Jeans instability of dark-baryonic matter model in the context of Kaniadakis' statistic distribution

Abstract

From a theoretical viewpoint, dark matter is a prerequisite entity in the formation processes of large-scale structures on the cosmological (galactic) scales of space and time. The behaviours of Jeans modes for gravitational systems composed of dark and baryonic matters are restudied in the framework of Kaniadakis' statistics and kinetic theory. The results show that the $\kappa$ parameter and density ratio of dark to baryonic matter ${\rho }_{\ast }$ have significant effects on the gravitational instabilities of such systems. As a test of the viability of this generalized context, we also prove that the dispersion relations for the Maxwellian case is recovered in the limitation of $\kappa \to 0$. Related results in the present work can provide scientific reference for structure formation on the cosmic Jeans scale.

Keywords:

• Jeans instability
• Kaniadakis' statistic
• dark matter

1. Introduction

The Jeans instability, firstly proposed by Jeans [1] and developed during the past several decades, is ubiquitous in the fields of astrophysics [2–6], plasma [7–13] and quantum plasma [14,15], etc. Also, it shows that the collapse of self gravitational systems will occur when its thermal pressure is too weak to resist its own gravity. In general, there are two paths to study the Jeans instability, namely, hydrodynamics model (or generalized hydrodynamics model and MHD model) and kinetic regime. In the former, various effects, such as rotation, magnetic field, shear viscosity or fluid viscosity, can be handily taken into account, which can be found elsewhere. For example, Prajapati et al. [16] investigated the effects of the magnetic field, shear viscosity and fluid viscosity on Jeans instability of self-gravitating magnetized strongly coupled plasma-based on generalized hydrodynamic model. Durrive et al. [17] proposed an MHD spectral theory approach to Jeans instability for the magnetized gravitating system. While in the latter, the Vlasov equation or Boltzmann equation coupled with the Poisson equation through different statistics distributions were adopted [18–20].

Until very recently, the Jeans instability still is considered the key mechanism to explain the gravitational formation of structures and their evolution in the linear regime. Based on the Jeans' pioneering work, Tsiklauri had theoretically researched the model of interstellar gas clouds with weakly interacting massive particles and found that the Jeans length decreased due to the presence of the dark matter, in turn, the Jeans mass of the interstellar gas clouds [21]. Kremer et al. have recently examined the effective role of dark matter in star formation based on the kinetics theory [19,22]. Furthermore, combining the observation data, Arun et al. also showed the dark matter may provide the seed for star formation in the early universe [23]. Note that all of these investigations emphasize the role of dark matter in the process of star formation, which means that the consideration of the presence of dark matter in the process of structure formation is indispensable, or generally speaking the dark matter coupled with baryonic matter existing in self-gravitational systems will occur some remarkable phenomenons. In order to get a more complete view of this key ingredient of the present work, dark matter, it is necessary to illustrate its development context. In fact, the dark matter, one of the hottest topics of present-day physics, is viewed as the most plausible explanation for various observations from galactic to cosmological scales, such as flat rotation curves of galaxies [24], formation in large scale structures [25,26] and cosmic microwave background radiation [27], etc. It indeed plays an auxiliary role in the process of structure formation and exists on all astrophysical scales [28,29]. Very recently, based on the generalized hydrodynamics model, Karmakar and his co-author [30–32] gave a series of comprehensive works on the gravitational instabilities of gravito-coupled complex gyratory astrofluids with the presence of dark matter and showed that the dark matter plays a significant dynamic role in the instabilities. On the other hand, by assuming that the dark matter is bosonic particles and the form of a Bose-Einstein condensate state occurs, Harko studied the Jeans instability and the gravitational collapse of the rotating dark matter halos [33]. Furthermore, based on such assumption, Chavanis studied the Jeans instability of an infinite homogeneous dissipative self-gravitating Bose-Einstein condensate with repulsive or attractive self-interaction [34]. Those works provide us an alternative way to get insight into the physics of dark matter and dynamics of large-scale structure formation in the presence of dark matter.

