Conditions for the establishment of small density perturbations in a self-gravitating two component fluid mixture are studied using a dynamical system approach.

It is shown that besides the existence of exponentially growing and decaying modes, which are present for values of the perturbation wave-number kݑ˜kitalic_k smaller than a critical value kMsubscriptݑ˜ݑ€k_{}_Mitalic_k start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_M end_FLOATSUBSCRIPT end_POSTSUBSCRIPT, two other, pure oscillatory, modes exist at all scales. For k
Due to the existence of a resonance between the baryonic and the dark perturbations, it is shown that the onset of structure formation in the post recombination epoch is substantially enhanced in a narrow scale band around another critical value kcsubscriptݑ˜ݑk_citalic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. For dark matter particles having a mass ∼30similar-toabsent30\sim 30∼ 30 eV, the corresponding critical mass scale for the establishment of density perturbations at the time of recombination is of the same order of magnitude as the galactic one.

Key words: gravitation - hydrodynamics - instabilities - galaxies: formation - dark matter

The study of the mechanisms responsible for the formation of structures in the universe was started by the pioneer work of Jeans (1902, 1928). Studying the conditions under which a cloud of gas (in a static background) becomes gravitationally unstable under its own gravity Jeans concluded that perturbations with masses smaller than a ”critical” value MJsubscriptݑ€ݐ½M_{}_Jitalic_M start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_J end_FLOATSUBSCRIPT end_POSTSUBSCRIPT (the Jeans mass) do not grow and behave like acoustic waves whereas perturbations with a mass Mݑ€Mitalic_M bigger than MJsubscriptݑ€ݐ½M_{}_Jitalic_M start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_J end_FLOATSUBSCRIPT end_POSTSUBSCRIPT grow under the effect of their self-gravity, leading to gravitationally bound structures.

Since then, the mechanism proposed by Jeans has been extensively applied as a criterion of stability in several models of galaxy formation. However, by the recombination epoch (which is believed to be the one after which baryonic perturbations could begin to grow), the critical scale predicted by the classical Jeans theory is not at all related to the galactic scale. Instead, it is well known that just before recombination the critical Jeans mass is ∼1016-17similar-toabsentsuperscript101617\sim 10^16-17∼ 10 start_POSTSUPERSCRIPT 16 - 17 end_POSTSUPERSCRIPTM⊙direct-product{}_\odotstart_FLOATSUBSCRIPT ⊙ end_FLOATSUBSCRIPT, (i.e. the mass of rich clusters or even superclusters of galaxies), and immediately after recombination the Jeans mass drops abruptly, more than 10 orders of magnitude, to a value ∼105-6similar-toabsentsuperscript1056\sim 10^5-6∼ 10 start_POSTSUPERSCRIPT 5 - 6 end_POSTSUPERSCRIPTM⊙direct-product{}_\odotstart_FLOATSUBSCRIPT ⊙ end_FLOATSUBSCRIPT, which is related to the mass of globular clusters (Weinberg 1972; Zel’dovich & Novikov 1983).

The origin of this discrepancy between the Jeans mass, before and after recombination, and the mass of a typical galaxy, a few 1011superscript101110^1110 start_POSTSUPERSCRIPT 11 end_POSTSUPERSCRIPTM⊙direct-product{}_\odotstart_FLOATSUBSCRIPT ⊙ end_FLOATSUBSCRIPT, has not been, up to now, well understood.

One should emphasize, however, that these values for the Jeans mass just before and just after recombination epoch are obtained assuming a baryon-dominated universe.

On the other hand, the observation of flat galactic rotation curves at large galactic radii as well as high galaxy velocity dispersions in clusters has led to the missing mass problem and to the dark matter conjecture to solve it. This fact has, in particular, led to the hypothesis that dark matter halos exist around galaxies.

The existence of dark matter therefore also implies that when studying the dynamics and the formation of structures in the universe one cannot use the simple Jeans stability criterion for a one component gas, but one has to study how does the gravitational instability arise in a mixture of at least two components. This formulation of the problem leads to a different Jeans mass, as well as to a better understanding of the origin of galactic mass spectrum. This will be the main goal of this paper.

