Statistical and Dynamical Decoupling of the IGM from Dark Matter

Abstract

Although the gravitational field in the universe is dominated by dark matter, cosmological observations show that the statistical properties of cosmic baryonic matter are significantly and systematically decoupled from that of the underlying dark matter. The dynamical reason of the decoupling is the difference of the nonlinear evolution of baryon fluid from that of collisionless dark matter. In highly nonlinear regime, the cosmic baryon fluid on scale free range evolves into the state of fully developed turbulence, of which the velocity field consists of shocks, vortices and complex structures. This scenario provides a coherent explanation of various phenomena referring to the statistical and dynamical decoupling of the IGM from dark matter.

1. Introduction

The mass field of the universe is dominated by dark matter, and only a tiny fraction of cosmic matter is in the form of baryonic particles. Gravitational clustering of dark matter is the major imputs of cosmic structure formation. Collapsed massive halos of dark matter are believed to be the hosts of various observed objects. On the other hand, light-emitting and absorbing objects are made up of baryonic matter.

Most of the baryonic matter in the universe actually is made of intergalactic gas or intergalactic medium (IGM), mainly consisting of neutral and ionized hydrogen and neutral and ionized helium [1–3]. For a long time, the mass field of the IGM, 𝜌_𝑏(𝐫), is believed to be proportional to dark matter field 𝜌_dm(𝐫). The similarity relation 𝜌_𝑏(𝐫)∝𝜌_dm(𝐫) should be hold, at least, on scales larger than the thermal diffusion or Jeans length of the IGM. In other words, the IGM mass field is considered as a point-by-point tracer of the dark matter distribution. Obviously, in this case, the baryon fraction 𝜌_𝑏(𝐫)/[𝜌_dm(𝐫)+𝜌_𝑏(𝐫)] is independent of 𝐫, and equals to the cosmic mean Ω_𝑏/(Ω_𝑏+Ω_dm)=0.17±0.01 [4, 5], where Ω_𝑏 and Ω_dm are, respectively, the cosmological parameters of baryonic and dark matter.

However, the similarity 𝜌_𝑏(𝐫)∝𝜌_dm(𝐫) is found to be seriously inconsistent with observation in collapsed objects. The baryon fractions in gravitational bound objects generally are much lower than the cosmic mean. It raises the so-called missing baryon problem. The baryon fraction is claimed to decrease monotonically with the mass of collapsed halos. For X-ray clusters and groups, the baryon fractions are smaller than the cosmic mean by a factor of 2–4 [6–8]. The baryon fraction of dwarf galaxies can be as low as about 1% of the cosmic mean [9].

Many works have been carried out to explain the deficit of baryon content in gravitational collapsed objects, including feedback due to supernovae [10] or Active Galactic Nuclei (AGN) [11, 12], which blows out hot gas from halos, and preheating [13, 14], which heats baryon gas in the IGM and slows down the accretion of the IGM into halos. However, cosmologically hydrodynamic simulations show that these mechanisms fail to produce the baryon fraction in halos revealed by observation [15–17].

Actually, the baryon-dark matter decoupling probably is due mainly to the nonlinear nature of the hydrodynamics of the velocity and mass fields of IGM. That is, even without extra heating and cooling, the hydrodynamical evolution can already provide mechanisms of separating the baryonic matter from dark matter. A possible dynamical mechanism driving the baryonic matter deviate from dark matter has been addressed in the pioneer work of Shandarin and Zeldovich [18]. As the dark matter particles are assumed to be collisionless, their velocities would be multivalued at the intersection of trajectories. However, the baryon matter has a single-valued velocity field as a compressible fluid. Discontinuities, such as, shocks or complex structures, will develop inevitably in the baryon fluid at these intersections. Shocks would then cause the underdense of baryonic matter in the postshock area. On the other hand, dark matter is not affected by the shocks. Thus, discontinunities might prevent baryon matter to follow dark matter. Consequently, the baryon fraction in the collapsed halos would be less than cosmic mean [19].

Therefore, the nonlinear evolution of IGM would be a major reason of the statistical and dynamical decoupling between the cosmic baryon fluid and dark matter. We report the results of this mechanism. This paper will be arranged as follows. Section 2 is on the linear theory of evolution of IGM perturbations. Section 3 presents the decoupling in moderate nonlinear regime. Section 4 studies the fully nonlinear development of the IGM. It shows the cosmic baryon fluid to be turbulent on small scales. Section 5 gives observation evidences. The conclusions will be in Section 6.

