter and leave the pseudohorizon. An eccentrically orbiting pulsar that enters and leaves the interior could reveal dramatic timing and spectral variations due to local redshift, regardless of lensing effects. They could also couple to the SIDM tidally. Their gravitational wave emissions will deviate from the ordinary scenario of a SMBHdominated vacuum. Such coupling was previously predicted for events around massive boson stars (Kesden et al., 2005; Macedo et al., 2013; Eda et al., 2013). The signatures of our SIDM envelope may differ significantly, e.g. because soft high fluid has a lower maximum sound speed.
If external tracers lead to an estimated horizon radius , under a very generic assumption that the object is a black hole, then it is possible that finer observations will reveal internal substructures smaller than or flaring events quicker than the timescale . Observationally, some AGN do show temporal variability on subhorizon scales (e.g. Aleksić et al., 2014). There are also peculiar eruptions in AGN with Xray lines that appear to be more deeply redshifted than is likely from a SMBH accretion disc (e.g. Bottacini et al., 2015). The Xray detected flares of some candidates for stellar tidal disruptions seem to imply detonations located at (Gezari, 2012, and references therein). Early VLBI observations indicate luminous structures slightly smaller than the expected shadow of Sgr A* (e.g. Doeleman et al., 2008; Johannsen et al., 2012). These features might be explained by disc and jet events occurring inside a pseudohorizon (c.f. bosonic models, Diemer et al., 2013; Vincent et al., 2015). More mundane explanations could invoke relativistic plasma flows outside a horizon, with compact coruscating bright spots due to beaming; or MHD shocks and reconnection in the inner jet (e.g. Younsi & Wu, 2015; Pu et al., 2015; Mizuno, 2013). Distinguishing these possibilities requires spatially resolved images much finer than the horizon size, which could be feasible in the near future. (Note however that a shadow is not definitive proof of a black hole event horizon; Vincent et al. 2015.)
Around black holes in vacuum, there is an innermost stable circular orbit (ISCO), beyond which the gas from the inner accretion disc is expected to plunge inwards so rapidly that there is little time for it to radiate away its energy. However, the ISCO is absent in many cases with a massive dark envelope, and also when the compact object is a nonsingular SIDM ball. This allows the gas to radiate while it gradually flows inwards into the centre of the gravitational well. Subject to Eddington (1918) radiation pressure limits, gas can continue swirling inwards forever, if the inner boundary is singular. The implied radiative efficiency of accretion is therefore higher than for a black hole, though we might expect cooler spectra due to the deep gravitational redshift.
Except the vicinity of a singularity spike, the pseudohorizon interior has a nearly constant SIDM density, and circular orbits have a uniform period (a classic ‘harmonic potential,’ Binney & Tremaine, 1987). If a gaseous accretion disc occupies this region, the lack of differential rotation will allay viscous heating. Without shear and without magnetic flaring, this becalmed zone may be darkened compared to outer annuli of the disc (). This unhidden but inactive central patch could give the illusion of the central gap due to ISCO in a spinning BH system (e.g. Laor, 1991).
Parametric models of a stable compact DM sphere and central singularity, built from an assumed mass profile, have been proposed (Joshi et al., 2011, 2014; Bambi & Malafarina, 2013), with the (anisotropic) pressure and effective equation of state derived retrospectively. Though these models were not derived from firstprinciples, they predict accretion disc properties qualitatively similar to those we expect for the polytropic DM model. Detailed modelling of accretion discs in the framework of SIDM models is beyond the scope of this paper, and we leave this exercise to a future study.
5.2 Distortion by visible matter
For the sake of investigating fundamental features, the above presented models consider idealised spheres of SIDM at rest, without any gravitational influence from other material. We note cautiously that extra constituents could break the model homologies, and perhaps alter some halo features.
Dark matter is apparent in many galaxy centres, as well as the halo. It accounts for several tens of percent of the mass within the halflight radius of elliptical galaxies (Loewenstein & White, 1999; Ferreras et al., 2005; Thomas et al., 2005, 2007; Bolton et al., 2008; Tortora et al., 2009, 2012; Saxton & Ferreras, 2010; Grillo, 2010; Memola et al., 2011; Bate et al., 2011; Norris et al., 2012; Grillo et al., 2013; Napolitano et al., 2014; Oguri et al., 2014; JiménezVicente et al., 2015). In theory, the stellar mass distribution can compress the kpcsized DM core somewhat, compared to DMonly models (e.g. figure 1 of Saxton, 2013).
