A study of 137 intermediate mass black hole candidates

G. Ter-Kazarian; S. Shidhani

doi:10.1007/s10509-019-3657-2

Astrophysics and Space Science

October 2019, 364:165 | Cite as

A study of 137 intermediate mass black hole candidates

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G. Ter-Kazarian
S. Shidhani

Original Article

First Online: 03 October 2019

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Abstract

This communication aims to expose further the assertions made in microscopic theory of black hole (MTBH) to the physics of intermediate mass black holes (IMBHs). The MTBH explores the most important spontaneous breaking of gravitation gauge symmetry above nuclear density. One of the most remarkable drawback of this framework is the fact that instead of infinite collapse and central singularity, an inevitable end product of the evolution of massive object is the stable superdense proto-matter core (SPC), subject to certain rules. The physical entropy is assigned to SPC as a measure of the large number of thermodynamical real microstates of proto-matter, which is compatible with a concept of ergodicity. The SPCs could be observed only in presence of accreting matter. As a corollary, this allows to construct microscopic models of accreting IMBHs. The mass estimates collected from the literature of all the observational evidence for 137 IMBH-candidates, even though there are still large uncertainties, allow us to undertake a large series of numerical simulations to obtain all their essential physical characteristics.

Keywords

Black hole physics Intermediate mass black holes Dwarf galaxies Globular clusters ULXs and HLXs

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Notes

Acknowledgements

The very helpful and knowledgable comments and positive feedback from the anonymous referee that have essentially improve the quality of the final manuscript are much appreciated.

Appendix A: The baryonic-quark proto-matter

In this section we develop as from first principles to discuss the physics of the n-p-e proto-matter at short nucleon-nucleon distances, in presence of ID, and the onset of melting down of hadrons when nuclear matter turns stepwise first to quark matter and then to quark proto-matter (Ter-Kazarian 2014, 2015a).

A.1 The one-component baryonic proto-matter

The state equation of baryonic proto-matter can be derived usually from the minimization of energy density incorporated with the conservation laws of baryonic and electric charges. The state equation of one-component degenerate Fermi proto-matter can be written (Ter-Kazarian 2014, 2015a) and references therein:

$$ \begin{aligned} &\widetilde{\rho }=\widetilde{m}c^{2}\frac{\chi (\widetilde{y})}{ \widetilde{\lambda }^{3}}+ \widetilde{n}_{b}\widetilde{U}( \widetilde{n}_{b}),\\ &\widetilde{P}=\widetilde{m}c^{2}\frac{\varphi (\widetilde{y})}{\widetilde{\lambda }^{3}}+ \widetilde{n}_{b}^{2}\frac{ \partial \widetilde{U}(\widetilde{n}_{b})}{\partial \widetilde{n}_{b}}, \end{aligned} $$

(7)

where $\widetilde{U}(\widetilde{n}_{b})$ is the potential energy per baryon. The following notational conventions are used throughout:

$$ \begin{aligned} &\chi (\widetilde{y})=\frac{1}{8\pi ^{2}} \bigl\{ \widetilde{y}\bigl(1+ \widetilde{y}^{2}\bigr)^{1/2}\bigl(1+2\widetilde{y}^{2}\bigr)\\ &\phantom{\chi (\widetilde{y})=}{}- \ln \bigl[\widetilde{y} +\bigl(1+\bigl(1+\widetilde{y}^{2}\bigr)^{1/2} \bigr] \bigr\} , \\ &\varphi (\widetilde{y})=\frac{1}{8\pi ^{2}} \biggl\{ \widetilde{y}\bigl(1+\widetilde{y}^{2}\bigr)^{1/2}\biggl(\frac{2}{3}\widetilde{y}^{2}-1\biggr)\\ &\phantom{\varphi (\widetilde{y})=}{}+ \ln \bigl[\widetilde{y} +\bigl(1+\bigl(1+\widetilde{y}^{2}\bigr)^{1/2} \bigr] \biggr\} ,\quad \widetilde{m}=\bigl(| \eta | \bigr)^{1/2},\\ &\eta =1-x^{2}-xy/\sqrt{3}-y^{2}x^{4} / 6\bigl(1+x^{2}\bigr),\\ &y=P_{F}/mc=\bigl(3\pi ^{2}\bigr)^{1/3}\lambda n^{1/3},\\ &\widetilde{y}=\widetilde{P}_{F}/\widetilde{m}c=\bigl(3\pi ^{2}\bigr)^{1/3} \widetilde{\lambda }\widetilde{n}^{1/3},\quad \widetilde{P_{F}}=P_{F}\zeta ^{1/2},\\ &\zeta =y^{2} \bigl[ 1-2x^{2}/3\bigl(1+x^{2}\bigr) \bigr] + 2xy/\sqrt{3}+x ^{2}, \end{aligned} $$