2. Linear Evolution of the Perturbations of IGM Mass Field

2.1. Basic Equations

Let us consider a flat universe having cosmic factor 𝑎(𝑡)∝𝑡^2/3, and dominated by dark matter. We describe the IGM by a mass density field 𝜌(𝐱,𝑡) and a peculiar velocity field 𝐯(𝐱,𝑡), where 𝐱 is the comoving coordinate. In hydrodynamical description, the equations of the IGM consist of the continuity and the momentum (or the Eular) equations as [20]𝜕𝛿𝜕𝑎𝐯𝜕𝑡+𝑎1∇⋅(1+𝛿)𝐯=0,(1)𝜕𝑡+(𝐯⋅∇)𝐯=−𝜌1∇𝑝−∇,(2) where the density perturbation of the IGM 𝛿(𝐱,𝑡)=[𝜌(𝐱,𝑡)−𝜌]/𝜌, and 𝜌 is the mean density of the IGM. 𝑝 is the pressure of the IGM.

The IGM can be considered as a “passive substance” with respect to the underlying dark matter. Namely, the evolution of the IGM mass field 𝜌(𝐱,𝑡) is governed by the gravity of total matter (baryonic plus dark matter). The peculiar gravitational potential 𝜙 in (2) satisfies∇²𝜙=4𝜋𝐺𝑎²𝜌_𝑡𝛿_𝑡,(3) where the operator ∇ is acting on the comoving coordinate 𝐱, and 𝛿_𝑡(𝐱,𝑡)=[𝜌_𝑡(𝐱,𝑡)−𝜌_𝑡]/𝜌_𝑡 is the perturbation of total density field, with mean density 𝜌_𝑡=1/6𝜋𝐺𝑡²∝𝑎⁻³. The potential 𝜙 will be zero (or constant) when density perturbation 𝛿_𝑡=0.

To sketch the gravitational clustering of the IGM, we will not consider the details of heating and cooling. Thermal processes generally are local, and therefore, it is reasonable to describe the thermal processes by a polytropic relation 𝑝(𝐱,𝑡)∝𝜌^𝛾(𝐱,𝑡). Thus (2) can be rewritten as𝜕𝑎𝐯𝜕𝑡+(𝐯⋅∇)𝐯=−𝛾𝑘^𝐵𝑇𝜇𝑚_𝑝∇𝛿(1+𝛿)−∇𝜙,(4) where the parameter 𝜇 is the mean molecular weight of the IGM particles, and m_𝑝 is the proton mass. In this case, we do not need energy equation, and the IGM temperature evolves as 𝑇∝𝜌^𝛾−1 or 𝑇=𝑇₀(1+𝛿)^𝛾−1. Exactly speaking, there is also a noise in the hydrodynamical (2) and (4), which are related to the viscous of the IGM fluid due to the fluctuation-dissipation theorem. Since the IGM is a “passive substance”, this noise would be ignorable in searching for the deviation of the IGM from the underlying dark matter.

2.2. Linear Solution of Growth Mode

The unperturbed solutions of the density and velocity fields of baryonic matter are 𝜌=[Ω_𝑏/(Ω_𝑏+Ω_dm)]𝜌_𝑡∝𝑎⁻³ and 𝐯=0. Therefore, the linearization of (1) and (4) yields𝜕𝛿𝜕𝑎𝐯𝜕𝑡+𝑎1∇⋅𝐯=0,𝜕𝑡=−𝛾𝑘^𝐵𝑇𝜇𝑚_𝑝∇𝛿−∇,(5) where the mean temperature 𝑇∝𝜌^𝛾−1∝𝑎^{−3(𝛾−1)}. In the Fourier space, we have𝜕²𝛿(𝐤,𝑡)𝜕𝑡²+2𝑎̇𝑎𝜕𝛿𝜕(𝐤𝑡,𝑡)+𝑡1²𝑘²𝑘^2𝐽𝜕𝜐(𝐤,𝑡)𝛿(𝐤,𝑡)=4𝜋𝐺𝜌𝛿_𝑡(𝐤,𝑡),𝜕𝑡+𝑎𝑎̇𝜐(𝐤,𝑡)=−𝑡²1𝑎𝑘1^2𝐽𝛿(𝐤,𝑡)+4𝜋𝐺𝜌𝑎𝑘²𝛿_𝑡(𝐤,𝑡),(6) where 𝐯(𝐤,𝑡)=𝑖𝐤𝜐(𝐤,𝑡), and the Jeans wavenumber 𝑘_𝑗 is given by 𝑘^2𝐽=(𝑎²/𝑡²)(𝜇𝑚_𝑝/𝛾𝑘_𝑏𝑇)∝𝑎^3𝛾−4. When 𝛾=4/3, 𝑘_𝐽 is time-independent. Here, we ignore the stochastic nature of (6), and the randomness of all the variables is given by the random initial conditions.

