Asymptotic solution for expanding universe with matter-dominated evolution
PPrepared for submission to JCAP
Asymptotic solution for expandinguniverse with matter-dominatedevolution
Ž. Mijajlović, N. Pejović, and V. Radović
Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, SerbiaE-mail: [email protected], [email protected], [email protected]
Abstract.
We applied the theory of regularly varying functions to the analysis of the cos-mological parameters for the Λ CDM model with the matter dominated evolution. Carroll etal. proved in 1992 that for this type of universe with the curvature k = 0 , − , the expression H ( t ) t ( H ( t ) is the Hubble parameter) depends solely on the density parameter Ω( t ) . Usingthis result and the theory of regular variation we infer for such universe the complete asymp-totics of all main cosmological parameters. More specifically, the following is derived. If thelimit ω = lim t →∞ Ω( t ) does exist and ω (cid:54) = 0 then the cosmological constant Λ is equal to .If ω = 0 then for the expansion scale factor a ( t ) we have a ( t ) ∼ e √ Λ / . On the other hand,if the limit lim t →∞ Ω( t ) does not exist then a ( t ) bounces between two power functions andtherefore has infinitely many flexion points. Hence, the deceleration parameter in this casechanges the sign infinitely many times. Corresponding author. a r X i v : . [ m a t h - ph ] N ov ontents Γ Ω( t ) and regular variation 8 lim t →∞ Ω( t ) lim t →∞ Ω( t ) In our papers Mijajlović et al. [27] and Mijajlović, Pejović & Marić [28], we applied the theoryof regularly varying functions in the asymptotic analysis of cosmological parameters of theexpanding universe. The main aim of the present paper is to apply this technique in thestudy of the Λ CDM model with a matter dominated evolution. As a result we obtainedthe complete asymptotics of solutions of Friedmann equations and the related cosmologicalparameters. Our secondary goal is to present further techniques of the theory of regularvariation as a natural method in the theoretical studies in cosmology.The paper is organized as follows. In the introduction we present some basic facts onFriedmann equations, short history and elements of regular variation needed in the rest ofthe paper. Also, properties of the constant Γ introduced in Mijajlović et al. [27] are reviewed.This constant and the related operator M will have to play the crucial role in our analysis ofasymptotics of cosmological parameters. Due to the variety of appearance in the literature,in the section notation and the meaning od cosmological parameters are fixed. The section is central in this paper. There we inferred complete asymptotics of cosmological parametersfor a matter dominated Λ CDM model. By complete inference we mean that we resolved theasymptotics when the limit lim t →∞ Ω( t ) exists, but in the opposite case, too. This analysisis based on Carroll, Press and Turner formulas for Ω( t ) [9] for this type of universe. In thelast section we demonstrated a use of the theory of regular variation in possible descriptionsof cosmological parameters of the dual universe.We remind that the cosmological parameters are solutions of Friedmann equations: (cid:32) ˙ RR (cid:33) = 8 πG ρ − kc R , Friedmann equation , ¨ RR = − πG (cid:18) ρ + 3 pc (cid:19) , Acceleration equation , ˙ ρ + 3 ˙ RR (cid:16) ρ + pc (cid:17) = 0 , Fluid equation . (1.1)– 1 –unctions appearing in these equations are the expansion scale factor R = R ( t ) , theenergy density ρ = ρ ( t ) and the pressure of the material in the universe p = p ( t ) . Weshow that all these parameters, including the Hubble parameter H = H ( t ) = ˙ R/R anddeceleration parameter q = q ( t ) = − R ¨ R/ ˙ R , are regularly varying functions under standardcircumstances. The scale factor R is often normalized in respect to the present epoch by R = R a ( t ) , where R = R ( t ) and t is a certain fixed time moment at the present. If itis assumed that R ( t ) is slowly varying in the present epoch then a ( t ) is the normalized scalefactor having the value one at the present. Observe that then the first Friedmann equationin a ( t ) looks like: (cid:18) ˙ aa (cid:19) = 8 πG ρ − kc R a , (1.2)while the fluid equation and the acceleration equation are invariant under the substitution R ( t ) → a ( t ) . In most cases we shall refer to a ( t ) instead of R ( t ) .The system (1.1) involves three unknown functions, but only two of them are indepen-dent, for example the first and the third equation. However the acceleration equation iscentral in our study of asymptotics of solutions of Friedmann equations for several reasons.For instance it does not depend from the curvature index k . Further, the theory of regu-larly varying solutions of such type of equations can be applied successfully in the analysis ofFriedmann equations even if the cosmological constant Λ is added: (cid:18) ˙ aa (cid:19) = 8 πG ρ − kc R a + Λ3 , ¨ aa = − πG (cid:18) ρ + 3 pc (cid:19) + Λ3 . (1.3)Namely, under the transformations ρ (cid:48) = ρ + Λ / (8 πG ) , p (cid:48) = p − Λ / (8 πG ) the Friedmannequations (1.1) are invariant, while the fluid equation is not affected by the parameter Λ .Theory of regularly varying functions is introduced mainly for studying behaviour ofreal functions at infinity and also the functions satisfying the power law. This theory wasstarted in the thirties of the last century by J. Karamata in Karamata [16]. Many otherauthors continued to develop it [5, 35]. At the present time, this theory is used in many areasof mathematics particularly in the asymptotic analysis of functions and probability theory.There were also some uses of this theory in cosmology, particularly in the study of asymptoticbehaviour of cosmological parameters, e.g. Mijajlović et al. [27], Mijajlović, Pejović & Marić[28], but also by Molchanov, Surgailis & Woyczynski [29], Stern [37]. Barrow [6] and Barrow& Shaw [7] had a similar approach in studies of asymptotic behaviour of solutions to theEinstein equations describing expanding universes. They used there a theory of Hardy andFowler which preceded the theory of regular variation. We give a short review of main notionsrelated to the regular variation and some extensions of this theory that we shall need in therest of the paper. A real positive continuous function L ( t ) defined for x > x which satisfies L ( λt ) L ( t ) → t → ∞ , for each real λ > , (1.4)– 2 –s called a slowly varying (SV) function. Continuing works of G.H. Hardy, J.L. Littlewoodand E. Landau, Karamata Karamata [16] originally defined and studied this notion for con-tinuous functions. Later this theory was extended to measurable functions. Due to physicalconstraints, we assume here that all functions are continuously differentiable. Definition 1.
