# Warm Dark Matter Galaxies with Central Supermassive Black Holes

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## Abstract

**:**

**both**DM regimes—classical (Boltzmann dilute) and quantum (compact)—do

**necessarily**co-exist generically in

**any**galaxy, from the smaller and compact galaxies to the largest ones. The ratio $\mathcal{R}\left(r\right)$ of the particle wavelength to the average interparticle distance shows consistently that the transition, $\mathcal{R}\simeq 1$, from the quantum to the classical region occurs precisely at

**the same point**${r}_{A}$ where the chemical potential vanishes. A

**novel halo structure**with three regions shows up: in the vicinity of the BH, WDM is

**always**quantum in a small compact core of radius ${r}_{A}$ and nearly constant density; in the region ${r}_{A}<r<{r}_{i}$ until the BH influence radius ${r}_{i}$, WDM is less compact and exhibits a clear classical Boltzmann-like behavior; for $r>{r}_{i}$, the WDM gravity potential dominates, and the known halo galaxy shows up with its astrophysical size. DM is a dilute classical gas in this region. As an illustration, three representative families of galaxy plus central BH solutions are found and analyzed: small, medium and large galaxies with realistic supermassive BH masses of ${10}^{5}{M}_{\odot}$, ${10}^{7}{M}_{\odot}$ and ${10}^{9}{M}_{\odot}$, respectively. In the presence of the central BH, we find a minimum galaxy size and mass ${M}_{h}^{min}\simeq {10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$,

**larger**($2.2233\times {10}^{3}$ times) than the one without BH, and reached at a minimal non-zero temperature ${T}_{min}$. The supermassive BH

**heats up**the DM and prevents it from becoming an exactly degenerate gas at zero temperature. Colder galaxies are smaller, and warmer galaxies are larger. Galaxies with a central black hole have large masses ${M}_{h}>{10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}>{M}_{h}^{min}$; compact or ultracompact dwarf galaxies in the range ${10}^{4}{M}_{\odot}<{M}_{h}<{10}^{7}{M}_{\odot}$

**cannot**harbor central BHs. We find

**novel**scaling relations ${M}_{BH}=D{M}_{h}^{\frac{3}{8}}$ and ${r}_{h}=C{M}_{BH}^{\frac{4}{3}}$, and show that the DM galaxy scaling relations ${M}_{h}=b\phantom{\rule{0.277778em}{0ex}}{\Sigma}_{0}{r}_{h}^{2}$ and ${M}_{h}=a\phantom{\rule{0.277778em}{0ex}}{{\sigma}_{h}}^{4}/{\Sigma}_{0}$ hold too in the presence of the central BH, ${\Sigma}_{0}$ being the constant surface density scale over a wide galaxy range. The galaxy equation of state is derived: pressure $P\left(r\right)$ takes huge values in the BH vicinity region and then sharply decreases entering the classical region, following consistently a self-gravitating perfect gas $P\left(r\right)={\sigma}^{2}\rho \left(r\right)$ behavior.

## 1. Introduction and Results

**main basic magnitudes**of galaxies (as masses and sizes) as well as main structural properties of density profiles and rotation curves. Baryons should give corrections to the pure DM results. For such reasons, we consider here warm dark matter galaxies with central supermassive black holes without including baryons as a first approximation.

**correct abundance**of substructures and solves the cold dark matter (CDM) overabundance of structures at small scales [21,22,23,24,25,26,27,28,29]. For scales larger than $\phantom{\rule{3.33333pt}{0ex}}100$ kpc, WDM yields the same results as CDM. Hence, WDM agrees with the small-scale as well as large-scale structure observations and CMB anisotropy observations.

**cored**to scales below the kiloparsec (kpc) [30,31,32,33,34,35,36,37,38]. On the other hand, N-body CDM simulations exhibit cusped density profiles with a typical $1/r$ behavior near the galaxy center $r=0$. Inside galaxy cores, below ∼100 pc, N-body classical physics simulations do not provide the correct structures for WDM because quantum effects are important in WDM at these scales. Classical, that is, non-quantum physics N-body WDM simulations which do not take into account the quantum WDM pressure, exhibit cusps or small cores with sizes smaller than the observed cores [39,40,41,42]. WDM predicts correct structures and cores with the right sizes for small scales (below kiloparsec) when the

**quantum**nature of the WDM particles, that is, the

**quantum pressure**of the fermionic WDM, is taken into account [9,10,11,12].

**self-consistent**and

**non-linear**Poisson equation

**solely**derived from the purely

**gravitational**interaction of the WDM particles and their

**fermionic**nature.

**both**regimes do appear. The strong gravitational field of the central black hole makes the WDM chemical potential large and positive near the center. This implies that the WDM behaves

