THE EXISTENCE OF STERILE NEUTRINO HALOS IN GALACTIC CENTERS AS AN EXPLANATION OF THE BLACK HOLE MASS–VELOCITY DISPERSION RELATION

Understanding the nature of dark matter remains a fundamental problem in astrophysics and cosmology. Since the discovery of neutrinos' nonzero rest mass (Fukuda et al. 1998; Bilenky et al. 1998), the possibility that neutrinos contribute to cosmological dark matter has become a hot topic again. In particular, the sterile neutrinos are a class of candidate dark matter particles with no standard model interaction. Although the recent MiniBooNE result challenges the liquid scintillator neutrino detector (LSND) result that suggests the existence of eV scale sterile neutrinos (Aguilar-Arevalo et al. 2007), more massive sterile neutrinos (e.g., keV) may still exist (Dodelson & Widrow 1994). The fact that active neutrinos have rest mass implies that right-handed neutrinos should exist, which may indeed be massive sterile neutrinos with rest mass m_s.

Recently, it was proposed that a degenerate sterile neutrino halo exists in the galactic center (Viollier et al. 1993; Munyaneza & Viollier 2002). Later, this model was ruled out with the availability of more precise data, particularly those on orbits of stars such as S2 near the Milky Way center (Schödel et al. 2002). Nevertheless, the existence of such a sterile neutrino halo at the centers of protogalaxies may still be possible. Sterile neutrinos decay into lighter neutrinos and photons, and provide mass to fuel the growth of supermassive black holes (BHs). However, since most of these sterile neutrinos may have either decayed or fallen into the supermassive BH, the total mass of the sterile neutrino halo at the galactic center becomes very small (≪10⁶M_☉). Moreover, the existence of decaying sterile neutrinos may help to solve the cooling flow problem in clusters (Chan & Chu 2007) as well as re-ionization in the universe (Hansen & Haiman 2004). Therefore, it is worthwhile to discuss the consequences of the existence of massive sterile neutrinos at the centers of protogalaxies, which decay into light neutrinos and photons.

On the other hand, recent observations have led to some tight relations between the central BH masses M_BH,f and velocity dispersions σ in the bulges of galaxies. These relations can be summarized as log(M_BH,f/M_☉) = αlog(σ/200 km s^-1) + β, where α was found to be 3.75 ± 0.3 (Gebhardt et al. 2000) and 4.8 ±  0.5 (Ferrarese & Merritt 2000) by two different groups. Tremaine et al. (2002) reanalyzed both sets of data and obtained α = 4.02 ±  0.32 and β = 8.13 ±  0.06. These results indicate that BH formation may be related to galaxy formation, which challenges existing galaxy formation theories (Adams et al. 2001).

This relation has been derived by recent theoretical models (Adams et al. 2001; MacMillan & Henriksen 2002; Robertson et al. 2005; Murray et al. 2005; King 2003, 2005; McLaughlin et al. 2006). We assume that a degenerate sterile neutrino halo exists in the center of a protogalaxy. There are two different decay modes for sterile neutrinos ν_s. The major decaying channel is ν_s → 3ν with decay rate (Barger et al. 1995; Boyarsky et al. 2008)

$$\begin{equation} \Gamma _{3 \nu }= \frac{G_F^2}{384 \pi ^3} \sin ^22 \theta m_s^5=1.77 \times 10^{-20} \sin ^22 \theta \left(\frac{m_s}{1 \rm keV} \right)^5 {\rm s^{-1}}, \end{equation} \tag{ 1 }$$

where G_F and θ are the Fermi constant and mixing angle of sterile neutrino with active neutrinos, respectively. The minor decaying channel is ν_s → ν + γ with decay rate (Barger et al. 1995; Boyarsky et al. 2008)

$$\begin{eqnarray} \Gamma &=& \frac{9 \alpha G_F^2}{1024 \pi ^4} \sin ^22 \theta m_s^5\nonumber\\ &=&1.38 \,\times\, 10^{-22} \sin ^22 \theta \left(\frac{m_s}{1 \rm keV} \right)^5 {\rm s^{-1}}\approx \frac{1}{128} \Gamma _{3 \nu }, \end{eqnarray} \tag{ 2 }$$

where α is the fine structure constant. It is quite difficult to detect the active neutrinos produced in the major decaying channel. Therefore, we focus on the observational consequence of the radiative decay of the sterile neutrinos with rest mass m_s ⩾ 10 keV. They emit high energy photons (≈m_s/2) which heat up the surrounding gas so that hydrostatic equilibrium of the latter is maintained. The sterile neutrino halo also provides mass to form a supermassive BH from a small seed BH. Without any further assumption, α ≈ 4 is consistent with the range of the decay rate obtained by observational data from cooling flow clusters. In this paper, we first give a brief review of three popular analytic models that explain the M_BH,f–σ relation. Then we give a detailed description of our model and compare it with the existing models.