We consider only the growth mode of the perturbation of dark matter, that is, 𝛿(𝑘,𝑡)∝𝑎. In the case of 𝛾=4/3, the solution of (6) is [21]𝛿(𝐤,𝑡)=𝛿^𝑡(𝐤,𝑡)1+3𝑘²/2𝑘^2𝐽+𝑐₁𝑡^{−(1+𝜖)/6}+𝑐₂𝑡^{−(1−𝜖)/6},(7) where 𝜖=(1−4𝑘²/9𝑘^2𝐽)^1/2, and constants 𝑐₁ and 𝑐₂ depend on the initial condition 𝛿(𝐤,0) and 𝜐(𝐤,0). Therefore, regardless of the initial condition of the IGM, after a long evolution we always have𝛿(𝐤,𝑡)=𝛿_𝑡(𝐤,𝑡)if𝑘≪𝑘_𝐽.(8) This means that the evolution of the IGM will completely follow dark matter, and the initial difference between 𝛿(𝐤,𝑡) and 𝛿_𝑡(𝐤,𝑡) will be forgotten. Equation (8) yields𝑃_𝑏(𝑘)=𝑃_dm(𝑘)if𝑘≪𝑘_𝐽.(9) The power spectrum of IGM field, 𝑃_𝑏(𝑘), has to be the same as that of dark matter 𝑃_dm(𝑘).

The linear solutions of the IGM have also been studied by using different assumptions of the IGM thermal processes other than that used in (4) [22, 23]. A common feature of these solutions is𝛿(𝐤,𝑡)=+(1d+ecdaeycinaygintgermtesrmisf)𝑘𝛿_𝑡≪(𝐤,𝑘𝑡_𝐽).(10) Only the decaying terms are affected by the initial conditions, and therefore, the solutions of equations (8) and (9) hold in general, regardless of the specific assumptions of the IGM thermal processes.

The physical explanation of the solutions (8) and (10) is straightforward. The IGM is only a tiny component in the cosmic mass field and its evolution is completely governed by the gravity of dark matter in the linear regime. After the decaying of the initial conditions, the IGM should follow the same trajectory as dark matter on scales larger than the Jeans length.

In linear approximation, we also have the linear relation as follows:𝛿(𝐱,𝑡)=−𝑓∇⋅𝐯(𝐱,𝑡),(11) where the parameter 𝑓 depends on dark matter models [24].

3. Moderate Nonlinear Regime of the IGM Evolution

3.1. Burgers' Equation for Cosmic Baryonic Fluid

In the moderate regime, we may use linear relation (11) for 𝛿 in (4). We have then𝜕𝑎𝐯𝜕𝑡+(𝐯⋅∇)𝐯=−𝜈𝑎∇²𝐯−∇𝜙,(12) where the coefficient 𝜈 is given by𝜈=𝛾𝑘^𝐵𝑇₀𝜇𝑚_𝑝(𝑑ln𝐷(𝑡)/𝑑𝑡),(13) in which 𝐷(𝑡) describes the linear growth behavior. The term with 𝜈 in (12) describes a diffusion which is characterized by the Jeans length 𝑘_𝐽.

To sketch the evolution of gravitational clustering, only the growth modes are of interest. For the growth modes, the velocity field is irrotational, and we can define a velocity potential by𝐯=−1𝑎∇𝜑.(14) Substituting (14) into (12), we have𝜕𝜑𝜕𝑡−21𝑎²(∇𝜑)²−𝜈𝑎²∇²𝜑=𝜙.(15) Equation (15) is the stochastic-force-driven Burgers' equations or the KPZ equations [25]. It is the simplest nonlinear Langevin equation for growth modes [26–28]. This equation can also be found with a coarse-graining approximation [29].

Thus, in moderate nonlinear regime, the cosmic baryonic fluid is similar to Burgers' fluid. The second term on the l.h.s. results from convection, and the third on the l.h.s. describes relaxation of the clustering by diffusion. The term on the r.h.s represents the gravitational potential of dark matter, which provides the initially random perturbations. Clustered structures will develop from the initial seeds via the competition of the convection and diffusion.

3.2. IGM as a Burgers' Fluid

The Burgers' equation contains two scales: the dissipation length or the Jeans length 1/𝑘_𝐽 and the correlation length 𝑟_𝑐 of the random force, 𝜙. The intensity of the random force 𝜙 can be characterized by the density contrast of dark matter 𝛿_dm. Many theoretical works on Burgers fluids have been done [30–34]. They show that turbulence will develop in Burger fluid when the following Reynolds number is large [33, 35]:𝑘_𝐽𝑟_𝑐^2/3𝛿²_dm^1/3>1.(16) This condition corresponds to the system in which the dark matter field has entered the nonlinear regime. In the linear regime, 𝛿_dm≪1, and therefore, perturbations on scale 𝑟_𝑐>1/𝑘_𝐽 will not cause Burgers' turbulence in the IGM. In the moderate nonlinear regime, 𝛿_dm≥1, and Burgers' turbulence on scales larger the Jeans length can develop. Since the dark matter field becomes nonlinear first on small scales and then on large scales, one can expect that Burgers' turbulence in the IGM develops from small to large scales.