**quantum mechanically**inside a small

**quantum core**with a nearly constant density.

- (i)
- We find that $\mu \left(r\right)$ takes large positive values in the inner regions as implied by Equation (28), then decreases until vanishing at $r={r}_{A}$, and becomes negative for $r>{r}_{A}$, as shown by our detailed resolution of the Thomas–Fermi equation (Section 3.3 and Figure 1). Therefore, ${r}_{A}$ is precisely the transition between the quantum and classical DM behaviors; ${r}_{A}$ plays the role of the
**quantum DM radius**of the galaxy for galaxies exhibiting a central black hole. Namely, inside ${r}_{A}$, the WDM gas is a self-gravitating**quantum**gas, while for $r\gtrsim {r}_{A}$ the WDM gas is a self-gravitating classical Boltzmann gas. The size ${r}_{A}$ of the quantum WDM core turns to be smaller for increasing galaxy masses and black hole masses. WDM inside a small core of radius ${r}_{A}$ is in a quantum gas high-density state, namely a Fermi nearly degenerate state with nearly constant density ${\rho}_{A}$. For the three representative families of galaxy solutions we find here, the values of ${r}_{A}$ and ${\rho}_{A}$ are given by Equations (75)–(77). The density ${\rho}_{A}$ is orders of magnitude larger than its values for $r>{r}_{A}$, where the WDM is in the classical Boltzmann regime. ${r}_{A}$ runs between 0.07 and 1.90 pc for galaxies with virial masses from ${10}^{16}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ to ${10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ (as shown in Section 3.3). In any case, ${r}_{A}$ is much**larger**than the Schwarzschild radius of the central black hole, which runs from ${10}^{-4}$ to ${10}^{-8}$ pc.This is an**important result**: in the vicinity of the central black hole, the fermionic WDM is always in a quantum regime, while far from the central black hole, the WDM follows a classical Boltzmann regime [12]. This is natural to understand: the strong attractive gravitational force near the central BH compacts the WDM, and its high density makes it to behave quantum mechanically. On the contrary, far from the BH, the gravitational forces are weak, the WDM is diluted, and it is then described by a classical Boltzmann gas. Ultracompact dwarf galaxies also exhibit WDM in a quantum regime [9,10,12]. - (ii)
- In addition, the black hole has an influence radius ${r}_{i}$. In the vicinity of the black hole, the gravitational force, due to the black hole, is larger than the gravitational force exerted by the dark matter. The influence radius of the black-hole ${r}_{i}$ is defined as the radius where both forces are of equal strength. Both forces point inward and always sum up. ${r}_{i}$ turns out to be larger than the radius ${r}_{A}$ where the chemical potential vanishes, ${r}_{i}>{r}_{A}$. The region ${r}_{A}<r<{r}_{i}$ is dominated by the central black hole, and the WDM exhibits there a classical behavior. For $r\lesssim {r}_{i}$, we see from Figure 1 and Figure 2 that both $\mu \left(r\right)$ and $\left|d\mu \right(r)/dr|$ (or equivalently, the dimensionless potential $\nu \left(\xi \right)$ and its derivative $\left|d\nu \right(\xi )/dx|$, x= ln $r/{r}_{h}$), follow the behavior dictated by the central black hole Equation (32), which produce straight lines on the left part of the logarithmic plots Figure 1 and Figure 2. Consistently, for $r\gtrsim {r}_{i},\phantom{\rule{0.277778em}{0ex}}\nu \left(\xi \right)$ and $\left|d\nu \right(\xi )/dx|$ are dominated by the WDM and exhibit a similar behavior to that of the Thomas–Fermi solutions without a central black hole [9,10,11,12].Figure 3 shows that the local density behavior is dominated by the black hole for $r\lesssim {r}_{i}$. For ${r}_{i}\lesssim r\lesssim {r}_{h}$, the WDM gravitational field dominates over the black hole field, and the galaxy core shows up. For medium and large galaxies, the core is seen as a plateau. At the same time, the chemical potential is negative for $r\gtrsim {r}_{i}>{r}_{A}$, and the WDM is a classical Boltzmann gas in this region.The surface density$${\Sigma}_{0}\equiv {r}_{h}\phantom{\rule{0.277778em}{0ex}}{\rho}_{0}\simeq 120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}/{\mathrm{pc}}^{2}\phantom{\rule{1.em}{0ex}}\mathrm{up}\phantom{\rule{3.33333pt}{0ex}}\mathrm{to}\phantom{\rule{0.277778em}{0ex}}10\u201320\%\phantom{\rule{0.277778em}{0ex}},$$
**constant**and independent of luminosity in different galactic systems (spirals, dwarfs irregular and spheroidal, and ellipticals) spanning over 14 magnitudes in luminosity and over different Hubble types [36,37]. It is therefore a useful physical characteristic scale in terms of which galaxy magnitudes are expressed. - (iii)
- We find the main galaxy magnitudes as the halo radius ${r}_{h}$, halo mass ${M}_{h}$, black hole mass ${M}_{BH}$, velocity dispersion, circular velocity, density, pressure and phase space density. Analytic formulae are derived for them and expressed in terms of the reference surface density ${\Sigma}_{0}$. Moreover, we can express the black hole mass as$${M}_{BH}=2.73116\times {10}^{4}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}\phantom{\rule{0.277778em}{0ex}}\frac{{\xi}_{0}}{{\left[{\xi}_{h}\phantom{\rule{0.277778em}{0ex}}{I}_{2}\left({\nu}_{0}\right)\right]}^{\frac{3}{5}}}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{\mathrm{pc}}^{2}}{120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{3}{5}}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{2\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}}{m}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{16}{5}}\phantom{\rule{0.277778em}{0ex}}$$${\xi}_{0}$ being the dimensionless central radius and ${I}_{2}\left({\nu}_{0}\right)$ the 2nd momentum of the distribution function Equation (21). The black hole mass ${M}_{BH}$ grows when ${\xi}_{0}$ grows. Notice that ${M}_{BH}$ does not simply grow linearly with ${\xi}_{0}$ due to the presence of the factor ${\left[{\xi}_{h}\phantom{\rule{0.277778em}{0ex}}{I}_{2}\left({\nu}_{0}\right)\right]}^{-\frac{3}{5}}$.
- (iv)
- We find in this approach explicit realistic galaxy solutions with central supermassive black holes and analyze three representative families of them: small size (mass) galaxies, intermediate size (mass) galaxies, and large size (mass) galaxies.For a fixed value of the surface density ${\Sigma}_{0}$, the solutions are parametrized by two truly physical parameters: the dimensionless central radius ${\xi}_{0}$ and the constant A characteristic of the chemical potential behavior Equation (32) at the center $\xi \to 0$. The dimensionless central radius ${\xi}_{0}$ is explicated in Equation (28). This is the ratio of the relevant physical parameters $(m,{M}_{BH},T)$ which appear in the chemical potential at the center. The constant A is truly physical too and characterizes the boundary condition of the chemical potential at the center in the presence of the central supermassive black hole, Equation (32). In the absence of the central SMBH: ${\xi}_{0}=0$, and the boundary condition at the center without BH, $\nu \left(0\right)=A$ is recovered.We derive an illuminating expression for the central radius ${r}_{0}$ for large galaxies ${M}_{h}\gtrsim {10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ explicitly in terms of the black hole mass ${M}_{BH}$, the halo mass ${M}_{h}$ and the reference surface density ${\Sigma}_{0}$. It follows from Equations (41), (43) and (66) that,$${r}_{0}={l}_{0}\phantom{\rule{0.277778em}{0ex}}{\xi}_{0}=126.762\phantom{\rule{0.277778em}{0ex}}\sqrt{\frac{{10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}{{M}_{h}}}\phantom{\rule{0.277778em}{0ex}}\frac{{M}_{BH}}{{10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}\phantom{\rule{0.277778em}{0ex}}\sqrt{\frac{120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}{{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{\mathrm{pc}}^{2}}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{pc}$$
- (v)
- We find from our extensive numerical calculations that the halo is thermalized at the uniform temperature ${T}_{0}$ and matches the circular temperature ${T}_{c}\left(r\right)$ by $r\sim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$. This picture is similar to the picture found in the absence of the central black hole which follows from the observed density profiles in the Eddington-like approach to galaxies [13]. We obtain here in the Thomas–Fermi approach and in the presence of a central supermassive black hole that the halo is thermalized at a uniform temperature ${T}_{0}$ inside $r\lesssim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$, which tends to the circular temperature ${T}_{c}\left(r\right)$ at $r\sim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$ as illustrated in Figure 4. The circular temperature is defined in terms of the circular velocity as ${T}_{c}\left(r\right)=\frac{m}{3}\phantom{\rule{0.277778em}{0ex}}{v}_{c}^{2}\left(r\right)$. The circular temperature is discussed in Section 3. We introduce the circular temperature ${T}_{c}\left(r\right)$ in terms of the circular (virial) velocity ${v}_{c}^{2}\left(r\right)$ in the same way the temperature $T\left(r\right)$ is defined in terms of the velocity dispersion $T\left(r\right)=m{v}^{2}\left(r\right)/3$. The circular velocity ${v}_{c}^{2}\left(r\right)$ is defined and found in Section 2. Near the central black hole, the space dependent temperature ${T}_{c}\left(r\right)$ is given by an equipartition and the virial theorem, as shown by Equations (68)–(70).From our extensive numerical calculations, we find that the galaxy mass increases and the galaxy size increases when the constant $\left|A\right|$ characteristic of the the central behavior of $\nu \left(\xi \right)$ for $\xi \to 0$ Equation (32) increases. This is similar to the case with the absence of central black holes, where $A=\nu \left(0\right)$ [9,10,12].
- (vi)
- We plot in Figure 4 the circular velocity given by Equation (58) vs. ${log}_{10}r/{r}_{h}$. For $r>{r}_{h}$, the circular velocity tends to the velocity dispersion as obtained from the Eddington equation for realistic density profiles [13]. For $r\to 0$, the circular velocity grows as in Equation (59) due to the central black hole field.
- (vii)
