# Using kinetic theory to examine a self-gravitating system composed of baryons and cold dark matter

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## Abstract

We examine the evolution of non-relativistic cold dark matter gravitationally coupled to baryons with modes deep inside the Hubble radius (sub-horizon regime) using a kinetic theory approach within the realm of Newtonian theory. We obtain the general solution for the total density perturbation and we also show that a baryon perturbation catches up with the dark matter perturbation at late times, which in turn makes possible the formation of bound structures. We extend the linear perturbation analysis by considering the turn-around event, collapse of matter, and its virialization process.

## 1 Introduction

Current lore associated with the standard cosmological model would indicate that on scales larger than 100 Mpc the universe is essentially homogeneous and isotropic, but on smaller scales there are some deviations from the mean density in the form of galaxies, galaxy clusters, amongst other configurations [1, 2]. A natural question is to ask: How do such structures grow in the universe? What is the basic mechanism to aggregate matter and make it collapse in the form of a bump? The first attempt to give an answer to that question was made by Jeans long time ago [3, 4]. He focused on the necessary condition under which small perturbations of a gas cloud could grow exponentially, leading to the collapse of the cloud and therefore ending in the formation of stars. In other words, the Jeans mechanism describes the gravitational instability of a self-gravitating gas cloud. Interestingly enough, there exist plenty of ways to understand the Jeans mechanism and to derive such a criterion. To be more precise, consider an initially stable and static cloud that can be initially perturbed by the environment such as a shock wave, passing spiral arms of the galaxy, etc. This configuration can collapse if the inwards directed gravitational force is bigger than the outwards directed pressure force. The critical maximal radius that allows stability depends essentially on the Newton constant, the speed of sound and and the matter density; in fact, it leads to the idea that the denser the clouds, the more unstable they become. During the collapse only a part of the gas ends up in stars, so many stars form out of one collapsing cloud, which means that young stars are born in clusters [2].

At the cosmological level, one way to get some insight in the structure formations is to look at a simplified treatment, which in this case corresponds to the Newtonian approach [2]. Indeed, the theory of Newtonian structure formation is sufficient to understand most of the processes which are well within the horizon. To do so, one must derive the full Newtonian hydrodynamics in an expanding universe and it turns out that the Boltzmann kinetic theory is the natural way to achieve such a goal [5, 6, 7].

The first step to understanding how the cooperative effects of baryons and dark matter may work in the process of structure formation is by inspecting a collisionless self-gravitating system composed of two components and then solving the coupled collisionless Boltzmann and Poisson equation together [8, 9]. A system composed of baryons and dark matter leads to a total Jeans mass which is smaller than the one associated with a single component, indicating that a smaller amount of mass is needed to ignite the collapsing process. One could expect that the bumps with masses greater than the Jeans mass initiate the collapsing process, but an over-dense region in an expanding universe eventually recollapses and virialises. In the case of a single component with an expanding background it turned out that the “swindle” proposal may be avoided, while the Jeans instability is expected to arise in the limit of large wavelengths [8]. Nevertheless, one must stress that the Jeans instability is not only restricted to the Newtonian (or General Relativity) realm and it can emerge within the context of alternative gravity theories as well [10, 11, 12, 13, 14].

This paper is organized as follows. In Sect. 2, one presents the kinetic theory formalism for dealing with a self-gravitating system of two components within the framework of Newtonian cosmology and by doing so one also examines the general conditions to achieve the Jeans instability. Besides, the perturbation of baryons and dark matter are studied along with the conditions under which the total matter starts to virialize. In Sect. 3, the conclusions are stated. We will use the metric convention \((+,-,-,-)\) and non-geometric units in which \(8\pi G\ne 1\) and \(c \ne 1\) unless stated otherwise.

