Gravitational Stability and Bulk Cosmology
aa r X i v : . [ g r- q c ] D ec Gravitational Stability and BulkCosmology
Nakia Carlevaro a, b and Giovanni Montani b, c, d, e a Department of Physics, Polo Scientifico – Universit`a degli Studi di Firenze,INFN – Section of Florence, Via G. Sansone, 1 (50019), Sesto Fiorentino (FI), Italy b ICRA – International Center for Relativistic Astrophysics,c/o Dep. of Physics - “Sapienza” Universit`a di Roma c Department of Physics - “Sapienza” Universit`a di Roma, Piazza A. Moro, 5 (00185), Rome, Italy d ENEA – C.R. Frascati (Department F.P.N.), Via Enrico Fermi, 45 (00044), Frascati (Rome), Italy e ICRANet – C. C. Pescara, Piazzale della Repubblica, 10 (65100), Pescara, Italy [email protected] [email protected]
Abstract:
We present a discussion of the effects induced by bulk viscosity either on the very early Universe stability and on the dynamics associated tothe extreme gravitational collapse of a gas cloud. In both cases the viscositycoefficient is related to the energy density ρ via a power-law of the form ζ = ζ ρ s (where ζ , s = const. ) and the behavior of the density contrast in analyzed.In the first case, matter filling the isotropic and homogeneous background isdescribed by an ultra-relativistic equation of state. The analytic expression of thedensity contrast shows that its growth is suppressed forward in time as soon as ζ overcomes a critical value. On the other hand, in such a regime, the asymptoticapproach to the initial singularity admits an unstable collapsing picture.In the second case, we investigate the top-down fragmentation process of anuniform and spherically symmetric gas cloud within the framework of a Newtonianapproach, including the negative pressure contribution associated to the bulkviscous phenomenology. In the extreme regime toward the singularity, we showthat the density contrast associated to an adiabatic-like behavior of the gas (whichis identified by a particular range of the politropic index) acquire, for sufficientlylarge viscous contributions, a vanishing behavior which prevents the formation ofsub-structures. Such a feature is not present in the isothermal-like collapse. Wealso emphasize that in the adiabatic-like case bulk viscosity is also responsible forthe appearance of a threshold scale (equivalent to a Jeans length) beyond whichperturbations begin to increase. PACS : 98.80.-k, 95.30.Wi, 51.20.+d 1 ntroduction
Both the extreme regime of a gravitational collapse and the very early stages of the Universeevolution are characterized by a thermodynamics which can not be regarded as settled downinto the equilibrium. At sufficiently high temperatures, cross sections of the micro-physicalprocesses are no longer able to restore the thermodynamical equilibrium. Thus we meetstages where the expansion and collapse induce non-equilibrium phenomena. The averageeffect of having such kind of micro-physics results into dissipative processes appropriatelydescribed by the presence of bulk viscosity ζ . As shown in [1, 2, 3] this kind of viscosity canbe phenomenologically described by a function of the energy density ρ as ζ = ζ ρ s where ζ , s = const .In the first part of this work we investigate the effects that bulk viscosity has on thestability of the isotropic Universe [4], i.e. the dynamics of cosmological perturbations isanalyzed when viscous phenomena affect the zeroth- and first-order evolution of the system.We consider a background corresponding to a Friedmann-Robertson-Walker model filled withultra-relativistic viscous matter, whose coefficient ζ corresponds to the choice s = / andthen we develop a perturbation theory which generalizes the original analysis performed byLifshitz [5, 6].As issue of our first analysis we find that two different dynamical regimes appear whenviscosity is taken into account and the transition from one regime to the other one takesplace when the parameter ζ overcomes a given threshold value. However in both thesestages of evolution the Universe results to be stable as it expands; the effect of increasingviscosity is that the density contrast begins to decrease with increasing time when ζ isover the threshold. It follows that a real new feature arises with respect to the non-viscous( ζ = 0) analysis when the collapsing point of view is addressed. In fact, if ζ overcomesits critical value, the density contrast explodes asymptotically and the isotropic Universeresults unstable approaching the initial singularity. Since a reliable