Archimedean-type force in a cosmic dark fluid: III. Big Rip, Little Rip and Cyclic solutions
aa r X i v : . [ g r- q c ] D ec ARCHIMEDEAN-TYPE FORCE IN A COSMIC DARK FLUID:III. BIG RIP, LITTLE RIP AND CYCLIC SOLUTIONS
Alexander B. Balakin and Vladimir V. Bochkarev
Kazan Federal University, Kremlevskaya str., 18, 420008, Kazan, Russia (Dated: August 26, 2018)We analyze late-time evolution of the Universe in the framework of the self-consistent model, inwhich the dark matter is influenced by the Archimedean-type force proportional to the four-gradientof the dark energy pressure. The dark energy is considered as a fluid with the equation of stateof the relaxation type, which takes into account a retardation of the dark energy response to theUniverse accelerated expansion. The dark matter is guided by the Archimedean-type force, whichredistributes the total energy of the dark fluid between two its constituents, dark energy and darkmatter, in the course of the Universe accelerated expansion. We focus on the constraints for thedark energy relaxation time parameter, for the dark energy equation of state parameter, and forthe Archimedean-type coupling constants, which guarantee the Big Rip avoidance. In particular,we show that the Archimedean-type coupling protects the Universe from the Big Rip scenario withasymptotically infinite negative dark energy pressure, and that the Little Rip is the fate of theUniverse with the Archimedean-type interaction inside the dark fluid.
PACS numbers: 04.20.-q, 04.40.-b, 04.40.NrKeywords: Dark matter, dark energy, Archimedean-type interaction, accelerated expansion
I. INTRODUCTION
The present time accelerated expansion of theUniverse discovered in the observations of the Super-novae Ia [1–3] revived the discussion about the fateof our Universe. A psychologically jeopardy variantof the Universe future is that the permanently in-creasing scale factor a ( t ) and the Hubble function H ( t ) can reach infinite values during a finite timeinterval. The life-time of such Universe is finite, andthe catastrophe of this type produces cosmic inertial(tidal) forces, which destroy the bounds in all physi-cal systems. Various aspects of the singular scenariaof the future stage of the Universe evolution werediscussed during last two decades (see, e.g., the re-view [4]). The specific term Big Rip (or Doomsday)entered the scientific lexicon after publication of thepaper [5]; nowadays this term indicates a new trendin theoretical cosmology (see, e.g., [6]-[15]). It seemsto be uncomfortable for physicist to think that theBig Rip is the fate of our Universe, probably, it isa reason that many authors consider the models, inwhich the Big Rip can be avoided . In particular, theBig Rip can be avoided in the F ( R ), f ( T )-gravitymodels and their modifications, in the models withChaplygin gas, in the model for the dark energywith various effective time-dependent equations ofstate (see, e.g., [16]-[27]). As a more optimistic vari-ant of the Universe behavior one could consider thecase, when the Universe life-time is infinite, and theHubble function tends asymptotically to a constant, H → H ∞ . The asymptotic regime of this type ap-pears, in particular, in the ΛCDM model, which converts at t → ∞ into the de Sitter model with H ∞ = q Λ3 (Λ is the cosmological constant). Moregeneral case, when H ( t ) → H ∞ = const , but H ∞ isnot necessary equal to q Λ3 , is indicated as PseudoRip in [28]. Various intermediate scenaria with theinfinite Universe’s life-time, in which the scale factor,the Hubble function and (probably) its time deriva-tive tend to infinity, belong to the class indicated bythe term Little Rip (see, e.g., [29]-[34] for references). A. On the classification of the models of thelate-time Universe behavior
In order to classify the models of three types men-tioned above in more detail, we use the terminology(see, e.g., [11]) based on the asymptotic properties ofthe scale factor a ( t ), Hubble function H ( t )= ˙ aa , andits time derivative ˙ H , which are the basic quanti-ties in the isotropic spatially homogeneous cosmolog-ical models of the Friedmann-Lemaˆıtre-Robertson-Walker (FLRW)-type with the metric ds = dt − a ( t )[( dx ) +( dx ) +( dx ) ] . (1)(Here and below we use the units with c =1). In factthis classification is originated from the idea thattwo bonded particles are influenced by the inertialforce proportional to the quantity¨ aa = ˙ H + H = − H q ( t ) , (2)where − q is the acceleration parameter. The inter-nal structure of the physical system is assumed