## 2 Kinetic theory and self-gravitating components

*t*. The Boltzmann equation in the absence of collisions between the particles but in the presence of a gravitational potential \(\Phi \) reads (see e.g. [8, 9, 10])

*a*(

*t*) is the so-called cosmic scale factor. If the universe is dominated by a perfect fluid then Einstein’s field equations are reduced to some coupled differential equations known as the Friedmann and acceleration equations, respectively:

*G*is the gravitational constant, while \(\rho \) and

*p*are the mass density and the pressure of the source that generates the gravitational field.

*a*(

*t*) for an epoch of post-recombination, namely

**q**. Thus, the factor 1 /

*a*(

*t*) in the wavenumber takes into account that the wavelength is stretched out in an expanding universe:

Here the source term corresponds to the total density and does not involve the perturbed density contrast. Equation (28) tells us that a dissipative effect enters through the usual friction term proportional to 2*H*. One must emphasize that the physical Jeans scale (length or wavelength) is obtained by demanding that the term \({\delta }_{t}\) vanishes, namely, the Jeans wavenumber is \(q_{J}=(\sqrt{3/2})Ha/ v_s\). The Jeans length can heuristically be derived by balancing the sound crossing time, \(t_{s} \propto a/v_sq_{J}\), with the gravitational free-fall time, \(t_\mathrm{ff}\propto 1/ \sqrt{G\rho _{t}}\), which yields the same result as mentioned above in an intuitive manner. Moreover, this scale seems to be sensitive to the thermal dispersion velocity of the particles through \( v_s\) and the total material content \(\rho _t\); notice that one considers the universe after decoupling, then one can focus on dark matter and baryons only. One should mention also that the Jeans wavenumber for each component in principle is different provided the propagation speeds are not the same. Here one is looking at the Jeans length of the composed system. In this context, the comoving Jeans wavenumber can be written in terms of the scale factor as \(q_{J}=(4\pi G \rho _{t}^0 a_0)^{1/2}a^{1/2}/ v_{s}^0\)—by considering that \( v_s= v_{s}^0(a_0/a)\) and \(\rho _t=\rho _t^0(a_0/a)^3\)—while the comoving Jeans length is simply \(\lambda _{J}=2\pi /q_{J}\). One can prove that perturbations can grow for \(q \le q_{J}\); otherwise they just oscillate ( \(q \ge q_{J}\)). The latter result is consistent with the behavior of the Jeans scale in a universe dominated by matter after decoupling time, \(q_{J} \propto a^{1/2}\) [1]. One final comment: the fact that \(\sigma _j=\sigma _{j}^0a^{-1}\) (which is equivalent to having a pressure \(p_{j}= \sigma _{j}^2\rho _{j} \propto a^{-5}\)) implies that the collisionless fluid may be treated as pressureless as long as \(q<q_{J}\). As one is trying to arrive at the general solution without demanding the latter condition the system of equations will be much harder to solve than the usual case. One will examine the situation where the condition \(q>q_{J}\) holds as well.

*t*for values of \(\left( \frac{4\sqrt{6}}{H_0t}\right) ^\frac{1}{3}\frac{q}{q^0_{J}}\) not too small. When the time increases, the factor \(\left( \frac{4\sqrt{6}}{H_0t}\right) ^\frac{1}{3}\frac{q}{q^0_{J}}\) becomes much smaller than unity and the total density contrast will grow provided that the cosine integral term will increase for large values of the cosmic time.

*R*and mass

*M*embedded into the otherwise homogeneous spatially flat and matter-dominated universe (EdS). Given the fact that it is over-dense, this configuration will reach a maximum radius and subsequently contract until collapse. Such a toy model is a reasonable approximation provided the distribution of the dark matter in the universe can be considered as composed of individual so-called halos, approximately spherical over-dense clouds of dark matter which can reach highly non-linear densities in their centers. This means that one can work with a Newtonian equation of motion for the radius,