A function F(t) is said to satisfy the generalized power law if F ( t ) = t r L ( t ) (1.5) where L ( t ) is a slowly varying function and r is a real constant. Logarithmic function ln( x ) and iterated logarithmic functions ln( . . . ln( x ) . . . ) are exam-ples of slowly varying functions. More complicated examples are provided in Bingham, Goldie& Teugels [5], Seneta [35] and Marić [25].A positive continuous function F defined for t > t , is a regularly varying (RV) functionof the index r , if and only if it satisfies F ( λt ) F ( t ) → λ r as t → ∞ , for each λ > . (1.6)It immediately follows that a regularly varying function F ( t ) has the form (1.5). Therefore F ( t ) is regularly varying if and only if it satisfies the generalized power law. By R α we denotethe class of regularly varying functions of index α . Hence R is the class of all slowly varyingfunctions. By Z we shall denote the class of zero functions at ∞ , i.e. ε ∈ Z if and only if lim t → + ∞ ε ( t ) = 0 . The following representation theorem [16] describes the fundamental propertyof these functions. Theorem 1.
Representation theorem L ∈ R is slowly varying function if and only if thereare measurable functions h ( x ) , ε ∈ Z and b ∈ R so that L ( x ) = h ( x ) e (cid:82) xb ε ( t ) t dt , x ≥ b, (1.7) and h ( x ) → h as x → ∞ , h is a positive constant. If h ( x ) is a constant function, then L ( x ) is called normalized. A regularly varyingfunction having normalized slowly varying part will be also called normalized. Let N denotethe class of normalized slowly varying functions. The next fact on N -functions will be usefulfor our later discussion. If L ∈ N and there is ¨ L , then ε in (1.7) has the first order derivative ˙ ε . This follows from the identity ε ( t ) = t ˙ L ( t ) /L ( t ) .There are various classes of positive measurable functions with similar asymptotic be-haviour to regularly varying function such as the class ER – extended regularly varying func-tions (or Matuszewska class of functions), OR – O -regularly varying functions, or recentlyintroduced [8] interesting class M . Here we shall particularly use the classes ER and OR.Extensive literature on these functions is available, e.g. Aljančić & Arandjelović [2], Bingham,Goldie & Teugels [5], Djurčić [11], Seneta [35]. We review here, following Bingham, Goldie &Teugels [5], their definitions and very basic notions and properties related to these functions.As in the case of regular variation we assume that all mentioned functions are continuous.The limit in (1.4) does not exist always for an arbitrary function, but the limit superiorand the limit inferior do exist. So let f ∗ ( λ ) = lim sup x →∞ f ( λx ) f ( x ) , f ∗ ( λ ) = lim inf x →∞ f ( λx ) f ( x ) (1.8)– 3 –ote that the difference f ∗ ( λ ) − f ∗ ( λ ) measures the oscillation of f ( λx ) /f ( x ) at infinity. Itis said that a real positive measurable function f belongs to the ER class if and only if thereare constants d and c such that λ d ≤ f ∗ ( λ ) ≤ f ∗ ( λ ) ≤ λ c , λ ≥ . (1.9)The class OR functions is the set of positive measurable f such that < f ∗ ( λ ) ≤ f ∗ ( λ ) < ∞ , λ ≥ . (1.10)We see at once that RV ⊆ ER ⊆ OR. There are examples that show that both extensions areproper. Note that f is RV if and only if f ∗ ( λ ) = f ∗ ( λ ) , λ ≥ . Therefore, RV functions areexactly OR (or ER) functions which do not oscillate at infinity. Proposition 1.
The following statements are equivalent for a real function f :1. f belongs to the class ER.2. f has the representation: f ( x ) = exp( C + η ( x ) + (cid:82) x ξ ( t ) dt/t ) , x ≥ , with C constant, η ( x ) → as x → ∞ , ξ bounded and ξ , η both measurable. Similar representation theorem holds for OR functions: f ∈ OR if and only if f ( x ) = exp( η ( x ) + (cid:90) x ξ ( t ) dt/t ) , x ≥ , (1.11)where η ( x ) and ξ ( x ) are measurable and bounded. Γ In our analysis an important role will have a constant Γ which we introduced in Mijajlović etal. [27]. This constant is related to the Friedmann equations, particularly to the accelerationequation which is considered there as a second order linear differential equation ¨ a + µ ( t ) t a = 0 . (1.12)Here µ ( t ) = 4 πG t (cid:18) ¯ ρ ( t ) + 3¯ p ( t ) c (cid:19) , (1.13)where ¯ ρ ( t ) and ¯ p ( t ) are some particular solutions of Friedmann equations. Then Γ is definedas the integral limit Γ = Γ µ = lim x →∞ x (cid:90) ∞ x µ ( t ) t dt ≡ M ( µ ) . (1.14)assuming the existence of the limit integral on the right-hand side of (1.14). The function M ( µ ) is a linear functional M : M → R , where M is the space of all real functions satisfyingthe above integral condition (1.14). M is called a Marić class of functions (see Mijajlović et al.[27]). If this integral limit does not exist, we say that Γ does not exist. The constant Γ appearsin description of cosmological parameters by use of the theory of regularly varying solutionsof linear second order differential equations, see Mijajlović et al. [27], Mijajlović, Pejović &Marić [28] and Marić [25]. Determining the values of Γ one can obtain the asymptotical– 4 –ehaviour of the solutions a ( t ) , ρ ( t ) and p ( t ) . The limit integral in (1.14) in general is noteasy to compute. However, as lim t →∞ µ ( t ) = Γ (1.15)implies (1.14), we see that (1.15) gives a useful sufficient condition for the existence of regularsolutions of the equation (1.12).The significance of the integral limit (1.14) is seen in the following facts. We have shownin Mijajlović et al. [27] that for any function µ ∈ M such that Γ µ < / , the Friedmannequations has a normalized regularly varying solutions and the universe modelled by thesesolutions must be spatially flat or open. On the other hand, if Γ µ > / then the universe isoscillatory. Therefore, Γ provides a kind of sharp threshold, or cut-off point, from which theoscillation of a ( t ) takes place [13]. It was also proved that Γ < / implies a weak form ofthe equation of state. The converse also holds. Namely, we show: Proposition 2.