estimation fixes theappearance of thermal bath into the equilibrium below temperatures O (10 GeV ) and thislimit corresponds to the pre-inflationary age, our result supports the idea that an isotropicUniverse outcomes only after a vacuum phase transition settled down.In the second part of this work, we study one of the most attractive challenge of relativisticcosmology concerning the research a self-consistent picture for processes of gravitationalinstability which connect the high isotropy of the cosmic microwaves background radiationwith the striking clumpiness of the actual Universe. An interesting framework is provided bythe top-down scheme of structure fragmentation which is based on the idea that perturbationscales, contained within a collapsing gas cloud, start to collapse (forming sub-structures)because their mass overcomes the decreasing Jeans value [7, 8, 9, 10] of the backgroundsystem. In a work by C. Hunter [11], a specific model for a gas cloud fragmentation wasaddressed and the behavior of sub-scales density perturbations, outcoming in the extremecollapse, was analytically described.In this respect, we investigate how the above picture is modified by including, in the gascloud dynamics, the presence of bulk viscosity effects [12]. We start describing the Lagrangianzeroth-order evolution by taking into account the force acting on the collapsing shell as aresult of the negative pressure connected to the presence of bulk viscosity. We constructsuch an extension requiring that the asymptotic dynamics of the collapsing cloud is notqualitatively affected by the presence of viscosity: in this respect we can assume the viscousexponent as s = / . Then we face the Eulerian motion of the inhomogeneous perturbationsliving within the cloud. The resulting dynamics is treated in the asymptotic limit to thesingularity.As a result, we show that the density contrast behaves, in the isothermal-like collapse ( i.e. in correspondence of a politropic index 1 γ < / ), as in the non-viscous case ζ = 0. Onthe other hand, the damping of perturbations increases monotonically as γ runs from / to2 / ; in fact, for such adiabatic-like case, we see that density contrasts asymptotically vanishand no fragmentation processes take place within the cloud when the viscous correctionsare sufficiently large. In particular, we observe the appearance of a threshold value for thescale of the collapsing perturbations depending on the values taken by the parameters ζ and γ ∈ ( / , / ]; such a viscous effect corresponds to deal with an analogous of the Jeans length,above which perturbations are able to collapse. However such a threshold value does notensure the diverging behavior of density contrasts which takes place, in turn, only when asecond (greater) critical length is overcome. According to our analysis, if the viscous effectsare sufficiently intense, the final system configuration is not a fragmented cloud as a cluster ofsub-structures but simply a single object (a black hole, in the present case, because pressureforces are taken negligible). In order to describe the temporal evolution of the energy density small fluctuation, we de-velop a perturbations theory on the Einstein equations. We limit our work to the studyof space regions having small dimensions compared with the scale factor of the Universe a [13]. According to this approximation, we can consider a 3-dimensional Euclidean (timedependent) metric as spatial component of the background line element ds = dt − a ( dx + dy + dz ) . (1)In linear approximation, perturbed Einstein equations write as δR νµ − δ νµ δR = 8 πGδT νµ , (2)where the term δT νµ represents the perturbation of the energy-momentum tensor which de-scribes the properties of the matter involved in the cosmological collapse. The perturbationsof the Ricci tensor δR νµ can be written in terms of metric perturbations h νµ = − δg νµ , startingfrom the general expression for the perturbed curvature tensor. For convenience let us nowintroduce a new temporal variable η , set by the relation dt = a dη , and use the symbol ( ′ ) forits derivatives; we moreover impose, without loss of generality, that the synchronous referencesystem is still preserved in the perturbations scheme: h = h α = 0.With the above assumptions, the perturbations of the Ricci tensor and the curvature scalarread: δR = − a h ′′ − a ′ a h ′ , (3a) δR α = 12 a ( h , α ′ − h α, β ′ β ) , (3b) δR βα = − a ( h γ, βα, γ + h β, γγ, α − h β, γα, γ − h , β, α )+ − a h β ′′ α − a ′ a h β ′ α − a ′ a h ′ δ βα , (3c) δR = − a ( h γ, αα, γ − h , γ, γ ) − a h ′′ − a ′ a h ′ . (3d)By using these expressions, we are able to rewrite the left-hand side of Einstein eqs. (2)through the metric perturbations h αβ . 