- We find in Equations (75)–(77) the WDM mass ${M}_{A}$ inside the quantum galaxy radius ${r}_{A}$. ${M}_{A}$ represents only a small fraction of the halo or virial mass of the galaxy but it is a significant fraction of the black hole mass ${M}_{BH}$. We see from Equations (75)–(77) that ${M}_{A}$ amounts to 20% of ${M}_{BH}$ for the medium and large galaxies and 45% for the small galaxy.
- (viii)
- We also measure the classical and quantum gas character of the galaxy plus the black hole system by means of the ratio $\mathcal{R}\left(r\right)$ between the particle de Broglie wavelength and the average interparticle distance. For $\mathcal{R}\lesssim 1$, the system is of a classical dilute nature while for $\mathcal{R}\gtrsim 1$, it is a macroscopic quantum system. We find $\mathcal{R}\left(r\right)$ in terms of the surface density and momenta of the gravitational or chemical potential in dimensionless units $\nu \left(\xi \right)$ Equations (80) and (81). Figure 5 shows ${log}_{10}\mathcal{R}$ vs. ${log}_{10}(r/{r}_{h})$ for the three representative galaxy solutions. The transition from the quantum to the classical regime occurs precisely at
**the same point**${r}_{A}$ where the chemical potential vanishes (see Figure 1), as it must be, showing the consistency and powerful of our treatment. This point defines the transition from the quantum to the classical behavior. - (ix)
- There is an
**important qualitative**difference between galaxy solutions with a black hole (${\xi}_{0}>0$), and galaxy solutions without a black hole (${\xi}_{0}=0$). In the absence of the central black hole, the halo mass ${M}_{h}$ reaches the minimal value ${M}_{h}^{min}$ Equation (82), which is the degenerate quantum limit at zero temperature ${T}_{0}^{min}=0$ [9,10,12]. In the presence of a central black hole, we find that the minimal temperature ${T}_{0}^{min}$ is always**non-zero**and that the halo mass takes as a minimal value$${M}_{h}^{min}=6.892\times {10}^{7}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{2\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}}{m}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{16}{5}}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{\mathrm{pc}}^{2}}{120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{3}{5}}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\mathbf{with}\phantom{\rule{3.33333pt}{0ex}}\mathbf{central}\phantom{\rule{3.33333pt}{0ex}}\mathbf{black}\phantom{\rule{3.33333pt}{0ex}}\mathbf{hole}.\phantom{\rule{0.277778em}{0ex}}$$$${M}_{h}^{min}=3.0999\times {10}^{4}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{2\phantom{\rule{0.166667em}{0ex}}\mathrm{keV}}{m}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{16}{5}}\phantom{\rule{0.277778em}{0ex}}{\left(\frac{{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{\mathrm{pc}}^{2}}{120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}}\right)}^{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\frac{3}{5}}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}{T}_{0}^{min}=0\phantom{\rule{0.277778em}{0ex}},\phantom{\rule{0.277778em}{0ex}}\mathbf{without}\phantom{\rule{3.33333pt}{0ex}}\mathbf{central}\phantom{\rule{3.33333pt}{0ex}}\mathbf{black}\phantom{\rule{3.33333pt}{0ex}}\mathbf{hole}.\phantom{\rule{0.277778em}{0ex}}$$The presence of the supermassive black hole**heats up**the dark matter gas and prevents it from becoming an exact degenerate gas at zero temperature. The minimal galaxy mass and size and most compact galaxy state with a central black hole is a nearly degenerate state at very low but non-zero temperature as seen from Equation (84). All matter studied in this paper is dark matter, and the only DM interaction is the gravitational interaction. The presence of the black hole naturally makes the DM particles acquire a higher velocity (and thus, a higher associated temperature), and in this sense, the SMBH does “heat” the dark matter around it. Gravitation self-consistently acts on such DM, and the SMBH adds too to such gravitational action. This is a very clean physical process, with a clean framework and clean conclusive results.This situation is clearly shown in Figure 6. The value of ${M}_{h}^{min}$ with a central black hole is $2.2233\times {10}^{3}$ times larger than without the black hole. Notice that the small galaxy solution Equation (75) is just 11% larger in halo mass than the minimal galaxy Equation (83) with a central black hole.**We conclude**that galaxies possessing a central black hole are in the dilute Boltzmann regime because of their large mass ${M}_{h}>{10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}>{M}_{h}^{min}$ [12]. On the contrary, compact galaxies, in particular, ultracompact galaxies in the quantum regime ${M}_{h}<2.3\times {10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$ [12],**cannot**harbor central black holes because the minimal galaxy mass with central black hole Equation (83) is always larger than $2.3\times {10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$. In other words, galaxies with masses ${M}_{h}<{M}_{h}^{min}$, namely ultracompact dwarfs,**necessarily**do not possess central black holes.The mass of the supermassive black hole ${M}_{BH}$ monotonically increases with the central radius ${r}_{0}$ or equivalently the dimensionless one ${\xi}_{0}$ at fixed A. In addition, for ${\xi}_{0}<0.3$, that is, for small supermassive black holes, and all A, the galaxy parameters, such as halo mass ${M}_{h}$, halo radius ${r}_{h}$, virial mass ${M}_{vir}$ and galaxy temperature ${T}_{0}$, become**independent**of ${\xi}_{0}$, showing a limiting galaxy solution. Only the BH mass depends on ${\xi}_{0}$ in this regime.Figure 7**displays our results for ${T}_{0}$**. Figure 8 displays our results for the black hole mass ${log}_{10}{M}_{BH}$ vs. the halo mass ${log}_{10}{M}_{h}$. We see that ${M}_{BH}$ is a**two-valued**function of ${M}_{h}$. For each value of ${M}_{h}$, there are two possible values for ${M}_{BH}$, which are quite close to each other. This two-valued dependence on ${M}_{h}$ is a direct consequence of the dependence of ${M}_{h}$ on A shown in Figure 6. The branch points on the left in Figure 8 correspond to the minimal galactic halo mass ${M}_{h}^{min}$ Equation (83) when the central supermassive black hole is present. At**fixed**${\xi}_{0}$, as shown in Figure 8, the central black hole mass ${M}_{BH}$**scales**with the halo mass ${M}_{h}$ as$${M}_{BH}=D\left({\xi}_{0}\right)\phantom{\rule{0.277778em}{0ex}}{M}_{h}^{\frac{3}{8}}\phantom{\rule{0.277778em}{0ex}},$$We find galaxy solutions with central black holes for arbitrarily small values of ${\xi}_{0}>0$ and a correspondingly, arbitrarily small central BH mass. There is no minimal central BH mass. The only minimal central BH mass possibility is zero (for $\xi =0$). - (x)
- We find that ${M}_{h}$
**scales**as ${r}_{h}^{2}$, which is the same scaling found in the Thomas–Fermi approach to galaxies in the absence of black holes [9,10,12]. We plot in Figure 10 the ordinary logarithm of the halo radius ${log}_{10}{r}_{h}$ vs. the ordinary logarithm of the halo mass ${log}_{10}{M}_{h}$ for galaxies with central black holes of many different masses. The halo mass in the absence of a central black hole behaves in the Thomas–Fermi approach as [12]$${M}_{h}=1.75572\phantom{\rule{0.277778em}{0ex}}{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{r}_{h}^{2}\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\mathbf{without}\phantom{\rule{3.33333pt}{0ex}}\mathbf{central}\phantom{\rule{3.33333pt}{0ex}}\mathbf{black}\phantom{\rule{3.33333pt}{0ex}}\mathbf{hole}.\phantom{\rule{0.277778em}{0ex}}$$The proportionality factor in this scaling relation is confirmed by the galaxy data [12]. In the presence of a central black hole, we find in the Thomas–Fermi approach an analogous relation$${M}_{h}=b\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{r}_{h}^{2}\phantom{\rule{1.em}{0ex}},\phantom{\rule{1.em}{0ex}}\mathbf{with}\phantom{\rule{3.33333pt}{0ex}}\mathbf{central}\phantom{\rule{3.33333pt}{0ex}}\mathbf{black}\phantom{\rule{3.33333pt}{0ex}}\mathbf{hole}\phantom{\rule{0.277778em}{0ex}}.$$**In summary**, the scaling relation Equation (86) and the coefficient b turn out to be**remarkably robust**. - (xi)
- We plot in Figure 12 the ordinary logarithm of the halo radius ${log}_{10}{r}_{h}$ versus the ordinary logarithm of the central black hole mass ${log}_{10}{M}_{BH}$ for many galaxy solutions. The halo radius ${r}_{h}$ turns out to be a double-valued function of ${M}_{BH}$. Remarkably, ${r}_{h}$ for
**fixed**${\xi}_{0}$ scales as$${r}_{h}=C\left({\xi}_{0}\right)\phantom{\rule{0.277778em}{0ex}}{M}_{BH}^{\frac{4}{3}}.$$The constant $C\left({\xi}_{0}\right)$ turns out to be a decreasing function of ${\xi}_{0}$. - (xii)
- We find the local pressure $P\left(r\right)$ as given by Equation (53). In Figure 13, we plot ${log}_{10}P\left(r\right)$ vs. ${log}_{10}(r/{r}_{h})$ for the three representative galaxy solutions. $P\left(r\right)$ monotonically decreases with r. The pressure $P\left(r\right)$ takes huge values in the quantum (high density) region $r<{r}_{A}$ and then it sharply decreases entering the classical (dilute) region $r>{r}_{A}$. In Figure 14, we plot the derived
**equation of state**${log}_{10}P\left(r\right)$ vs. ${log}_{10}\rho \left(r\right)/{\rho}_{0}$ for the three galaxy solutions we find here with central SMBH. The three curves almost coincide and are almost straight lines of unit slope. That is, the equation of state is in very good approximation of a perfect gas equation of state $P\left(r\right)={\sigma}^{2}\rho \left(r\right)$, which stems from the fact that galaxies with central black holes have halo masses ${M}_{h}>{M}_{h}\gtrsim {10}^{6}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}>{M}_{h}^{min}$, Equation (83), and therefore necessarily belong to the dilute Boltzmann classical regime [12]. The equation of state turns out to be a local (r-dependent) perfect gas equation of state because of the gravitational interaction (WDM self-gravitating perfect gas). Indeed, for galaxies with central black holes, the WDM is in a quantum (highly compact) regime inside the quantum radius ${r}_{A}$. However, because ${r}_{A}$ is in the parsec scale or smaller (see Equations (75)–(77)), the bulk of the WDM is in the Boltzmann classical regime, which is consistently reflected in the perfect gas equation of state behavior.