For any normalized regularly varying solution of Friedman equations theintegral limit (1.14) must exist and Γ ≤ / . Proof . So suppose a ( t ) = t α L is a solution of Friedmann equations where L is anormalized slowly regular function and let β = 1 − α . In accordance with the representation(1.7), Theorem 1, of slowly varying functions and as L is normalized, we have ˙ L = εL/t , ε ∈ Z . From there immediately follows the next derivation: ˙ a = αt α − L + t α ˙ L = ( α + ε ) t α − L, ¨ a = ˙ εt α − L − ( α + ε )( β − ε ) t α − L, wherefrom we obtain ¨ aa = ( ˙ εt − ( α + ε )( β − ε )) /t . As a ( t ) is a solution of Friedmann equations, in particular of the equation (1.12), it must be µ = − ˙ εt + ( α + ε )( β − ε ) and also M ( µ ) = M ( − ˙ εt + ( α + ε )( β − ε )) = M ( − ˙ εt ) + M (( α + ε )( β − ε )) . Further, M ( − ˙ εt ) = 0 (see proof of Theorem 2.2 in Mijajlović et al. [27]). Since ( α + ε )( β − ε ) → αβ as t → ∞ , it follows M (( α + ε )( β − ε )) = αβ . Hence M ( µ ) = αβ and so Γ = αβ , whatproves the existence of Γ . Further, if α, β ≥ then √ αβ ≤ ( α + β ) / / , hence Γ ≤ / .If α < then β > , hence Γ < ≤ / .Observe that α and β are roots of the equation x − x + Γ = 0 which defines thefundamental solutions of the acceleration equation, see Marić [25], Mijajlović et al. [27]. In this section we review some basic cosmological parameters and constants and their mostlywell known properties. Our only aim here is to establish their exact notation and meaningused in the rest of the paper.A cosmological parameter basically is a real function P = P ( t ) of the time variable t andit usually represents an essential physical value related to the standard cosmological model.– 5 –n the literature there are variations of their notation and even definitions that may lead toambiguities. Therefore, we fix here their notation, at least for some of the parameters, butwe assume their standard meaning. Also, for the simplicity of the notation from now on forthe speed of light we shall take c = 1 . ρ c = ρ c ( t ) = 3 H πG critical density , Ω = Ω( t ) = ρ/ρ c density parameter , Ω Λ = Ω Λ ( t ) = Λ3 H Λ density parameter , Ω k = Ω k ( t ) = − kR H curvature density parameter ,q = q ( t ) = − ¨ aaH deceleration parameter. (2.1)Derived constants are obtained from cosmological parameters fixing their values at t = t .The time value t usually stands for a certain moment in the present epoch. If P is acosmological parameter then we often write shortly P instead of P ( t ) . Derived constantsinherit their names from the parameters from which they were obtained, e.g. Hubble constant H . Here are some examples of derived constants. Ω = 8 πG H ρ , Ω Λ0 = Λ3 H , Ω k = − kR H (2.2)The following form of Fiedmann equation is useful in determining relations linking basiccosmological parameters for pressureless universe [9]. It’s usefulness follows from the explicitappearance of a term which relates the scale factor and the density for this type of universe.This theorem and it’s proof also illustrate the use of derived cosmological constants as thederived equation is in the parametrized form. For the simplicity of the computation we tookthe speed of light for unit, hence c = 1 . Theorem 2.
The first Friedmann equation with non-zero cosmological constant Λ is equiva-lent to the following equation: ˙ a + πG ( ρ /a − ρ ) a = H (1 + Ω (1 /a −
1) + Ω Λ0 ( a − . (2.3) Proof If L = ˙ a + 8 πG ρ /a − ρ ) a then using (1.3) and definitions (2.2) of Ω and– 6 – Λ0 we have L = a (cid:18) πG · ρ a + ˙ a a − πG ρ (cid:19) = a (cid:18) πG · ρ a − kR a + Λ3 (cid:19) = H a + 1 a (cid:18) πG ρ − H (cid:19) − kR + Λ3 a = H a + 1 a (cid:18) kR − Λ3 (cid:19) − kR + Λ3 a = H + (cid:18) H + kR − Λ3 (cid:19) (cid:18) a − (cid:19) + Λ3 a − Λ3= H + 8 πG ρ (1 /a −
1) + Λ3 ( a − H (cid:18) πG H ρ (1 /a −
1) + Λ3 H ( a − (cid:19) = H (1 + Ω (1 /a −
1) + Ω Λ0 ( a − . (cid:3) It is well known that a universe is pressureless (matter dominated) if and only if ρ = ρ /a . Therefore, in this case the second term in L vanishes and we have the followingstatement. Corollary [see 9] If matter dominated universe is assumed, then the first Friedmann equa-tion reduces to the following equation with the constant coefficients: ˙ a H = 1 + Ω (1 /a −
1) + Ω Λ0 ( a − (2.4)If a flat universe is assumed, i.e. Ω k = 0 , due to the formula Ω + Ω Λ + Ω k = 1 therelation (2.4) obviously obtains more simple form: ˙ a H = Ω a − + (1 − Ω ) a . (2.5) Note . By careful examination of the proof of Theorem 2, we see that no infinitesimaltransformation is applied on Λ , i.e. only algebraic transformations were used in this proof.We have the same conclusion for formulas (2.4) and (2.5).It is interesting that from (2.4) a rough estimation of a ( t ) can be obtained in regardsto the various epochs in the development of the universe. If the gravitation dominationis assumed, i.e. the rate of the universe expansion is small then the scale factor a is small.Therefore the term /a dominates on the right-hand side of the equation (2.4) and so ˙ a ∝ a − .From this relation one obtains at once a ( t ) ∝ t , a well known asymptotics for a matterdominated universe. On the other side if the dark energy prevails, then the rate of the universeexpansion is large, i.e. now the term a dominates the right-hand side of the equation (2.4).Hence ˙ a /H ∼ Ω Λ0 a , i.e. ( ˙ a/a ) ∼ Λ / , wherefrom we obtain a ( t ) ∼ exp (cid:18)(cid:113) Λ3 t (cid:19) , a wellknown result for this epoch of the universe evolution.– 7 – Density parameter Ω( t ) and regular variation As already noted the two Friedmann equations are not enough to fully solve for the energydensity, the pressure and the scale factor. Hence we need an additional relation which connectsthese parameters. Almost without exceptions equation of state p = f ( ρ ) is assumed, usuallyin the form p = wρ , or p i = w i ρ , if it assumed that the universe is composed from severalcomponents i (see page 6 in "The accelerating universe" ). However, if the pressureless Λ CDMmodel is assumed, the regular variation property immanent to the solutions of the system (1.1)and to other cosmological parameters as well, enables us to use another approach. Namely,we can give without additional assumptions a complete asymptotics for this type of universeusing the theory of regularly varying functions. This section, including the subsections 3.1and 3.2 are devoted to this analysis. Hence, from now on we shall assume a pressurelessuniverse with nonzero cosmological constant Λ . For this type of universe S.M. Carroll, W.H.Press and E.L. Turner developed in Carroll, Press & Turner [9] formulas of the form H ( t ) t = F (Ω) , Ω = Ω( t ) is a density parameter , (3.1)Our aim in this section is to show that under this assumption and from these relation onecan infer asymptotics for cosmological parameters regardless of the convergence of the limitintegral (1.14). The functions F (Ω) are defined as follows. F (Ω) for flat Universe ( k = 0 ): F (Ω) = 23 (1 − Ω) − ln (cid:18) √ − Ω √ Ω (cid:19) (3.2) F (Ω) for open Universe ( k = − ): F (Ω) = 11 − Ω − Ω2(1 − Ω) cosh − (cid:18) − ΩΩ (cid:19) (3.3)Observe that F (Ω) is a dimensionless physical quantity.We note that the functions F (Ω) are easily inferred from (2.4). Acceleration equationwith Λ is written as follows: ¨ aa = − µt + Λ3 (3.4)Taking in (1.13) ¯ ρ = ρ ∗ − Λ8 πG , ¯ p = p ∗ + Λ8 πG . (3.5)the acceleration equation is reduced to the standard form (1.1) where µ is replaced by µ Λ = (3 µ − Λ t ) / πGt ( ρ ∗ + 3 p ∗ ) / . (3.6)In other words ¨ aa = − µ Λ t and q = µ Λ F (Ω) . (3.7) – 8 –t is well known [see 21, and link https://ned.ipac.caltech.edu/level5/Carroll2] that inthe matter dominated universe with the cosmological constant the following holds q = Ω / − Ω Λ . (3.8)Thus, for such universe, by definition of q ( t ) , (3.4) and (3.8) we have Ω / − Ω Λ = q = µ ( tH ) − Λ3 H = µF (Ω) − Ω Λ . k= 0k= -1 ! y Figure 1 . Graphs of y = F (Ω) k= 0k= -1 ! y Figure 2 . Graphs of µ (Ω) = Ω2 F (Ω) Hence (see Figure 2) µ = Ω2 F (Ω) . (3.9)We show how to infer asymptotic formulas for the scale factor a ( t ) using F (Ω) . For this, wenote that the Hubble parameter is the logarithmic derivative of a ( t ) : H ( t ) = ˙ a ( t ) a ( t ) = d ln a ( t ) dt . – 9 –ence a ( t ) = exp (cid:16)(cid:82) tt H ( t ) dt (cid:17) = exp (cid:16)(cid:82) tt H ( t ) tt dt (cid:17) = exp (cid:16)(cid:82) tt F (Ω) t dt (cid:17) . (3.10)We shall prove the following crucial characteristic of the scale factor a ( t ) : Proposition 3.