tobe destroyed, when this inertial force exceeds theinternal (e.g., intermolecular) forces. We think thatsuch force can also be indicated as a tidal force, ifwe take into account the following motives. Let usstart with the well-known equation of the world-linedeviation (see, e.g., [35]) D n i Dτ = − R iklm U k U m n l + F i , (3)where U k = dx k dτ is the velocity four-vector, n i = dx i dλ isthe deviation four-vector, τ is the world-line param-eter and λ is the parameter describing the world-line family. The second term in the right-hand sideof (3) describes the contribution of the force of thenon-gravitational origin (its structure for the electro-magnetic interactions is described, e.g., in [36]). Thefirst term in the right-hand side of (3) is interpretedas the tidal (curvature induced) force proportionalto the convolution of the Riemann tensor R iklm withthe deviation four-vector. Taking into account thatin the FLRW model the velocity four-vector is of theform U i = δ i , we can choose arbitrary direction, say0 x , and calculate the corresponding tidal force com-ponent ℜ = − R n . The result is ℜ = ¨ aa n , thusthe interpretation in terms of tidal force is equiva-lent to the one in terms of inertial force. This forcetends to infinity, when ˙ H or H (or both of them)tend to infinity. The standard classification uses thefollowing terminology:1. Power-law inflation, if H = ht , H ( t → ∞ ) → H = H Λ ≡ q Λ3 , where Λ is thecosmological constant;3. Pseudo Rip, if H ( t → ∞ ) → H ∞ < ∞ ;4. Little Rip, when H ( t → ∞ ) → ∞ , e.g., as h ∗ e ξt ;5. The future singularities at finite time t s , when H ( t → t s ) → ∞ .The fifth subclass can be specify, for instance, asfollows (see [11]):5.1. Big Rip (Type I), if a → ∞ , e.g., as a → t − t s ) ν ;5.2. Sudden singularity (Type II), if the pressureΠ tends to infinity, but a ( t ) and energy density ρ remain finite;5.3. Singularity of the Type III, if Π → ∞ and ρ → ∞ , but a ( t ) is finite;5.4. Singularity of the Type IV, if only ˙ H → ∞ . B. The goal of this paper
This paper is considered to be the third (final)one describing the fundamentals of the model of theArchimedean-type interaction between the dark en-ergy (DE) and dark matter (DM). In [37] we estab-lished the self-consistent model in which the dark matter is influenced by the force proportional to thefour-gradient of the dark energy pressure. In [38] weclassified the corresponding scenaria of the Universeevolution with respect to a number of the transitionpoints, in which the acceleration parameter − q ( t )takes zero values, and thus the epochs of deceleratedexpansion is replaced by the epoch of the acceleratedexpansion. In [39] we considered the application ofthe model to the problem of light propagation withnon-minimal coupling in a cosmic dark fluid withan Archimedean-type interaction between the DEand DM, and described the so-called unlighted cos-mological epochs, for which the effective refractionindex n ( t ), the phase and group velocities, V ph ( t )and V gr ( t ), respectively, were imaginary functions oftime.In [37–39] we focused on the solutions with asymp-totically finite Hubble function and finite accelera-tion parameter, and claimed that the case of infinite H and q is the topic of special discussion. Now wesystematically consider the last case in terms of BigRip and Little Rip solutions. Why this consider-ation might be interesting for readers? There areat least three motives to complete our investigation.The first motif is the following. The dark energy isconsidered in [37, 38] as a fluid with the simplestrheological property: the equation of state is of therelaxation type, i.e., it takes into account a retar-dation of the dark energy response, and includes anextra parameter, ξ . The first question arises: is itpossible to avoid the Big Rip by the appropriatechoice of the relaxation parameter ξ ? The secondmotif is connected with the effective constant of theArchimedean-type coupling, ν ∗ . When this couplingis absent, the energy density and pressure of the darkmatter decrease in the course of the Universe expan-sion, so that the dark matter does not play any rolein the asymptotic regime. However, when the pres-sure of the dark energy, being negative, becomes in-finite in the course of accelerated expansion of theUniverse, the Archimedean-type coupling leads toan effective heating of the dark matter, so that theDE and DM contributions happen to be of the sameorder. The second question arises: for what val-ues of the coupling parameter ν ∗ the Archimedean-type