(8)

provided, $\widetilde{P}_{F}$ and $P_{F}$ are distorted and ordinary Fermi momenta, $\widetilde{n}$ is the distorted concentration of particles, $\lambda =\hbar /mc$, $\widetilde{\lambda } =\hbar / \widetilde{m}c $. To simplify the problem in (8), we approximately set $P_{1}=P_{2}=P_{3}=P/\sqrt{3}=| P | / \sqrt{3}$, $P/mc \simeq x/2$.

A.2 The n-p-e proto-matter

Suppose that the free neutrons, protons and electrons of n-p-e proto-matter, at high densities $\rho \ge \rho _{d}$, are in complete $\beta $-equilibrium. That is, $\widetilde{\mu }_{e}+\widetilde{\mu } _{p}=\widetilde{\mu }_{n}$ and $\widetilde{\mu }_{\nu }=\mu _{\nu }=0$. Then,

$$ \widetilde{m}_{e}(1+y_{e})^{1/2}+\widetilde{m}_{p}(1+y_{p})^{1/2}= \widetilde{m}_{n}(1+y_{n})^{1/2}, $$

(9)

where $m_{e}$, $m_{p}$ and $m_{n}$ are the ordinary masses at rest of electron, proton and neutron. The electrical charge neutrality implies $\widetilde{n}_{e}=\widetilde{n}_{p}$, i.e. $\widetilde{m}_{e}\widetilde{y}_{e}= \widetilde{m}_{p}\widetilde{y}_{p}$. By virtue of (7) and (8), the energy density and internal pressure read

$$\begin{aligned} &\widetilde{\rho }= \widetilde{m}_{e}c^{2}\chi (\widetilde{y}_{e})/ \widetilde{\lambda }^{3}_{e}+ \widetilde{m}_{p}c\chi (\widetilde{y} _{p})/ \widetilde{\lambda }^{3}_{p}+ \widetilde{m}_{n}c^{2}\chi (\widetilde{y}_{n})/ \widetilde{\lambda } ^{3}_{n}, \\ &\widetilde{P}= \widetilde{m_{e}}c^{2}\varphi (\widetilde{y}_{e})/ \widetilde{\lambda }^{3}_{e}+ \widetilde{m_{p}}c^{2}\varphi ( \widetilde{y}_{p})/ \widetilde{\lambda }^{3}_{e}+ \widetilde{m_{n}}c^{2}\varphi (\widetilde{y}_{n})/ \widetilde{\lambda }^{3}_{e}. \end{aligned}$$

(10)

Further simplification gives

$$ \widetilde{y}_{p}= \biggl( \frac{\widetilde{a}_{1}+\widetilde{a}_{2} \widetilde{y}_{n}^{2}+ \widetilde{a}_{3}\widetilde{y}_{n}^{4}}{1+ \widetilde{y}_{n}^{2}} \biggr)^{1/2}, $$