On the other hand, some investigations [20,35–37] involving the statistical description of various physical models indicate that $\kappa$-deformed Kaniadakis velocity distribution has significant effect on the Jeans instability of self-gravitational systems. For example, Abreu et al. recently studied the Jeans instability in the context of Kaniadakis' statistic and found a new Jeans' criterion related to the parameter κ. Based on the work of Abreu et al. [36], Chen et al. [20] have further researched the effect of $\kappa$-deformed Kaniadakis velocity distribution on the self-gravitational systems and showed that it modifies the range of unstable modes of Jeans instability. As a matter of fact, the $\kappa$-deformed Kaniadakis velocity distribution, proposed by Kaniadakis [38] in 2001 and developed over several decades, is one of the extensions of the standard Maxwellian distribution, which has been widely applied to the field of black-body radiation [39], quantum entanglement [40], cosmic rays [41], quark-gluon plasma formation [42], the financial systems [43], and especially for the self-gravitational matter systems, etc. For brevity, we call the $\kappa$-deformed Kaniadakis velocity distribution as Kaniadakis distribution.

With the above-mentioned contents, one can conclude that the effect of Kaniadakis distribution on Jeans instability of self-gravitational systems has been widely researched, and the related conclusions also provide important physical significance for the Jeans instability in the interstellar could. However, to the best of author's knowledge, no investigations have been made to study the role of dark matter on the self-gravitational systems with Kaniadakis distribution, despite dark matter is viewed as an invisible hand promoting the formation of structure. Thus, a revision of the classical theory of Jeans instability for self-gravitational systems composed by dark and baryonic matter in the context of Kaniadakis distribution, or generally speaking, the consideration of dark matter in such self-gravitational systems, seems to be of considerable importance. In this sense, the present work is devoted to studying what extent the Kaniadakis effect modifies the Jeans modes of such systems.

The present manuscript is organized as follows. The modified Vlasov-Poisson equations, the form of Kaniadakis velocity distribution, and finally the analysis of general dispersion relation are given in Section 2. The effect of Kaniadakis on the self-gravitational systems is discussed in Section 3. And the importance of dark matter on the process of structure formation is analysed in Section 4. Conclusions are presented in Section 5.