The inner tens of parsecs of bright galaxies are presumably dominated by visible gas and stars. By conventional assumption, any invisible mass at small radii is attributed to the SMBH (though an unknown portion may actually be dense DM). In stellar dynamical theory, when a SMBH is surrounded by a collisional population of stars, the stellar density evolves a powerlaw cusp, (e.g. Bahcall & Wolf, 1976). The compact elliptical galaxy M32 contains one the densest stellar nuclei known: and still rising within pc, (Lauer et al., 1992; van der Marel et al., 1998). The profile is steep () and some kinematic models indicate a heavy central object (). The centre of the Milky Way also appears cuspy (), till the density peaks in the nuclear cluster () and then dips at smaller radii (Becklin & Neugebauer, 1968; Kent, 1992; Zhao, 1996; Figer et al., 2003; Genzel et al., 2003; Schödel et al., 2007; Zhu et al., 2008; Schödel et al., 2009; Buchholz et al., 2009). Orbital motions of the innermost stars appear consistent with a dominant compact mass, but may also be consistent with a dark spike within 10mpc (e.g Mouawad et al., 2005; Zakharov et al., 2007; Ghez et al., 2008; Gillessen et al., 2009; Schödel et al., 2009; Zakharov et al., 2010; Iorio, 2013). However these observations only indicate the total nonluminous mass within the inner stellar orbits, not the partitioning between stellar remnants, the DM spike, and the SMBH or exotic alternative.
The sharp concentration of the stellar cusp in galaxy nuclei might pinch the DM distribution inwards via ‘adiabatic contraction,’ enforceing a DM spike, and perhaps altering traits such as the pseudohorizon radius . Assessing the possible effects on the roots or SMBH/galaxy scaling relations (Saxton et al., 2014) requires detailed multiparameter calculations. Nevertheless, at sufficiently small radii — at least within the innermost star’s orbit — the stars cannot directly affect the profile of the central massive object and its dark envelope. Space inside the radius of stellar tidal disruptions by the central object (Hills, 1975; Young et al., 1977; Ozernoi & Reinhardt, 1978) will obviously be free of stars. Unless this nuclear environment is dominated by gas, rotation, or swarms of stellar remnants, its inner features should resemble our SIDMonly model.
Luminous gas accumulating inside the dark envelope and pseudohorizon might also become influential. In principle, accumulating baryonic matter could eventually distort the potential towards the limit of SMBH formation (e.g. Lian & Lou, 2014). Alternatively, if a compact stellar remnant enters the pseudohorizon and accretes DM, it might devour the supermassive object from within. This was proposed in the context of supermassive fermion balls (e.g. Munyaneza & Biermann, 2005; Richter et al., 2006) and boson balls (e.g. Torres et al., 2000; Kesden et al., 2005). In this way, the supermassive SIDM ball could incubate a seed BH to form a SMBH, predetermining the mass of the final object. This nonluminous growth process evades the Soltan (1982) limit, enabling modernsized SMBH to arise early in cosmic history.
5.3 Discontinuous halo profiles
Our calculations assume that the pseudoentropy (), phasespace density () and degrees of freedom () are spatially constant. If the adiabatic fluid were a certain kind of boson condensate then these values could be universal and derivable from the properties of the fundamental particle. In such theories, a universal value of could imply a maximum halo mass limit. If however SIDM is a degenerate fermion medium, then Pauli exclusion sets a lower bound on , forbidding regions below some line in the plane.
If the halo is a dark fluid, then and are local thermodynamic variables, and could vary spatially. Major galaxy mergers might shock and mix the halo, justifying the uniform assumption. A gentler history (with less mixing) could deposit concentric layers with different ( values. Buoyant stability requires and everywhere. Stable composite models could embed a high centre under low outskirts, with discontinuities or gradients between. Compared to our homogenous models, stratified haloes could host a smaller compact object than expected from the outer profile.
The universality of the effective degrees of freedom () depends on the underlying dark matter microphysics. Phase changes could alter suddenly. If the normally large values are due to bound ‘dark molecules,’ high densities favour more complex bound state formation (increasing ), while high temperatures might favour dissociation () near the horizon. Which effect wins is modeldependent. If however the large were due to DM experiencing extra compact spatial dimensions, then these might remain accessible in all conditions. If the value derives from a theory like Tsallis thermostatistics, then it might differ from system to system.
