Dynamic model of spherical perturbations in the Friedman universe. III. Automodel solutions
aa r X i v : . [ g r- q c ] J a n Dynamic model of spherical perturbations in theFriedman universe. III. Automodel solutions.
Yu.G. Ignatyev, N. ElmakhiTatar State Humanitarity Pedagocical University1 Mezhlauk St., Kazan 420021, Russia
Abstract
A class of exact spherically symmetric perturbations of retarding auto-model solutions linearized around Friedman background of Einstein equa-tions for an ideal fluid with an arbitrary barotrope value is obtained andinvestigated.
In the previous works of the authors [1,2] a class of exact retarding solutionsfor linear spherical perturbations of the Friedman universe with an ultrarela-tivistic equation of state of the ideal fluid filling it, corresponding to the centralsingular source presence and having the sight of polinoms by radial variable,was obtained. At that it was noted, by zero boarding conditions at the soundhorizon for the C class metrics perturbations the energy dense perturbationshave the first genus break at the sound horizon. In this paper we shall studythe retarding solutions in detail by extending the investigations range to theequations of state of the fluid with an arbitrary barotrope coefficient κ .So we shall investigate the retarding solutions of the evolutionary equationfor the spherical perturbations [1] (84):¨Ψ + 2 η ˙Ψ − κ )(1 + 3 κ ) Ψ η − κ Ψ ′′ = 0 . (1)with boarding conditions at the sound horizon corresponding to the zero valuesof the metrics perturbations and its first radial derivativesΣ : r = √ κη. (2)Ψ( r, η ) | r √ κη = µ ( η ); Ψ ′ ( r, η ) | r √ κη = 0 , (3)where µ ( η ) is the central source singular mass, so that the metrics componentperturbation g is equal to δg = a ( η ) δν, δν = 2 Φ( r, η ) ar = 2 Ψ( r, η ) − µ ( η ) ar , (4)moreover the function Ψ( r, η ) corresponds to the nonsingular part of the poten-tial: Ψ(0 , η ) = 0 . (5)1emporal mass evolution is described by the equation¨ µ + 2 η ˙ µ − κ )(1 + 3 κ ) µη = 0 , (6)which has the solution µ = µ + η κ + µ − η − κ )1+3 κ , (1 + κ ) = 0; (7)(for detail see the previous works [1,2]).The potential function Ψ( r, η ) and the scalar µ ( η ) completely determine theenergy density perturbations and the fluid velocity δεε = − πra ε (cid:18) aa ˙Φ − Ψ ′′ (cid:19) , (8)(1 + κ ) v = − πra ε ∂∂r ˙Φ r , (9)where ε ( η ) is a non-perturbed energy density of the Friedman universe ε ∼ η − κ )1+3 κ ; a ∼ η κ ; ε a ∼ η − κ κ . (10) So we shall look for solutions of the evolutionary equations for the perturbationswith zero boarding conditions at the zero sound horizon (2). Supposing that in(1): Ψ( r, η ) = η α G α ( z ) , (11)where z = r √ κη , (12)we come to the automodel solutions class and get a common differential equationfor the function G α ( z ):(1 − z ) G ′′ α ( z ) + 2 αz G ′ α ( z ) + (cid:20) κ )(1 + 3 κ ) − α (1 + α ) (cid:21) G ( z ) α = 0 . (13)The potential function Ψ( r, η ) must be a combination of the private solutionsΨ( r, η ) = X α η α G α ( z ) . (14)2rom (3) and (7) it follows that by α = 0 this combination can have two membersonly Ψ( r, η ) = G + ( z ) η κ + G − ( z ) η − κ )1+3 κ , (15)at that in consequence of the boarding conditions (3) the functions G ± ( z ) mustsatisfy the following boarding conditions G ± (1) = µ ± ; G ′± (1) = 0 . (16)In particular from (3) and (8) it follows immediately that in the case α = 0the zero mass of the singular source corresponds to the pointed out class ofsolutions.At the parameter α arbitrary values the general solution of the linear differentialequation (13) is expressed by the Legendre functions, P µν ( z ), and the adjointLegendre functions Q µν : G ( z ) = (cid:18) − z z (cid:19) α +12 " C P α +13(1 − κ )2(1+ κ ) ( z ) + C Q α +13(1 − κ )2(1+ κ ) ( z ) . (17) µ = 0 ) The mentioned above is just for the formal general solution (17) at an arbitraryvalue of the parameter α . However, in our particular case (15) the values of theparameter α : α = (cid:18)
21 + 3 κ , − κ )1 + 3 κ (cid:19) (18)are the quadratic equation roots simultaneously6(1 + κ )(1 + 3 κ ) − α (1 + α ) = 0 . (19)Therefore in this case the equation (13) degenerates into a simpler one(1 − z ) G ′′ α ( z ) + 2 αz G ′ α ( z ) = 0 , (20)However, integrating it we obtain the following G ′ α ( z ) = C (1 − z ) α . (21)Comparing the second boarding condition (16) with the expression (21) wesee that for the fulfillment of the zero conditions for the first derivative of thepotential at the zero sound horizon it is necessary that α >
0, that is The For example, see [7] and [8]. η = 0; thus, =1 to provide a smooth sewing together of the solutionwith the Friedman one at the zero sound horizon it is necessary that µ − = 0 . (22)In the case of α > α > C class at the sound horizon.Now integrating the equation (21) formally and considering the conditions atthe beginning of the coordinates (5), according to which G (0) = 0 , (23)we find its formal solution within the whole interval of the values r = [0 , + ∞ ): G ( κ, z ) = C zF (cid:16) , − κ , , z (cid:17) , ( z √ π Γ (cid:16) κ + 1 (cid:17) (cid:16) κ + (cid:17) + Z ln( z + √ z − κ +1 xdx, ( z > , (24)where F ( a, b, c, x ) is a hyper-geometrical function (for example ,see [7]): F ( α, β, γ, z ) = Γ( γ )Γ( β )Γ( γ − β ) Z t β − (1 − t ) γ − β − (1 − tz ) − α , (25) ℜ ( γ ) > ℜ ( β ) > | arg(1 − z) | < π. At that the useful limitary relation is just [7]:lim z → − F ( α, β, γ, z ) = Γ( γ )Γ( γ − α − β )Γ( γ − α )Γ( γ − β ) , ℜ ( γ − α − β ) >
0; (26)where Γ( x ) is Γ-function. In Fig. 1 the solutions (24) for values series of thebarotrope coefficient are shown. 4 ig. 1. Normalized function G ( k, z ) calculated with the formula(24) at the value (27) of the constant C and µ = 1. In the left partof the figure from bottom up κ = 0; ; ; ; 1.In terms of the obtained solution (24) it is easy to see G ( − z ) = G ( z ), that isthe function Ψ( r, η ) is an odd function of a radial variable really, it was shownin Ref. [2]. Considering the first boundary condition (16), we find the constant C : C = µ + (cid:16) κ + (cid:17) √ π Γ (cid:16) κ + 1 (cid:17) (27)and thus, finally we obtain the automodel C class solution corresponding to thezero boundary conditions at the sound horizonΨ( r, η, κ ) = µ + η κ × (cid:16) κ + (cid:17) √ π Γ (cid:16) κ + 1 (cid:17) r √ κη F (cid:18) , −
21 + 3 κ , , r κη (cid:19) , ( r √ κη );1 , ( r > √ κη ) , (28)Taking into consideration the relation (28) according to (4) we obtain the scalarfunction of the metrics perturbation δν :5 ν ( r, η ) = − µ + r × − (cid:16) κ + (cid:17) √ π Γ (cid:16) κ + 1 (cid:17) r √ κη F (cid:18) , −
21 + 3 κ , , r κη (cid:19) χ ( √ κη − r ) , (29)where χ ( x ) is the Heavyside function: χ ( x ) = { , x