(11)

where

$$ \begin{aligned} &\widetilde{a}_{1}= (1/4 ) \bigl[ ( \widetilde{Q}/ \widetilde{m}_{p})^{2}- ( \widetilde{m}_{e} / \widetilde{m}_{p} )^{2} \bigr]\\ &\phantom{\widetilde{a}_{1}=}{}\times\bigl[( 1+\widetilde{m}_{p} /\widetilde{m}_{n} ) ^{2} - (\widetilde{m}_{e} /\widetilde{m}_{n}) ^{2} \bigr], \\ &\widetilde{a}_{2}= (1/2 )\bigl[ ( \widetilde{m} _{n}/\widetilde{m}_{p})^{2}-1- ( \widetilde{m}_{e}/\widetilde{m}_{p})^{2} \bigr], \\ &\widetilde{a}_{3}= ( \widetilde{m}_{n} /2\widetilde{m}_{p}) ^{2}, \quad \widetilde{Q}=\widetilde{m}_{n}-\widetilde{m}_{p}. \end{aligned} $$

(12)

The ratio of proton-neutron distorted concentrations takes form

$$ \frac{\widetilde{n}_{p}}{\widetilde{n}_{n}}= \biggl( \frac{ \widetilde{m}_{p}}{\widetilde{m}_{p}\widetilde{y}_{n}}\biggr)^{3} \bigl[ \bigl( \widetilde{a}_{1}+\widetilde{a}_{2}\widetilde{y}_{n} ^{2}+ \widetilde{a}_{3}\widetilde{y}_{n}^{4} \bigr) \bigl( 1+ \widetilde{y}_{n}^{2} \bigr) \bigr] ^{3/2}. $$

(13)

For the intermediate density domain of regular n-p-e gas in absence of ID, according to Shapiro and Teukolsky (1983), the proton-neutron ratio initially decreases, as the density increases, and reaches a maximum value of 0.0026 at $\rho _{0} \simeq 7.8 \times 10^{11}~\mbox{g}\,\mbox{cm}^{-3}$, and afterwards rises monotonically to $1/8$ for high densities.

A.3 Domain of string flip-flop of the quark proto-matter

A calculations of processes of rearrangement of quark string connections or flip-flop in dense matter, without QCD-corrections and the effects of pions, have been carried out by Miyazawa (1979). In this subsection similar calculations are carried out for the quark proto-matter. We are interested in the individual particle approximation (Hartree approximation), where the Hartree potential is almost linearly proportional to the string length. The $Y$ shape string is the most convenient for calculations, because the center of it almost equals to the center of gravity. At very first, we shall study the classical strings (Ter-Kazarian 2014, 2015a). The bispinor field equation for a massless quark found in the ID-region of space-time continuum under the rising linear potential $\widetilde{V}$ can be written in the form

$$ (\vec{\alpha } \widetilde{\vec{P}}+\widetilde{V}) \psi = \widetilde{E}\psi , $$

(14)

where $\vec{\alpha }$ are the Dirac’s matrices, $\widetilde{\vec{P}}$ and $\widetilde{E}$ are distorted momentum and energy of quark. In analogy to the case of ordinary quark matter, one may readily show that in order to have bound state the rising potential should be a scalar. In a string model the potential energy in Hartree approximation is given

$$ \widetilde{V}=a_{0}\min \sum _{i} \widetilde{r}^{4}_{i}\beta ^{(i)}, $$

(15)

where $\widetilde{r}_{i}$ is the distance from the center, $a_{0}$ is the string tension estimated from charmonium spectroscopy $a_{0} \simeq 0.1~\mbox{GeV}$. Similar to ordinary case a red quark searches for the nearest center and joins with it by a string and so on. One simplifies the calculations by assuming that the centers are uniformly distributed with a concentration $\widetilde{n}_{b}$. Hence, the average of distance to the nearest center equals to

$$ \widetilde{\bar{r}}=\varGamma (4/3)\biggl(\frac{4\pi }{3} \widetilde{n} _{b}\biggr) ^{-1/3}, $$

(16)

where $\varGamma $ is the gamma function. Since we consider the $\varLambda $-like proto-matter of relativistic quarks, then the average of string length equals $\widetilde{\bar{r}}=1.36\hbar / \widetilde{P}_{Fb} $. The potential energy per quark reads

$$ \widetilde{V}_{\text{per q}}=a_{0}\widetilde{\bar{r}}\beta = \widetilde{M}(\widetilde{P}_{Fb})\beta , $$

(17)

where

$$ \widetilde{M}(\widetilde{P}_{Fb})=1.36 a_{0}/c^{3}\widetilde{P}_{Fb}= \frac{0.1541}{y_{b}}m_{n}, $$