2. The modified Jeans modes

We focus our attention on the kinetic behaviour of the gravitational system, which is composed of collisionless dark and baryonic matter within the background of Kaniadakis distribution. The collisionless Boltzmann equations for dark and baryonic matter, or so-called Vlasov equations, are shown as follows (1) $\begin{array}{rl}& \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }t}+\mathit{v}\cdot \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }\mathit{r}}-\mathrm{\nabla }\varphi \cdot \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }\mathit{v}}=0,\end{array}$(1) (2) $\begin{array}{rl}& \frac{\mathrm{\partial }{f}_d}{\mathrm{\partial }t}+\mathit{v}\cdot \frac{\mathrm{\partial }{f}_d}{\mathrm{\partial }\mathit{r}}-\mathrm{\nabla }\varphi \cdot \frac{\mathrm{\partial }{f}_d}{\mathrm{\partial }\mathit{v}}=0,\end{array}$(2) where the indexes $b$ and $d$ denote baryonic and dark matter, respectively. The expression $f_{b(d)}(\equiv f_{b(d)}(\mathit{v},\mathit{r},t))$ represents the distribution function of the matter-particles. Considering the completeness of equations, the gravitational Poisson equation is written as (3) $\begin{array}{l}{\mathrm{\nabla }}^2\varphi =4\pi G(\int f_b\mathrm{d}\mathit{v}+\int f_d\mathrm{d}\mathit{v})\\ =4\pi G({\rho }_b+{\rho }_d),\end{array}$(3) where ${\rho }_{b(d)}$ is the density of the matters, and $\varphi$ is gravitational potential, respectively. On the line of kinetic theory, the distribution function and gravitational potential are expanded as (4) $\begin{array}{rl}& f=f_{b(d)}^0(\mathit{v})+f_{b(d)}^1(\mathit{v},\mathit{r},t),\end{array}$(4) (5) $\begin{array}{rl}& \varphi ={\varphi }_0(\mathit{r})+{\varphi }_1(\mathit{r}),\end{array}$(5) Here, the terms ${\varphi }_1(\mathit{r})$ and $f_{b(d)}^1(\mathit{v}\mathbf{,}\mathbf{r}\mathbf{,}\mathit{t})$ are the perturbations around equilibrium state, i.e. ${\varphi }_0(\mathit{r})$ and $f_{b(d)}^0(\mathit{v})$. Involving Jeans swindle [18], the equilibrium for a homogeneous system is achieved by setting ${\varphi }_0(\mathit{r})=0$. Then, substituting Equations (4(4) $\begin{array}{rl}& f=f_{b(d)}^0(\mathit{v})+f_{b(d)}^1(\mathit{v},\mathit{r},t),\end{array}$(4) )–(5(5) $\begin{array}{rl}& \varphi ={\varphi }_0(\mathit{r})+{\varphi }_1(\mathit{r}),\end{array}$(5) ) into the Equations (1(1) $\begin{array}{rl}& \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }t}+\mathit{v}\cdot \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }\mathit{r}}-\mathrm{\nabla }\varphi \cdot \frac{\mathrm{\partial }{f}_b}{\mathrm{\partial }\mathit{v}}=0,\end{array}$(1) )–(3(3) $\begin{array}{l}{\mathrm{\nabla }}^2\varphi =4\pi G(\int f_b\mathrm{d}\mathit{v}+\int f_d\mathrm{d}\mathit{v})\\ =4\pi G({\rho }_b+{\rho }_d),\end{array}$(3) ) and using the Fourier analysing with plane wave ansatz, i.e. $(f_{b(d)}^1,{\varphi }_1)\propto \exp [-i(\omega t-\mathit{k}\cdot \mathit{r})]$, one can get the dispersion equation for longitudinal modes in self-gravitational matter systems as [19] (6) $\begin{array}{rl}& 1+\frac{4\pi G}{k^2}(\int \frac{\mathit{k}\cdot \frac{\mathrm{\partial }{f}_b^0(\mathit{v}\mathbf{)}}{\mathrm{\partial }{\mathit{v}}_b}}{\mathit{k}\cdot {\mathit{v}}_b-\omega }\mathrm{d}{\mathit{v}}_b+\int \frac{\mathit{k}\cdot \frac{\mathrm{\partial }{f}_d^0(\mathit{v})}{\mathrm{\partial }{\mathit{v}}_d}}{\mathit{k}\cdot {\mathit{v}}_d-\omega }\mathrm{d}{\mathit{v}}_d)\\ & =0.