**independent**of any WDM particle physics model. It depends only on the fermionic WDM nature and gravity. The results presented in this paper do not depend on the precise value of the WDM particle mass m but only on the fact that m is in the kiloelectron volt scale, namely keV $2\lesssim m\lesssim 10$ keV, for example.

## 2. Galaxy Structure with Central Supermassive Black Holes in the WDM Thomas–Fermi Approach

**nonlinear**differential equation that determines

**self-consistently**the chemical potential $\mu \left(r\right)$ and constitutes the Thomas–Fermi approach [9,10,12] (see also Refs. [45,46,47]). This is a semi-classical approach to galaxy structure in which the quantum nature of the DM particles is taken into account through the quantum statistical distribution function $f\left(E\right)$.

#### 2.1. Thomas–Fermi Equations with a Central Black Hole

#### 2.2. Central Galactic Black Hole and Its Influence Radius

**important result**: in the vicinity of the central black hole, the fermionic WDM is always in a quantum regime, while far from the central black hole, the WDM follows a classical Boltzmann regime [12]. This is natural to understand: the strong attractive gravitational force near the central BH compacts the WDM, and its high density makes it to behave quantum mechanically. On the contrary, far from the BH, the gravitational forces are weak, the WDM is diluted, and it is then described by a classical Boltzmann gas.

**quantum DM radius**of the galaxy for galaxies exhibiting a central black hole. Namely, inside ${r}_{A}$, the WDM gas is a

**quantum**gas, while for $r\gtrsim {r}_{A}$, the WDM gas is a classical Boltzmann gas.

**quantum core**of DM forms around the central black hole. The size ${r}_{A}$ of the quantum core turns to be smaller for increasing galaxy masses and black-hole masses, because the larger the black hole mass, the larger its gravitational attraction on the WDM, which is thus more compact, and, hence, the smaller the quantum radius core ${r}_{A}$.

**larger**than the Schwarzschild radius of the central black hole which runs from ${10}^{-4}$ to ${10}^{-8}$ pc.

#### 2.3. Main Physical Magnitudes of the Galaxy plus Central Black Hole System

**constant**and independent of luminosity in different galactic systems (spirals, dwarfs irregular and spheroidal, and ellipticals) spanning over 14 magnitudes in luminosity and over different Hubble types. More precisely, all galaxies seem to have the same value for ${\Sigma}_{0}$, namely ${\Sigma}_{0}\simeq 120\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}/{\mathrm{pc}}^{2}$ up to 10–20% [36,37,38]. It is remarkable that, at the same time, other important structural quantities, such as ${r}_{h},\phantom{\rule{0.277778em}{0ex}}{\rho}_{0}$, the baryon fraction and the galaxy mass vary by orders of magnitude from one galaxy to another.

#### 2.4. Galaxy Properties in the Diluted Boltzmann Regime

## 3. Explicit Thomas–Fermi Galaxy Solutions with Central Supermassive Black Holes

#### 3.1. Local Thermal Equilibrium in the Galaxy

**(i)**a constant temperature ${T}_{0}$ for $r\lesssim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$, and

**(ii)**a space-dependent temperature $T\left(r\right)$ for $3\phantom{\rule{0.277778em}{0ex}}{r}_{h}<r\lesssim {R}_{virial}$, which slowly decreases with r. $T\left(r\right)$ outside the halo radius nicely follows the decrease in the circular velocity squared ${T}_{c}\left(r\right)$ [13]. These results are physically understood because thermalization is more easy achieved in the inner regions due to the fact that the gravitational interaction is stronger than in the external regions where instead virialization occurs. The slow decreasing in the temperature $T\left(r\right)$ with the halo radius consistently corresponds to a transfer flux of the kinetic energy into potential energy. These results were derived from empirical observed density profiles, which do not have information of the regions near the central black hole.

- Near the central black hole, the space-dependent temperature is given by equipartition and the virial theorem$${T}_{c}\left(r\right)=\frac{m}{3}\phantom{\rule{0.277778em}{0ex}}{v}_{c}^{2}\left(r\right)=\frac{G\phantom{\rule{0.277778em}{0ex}}m}{3\phantom{\rule{0.277778em}{0ex}}r}\phantom{\rule{0.277778em}{0ex}}{M}_{BH}=\frac{{T}_{0}\phantom{\rule{0.277778em}{0ex}}{\xi}_{0}}{3\phantom{\rule{0.277778em}{0ex}}\xi}$$
- For $\xi \ge {\xi}_{0}/3$ we set$${T}_{c}\left(r\right)={T}_{0}\phantom{\rule{0.277778em}{0ex}}.$$Here, the circular temperature ${T}_{c}\left(r\right)$ associated to the velocity squared is given by$${T}_{c}\left(r\right)=\frac{m}{3}\phantom{\rule{0.277778em}{0ex}}\frac{G\phantom{\rule{0.277778em}{0ex}}[M\left(r\right)+{M}_{BH}]}{r}\phantom{\rule{0.277778em}{0ex}},$$$${T}_{c}\left(r\right)=\frac{1}{3}\phantom{\rule{0.277778em}{0ex}}\xi \phantom{\rule{0.277778em}{0ex}}\left|{\nu}^{\prime}\left(\xi \right)\right|\phantom{\rule{0.277778em}{0ex}}{T}_{0}\phantom{\rule{0.277778em}{0ex}}.$$
- We find from our extensive numerical calculations that the halo is thermalized at the uniform temperature ${T}_{0}$ and matches the circular temperature ${T}_{c}\left(r\right)$ by $r\sim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$. This picture is similar to the picture found in the absence of the central black hole which follows from the observed density profiles in the Eddington-like approach to galaxies [13]. We obtain here in the Thomas–Fermi approach and in the presence of a central supermassive black hole that the halo is thermalized at a uniform temperature ${T}_{0}$ inside $r\lesssim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$ which matches the circular temperature ${T}_{c}\left(r\right)$ at $r\sim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$ (see Figure 4).
- In summary, each galaxy solution with a central black hole depends only on
**two free parameters**: the dimensionless constants ${\xi}_{0}$ and A in Equation (32). We have a two-parameter family of Thomas–Fermi galaxy solutions with a central supermassive black hole parametrized by ${\xi}_{0}$ and A.The black hole mass ${M}_{BH}$ grows when ${\xi}_{0}$ grows as shown by Equation (45). Notice that ${M}_{BH}$ does not simply grow linearly with ${\xi}_{0}$ due to the presence of the factor$${\left[{\xi}_{h}\phantom{\rule{0.277778em}{0ex}}{I}_{2}\left({\nu}_{0}\right)\right]}^{-\frac{3}{5}},$$