2.1. Super-Eddington Accretion Model

King (2003) presented a model to explain the M_BH,f − σ relation. He assumed that the gas density profile of a protogalaxy is isothermal (ρ ∼ r⁻²; King 2003, 2005). Therefore the gas mass inside radius R is

$$\begin{equation} M(R)=4 \pi \int _0^R \rho r^2dr= \frac{2f_g \sigma ^2 R}{G}, \end{equation} \tag{ 3 }$$

where f_g ≈ 0.16 is the cosmological ratio of baryon to total mass, assumed to be the same throughout a galaxy, and the Virial theorem is used. Consider a super-Eddington accretion onto a seed BH. The accretion feedback produces a momentum-driven superbubble that sweeps ambient gas into a thin shell which expands to the galaxy. The equation of motion is

$$\begin{equation} \frac{d}{dt}[M(R) \dot{R}]\,+\, \frac{GM(R)[M_{{\rm BH}}(t)+M(R)]}{R^2}=\frac{L_{\rm edd}}{c}, \end{equation} \tag{ 4 }$$

where L_edd = 4πGM_BH(t)c/κ, with κ the opacity and M_BH(t) the mass of the central BH at time t. Integrating twice and assuming R ≫ GM_BH,f/σ², one gets

$$\begin{equation} R^2=R_0^2\,+\,2R_0 \dot{R}_0t\,-\, \sigma ^2 \left(1- \frac{M_{\rm BH}(t)}{M_{\sigma }} \right)t^2, \end{equation} \tag{ 5 }$$

where $\dot{R}_0=\dot{R}$ at R = R₀, with R₀ some large radius (≫GM_BH,f/σ²), and M_σ ≡ f_gκσ⁴/πG². Therefore, the maximum radius R_max is given by

$$\begin{equation} \frac{R_{\rm max}^2}{R_0^2}=1+ \frac{\dot{R}_0^2}{2 \sigma ^2(1-M_{\rm BH}(t)/M_{\sigma })}. \end{equation} \tag{ 6 }$$

When M_BH(t) approaches M_σ, R_max becomes very large, because such that the cooling of the shocked wind is inefficient because the cooling time t_cooling ∝ R² and the accretion is stopped because the shell can escape the galaxy entirely by gas pressure (King 2003). Therefore, given an adequate mass supply (such as in a merger), we get (King 2005)

$$\begin{equation} M_{{\rm BH},f}= \frac{f_g \kappa }{ \pi G^2} \sigma ^4. \end{equation} \tag{ 7 }$$

Here, the proportionality constant f_gκ/πG² lies within the observational constraints. To summarize, the M_BH,f–σ relation is obtained with three important assumptions: (1) isothermal gas density distribution throughout the galaxy formation, (2) super-Eddington accretion, and (3) an adequate mass supply.

2.2. The Self-similar Model

MacMillan & Henriksen (2002) obtained a relation between M_BH,f and σ by assuming that the density and velocity distributions of matter are self-similar. They assumed that the galaxy is formed by the extended collapse of a halo composed of collisionless matter. The central BH is grown proportionally to the halo as matter continues to fall in. The relation is given by MacMillan & Henriksen (2002)

$$\begin{equation} \log M_{{\rm BH},f} \propto \left(\frac{3 \delta / \alpha -2}{\delta / \alpha -1} \right) \log \sigma, \end{equation} \tag{ 8 }$$

where δ and α are scales in space and time, respectively, and their ratio is related to the power-law index of the initial density perturbation in the spherical infall model of halo growth (Henriksen & Widrow 1999):

$$\begin{equation} \frac{\delta }{\alpha }= \frac{2}{3} \left(1+ \frac{1}{\epsilon } \right). \end{equation} \tag{ 9 }$$

The power-law index = (n + 3)/2, where n is the index of the primordial matter power spectrum P(k) ∝ kⁿ. If n = −2, Equation (8) agrees with the observation M_BH,f ∝ σ⁴. This model involves a relation (Equation (9)) which is quite model dependent.