When Burgers turbulence develops in the IGM, its velocity and mass density fields will be intermittent and contain discontinuities or shocks. The probability distribution function (PDF) of velocity field is long tailed [33]. Since dark matter is not affected by Burgers turbulence, the IGM velocity field will dynamically decouple from the dark matter field on scales larger than the Jeans length once Burgers turbulence is developed. The intermittence of the IGM mass field, as a result of the Burgers equation, has also been emphasized in [23].

Jeans diffusion will also lead to the decoupling between the velocity fields of the IGM 𝐯 and dark matter 𝐯_dm. However, the decoupling given by the Jeans diffusion is very different from that results from the Burgers turbulence shocks. For the former, the distribution 𝐯 and Δ𝐯≡𝐯(𝐱+𝐫)−𝐯(𝐱−𝐫) is symmetric with respect to the transformation 𝐯_dm→−𝐯_dm, that is, the velocity PDF with Δ𝐯⋅𝐯_dm/|𝐯_dm|>0 (acceleration in the direction of 𝐯_dm/|𝐯_dm|) is the same as the PDF with Δ𝐯⋅𝐯_dm/|𝐯_dm|<0 (deceleration in the direction of 𝐯_dm/|𝐯_dm|). For Burgers turbulence, the shocks consist of an acceleration ramp followed by a rapid deceleration. Therefore, the IGM is not symmetric between the sections accelerating and decelerating. Since the acceleration is due to the gravity of dark matter, the acceleration is generally in the direction of 𝐯_dm. Thus, the IGM velocity field 𝐯 is asymmetric with respect to the transformation 𝐯_dm→−𝐯_dm. The PDF of Δ𝐯 with Δ𝐯⋅𝐯_dm/|𝐯_dm|>0 will not be the same as the PDF of Δ𝐯 with Δ𝐯⋅𝐯_dm/|𝐯_dm|<0.

If 𝜈=0, Burgers equation (15) is scale-free when the random force 𝜙 is scale-free. In this case, the velocity field of the Burgers fluid is self-similar and the PDF of the velocity difference Δ𝐯 is scale-invariant.

In summary, if the IGM can be described as a Burgers' fluid, its velocity field should show the following properties: (1) the peculiar velocity of the IGM at a given point will generally be lower than that of dark matter at the same point, (2) the PDF of Δ𝐯_𝑟⋅𝐯_dm/|𝐯_dm|>0 will not be the same as the PDF of Δ𝐯_𝑟⋅𝐯_dm/|𝐯_dm|<0, and (3) the PDF of Δ𝐯_𝑟 is scaling. These predictions have been proven by cosmologically hydrodynamic simulation samples [36].

4. Fully Developed Nonlinear Regime

4.1. Vorticity

Burgers turbulence is qualitatively different from turbulence described by the Navier-Stokes equations. The latter generally consists of vortices on various scales, while the former is a collection of shocks. These features arise because growth modes of the perturbations of velocity fields are potential, and the velocity field is irrotational. Moreover, from the condition (16), we see that the Burgers turbulence can only develop in dense areas of dark matter 𝛿_dm>1.

However, in fully developed nonlinear regime, the cosmic baryon fluid will on longer be potential, and the turbulence of Navier-Stokes fluid can also be developed in high as well as low density regions. This point can be seen with the vorticity, which is defined by 𝜔_𝑖=(1/2)(𝜕_𝑖𝑣_𝑗−𝜕_𝑗𝑣_𝑖) or 𝜔⃗=∇×𝐯, where 𝑖=1,2,3. The dynamical equation of the vorticity 𝜔⃗ of cosmic baryon fluid can be derived from Navier-Stokes equations as follows [37]:𝐷𝜔⃗𝐷𝑡≡𝜕^𝑡𝜔⃗+1=1𝑎𝐯⋅∇𝜔⃗𝑎𝐒⋅𝜔⃗−𝑑𝜔⃗+𝜌1²∇𝜌×∇𝑝−𝑎𝜔̇⃗,(17) where 𝑑=𝜕_𝑖v_𝑖 is the divergence of the velocity field. Tensor 𝐒 is called strain rate defined as 𝑆_𝑖𝑗=(1/2)(𝜕_𝑖𝑣_𝑗+𝜕_𝑗𝑣_𝑖), and the vector [𝐒⋅𝜔⃗]_𝑖=𝑆_𝑖𝑗𝜔_𝑗.

A remarkable property of (17) is the absence of the term of gravity of mass fields, indicating that the gravitational field of dark matter is not the sources of the vorticity. This point is obvious, as gravity is curl-free in nature. In this sense, the vorticity is “nongravitational”.