The function a ( t ) belongs to the class ER . (3.11) Proof
This property will follow from (3.10) if we show that F (Ω( t )) is bounded on aninterval [ t , ∞ ] for some t > . So first suppose that the universe is open. Then the function F (Ω( t )) is obviously bounded, see Figure 1. If the universe is flat, then Ω ∞ = 1 and so F (Ω( t )) is bounded at the infinity, too. Further, the value Ω = 0 . is widely taken forthe present epoch (see e.g. Liddle & Lyth [21]) and is close to the value preferred by theobservation. If we assume that the energy density ρ becomes lower as the age of the universebecomes older, we may suppose that the possible range for the constant Ω ∞ is the interval [0 . , and so F (Ω( t )) is bounded, too. Therefore in all cases we may take that F (Ω( t )) isbounded on the every interval [ t , ∞ ] for every t > . Hence, due to (3.10), is extendedregularly varying function. (cid:3) In deriving asymptotic formula for a ( t ) and other cosmological parameters we shalldistinguish the following cases. The first one is when there is lim t →∞ Ω( t ) . The second one iswhen this limit does not exist. lim t →∞ Ω( t ) In the greatest part of this section we shall discuss cosmological parameters for a pressurelessuniverse assuming that Ω( t ) converges as t → ∞ and for flat universe we shall obtain theircomplete asymptotics. We shall briefly discuss asymptotics for an arbitrary universe, but notassuming the equation od state.Suppose first a pressureless universe and that there is a limit lim t →∞ Ω( t ) = ω . From(3.2) we see that for the flat universe it must be ≤ ω ≤ . We shall distinguish the cases < ω ≤ and ω = 0 . Suppose first < ω ≤ . Then by the continuity of F (Ω) , there exists lim Ω → ω F (Ω) = lim t →∞ F (Ω( t )) = α and by (3.1) H ( t ) = αt + εt , ε ∈ Z . (3.12)Then, using (3.10) it follows a ( t ) a ( t ) = exp (cid:18)(cid:90) tt αt dt + (cid:90) tt εt dt (cid:19) = (cid:18) tt (cid:19) α exp (cid:18)(cid:90) tt εt dt (cid:19) . (3.13)Hence a ( t ) = a ( t ) (cid:18) tt (cid:19) α L ( t ) , t > t > . (3.14)where L ( t ) = exp( (cid:82) tt ε ( t ) t dt ) , ε ( t ) → if t → ∞ . Therefore L ( t ) is slowly regular, and so a ( t ) is a normalized regularly varying function.Note that the functions F (Ω) are defined for < Ω < . If Ω ∼ , then:– 10 – (Ω) = 23 (cid:18) √ − Ω1 + √ Ω (cid:19) + o (1 − Ω) , ( k = 0) .F (Ω) = 2Ω − + 2 √ − ΩΩ + o (1 − Ω) , ( k = − . Hence, in both cases lim Ω → F (Ω) = 2 / as Ω → . It is also easy to see that F (Ω) → + ∞ as Ω → + ( k = 0) ,F (Ω) → → + ( k = − . Therefore, if ω = lim t →∞ Ω( t ) exists where ω > and lim t →∞ F (Ω( t )) = α , then thescale factor a ( t ) satisfies generalized power law: a ( t ) = t α L ( t ) , L ( t ) is slowly varying . According to Mijajlović et al. [27] then all cosmological parameters are uniquely determinedand a form of equation of state holds.We shall consider in more details these parameters for the curvature index k = 0 . Hencewe assume until the end of this section k = 0 and in this analysis we follow definitions (2.1).It will appear that the convergence of Ω( t ) to ω (cid:54) = 0 at infinity is a strong assumption. It’sconsequence is that the cosmological constant is equal to 0. To see this, first observe thatfrom the identity Ω + Ω Λ = 1 it follows that there exists ω Λ = lim t →∞ Ω Λ . Hence ω + ω Λ = 1 . (3.15)As Λ = 3Ω Λ H , by (3.12) there is a zero function ξ = ξ ( t ) such that Λ = 3( ω λ α + ξ ) t . (3.16)Hence Λ = 0 , ω Λ = 0 and ω = 1 since ω λ α + ξ ) t → t → ∞ . (3.17)As lim Ω → F (Ω) = 2 / , by (3.1) we obtain H ( t ) ∼ t and so a ( t ) ∼ ( t/t ) as t → + ∞ ,as expected a well-known result for a matter dominated universe with Λ = 0 .Now we shortly discuss asymptotics fo more interesting case, an arbitrary universe i.e.without assumptions of the matter dominance. However we assume
Λ = 0 , but in thisderivation we do not assume equation of state. In fact we shall obtain a weak form of it.First we note that the threshold constant Γ (see section 1.2) is in relation to accelerationequation and it is defined by (1.14). In Mijajlović et al. [27] is shown that the exponent α ofthe expansion scale factor (3.14) is a solution of x − x + Γ = 0 , hence Γ = α (1 − α ) . (3.18)For the curvature index k = 0 , − it must be Γ ≤ / (see Proposition (2) and commentsthat precedes this proposition). By (3.18) this condition is fulfilled since α ≤ / . Followingideas in Mijajlović et al. [27], we define the equation of state constant w by w = 23 α − . (3.19)– 11 –hen the formulas for other cosmological parameters can be written as follows (see Theorem3.5 in Mijajlović et al. [27]): α = 23(1 + w ) , a ( t ) = a t w ) L ( t ) H ( t ) ∼ w ) t , M ( q ) = 1 + 3 w (3.20)We see that these formulas give the standard solutions of Friedman equations for the case k = 0 . Also, there are functions ˆ w ( t ) , ξ ( t ) and ζ ( t ) such that p = ˆ wρc , (equation of state) (3.21)where ˆ w ( t ) = w − t ˙ ξ + ζ , ξ, ζ ∈ Z . Hence the weak form of the