We must note that, after we finished this work, it was pointed out to us by Dr. Varun Sahni that this problem had already been studied, using a different formalism, by some russian authors like Grishchuk & Zel’dovich (1981) or Polyachenko & Fridman (1981). Although having reached some of our conclusions, such studies lack most of the features present in this work. In particular they do not refer to the existence of the resonance we shall point out in this paper.

Since it is generally accepted that dark matter is constituted by WIMPs, (Weakly Interacting Massive Particles) in this study we shall consider the case for which the two components of the cosmic fluid interact only gravitationally.

The two fluid components that we shall consider will be a baryonic one and a hot dark matter (HDM) one, hereafter component BݐµBitalic_B and Dݐ·Ditalic_D respectively. The latter being assumed to be made of neutrino-like particles.

Although the numerical results obtained in this paper refer to the particular case of HDM, one should point out that the formalism here developed is quite general and applies to any type of a two-component mixture.

In Sect. 2 we shall establish the linear coupled equations which describe the dynamical system for the density perturbations in the two-component fluid we wish to study. We shall then develop, in Sect. 3, a qualitative analysis of the dynamical system.

In Sect. 4 we shall study the behaviour, in phase-space, of the trajectories representing the general solution of the dynamical system. This analytical study will be complemented with some numerical plots in order to clarify some of the qualitative results obtained.

For numerical purposes we shall consider that the onset of gravitational instability occurs in a flat (Ω=1)Ω1(\Omega=1)( roman_Ω = 1 ) universe at recombination epoch, after the decoupling between radiation and baryonic matter, when the radiation temperature is ∼3000similar-toabsent3000\sim 3000∼ 3000 K (Kolb & Turner 1990). We shall assume that the density parameter ΩBsubscriptΩݐµ\Omega_Broman_Ω start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT has a value in agreement with the limits imposed by standard nucleosynthesis (Walker et al. 1991). Assuming 0.5≤0.5absent0.5\leq0.5 ≤ h ≤1absent1\leq 1≤ 1, where h is the Hubble constant in units of 100100100100 Km/s/Mpc, i.e. h = H/0subscript0{}_0/start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT /(100100100100 Km/s/Mpc), such a value is ∼0.05similar-toabsent0.05\sim 0.05∼ 0.05.

The neutrino-like particles which constitute the dark matter component are assumed to have a mass compatible with the assumption of being already non-relativistic by the recombination epoch. It will be shown later in this paper that this value for the mass of the dark matter particles seems to provide a good fit for the galactic mass spectrum.

In Sect. 5 we shall describe the occurence of a resonance which, in our opinion, is responsible for the formation of structure at the typical galactic scales.

The relation between the existence of the resonance, the galactic scale and the mass of dark matter particles is discussed in Sect. 6.

Finally in Sects. 7 and 8 we shall point out the main results presented in this paper and outline some open questions as well as the research paths we intend to follow in the near future.

2 Dynamical equations for a two component fluid

The dynamics of a non-relativistic self-gravitating fluid can be described by the usual set of hydrodynamical equations: continuity equation, Euler equation and Poisson’s equation.

In this paper we shall neglect the expansion of the universe, since we shall be interested only in the qualitative behaviour, at the onset of the process of growth of density perturbations in the post recombination epoch, and not in the exact dynamics of the process.

In this case, for a fluid made of two components, Dݐ·Ditalic_D and BݐµBitalic_B, the above set of equations becomes:

∂ρD∂t+∇⋅(ρDݒ—D)=0subscriptݜŒݐ·ݑ¡bold-⋅∇subscriptݜŒݐ·subscriptݒ—ݐ·0\frac\partial\rho_{}_D\partial t+\nabla\mbox\boldmath$\cdot$\,(\rho_% {}_D\mbox\boldmath$v$_{}_D)=0divide start_ARG ∂ italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_t end_ARG + ∇ bold_⋅ ( italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ) = 0 (1)