In the linear regime of perturbations, both the terms 𝐒⋅𝜔⃗−𝑑𝜔⃗ and ∇𝜌×∇𝑝 would be zero. Vorticity 𝜔⃗ will decay as 𝑎⁻¹, given by the term 𝑎𝜔̇⃗. Therefore, as expected, in the linear regime the velocity field of baryon matter is irrotational. In other words, the linear evolution of the velocity field of baryon fluid is fully governed by the gravity.

However, in the fully developed nonlinear regime, the vorticity will no longer be zero. First, it has been shown that at the moderate nonlinear regime, Burgers' fluid will inevitably lead to the development of shocks and complex structures, which can yield nonzero term ∇𝜌×∇𝑝 of (17), being called baroclinity. Thus, the vorticity is excited, regardless of the gravity of dark matter. Physically, once multistreaming has developed, complex structures, like curved shocks, will lead to deviation of the direction of ∇𝑝 from that of ∇𝜌. Consequently, the density pressure relation cannot be simply given by a single-variable function equation as 𝑝=𝑝(𝜌) [38] and the baroclinity term will no longer be zero.

Once vorticity is initiated, (17) shows that the vorticity of baryon velocity field can be maintained by itself, that is, the system actually is nonlinearly self-excited and self-maintained. The divergence 𝑑=𝜕_𝑗𝑣_𝑗 is generally negative in regions of clustering. Therefore, the term −𝑑𝜔 of (17) will lead to an amplification of vorticity in overdense regions. On the other hand, the vorticity will be stretched by the nonlinear term 𝐒⋅𝜔⃗. Therefore, vorticity can also develop in low mass density region. This picture is very different from the Burgers' turbulence.

Vortices are considered as a fundamental ingredient of turbulence. The fluctuations of the vorticity field is an important measurement of the turbulence of fluid [39, 40]. Therefore, the dynamics of vorticity provides a non-gravitational mechanism of the decoupling between baryon matter from dark matter.

The turbulence of cosmic baryon fluid can also be seen with the spectrum tensors Φ_𝑖𝑗(𝐤) and Ω_𝑖𝑗(𝐤), which are, respectively, the Fourier counterparts of the two-point correlation tensors of velocity ⟨𝑣_𝑖(𝐱+𝐫)𝑣_𝑗(𝐱)⟩ and vorticity ⟨𝜔_𝑖(𝐱+𝐫)𝜔_𝑗(𝐱)⟩. That is,Φ_𝑖𝑗(𝐤)=1(2𝜋)³𝑣^𝑖Ω(𝐱+𝐫)𝑣_𝑗(𝐱)𝑒^{−𝑖𝐤⋅𝐫}𝑑𝐫,_𝑖𝑗(𝐤)=1(2𝜋)³𝜔^𝑖(𝐱+𝐫)𝜔_𝑗(𝐱)𝑒^{−𝑖𝐤⋅𝐫}𝑑𝐫,(18) respectively, where ⟨⋯⟩ denotes average over spatial coordinates 𝐱. For a homogeneous turbulence, we have [39]Ω_𝑖𝑗(𝐤)=𝛿_𝑖𝑗𝑘²−𝑘_𝑖𝑘_𝑗Φ_𝑙𝑙(𝐤)−𝑘²Φ_𝑖𝑗(𝐤),(19) and hence,Ω_𝑖𝑖(𝐤)=𝑘²Φ_𝑖𝑖(𝐤).(20) The power spectra of velocity and vorticity fields are defined, respectively, as𝑃_𝑣(𝑘)=12Φ^𝑖𝑖𝑃(𝐤)𝛿||𝐤||−𝑘𝑑𝐤,_𝜔(𝑘)=12Ω^𝑖𝑖(𝐤)𝛿||𝐤||−𝑘𝑑𝐤.(21) Combining (20) and (21) yields𝑃_𝜔(𝑘)=𝑘²𝑃_𝑣(𝑘).(22) This relation can be used to measure the developed level of turbulence. If the velocity and vorticity fields of a fluid satisfy the relation given by (22), it should be in the state of fully developed homogeneous turbulence. Otherwise, it would be less developed. The samples of cosmological hydrodynamic simulation have showed that the relation (22) is perfectly hold on scales less than 3h⁻¹ Mpc at 𝑧∼0 [37]. Actually, the dynamics of (17) reveals that the cosmic baryonic fluid underwent a scale-free evolution, which leads to a state of fully developed turbulence.