#### 3.2. Thomas–Fermi Equations with r-Dependent Temperature Tc(r)

#### 3.3. Examples of Thomas–Fermi Galaxy Solutions with a Central Supermassive Black Hole

**Quantum to classical behavior**: The central black hole strongly attracts the WDM and makes its density very high for $r<{r}_{A}$, where a compact quantum core gets formed. The dimensionless chemical potential $\nu \left(\xi \right)$ vanishes at $r={r}_{A}$ and becomes negative for $r>{r}_{A}$. The density $\rho \left(r\right)$ drops several orders of magnitude immediately after ${r}_{A}$ as shown in Figure 3. $\nu \left(\xi \right)$ is negative for $r>{r}_{A}$, and the WDM exhibits there a classical Boltzmann behavior while the WDM exhibits a quantum behavior for $r<{r}_{A}$, where the chemical potential is large and positive. Therefore, the point ${r}_{A}$ where the chemical potential vanishes**defines the transition from the quantum to classical behavior**. In the quantum region $r<{r}_{A}$, the density exhibits a constant plateau as shown in Figure 3. Notice from Equation (75) that ${r}_{A}$ turns to be much larger than the Schwarzschild radius of the central black hole ${r}_{A}\gg {r}_{BH}^{Schw}$.**Black hole influence radius ${r}_{i}$**: For $r<{r}_{i}$, the black hole gravitational field dominates over the dark matter gravitational field. The influence radius ${r}_{i}={l}_{0}\phantom{\rule{0.277778em}{0ex}}{\xi}_{i}$ is defined by Equation (31). The black hole influence radius turns out to be larger than the radius ${r}_{A}$ where the chemical potential vanishes, ${r}_{i}>{r}_{A}$. The region ${r}_{A}<r<{r}_{i}$ is dominated by the central black hole and the WDM exhibits there a classical behavior. For $r\lesssim {r}_{i}$, we see from Figure 1 and Figure 2 that both $\nu \left(\xi \right)$ and $\left|d\nu \right(\xi )/dx|$ follow the behavior dictated by the central black hole. That is, from Equation (32)$$\nu \left(\xi \right)\simeq {\xi}_{0}\phantom{\rule{0.277778em}{0ex}}{e}^{-x}+A={\xi}_{0}\phantom{\rule{0.277778em}{0ex}}\frac{{r}_{h}}{r}+A,\phantom{\rule{1.em}{0ex}}|d\nu \left(\xi \right)/dx|\simeq {\xi}_{0}\phantom{\rule{0.277778em}{0ex}}{e}^{-x}={\xi}_{0}\phantom{\rule{0.277778em}{0ex}}\frac{{r}_{h}}{r},\phantom{\rule{1.em}{0ex}}x\equiv ln\frac{r}{{r}_{h}}\phantom{\rule{0.277778em}{0ex}},$$Figure 3 shows that the local density behavior is dominated by the black hole for $r\lesssim {r}_{i}$. Coherently, for $r\gtrsim {r}_{i}$ the WDM gravitational field dominates over the black hole field and the galaxy core shows up for ${r}_{i}\lesssim r\lesssim {r}_{h}$ in Figure 3. For medium and large galaxies, the core is seen as a plateau. At the same time, the chemical potential is negative for $r\gtrsim {r}_{i}>{r}_{A}$, and the WDM is a classical Boltzmann gas in this region.**Halo radius ${r}_{h}$**: Finally, we see in Figure 3 the tail of the WDM density profile for $r\gtrsim {r}_{h}$, which exhibits a similar shape for all three galaxy solutions.**WDM thermalization**: As shown by Figure 4, the velocity dispersion $<{v}^{2}>\left(r\right)$ is constant as a function of r, indicating a thermalized WDM with temperature$${T}_{0}=\frac{1}{3}\phantom{\rule{0.277778em}{0ex}}m\phantom{\rule{0.277778em}{0ex}}<{v}^{2}>\phantom{\rule{0.277778em}{0ex}}.$$WDM is thermalized as in the absence of the central black hole [12]. This is consistent with the use of a thermal Fermi–Dirac distribution function for $r\ge {r}_{0}/3$.- We also plot in Figure 4 the circular velocity given by Equation (58) vs. ${log}_{10}r/{r}_{h}$. For $r>{r}_{h}$, the circular velocity tends to the velocity dispersion as obtained from the Eddington equation for realistic density profiles [13]. For $r\to 0$, the circular velocity grows as in Equation (59) due to the central black hole field.
- WDM inside a small core of radius ${r}_{A}$ is in a quantum gas high density state, namely, a Fermi nearly degenerate state with nearly constant density ${\rho}_{A}$. For the three galaxy solutions, the values of ${r}_{A}$ and ${\rho}_{A}$ are given by Equations (75)–(77). Notice that the density ${\rho}_{A}$ is orders of magnitude larger than its values for $r>{r}_{A}$, where the WDM is in the classical Boltzmann regime.
- We also give in Equations (75)–(77) the WDM mass ${M}_{A}$ inside ${r}_{A}$. ${M}_{A}$ represents only a small fraction of the halo or virial mass of the galaxy, but it is a significant fraction of the black hole mass ${M}_{BH}$. We see from Equations (75)–(77) that ${M}_{A}$ amounts to 20% of ${M}_{BH}$ for the medium and large galaxies and 45% for the small galaxy.

#### 3.4. Quantum Physics in Galaxies

## 4. Systematic Study of the Thomas–Fermi Galaxy Solutions with a Central Supermassive Black Hole

**two free parameters**: ${\xi}_{0}$ and A defining the boundary conditions near the center (see Equation (32)), ${\xi}_{0}$ being the dimensionless central radius and A characterizing the central chemical potential behavior.

**important qualitative**difference between galaxy solutions with a black hole (${\xi}_{0}>0$ ), and galaxy solutions without a black hole (${\xi}_{0}=0$). In the absence of the central black hole, the halo mass ${M}_{h}$ monotonically decreases when A increases until ${M}_{h}$ reaches a minimal value, which is the degenerate quantum limit at zero temperature [9,10,12]:

**cannot**harbor central black holes.

**heats up**the dark matter gas and prevents it from becoming an exact degenerate gas at zero temperature. The minimal mass and size and most compact galaxy state with a supermassive black hole is a nearly degenerate state at very low temperature as seen from Equation (84).

**independent**of ${\xi}_{0}$, showing a limiting galaxy solution. Only the BH mass depends on ${\xi}_{0}$ in this regime.

**two-valued**function of ${M}_{h}$. For each value of ${M}_{h}$, there are two possible values for ${M}_{BH}$. These two values of ${M}_{BH}$ for a given ${M}_{h}$ are quite close to each other. This two-valued dependence on ${M}_{h}$ is a direct consequence of the dependence of ${M}_{h}$ on A shown in Figure 6.

**fixed**${\xi}_{0}$, as shown in Figure 8, the central black hole mass ${M}_{BH}$ scales with the halo mass ${M}_{h}$ as

- We plot in Figure 9 the halo galaxy mass ${log}_{10}{M}_{h}$ vs. the galaxy temperature ${log}_{10}{T}_{0}/\mathrm{K}$. The halo mass ${M}_{h}$ grows when ${T}_{0}$ increases. Colder galaxies are smaller. Warmer galaxies are larger.
- We find galaxy solutions with central black holes for arbitrarily small values ${\xi}_{0}>0$ and correspondingly arbitrarily small central BH mass. There is no emergence of a minimal mass for the central black hole.

#### 4.1. Universal Scaling Relations in the Presence of Central Black Holes

**all cases**that ${M}_{h}$ scales as ${r}_{h}^{2}$. The same scaling was found in the Thomas–Fermi approach to galaxies in absence of black holes [9,10,12].