\end{array}$(6) Along the standard procedure, choosing ${\mathit{v}}_{b(d)}$ in the direction of $x$ and setting $\mathit{k}=k{\hat{e}}_x$, then Equation (6(6) $\begin{array}{rl}& 1+\frac{4\pi G}{k^2}(\int \frac{\mathit{k}\cdot \frac{\mathrm{\partial }{f}_b^0(\mathit{v}\mathbf{)}}{\mathrm{\partial }{\mathit{v}}_b}}{\mathit{k}\cdot {\mathit{v}}_b-\omega }\mathrm{d}{\mathit{v}}_b+\int \frac{\mathit{k}\cdot \frac{\mathrm{\partial }{f}_d^0(\mathit{v})}{\mathrm{\partial }{\mathit{v}}_d}}{\mathit{k}\cdot {\mathit{v}}_d-\omega }\mathrm{d}{\mathit{v}}_d)\\ & =0.\end{array}$(6) ) is rewritten as (7) $\begin{array}{rl}& 1+\frac{4\pi G}{k^2}(\int \frac{k\frac{\mathrm{\partial }{f}_{bx}^0}{\mathrm{\partial }{v}_{bx}}}{k{v}_{bx}-\omega }\mathrm{d}{v}_{bx}+\int \frac{k\frac{\mathrm{\partial }{f}_{dx}^0}{\mathrm{\partial }{v}_{dx}}}{k{v}_{\mathrm{d}x}-\omega }\mathrm{d}{v}_{\mathrm{d}x})\\ & =0,\end{array}$(7) where that the one-dimensional distribution functions $f_{bx}^0$ and $f_{dx}^0$ are defined as (8) $f_{b(d)x}^0=\int f_{b(d)}^0(\mathit{v})\mathrm{d}{v}_{b(d)y}\mathrm{d}{v}_{b(d)z}.$(8) Consider now the Kaniadakis' statistics framework proposed by Kaniadakis, the Kaniadakis distribution function can be expressed by [20,38,41] (9) $f_{b(d)}(\mathit{v})=A_{\kappa }{(\sqrt{1+{\kappa }^2\frac{v^4}{4{\sigma }_{b(d)}^4}}-\kappa \frac{v^2}{2{\sigma }_{b(d)}^2})}^{\frac{1}{\kappa }},$(9) where the velocity dispersion ${\sigma }_{b(d)}$ is viewed as the thermal velocity of the matter systems, and through the normalized condition ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}_{b(d)}^0(\mathit{v})\mathrm{d}\mathit{v}={\rho }_{b(d)}^0$, the normalized coefficient $A_{\kappa }$ is presented as (10) $A_{\kappa }={\rho }_{b(d)}^0\frac{{\left|\frac{\kappa }{\pi }\right|}^{\frac{3}{2}}}{{\sigma }_{b(d)}^3}(1+\frac{3}{2}|\kappa |)\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{3}{4})}.$(10) It should be pointed out that there are several studies that reported Boltzmann-Gibbs statistics or standard Maxwellian distribution cannot be applied rigorously, due to the fact that there is a fundamental difference between stars in a galaxy governed by long-range gravitational interactions and the short-range correlations between molecules in a gas [44,45]. Beside the non-extensive statistics, the Kaniadakis statistics is also an optional candidate for cosmological models [46]. And as shown previously, the Kaniadakis distribution function can be viewed as a generalized form of Maxwellian function and has been widely adopted in several systems. Now, introducing the second velocity moment in a usual way, (11) $\begin{array}{rl}⟨v^2⟩& \equiv 3{\sigma }_{\mathrm{eff}}=\frac{3}{2}\frac{1+\frac{3}{2}|\kappa |}{(1+\frac{5}{2}|\kappa |)\kappa }\\ & \times \frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{5}{4})}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{3}{4})}{\sigma }^2,\end{array}$(11) which is an important parameter embodying the relationship between the effect velocity and $\kappa$ parameter. Where the root-mean-square speed ${\sigma }_{\mathrm{eff}}$ can be viewed as the effect velocity in the Kaniadakis distribution function. In the study of Mace et al. [47], the critical wave number for the Jeans instability was defined as $k_J\equiv \sqrt{{\mathrm{\Omega }}_G/{\sigma }_{\mathrm{eff}}}$. Then one can find that in the framework of Kaniadakis's statistics, the critical wave number strongly depends on the distribution index $\kappa$ (see Equation (11(11) $\begin{array}{rl}⟨v^2⟩& \equiv 3{\sigma }_{\mathrm{eff}}=\frac{3}{2}\frac{1+\frac{3}{2}|\kappa |}{(1+\frac{5}{2}|\kappa |)\kappa }\\ & \times \frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{5}{4})}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{3}{4})}{\sigma }^2,\end{array}$(11) )).