2.3. The Ballistic Model

Adams et al. (2001) assume the dark matter and baryons to be unsegregated and the isothermal initial mass density distribution (M_t(r) ∝ r). The specific orbital energy is conserved when the particles fall into the small seed BH:

$$\begin{equation} E= \frac{1}{2}v_r^2+ \frac{j^2}{2r^2}- \frac{GM_t(r)}{r}, \end{equation} \tag{ 10 }$$

where v_r and j are the radial velocity and angular momentum per unit mass. When the particles fall into the equatorial plane, their pericenters are

$$\begin{equation} p= \frac{j^2}{2GM_t(r)}=\frac{(GM_t(r))^3 \Omega ^2}{2 \sigma ^8}, \end{equation} \tag{ 11 }$$

where Ω is the average angular speed of the particles. In the early stage, all the particles fall into the BH until the BH mass reaches a critical point that corresponds to p = 4R_s, where M_BH(t) = M_t(R_s), with R_s = 2GM_BH(t)/c² the Schwarzschild radius. This gives the final relation (Adams et al. 2001)

$$\begin{equation} M_{{\rm BH},f}= \frac{4 \sigma ^4}{Gc \Omega }. \end{equation} \tag{ 12 }$$

This model is based on the assumption of the isothermal distribution of matter, and there is a free parameter Ω, which is assumed to have the same value for all galaxies.

Suppose the sterile neutrino halo dominates the mass in the protogalactic center, most of the mass in the BH comes from the sterile neutrino halo with radius $\tilde{R}$, which was formed in the very early universe t ∼ 0 (Munyaneza & Biermann 2005; Chan & Chu 2007). The total mass of the degenerate sterile neutrino halo at time t_b is

$$\begin{equation} M_s(t_b)=4 \pi \int _0^{\tilde{R}} \rho _sr^2dr=M_{s0}e^{-\Gamma _{3 \nu } t_b}, \end{equation} \tag{ 13 }$$

where ρ_s is the mass density of the sterile neutrino halo. Assume that a seed BH with mass M_BH,0 of order solar mass is formed at t_b, long after the formation of the sterile neutrino halo. It would grow by accreting mass of the sterile neutrino halo to mass M_BH(t) at time t. As some sterile neutrinos would be accreted by the seed BH, the degenerate pressure is decreased and more sterile neutrinos will fall into the BH as their Fermi speed is less than their escape speed (Munyaneza & Biermann 2005). The falling timescale at distance r from the BH in a free falling model is given by (Phillips 1994)

$$\begin{equation} t_{ff}=\frac{\pi }{2} \sqrt{ \frac{r^3}{2G[M_{{\rm BH},0}+M_s(r,t)]}}. \end{equation} \tag{ 14 }$$

For M_s(t) ⩾ 10⁶M_☉, m_s ⩾ 10 keV and $\tilde{R} \le 0.04$ pc, t_ff ⩽ 160 years for $r \le \tilde{R}$, which is much shorter than the Hubble time. Therefore, we do not need any intermediate mass BHs since the small seed BH can grow to a 10⁶–10⁹M_☉ supermassive BH rapidly as long as there is enough mass in the initial sterile neutrino halo. In the following, we assume that all sterile neutrinos fall into the BH and decay into active neutrinos and photons, and so $M_{{\rm BH},f}=M_s(t_b)+M_{{\rm BH},0} \approx M_{s0}e^{-\Gamma _{3 \nu } t_b}$.

The photons emitted by the original decaying sterile neutrinos provide energy to the gas in the protogalaxies by Compton scattering. The optical depth of a decayed photon in the bulge is τ = n_eσ_TR_e, where R_e is the J-band effective bulge radius (Marconi & Hunt 2003), σ_T is the Compton scattering cross section, and n_e is the mean number density of the gas. In equilibrium, the heating rate is equal to the cooling rate by bremsstrahlung radiation Λ_B, recombination Λ_R, and adiabatic expansion Λ_a. We have (Katz et al. 1996)

$$\begin{eqnarray} L(1-e^{-\tau })&=& \Lambda _B+ \Lambda _R+ \Lambda _a\nonumber\\ &=& \left[ \Lambda _{B0}n_e^2T^{0.5}+ \Lambda _{R0}n_e^2T^{0.3} \left(1+ \frac{T}{10^6 {\rm K}} \right)^{-1} \right]V\nonumber\\ &&+PV^{2/3}c_s, \end{eqnarray} \tag{ 15 }$$