To obtain the parameters that characterize the collapsing phase, one begins with the behavior of \(\tau \) at early times. Making an expansion of \(\tau [y,\zeta ]\) at the lowest order possible in *y* around \(y=0\) yields \(\tau \simeq (8/9\pi )y^{2/3}(1+ 3y/10)\). Furthermore, one defines the over-density inside the halo in relation to the background density as \(\Delta \equiv \rho _{\mathrm{{halo}}}\zeta /\rho _{\mathrm{{bg}}}=\zeta (x/y)^{3}\). Replacing the expansion for \(\tau \) into the definition of over-density inside the halo leads to \(\Delta = 1+ 3y/5\). Then the linear density contrast inside the halo is \(\delta =\Delta -1=3y/5\). By extrapolating this formula linearly until the turn-around event one gets \(\delta _{\mathrm{{ta}}}=\delta (y)/x \simeq 3y/5x\) but \( x[\tau (y)]\simeq \zeta ^{-1/3} y\) so the contrast linear density at the turn-around is \(\delta _{\mathrm{{ta}}}\simeq (3/5) \zeta ^{-1/3} \simeq 1.06\). In the subsequent phase after the turn-around, when the collapse is reached, the inner contrast linear density is \(\delta _{\mathrm{{coll}}}\simeq x_{\mathrm{{coll}}} (3/5) \zeta ^{-1/3} \simeq 1.69\). In other words, the halo of dark matter has already collapsed when its expected linear density reaches the value \(\delta _{\mathrm{{coll}}}\simeq 1.69\). Notice that the previous result is independent of the mass M, the initial over-density, and the epoch of virialization. The next step in the evolution is to consider what happens when the halo reached the virial equilibrium: essentially the potential energy of the halo must be twice that at the turn-around event and its radius must decrease at \(y_{\mathrm{{v}}}=1/2\), implying that \(\Delta _{\mathrm{{v}}}=(2x_{\mathrm{{coll}}}/)^{3}\zeta \simeq 178\). Therefore, a halo in virial equilibrium is expected to have a mean density nearly 178 times higher than the background although in numerical simulations the density contrast is fixed at the value 200, furnishing in this way a natural definition of the virial radius of a virialized object [2]. The lesson from this simple analysis is that density perturbations can form bound structures generated by gravitational collapse after they become 200 times as dense as the background. Such a result seems to be consistent with the full results from N-body simulations where galaxies and clusters of galaxies separate out as distinct gravitationally bound structures when their densities are at least 100 times greater than the background density [2]. However, this must be thought of just as a heuristic rule provided one is using the fact that the linear theory is valid until \(\Delta _{\mathrm{{coll}}} \ge 1\), where the process of virialization cannot be stopped.

Having mentioned the ideal picture of halos, one must also say that the halo is not necessarily isolated from the background and there will take place a constant inflow of material into the halo, or the halo might even merge with another halo. Thus, the evolution of a halo after its formation is quite non-trivial so it cannot be easily analyzed within this simplistic model. However, if one assumes that the continuous inflow of material and the merging with other halos produce new halos which are still characterized by the same density ratio \(\Delta _{\mathrm{{v}}}=(2x_{\mathrm{{coll}}}/)^{3}\zeta \simeq 178\), then one would expect that the mean density of a typical halo with a given mass M scales with redshift like \(\rho _{\mathrm{{v}}}=\Delta _{\mathrm{{v}}}\rho _{\mathrm{{bg}}}\propto (1 + z)^3\), whereas its physical radius should go as \(r_{\mathrm{{v}}} \propto \rho ^{-1/3}_{\mathrm{{v}}}\). Notice that the over-density does not depend on the mass of the perturbation, on the initial over-density, nor on the epoch of virialization \(t_{\mathrm{{v}}}\). Thus, whenever one observes an over-density of the order of \(\delta _{\mathrm{{v}}}\), one positively assumes that the corresponding structure is virialized (or close to virialization) irrespective of its mass or formation history.