Putting Equation (9(9) $f_{b(d)}(\mathit{v})=A_{\kappa }{(\sqrt{1+{\kappa }^2\frac{v^4}{4{\sigma }_{b(d)}^4}}-\kappa \frac{v^2}{2{\sigma }_{b(d)}^2})}^{\frac{1}{\kappa }},$(9) ) into Equation (8(8) $f_{b(d)x}^0=\int f_{b(d)}^0(\mathit{v})\mathrm{d}{v}_{b(d)y}\mathrm{d}{v}_{b(d)z}.$(8) ), the one-dimensional Kaniadakis distribution function is obtained as (12) $\begin{array}{rl}{f}_{b(d)x}^0& =-\frac{2{\sigma }_{b(d)}^2{A}_{\kappa }\pi }{{\kappa }^2-1}\\ & \times {[-\kappa \frac{v_x^2}{2{\sigma }_{b(d)}^2}+\sqrt{1+{\kappa }^2{\left(\frac{v_x^2}{2{\sigma }_{b(d)}^2}\right)}^2}]}^{\frac{1}{\kappa }}\\ & \cdot [{\kappa }^2\frac{v_x^2}{2{\sigma }_{b(d)}^2}+\sqrt{1+{\kappa }^2{\left(\frac{v_x^2}{2{\sigma }_{b(d)}^2}\right)}^2}].\end{array}$(12) Furthermore, substituting $\frac{\mathrm{\partial }{f}_{b(d)x}^0}{\mathrm{\partial }{v}_x}$ into Equation (7(7) $\begin{array}{rl}& 1+\frac{4\pi G}{k^2}(\int \frac{k\frac{\mathrm{\partial }{f}_{bx}^0}{\mathrm{\partial }{v}_{bx}}}{k{v}_{bx}-\omega }\mathrm{d}{v}_{bx}+\int \frac{k\frac{\mathrm{\partial }{f}_{dx}^0}{\mathrm{\partial }{v}_{dx}}}{k{v}_{\mathrm{d}x}-\omega }\mathrm{d}{v}_{\mathrm{d}x})\\ & =0,\end{array}$(7) ) and using the following integral relation [48] (13) $\begin{array}{rl}& {\int }_0^{\mathrm{\infty }}{x}^{r-1}{(-\kappa x+\sqrt{1+{\kappa }^2{x}^2})}^{1/\kappa }\mathrm{d}x\\ & =\frac{|2\kappa {|}^{-r}}{1+r|\kappa |}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{r}{2})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{r}{2})}\mathrm{\Gamma }(r),\end{array}$(13) the general dispersion relation is obtained (14) $\begin{array}{r}1-\frac{1}{C_{\kappa }{k}^2}[k_{Jb}^2(B_{\kappa }-Z\left({\xi }_b\right))+k_{Jd}^2(B_{\kappa }-Z\left({\xi }_d\right))]=0.\end{array}$(14) The expression $k_{Jb(d)}^2(\equiv 4\pi G{\rho }_{b(d)}^0/{\sigma }_{b(d)}^2)$ denotes the corresponding classical Jeans wave numbers. And the coefficient $B_{\kappa }$, $C_{\kappa }$ are written by (15) $\begin{array}{rl}{B}_{\kappa }& =\frac{|2\kappa {|}^{-\frac{1}{2}}}{1+\frac{1}{2}|\kappa |}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{1}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{1}{4})}\sqrt{\pi },\end{array}$(15) (16) $\begin{array}{rl}{C}_{\kappa }& =\frac{|\kappa {|}^{-\frac{3}{2}}}{1+\frac{3}{2}|\kappa |}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{3}{4})}\frac{\sqrt{\pi }}{2\sqrt{2}},\end{array}$(16) and the modified dispersion function is shown as (17) $\begin{array}{rl}Z\left({\xi }_{\alpha }\right)& =\int \frac{{\xi }_{\alpha }}{X_{\alpha x}-{\xi }_{\alpha }}\\ & \times {(-\kappa X_{\alpha x}^2+\sqrt{1+{\kappa }^2{\left(X_{\alpha x}^2\right)}^2})}^{\frac{1}{\kappa }}\mathrm{d}{X}_{\alpha x}\end{array}$(17) with its argument ${\xi }_{\alpha }=\omega /\sqrt{2}k{\sigma }_{\alpha }$.