∂ρB∂t+∇⋅(ρBݒ—B)=0subscriptݜŒݐµݑ¡bold-⋅∇subscriptݜŒݐµsubscriptݒ—ݐµ0\frac\partial\rho_{}_B\partial t+\nabla\mbox\boldmath$\cdot$\,(\rho_% {}_B\mbox\boldmath$v$_{}_B)=0divide start_ARG ∂ italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_t end_ARG + ∇ bold_⋅ ( italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ) = 0 (2)

∂ݒ—D∂t+(ݒ—D⋅∇)ݒ—D=-1ρD∇pD-∇Φsubscriptݒ—ݐ·ݑ¡bold-⋅subscriptݒ—ݐ·∇subscriptݒ—ݐ·1subscriptݜŒݐ·∇subscriptݑݐ·∇Φ\frac\partial\mbox\boldmath$v$_{}_D\partial t+(\mbox\boldmath$v$_% {}_D\mbox\boldmath$\cdot$\,\nabla)\,\mbox\boldmath$v$_{}_D=-\frac1% \rho_{}_D\nabla p_{}_D-\nabla\Phidivide start_ARG ∂ bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_t end_ARG + ( bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT bold_⋅ ∇ ) bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG ∇ italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT - ∇ roman_Φ (3)

∂ݒ—B∂t+(ݒ—B⋅∇)ݒ—B=-1ρB∇pB-∇Φsubscriptݒ—ݐµݑ¡bold-⋅subscriptݒ—ݐµ∇subscriptݒ—ݐµ1subscriptݜŒݐµ∇subscriptݑݐµ∇Φ\frac\partial\mbox\boldmath$v$_{}_B\partial t+(\mbox\boldmath$v$_% {}_B\mbox\boldmath$\cdot$\,\nabla)\,\mbox\boldmath$v$_{}_B=-\frac1% \rho_{}_B\nabla p_{}_B-\nabla\Phidivide start_ARG ∂ bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG ∂ italic_t end_ARG + ( bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT bold_⋅ ∇ ) bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_ARG ∇ italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT - ∇ roman_Φ (4)

∇2Φ=4πG(ρD+ρB)superscript∇2Φ4ݜ‹ݐºsubscriptݜŒݐ·subscriptݜŒݐµ\nabla^2\Phi=4\pi G\,(\rho_{}_D+\rho_{}_B)∇ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT roman_Φ = 4 italic_π italic_G ( italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ) (5)
where ΦΦ\Phiroman_Φ is the total self-gravitational potential, ρjsubscriptݜŒݑ—\rho_jitalic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, ݒ—jsubscriptݒ—ݑ—\mbox\boldmath$v$_jbold_italic_v start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and pjsubscriptݑݑ—p_jitalic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are respectively the density, velocity and pressure fields for fluid elements of component jݑ—jitalic_j, (the index jݑ—jitalic_j taking the values Dݐ·Ditalic_D or BݐµBitalic_B) .

Using a perturbative analysis, to first order in the perturbation, one can write the dynamical variables as follows:

ρj=ρ0j+δρjݒ—j=ݒ—0j+δݒ—jpj=p0j+δpjΦT=Φ0T+δΦTsubscriptݜŒݑ—subscriptݜŒsubscript0ݑ—ݛ¿subscriptݜŒݑ—subscriptݒ—ݑ—subscriptݒ—subscript0ݑ—ݛ¿subscriptݒ—ݑ—subscriptݑݑ—subscriptݑsubscript0ݑ—ݛ¿subscriptݑݑ—subscriptΦݑ‡subscriptΦsubscript0ݑ‡ݛ¿subscriptΦݑ‡\beginarray[]l\rho_{}_j=\rho_{}_j{}_{}_0+\delta\rho_{}_j% \\ \mbox\boldmath$v$_{}_j=\mbox\boldmath$v$_{}_j{}_{}_0+\delta% \mbox\boldmath$v$_{}_j\\ p_{}_j=p_{}_j{}_{}_0+\delta p_{}_j\\ \Phi_{}_T=\Phi_{}_T{}_{}_0+\delta\Phi_{}_T\endarraystart_ARRAY start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + italic_δ italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + italic_δ bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + italic_δ italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL roman_Φ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_T end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = roman_Φ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_T end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + italic_δ roman_Φ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_T end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL end_ROW end_ARRAY (6)
where the index ”00” indicates the unperturbed value, and δݛ¿\deltaitalic_δ the first order perturbation.