3 ynamical Representation of Perturbations Since we use an Euclidean background met-ric (1), we can expand the perturbations in plane waves of the form e i q · r , where q is thedimensionless comoving wave vector being the physical one k = q /a . Here we investigate thegravitational stability properly described by the behavior of the energy density perturbationsexpressible only by a scalar function; in this sense we have to choose a scalar representationof the metric perturbations [6, 13]. Such a picture is made by the scalar harmonics Q = e i q · r ,from which the following tensor Q βα = δ βα Q , P βα = [ δ βα − q α q β q ] Q , (4)can be constructed. We can now express the time dependence of the gravitational perturba-tions through two functions λ ( η ), µ ( η ) and write the tensor h βα in the form h βα = λ ( η ) P βα + µ ( η ) Q βα , h = µ ( η ) Q . (5)
The presence of dissipative processes within the Universe dynamics, as it is expected attemperatures above O (10 GeV ), can be expressed by an additional term in the standardideal fluid energy-momentum tensor used in the original works by Lifshitz and Khalatnikov[5, 6]. Using the conservation law T νµ ; ν = 0 we can express the viscous energy-momentumtensor as T µν = (˜ p + ρ ) u µ u ν − ˜ p g µν , ˜ p = p − ζ u ρ ; ρ , (6)where p denotes the usual thermostatic pressure and ζ is the bulk viscosity coefficient. In thehomogeneous models this quantity depends only on time, and therefore we may consider it asa function of the Universe energy density ρ . According to literature developments [1, 2, 3, 14]we assume that ζ depends on ρ via a power-law of the form ζ = ζ ρ s , (7)where ζ > s is a dimensionless parameter. Furthermore, in a comovingsystem the 4-velocity can be expressed as u = / a , u α = 0 and the viscous pressure ˜ p assumes the form ˜ p = p − ζ ρ s a ′ / a . (8)Let us now consider the earlier stages of a flat Universe corresponding to η ≪ p = ρ/
3. The Universe zeroth-order dynamics isdescribed by the energy conservation equation and the Friedmann one, which are respectively ρ ′ + 3 a ′ a ( ρ + ˜ p ) = 0 , a ′ a = p / πGρ . (9)In this analysis we assume s = / in order to deal with the maximum effect that bulk viscositycan have without dominating the Universe dynamics since it corresponds to a phenomeno-logical issue of perturbations to the thermodynamical equilibrium [15, 16]. The solutions ofthe zeroth-order dynamics, for s = / , assume the form ρ = Ca − (2+2 ω ) , a = a η /ω , ω = 1 − χ ζ , (10)being C an integration constant, χ = √ πG and the parameter a = (8 ω πCG/ / ω .Since we consider an expanding Universe, the factor a must increase with positive power ofthe temporal variable ( i.e. ω >
0) thus we obtain the constraint 0 ζ < / χ which ensuresthis feature. 4 .3 Perturbation Theory in the Early Universe Let us now perturb the viscous energy-momentum tensor. Using the synchronous characterof the perturbed metric we get the following expressions δT = δρ , δT α = a (˜ p + ρ ) δu α , (11) δT βα = δ βα (cid:2) − Σ δρ + ζ (cid:0) δu γ,γ + h ′ / a (cid:1)(cid:3) , (12)here Σ ≡ v s − ζ sρ s − a ′ /a and v s is the sound speed of the fluid given by v s = δp/δρ .The presence of viscosity does not influence the expression of the Ricci tensor and itsperturbations, thus we can still keep expressions (3) and use the perturbed form of theenergy-momentum tensor to build up the equations which describe the dynamics of h βα and δρ . It is convenient to choose, as final equations, the ones obtained from the Einstein onesfor α = β and for contraction over α and β , which read respectively (cid:0) h γ, βα, γ + h β, γγ, α − h β, γα, γ − h , β, α (cid:1) + h β ′′ α + a ′ a h β ′ α = 0 , (13) (cid:0) h γ, αα, γ − h , γ, γ (cid:1)(cid:0) (cid:1) + h ′′ ++ a ′ a (cid:0) − πG aa ′ ζ (cid:1) h ′ + − ζ a (˜ p + ρ ) (cid:0) h , α ′ , α − h γ, α ′ α, γ (cid:1) = 0 . (14)Furthermore, taking the 00-components of gravitational eqs., we get the expression of thedensity perturbations δρ = 116 πGa ( h γ, αα, γ − h , α, α + a ′ a h ′ ) . (15)Substituting in eqs. (13),(14) the zeroth-order solutions (10) and the scalar representationof the metric perturbations (5), we can get, respectively, two equations for λ , µ which read λ ′′ + ωη λ ′ − q ( λ + µ ) = 0 , (16) µ ′′ + (cid:16) ωη (cid:17) µ ′ − (cid:16) π √ CGζ a ω η /ω (cid:17) µ ′ ++ q ( λ + µ ) (cid:0) (cid:1) + q ζ η ( µ ′ + λ ′ ) √ C / a ω − ζ / ω = 0 . (17)It is worth nothing that, among the solutions, there are some which can be removed bya simple transformation of the reference system, compatible with its synchronous character.They do not represent any real physical change in the metric and we must exclude themfrom the dynamics. Such fictitious solutions, in the ultra-relativistic limit, assume the form λ − µ = const. and λ + µ ∼ /η [13]. Let us now study the gravitational collapse dynamics of the primordial Universe near theinitial