For Jeans modes, the boundary between stable and unstable modes is defined as $\omega =0$, i.e. ${\xi }_{\alpha }=0$ in Equation (14(14) $\begin{array}{r}1-\frac{1}{C_{\kappa }{k}^2}[k_{Jb}^2(B_{\kappa }-Z\left({\xi }_b\right))+k_{Jd}^2(B_{\kappa }-Z\left({\xi }_d\right))]=0.\end{array}$(14) ) and thus, the modified critical wave number $k_{\kappa }$ is given by (18) $k_{\kappa }^2=\frac{B_{\kappa }}{C_{\kappa }}(k_{Jb}^2+k_{Jd}^2).$(18) One can find that when ignoring the effect of dark matter, that is, $k_{Jd}=0$, the modified critical wave number $k_{\kappa }$ will go back to the results of Abreu et al. [36].

3. The effect of Kaniadakis distribution

In the following, we will study the properties of unstable modes incorporating the effects of Kaniadakis distribution. Setting $\omega =i\gamma$, where $\gamma$ is real positive, and putting it into Equation (14(14) $\begin{array}{r}1-\frac{1}{C_{\kappa }{k}^2}[k_{Jb}^2(B_{\kappa }-Z\left({\xi }_b\right))+k_{Jd}^2(B_{\kappa }-Z\left({\xi }_d\right))]=0.\end{array}$(14) ), the dispersion relation with $\omega =i\gamma$ can be re-written by (19) $\begin{array}{rl}{k}_{\ast }^2& =\frac{B_{\kappa }}{C_{\kappa }}\frac{{\sigma }_d^2{\rho }_b^0}{{\sigma }_b^2{\rho }_d^0}-\frac{1}{C_{\kappa }}Z\left(\frac{{\sigma }_d}{{\sigma }_b}\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)\frac{{\sigma }_d^2{\rho }_b^0}{{\sigma }_b^2{\rho }_d^0}+\frac{B_{\kappa }}{C_{\kappa }}\\ & -\frac{1}{C_{\kappa }}Z\left(\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)\end{array}$(19) where the expressions $k_{\ast }(\equiv \frac{{\sigma }_{d}k}{\sqrt{4\pi G{\rho }_d^0}})$ and ${\omega }_{\ast }(\equiv \frac{\gamma }{\sqrt{4\pi G{\rho }_d^0}})$ are the normalized wave number and frequency, respectively, as well as the modified dispersion function should be transformed to the following form, (20) $\begin{array}{rl}Z\left({\beta }_{\alpha }\right)& ={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}\frac{{\beta }_{\alpha }^2}{X^2+{\beta }_{\alpha }^2}\\ & \times {(-\kappa X_{\alpha x}^2+\sqrt{1+{\kappa }^2{\left(X_{\alpha x}^2\right)}^2})}^{\frac{1}{\kappa }}\mathrm{d}{X}_{\alpha x}\end{array}$(20) where ${\beta }_{\alpha }=\gamma /\sqrt{2}k{\sigma }_{\alpha }$. Under the limitation $\kappa \to 0$, one has $B_{\kappa },C_{\kappa }\to \sqrt{\pi }$ [20] and $(-\kappa X_{\alpha x}^2+\sqrt{1+{\kappa }^2{X}_{\alpha x}^4}{)}^{\frac{1}{\kappa }}\to \exp (-x^2)$ [48]. With the aid of the following integral function[49] (21) ${\int }_0^{\mathrm{\infty }}\frac{x^2{e}^{-{\mu }^2{x}^2}}{x^2+{\beta }^2}\mathrm{d}x=\frac{\sqrt{\pi }}{2\mu }-\frac{\pi \beta }{2}{e}^{{\beta }^2}\mathrm{erfc}(\beta \mu ),$(21) then the modified dispersion function is rewritten as (22) $Z({\beta }_{\alpha })=\pi {\beta }_{\alpha }{e}^{{\beta }_{\alpha }^2}\mathrm{erfc}({\beta }_{\alpha }).$(22) Substituting Equation (22(22) $Z({\beta }_{\alpha })=\pi {\beta }_{\alpha }{e}^{{\beta }_{\alpha }^2}\mathrm{erfc}({\beta }_{\alpha }).$(22) ) into Equation (19(19) $\begin{array}{rl}{k}_{\ast }^2& =\frac{B_{\kappa }}{C_{\kappa }}\frac{{\sigma }_d^2{\rho }_b^0}{{\sigma }_b^2{\rho }_d^0}-\frac{1}{C_{\kappa }}Z\left(\frac{{\sigma }_d}{{\sigma }_b}\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)\frac{{\sigma }_d^2{\rho }_b^0}{{\sigma }_b^2{\rho }_d^0}+\frac{B_{\kappa }}{C_{\kappa }}\\ & -\frac{1}{C_{\kappa }}Z\left(\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)\end{array}$(19) ), and thus the dispersion relation for Maxwellian case can be recovered, which has been studied by Kremer et al. [19] (23) $\begin{array}{rl}{k}_{\ast }^2& =1-\sqrt{\frac{\pi }{2}}\frac{{\omega }_{\ast }}{k_{\ast }}\exp \left(\frac{{\omega }_{\ast }^2}{2k_{\ast }^2}\right)\mathrm{erfc}\left(\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)+\frac{{\rho }_b^0{\sigma }_d^2}{{\rho }_d^0{\sigma }_b^2}\\ & \cdot [1-\sqrt{\frac{\pi }{2}}\frac{{\sigma }_d}{{\sigma }_b}\frac{{\omega }_{\ast }}{k_{\ast }}\exp \left(\frac{{\sigma }_d^2}{{\sigma }_b^2}\frac{{\omega }_{\ast }^2}{2k_{\ast }^2}\right)\\ & \times \mathrm{erfc}\left(\frac{{\sigma }_d}{{\sigma }_b}\frac{{\omega }_{\ast }}{\sqrt{2}{k}_{\ast }}\right)].\end{array}$(23)