0; 1 , x > . (30)In the private values series of the barotrope coefficient κ the obtained solutionis expressed in elementary functions κ = 1 / G (1 / , z ) = z − z ;Ψ = 32 µ + η (cid:18) z − z (cid:19) ; δν = − µ + r " − r √ η + 12 (cid:18) r √ η (cid:19) ; (31)Being auto-model the solution (31) coincides with the general solution in theform of the degree series, earlier obtained in the works [3,4,5,6], it confirms thecorrectness of the proved in [2] theorem about the retarding solution uniquenessin the case of the ultra-relativistic state equation once more. κ = 1: fluid withextremely stiff state equation G (1 , z ) = 12 z p − z + 12 arcsin z ;Ψ = 2 µ + π √ η rη s − r η + 2 arcsin rη ! ; δν = − µ + r " − π rη s − r η + arcsin rη ! . (32)Note, the first derivative conversion by a radial variable at the zero soundhorizon in the obtained solutions at α > Now supposing that in (11) α = 0, we get the following equation instead of (13)(1 − z ) G ′′ ( z ) + 6(1 + κ )(1 + 3 κ ) G ( z ) = 0 . (33)The general solution of this equation is a linear combination of the hyper-geometrical functions 6 ( κ, z ) = (1 − z ) × (cid:20) zC F (cid:18) κ κ , κ κ , , z (cid:19) + C F (cid:18) κ κ , κ −
12 + 6 κ , , z (cid:19)(cid:21) . (34)If the geometrical functions F (cid:18) κ κ , κ κ , , z (cid:19) and F (cid:18) κ κ , κ −
12 + 6 κ , , z (cid:19) remained final at the sound horizon z = 1, the general solution (33) would auto-matically turn into zero at the sound horizon, then the zero boundary conditionsat the sound horizon (16) would be in principle satisfied by the choice of con-stants in the general solution (33). However the pointed out hyper-geometricalfunctions have some peculiarities at the sound horizon.Really, as it is easy to see, the parameters α, β, γ of the hyper-geometrical func-tions of this linear combination satisfy the condition (See the hyper-geometricalfunction definition (26).) α + β − γ = 1 , (35)At that for the first member of this combination the additional condition isfulfilled γ − β = −
11 + 3 κ , (36)at the same time for the second one it is γ − β = + 11 + 3 κ . (37)In this case it is necessary to use the functional relation for the hyper-geometricalfunction [7], which conformably to (35) is given with the formula F ( α, β, α + β − , z ) = 1(1 − z ) F ( α − , β − , α + β − N (38)Using this relation in the formula (34) let us rewrite the general solution in theform G ( κ, z ) = (1 + z ) (cid:20) zC F (cid:18) −
11 + 3 κ , κ )2(1 + 3 κ ) , , z (cid:19) + C F (cid:18)
11 + 3 κ , κ )2(1 + 3 κ ) , − , z (cid:19)(cid:21) . (39)7alculating the hyper-geometrical functions values in the right part of Eq. (39)at the sound horizon we see these functions have peculiarities at the soundhorizon. Therefore the auto-model solution satisfying the zero boundary condi-tions at the sound horizon in the case under investigation is the trivial solution C = C = 0 only. Summarizing this subsection let us formulate the theorem Theorem.
There are no retarding spherically-symmetric auto-model solutionsof the equation (1) satisfying the zero boundary conditions (16) at the soundhorizon without the central particle-like source ( µ = 0 ). The proved theorem is analogues to the theorem about the analytical solutionof the Laplace equation in the spherical symmetry case.