Big-Bang in the limit η ≪
1. As in Lifshitz work [6], we analyze the case of perturba-tions scale sufficiently large to use the approximation ηq ≪
1. In our scheme eqs. (16) and(17) admit asymptotic analytic solutions for the functions λ and µ ; in the leading order λ takes the form λ = C η /ω − + C , (18)5here C , C are two integration constants. Substituting this expression in eq. (17) we get,in the same order of approximation, the behavior of the function µ as µ = ˜ C η /ω − + C , (19)where we have excluded the non-physical solutions introduced above. The constant ˜ C isgiven by the expression ˜ C = A / B (3 − / ω ), A and B being constants having the form A = C q ` ´ + C (1 − /ω ) q ζ √ C/ a ω − ζ /ω , B = 12 π √ CGζ a ω . Let us now write the final form of perturbations pointing out their temporal dependencein the viscous Universe. The metric perturbations (5) become h βα = C η /ω − P βα + ˜ C η /ω − Q βα + C (cid:0) Q βα + P βα (cid:1) , (20)and, by (15) and (10), the density contrast δ = δρ / ρ reads δ = F [ C η − / ω + C η + C η − / ω + ˜ C η − / ω ] , (21)where C = 3 A/q ωB , and F = ω Qq / η . The inequalities h βα ≪ δ ≪ C ≪ η /ω − , C ≪ , (22)for any ω -value within the interval (0 , q and the integration constant C ; in particular a roughestimation for ω < / C ≪ η /ω − and C ≪ η /ω − yields the condition q ≪ ( GCη ) − / ω which ensures the smallness of the cosmological perturbations.Using the hypothesis η ≪ η can be positive or negative accordingto the value of the viscous parameter ω ( ζ ). This behavior produces two different regimes ofthe density contrast evolution: Case ζ < / χ Here perturbations increase forward in time. This behavior corre-sponds qualitatively to the same picture of the non-viscous Universe (obtained setting ζ = 0)in which the expansion can not, nevertheless, imply the gravitational instability: if we con-sider the magnitude order η ∼ /q , the constraints on C , C imply that δ remains smalleven in the higher order of approximation. This behavior yields the gravitational stability ofthe primordial Universe. Case / χ < ζ < / χ In this regime the density contrast is suppressed behaving likea negative power of η . When the density contrast results to be increasing, the presence ofviscosity induces a damping of the perturbation evolution in the direction of the expandingUniverse, so the cosmological stability is fortified since the leading η powers are smaller thanthe non-viscous ones obtained setting ζ = 0.In this case the density fluctuation decreases forward in time but the most interesting resultis the instability which the isotropic and homogeneous Universe acquires in the direction ofthe collapse toward the Big-Bang. For ζ > / χ the density contrast diverges approachingthe cosmological singularity, i.e. for η →
0. In this regime, scalar perturbations destroyasymptotically the primordial Universe symmetry. The dynamical implication of this issue isthat an isotropic and homogeneous stage of the Universe can not be generated, from genericinitial conditions, as far as the viscosity becomes smaller than the critical value ζ ∗ = / χ .6 Spherically Symmetric Gas Cloud Fragmentation
In this second part, we present an hydrodynamical analysis of a spherically symmetric gascloud collapse. This model was firstly proposed in a work by C. Hunter [11], where hesupposed that a perfect fluid cloud becomes unstable with respect to its own gravitation andbegin to condense. The collapsing cloud is assumed to be the dynamical background on whichstudying, in a Newtonian regime, the evolution of density perturbations generated on thisbasic flow. Such an analysis is suitable for the investigation of the cosmological structuresformation in the top-down scheme [17, 18] since it deals with the sub-structures temporalevolution compared with the basic flow of the gravitational collapse.