4.2. She-Leveque Intermittence

The velocity field of a fully developed turbulence is highly intermittent, which is measured by the structure functions 𝑆_𝑝(𝑟) defined as𝑆_𝑝(𝑟)≡𝛿𝑣^𝑝𝑟∼𝑟^𝜁𝑝,(23) where 𝛿𝑣_𝑟≡[(𝐯(𝐱+𝐫)−𝐯(𝐱))⋅𝐫/𝑟] and 𝜁_𝑝 is called intermittent exponent. Based on dimensional argument of hierarchical evolution, Kolmogorov in 1941 [41] predicted that for fully developed turbulence, the intermittent exponent is 𝜁_𝑝=𝑝/3 on scales of inertial range. Experimental and numerical results do not, however, support the 𝑝/3 law. It should be attributed to intermittency, that is, turbulence field in inertial range is characterized by stronger non-Gaussianity on smaller scales. A remarkable development was made by She and Leveque (SL hereafter) [42]. They proposed that the non-Gaussian behavior of fully developed turbulence is determined by the hierarchical structure originated from the Navier-Stokes equation, and the 𝑝/3 law should be replaced by𝜁_𝑝/𝜁₃=1−𝐶1−𝛽³𝑝3+𝐶(1−𝛽^𝑝),(24) which contains only two dimensionless parameters 𝐶 and 𝛽. 𝐶 is the Hausdorff dimension of the most dissipative structures, and parameter 𝛽 is given by the hierarchical evolution. SL formula (24) is in excellent agreement with various experiments of turbulence, including turbulence in compressible fluid. The SL scaling law of structure function is considered to be universal for characterizing the fully developed turbulence.

With the hydrodynamic simulation sample of the concordance ΛCDM universe, the intermittent exponent of the velocity field of cosmic baryon fluid at redshift 𝑧=0 in the scale range from the Jeans length to about 16 h⁻¹ Mpc is found to be extremely well described by the She-Leveque's universal scaling formula (24) with 𝐶=1 and 𝛽³=1/3 [43]. It indicates that the dissipative structures are dominated by sheets. These results imply once again that the evolution of highly evolved cosmic baryon fluid is similar to a fully developed turbulence.

When a fluid is turbulent, the kinetic energy passes from large to the smallest eddies, and finally dissipates into thermal motion. For cosmic baryonic gas, the evolution of vorticity is also hierarchical. The vorticity evolves from large scales to small scales, and finally falls into massive halos to form structures, including light-emitting objects. This result strongly indicates that the highly nonlinear evolution would lead to the cosmic baryon fluid reaching a statistically quasiequilibrium state satisfying the universal scaling as that of fully developed turbulence.

In view of this picture, we can say that in the highly nonlinear regime, the statistical properties, especially the intermittent behavior, of the velocity fluctuations are actually independent of the details of the dissipative processes. The state depends only on the dimension of dissipative structures and the hierarchical relation index.

5. Observable Effects

The nonlinearly evolved IGM provides a coherent and uniform explanation of various observations, including the intermittence and non-Gaussianity of Ly𝛼 transmitted fllux of quasar absorption spectrum [44, 45], Ly𝛼 leaks of absorption spectrum of high redshift objects [46], the turbulence broadening [47], Scaling relations between SZ effect and X-ray luminosity of X-ray clusters [38, 48, 49], baryon missing in collapsed halos [19], and so forth. Here we address only two of them.

5.1. Non-Gaussianity of Log-Poisson Hierarchy

It has been shown that the SL scaling is yielded from the Log-Poisson hierarchy process, which is related to the so-called generalized scale covariance of the Navier-Stokes equations [50]. Therefore, the statistical behavior of the mass field of cosmic baryon matter should be given by the log-Poisson random multiplicative processes (RMP).

The log-Poisson RMP assumes that, in the scale-free range, the variables 𝛿𝜌_𝑟≡𝜌(𝐱+𝐫)−𝜌(𝐱), 𝑟=|𝐫|, on different scales 𝑟 are related from each other by a statistically hierarchy relation as||𝛿𝜌_𝑟2||=𝑊_𝑟1𝑟2||𝛿𝜌_𝑟1||,(25) where𝑊_𝑟1𝑟2=𝛽^𝑚𝑟𝑟¹₂^𝛾,(26) which describes how the fluctuation |𝛿𝜌_𝑟1| on the larger scale 𝑟₁ is related to fluctuations |𝛿𝜌_𝑟2| on the smaller scale 𝑟₂. In (26), 𝑚 is a Poisson random variable with the PDF𝑃(𝑚)=exp−𝜆_𝑟1𝑟2𝜆^{𝑚𝑟1𝑟2}𝑚!.(27) To insure the normalization ⟨𝑊_𝑟1𝑟2⟩=1, where ⟨⋯⟩ is over m, the mean 𝜆_𝑟1𝑟2 of the Poisson distribution should be𝜆_𝑟1𝑟2=𝛾ln𝑟¹/𝑟₂1−𝛽.(28) It is enough to consider only |𝛿𝜌_𝑟|, as the distribution of positive and negative 𝛿𝜌_𝑟 is symmetric.