(18)

so,

$$\begin{aligned} &\widetilde{M}(\widetilde{P}_{Fb})=M(P_{Fb})/\widetilde{\zeta }_{b} ^{2}, \\ &\quad \widetilde{\zeta }_{b}=\zeta _{b}/y_{b}^{2}=1- 2x^{2}/3\bigl(1+x^{2}\bigr)+ 2x/\sqrt{3}y_{b}+x^{2}/y^{2}_{b}, \\ &\quad y_{b}=P_{Fb}/m_{n}c=\bigl(3\pi ^{2}\bigr)^{1/3}\hbar n_{b}^{1/3}/m_{n}c, \end{aligned}$$

(19)

with $m_{n}$ is the neutron mass at rest, $\widetilde{M}(\widetilde{P}_{Fb})$ is the effective distorted mass of the quark. We assume that quarks have small ordinary mass $m_{i}\simeq m_{u}=5~\mbox{MeV}$. Then,

$$ \widetilde{m}=\widetilde{m}_{u} + \widetilde{M}(\widetilde{P}_{Fb})= \widetilde{m}_{u} \widetilde{\eta }_{u}^{1/2}, $$

(20)

where

$$\begin{aligned} &\widetilde{\eta }_{u}=\bigl(| \eta _{u}| ^{1/2}+ 5441.57/\zeta _{u}^{1/2} \bigr) ^{2}, \\ &\quad\eta _{u}=1-x^{2}-xy_{u}/\sqrt{3}-y_{u}^{2}x^{4}/6\bigl(1+x^{2}\bigr), \\ &\quad\zeta _{u}=y_{u}^{2}\bigl[ 1-2x^{2}\big/3\bigl(1+x^{2}\bigr)\bigr]+ 2xy_{u}/ \sqrt{3}+x^{2}. \end{aligned}$$

(21)

The total distorted energy of the free ground state of quark proto-matter made of classical strings is $\widetilde{E}_{\text{tot}}= \sum_{i}\sqrt{c^{2}\widetilde{P}_{i}^{2}+ \widetilde{m}^{2}c^{4}} $, which gives the following expression of distorted energy per quark

$$ \widetilde{E}_{\text{per q}}=\frac{3\int _{0}^{ \widetilde{P}_{Fb}} \sqrt{c ^{2}\widetilde{P}^{2}+ \widetilde{m}^{2}c^{4}}d \widetilde{P}}{ \int d^{3} \widetilde{P}}, \quad | \widetilde{P} | < \widetilde{P}_{Fb}, $$

(22)

while

$$ \widetilde{E}_{\text{per q}}= \left \{ \textstyle\begin{array}{l@{\quad}l} \widetilde{M}(\widetilde{P}_{Fb}) & \widetilde{P}_{Fb}\ll \widetilde{M}(\widetilde{P}_{Fb}),\\ \frac{3}{4}\widetilde{P}_{Fb} & \widetilde{P}_{Fb}\gg \widetilde{M}( \widetilde{P}_{Fb}). \end{array}\displaystyle \right . $$

Next, we explore a tunneling effect of quantum fluctuations of string, and the negative potential energy caused by such a quantum jump. The basic technique adopted for calculation of transition matrix element $\widetilde{K}$ is the instanton technique (semi-classical treatment). Due to quantum string flip-flop, an attractive interaction between quarks is presented, when during the quantum transition from a state $\psi _{1}$ of energy $\widetilde{E}_{1}$ to another one $\psi _{2}$ of energy $\widetilde{E}_{2}$, the lowering of energy of system occurs

$$ \Delta \widetilde{E}= \sqrt{ \biggl( \frac{\widetilde{E}_{1}- \widetilde{E}_{2} }{2} \biggr) ^{2}+ \widetilde{K}^{2} }. $$

(23)