The unperturbed flow is supposed to be homogeneous, spherically symmetric and initiallyat rest. Furthermore the gravitational forces are assumed to be very much greater than thepressure ones, which are therefore neglected in the zeroth-order analysis. In such an approachthe gas results to be unstable, since there are no forces which can contrast the collapse, andthe condensation starts immediately.The basic flow is governed by the Lagrangian motion equation of a spherically symmetricgas distribution which collapses under the only gravitational action. As in the previousapproach (see eqs. (6), (7)), in order to include bulk viscosity effects in the dynamics, weintroduce the bulk pressure ˜ p = p − ζ ρ s u µ ; µ , (23)where u µ = (1 , ) is the shell comoving 4-velocity. In the Newtonian limit we consider, themetric can be assumed as a flat Minkowskian one expressed in the usual spherical coordinatesand the metric determinant g becomes g = − r sin θ . In this case, for the 4-divergence u µ ; µ we immediately obtain: u µ ; µ = 2 ˙ r/r .Considering the basic flow density as ρ = M/ ( / πr ) and the pressure force acting on thecollapsing shell of the form F ˜ p = ˜ p πr , the Lagrangian motion equation for a viscous fluidbecomes now ∂ r∂t = − GMr − Cr s − ∂r∂t , (24)where C = 8 πζ (3 M/ π ) s . The origin O is taken at the center of the gas, r is the radialdistance, G the gravitational constant and M the mass of the gas inside a sphere of radius r .In what follows, we shall suppose that the gas was at a distance a from O in correspondence tothe initial instant t ; this distance a identifies a fluid particle and will be used as a Lagrangianindependent variable so r = r ( a, t ) and the following parametrization can be introduced r = a cos β . (25)Here β = β ( t ) is a time dependent function such that β ( t ) = 0 and β (0) = π/ t = 0 when r = 0 and t takes negative values.Eq. (24) must be integrated to obtain the evolution of the radial velocity v = ∂r / ∂t and thedensity ρ of the unperturbed flow. Let us now require that the viscosity does not influencethe final form of the velocity which in the non-viscous Hunter analysis ( i.e. ζ = 0) isproportional to / √ r [11]. Substituting v = B / √ r into eq. (24) we see that, in correspondenceto the choice s = / , it is again a solution as soon as B = C − √ C + 2 GM , where B assumes only negative values. Although this dynamics is analytically integrable only for theparticular value s = / , the obtained behavior v ∼ / √ r remains asymptotically (as r → s < / is satisfied. 7sing such a solution we are able to build an explicit form of the quantity β defined by(25). After standard manipulations we obtain the relationcos β = 3 A ( − t ) , (26)where A is defined to be A = − B/ a / .It is more convenient to use an Eulerian representation of the flow field. To this end,once solving the well known Poisson equation for the gravitational potential φ , we obtainthe unperturbed solutions describing the background motion; all these quantities take theexplicit forms v = [ v, , , v = − r ˙ β tan β , (27a) ρ = ρ cos − β , (27b) φ = − πρ G (cid:0) a − r / (cid:1) cos − β , (27c)where (˙) denotes the derivate with respect to time and the non-radial components of velocitymust vanish since we are considering a spherical symmetry. The zeroth-order motion of a viscous basic flow which collapses under the action of its owngravitation was discussed above. We shall now suppose that small disturbances appear on thisfield; the perturbations evolution can be described by the Eulerian equations in presence ofdissipative processes which come out from the thermodynamical irreversibility of the collapseprocess and are due to the microphysics of non-equilibrium [19].Perturbations are investigated in the Newtonian limit starting from the well known conti-nuity, Euler-Navier-Stokes and Poisson equations. In such a picture, we are now interestedto study the effects of the thermostatic pressure on the perturbations evolution. We shalltherefore consider terms due to the pressure p in the motion equations, which read˙ ρ + ∇ · ( ρ v ) = 0 , (28a)˙ v + ( v · ∇ ) · v = −∇ φ − / ρ ∇ p + ζ / ρ ∇ ( ∇ · v ) , (28b) ∇ φ = 4 πGρ . (28c)The gas is furthermore assumed to be barotropic, i.e. the pressure depends only