To get the numerical solutions of dispersion relation with different indexes $\kappa$, we adopt the dispersion velocities ratio ${\sigma }_{\ast }={\sigma }_d/{\sigma }_b\approx 1.83$ [50], and the density ratio ${\rho }_{\ast }={\rho }_{d0}/{\rho }_{b0}\approx 5.47$ [51]. The dependence of the growth rate of Jeans instability on the $\kappa$ parameter is depicted in Figure 1. The black curve with $\kappa =0$ corresponds to the Maxwellian case, which is identical to the blue curve in Figure 1 of Kremer and Andr [19]. And the green curve with $\kappa =0.01$ highly overlaps with the black curve in Figure 1, which means the growth rate curve for the Maxwellian case are recovered in the limitation $\kappa \to 0$, as expected. Furthermore, one can find from Figure 1 that as the value of $\kappa$ increased, the ranges and growth rates of the unstable mode decrease. One thus can arrive a conclusion that the Jeans instability of gravitational systems composed by collisionless dark and baryonic matter is suppressed in the background of the Kaniadakis distribution comparing with the Maxwellian case. In other word, the gravitational collapse of such systems is disfavoured with respect to the Kaniadakis distribution. Physically, Jeans instability occurs because the thermal pressure is too weak to prevent gravitational collapse of the matter systems. One can find from Equation (11(11) $\begin{array}{rl}⟨v^2⟩& \equiv 3{\sigma }_{\mathrm{eff}}=\frac{3}{2}\frac{1+\frac{3}{2}|\kappa |}{(1+\frac{5}{2}|\kappa |)\kappa }\\ & \times \frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{5}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{5}{4})}\frac{\mathrm{\Gamma }(\frac{1}{|2\kappa |}+\frac{3}{4})}{\mathrm{\Gamma }(\frac{1}{|2\kappa |}-\frac{3}{4})}{\sigma }^2,\end{array}$(11) ) that the effect velocity ${\sigma }_{\mathrm{eff}}$ and its related thermal pressure augments with the increase of $\kappa$, it is therefore clear why both of the ranges of unstable mode and growth rates will decrease as $\kappa$ increased.

Figure 1. The normalized frequency ${\omega }_{\ast }=\gamma /\sqrt{4\pi G{\rho }_d^0}$ versus the normalized wave number $k_{\ast }=k/k_{Jd}$ with different $\kappa$ and the fixed value of ${\sigma }_{\ast }=1.83$, ${\rho }_{\ast }=5.47$.