Hence,

$$ \widetilde{E}= \left \{ \textstyle\begin{array}{l@{\quad}l} -\widetilde{K}, &\widetilde{E}_{1}-\widetilde{E}_{2} \ll 2\widetilde{K},\\ \frac{\widetilde{K}^{2}}{\widetilde{E}_{1}-\widetilde{E}_{2}} & \widetilde{E}_{1}-\widetilde{E}_{2} \ll 2\widetilde{K}. \end{array}\displaystyle \right . $$

The quark matter acquires $\Delta \widetilde{E}$ correction to the classical string energy (23), such that the flip-flop energy lowers the energy of quark matter, consequently by lowering the critical density or critical Fermi momentum. If one, for example, looks for the string flip-flop transition amplitude of simple system of $q\bar{q} q \bar{q}$ described by the Hamiltonian $\widetilde{H}$ and invariant action $\widetilde{S}$, then one has

Open image in new window

(24)

where $T$ is a (imaginary) time interval, $[d \widetilde{\sigma } ]$ is the integration over all the possible string motion. The action $\widetilde{S}$ is proportional to the area $\widetilde{A}$ of the surface swept by the strings in the finite region of ID-region of $V_{4}$. The strings are initially in the Open image in new window

configuration and finally in the Open image in new window

configuration. Note that the maximal contribution to the path integral (25) comes from the surface $\sigma _{0}$ of the minimum surface area (‘instanton’). A computation of the transition amplitude is straightforward by summing over all the small vibrations around $\sigma _{0}$. We forbear to write further calculations out as they can be easily carried out in analogy with Miyazawa (1979). Hence

$$ \widetilde{K}\simeq \pi \sqrt{a_{0}}e^{-a_{0}\Delta \widetilde{A}}, $$

(25)

where $\Delta \widetilde{A}$ is the increase of the 1-instanton surface area over the 0-instanton one, which actually equals to the area the string sweeps in making the transition. Note that string has a finite thickness $d$, and the width of the area $\Delta \widetilde{A}$ cannot be less than $d$. This cutoff introduces a factor $\exp (-a_{0}d r _{NN})$, (where $r_{NN}$ is the distance between two separated centers) in the amplitude $\widetilde{K}$ resulting in the finite-ranged potential. The interaction energy between two centers has a range of order $2\widetilde{\bar{r}}$ due to overlap of wave functions. For $r_{NN} < \widetilde{\bar{r}}$ the interaction potential reads

$$ \widetilde{V_{cc}}= \left \{ \textstyle\begin{array}{l@{\quad}l} -\sqrt{a_{0}} & r_{NN}< \widetilde{\bar{r}},\\ 0 & \mbox{otherwise}, \end{array}\displaystyle \right . $$

(26)

which leads to expression of the interaction energy per center (or per baryon)

$$ \widetilde{U}(\widetilde{n}_{b})=\frac{1}{2}\widetilde{n}_{b} \int \widetilde{V_{cc}}d^{3} r_{NN}\simeq -\frac{1}{2}\widetilde{n} _{b} \frac{4\pi }{3}\widetilde{\bar{r}}^{3}. $$

(27)

The potential energy, as usual, is of the Wigner type and caused a surface tension to the quark proto-matter

$$ \mbox{Surface tension} \simeq \frac{\pi }{8}\widetilde{n}_{b}^{2} \widetilde{\bar{r}}^{4}\sqrt{a_{0}}. $$

(28)

A string thickness $d$ can be estimated to be $0.25~\mbox{fm}$. Making use of the expression of $\widetilde{\bar{r}}$ and $\sqrt{a_{0}}\simeq 0.1~\mbox{GeV}$, one gets

$$ \widetilde{U}(\widetilde{n}_{b})\simeq -56.29~\mbox{MeV}, \qquad \frac{d\widetilde{U}}{d\widetilde{n}_{b}}=0. $$

(29)

Thus, in terms of (7) and (8), the macroscopic density of energy and internal pressure of quark proto-matter read

$$ \widetilde{\rho }=3\widetilde{m}c^{2}\frac{\chi (\widetilde{y})}{ \widetilde{\lambda }^{3}}-56.29~\mbox{MeV} \widetilde{n}_{b} ,\qquad \widetilde{P}=3\widetilde{m}c^{2}\frac{\varphi (\widetilde{y})}{\widetilde{\lambda }^{3}}. $$