by thebackground density by p = κρ γ , (29)where κ , γ are constants and 1 γ / . By this expression we are able to distinguish a setof different cases related to different values of the politropic index γ . The asymptotic value γ = 1 represents an isothermal behavior of the gas cloud; the case γ = / describes, instead,an adiabatic behavior and it will be valid when changes are taking place so fast that no heatis transferred between elements of the gas. We can suppose that intermediate values of γ willdescribe intermediate types of behavior between the isothermal and adiabatic ones.In this model, zeroth-order solutions (27) are already verified since the pressure gradient, inthe homogeneity hypothesis, vanishes and the pressure affects only the perturbative dynamics.Let us now investigate first-order fluctuations around the unperturbed solutions, i.e. wereplace the perturbed quantities: ( v + δ v ), ( ρ + δρ ), ( φ + δφ ) and ( p + δp ) in eqs. (28).Following the line of the Hunter work, equations for the perturbed quantities δ v , δρ and δφ can be built. Skipping many technicality, we are able to manipulate the system of themotion eqs. and write down an unique equation which which governs the evolution of density8erturbations δρ in the viscous regime. Taking into account the parameterization (25) weget (cos β ˙ δρ − β cos β ˙ βδρ ) ˙ − πGρ cos βδρ == ( v s cos β − ζ / ρ sin β cos β ˙ β ) D δρ ++ ζ / ρ cos β D ˙ δρ . (30)Here time differentiation is taken at some fixed comoving radial coordinate, v s is the soundspeed v s = ∂p/∂ρ and D is the Laplace operator as written in comoving spherical coordi-nates.In order to study the temporal evolution of density perturbations, we assume to expand δρ in plane waves of the form δρ ( r , t ) → δρ ( t ) e − i k · r , (31)where 1 /k (with k = | k | ) represents the initial length scale of the considered fluctuation.According to this assumption, we are able to write the asymptotic form of eq. (30) near theend of the collapse as ( − t ) →
0. In this case, we can make the approximation sinβ ≈ cosβ is given by (26).The background motion equations were derived for a particular value of the viscosity pa-rameter s = / : substituting the basic flow density given by (27b) in the standard expressionof the bulk viscosity (7), we obtain ζ = ζ ρ / cos − β . With these assumptions, eq. (30)now reads ( − t ) ¨ δ̺ − (cid:20) − λ A (cid:21) ( − t ) ˙ δ̺ ++ (cid:20) − πGρ A + v k ( − t ) / − γ (3 A ) γ − / − λ A (cid:21) δ̺ = 0 , (32)where v = κγρ γ − and the viscous parameter λ is given by λ = ζ ρ − / k . (33) A complete solution of eq. (32) involves Bessel functions of first and second species J and Y respectively and it explicitly reads δ̺ = C G ( t ) + C G ( t ) , (34)where C , C are integration constants and the functions G and G are defined to be G ( t ) = ( − t ) − + λ A J n (cid:2) q ( − t ) / − γ (cid:3) , (35) G ( t ) = ( − t ) − + λ A Y n (cid:2) q ( − t ) / − γ (cid:3) , (36)having set the Bessel parameters n and q as n = [ A − λA + λ + 16 πGρ ] / (6 A ( / − γ )) ,q = − kv (3 A ) / − γ / (4 / − γ ) . We now proceed, in order to study the asymptotic evolution of the solution (34), analyzingthe cases 1 γ < / and / < γ / separately, since Bessel functions have different limits9onnected to the magnitude of their argument. In the asymptotic limit to the singularity, theisothermal-like case is characterized by a positive time exponent inside Bessel functions so q ( − t ) / − γ ≪
1, on the other hand, in the adiabatic-like behavior, we obtain q ( − t ) / − γ ≫ J and Y behave like a power-law of theform J n ( x ) ∼ x + n , Y n ( x ) ∼ x − n , for x ≪
1. And in the adiabatic-like case argument becomesmuch gather than unity and they assume an oscillating behavior like J n ( x ) ∼ x − / cos ( x ) , Y n ( x ) ∼ x − / sin ( x ) , for x ≫ Isothermal-like Case
In this first regime, an asymptotic form of functions G can be foundas follow G ISO , = c ( − t ) − + λ A ± ( − γ ) n (38)where c and c are constants quantities. The condition which implies the density pertur-bations collapse is that at least one of G functions diverges as ( − t ) →