equation of state holds. If t ˙ ξ → as t → ∞ , then ˆ w ( t ) ≈ w , what leads to p = wρc , the standard equation of state andclassical asymptotics for cosmological parameters. In Mijajlović et al. [27] is also found M ( µ ) = Γ = 29 · w (1 + w ) . (3.22)However, there are several evidences measured in the last two decades that are againstof Λ = 0 . The first one is the estimated large value Ω Λ = 0 . ± . , accordingto results published by the Planck Collaboration in 2016 [31], and the second one is theaccelerated universe [32–34]. Hence, it remains to consider other two possibilities, ω = 0 andthe divergent Ω . We shall discuss first Case lim t →∞ Ω( t ) = 0 . Under this assumption and (3.2) we find F (Ω) ∼ (ln(2) − ln(Ω)) for Ω → +0 , or F (Ω) ∼ −
13 ln(Ω) as Ω → +0 . (3.23)Hence, by (3.1) we have immediately H ( t ) ∼ − t ln(Ω) as t → ∞ . (3.24)As H = ˙ a/a we also find for large ta ( t ) = a ( t ) e − (cid:82) tt t dt . (3.25)We can find more specific formula for the scale factor a ( t ) . By formulas (3.9) and (3.23) wesee that µ ∼ Ω ln(Ω) / as Ω → +0 . Therefore (see also Figure 2) lim Ω → +0 µ = 0 . (3.26)Hence, the term µ/t may be neglected in the equation (3.4). Assuming Λ > , the funda-mental solutions of so simplified equation ¨ a/a = Λ / are exp( (cid:112) Λ / t ) and exp( − (cid:112) Λ / t ) . Asthe first fundamental solution dominates at infinity the second one, we find: a ( t ) ∼ exp( (cid:112) Λ / t ) as t → + ∞ . (3.27)Interestingly, we obtained the same asymptotics for the scale factor a ( t ) as in the case ofcosmic inflation, perfectly consistent with the ultimate fate of the universe such as Big Rip.Now we turn to the next section where we consider the non-convergent Ω( t ) at infinity.– 12 – .2 There is no limit lim t →∞ Ω( t ) In this section we shall suppose that the limit lim t →∞ Ω( t ) does not exist. However, by (3.11) a ( t ) belongs to the class ER, hence we shall apply theory of these class of function to thestudy of the asymptotic of a ( t ) .The following simple statement will be useful in the next consideration. Proposition 4.
Let g ( t ) , f ( t ) , h ( t ) be real functions defined on a real interval [ t , ∞ ] andsuppose they satisfy: g ( t ) ≤ f ( t ) ≤ h ( t ) , t ∈ [ t , ∞ ] . Then there is a real function u ( t ) such that f ( t ) = g ( t ) cos( u ( t )) + h ( t ) sin( u ( t )) , t ∈ [ t , ∞ ] . (3.28) If f , g and h are continuously differentiable functions such that g ( t ) < h ( t ) , t ∈ [ t , ∞ ] , then u ( t ) is also continuously differentiable on [ t , ∞ ] . Proof
For any triple a ≤ x ≤ b of real numbers there are α, β ≥ such that α + β = 1 and x = αa + βb . But then there is u such that α = cos( u ) and hence β = sin( u ) . Taking foreach t , a = g ( t ) and b = h ( t ) we obtain (3.28). Further, it is easy to see that from (3.28)follows f = 12 ( g + h ) + 12 ( g − h ) cos(2 u ) (3.29)wherefrom u = 12 arccos (cid:18) h + g − fh − g (cid:19) , (3.30)hence u ( t ) is a continuously differentiable function. (cid:3) We have shown that a ( t ) belongs to the class ER and that F (Ω) is a bounded functionfor t ≥ t for t > . Hence there are real numbers α and β such that α < β and α ≤ Ω( t ) ≤ β .By (3.11) and the representation (3.10) we immediately infer ( t/t ) α ≤ a ( t ) ≤ ( t/t ) β , t ≥ t . (3.31)Then by the previous proposition there is a continually differential function u ( t ) such that a ( t ) = 12 (cid:32)(cid:18) tt (cid:19) α + (cid:18) tt (cid:19) β (cid:33) + 12 (cid:32)(cid:18) tt (cid:19) α − (cid:18) tt (cid:19) β (cid:33) cos(2 u ( t )) (3.32)or by normalizing, i.e putting t = 1 , we obtain somewhat simpler form a ( t ) = 12 (cid:16) t α + t β (cid:17) + 12 (cid:16) t α − t β (cid:17) cos(2 u ) ≡ t α cos( u ) + t β sin( u ) . (3.33)Depending on the nature of the function u ( t ) , there are several possible cases in the evolutionof the scale factor a ( t ) . These cases are depicted on the Figures 3. and 4.If α ≤ or β ≤ then the term in (3.33) containing the negative exponent may beomitted due to the largeness of t . Therefore, in inferring above possibilities we suppose < α < β . Then we have ˙ a = αt α − cos( u ) + βt β − sin( u ) + ( t β − t α ) ˙ u sin(2 u ) , (3.34)– 13 – niverse with increasingscale factor a(t) y= t b y= t a a= a(t)
200 400 600 800 1000 t Hesitating universe y= t b a y= ta= a(t) t ! ! ! ! ! ! y Figure 3 . Evolution of the universe with increasing (top) and hesitating (bottom) scale factor a ( t ) . Expanding universe withoscillating scale factor a(t)on small time scale
Strip AStrip B
40 60 80 t a Expanding universe withoscillating scale factor a(t)on large time scale visitingstrips A and B
Strip AStrip B t ! ! ! ! a Figure 4 . Evolution of the oscillating scale factor a ( t ) on small (top) and large (bottom) timescale. ¨ a = α ( α − t α − cos( u ) + β ( β − t β − sin( u ) +2( βt β − − αt α − ) ˙ u sin(2 u ) + ( t β − t α )(2 ˙ u cos(2 u ) + ¨ u sin(2 u )) (3.35)Looking at (3.32) for large t we can conclude: Proposition 5.