Let us come over to the analyses of the obtained auto-model solutions in theparticle-like source case. For that let us use the expression (8) of the relativeenergy density of perturbation and the expression (9) of the radial mediumvelocity in the perturbation. At that we shall need the expressions for the firstand second derivatives of the potential functions. Considering the definitions(11) and (12) and the relations (20) and (27) we shall get expressions for thefirst and second radial derivatives of the potential functions Ψ( r, η )Ψ ′ r ( r, η ) = µ + η − κ κ (cid:16) κ + (cid:17) √ κπ Γ (cid:16) κ + 1 (cid:17) (1 − z ) κ ; (40)Ψ ′′ rr ( r, η ) = − µ + η − κ κ √ π Γ (cid:16) κ + (cid:17) κ (1 + 3 κ )Γ (cid:16) κ + 1 (cid:17) (1 − z ) − κ κ , (41)and for the temporal derivative of the function Φ( r, η ) also˙Φ( r, η ) = η − κ κ G ( k, z ) − µ + κ − µ + Γ (cid:16) κ + (cid:17) √ π Γ (cid:16) κ + 1 (cid:17) z (1 − z ) κ , (42)here the function G ( κ, z ) is determined by the relation (24) with the constant C from (27).From the given expressions it follows: by κ > − / r, η ) and Φ( r, η ) turn intozero at the sound horizon ∂∂r Ψ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = ∂∂r Φ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = 0; (1 + 3 κ > ∂η Ψ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = ∂∂η Φ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = 0; (1 + 3 κ > . (44)At κ < / ∂ ∂r Ψ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = ∂ ∂r Φ( r, η ) (cid:12)(cid:12)(cid:12)(cid:12) r = √ kη = 0; (1 − κ > . (45)By κ = 1 / κ > / Calculating the relative density of the perturbation energy by the formula (8)considering the relations (10) and (40),(41) we get finally δεε = − zηπ √ κ (1 + 3 κ ) × G ( k, z ) − µ + κ − µ + Γ (cid:16) κ κ ) (cid:17) √ π Γ (cid:16) κ )1+3 κ + 1 (cid:17) z (1 − z ) κ ++ 2 µ + Γ (cid:16) κ κ ) (cid:17) √ π Γ (cid:16) κ )1+3 κ + 1 (cid:17) (1 − z ) − κ κ χ (1 − z ) = µ + η ∆( z ) χ (1 − z ) , (46)where the reduced relative density of the perturbation energy ∆( z ) is introduced.It can be strictly shown that ∆( z ) > . From this expression it is seen that by the time the profile form of theperturbation energy density does not change relatively the dimensionless radialvariable z = r/ √ kη and the relative density of the perturbation energy decreasesin inverse proportion to the temporal variable η (Fig.2).At that in the terms of the common radial variable r the energy densityperturbation profile is deformed. For example in Fig.3 the evolution of therelative energy density perturbation profile at the barotrope index κ = 1 / κ < / κ = 1 / κ > / ig. 2. It is the evolution of thereduced relative density of the per-turbation energy ∆(z) calculated bythe formula (45) as the function z .From bottom up in fine line κ =1 /
6; 1 /
5; 1 /
4; 1 /
3; the heavy line cor-responds to κ = 1 / Fig. 3.
It is the evolution ofthe reduced relative density of theenergy ∆(z) calculated by the for-mula (45) as the function r at κ =1 /
6. From the left to the right η =1; 2; 4; 8; 12; 16.Let us find out the physical sense of the obtained solution. The energy perturba-tion corresponding to the nonsingular part of the potential function is describedby the formula δE = 4 πa Z √ κη δεr dr. Hence considering (10) we get δE = 4 πη − κ κ Z √ κη δεε r dr. Coming over to the dimensionless variable z in the integral by the formula r = √ κηz, we bring it to the form δE = 4 πµ + η κ κ / Z ∆( z ) dz ∼ − m ( η ) , (47)where (see (7)): m ( η ) = µ + η κ (48)is the mass of the central singular particle-like source. Thus, the full energyin the included in the nonsingular mode of the perturbation is negative andproportional to the mass of the particle-like source.10 .3 Evolution of the fluid radial velocity in the sphericalperturbation Fulfilling the analogues calculations we get the expression for the radial velocityfrom (9) v = 38 πη κ / (1 + 3 κ ) (cid:20) G ( k, z ) − µ + z − µ + Γ (cid:16) κ κ ) (cid:17) √ π Γ (cid:16) κ )1+3 κ + 1 (cid:17) (1 − z ) κ z − µ + Γ (cid:16) κ κ ) (cid:17) √ π Γ (cid:16) κ )1+3 κ + 1 (cid:17) (1 − z ) − κ κ = 1 η Υ( z ) . (49)From this expression it is also seen that the radial velocity is negative and itsprofile remains constant within the scale z , and the absolute value of the velocitydrops in inverse proportion to η . Fig. 4.
Dependence of the profile of the perturbation radialvelocity Υ( z ) on the barotrop coefficient at κ = 1 /
6. From bottomup κ = 1 /
6; 1 /
4; 1 /
3; 1 /
2; 1