0. An analysis oftime exponents yields that G diverges if λ < A − πGρ / A but, on the other hand, G is always divergent for all λ . These results imply that, in the isothermal case, perturbationsalways condense.Let us now compare this collapse with the basic flow one; the background density evolveslike cos − β (see (27b)) that is, using (26) ρ ∼ ( − t ) − . (39)In the non-dissipative case ( λ = 0) perturbations grow more rapidly with respect to thebackground density involving the fragmentation of the basic flow independently on the valueof γ ; in presence of viscosity the density contrast assumes the asymptotic form δ ISO ∼ ( − t ) − + λ A − A √ [ A − λA + λ +16 πGρ ] . (40)Here the exponent is always negative and it does not depend on γ , this implies that δ ISO diverges as the singularity is approached and real sub-structures are formed involving thebasic flow fragmentation. This issue means that the viscous forces do not have enoughstrength to contrast an isothermal perturbations collapse in order to form of an uniquestructure.
Adiabatic-like Case
For / < γ / , J and Y assume an oscillating behavior. In thisregime functions G read G ADB , = ˜ c , (cid:2) q ( − t ) / − γ (cid:3) ( − t ) γ − + λ A , (41)where ˜ c , are constants. Following the isothermal approach, we shall now analyze the timepower-law exponent in order to determine the collapse conditions. G functions diverge,involving perturbations condensation, if the parameter λ is less than a threshold value: thiscondition reads λ < A − Aγ (for a given value of the index γ ). Expressing λ in function ofthe wave number (33), we outline, for a fixed viscous parameter ζ , a constraint on k whichis similar to the condition appearing in the Jeans model [7], [8]. The threshold value for thewave number is given by the relation k C = (17 A − γA ) ρ / / ζ (42)and therefore the condition for the density perturbations collapse, i.e. δρ ADB → ∞ , reads k < k C , recalling that, in the Jeans model for a static background, the condition for thecollapse is k < k J = [4 πGρ / v s ] / . It is to be remarked that, in absence of viscosity( ζ = 0), expression (42) diverges implying that all perturbations scales can be conducted10o the collapse. On the other hand, if we consider perturbations of fixed wave number, theyasymptotically decrease as ( − t ) → λ > A − Aγ . Thus for each k there is a valueof the bulk viscosity coefficient over which the dissipative forces contrast the formation ofsub-structures.If k < k C , perturbations oscillate with ever increasing frequency and amplitude. For anon-zero viscosity coefficient, the density contrast evolves like δ ADB ∼ ( − t ) γ − + λ A . (43)A study of the time exponent yields a new threshold value. If λ < A − Aγ , i.e. the viscosityis enough small, sub-structures form; on the other hand, when the parameter ζ , or the wavenumber k , provides a λ -term overcoming this value, the perturbations collapse is so muchcontrasted that no fragmentation process occurs. In other words, if λ > A − Aγ we get δ ADB → i.e. for a given γ there is a viscous coefficient ζ enough large ables to preventsthe sub-structures formation.It is remarkable that in the pure adiabatic case, γ = / , dissipative processes, of anymagnitude order, contrast the fragmentation because, while the Jeans-like length survives,the threshold value for sub-structures formation approaches infinity. We can conclude that,in the case / < γ / , the fragmentation in the top-down scheme is deeply unfavored bythe presence of bulk viscosity which strongly contrasts the density perturbations collapse. To complete the analysis of the gravitational stability in presence of dissipative affects, wenow point out some relevant feature about the validity of our picture concerning both therelativistic analysis and the Newtonian fragmentation:
Shear Viscosity
In our approach, the hypothesis of the Universe isotropy and the factthat the shell results comoving with the collapsing background, imply that there are nodisplacements between parts of fluid with respect to ones other in both cases. Dissipativeprocesses are therefore related only to thermodynamical properties of the fluid compressionand can be phenomenologically described by the presence of bulk viscosity. Furthermore,in this model we are able to neglect the so-called shear viscosity since it is connected withprocesses of relative motion among different parts of the fluid.