interaction is able to protect the Universe fromthe Big Rip singularity? The third motif is thatthe pressure of the dark energy, Π, is the key el-ement in our model, which guides the behavior ofboth state functions of the dark matter, the energy-density E and pressure P (via the Archimedean-typeforce), the energy-density ρ of the dark energy (viathe equation of state), thus predetermining the be-havior of the Hubble function H and its derivative˙ H (via the Einstein equations). This means thatin the model under consideration one can classifythe asymptotic types of the Universe behavior us-ing only one element, just the function Π( t ). Thisfunction, Π( t ), satisfies the key equation, which isthe nonlinear differential equation of the second or-der and includes six guiding parameters: ξ , σ and ρ coming from the three-parameter equation of statefor the dark energy, the effective coupling constantof the Archimedean-type interaction ν ∗ , the startingenergy, E ∗ , and temperature, T ∗ , of the dark matter.Thus, (keeping in mind the initial data Π( t ) and˙Π( t )) we have a possibility to classify all the typesof asymptotic behavior of the model using eight pa-rameters. The question arises: in what domains ofthe 8-dimensional effective space of the guiding pa-rameters the Big Rip is reliably avoided?In order to answer these questions we organizedthis paper as follows. In Section II we remindbriefly the master equations for the DE and DMevolution in the framework of the Archimedean-typemodel (Subsections IIA-IIC). In Subsection IID wedescribe our classification scheme, which is basedon eight parameters and includes three bifurcationpoints: ν ∗ =0 ( ν ∗ is the Archimedean-type couplingconstant); ξ =0 ( ξ is the DE relaxation parameter); σ = −
1, the phantom-crossing point. In Section IIIwe focus on the analysis of the models indicated asmodels with asymptotic DE domination, for whichthe contribution of the DM to the Universe expan-sion is asymptotically negligible. In Subsection IIIAwe describe the Big Rip, Pseudo Rip, Cyclic solu-tions with ν ∗ =0 and ξ =0 in terms of two guidingparameters σ , ρ , and the initial value of the DEpressure. In Subsection IIIB we focus on the mod-els with ν ∗ =0 and ξ = 0, thus analyzing the roleof the DE relaxation parameter in various scenariaof the Big Rip and Little Rip formation (in termsof three guiding parameters ξ , σ , ρ , and two ini-tial data for the DE). In Section IV we considerthe models with ν ∗ = 0 and ξ =0 (in terms of threeguiding parameters ν ∗ , σ , ρ and the initial value ofthe DE pressure), and show that the Archimedean-type coupling of the DM with the barotropic DEavoids the Big Rip regimes in the Universe expan-sion. In Section V we discuss the general case ν ∗ = 0and ξ = 0 qualitatively and numerically, and showthat the Archimedean-type coupling of the DM withthe DE of the rheological type converts the Big Ripregimes into the Little Rip, Pseudo Rip and Cyclicregimes of the Universe expansion. II. MASTER EQUATIONS
Let us remind briefly the master equations of themodel of the Archimedean-type coupling betweenthe dark energy and dark matter. These equationsare derived in [37] and now we use them to analyzethe late-time behavior of the Universe in the frame-work of this model.
A. Equations for gravity field
The FLRW-type isotropic spatially homogeneouscosmological model with two interacting con-stituents: the dark energy and dark matter areknown to be described by two nontrivial Einsteinequations ˙ H = − πG [ ρ + E + Π + P ] , (4) H = 8 πG ρ + E ) . (5)Here the dot denotes the derivative with respect totime, the functions ρ ( t ) and Π( t ) are the energy-density and pressure of the dark energy, respectively, E ( t ) and P ( t ) describe the corresponding state func-tions of the dark matter. The cosmological con-stant Λ is considered to be incorporated into theDE energy density ρ and DE pressure Π. As usual, H ( t ) ≡ ˙ aa is the Hubble function. The conservationlaw as the compatibility equation for the set (4)-(5)has the form˙ ρ + ˙ E + 3 H ( ρ + E + Π + P ) = 0 . (6) B. Dark matter description
The energy density and pressure of the one-component dark matter can be effectively presented,respectively, by the integrals E ( x ) = E ∗ x Z ∞ q dq p q F ( x ) e − λ ∗ √ q , (7) P ( x ) = E ∗ x Z ∞ F ( x ) q dq p q F ( x ) e − λ ∗ √ q . (8)The auxiliary function F ( x ) of the dimensionlessvariable x ≡ a ( t ) a ( t ) F ( x ) = 1 x exp { ν ∗ [Π(1) − Π( x )] } , (9)and convenient parameters E ∗ ≡ N ∗ m ∗ λ ∗ K ( λ ∗ ) , λ ∗ ≡ m ∗ k (B) T ∗ , (10)contain the effective Archimedean-type couplingconstant ν ∗ , an effective number density N ∗ , themass m ∗ and temperature T ∗ of the leading sort ofthe dark matter particles. The term K s ( λ ∗ ) ≡ Z ∞ dz cosh sz · exp [ − λ ∗ cosh z ] , (11)is the modified Bessel function, k (B) is the Boltz-mann constant. C. Dark energy dynamics