5.4 Dark accretion flow & SMBH growth
Our spherical solutions are stationary by construction: hydrostatic pressure supports every layer at rest, all the way down to the origin, or else a bottomless and timeless abyss where . However, quasistationary inflow/outflow solutions are also conceivable. If the pressure were raised above the static solution, the halo might excrete unbound dark matter outwards. If the central pressure were deficient, a contraction and inflow of DM ensues, ultimately accreting from the cosmic background. The accretion rate () could take any value from zero (our hydrostatic profiles) continuously up to the ideal Bondi (1952) rate applicable at the halo surface. Previous selfgravitating GR accretion modelling investigated maximal inflow cases with a ‘sonic point’ (Karkowski et al., 2006; Kinasiewicz et al., 2006; Mach, 2009). Over a lifetime , each instantaneous inflow solution evolves into another case with adjacent .
Our equilibrium profiles share several features with previous models of DMfed BH growth, with nonrelativistic, gravitationally negligible, or collisionless conditions. Spikes appear universally. Gravitational scattering of DM by circumnuclear stars confers a kind of indirect collisionality, producing a fluidlike spike ( with for pointlike particles) even if the DM theory were collisionless on cosmic scales (Ilyin et al., 2004; Gnedin et al., 2004; Merritt, 2004; Zelnikov & Vasiliev, 2005; Vasiliev & Zelnikov, 2008; Merritt, 2010). Models of a SMBH growing by adiabatic accretion of collisionless DM or stars (from an initially uniform background) will also tend to produce this form of spike (Young, 1980; Ipser & Sikivie, 1987; Quinlan et al., 1995; Gondolo & Silk, 1999; Ullio et al., 2001; MacMillan & Henriksen, 2002; Peirani et al., 2008). Initially cusped collisionless CDM haloes evolve sharper spikes than an initially cored halo (Quinlan et al., 1995; Gondolo & Silk, 1999).
The observation that real SMBH candidates haven’t overgrown and devoured their host haloes (via runaway DM accretion) may imply that DM is not collisionless and/or the haloes were never cuspy in the first place (MacMillan & Henriksen, 2002; Hernandez & Lee, 2010). This of course is consistent with SIDM expectations. Nonetheless, in some investigations of BH growth, implicitly or explicitly fluidlike SIDM could contribute significantly. (Hernandez & Lee, 2010; Pepe et al., 2012). To prevent IMBH in globular clusters from growing larger than observed, DM may require sound speeds in large galaxy haloes (in the model of Pepe et al., 2012). Guzmán & LoraClavijo (2011a, b) simulated GR accretion without selfgravity; they found runaway growth from collisionless DM, and minor growth of the SMBH for a fluid with . LoraClavijo et al. (2014) included selfgravity, and found that SIDM accretion was still only a minor source of SMBH growth. We speculate that a condition with and more galaxylike densities might boost DMfed growth, as in the (newtonian gas) cooling inflow models of Saxton & Wu (2008, 2014).
Quasistationary spherical accretion is not the only possible channel for SMBH growth from SIDM. If the matter is only semifluid, but the meanfreepath is long enough to enable thermal conduction on short cosmic timescales, then a gravothermal catastrophe might feed the central object. This possibility was explored in spherical timedependent PDE calculations (Ostriker, 2000; Hennawi & Ostriker, 2002; Balberg & Shapiro, 2002; Balberg et al., 2002; Pollack et al., 2015). In our scenario of fully fluidlike SIDM with , the nuclear spike could be perturbed into a local dynamical collapse, spawning a SMBH directly via ‘dark gulping’ (in cluster contexts, Saxton & Wu, 2008, 2014). The ‘skotoseismology’ of elliptical galaxies implies collapse modes when the density ratio of stars to SIDM is abnormal (Saxton, 2013). These analytically inferred processes await exemplification in nonlinear timedependent simulations.
6 Conclusions
We selfconsistently obtain the equilibrium spherical structures of selfgravitating adiabatic selfinteracting dark matter, from the halo outskirts to the relativistic central region. Lowentropy solutions resemble the cored haloes of primordial galaxies that have not formed a distinct nucleus. There also exist solutions that are pressuresupported all the way down to a fuzzyedged massive central object or else a naked singularity. For SIDM theories that naturally provide the most realistic core and halo profiles (with thermal degrees of freedom ) there exist discretised solutions where the radial origin is exposed. Among galaxylike solutions of specified gravitational compactness () the special internal configurations can be labelled by their dimensionless phasespace densities (), or their entropies.