The log-Poisson hierarchy given by (26) depends only on the ratio 𝑟₁/𝑟₂, which is obviously scale invariant. The hierarchy is determined by two dimensionless positive parameters: 𝛽 and 𝛾, describing, respectively, the intermittence and singularity of the random fields. Equation (26) relates 𝛿𝜌_𝑟 on different scales by multiplying a random factor 𝑊, which generally yields a non-Gaussian field even if the field originally is Gaussian [51].

The cascade from scale 𝑟₁ to 𝑟₂, and then to 𝑟₃ is identical to the cascade from 𝑟₁ to 𝑟₃. It is because 𝑊_𝑟1𝑟3=𝑊_𝑟1𝑟2𝑊_𝑟2𝑟3=𝛽^𝑁(𝑟₁/𝑟₃)^𝛾, where 𝑁 is again a Poisson random variable with 𝜆_𝑟1𝑟3=𝜆_𝑟1𝑟2+𝜆_𝑟2𝑟3=𝛾[ln(𝑟₁/𝑟₃)]/(1−𝛽). Therefore, the log-Poisson hierarchy removes an arbitrariness in defining the steps of cascade from 𝑟₁ to 𝑟₂ or 𝑟₂ to 𝑟₃. The log-Poisson hierarchy is discrete in terms of the discrete random number 𝑚. However, the scale 𝑟 is infinitely divisible. Namely, there is no lower limit on the difference 𝑟₁−𝑟₂. It can be infinitesimal, and the hierarchical process is of infinite divisibility.

With the log-Poisson hierarchy (25)–(28), the density field should have the non-Gaussianities, which can be described by the structure function of density field 𝑆_𝑝(𝑟)≡⟨|𝛿𝜌_𝑟|^𝑝⟩ as follows [52].

(1) Intermittent Exponent
The structure function should be a function of power law of 𝑟 as 𝑆_𝑝(𝑟)∝𝑟^𝜉(𝑝),(29) and the intermittent exponent 𝜉(𝑝) is given by 𝜉(𝑝)=−𝛾𝑝−(1−𝛽^𝑝)1−𝛽.(30) This is actually the SL scaling formula for density field.

(2) Hierarchical Relation
Defining 𝐹_𝑝(𝑟)≡𝑆^𝑝+1(𝑟)𝑆_𝑝(𝑟),(31) the log-Poisson RMP gives hierarchical relation as 𝐹_𝑝(𝑟)𝐹_∞(𝑟)=𝐹^𝑝+1(𝑟)𝐹_∞(𝑟)^1/𝛽.(32) Relation (32) is invariant with respect to a translation in 𝑝.

(3) Moments
The ratio between high-order and 2nd-order moments, ⟨𝛿𝜌^𝑟2𝑝⟩/⟨𝛿𝜌^2𝑟⟩^𝑝, should satisfy ln𝛿𝜌^𝑟2𝑝𝛿𝜌^2𝑟^𝑝=𝐾_𝑝ln𝑟+const(33) with 𝐾_𝑝=−𝛾𝑝1−𝛽²−1−𝛽^2𝑝1−𝛽.(34)

(4) Scale-Scale Correlation
The so-called scale-scale correlation is defined as 𝐶^{𝑟𝑝,𝑝}_1,𝑟2≡𝛿𝜌^𝑝𝑟1𝛿𝜌^𝑝𝑟₂𝛿𝜌^𝑝𝑟₁𝛿𝜌^𝑝𝑟₂.(35) Obviously, for a Gaussian field, 𝐶^{𝑟𝑝,𝑝}_1,𝑟2=1. If the ratio 𝑟₂/𝑟₁ is fixed, the log-Poisson model predicts the scale-scale correlation to be 𝐶^{𝑟𝑝,𝑝}_1,𝑟2=𝐵𝑟²𝑟₁𝑟^{1𝜉(2𝑝)−2𝜉(𝑝)},(36) where the coefficient 𝐵(𝑟₂/𝑟₁) depends only on the ratio 𝑟₂/𝑟₁, as the log-Poisson model is invariant of the dilation.

It should be emphasized again that all above-mentioned non-Gaussianities depend only on two dimensionless parameters 𝛽 and 𝛾. It has been revealed that the fields of Ly𝛼 transmitted flux fluctuations of observed high resolution and high S/N Ly𝛼 absorption spectra of quasars at moderate redshift 𝑧∼2-3 and high redshift 𝑧∼5-6 do show all the log-Poisson non-Gaussianities [53, 54].

5.2. Turbulence Pressure and Baryon Missing

As shown in Section 2.2, the Jeans length 𝜆_𝐽=2𝜋/𝑘_𝐽 is 𝜆_𝐽=𝑐_𝑠√𝜋/𝐺𝜌, with 𝑐_𝑠 being the sound speed. The effect of turbulent motions on the Jeans length can be estimated by replacing 𝑐_𝑠 by an effective sound speed𝑐²_s,eff=𝑐^2𝑠+13𝑣²,(37) where ⟨𝑣²⟩ is the rms velocity dispersion due to turbulent motion [55, 56]. Namely, the random velocity field of turbulence contributes to an extra pressure, the turbulence pressure, 𝑝_tub=𝜌⟨𝑣²⟩ to resist the gravitational collapsing.