To describe the dark energy fluid we use the linearthree-parameter equation of state ρ ( t ) = ρ + σ Π + ξH ( t ) ˙Π . (12)When σ =0 and ξ =0, Eq. (12) introduces the modelin which the dark energy relates to the cosmologicalconstant Λ. When ρ =0 and ξ =0, Eq. (12) gives thewell-known linear relation Π= wρ with w ≡ σ . Theretardation of response in the dark energy evolutionis taken into account by inserting the term contain-ing the first derivative of the pressure ˙Π. An equiv-alent scheme is widely used in the extended ther-modynamics and rheology (see, e.g., [40]), in whichthe extended constitutive equation for the thermo-dynamically coupled variables X and Y has the form τ ˙ X + X = w Y . (13)Here τ is a relaxation time, a new coupling param-eter of the model. In the cosmological context τ isgenerally the function of time, τ ( t ). We assume that τ ( t )= ξσH ( t ) , i.e., this relaxation time can be mea-sured in natural cosmological scale [41]. Our ansatzhere is that the dimensionless parameter ξ is con-stant.When the quantities ρ ( t ) and Π( t ) depend on timethrough the scale factor a ( t ) only, i.e., ρ = ρ ( a ( t )),Π = Π( a ( t )), the so-called x -representation is con-venient, which is based on the following relations ddt = xH ( x ) ddx , (14) t − t = Z a ( t ) a ( t dxxH ( x ) . (15) In these terms the balance equation (6) and the con-stitutive equation (12) give the key equation for thepressure of the dark energy Π( x ) ξx Π ′′ ( x )+ x Π ′ ( x ) (4 ξ + σ ) +3(1+ σ )Π+3 ρ = J ( x ) . (16)The prime denotes the derivative with respect to x .The source J ( x ) is defined as follows: J ( x ) ≡ − E ∗ (cid:2) x F ( x ) (cid:3) ′ x Z ∞ q dqe − λ ∗ √ q p q F ( x ) . (17)The quantity J ( x ) vanishes, when the Archimedean-type coupling parameter vanishes, i.e., ν ∗ = 0. Wedeal with the differential equation of the second-order linear in the derivatives but nonlinear in theunknown function Π( x ). D. Classification scheme based on the analysisof the DE pressure Π The function Π plays the key role in the analysisof the whole model behavior. Indeed, when Π( x ) isfound from the key equation (16), we can reconstructthe DE energy-density ρ using the formula ρ ( x ) = ρ + σ Π( x ) + ξx Π ′ ( x ) , (18)then calculate the DM state functions E ( x ) and P ( x ) using (7) and (8), then find H ( x ) from theEinstein equation (5), and finally, reconstruct thescale factor as the function of time using (15).We are interested to discuss the late-time periodof evolution, i.e., the period, when t → t ∞ . We con-sider both cases: t ∞ = t s (future finite time singu-larity) and t ∞ = ∞ . Clearly, there are three possibletypes of the asymptotic behavior of the dark fluidcomposed of the dark energy and dark matter.1. The first type refers to the asymptote Π( t → t ∞ ) → Π ∞ = const . This case includes the submodelwith Π ∞ =0). Since the Archimedean-type force isproportional to the four-gradient of the dark energypressure Π, then asymptotically the dark matter de-couples from the dark energy, its energy-density E decreases as a and we obtain the case indicated asan asymptotic DE domination.2. The second type of behavior refers to the caseΠ( t → t ∞ ) → + ∞ . Now the DM becomes frozen,since the function F ( x ) (9) tends to zero exponen-tially, the DM decouples from the DE, and again wedeal with the model of the DE domination.3. The third and the most interesting case, refersto the asymptotic behavior Π( t → t ∞ ) → −∞ , forwhich the function F ( x ) tends to infinity, the DMbecomes effectively ultrarelativistic and thus playsan active role in the energy redistribution process.The model as a whole contains four effective guid-ing parameters ( ξ , σ and ρ describing the DE equa-tion of state, and ν ∗ , the effective Archimedean-type coupling constant) and four initial parameters(Π( t ) and ˙Π( t ), initial data for the DE pressure,and E ∗ , T ∗ initial data for the DM). The classifica-tion scheme includes three bifurcation points: thefirst, ν ∗ =0 indicates whether the Archimedean-typecoupling is switched on or not; the second, ξ =0 in-dicates whether the DE possesses the simplest rheo-logical property or not, the third, σ = −
1, relates tothe phantom-crossing point.