4. The role of dark matter

To study the role of dark matter on the self-gravitational systems with Kaniadakis distribution, we introduce a new normalized wave number $K_{\ast }(=\frac{{\sigma }_{b}k}{\sqrt{4\pi G{\rho }_b^0}})$ and frequency ${\mathrm{\Omega }}_{\ast }(=\frac{\gamma }{\sqrt{4\pi G{\rho }_b^0}})$, then the general dispersion relation should be rewritten as (24) $\begin{array}{rl}{K}_{\ast }^2& =\frac{B_{\kappa }}{C_{\kappa }}\frac{{\sigma }_b^2{\rho }_d^0}{{\sigma }_d^2{\rho }_b^0}-\frac{1}{C_{\kappa }}Z\left(\frac{{\sigma }_b}{{\sigma }_d}\frac{{\mathrm{\Omega }}_{\ast }}{\sqrt{2}{K}_{\ast }}\right)\frac{{\sigma }_b^2{\rho }_d^0}{{\sigma }_d^2{\rho }_b^0}+\frac{B_{\kappa }}{C_{\kappa }}\\ & -\frac{1}{C_{\kappa }}Z\left(\frac{{\mathrm{\Omega }}_{\ast }}{\sqrt{2}{K}_{\ast }}\right).\end{array}$(24) The numerical results of Equation (24(24) $\begin{array}{rl}{K}_{\ast }^2& =\frac{B_{\kappa }}{C_{\kappa }}\frac{{\sigma }_b^2{\rho }_d^0}{{\sigma }_d^2{\rho }_b^0}-\frac{1}{C_{\kappa }}Z\left(\frac{{\sigma }_b}{{\sigma }_d}\frac{{\mathrm{\Omega }}_{\ast }}{\sqrt{2}{K}_{\ast }}\right)\frac{{\sigma }_b^2{\rho }_d^0}{{\sigma }_d^2{\rho }_b^0}+\frac{B_{\kappa }}{C_{\kappa }}\\ & -\frac{1}{C_{\kappa }}Z\left(\frac{{\mathrm{\Omega }}_{\ast }}{\sqrt{2}{K}_{\ast }}\right).\end{array}$(24) ) are shown in Figure 2, where the normalized growth rate ${\mathrm{\Omega }}_{\ast }$ vs normalized wave number $K_{\ast }$ are presented for three different case, i.e. ${\rho }_{\ast }=0$ for only baryonic matter considered (red line), and ${\rho }_{\ast }=1$ (green line) and ${\rho }_{\ast }=5.47$ (black lines), respectively. It should be pointed out that the red line with only baryonic matter is identical to the green curve in Figure 2 of Chen et al. [20]. And the comparison between the red and black lines indicates that the presence of dark matter boosts the growth rate of Jeans instability for self-gravitational systems. It also can be found that an increase in density ratio ${\rho }_{\ast }$ (that is the augmentation of dark matter) leads to the increase the growth rate of Jeans instability. More interested is to discuss the physical significance of such a phenomenon, which may reveal the role of dark matter on the process of structure formation. In fact, the conclusion in the present work is not incompatible with the model of cold dark matter. The dark matter, an auxiliary matter in a process of structure formation, couples the baryonic matter only via gravity fields. For the star dominated by dark matter, the gravity fields produced by dark matter are more stronger than that star only having baryonic matter, and in the effect of such gravity field, more baryonic matter will be attracted by an initial conglomeration composed of dark matter. Furthermore, more important thing is that dark matter with Kaniadakis distribution leads to a new distribution of gravity field different from the Maxwellian case. Such spectacle explains why the growth rate is increased with the enhancement of density ratio ${\rho }_{\ast }$, whereas it decreased with the increased $\kappa$ parameter.

Figure 2. The normalized frequency ${\mathrm{\Omega }}_{\ast }=\gamma /\sqrt{4\pi G{\rho }_b^0}$ versus the normalized wave number $K_{\ast }=k/k_{Jb}$ with different ${\rho }_{\ast }$ and the fixed value of $\kappa =0.2$, ${\sigma }_{\ast }=1.83$.

5. Conclusion

The Jeans instability of gravitational systems composed by dark and baryonic matter in the context of Kaniadakis' statistics has been analysed in the present work. In this extended kinetic framework, we have made a step forward by converting the Gaussian phase space density to a family of power-law distributions parameterized by the $\kappa$ parameter for studying the effect of Kaniadakis distribution on the criterion of Jeans modes. The analysis of the dispersion equation derived by modified Vlasov-Poisson equations shows that in comparison with the Maxwellian distribution, the Jeans instability of gravitational systems composed by dark and baryonic matter is suppressed in the background of the Kaniadakis distribution and, therefore, opposes the gravitational collapse. Also, one also finds that a new critical value of Jeans wave number depends explicitly on the $\kappa$ parameter and dark matter. Additionally, the presence of dark matter makes the self-gravitating gas cloud more unstable against gravitational collapse, which has also been reported previously [19,21].

As a more generalized statistics, the Kaniadakis' statistics provides a new insight into the investigation of Jeans instability. The results of the present work provide scientific reference for further study and exploitation of the Jeans instability of self-gravitational systems. However, the present work can be possibly extended with the considerations of other research in non-linear partially ionized dusty plasma. Furthermore, given the fact that Bose-Einstein statistics is reasonable for the dark matter [33,34], then the Bose-Einstein distribution function can be adapted to study the effect of dark matter on the dynamics of Jeans modes within the kinetic regime. We will make a plan to investigate the related works.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant numbers 11763006, 11863004], the Jiangxi Province Key Laboratory of Fusion and Information Control [grant number 20171BCD40005].