(30)

A.4 Domain of asymptotic freedom

The QCD vacuum has a complicated structure due to the gluon-gluon interactions. The confinement of quarks is a natural feature of the exercising a pressure $B$ on the surface of the local region of the perturbative vacuum to which quarks are confined. This is just the main idea of phenomenological MIT quark bag model (Chodos et al. 1974), where quarks are assumed to be confined in a bag. Due to the screening of strong forces, the quarks are considered to be free inside the bag and to interact only in the surface region. The surface energy is estimated to be proportional to quark density. The stability of the hadron is ensured by the vacuum pressure $B$ and surface tension. The surface energy is estimated to be proportional to quark density. In most applications, sufficient accuracy is obtained by assuming that all the quarks are almost massless inside a bag. Now, our purpose is to convert this picture to the medium of quark proto-matter. The quark proto-matter is in overall color singlet ground state, which is a non-interacting relativistic Fermi gas found in the ID-region of the spacetime continuum, at $r_{NN}\le 0.25~\mbox{fm}$. We consider the quark proto-matter of $u$, $d$ and $s$ flavors, in complete $\beta$-equilibrium (Ter-Kazarian 2014, 2015a):

$$ d \rightarrow u+l+\bar{\nu },\qquad s \rightarrow u+l+\bar{\nu }, $$

(31)

where $l$ denotes electron $e^{-}$ or muon $\mu ^{-}$, $\nu ^{-}$ is the associated antineutrino. Assuming that the neutrinos are not retained in the medium, we get for the generalized chemical potentials $\widetilde{\mu}=[(\widetilde{P}_{F} c)^{2}+ (\widetilde{m} c ^{2})^{2}] ^{1/2}$:

$$ \widetilde{\mu }_{d}\rightarrow \widetilde{\mu }_{u}+l+ \widetilde{\mu }_{l},\qquad \widetilde{\mu }_{s}\rightarrow \widetilde{\mu }_{u}+ \widetilde{\mu }_{l}, $$

(32)

provided, $\widetilde{\mu }_{\nu }=\mu _{nu}=0$. For an extremely relativistic degenerate Fermi gas, the relation $\widetilde{\nu } _{i} \propto g_{i}\widetilde{\mu }_{i} $ holds, where $g_{i}=2$ for $i=l$, and $g_{i}=6$ for each type of quark $i=u,d,s$ in two spin and three color states. This leads to $\widetilde{\mu }_{d}= \widetilde{\mu }_{s}$ and $\widetilde{\nu }_{d}= \widetilde{\nu }_{s} $, while the equilibrium between $\mu ^{-}$ and $e^{-}$ yields $\widetilde{\mu }_{\mu }=\widetilde{\mu }_{e}\equiv \widetilde{\nu }_{l}$ and $\widetilde{\nu }_{\mu }= \widetilde{\nu }_{e}\equiv \widetilde{\nu }_{l} $. The charge neutrality condition gives $\frac{2}{3}\widetilde{\nu }_{u}-\frac{1}{3} \widetilde{\nu }_{d}- \frac{1}{3}\widetilde{\nu }_{s}- \widetilde{\nu }_{\mu }-\widetilde{\nu }_{e}=0 $, or $\widetilde{\nu }_{u}-\widetilde{\nu }_{s}- 3\widetilde{\nu }_{e}=0 $. Whence $( \frac{\widetilde{\nu }_{s}}{6} )^{1/3}= ( \frac{\widetilde{\nu }_{u}}{6} )^{1/3}+ ( \frac{ \widetilde{\nu }_{e}}{6} )^{1/3} $. In terms of new variables $Z_{e}=\frac{\widetilde{\nu }_{e}}{\widetilde{\nu }_{s}}$ and $Z_{u}=\frac{\widetilde{\nu }_{u}}{\widetilde{\nu } _{s}} $ one then gets

$$ Z_{u}-1=3Z_{e}, \quad 1= Z_{u}^{1/3}+(3Z_{e})^{1/3}. $$

(33)