Remark on the Big Rip symptom: when the integral in the right-hand side of (15) con-verges on the upper limit, the infinite value of thescale factor a ( t ) can be reached at finite time t (s) ,thus we deal with the Big Rip. III. BIG RIP SOLUTIONS IN THE MODELSWITH ASYMPTOTIC DARK ENERGYDOMINATIONA. Big Rip solutions at ξ =0 and ν ∗ =0 This first model relates to the case, when the darkenergy does not possess rheological properties, andthere is no Archimedean-type coupling between DEand DM. The contribution of the DM is asymptot-ically vanishing, and in this sense we deal with thecase of the DE domination. The main results forthis model are well-known and our goal is to recoverthem as the limiting case of our model in the corre-sponding terminology.
1. Power-law expansion with ρ =0 Let us recover, first, the well-known results pre-senting the Big Rip solutions, when ξ =0, ρ =0 and ν ∗ =0. The key equation (16) takes the form σx Π ′ ( x )+3(1+ σ )Π=0 , (19)and the solution isΠ( x ) = Π(1) x − (1+ σ ) σ . (20)It is convenient to introduce the new dimensionlessparameter α ≡ − σ σ ) , (21) since it plays an important role below. For the DEenergy-density ρ ( x ) we immediately obtain ρ ( x ) = σ Π(1) x α , (22)so that H ( x ) = 13 8 πGσ Π(1) x α . (23)Clearly, the real solution for the Hubble function ex-ists, when the product σ Π(1) is positive. Using theremark on the Big Rip symptom one can concludethat the integral in (15) converges, when α is pos-itive, i.e. − < σ <
0. This simplest submodelallows us to verify this fact directly by computingthe scale factor analytically. Indeed, according to(15) the scale factor has the form a ( t ) a ( t ) = (cid:20) − α p πGσ Π(1)( t − t ) (cid:21) − α . (24)When − < σ < w = σ < −
1, one obtains that the parameter α is positive.This means that the scale factor tends to infinity atsome finite time value t = t ∗ , given by t ∗ = t + 3 α p πGσ Π(1) > t . (25)When σ <
0, the solution is real, when the initialvalue of the DE pressure is negative Π(1) <
0. Thuswe deal with the well-known Big Rip solution, whichis characterized by the following asymptotes at t → t ∗ : Π → −∞ , ρ → + ∞ , a → ∞ , H → ∞ and˙ H → ∞ . Let us add that when the parameter σ isnegative and belongs to the interval − < σ < α is positive (see (21)). There aretwo special cases. When σ → −
1, the parameter α becomes infinite, α → + ∞ ; when σ →
0, theparameter α tends to zero also. Finally, when α isnegative, i.e., σ > σ < −
1, the solution (24)describes power-law expansion of the Universe. TheLittle Rip solutions do not appear in this submodel.
2. Solutions with ρ = 0 When ρ = 0, the solution of the key equation σx Π ′ ( x )+3(1+ σ )Π+3 ρ =0 (26)has the formΠ( x )= − ρ σ + x α (cid:20) Π(1)+ ρ σ (cid:21) . (27)The square of the Hubble function reads now H ( x )= 8 πGρ σ ) h A x α i , (28)where A ≡ σ (cid:20)
1+ (1+ σ )Π(1) ρ (cid:21) . (29)The signs of the parameters A , ρ , α and σ provideimportant details of the classification. In order toclassify the models we, first, distinguish three (prin-cipal) cases and then show that all other cases canbe reduced to these three ones. (1) The model with A > , − < σ < , ρ > . Let us consider, first, the case when − < σ < A >
0. The corresponding initial value Π(1) shouldbe negative, and the inequality Π(1) < − ρ σ has tobe satisfied. The solution for the scale factor has theform a ( t ) a ( t ) = h cosh µ ( t − t ) −√ A sinh | µ | ( t − t ) i − α , (30)where µ ≡ σ p πGρ (1+ σ ) . (31)There exists a moment t = t ∗ , for which a ( t ∗ )= ∞ . At t → t ∗ the plot of a ( t ) has a vertical asymptote. Thiscritical moment of time, t ∗ , can be found from theequation tanh | µ | ( t ∗ − t ) = 1 √ A , (32)and has the form t ∗ = t + 12 | µ | log (cid:18) √ A +1 √ A− (cid:19) > t . (33)Again we deal with the Big Rip solution, character-ized by the asymptotes Π → −∞ , ρ → + ∞ , a → ∞ , H → ∞ , ˙ H → ∞ , but now, in the case ρ = 0,we can indicate the Big Rip singularity as that ofhyperbolic type in contrast to the one of the power-law type, presented in the previous subsection (with ρ =0). Remark 1:
Two submodels with negative α , namely, A > σ > ρ > A > σ < − ρ <
0, give formallythe same scale factor (30). However, the essentialdifference is that at α < H (28) tends to the constant H → H ∞ = q πGρ σ ) , andwe deal with the Pseudo Rip instead of the Big Rip. (2) The model with A < , − < σ < , ρ > . When − < σ < A <
0, one obtains from(28), that there is a critical value x = x ∗ x ∗ ≡ |A| − α , (34)for which the function H ( x ) H ( x )= 8 πGρ σ ) " − (cid:18) xx ∗ (cid:19) α (35)takes zero value. Let us assume that x ∗ >
1, i.e.,according to (34) |A| < − < A <
0. Itis possible when − ρ (1+ σ ) < Π(1) < ρ | σ | . Since themodel solutions for H ( x ) obtained from (35) can notbe prolonged for x > x ∗ , x starts to decrease afterthe moment t = t (max) . The corresponding solutionfor the scale factor has now the form a ( t ) a ( t ) = (cid:20) cosh µ ( t − t (max) )cosh µ ( t − t (max) ) (cid:21) α , (36)where t (max) = t + 12 | µ | log p −|A| p |A| ! > t . (37)Clearly, the plot of this function is symmetric in ref-erence to the moment t (max) ; the formula (36) givesthe maximal value (34) a ( t ∗ )= a ( t ) |A| − α at this mo-ment. At t → ∞ the scale factor a ( t ) tends to zeroas a ( t ) ∝ exp {− H ∞ t } , H ∞ ≡ s πGρ σ ) . (38)The Hubble function is described by the finite func-tion H ( t ) = − H ∞ tanh (cid:2) | µ | ( t − t (max) ) (cid:3) , (39)which changes the sign at t = t (max) . This means thatthe Universe expansion turns into a collapse at thismoment. The acceleration parameter − q ( t )= ¨ aaH =1 − α sinh − µ ( t − t (max) ) (40)takes infinite value at t = t (max) , since the Hubblefunction in the denominator vanishes at this mo-ment. Clearly, the Universe passes two eras with − q > − q < − < σ < Remark 2:
Two submodels with negative α , namely, A < σ > ρ < A < σ < − ρ >
0, give formallythe same expression for the scale factor (36). Sincenow α is negative, the Hubble function H (28) tendsagain to the constant H → H ∞ = q πGρ σ ) , if werequire x ∗ < (3) The model with A < , − < σ < , ρ < . Now the scale factor can be expressed in terms oftrigonometric functions a ( t ) a ( t ) = h cos | µ | ( t − t ) − p |A|− | µ | ( t − t ) i − α , (41)the parameter A has to satisfy the inequality A < −
1, thus, the initial value Π(1) is restricted byΠ(1) < − | ρ || σ | . Again, there exists a moment t ∗ t ∗ = t + 1 | µ | arctg p |A|− ! > t , (42)for which the plot of the scale factor has the verticalasymptote typical for the Big Rip with Π → −∞ , ρ → + ∞ , a → ∞ , H → ∞ , ˙ H → ∞ . It is con-venient to indicate such singularity as the Big Ripsingularity of the trigonometric type. Remark 3:
Two submodels with negative α , namely, A < σ > ρ < A < σ < − ρ >
0, giveformally the same scale factor (41). Since now theparameter µ becomes imaginary (see (31)), at α < B. Big Rip and Little Rip solutions at ξ = 0 and ν ∗ =0 When we consider this submodel we assume thatthe Archimedean-type coupling is switched of, butthe dark energy possesses the simplest rheologicalproperties. Since ν ∗ =0, the asymptotic regime againwill be characterized by the dark energy domination.In order to analyze this model let us introduce thenew unknown function Z ( x ) as follows:Π( x )= − ρ σ + Z ( x ) , (43)and define the initial value Z (1) as Z (1) = Π(1) + ρ σ . (44) The function Z ( x ) satisfies the Euler equation ξx Z ′′ ( x ) + xZ ′ ( x ) (4 ξ + σ ) + 3(1 + σ ) Z = 0 . (45)The characteristic polynomial of this Euler equationhas two roots s , = 12 ξ h − ( σ + 3 ξ ) ± p ( σ − ξ ) − ξ i , (46)which can be real or complex depending on the val-ues of the parameters ξ and σ . One can distinguishthree subcases.