where Λ_B0 = 1.4 × 10⁻²⁷ erg s⁻¹, Λ_R0 = 3.5 × 10⁻²⁶ erg s⁻¹, c_s, P, and V are the sound speed, pressure, and total volume of the gas within R_e, respectively. The M_BH,f–σ relation can be obtained by using Equation (15) and the Virial theorem numerically. Nevertheless, we first illustrate the idea by obtaining analytic relations in two different regimes. Suppose M_s(t_b) ⩾ 10⁶M_☉; if τ ≫ 1, the resulting temperature is above 10⁶ K and the total cooling rate is dominated by Λ_a. For τ ⩽ 1, the resulting temperature is lower and the total cooling rate is dominated by Λ_B and Λ_R (see Figure 1).

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In the optically thick regime, τ ≫ 1 and Λ_a ≫ Λ_B + Λ_R, and we get

$$\begin{equation} kT= \left(\frac{m_g}{\gamma } \right)^{1/3}V^{-4/9}L^{2/3}n_e^{-2/3}, \end{equation} \tag{ 16 }$$

where m_g is the mean mass of a gas particle. By using the Virial theorem kT = f₁GM_Bm_g/3R_e, where M_B is the effective bulge mass of the protogalaxy within R_e, and substituting L ≈ M_sΓc²/2, we get

$$\begin{equation} M_s= \frac{2 \gamma ^{1/2}f_1^{3/2}G^{3/2}e^{-\Gamma _{3 \nu } t}}{3^{7/6}(4 \pi)^{1/3} \Gamma c^2} \left(\frac{M_B}{R_e} \right)^{5/2}, \end{equation} \tag{ 17 }$$

where γ is the adiabatic index of the gas and f₁ ∼ 1 is a constant that depends on the density distribution of the protogalaxy. As time passes, the energy gained by the gas would decrease gradually and the mass distribution at the center would also change slightly. If the supermassive BH was formed when the galaxy formation was nearly completed (t_b = 10¹⁶–10¹⁷ s), the ratio M_B/R_e and the velocity dispersion do not change significantly. According to Equation (17), the ratio M_B/R_e is fixed by M_s and Γ. By using the Virial theorem again and assuming spherical symmetry, one can relate this ratio to the final bulge velocity dispersion after the supermassive BH is formed,

$$\begin{equation} \sigma ^2=f_2 \frac{GM_B}{R_e}, \end{equation} \tag{ 18 }$$

where f₂ ∼ 1 is a constant that depends on the mass distribution at present. Combining Equations (17) and (18), we get

$$\begin{equation} M_{{\rm BH},f}=M_s(t_b)= \frac{2 \gamma ^{1/2}f_1^{3/2}e^{-\Gamma _{3 \nu } t_b}}{3^{7/6}(4 \pi)^{1/3}f_2^{5/2}G \Gamma c^2} \sigma ^5. \end{equation} \tag{ 19 }$$

In the optically thin regime, τ ⩽ 1 and Λ_R + Λ_B ≫ Λ_a, the total power absorbed by the gas in the protogalaxies within R_e is ≈Ln_eσ_TR_e. If the cooling rate is dominated by bremsstrahlung radiation, in equilibrium, we get

$$\begin{equation} kT=k \left(\frac{L \sigma _TR_e}{\Lambda _0n_eV} \right)^2. \end{equation} \tag{ 20 }$$

By using the Virial theorem and Equation (18), we obtain

$$\begin{equation} M_{{\rm BH},f}= \frac{2 \Lambda _{B0}e^{- \Gamma _{3 \nu } t_b}}{\sigma _Tf_2^{3/2}G \Gamma c^2} \left(\frac{f_1}{3km_g} \right)^{1/2} \sigma ^3. \end{equation} \tag{ 21 }$$

If the cooling rate is dominated by recombination, in equilibrium and for T ⩽ 10⁶ K, we get

$$\begin{equation} kT=k \left(\frac{L \sigma _TR_e}{\Lambda _0n_eV} \right)^{10/3} \end{equation} \tag{ 22 }$$

and

$$\begin{equation} M_{{\rm BH},f}= \frac{2 \Lambda _{R0}e^{-\Gamma _{3 \nu } t_b}}{m_g^{7/10} \sigma _Tf_2^{13/10}G \Gamma c^2} \left(\frac{f_1}{3km_g} \right)^{3/10} \sigma ^{2.6}. \end{equation} \tag{ 23 }$$

Therefore, M_BH,f and σ are closely related in both optically thick and thin regimes.