Some solution profiles have more than one core of nearuniform density, nested concentrically across orders of magnitude in radius. In many models, a dense part of the inner mass profile has a pseudohorizon, at scales compatible with astronomical SMBH candidates. The relativistic supermassive SIDM ball has interior regions that remain visible from the outside Universe. Gravitational redshifts can reach or more, depending on (specific) galaxy properties and the (universal) DM heat capacity. There may be testable consequences. The lack of a perfect horizon means that the effective stronglensing silhouette of the central structure may differ significantly from SMBH predictions. We present raytracing calculations (as described in Younsi et al., 2012; Younsi & Wu, 2015) of the timing anomalies of pulsar signals emitted from the vicinity of the central object, which can potentially distinguish these horizonless softedged objects from an ordinary supermassive black hole in vacuum.
Acknowledgments
ZY is supported by an Alexander von Humboldt Fellowship and acknowledges support from the ERC Synergy Grant ‘BlackHoleCam – Imaging the Event Horizon of Black Holes’ (Grant 610058). Numerical calculations employed mathematical routines from the Gnu Scientific Library. This publication has made use of code written by James R. A. Davenport.^{5}^{5}5http://www.astro.washington.edu/users/jrad/idl.html Specifically, the Fig. 9 colour scheme^{6}^{6}6http://www.mrao.cam.ac.uk/~dag/CUBEHELIX/ was modified from one developed by Green (2011). This research has made use of NASA’s Astrophysics Data System.
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Appendix A Scaling Homologies
The speed of light is an absolute scale. All ratios of velocities to must remain fixed in a homologous transformation of a particular model, and is not allowed to rescale within homologous families of models. Therefore we can only accept rescaling factors
(20) 
By dimensional analysis of both sides of the temperature equation (12), we see that the masses scale in proportion to radial measurements,
(21) 
Dimensional analysis of the polytropic equation of state (7) yields:
(22) 
(23) 
(24) 
By construction, the dimensionless constants and are invariant under all the valid homology transformations, .
Appendix B Energetics & Stability
The energies characterising each model solution are obtained from supplementary ODEs, solved simultaneously with those for the radial profile (e.g. Iben, 1963; Tooper, 1964). For diagnostic interest, we record the total mass (), rest mass (), thermal energy (), proper energy () between the inner and outer radii:
(25) 
(26) 
(27) 
(28) 
The total energy of the system is
(29) 
where the gravitational potential energy is . Binding energy () refers to the hypothetical initial configuration in which the uncollapsed rest mass was dispersed widely, at zero density and zero pressure. In the absence of detailed mode analyses, a positive binding energy is traditionally interpreted as a sign of secular stability in vacuum conditions, while negative binding energy was seen as a sign of secular instability. We explain below that real stability criteria are not so simple.
In our results for models, the binding energy (relative to a vacuum) is positive for , and negative for . At fixed , the magnitude of is greater for the lowest eigenmodels (most concentrated, highest entropy) and lowest for the higher eigenmodels (largest core, lowest entropy). Specifically, the maximum solution has binding energy , which is insignificant (in magnitude) compared to the massenergy of a galaxysized object. For , the three lowest models have large fractional binding energies: (and in terms of restmass). Thus for haloes, the cored states are lowentropy (primordial?) configurations, and could degrade into singular profiles through dissipative events. However while rising entropy favours concentrated states, binding energy favours the cored states.
Galaxies and clusters with astronomically realistic core sizes and inner mass concentrations may require (Saxton & Wu, 2008; Saxton & Ferreras, 2010), which suggests negative binding energies (at least for the dark halo). Can such a structure condense naturally? The real Universe has a positive mean density, (Hinshaw et al., 2013). This value is a more appropriate reference background than an ideal vacuum. The binding energies of cosmic voids are opposite in sign to selfbound haloes. An initially uniform medium of volume can differentiate into galaxies and void matter, in some ratio such that where . In principle, the measurable cosmic fractions of voids and haloes could constrain the effective universal value of .