It’s common to consider the zero order solution to be an infinite fluid at rest, (ݒ—0=0subscriptݒ—00\mbox\boldmath$v$_{}_0=0bold_italic_v start_POSTSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = 0), for which the density and pressure are the same everywhere (unperturbed fluid). We must point out, as it is well known, that this static solution for the unperturbed state of the above equations is not mathematically correct, except for ρ0=0subscriptݜŒ00\rho_{}_0=0italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = 0. However, since this problem is removed when considering the more realistic case of a fluid embedded in the expanding universe, (which we shall discuss in a forthcoming paper), one can start perturbing this static configuration in order to obtain some qualitative information on the fluid behaviour.

We shall assume that the solutions of the above coupled system of Eqs. (1) - (5) are square integrable functions of the spatial variables and therefore can be Fourier analysed with respect to these variables. To first order in the perturbation, after the Fourier analysis in the spatial variables and some manipulation, the set of Eqs. (1) - (5) leads to the following equations:

{Δ¨D+(cD2k2-W)DΔD-WΔBB=0Δ¨B+(cB2k2-W)BΔB-WΔDD=0casesfragmentssubscript¨Δݐ·fragments(superscriptsubscriptݑݐ·2superscriptݑ˜2Wsubscript)ݐ·subscriptΔݐ·WsubscriptsubscriptΔݐµݐµ0missing-subexpressionmissing-subexpressionmissing-subexpressionfragmentssubscript¨Δݐµfragments(superscriptsubscriptݑݐµ2superscriptݑ˜2Wsubscript)ݐµsubscriptΔݐµWsubscriptsubscriptΔݐ·ݐ·0\left\{\begin{array}[]{lll}\ddot{\Delta}_{{}_{D}}+\left(c_{{}_{D}}^{2}\,k^{2}-% W\!{{}_{{}_{D}}}\right)\Delta_{{}_{D}}-W\!{{}_{{}_{B}}}\,\Delta_{{}_{B}}&=&0\\ \\ \ddot{\Delta}_{{}_{B}}+\left(c_{{}_{B}}^{2}\,k^{2}-W\!{{}_{{}_{B}}}\right)% \Delta_{{}_{B}}-W\!{{}_{{}_{D}}}\,\Delta_{{}_{D}}&=&0\end{array}\right.{ start_ARRAY start_ROW start_CELL over¨ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + ( italic_c start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT ) roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT - italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL over¨ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT + ( italic_c start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT ) roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT - italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY (7)
where the double dot stands for second time derivative, kݑ˜kitalic_k is the wave number of the perturbation. For each of the components jݑ—jitalic_j, the parameter WjsubscriptݑŠݑ—W\!_{j}italic_W start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is given by Wj=4πGρ0jsubscriptݑŠݑ—4ݜ‹ݐºsubscriptݜŒsubscript0ݑ—W\!_{j}=4\pi G\rho_{{}_{j}{{}_{{}_{0}}}}italic_W start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 4 italic_π italic_G italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT and ΔjsubscriptΔݑ—\Delta_{j}roman_Δ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT is the density contrast δρj/ρ0jݛ¿subscriptݜŒݑ—subscriptݜŒsubscript0ݑ—\delta\rho_{{}_{j}}/\rho_{{}_{j}{{}_{{}_{0}}}}italic_δ italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT / italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT 0 end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_POSTSUBSCRIPT and cjsubscriptݑݑ—c_{j}italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT the adiabatic sound speed given by cj=(∂pj/∂ρj)1/2subscriptݑݑ—superscriptsubscriptݑݑ—subscriptݜŒݑ—12c_{{}_{j}}=\left(\partial p_{{}_{j}}/\partial\rho_{{}_{j}}\right)^{1/2}italic_c start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT = ( ∂ italic_p start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT / ∂ italic_ρ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_j end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT.