1. Two different real roots [ ( σ − ξ ) > ξ ] (i) ξ > Z ( x )= x − γ (cid:26) Z (1)2 (cid:0) x Γ + x − Γ (cid:1) ++ [ Z ′ (1)+ γZ (1)]2Γ (cid:0) x Γ − x − Γ (cid:1)(cid:27) , (47)where γ ≡ σ + 3 ξ ξ , Γ ≡ ξ p ( σ − ξ ) − ξ . (48)According to the remark about the Big Rip symptomwe can conclude that the integral in (15) convergesnow, when Γ > γ , thus this inequality guarantees theBig Rip existence. In order to illustrate the existenceof the Big Rip by exact solutions, let us assume thatinitial data satisfy the equality Z ′ (1) = Z (1)(Γ − γ ) = (Γ − γ ) (cid:20) Π(1) + ρ σ (cid:21) . (49)Then the solutions can be simplified as follows Z ( x )= Z (1) x Γ − γ , (50)and Z ( x ) becomes proportional to the required lead-ing order term exactly. Now the DE state functionsare of the formΠ( x )= − ρ σ + Z (1) x Γ − γ , (51) ρ ( x )= ρ σ + Ω( ξ ) Z (1) x Γ − γ , (52)where the new parameterΩ( ξ ) ≡ σ + ξ (Γ − γ ) (53)is introduced. In the case of the DE domination thesquare of the Hubble function H ( x )= 8 πGρ σ ) (cid:20)
1+ Ω( ξ )(1+ σ ) ρ Z (1) x Γ − γ (cid:21) , (54)formally coincides with (28), if we use the substitu-tions A → Ω( ξ )(1+ σ ) ρ Z (1) , α → Γ − γ . (55)Based on the results of the previous section one canconclude that the Big Rip takes place, if Γ > γ .There are two domains on the plane of the parame-ters σ ξ , where this inequality holds at ξ >
0: first, − < σ < ξ − √ ξ , ξ < , second, σ < −
1. Thefirst domain is a natural extension of the interval − < σ <
0, which is obtained if we put ξ =0 intothe right-hand side of the inequality; let us remindthat just for the case − < σ < ξ =0. The second interval for the BigRip existence, σ < − ξ >
0, has no analog at ξ =0.The parameter ρ distinguishes three situationsadmitting the Big Rip at ξ > ρ =0.In both domains the singularity of a ( t ) is of thepower-law type.(2) ρ > − < σ < ξ − √ ξ , ξ < , the singularity is of the hyperbolic type. Inthe second domain, where σ < −
1, the singularity isof the trigonometric type.(3) ρ < − < σ < ξ − √ ξ , ξ < , the singularity is of the trigonometric type.In the second domain, where σ < −
1, the singularityis of the hyperbolic type.More detailed constraints for the guiding param-eters follow from the inequalities Ω( ξ ) Z (1) > ξ ) Z (1) <
0, respectively. (ii) ξ < ξ is negative, we use the fol-lowing scheme of analysis. We put Z ′ (1) = − Z (1)(Γ + γ ) (56)and obtain that Z ( x )= Z (1) x − (Γ+ γ ) . (57)The DE state functions take now the formΠ( x )= − ρ σ + Z (1) x − (Γ+ γ ) , (58) ρ ( x )= ρ σ + Ω( ξ ) Z (1) x − (Γ+ γ ) , (59)and the square of the Hubble function reads H ( x )= 8 πGρ σ ) (cid:20)
1+ Ω( ξ )(1+ σ ) ρ Z (1) x − (Γ+ γ ) (cid:21) . (60)Again, these functions can be obtained asymptoti-cally at x → ∞ , since now Γ < x Γ →
0. The sum Γ+ γ should benegative, and the product Ω( ξ ) Z (1) should be pos-itive, if we try to find the analogs of the Big Rip.This is possible, when σ > −
1. Clearly, we obtainthe power-law singularity, when ρ =0, the hyper-bolic singularity, when ρ >
0, and the trigonomet-ric singularity, when ρ >
2. Double real roots [ ( σ − ξ ) =12 ξ ] The discriminant can take zero value, only when ξ >
0. For the positive ξ there are three interestingcases: ξ < , ξ > and ξ = . (i) ξ < .The corresponding solution for Z ( x ) is Z ( x ) = x − γ { Z (1)+ log x [ Z ′ (1)+ γZ (1)] } , (61)where γ =3+ q ξ , if σ =3 ξ +2 √ ξ , and γ =3 − q ξ , if σ =3 ξ − √ ξ . According to the mentioned symptom,the Big Rip is possible, when γ is negative; clearly, itis possible only if we deal with the second solution,i.e., when ξ < and σ =3 ξ − √ ξ . These constraintsrequire that − < σ <