Early Universe Dynamics
In the fist Universe analysis, we fix the value s = / in or-der to deal with the maximum effect that bulk viscosity can have without dominating thedynamics. In fact, the notion of this kind of viscosity corresponds to a phenomenologicalissue of perturbations to the thermodynamical equilibrium. In this sense, we remark that,if s > / , the dissipative effects become dominant and non-perturbative. Moreover, if weassume the viscous parameter s < / , the dynamics of the early Universe is characterized byan expansion via a power-law a ( t ) ∼ t / γ starting from a perfect fluid Friedmann singularityat t = 0 (here γ is identify by the relation p = ( γ − ǫ ). After this first stage of evolu-tion, where viscosity does not affect at all the dynamics, the Universe inflates in the limit t → ∞ ( i.e. out of our approximation scheme) to a viscous deSitter solution characterizedby a ( t ) ∼ e H t , H = √ ǫ / / ( ζ √ /γ ) / (1 − s ) [15, 16]. Since, in this work, we deal withthe asymptotic limit t →
0, we only treat the case s = / in order to quantitatively includedissipative effects in the primordial dynamics. Zeroth-order Cloud Dynamics
We now clarify why the choice s = / is appropriate toa consistent treatment of the asymptotic viscous collapse. We start by observing that bulk11iscous effects can be treated in a predictive way only if they behave as small corrections tothe thermodynamical system. To this respect we have to require that the asymptotic collapseis yet appropriately described by the non viscous background flow. Taking into account eq.(24) it is easy to infer that in the asymptotic limit as r →
0, the non viscous behavior v ∼ / √ r is preserved only if s / ; in fact, in correspondence to this restriction, the viscouscorrection, behaving like O ( r − s +1 / ), is negligible with respect to the leading order O ( r − )when the singularity is approached. On the other hand, it is immediate to verify that, as( − t ) →
0, the viscous terms in ˙ δρ and in δ̺ respectively are negligible and the perturbationdynamics (32) matches asymptotically the non-viscous Hunter result if s < / . Matchingtogether the above considerations for the zeroth- and first-order respectively, we see that s = / is the only physical value which does not affect the background dynamics but makesimportant the viscous corrections in the asymptotic behavior of the density contrasts. Validity of the Newtonian Approximation
Since the analysis of the cloud collapse ad-dresses Newtonian dynamics while the cloud approaches the extreme collapse, it is relevantto precise the conditions which ensure the validity of such a scheme. The request that theshell corresponding to the radial coordinate r lives in the Newtonian paradigm leads to im-pose that it remains greater than its own Schwarzschild Radius , r ( t ) ≫ GM ( a ) , where M ( a ) = ρ ( πa ). About the dynamics of a physical perturbations scale l = (2 π/k ) cosβ (here cosβ plays the same role of a cosmic scale factor), its Newtonian evolution is ensuredby the linear behavior, as soon as, the Schwarzschild condition for the background holds.More precisely a perturbations scale is Newtonian if its size is much smaller than the typicalspace-time curvature length, but for a weak gravitational field this requirement must haveno physical relevance. To explicit such a condition, we require that the physical perturba-tions scale is much greater than its own Schwarzschild Radius, which leads to the inequality k ≫ χ ( − t ) − / , where χ = (cid:2) (2 π ) Gρ (3 A ) − / (cid:3) / . Such a condition tells us which modesare Newtonian within the shell. References [1] V.A. Belinskii and I.M. Khalatnikov,
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