**Remarkably**, ${r}_{h}$ scales for

**fixed**${\xi}_{0}$ as

#### 4.2. Pressure and Equation of State in the Presence of Central Black Holes

## 5. Conclusions

- We presented here a novel study of galaxies with central supermassive black holes, which shows itself as fruitful and enlightening. This framework stresses the key role of gravity and warm dark matter in structuring galaxies with their central supermassive black holes and provides correctly the major physical quantities to be first obtained for the galaxy–black-hole system: the masses, sizes, densities, velocity dispersions, and their internal physical states. This also yields a physical and precise characterization of whether they are compact, ultracompact, low density or large dilute galaxies, encompassed with their classical physics and quantum gas physical properties.
- We thus found different regions structuring internally the halo of the galaxy from the vicinity of the supermassive central black hole region to the external regions or virial radius. For all galaxies harboring a central black hole, there is a transition from the quantum to the classical regime going from the more compact inner regions, which are in a quantum gas state, to the classical dilute regions in a Boltzmann-like state. This is accompanied by a decreasing in the local temperature from the central warmer regions to the colder external ones. The SMBH heats the DM near around and prevents it from becoming exactly degenerated at zero temperature. Although the inner DM quantum core is highly compact in a nearly degenerate quantum gas state, it is not at zero temperature. Inside $r\lesssim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$, the halo is thermalized at a uniform or slowly varying local temperature ${T}_{0}$, which tends to the circular temperature ${T}_{c}\left(r\right)$ at $r\sim 3\phantom{\rule{0.277778em}{0ex}}{r}_{h}$.
- We formulated the problem of galaxy structure with central supermassive black holes in the WDM Thomas–Fermi approach and found the main physical magnitudes and properties of the galaxy plus black hole system. We solved the corresponding equations and boundary conditions, found three representative families of realistic galaxy solutions (small, medium and large size galaxies) with central supermassive black holes, and provided a systematic analysis of the new quantum and classical physics properties of the system. The approach naturally incorporates the quantum pressure of the self-gravitating dark matter fermions, showing its full power and clearness to treat the galaxy plus supermassive black hole system. The realistic astrophysical masses of supermassive black holes are naturally obtained in this framework.
- We found the main important physical differences between galaxies with and without the presence of a central black hole. In the presence of a central black hole, both the quantum and classical behaviors of the dark matter gas do co-exist generically in any galaxy from the compact small galaxies to the dilute large ones, and a novel galaxy halo structure with three regions show up.
- The transition from the quantum to classical regime occurs at the point ${r}_{A}$, where the chemical potential vanishes and which is, in addition, precisely and consistently, the point where the particle wavelength and the interparticle distance are equal (their ratio being a measure of the quantum or classical properties of the system). The quantum radius ${r}_{A}$ is larger for the smaller and more compact galaxies and diminishes with increasing galaxy and black hole masses for the large dilute galaxies. The WDM mass ${M}_{A}$ inside the quantum galaxy radius ${r}_{A}$ represents only a small fraction of the halo mass ${M}_{h}$ or virial mass of the galaxy, but it is a significant fraction of the black hole mass ${M}_{BH}$. ${M}_{A}$ amounts to 20% of ${M}_{BH}$ for the medium and large galaxies and 45% for the small galaxies.
- The minimal mass ${M}_{h}^{min}$ that a galaxy should have in order to harbor SMBHs was found, which shows, among other features, why compact or ultracompact galaxies (in the range ${10}^{4}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}<{M}_{h}<{10}^{7}\phantom{\rule{0.277778em}{0ex}}{M}_{\odot}$) cannot harbor necessarily central black holes.
- Novel universal scaling relations in the presence of a central supermassive black hole were derived: black hole mass ${M}_{BH}$, halo radius ${r}_{h}$ and halo mass ${M}_{h}$ relations. The black hole mass ${M}_{BH}$ turned out to be a two-valued function of the halo mass ${M}_{h}$ and size ${r}_{h}$, and we found the local pressure and equation of state of the galaxy–black-hole system and its different regimes.
- A more detailed quantitative account of the main features and results of this paper is presented in the Introduction, Section 1.
- The circular velocities, galactic rotation curves in the WDM halo with central SMBH are discussed, self-consistently computed and plotted in Section 2, Equations (56)–(59), (63) and (64) and Figure 4 of this paper, together with the obtained velocity dispersions. These results are presented for the three family of galaxy solutions with SMBHs obtained here with this approach: small or dwarf galaxies, medium galaxies and large galaxies. They remarkably encompass the other relevant physical magnitudes obtained for these systems in this paper with the same approach. Toward the central regions, the circular velocity grows as in Equation (59) due to the central black hole field. As seen from Figure 4, the dispersion velocity is constant in the Boltzmann (outer or classical) region and in the quantum (inner or compact) region, indicating WDM thermalization. For $r>{r}_{h}$, the circular velocity tends to the velocity dispersion. Remarkably, this result confirms the same behavior we obtained independently with a different approach (the inverse problem or the Eddington integral equation for galaxies which we developed in Ref. [13]), namely, given the observed density profiles as input, the velocities, pressure and other galaxy magnitudes are obtained and analyzed as output. The observed density profiles being, by definition, real realistic data, the obtained results from them are valid, realistic magnitudes. Moreover, another robust verification of the kiloelectron volt WDM Thomas–Fermi approach is the 10 independent sets of observational data we used in Ref. [11] for galaxy masses from $5\times {10}^{9}{M}_{\odot}$ to $5\times {10}^{11}{M}_{\odot}$. And also for many other different observational data sets [48,49,50,51,52,53,54,55,56,57,58,59,60]. The theoretical and observational rotation curves do agree. In addition, they agree extremely well with the observational rotation curves described by the empirical Burkert profile for $r\ge 2{r}_{h}$ (they differ from each other by only $2.4$ percent). These results show the success of the kiloelectron volt WDM Thomas–Fermi approach to correctly describe the galaxy structures.
- We first investigated pure WDM galaxies with their central black holes because DM is, on average, the over-dominant component in galaxies, and it is reasonable then to investigate first the effects of gravity plus WDM. This is thus a first approximation, more precisely the zero order of a first approximation in which the visible matter component, baryons, can be incorporated to provide a most accurate and complete picture. We observed that these zero order results found here are already realistic, very good and robust results, and they set the basis and the direction for improvements and a more complete understanding.
- Baryons will provide corrections to this picture and will allow to study other processes in which ordinary matter naturally plays a role as the gas and star components, but baryons will not change drastically the pure WDM results found here, which are the structural galaxy and black hole properties, masses, sizes, their scaling and relations, density profiles, the classical and quantum natures of the halo regions and their physical, high-density, medium-density or dilute states, the halo thermalization and virialization.
- This predictive theory and the obtained classes of solutions include very well the different galaxy types through their generic and important physical quantitive properties, such as the pressure, density, equation of state, mass, halo structure, and central black holes. Thus, we have primarily three galaxy classes: large dilute galaxies, intermediate galaxies, and small compact galaxies, whatever their astronomical empirical/historical name. The Milky Way galaxy is one of the galaxies in the large dilute galaxy class we found with all the specific properties of this class, mass, structure and central SMBH. Messier 87 is a larger (“supergiant”) galaxy within the large class of galaxies we found, hosting, consequently, a bigger central SMBH (M87).
- As explained in the paper, the central quantum WDM gaz is relevant for the presence of the obtained central non-cusped cores and their correct sizes, and for the presence of the central SMBHs and their realistic mass values without any ad hoc prescription. Recall, for instance, Figure 3 of the paper, which displays the density $\rho \left(r\right)$ normalized at the influence radius ${r}_{i}$, vs. $r/{r}_{h}$ for the three family of galaxy solutions with central SMBHs we found, large dilute galaxies, intermediate galaxies, and small compact galaxies, covering the different types of galaxies with their central SMBHs. The Milky Way is within the large dilute galaxy class we found with all the characteristic properties of this class: mass, structure and central SMBH, namely ${M}_{BH}=4.100\times {10}^{6}{M}_{\odot}$, galaxy mass $M=(0.8-1.5)\times {10}^{12}{M}_{\odot}$ and ${r}_{h}=580+/-120\text{}\mathrm{kpc})$. Notice that in the quantum WDM gas region $r<{r}_{A}$, the density is constant, clearly exhibiting a plateau behavior corresponding to the quantum macroscopic Fermi DM gas behavior in such a region. Figure 3 shows that the local density behavior is dominated by the black hole for $r\lesssim {r}_{i}$. Coherently, for ${r}_{i}\lesssim r\lesssim {r}_{h}$, the WDM gravitational field dominates over the black hole field, and the galaxy core shows up. For medium and large galaxies, such as the Milky Way, the core is seen as a plateau. At the same time, the chemical potential is negative for $r>{r}_{i}>{r}_{A}$ and the WDM is a Boltzmann gas in this region.The first or primary “signatures” are the set of galactic physical magnitudes and structural properties: sizes, masses, cored density profiles and their correct sizes. In particular, dwarf galaxies appear to be a full quantum macroscopic system. Dwarf galaxies are really interesting to observe in this respect, as tracers of the quantum kiloelctron volt WDM nature in nearly degenerated states, their temperatures and properties. These are important features all found and provided by the same and one single approach, without tailored prescriptions, and without considering different approaches for each of the different computed magnitudes. Therefore, these are all “signatures” for this approach.These results consistently encompass the ones shown in Figure 2: the derivative of the chemical potential vs. $(r/rh)$ for the three families of galaxy solutions with central SMBH. For $r\lesssim {r}_{i}$, the behavior is dictated by the central black hole. For $r>{r}_{i}$, they are dominated by the WDM and in this region exhibit a similar behavior to the Thomas-Fermi galaxy solutions without a central SMBH [9,10,11,12]. For galaxies with central black holes, the WDM is in a quantum (highly compact) regime inside the quantum radius ${r}_{A}$. Because ${r}_{A}$ is in the parsec scale or smaller (see Equations (75)–(77)), the bulk of the WDM is then in the Boltzmann regime, e.g., Figure 13 and Figure 14. In the quantum gas (dense) region, the equation of state becomes steeper than the perfect gas. Notice the huge values of $P\left(r\right)$ in the quantum (high density) region $r<{r}_{A}$ and its sharp decrease entering the classical (dilute) region $r>{r}_{A}$, all consistent with the other results we found.
- In all the obtained results, and in the Introduction, we carefully compared the results and solutions we obtained in this paper for galaxy systems with a central black hole and without a central black hole. From our results here, we recover, in particular, the galaxy structures, the cores of quantum WDM and their right sizes, the velocity dispersions, the scaling relations, the equation of state and the other related results in the absence of black holes, which we already discussed in our previous works [9,10,11,12,13] in which the careful check for rotation curves, masses, scaling relations, velocities, are in full agreement with observations for the whole set of properties.Cored density profiles and their right size, halo masses, are in full agreement with the observations. The quantum DM nature in the central regions is not an exotic property: it is the quantum nature of the degenerate or nearly degenerate gaz of DM particles. Interaction is fully gravitational, namely, a self-gravitating and self-consistent WDM gas. The first or primary “ signatures” are, therefore, the set of galactic physical magnitudes and structural properties—the sizes, masses, central cored profiles, velocity distributions, and surface density we found and confronted to real astronomical observations. Other effects, such as the influence of such DM structures, could in turn exert on the propagation of generated gravitational waves, on the accretion processes, which are superimposed effects, or on the secondary dark matter processes or secondary signatures, a problem which would require individual analysis and is clearly beyond the scope of the present paper, which is devoted to the primary dark matter effects, namely the dark matter galactic structures. Those secondary effects, such as the orbits, diffusion and absorption in the different regions and regimes around the BH require the interaction in propagation with other non dark matter components, as electromagnetic effects and accretion plasmas are not the subject of this paper.
- For the primary objectives of obtaining the galaxy structural magnitudes, e.g., the realistic astrophysical masses of the galaxies, the realistic SMBH central masses, their sizes, velocities, cored density and pressure profiles, the Newtonian treatment is largely enough. Recall that the Thomas–Fermi approach is a statistical many body approach. Near the black hole horizon, there will appear effects of spiraling, orbiting, or a glory effect (180 degrees backscattering) but it does not truly affect the properties and magnitudes of the galaxy–black-hole system (and this paper is not devoted to test GR black hole, horizon or baryonic effects). The values of the relevant radii are as follows (besides the halo radius): the quantum galaxy radius ${r}_{A}$, the BH influence radius ${r}_{i}$, and the horizon black hole radius ${r}_{BH}^{Schw}$ are given by Equations (75)–(77). The horizon radius is always extremely small with respect to the other radii. For galaxies with virial masses from ${10}^{16}$ to ${10}^{7}{M}_{\odot}$, ${r}_{A}$ runs between $0.07$ and $1.90$ pc, respectively (as shown in Section 3.3), while the horizon radius of the central black hole runs from ${10}^{-4}$ to ${10}^{-8}$ pc for such range of galaxy masses, respectively; ${r}_{i}$ is larger than ${r}_{A}$:${r}_{i}>{r}_{A}>>{r}_{BH}^{Schw}$. The important point in order to account for both the realistic galaxy and their central SMBH masses, their sizes, velocities, pressure profiles, density profiles and the core sizes, is the DM nature: kiloelectron volt WDM with its quantum and its relativistic treatment.Newtonian black holes have many common properties with general relativity black holes, and most importantly, they both have the same size. Recall that Newtonian and post-Newtonian approximation have proven to be remarkably effective, even in describing strong-field systems and astrophysical black hole systems (e.g., binary bhs) in spiraling toward a final merger (e.g., Ref. [61] and references therein). Of course, a fully GR treatment is needed to account for a causal space–time structure, central classical space–time curvature singularity, and precise tests of GR of the horizon or of the “no hair theorems”, for which inner orbits at milliparsec (mpc) distances need to be considered but not for the magnitudes of the galactic masses, sizes, or of their central SMBHs. The GR treatment minimally affects the obtained huge mass magnitude values. A high merit of the kiloelectron volt WDM approach is that it accounts naturally (with dark matter only) for the realistic astrophysical masses, sizes, density and velocity profiles, rotation curves, equation of state and structural properties of both galaxies and their central SMBHs.