*V*being the gravitational potential,

*K*the kinetic energy, and the brackets indicate the average value. From the latter fact, one gets \(M_{\mathrm{{v}}}\sigma ^{2}_{\mathrm{{d}}}=GM^{2}_{\mathrm{{v}}}/R_{\mathrm{{v}}}\) and therefore the radius is \(R_{\mathrm{{v}}}=GM_{\mathrm{{v}}}/\sigma ^{2}_{\mathrm{{d}}}\). Inserting the latter result into Eq. (45) one obtains

*R*enclosing a mass

*m*and its dynamic is governed by Newton equation:

*R*can be obtained by integrating Eq. (49). In doing so, we consider that the differential mass is \(\mathrm{d}m=\rho 4\pi r^{2}\mathrm{d}r\); then the total mass becomes \(M=4\pi R^{3} \rho /3\) when the total density is constant.

*x*we find that the ratio \(R_{\mathrm{{v}}}/R_{\mathrm{{ta}}}\) is given by

*N*-body numerical simulations show that \(R_{\mathrm{{v}}}\simeq 0.483R_{\mathrm{{ta}}}\) for a non-vanishing \(\Lambda \) [21].

*G*being the Newton constant. The latter fact confirmed that idea that the cosmological constant introduces a small deviation in the dispersion velocity obtained from the virial theorem.

## 3 Summary

In this work, we analyzed the dynamics and the collapse of a collisionless self-gravitating system composed of dark matter and baryonic matter. This system is described by two Boltzmann equations, one for each component, gravitationally coupled through the Poisson equation. First, we derived the general solution for the total density perturbation and then we solved numerically the coupled master equation for both components in different cases, showing that for modes deep inside the Hubble horizon and under the condition \(q \ll q_{J0}\) the density of matter tends to grow. In fact, we showed that a baryon perturbation catches up with the dark matter perturbation at late times, so dark matter is the driving force which provides the gravitational potential wells for the baryons to fall into, allowing them to create bound structures. Besides, we also examined in broad terms what happens with this toy model in the non-linear regime by working within the realm of Newtonian theory. We considered the formation of non-linear structures in the form of a spherical over-density of radius *R* and mass *M* embedded into the homogeneous spatially flat and matter-dominated universe. We explored the linear regime until the turn-around event followed by the collapse and the subsequently virialization process. The viral theorem along with the fact that the virialization criterion implies that galaxies were formed at low redshift, say less than ten. We also discussed the case with non-zero cosmological constant.

It is important to mention the main elements that we should introduce in the kinetic theory to study a composed system with three components (dark matter, baryons, and dark energy) within the Newtonian approach without going to the full relativistic formulation. We have basically three distribution functions with their corresponding Boltzmann equations along with one Poisson equation that couples gravitationally all the aforesaid components. We may consider the case of dark energy decoupled from the baryon plus dark matter system. The possibility of an extra-coupling between dark matter and baryons has been suggested by the Experiment to Detect the Global Epoch of Reionization Signature (EDGES) measuring the 21-cm absorption signal from primordial neutral hydrogen at redshift \(z\simeq 17\) [22, 23, 24]. This signal is considerably stronger than what is expected from the vanilla cosmic model and contains potentially new information about the true nature of dark matter. If that is the case under study, the interaction between dark matter and baryons must be specified through the collision operator. Such an interaction will affect the structure formation due to the exchange of momentum, energy and flux of heat; however, the specific result will strongly depend on the collision operator and therefore the specific interactions considered (say decay, annihilation, etc.). In fact, an interaction could potentially suppress the growth of structure in the early universe. Of course, we could expect that the structure formation will be totally ineffective at late times due to the overall expansion of the universe. Besides, one possible extension of the current work is to consider also the kinetic theory with a collision operator that takes into account the interaction between dark matter and dark energy in order to explain the current excess of brightness in the 21 cm line as proposed by several authors in the literature [25, 26, 27, 28]. We will address the latter possibility within our formalism in the near future.

## Notes

### Acknowledgements

G.M.K. is supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)-Brazil. M.G.R. is supported by a FAPES/CAPES Grant under the PPGCosmo Fellowship Programme. E.M.S. is supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)-Brazil.

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