We use 500 random data in the ranges of τ = 0.01–10,000, f₁ = 0.6–3, f₂ = 0.6–3, t_b = 10¹⁶–10¹⁷ s, 10⁹M_☉ ⩽ M_B ⩽ 10¹²M_☉, and 0.1 kpc ⩽R_e ⩽ 10 kpc to generate the values of M_BH,f and σ by using Equations (15) and (18) (see Figure 2). By using Γ_3ν = (5 ± 1) × 10⁻¹⁷ s⁻¹ which solves the cooling flow problem (Chan & Chu 2007)¹ and Γ_3ν = 128Γ, we get

$$\begin{eqnarray} \log \left(\frac{M_{{\rm BH},f}}{M_{\odot }} \right)&=&(3.98 \,\pm\, 0.29) \log \left(\frac{\sigma }{200 \; \rm km \;s^{-1}} \right)\nonumber\\ &&\,+\,(8.04 \,\pm\, 0.20), \end{eqnarray} \tag{ 24 }$$

which agrees with the recent observation: α = 4.02 ± 0.32 and β = 8.13 ± 0.06 (Tremaine et al. 2002).

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As an order of magnitude estimate, sin θ ∼ m_D/m_s, where m_D is the neutrino Dirac mass. In the standard see-saw mechanism, m_D is the geometric mean of the light neutrino mass-scale m_ν and m_s (Mohapatra & Senjanović 1980).² Therefore, $\sin \theta \sim \sqrt{m_{\nu }/m_s}$. From Equation (1), we get

$$\begin{equation} \Gamma _{3 \nu } \sim 10^{-23} \left(\frac{m_{\nu }}{\rm 1 eV} \right) \left(\frac{m_s}{\rm 1 keV} \right)^4 {\rm s^{-1}}. \end{equation} \tag{ 25 }$$

For Γ_3ν ∼ 10⁻¹⁷ s⁻¹, m_s ∼ 30 keV for m_ν ∼ 1 eV, which is consistent with our assumption (m_s ⩾ 10 keV).

We assume the existence of a degenerate neutrino halo (m_s∼ keV) at the protogalactic center with a parameter, Γ_3ν, which is universal and can be inferred from observation of cluster hot gas (Chan & Chu 2007). Without any assumptions of the protogalaxy and existence of any intermediate mass BH, α ≈ 4 and β ≈ 8 require the total decay rate to be Γ_3ν = (5 ± 1) × 10⁻¹⁷ s⁻¹, which is consistent with the observational data from cooling flow clusters (Chan & Chu 2007). We also reviewed several models to explain the M_BH,f–σ relation. These models require several assumptions or free parameters which may not be true for all galaxies. For example, King's model assumes the isothermal density profile (ρ ∼ r⁻²) for all galaxies during all the time of the BH formation. If the density profile changes into the form ρ ∼ r⁻¹, then $\sigma \propto \sqrt{r}$ which is not a constant. This problem also exists in the ballistic model which is based on the isothermal distribution of matter. However, in the self-similar model, there are several free parameters which are model dependent. Also, one cannot obtain the proportionality constant β of the M_BH,f–σ relation.

We have considered a wide range of τ = 0.1–10, 000, f₁ = 0.6–3, and f₂ = 0.6–3 encompassing almost all possibilities in galaxies. In our model, we assume that the supermassive BH was formed at the epoch when the galaxy formation was nearly completed (t_b = 10¹⁶–10¹⁷ s) so that the velocity dispersion does not change significantly between t_b and present. Therefore our result is valid only for supermassive BHs formed nearly at the end of the galaxy formation, as in the super-Eddington accretion and ballistic model. The assumed existence of a decaying sterile neutrino halo inside each galactic center provides enough mass to form the supermassive BH. It can also solve the cooling flow problem in clusters (Chan & Chu 2007) and explain the re-ionization of the universe (Hansen & Haiman 2004), all with the same decay rate Γ_3ν = (5 ± 1) × 10⁻¹⁷ s⁻¹ and m_s ⩾ 10 keV, which are consistent with the standard see-saw mechanism.

This work is partially supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (Project No. 400805).