1. The scale factor a ( t ) bounces between bounding functions t α and t β .2. If sin( u ) ≈ , i.e. u ≈ kπ for some integer k then a ( t ) ≈ t α .3. If sin( u ) (cid:54)≈ , then a ( t ) ≈ t α sin( u ) .4. Every jump of a ( t ) between t α and t β contains an inflection point, i.e. a ( t ) change fromconcave to convex, or vice versa. Therefore, ¨ a changes the sign in the future infinitelymany times and so does the deceleration parameter q ( t ) . Hence the universe’s expansionalternatively decelerates and accelerates. – 14 –urther we discuss the monotonicity of a ( t ) at infinity (i.e. for large t ). If in somepoint t , ˙ u ( t ) = 0 , or sin(2 u ( t )) = 0 , then by ( . we have a ( t ) = αt α − cos( u ( t )) + βt β − sin( u ( t )) , so αt α − ≤ ˙ a ( t ) ≤ βt β − . Therefore ˙ a ( t ) > and a ( t ) is increasing in a neighbourhood of t .If ˙ u ( t ) sin(2 u ( t )) (cid:54)≈ , then ˙ a ( t ) ≈ t β ˙ u ( t ) sin(2 u ( t )) . Suppose u ( t ) is a time-like function, i.e. monotonously increasing and unbounded. Then ˙ u ( t ) > and so the sign of ˙ a ( t ) depends solely on the term sin(2 u ( t )) . As u ( t ) is unbounded, sin(2 u ( t )) certainly change the sign and so does ˙ a ( t ) . Hence a ( t ) oscillates between t α and t β and the universe alternatively slows down and increases the expansion. But if u ( t ) is aperiodic function with the period of π , the same one as the function sin(2 u ) , But if ˙ u ( t ) and sin(2 u ( t )) are functions of the same sign then due to (3.34) for the sufficiently large t we have ˙ a ( t ) > , i.e. a ( t ) is increasing.According to the previous discussion, all of the following cases are possible for the scalefactor a ( t ) :1. For large t , a ( t ) is increasing.2. At some intervals ˙ a ( t ) ≈ , i.e. a ( t ) is an almost constant function on these intervals.This case corresponds to the so called Lemaitre hesitating universe.3. The scale factor a ( t ) oscillatory varies between t α and t β .All these situations are depicted on the diagrams of Figures 3. and 4. We also note that if u cos(2 u ) + ¨ u sin(2 u ) (cid:54)≈ , then for large t ¨ a ≈ t β (2 ˙ u cos(2 u ) + ¨ u sin(2 u )) . (3.36)The equation (3.1) tell us that F (Ω) = H ( t ) t . Since H = ˙ a/a , by (3.33) and (3.34) taking γ = α − β we have F (Ω) = αt γ cos( u ) + β sin( u ) + t (1 − t γ ) ˙ u sin(2 u ) t γ cos( u ) + sin( u ) (3.37)For large t we have t γ ≈ as γ < . Therefore, if u (cid:54)≈ kπ , k is an integer, then we have F (Ω) = β + 2 t ˙ u cot( u ) (3.38)On the other hand if u ≈ kπ , then we immediately find F (Ω) ≈ α . Further, if we integratethe equation (3.38) we find ln | sin( u ) | = 12 (cid:90) tt F (Ω) t dt − β ln( t ) , (3.39)wherefrom and (3.10) we obtain a ( t ) ≈ t β sin( u ) for u (cid:54)≈ kπ , an already noted result inProposition 5.3.Finally let us mention the following useful result on ER functions. According to thetheory of regularly varying functions [5] for an extended regularly varying function f , there– 15 –re numbers c (cid:48) ( f ) and d (cid:48) ( f ) , so called Karamata indices and α (cid:48) ( f ) and β (cid:48) ( f ) , so calledMatuszewska indices so that [see 5, , Theorem 2.1.8] λ d (cid:48) ( f ) ≤ f ∗ ( λ ) ≤ λ β (cid:48) ( f ) ≤ λ α (cid:48) ( f ) ≤ f ∗ ( λ ) ≤ λ c (cid:48) ( f ) , λ ≥ . (3.40)Using this property and definition (1.8) of functions f ∗ and f ∗ is is easy to show that thereare numbers α , β , a and b which naturally define strips A and B in the euclidean plane: A = { ( t, y ) ∈ R : t ≥ , t b ≤ y ≤ t β } ,B = { ( t, y ) ∈ R : t ≥ , t a ≤ y ≤ t α } . so that f ( t ) visits (intersects) strips A and B infinitely many times . (3.41)As the scale factor a ( t ) belongs to the class ER, we see that a ( t ) has the property (3.41).This characteristic of a ( t ) is depicted for a short time scale and a large time scale on graphs,the Figure 4.Even if the auxiliary function u ( t ) in the previous analysis is unknown, it has usefulproperties forced by the ER property of a ( t ) an the bounding functions t a , t α , t b and t β . Theseproperties were enough to deduce for divergent case of Ω( t ) , that a ( t ) is oscillating between thetwo indicated strips, has infinitely many flection points and that the deceleration parameter q ( t ) changes the sign infinitely many times. These types of variation of the cosmologicalparameters is difficult to explain by some inner evolution processes of the universe or therelicts from the early universe as it was indicated in the convergent case. Possible explanationis given in the next section. Here we shall discuss certain physical hypotheses for the universe models that might explainvariation of cosmological parameters we described in previous sections. We briefly indicatedsome of these possibilities in [28]. For example, one interesting possibility for the variationsof q ( t ) and p ( t ) for the convergent Ω( t ) could be a kind of reverberation due to the extremelyrapid expansion of the Universe which appeared in the inflationary epoch, about − secondsafter the Big Bang. One can speculate that these variations are the consequences of an echoeffect due to thermalization which appeared when the inflation epoch ended. However adigressive variation for divergent Ω( t ) of a ( t ) and q ( t ) on large time scale is difficult to explainonly with inner processes in the course of the universe evolution. We prefer the idea that thisvariation is an effect of the existence of the dual universe and that it is a resultant of theirmutual interference. We shall discuss this possibility in more details.One of the concepts of modern string theory and hence M-theory is that the big bangwas a collision between two membranes [14, 17, 19]. The outcome was the creation of twouniverses, one in the surface of each membrane. Using the Large Hadron Collider (LHC)located in CERN, some data are collected that might lead to the conclusion that the paralleluniverse exist. Specifically, if the LHC detects the presence of miniature black holes at certainenergy levels, then it is believed [3] that these would be the fingerprints of multiple universes.Collected data are still analyzed, but there are also other research in this direction, e.g.Aguirre, Johnson & Shomer [1].We will not enter here into the full discussion on the existence of the multiverse. But if itis assumed that the parallel universe exists, we can explicitly find, at least for regularly varying– 16 –osmological parameters, a set of formulas that might represent cosmological