If the collapsed halo is on scale 𝑅, the gravitational collapsing will not be affected by the rms velocity dispersion on scales larger than 𝑅, and the fluctuation of velocity on the scales 𝑘<2𝜋/𝑅 does not contribute to the turbulent pressure in terms of resist gravitational collapsing on scales that larger than 𝑅. Thus, the effect of turbulent pressure on gravitational clustering should be given by [57]𝑝_tur𝑘_𝑅=^𝑘max_𝑘𝑅𝐸(𝑘)𝑑𝑘,(38) where 𝐸(𝑘) is the power spectrum of kinetic energy of the turbulence, and 𝑘_𝑅=2𝜋/𝑅. Wavenumber 𝑘_max=2𝜋/𝑙_diss corresponds to the minimal scale 𝑙_diss below which the turbulence decays due to energy dissipation or virialization. The turbulent pressure is dynamical, not thermal. It can be as strong as a thermal pressure of temperature 10^5-6 K, while the temperature of the IGM is still remained in the range of 10^4-5 K.

Obviously, the bulk velocity will not contribute to the ⟨𝑣²⟩, as it can be given by the motion of falling into a gravitational well. Therefore, we should pick up the component of velocity field that is regardless of gravity. It is mainly given by the vorticity of velocity field. In other words, the turbulent vorticity field can prevent the collapsing of baryon matter into the halos of dark matter. It leads to the deviation of the mass density fields of baryon matter from that of dark matter.

With the samples produced by cosmological hydrodynamical simulation, we find that the turbulence pressure will play an important role of preventing the IGM clustering. In the IGM regions with 10 times mean overdensity, the turbulent pressure is equivalent to the thermal pressure of the baryon gas with temperature of about 1×10⁵ K. The turbulent pressure is dynamical and nonthermal. When the turbulence dissipated, the kinetic energy of turbulence becomes the thermal energy. It yields the entropy in halos [38].

The baryon fraction of dark matter halos is found to be a strong function of the halo mass 𝑀₂₀₀, the mass within the range of total mass density larger than 200𝜌_𝑡. The baryon fraction is significantly decreasing with the mass of dark halos. For halos with mass 𝑀₂₀₀=3×10¹⁰ M_⊙, the baryon fraction is only about 0.1 of the cosmic mean of the concordance ΛCDM universe, and it further decreases to 0.03 for 𝑀₂₀₀=10¹⁰M_⊙ halos. This result is well consistent with observed baryon content of galaxies and galaxy groups. Therefore, the turbulent velocity field of baryon fluid would be the major factor leading to the absence of baryon matter in halos with mass less than 10^11.5M_⊙. With this evolution picture, no preheating or reheating mechanisms are needed for explaining baryon missing.

6. Conclusion

The nonlinear evolution of cosmic baryon fluid is a central problem of cosmology. The concordance ΛCDM model provides an effective paradigm of formation and evolution of cosmic structures. The dynamical behavior of the CDM component has been successfully revealed with analytical study and N-body numerical simulation. On the other hand, the dynamics of baryon fluid is less understood, especially in the nonlinear regime, because of the involvement of complicated physical processes, such as cooling, heating, ionization, shock, and turbulence. On the other hand, many observations are directly related to the nonlinear evolution of baryon fluid, such as, the absorption spectra of quasars, galaxies and GRB at high redshift, Sunyaev-Zeldovich effect, the polarization of cosmic microwave background (CMB), X-ray background and so forth. One of the biggest challenges in galaxy formation is also on the physical origin of the nonlinear relation between baryon gas and low-mass dark halos.

Thanks to the development of reliable cosmological hydrodynamical simulation in the last decade, significant progress has been made in understanding the nonlinear dynamics of cosmic baryon fluid. The turbulence behavior of cosmic baryon fluid in scale-free regime is the key of understanding nonlinear evolution of the mass density and velocity fields. An important consequence of the turbulence is to yield statistical decoupling of the mass and velocity fields of baryon fluid from the underlying dark matter field. This result is extremely valuable to understand the low baryon fraction in galaxy clusters, the non-Gaussian features of quasar Ly𝛼 transmitted flux, the Ly𝛼 leaking of absorption spectra of high redshift objects, the shock heating of previrialized structures, and the strongly nonlinear dependence of the star formation efficiency on halo mass, and so forth. Considering these complexities, the nonlinear dynamics of the IGM also is essential for understanding the epoch of reionization and related observations.