The latter has a solution $Z_{u}=1$ and $Z_{e}=0$. Asymptotically $(\widetilde{P_{F}}\gg \widetilde{m}c)$,

$$ \begin{aligned} &\widetilde{\nu }_{u}=\widetilde{\nu }_{s}=\widetilde{\nu }_{d}\equiv \widetilde{\nu }_{b}, \qquad \widetilde{P}_{Fu}=\widetilde{P}_{Fs}= \widetilde{P}_{Fd}\equiv \widetilde{P}_{Fb}, \\ &\widetilde{\nu }_{e}=\widetilde{\nu }_{\mu }=0, \end{aligned} $$

(34)

i.e., no leptons are presented in quark proto-matter in $\beta $-equilibrium. Such a system is $\varLambda $-like proto-matter with energy density $\widetilde{E}_{0}=\frac{3}{4}(\widetilde{\nu }_{u} \widetilde{P}_{Fu}+ \widetilde{\nu }_{d}\widetilde{P}_{Fd}+ \widetilde{\nu }_{s}\widetilde{P}_{Fs}) $. Now, let discuss the QCD interaction effects in approximation at hand, with extension to quark proto-matter. The first effect is the shift of the vacuum energy per unit volume. The bag constant $B\simeq 55~\mbox{MeV}/\mbox{fm}^{3}$ of the MIT bag model must be added to the kinetic energy density. Including the gluon exchange perturbative interactions the energy density of quark proto-matter is then given by the non-interacting Fermi contribution plus bag constant

$$ \widetilde{E}_{0}=\sum _{i}\frac{3}{4}\widetilde{\nu }_{i} \widetilde{P}_{Fi} \widetilde{b}_{I}(\widetilde{N}, \alpha _{c}) +B, $$

(35)

where $\widetilde{P}_{Fi}$ is the distorted Fermi momentum of $i$ flavor, $\widetilde{N}$ is the number of flavors present. The $\widetilde{N}$ and running coupling constant $\alpha _{c}$ takes into account the QCD perturbative interactions. The first correction to the free ground state is the ordinary exchange energy corresponding to the second order closed loop diagrams (e.g. Baym and Chin 1976). Next correction is coming up from the sum of different ring diagrams. With equal numbers of quarks of each flavor presented, according to Freedman and McLerran (1977), Baluni (1978), the modified function $\widetilde{b}_{I} (\widetilde{N}, \alpha _{c})$ reads

$$\begin{aligned} &{\widetilde{b}_{I}(\widetilde{N}, \alpha _{c})} \\ &{\quad=\widetilde{N} \biggl[ 1+\frac{2 \alpha _{c}}{3\pi }+\frac{\alpha _{c}^{2}}{3\pi ^{2}} \biggl( \widetilde{N}\ln \frac{\alpha _{c} \widetilde{N}}{\pi }+ 0.02 \widetilde{N}+6.75 \biggr) \biggr].} \end{aligned}$$

(36)

For numerical calculations it is sufficient to make use of the value $\alpha _{c} \simeq 2.2$ fitting the MIT bag model. Thus, according to (7) and (8), the macroscopic density of energy and internal pressure of quark proto-matter read

$$ \widetilde{\rho }=\sum _{i}\widetilde{m}c^{2}_{i} \frac{\chi ( \widetilde{y_{i}})\widetilde{b}_{I}}{\widetilde{\lambda }^{3}_{i}}+ B, \qquad \widetilde{P}=\sum _{i}\widetilde{m_{i}}c^{2}\frac{\varphi ( \widetilde{y_{i}}) \widetilde{b}_{I}}{\widetilde{\lambda }^{3}_{i}}, $$

(37)

where the quarks will be taken fully relativistic $m_{i}\rightarrow 0$.