parameters of adual universe. Our argument for supporting this approach we found in the next theorem andthe symmetry which exists between formulas expressing cosmological parameters for this pairof universes. The symmetry is represented by the Galois group of the associated algebraicequation (4.1). This group and the translation of the set of formulas of the primary universeto it’s dual is also described in this section. We obtain them using the second fundamentalsolution L ( t ) in Howard - Marić theorem [see 25, 27]) applied to the acceleration equation. Theorem 3. (Howard-Marić) Let −∞ < Γ < / , and let α < α be two roots of theequation x − x + Γ = 0 . (4.1) Further let L i , i=1,2 denote two normalized slowly varying functions. Then there are twolinearly independent regularly varying solutions of ¨ y + f ( t ) y = 0 of the form y i ( t ) = t α i L i ( t ) , i = 1 , , (4.2) if and only if lim x →∞ x (cid:90) ∞ x f ( t ) dt = Γ . Moreover, L ( t ) ∼ − α ) L ( t ) . The second fundamental solution and so the dual set of these formulas is determinedby the second root β = 1 − α of the quadratic equation x − x + Γ = 0 appearing in thistheorem. To avoid singularities, we assume α, β (cid:54) = 0 . Now we use β instead of α for the indexof RV solution a ( t ) - deceleration parameter and for determination of other constants andcosmological parameters. As in (3.19) we introduce w β = β − . Then we have the followingsymmetric identity for equation of state parameters: w α + w β + 3 w α w β = 1 (4.3)For our universe we have w = w α , while for the dual universe the corresponding equation ofstate parameter is w β . Then the dual formulas are obtained by replacing α with β and w α with w β in (3.12), (3.21) and (3.20). If one wants to give any physical meaning to the so obtaineddual set of functions, it is rather natural to interpret them as the cosmological parameters ofthe dual universe.As we shall see these two universes are isomorphic in the sense that there is an isomor-phism which maps cosmological parameters into their dual forms. Hence, this isomorphismintroduces a symmetry between cosmological parameters and their dual forms. It wouldmean that both universes have same or similar physics as anticipated in Cai et al. [14]. Inthis derivation we use some elements of the Galois theory. For the basics of this theory thereader may consult for example [20].Our assumption Γ < and that the solutions α and β of the equation (4.1) differ, say α < β , introduces the following kind of symmetry. Let F = R ( t, Γ) be the extension algebraicfield where R is the field of real numbers and t and Γ are letters (variables). It is easy tosee that for such Γ the polynomial x − x + Γ is irreducible over the field F . Hence, theGalois group G of the equation (4.1) is of the order 2 and has a nontrivial automorphism σ . Let α and β be the roots of the polynomial x − x + Γ . Then σ ( α ) = β and σ ( β ) = α .Further, let Γ = · w (1+ w ) where w is a parameter. Then we can take α = w ) and β = w w ) . Let w α ≡ w and w β ≡ − w w . Then σ ( w α ) = w β since w α and w β are rationalexpressions respectively in α and β . Further, the time t and the constant Γ are invariant– 17 –nder σ i.e. σ ( t ) = t and σ (Γ) = Γ since t and Γ are the elements of the ground field F .The cosmological parameters (3.12) and (3.20) are rational expressions of w so if P α is thecorresponding parameter to the solution α , then σ ( P α ) = P β . For example, for the Hubbleparameters we have σ ( H α ) = H β . Hence, not only solutions (isomorphic via σ ) come into thepairs but the sets of all cosmological parameters come as well. At this point one may speculateabout two dual mutually interacting universes having the same time t and the constant Γ andthe conjugated parameters w α and w β connected by the relation (4.3).Of course, there is a question what are the values of the constants appearing in cosmo-logical parameters, for example of w = w α . During the last two decades [30] there was a greatprogress in fundamental measurements of cosmic data and estimation of cosmological con-stants. In particular collected precision data in WMAP (Wilkinson Microwave AnisotropyProbe) mission. enabled accurate testing of cosmological models. It was found that theemerging standard model of cosmology, a flat Λ -dominated universe seeded by a nearly scale-invariant adiabatic Gaussian fluctuations, fits the WMAP data [36]. According to theseobservations, the value of w , also called equation of state of cosmological constant (or thedark energy equation of state), is near − . More precisely, the collected data are consistentwith the density being time-independent as for a simple cosmological constant ( w = − ), withuncertainties in w at the 20% level [38]. Other results from experimental cosmology, suchas the Baryon Oscillation Spectroscopic Survey (BOSS) of Luminous Red Galaxies (LRGs)in the Sloan Digital Sky Survey (SDSS) are also in the favor of w = − , [4], Particle DataGroup: http://pdg.lbl.gov. However, the value w = − yields singularity in (3.20). For such w there is no corresponding α neither Γ . Equation of state is p = − ρc and then by fluidequation we have ˙ ρ = 0 , i.e ρ is constant. This case corresponds to the cosmological constant,so ρ = ρ Λ = Λ8 πG . In the absences of α and β for dual w β of w = w α we may take (4.3) fordefining relation . Putting w α = − in this identity we obtain w β = − . Hence, if the Λ CDMmodel is assumed, dual universe is also equipped wit cosmological constant and its expansionis also governed by dark energy.The other values of w are also considered. For example if w = 1 / then α = β = 1 / , Γ = 1 / and in this case Howard-Marić theorem cannot be applied since functions L ( t ) and L ( t ) from this theorem are not fundamental solutions. But there is a variant of this theoremappropriate for this case [25], and applying it one can show that a ( t ) is regularly varying ofindex if and only if w ∼ as t → ∞ , i.e. p ∼ c ρ holds asymptotically. This is the secondclassic cosmological solution. For more details one can consult Mijajlović et al. [27]. We analysed Friedmann equations and cosmological parameters from the point of view of reg-ular variation. The central role in this analysis had the acceleration equation since it can beconsidered as a linear second order differential equation and that the theory of regularly vary-ing solutions of such equations is well developed [25]. Particular attention in our discussionwas given