While the energy of cosmic voids compensates for haloes forming with , the pressure from the ambient cosmic sea of unbound DM may stabilise galaxies better than in the naïve vacuum assumption. Dynamical stabilisation by external pressure is well known in the analogous situation of a gaseous star confined by a dense interstellar medium (e.g. McCrea, 1957; Bonnor, 1958; Horedt, 1970; Umemura & Ikeuchi, 1986). In a newtonian stability condition by Bonnor (1958), the isobaric interface between a radially truncated halo and the external medium must occur within a critical radius () where the indicator
(30) 
changes sign ( in unstable outskirts). For our models with galaxylike compactness, occurs far outside the core, where the density index is steep (bottom panel, Fig. 10) and encloses most of the ideal complete polytrope’s mass (always : top panel, Fig. 10). The ratio is large for high models (cored; low entropy) and the lowest values shown in Fig. 10 are only the extreme low cases (sharply concentrated structures). The distribution of the Bonnor limit across polytropes of diverse appears not very sensitive to , for soft equations of state ().
Surveys and collisionless cosmological theories suggest bulk flows and velocity dispersions of a few hundred between galaxies that aren’t in larger structures (Jing et al., 1998; Strauss et al., 1998; Zehavi et al., 2002; Li et al., 2006; Nusser & Davis, 2011; Hellwing et al., 2014; Scrimgeour et al., 2016). If the intergalactic velocity dispersion (say ) is representative of thermal conditions in the unbound SIDM sea, then the cosmic mean pressure () constrains the absolute mass scale of any stable Bonnortruncated halo model. For a realistic galaxy, truncation must occur well outside the slope radius of the halo core. Fig. 11 depicts the relation between physical values of mass () and radius () of Bonnorstable halo models satisfying this constraint (). The occupied swathe of conditions are consistent with observable galaxy masses. The approximate trend is . Since the peak circular velocity of orbits in the halo is to within some form factor given by , and if the baryonic fraction varies little among galaxies, then this explains the origin of the observed Tully & Fisher relations, (Tully & Fisher, 1977; Freeman, 1999; McGaugh et al., 2000; McGaugh, 2012; Lelli et al., 2016; Papastergis et al., 2016).
The spherical SIDMonly halo model suffices to describe the interesting basic physics linking the galaxy halo and the relativistic central mass. Including the details of stellar and gaseous components may compress the DM core slightly (subsection 5.2, at the price of a wider parameter space. We expect an enlarged range of stable models. The mingling of the collisionless stellar matter imparts stability in nonsingular elliptical galaxies where the SIDM fraction inside the halflight radius is a few tens of percent (Saxton, 2013).
Appendix C PowerLaw Singularity
One of the singular solutions () exhibits a simple asymptotic behaviour near the origin. A suitable redefinition of the TOV model in composite variables will ensure finite values everywhere including the origin:
(31)  
(32)  
(33)  
(34) 
We choose a logarithmic radial coordinate and rewrite the ODEs:
(35) 
(36) 
(37) 
The inner boundary conditions are
(38) 
and . A similar asymptotic form was implied by de Felice et al. (1995), who assumed a different equation of state ( in our notation).
In our formulation and calculations, the radial profile can be integrated numerically as an initial value problem, starting at the origin with a large temperature () and integrating outwards. When the code reaches a low temperature (e.g. ) we switch to an integrator in the usual variables and ODEs until the outer boundary limit . After calculating the full radial profile, the values can be normalised retrospectively to match the Schwarzschild outer boundary condition. At given there is a unique pair of values consistent with the extreme powerlaw spike (Fig. 12). When , these values are compatible with the range of realistic galaxies or clusters, but lower gives compactness too high, and greater gives compactness too low (even when measured at the halfmass surface).
The other singular solutions, near the more astronomically relevant eigenvalues, involve a density spike that is steeper than a powerlaw. We don’t find any general analytic expressions for those cases. We can only obtain them via numerical integration.
Appendix D Orbits in the Envelope
The motion of the particle is determined by the EulerLagrange equation:
(39) 
where . The Lagrangian is given by , where is the spacetime metric. For a massless particle ; for a particle with mass we may set . Time translational symmetry and rotation symmetry are preserved in a spacetime whose metric has no explicit dependence on and . This gives the energy and angular momentum conservation conditions:
(40) 
and
(41) 
respectively, where and are constants. Conservation of angular momentum implies a planar orbit for the particle. As , we may set the particle orbit in the