Appendix B: The state equation specified for each domain

We use the Oppenheimer and Volkoff (OV)-units, where a length $\mbox{unit} =1.368 \times 10^{6}~\mbox{cm}$, a time $\mbox{unit}=4.564\times 10^{-5}~\mbox{s}$, a mass $\mbox{unit} =1.843 \times 10^{34}~\mbox{g}$, and energy $\mbox{unit}=1.656 \times 10^{55}~\mbox{erg}$. We also introduce a new variable $\nu $ as $n_{OV}=7.96178 \times 10^{55}e^{\nu }$, and rewrite the hydrostatic equilibrium equation in the form

$$ \nu '=-(s_{1}+s_{2})\frac{1}{2} (\ln g_{00} )', $$

(38)

where $(')$ means $\partial /\partial r$, $s_{1}=\widetilde{P}_{OV}( \nu ' /P_{OV}')$ and $s_{2}= \widetilde{\rho }_{OV}( \nu '/\rho _{OV}')$.

The resulting state equation is specified below for each domain step-by-step away from the domain of lower density up to the domain of higher density.

I-Class Configurations. The simple semiempirical formula of state equation is given by Harrison and Wheeler (see e.g. Shapiro and Teukolsky 1983).

Domain: $-27.2\le \nu < -15.5$,

$$\begin{aligned} &{P_{OV}=4.68\times 10^{-25} \bigl( 1.93\times 10^{5}\rho _{OV}^{1/3}-1.44 \bigr) ^{5}} \\ &{\phantom{P_{OV}=}{}-2.32\times 10^{-26},} \\ &{s_{1}=1.54\times 10^{-7}\rho _{OV}^{-1/3} \frac{ [ ( 1.93\times 10^{5}\rho _{OV}^{1/3}-1.44 ) ^{5}-1 ] }{ ( 1.93\times 10^{5}\rho _{OV}^{1/3}-1.44 ) ^{4}},} \\ &{s_{2}=6.64\times 10^{18}\rho _{OV}^{2/3} \bigl( 1.93\times 10^{5}\rho _{OV}^{1/3}-1.44 \bigr) ^{-4}.} \end{aligned}$$

(39)

Domain: $-15.5\le \nu < -2.8$,

$$ \begin{aligned} &P_{OV}=0.03 \rho _{OV}^{5/4} \bigl( 1+2.82\times 10^{-5}\rho _{OV}^{-1/2} \bigr) ^{-5/6}, \\ &s_{1}=\frac{0.08 ( 1+2.82\times 10^{-5}\rho _{OV}^{-1/2} ) }{ ( 1+3.99\times 10^{-5}\rho _{OV}^{-1/2} ) }, \\ &s_{2}=3.17\rho _{OV}^{-1/4} \bigl( 1+2.82\times 10^{-5}\rho _{OV}^{-1/2} \bigr). \end{aligned} $$

(40)

Domain: $-2.8\le \nu < -0.1$,

$$ \begin{aligned} &P_{OV}=1.79\times 10^{-5}\rho _{OV}^{2/3} \bigl( 1+1.39\rho _{OV}^{1/6}\bigr)^{6}, \\ &s_{1}=\frac{1.50 ( 1+1.40\rho _{OV}^{1/6} ) }{ ( 1+3.50 \rho _{OV}^{1/6} ) }, \\ &s_{2}=8.36\times 10^{4}\rho _{OV}^{1/3} \bigl( 1+1.40\rho _{OV}^{1/6} \bigr) ^{-5} \bigl( 1+3.50\rho _{OV}^{1/6} \bigr) ^{-1}. \end{aligned} $$

(41)

Domain: $-0.1\le \nu < 8.5$,

the regular n-p-e gas (ID is absent).

Domain: $8.5\le \nu $,

the state equation of the n-p-e proto-matter (10).

II-Class Configurations. For the II-class configurations, up to the density range $\nu =8.5$, one has the same domains of I-class configurations. At distances $0.25~\mbox{fm} < r_{NN}\le 0.4~\mbox{fm}$, in the

Domain: $8.5 \le \nu < 9.9$,

the string flip-flop state in ID regime (30).

Domain: $\nu \ge 9.9$,

the $\varLambda $-like system of quark proto-matter (37).

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G. Ter-Kazarian
- 1
Email author
S. Shidhani
- 2

1.Byurakan Astrophysical ObservatoryByurakanArmenia
2.Phys. Dept./College of ScienceSQUAl-KhoudOman

A study of 137 intermediate mass black hole candidates

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