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Analytic Evaluation of the Density and the Pressure

#### Appendix A.1. The Quantum (High Density) Regime

#### Appendix A.2. The Classical Boltzmann Regime

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**Figure 1.**The dimensionless chemical potential ${log}_{10}\left|\nu \left(\xi \right)\right|$ vs. ${log}_{10}(\xi /{\xi}_{h})={log}_{10}(r/{r}_{h})$ for the three illustrative galaxy solutions with central SMBH defined by Equation (74). $\nu \left(\xi \right)$ is negative for $r>{r}_{A}={l}_{0}\phantom{\rule{0.277778em}{0ex}}{\xi}_{A}$, and WDM exhibits there a classical dilute Boltzmann gas behavior, while WDM exhibits a compact quantum gas behavior for $r<{r}_{A}$, where the chemical potential is positive. The point ${r}_{A}$ where the chemical potential vanishes

**defines the transition from the quantum to the classical galaxy WDM behavior**. ${r}_{A}$ is at the downward spike of ${log}_{10}\left|\nu \left(\xi \right)\right|$ where $\nu \left(\xi \right)$ vanishes.

**Figure 2.**The derivative of the dimensionless chemical potential ${log}_{10}\left[\right|d\nu \left(\xi \right)/dx|/3]$ vs. ${log}_{10}(\xi /{\xi}_{h})={log}_{10}(r/{r}_{h})$ for the three galaxy solutions with central SMBHs defined by Equation (74). For $r\lesssim {r}_{i}$, both $\nu \left(\xi \right)$ and $\left|d\nu \right(\xi )/dx|$ follow the behavior dictated by the central black hole. ${r}_{i}$ is the influence radius of the BH defined by Equation (31). For $r\gtrsim {r}_{i},\phantom{\rule{0.277778em}{0ex}}\nu \left(\xi \right)$ and $\left|d\nu \right(\xi )/dx|$, they are dominated by WDM and exhibit a similar behavior to that for the Thomas–Fermi galaxy solutions without a central black hole [9,10,11,12].