The value of the parameter WjsubscriptݑŠݑ—W\!_{j}italic_W start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, at the cosmic time of recombination, trecsubscriptݑ¡ݑŸݑ’ݑt_{rec}italic_t start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT, can be related to the density parameter, ΩjsubscriptΩݑ—\Omega_{j}roman_Ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, of component jݑ—jitalic_j, by

Wj= 23Ωjtrec2subscriptݑŠݑ—23subscriptΩݑ—superscriptsubscriptݑ¡ݑŸݑ’ݑ2W\!_{j}=\frac{\,2\,}{3}\,\frac{\Omega_{j}}{t_{rec}^{2}}italic_W start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 2 end_ARG start_ARG 3 end_ARG divide start_ARG roman_Ω start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG start_ARG italic_t start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (8)

The adiabatic sound speed in component jݑ—jitalic_j at recombination, cjrecsubscriptݑsubscriptݑ—ݑŸݑ’ݑc_{j_{rec}}italic_c start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT, is the sound speed of a monoatomic ideal gas, which is given by:

cjrec=( 53KBolTjrecmj)1/2subscriptݑsubscriptݑ—ݑŸݑ’ݑsuperscript53subscriptݐ¾ݐµݑœݑ™subscriptݑ‡subscriptݑ—ݑŸݑ’ݑsubscriptݑšݑ—12c_{j_{rec}}=\left(\frac{\,5\,}{3}\,\frac{K_{{}_{Bol}}T_{j_{rec}}}{m_{j}}\right% )^{1/2}italic_c start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT = ( divide start_ARG 5 end_ARG start_ARG 3 end_ARG divide start_ARG italic_K start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B italic_o italic_l end_FLOATSUBSCRIPT end_POSTSUBSCRIPT italic_T start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT end_ARG start_ARG italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT (9)
where KBolsubscriptݐ¾ݐµݑœݑ™K_{{}_{Bol}}italic_K start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B italic_o italic_l end_FLOATSUBSCRIPT end_POSTSUBSCRIPT is the Boltzman constant and Tjrecsubscriptݑ‡subscriptݑ—ݑŸݑ’ݑT_{j_{rec}}italic_T start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT italic_r italic_e italic_c end_POSTSUBSCRIPT end_POSTSUBSCRIPT and mjsubscriptݑšݑ—m_{j}italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are respectively the temperature at recombination and mass of the particles of component jݑ—jitalic_j.

One should emphasize, that from the decoupling between the neutrino-like particles and radiation until the epoch at which these particles become non-relativistic, (which occurs for a red-shift zNR∼6×104similar-tosubscriptݑ§ݑݑ…6superscript104z_{{}_{NR}}\sim 6\times 10^{4}italic_z start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_N italic_R end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ∼ 6 × 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, Doroshkevich et al. 1988), their temperature TDݐ·{}_{D}start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT is given by T=D(4/11)1/3fragmentssubscriptݐ·superscriptfragments(411)13{}_{D}=(4/11)^{1/3}\,start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT = ( 4 / 11 ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPTTγݛ¾{}_{\gamma}start_FLOATSUBSCRIPT italic_γ end_FLOATSUBSCRIPT, where Tγݛ¾{}_{\gamma}start_FLOATSUBSCRIPT italic_γ end_FLOATSUBSCRIPT is the radiation temperature. Since then, it decays with a-2superscriptݑŽ2a^{-2}italic_a start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT, where aݑŽaitalic_a is the scale factor of the universe (Gao & Ruffini 1981; Padmanabhan 1993). One can therefore estimate that, at the recombination epoch, such a dark component Dݐ·Ditalic_D will have a temperature T∼Drec40fragmentssubscriptsimilar-tofragmentsDݑŸݑ’ݑ40{}_{D{{}_{rec}}}\sim 40start_FLOATSUBSCRIPT italic_D start_FLOATSUBSCRIPT italic_r italic_e italic_c end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT ∼ 40 K

The above system (7) of two linear differential equations of second order can be transformed into the following autonomous dynamical system of four first order differential equations which one can write in a matrix form as:

ݒ™˙=ݑ¨ݒ™˙ݒ™ݑ¨ݒ™\dot{\mbox{\boldmath$x$}}=\mbox{\boldmath$A\,x$}over˙ start_ARG bold_italic_x end_ARG = bold_italic_A bold_italic_x (10)
where ݒ™≡(x1,x2,x3,x4)T≡(Δ˙D,ΔD,Δ˙B,ΔB)Tݒ™superscriptsubscriptݑ¥1subscriptݑ¥2subscriptݑ¥3subscriptݑ¥4Tsuperscriptsubscript˙Δݐ·subscriptΔݐ·subscript˙ΔݐµsubscriptΔݐµT\mbox{\boldmath$x$}\equiv(x_{1},x_{2},x_{3},x_{4})^{\rm T}\equiv(\dot{\Delta}_% {{}_{D}},\Delta_{{}_{D}},\dot{\Delta}_{{}_{B}},\Delta_{{}_{B}})^{\rm T}bold_italic_x ≡ ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT ≡ ( over˙ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT , roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT , over˙ start_ARG roman_Δ end_ARG start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT , roman_Δ start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT roman_T end_POSTSUPERSCRIPT, the superscript TT{\rm T}roman_T standing for transpose. ݑ¨ݑ¨Abold_italic_A is a matrix whose components are only functions of the parameter kݑ˜kitalic_k and has the form:

ݑ¨≡ݑ¨(k)=(0W-DcD2k20WB10000WD0W-BcB2k20010)ݑ¨ݑ¨ݑ˜0fragmentsWsubscriptݐ·superscriptsubscriptݑݐ·2superscriptݑ˜20fragmentsWݐµ10000fragmentsWݐ·0fragmentsWsubscriptݐµsuperscriptsubscriptݑݐµ2superscriptݑ˜20010\mbox{\boldmath$A$}\equiv\mbox{\boldmath$A$}(k)=\left(\begin{array}[]{cccc}0&W% \!{{}_{{}_{D}}}-c_{{}_{D}}^{2}\,k^{2}&0&W\!{{}_{{}_{B}}}\\ 1&0&0&0\\ 0&W\!{{}_{{}_{D}}}&0&W\!{{}_{{}_{B}}}-c_{{}_{B}}^{2}\,k^{2}\\ 0&0&1&0\end{array}\right)bold_italic_A ≡ bold_italic_A ( italic_k ) = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT - italic_c start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_D end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT end_CELL start_CELL 0 end_CELL start_CELL italic_W start_FLOATSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_FLOATSUBSCRIPT - italic_c start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_B end_FLOATSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_k start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 end_CELL start_CELL 0 end_CELL end_ROW end_ARRAY ) (11)

We shall be particularly interested in the solutions which represent the density perturbations in components Dݐ·Ditalic_D and BݐµBitalic_B. These solutions are given by the second and fourth components, x2(t)subscriptݑ¥2ݑ¡x_{2}(t)italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ( italic_t ) and x4(t)subscriptݑ¥4ݑ¡x_{4}(t)italic_x start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ( italic_t ), of the general solution ݒ™(t)ݒ™ݑ¡\mbox{\boldmath$x$}(t)bold_italic_x ( italic_t ), of the dynamical system (10).

Using methods of qualitative theory of differential equations one can analyse the dynamical system and obtain some qualitative information on the behaviour of the perturbations. This will help understanding the stability, against density perturbations, of the two-component system. Even in a case like this one where an explicit solution exists, this is worth doing.

Alternatively the autonomous dynamical system can be numerically studied provided we specify the initial condition ݒ™0≡ݒ™(t0)subscriptݒ™0ݒ™subscriptݑ¡0\mbox{\boldmath$x$}_{0}\equiv\mbox{\boldmath$x$}(t_{0})bold_italic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≡ bold_italic_x ( italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ), where t0subscriptݑ¡0t_{0}italic_t start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT is the cosmic time at which the equilibrium state of the fluid is perturbed. We shall analyse the system qualitatively and, whenever it proves convenient, to clarify any result, we shall illustrate this study with some numerical plots.