to a pressureless universe with non-zero cosmological constant Λ . For this type ofuniverse S.M. Carroll, W.H. Press and E.L. Turner introduced in Carroll, Press & Turner [9]a function F (Ω) by which the other cosmological parameters can be expressed. This allowedus, using the theory of regular variation, to describe complete asymptotics for cosmologicalparameters when ω = lim t →∞ Ω does exist, but also in the case when this limit does not exist.As a conclusion we obtained for the convergent Ω( t ) at infinity and ω (cid:54) = 0 that Λ = 0 . We https://lambda.gsfc.nasa.gov/product/map/dr5/ – 18 –lso found asymptotics for all main cosmological parameters if ω = 0 . On the other hand,for divergent Ω we got that the expansion scale factor a ( t ) varies between two strips boundedby power functions and that it changes it’s convexity and concavity infinitely many times.Hence, in this case the deceleration parameter changes it’s sign infinitely many times and a ( t ) must accelerate and decelerate infinitely many times, too. There are strong evidences infavour of the existence of the dark energy. For example Tegmark et al. [38], list three suchobservational proofs: supernovae of type Ia, power spectrum analysis as done in Tegmarket al. [38] and the late ISW effect (Integrated Sachs–Wolfe effect). Therefore if the Λ CDMmodel is assumed, the case ω (cid:54) = 0 is excluded. We also discussed the threshold constant Γ andthe formally introduced equation of state parameter w . Both constants have an importantrole in describing asymptotics of cosmological parameters and evolution of the Universe. Wealso presented asymptotic formulas that might represent the cosmological parameters of theinteracting dual universe. Acknowledgments
This work was supported by the Serbian Ministry of Science, grant number III44006
References [1] Aguirre A., Johnson M. C., Shomer A.,
Towards observable signatures of other bubbleuniverses , PhRvD, 76, (2007), 063509[2] Aljančić, S., Arandjelović, D., O -regularly varying functions , Pub. Inst. Math., 22(36), (1977),5-22s[3] Ali A. F., Faizal M., Khalil M. M., Absence of black holes at LHC due to gravity’s rainbow ,PhLB, 743 (2015), 295[4] Anderson L., et al.,
The clustering of galaxies in the SDSS-III Baryon Oscillation SpectroscopicSurvey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples , MNRAS,441, (2014), 24[5] Bingham, N.H., Goldie, C.M., Teugels, J.L.,
Regular variation , Cambridge Univ. Press,Cambridge (1987)[6] Barrow J. D.,
Varieties of expanding universe , CQGra, 13 (1996), 2965[7] Barrow J. D., Shaw D. J.,
Some late-time asymptotics of general scalar tensor cosmologies ,CQGra, 25 (2008), 085012[8] Cadena, M., Kratz, M.,
An extension of the class of regularly varying functions ,https://hal-essec.archives-ouvertes.fr/hal-01097780 (2014), 35p[9] Carroll S. M., Press W. H., Turner E. L.,
The cosmological constant , ARA&A, 30 (1992), 499[10] Carroll S. M.,
The Cosmological constant , LRR, 4 (2001), 1[11] Djurčić, D., O -regularly varying functions and some asymptotic relations , Pub. Inst. Math. 61(1997), 44-52[12] Friedmann A., Über die Möglichkeit einer Welt mit konstanter negativer Krümmung desRaumes , ZPhy, 21 (1924), 326[13] Friedmann A.,
Non-oscillation theorems , Trans. Amer. Math. Soc., 64 (1948), 234[14] Cai Y.-F., Li H., Piao Y.-S., Zhang X.,
Cosmic duality in quintom Universe , PhLB, 646 (2007),141 – 19 –
15] Howard, H.C., Maric, V.,
Regularity and nonoscillation of solutions of second order lineardifferential equations , Bull. T. CXIV de Acad. Serbe Sci et Arts, Classe Sci. mat. nat, 22(1997), 85-98[16] Karamata, J,
Sur une mode de croissance réguliere fonctions , Math. (1930)[17] Khoury J., Ovrut B. A., Steinhardt P. J., Turok N.,
The Ekpyrotic Universe: Colliding Branesand the Origin of the Hot Big Bang , PhRvD, 64 (2001), 123522[18] Kusano, T., Marić, V.,
Regularly varying solutions of perturbed Euler differential equations andrelated functional differential equation , Publ. Inst. Math., 1 (2010), 88(102)[19] Lehners J.-L., McFadden P., Turok N.,
Colliding branes in heterotic M theory , PhRvD, 75(2007), 103510[20] Lang, S.,
Algebra , Springer (2002)[21] Liddle A. R., Lyth D. H.,
Cosmological Inflation and Large-Scale Structure , cils.book, 414(2000)[22] Liddle A.,
An Introduction to Modern Cosmology , imcs.book, 188 (2003)[23] Liddle A., Murdin P.,
The Cosmological Constant and its Interpretation , EAA, Murdin P., ed.IOP Publishing Ltd (2006)[24] Marić, V., Tomić, M.,
A classification of solutions of second order linear differential equationsby means of regularly varying functions , Publ. Inst. Math. (Belgrade), 58 (1990), 199[25] Marić, V.,
Regular Variation and Differential Equations , Springer, Berlin (2000)[26] Mijajlović, Ž., Pejović, N. & Ninković, S.,
Nonstandard Representations of Processes inDynamical Systems , AIP Conf. Proc, 934 (2007), 151[27] Mijajlović, Ž., Pejović, N., Šegan, S., Damljanović, G.,
On asymptotic solutions of Friedmannequations , Appl. Math and Computation, 219 (2012), 1273–1286[28] Mijajlović Ž., Pejović N., Marić V.,
On the ε cosmological parameter , SerAJ, 190 (2015), 25[29] Molchanov S.A., Surgailis D., Woyczynski, W.A., The large-scale structure of the universe andquasi-Voronoi tessalation of schock fronts in forced Burgers turbulence in R d , Ann. Appl.Probability, 7 (1997), 200[30] Peebles P. J. E., Growth of the nonbaryonic dark matter theory , NatAs, 1 (2017), 0057[31] Planck Collaboration, et al.,
Planck 2015 results. XIII. Cosmological parameters , A&A, 594(2016), A13[32] Perlmutter S., et al.,
A supernova at Z = 0.458 and implications for measuring thecosmological deceleration , ApJ, 440 (1995), L41[33] Riess A. G., et al.,
Type Ia Supernova Discoveries at z > 1 from the Hubble Space Telescope:Evidence for Past Deceleration and Constraints on Dark Energy Evolution , ApJ, 607 (2004),665[34] Schmidt B. P., et al.,
The High-Z Supernova Search: Measuring Cosmic Deceleration andGlobal Curvature of the Universe Using Type IA Supernovae , ApJ, 507 (1998), 46[35] Seneta E.,
Regularly varying functions , Springer, Berlin (1976)[36] Spergel D. N., et al.,
First-Year Wilkinson Microwave Anisotropy Probe (WMAP)Observations: Determination of Cosmological Parameters , ApJS, 148 (2003), 175[37] Stern I.,
The Effect of Lacunarity on the Convergence of Algorithms for Scaling Exponents ,ASPC, 125 (1997), 222[38] Tegmark M., et al.,
Cosmological parameters from SDSS and WMAP , PhRvD, 69 (2004),103501, PhRvD, 69 (2004),103501