**Figure 3.**The density $\rho $ normalized at the influence radius ${r}_{i}$, ${log}_{10}(\rho \left(r\right)/{\rho}_{0})$ vs. ${log}_{10}(r/{r}_{h})$ for the three galaxy solutions with central SMBHs. Notice that in the quantum gas WDM region $r<{r}_{A}$, the density is constant clearly exhibiting a plateau behavior corresponding to the quantum Fermi gas behavior in such a region.

**Figure 4.**The velocity dispersion $<{v}^{2}>\left(r\right)$ and the circular velocity ${v}_{c}^{2}\left(r\right)$ for the three representative galaxy solutions with central SMBH vs. ${log}_{10}(r/{r}_{h})$. The velocity dispersion is constant in the Boltzmann and in the quantum regions, indicating a thermalized WDM with two different temperatures, ${T}_{0}=\frac{1}{3}\phantom{\rule{0.277778em}{0ex}}m\phantom{\rule{0.277778em}{0ex}}<{v}^{2}>\left(r\right)$. For $r>{r}_{h}$, the circular velocity tends to the velocity dispersion [13]. These results are in agreement with the DM thermalization found in the absence of a central BH [12,13].

**Figure 5.**The ratio $\mathcal{R}$ of the particle de Broglie wavelength to the interparticle distance in the galaxy as a function of r for the three representative galaxy solutions with central SMBH: small galaxy (red), medium galaxy (green), and large galaxy (blue). For $\mathcal{R}\lesssim 1$, the galaxy plus SMBH system is of a classical nature, while for $\mathcal{R}\gtrsim 1$, the system is quantum. The transition from the quantum to the classical regime occurs precisely at

**the same point**${r}_{A}$ where the chemical potential vanishes (see Figure 1), showing the consistency and power of our treatment. This point defines the transition from the quantum to the classical behavior.

**Figure 6.**The halo mass ${log}_{10}{M}_{h}$ vs. the constant A of the chemical potential behavior at the origin for fixed values of ${\xi}_{0}$. The halo mass ${M}_{h}$ increases with ${\xi}_{0}$ at fixed A. ${M}_{h}$ increases when the absolute value of A increases at fixed ${\xi}_{0}>0$. In the absence of the central black hole, the halo mass monotonically decreases when A increases until ${M}_{h}$ reaches its minimal value, Equation (82) at the degenerate quantum limit at zero temperature [9,10,12]. In the presence of a central black hole, we find a larger minimal value for the halo mass ${M}_{h}^{min}$ Equation (83) with a non zero minimal temperature ${T}_{0}^{min}$ Equation (84). Therefore, there is an

**important qualitative**difference between galaxy solutions with a black hole ${\xi}_{0}>0$, and galaxy solutions without a black hole ${\xi}_{0}=0$.

**Figure 7.**The galaxy temperature ${log}_{10}({T}_{0}/\mathrm{K})$ vs. the constant A of the chemical potential behavior at the origin, for fixed values of ${\xi}_{0}$. As for the halo mass ${M}_{h}$, the galaxy temperature ${T}_{0}$ increases with ${\xi}_{0}$ at fixed A. ${T}_{0}$ increases when the absolute value of A increases at fixed ${\xi}_{0}>0$. In the absence of a black hole, the galaxy temperature ${T}_{0}$ tends to zero for $A\to \infty $ (at the exact Fermi degenerate state), while in the presence of a central black hole, we find that ${T}_{0}$ is always larger than a minimal non-zero value${T}_{0}^{min}$ given by Equation (84).

**Figure 8.**The black hole mass ${log}_{10}{M}_{BH}$ vs. the halo mass ${log}_{10}{M}_{h}$. The black hole mass ${M}_{BH}$ turns out to be a

**two-valued**function of ${M}_{h}$. For each value of ${M}_{h}$, there are two values for ${M}_{BH}$. These two values of ${M}_{BH}$ for a given ${M}_{h}$ are quite close to each other. This two-valued dependence on ${M}_{h}$ is a direct consequence of the dependence of ${M}_{h}$ on the central chemical potential behavior characterized by the constant A as shown in Figure 6.

**Figure 9.**The halo galaxy mass ${log}_{10}{M}_{h}$ vs. the galaxy temperature ${log}_{10}{T}_{0}/\mathrm{K}$. ${M}_{h}$ turns to be a

**two-valued**function of ${T}_{0}$. The halo mass ${M}_{h}$ grows when ${T}_{0}$ increases. Colder galaxies are smaller, while warmer galaxies are larger. We see at the branch points in Figure 9 the minimal galaxy temperature ${T}_{0}^{min}$ Equation (84) when a supermassive black hole is present.

**Figure 10.**The halo radius ${log}_{10}{r}_{h}$ vs. the common logarithm of the halo mass ${log}_{10}{M}_{h}$ for galaxies with supermassive central black holes of many different masses. ${r}_{h}$ turns out to be a

**two-valued**function of ${M}_{h}$. We see that ${M}_{h}$ accurately scales as ${r}_{h}^{2}$. The same scaling was found in the Thomas–Fermi approach for galaxies in the absence of black holes [9,10,12].

**Figure 11.**The scaling amplitude $b\equiv {M}_{h}/\left[{\Sigma}_{0}\phantom{\rule{0.277778em}{0ex}}{r}_{h}^{2}\right]$ as a function of the halo mass ${M}_{h}$. Except for halo masses near the minimum halo mass ${M}_{h}^{min}$ Equation (83), b in the presence of a central black hole takes values up to 10% below its value $1.75572$ in the absence of a central black hole Equation (85). The continuous red horizontal line $b=1.75572$ corresponds to galaxies without central black holes (Equation (85)).

**Figure 12.**The common logarithm of the halo radius ${log}_{10}{r}_{h}$ vs. the common logarithm of the central black hole mass ${log}_{10}{M}_{BH}$ for many galaxy solutions. The halo radius ${r}_{h}$ turns out to be a

**double-valued function**of ${M}_{BH}$. Remarkably, ${r}_{h}$ scales with the black hole mass for

**fixed**${\xi}_{0}$ as ${r}_{h}=C\left({\xi}_{0}\right)\phantom{\rule{0.277778em}{0ex}}{M}_{BH}^{\frac{4}{3}}$, where the constant $C\left({\xi}_{0}\right)$ is a decreasing function of ${\xi}_{0}$.

**Figure 13.**The logarithm of the local pressure ${log}_{10}P\left(r\right)$ vs. ${log}_{10}(r/{r}_{h})$ for the three galaxy solutions with central SMBH. Notice the huge values of $P\left(r\right)$ in the quantum (high density) region $r<{r}_{A}$ and its sharp decrease entering the classical (dilute) region $r>{r}_{A}$.

**Figure 14.**The obtained equation of state of the galaxy plus central SMBH system: the logarithm of the local pressure ${log}_{10}P\left(r\right)$ vs. ${log}_{10}\rho \left(r\right)/{\rho}_{0}$. In all the cases, we find almost straight lines of unit slope. The equation of state is a perfect gas equation of state in the Boltzmann classical region. In the quantum gas (dense) region, the equation of state becomes steeper than the perfect gas. Galaxies with central black holes are in the dilute Boltzmann regime because their halo masses are ${M}_{h}>{M}_{h}^{min}$, Equation (83). This explains the perfect gas equation of state.

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de Vega, H.J.; Sanchez, N.G.
Warm Dark Matter Galaxies with Central Supermassive Black Holes. *Universe* **2022**, *8*, 154.
https://doi.org/10.3390/universe8030154

**AMA Style**

de Vega HJ, Sanchez NG.
Warm Dark Matter Galaxies with Central Supermassive Black Holes. *Universe*. 2022; 8(3):154.
https://doi.org/10.3390/universe8030154

**Chicago/Turabian Style**

de Vega, Hector J., and Norma G. Sanchez.
2022. "Warm Dark Matter Galaxies with Central Supermassive Black Holes" *Universe* 8, no. 3: 154.
https://doi.org/10.3390/universe8030154