3 Qualitative study of the dynamical system

The above matrix ݑ¨(k)ݑ¨ݑ˜\mbox{\boldmath$A$}(k)bold_italic_A ( italic_k ) has 4444 different eigenvalues for all values of kݑ˜kitalic_k except for k=0ݑ˜0k=0italic_k = 0 and for a critical value k=kMݑ˜subscriptݑ˜ݑ€k=k_{{}_{M}}italic_k = italic_k start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_M end_FLOATSUBSCRIPT end_POSTSUBSCRIPT, which shall be defined below. Therefore for k>0ݑ˜0k>0italic_k >0 and k≠kMݑ˜subscriptݑ˜ݑ€k\neq k_{{}_{M}}italic_k ≠italic_k start_POSTSUBSCRIPT start_FLOATSUBSCRIPT italic_M end_FLOATSUBSCRIPT end_POSTSUBSCRIPT there are 4444 linearly independent eigenvectors. In this case the general solution ݒ™(t)ݒ™ݑ¡\mbox{\boldmath$x$}(t)bold_italic_x ( italic_t ) of the system (10) is well known (Arnold 1973).

If all the eigenvalues are real ones, that solution can be written in the form of a linear expansion of particular solutions eλjtݝƒjsuperscriptesubscriptݜ†ݑ—ݑ¡subscriptݝƒݑ—{\rm e}^{\lambda_{j}t}\,\mbox{\boldmath$\xi$}_{j}roman_e start_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT bold_italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT, i.e.:

ݒ™(t)=∑j=14αjeλjtݝƒjݒ™ݑ¡superscriptsubscriptݑ—14subscriptݛ¼ݑ—superscriptesubscriptݜ†ݑ—ݑ¡subscriptݝƒݑ—\mbox{\boldmath$x$}(t)=\sum_{j=1}^{4}\alpha_{j}\,{\rm e}^{\lambda_{j}t}\,\mbox% {\boldmath$\xi$}_{j}bold_italic_x ( italic_t ) = ∑ start_POSTSUBSCRIPT italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT roman_e start_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_t end_POSTSUPERSCRIPT bold_italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT (12)
where αjsubscriptݛ¼ݑ—\alpha_{j}italic_α start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are real-valued functions of the the parameter kݑ˜kitalic_k, which are determined by the initial conditions, and λjsubscriptݜ†ݑ—\lambda_{j}italic_λ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT and ݝƒjsubscriptݝƒݑ—\mbox{\boldmath$\xi$}_{j}bold_italic_ξ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT are respectively the eigenvalues of the matrix ݑ¨(k)ݑ¨ݑ˜\mbox{\boldmath$A$}(k)bold_italic_A ( italic_k ) and its associated eigenvectors.

If some of the eigenvalues are complex conjugates, one can easily see that the real and imaginary parts of the complex function eλjtݝƒjsuperscriptesubscriptݜ†ݑ—ݑ¡subscriptݝƒݑ—{\rm e}^{\lambda_{j}t}\,\mbox{\boldmath$\xi$}_{j}roman_e start_POSTSUPERSCRIPT italic_λ start_POSTSUBSCRIPT italic_j

Costway US $259.00 3 pcs Outdoor Patio Table Chairs Set Acacia Wood Loveseat-Red Costway US $244.00 Outdoor Gliding Loveseat Chair with Tempered Glass Coffee Table-2 Pieces Costway US $323.00 Modern Loveseat Sofa with 2 Bolsters and Side Storage Pocket-Gray Costway US $140.00 Loveseat Cushioned Patio Steel Frame Swing Glider -Beige Costway US $323.00 Modern Loveseat Sofa with 2 Bolsters and Side Storage Pocket-Pink Costway US $236.00 Outdoor Loveseat Chair Set with Coffee Table and Seat Back Cushions-Navy Costway US $484.00 4 Pieces Patio Acacia Wood Thick Cushion Loveseat Sofa Set-White Costway US $323.00 Modern Loveseat Sofa with 2 Bolsters and Side Storage Pocket-Light Gray Costway US $314.00 Modern Loveseat Sofa Upholstered Dutch Velvet Sofa Couch-Green Costway US $175.00 Swing Glider Chair 48 Inch Loveseat Rocker Lounge Backyard-Black