0. In order to illustrate thebehavior of the scale factor, let us assume that, first, ρ =0, second, Z ′ (1)= − Z (1) σξ and, third, Z (1) < ρ ( x ) = 3 | Z (1) | x | γ | log x , (62) | γ | = r ξ ( p ξ − . (63)The square of the Hubble function reads H ( x ) = 8 πG | Z (1) | x | γ | log x , (64)and the integral in (15) p G | γZ (1) | ( t − t ) = erf "r | γ | log x , (65)happens to be reduced to the error-functionerf[ z ] ≡ √ π Z z du e − u . (66)The error-function erf[ z ] takes finite value erf[ ∞ ]=1,when the upper limit in the integral goes to the in-finity, thus, the value x → ∞ can be reached at themoment t s = t + √ G | γZ (1) | . The scale factor has theform a ( t )= a ( t ) exp (cid:26) | γ | h erf − [ p G | γZ (1) | ( t − t )] i (cid:27) , (67)where the symbol erf − stands for the function in-verse to the error-function. Thus, when ξ < theBig Rip is possible, if σ =3 ξ − √ ξ . (ii) ξ > .This case corresponds to the positive parameter γ .This means that Z ( x ) (see (61)) asymptotically van-ishes, the Hubble function tends to constant, and wedeal with the Pseudo Rip. (iii) ξ = .Double real roots appear now, when ( σ − =4, i.e.,when σ =3 or σ = −
1. The first root, σ =3, corre-sponds to the solution (61) with γ = 6, and we againdeal with the Pseudo Rip. The second root, σ = − ρ σ , and we have to return to the key equation (16).As we emphasized in [37], in this special model theDE pressure and the DE energy density contain thelogarithmic terms squared. In particular, for the ini-tial data, which satisfy the equality ρ (1)+Π(1)=0,these DE state functions areΠ( x )=Π(1) − ρ log x − ρ log x , (68) ρ ( x ) = ρ (1) + 92 ρ log x . (69)The scale factor has the form a ( t ) a ( t ) = exp (s ρ (1)9 ρ sinh hp πGρ ( t − t ) i) , (70)it describes the super-exponential expansion. TheHubble function is monotonic H ( t )= r πGρ (1)3 cosh hp πGρ ( t − t ) i (71)and increases infinitely. The function ˙ H ˙ H ( t )=4 πG p ρ ρ (1) sinh hp πGρ ( t − t ) i (72)also tends to infinity at t → ∞ . The accelerationparameter − q ( t ) = 1+ s ρ ρ (1) sinh (cid:2) √ πGρ ( t − t ) (cid:3) cosh (cid:2) √ πGρ ( t − t ) (cid:3) (73) starts with − q ( t )=1, reaches the maximum − q (max) =1+ q ρ ρ (1) at t = t + log (1+ √ √ πGρ , and tendsasymptotically to − q ( ∞ )=1. This is the explicit ex-ample of the Little Rip.
3. Complex roots [ ( σ − ξ ) < ξ ] Complex roots appear at positive ξ . The solutionis quasiperiodic Z ( x ) = x − γ { Z (1) cos ( β log x )++ [ Z ′ (1)+ γZ (1)] β sin ( β log x ) (cid:27) , (74)where γ ≡ σ + 3 ξ ξ , β ≡ ξ p ξ − ( σ − ξ ) . (75)In this section it is convenient to use the variable τ = log x and to present the square of the Hubblefunction as follows H ( τ )= 8 πGρ σ ) (cid:8) e − γτ [ A cos βτ + B sin βτ ] (cid:9) , (76)where the parameters A = σZ (1) + ξZ ′ (1) , (77) B = 1 β ( σ − γξ ) [ Z ′ (1)+ γZ (1)] − ξβZ (1) (78)depend on the initial data Z (1) and Z ′ (1); for theillustration one can choose them so that B =0. Againthere are three interesting cases. (i) H ( τ ) > for arbitrary τ . For instance, when γ > B =0 and |A| < H ( τ ) >
0. Then the Hubble function H ( τ ) is real and tends to H ∞ = q πGρ σ ) at τ → ∞ ;we deal with the Pseudo Rip. (ii) H ( τ ) ≥ . To illustrate the situation, for which H is nonneg-ative and has a number of zeros, let us choose, first,that γ =0, second, that A =1, B =0, providing H ( τ )= 16 πGρ σ ) cos βτ ≥ . (79)The Hubble function is periodic with maximal value H (max) = q πGρ σ ) . The corresponding scale factor a ( t )= a ( t ) exp ( β arcsin " tanh β s πGρ σ ) ( t − t ) (80)0takes finite values satisfying the inequalities e − πβ ≤ a ( t ) a ( t ) ≤ e πβ . (81)Thus, there are neither Big Rip, nor Little Rip, de-spite − < σ < ξ < ). (iii) H changes the sign. When γ <
0, the quasiperiodic function H ( τ ) givenby (76) inevitably reaches zero value at some mo-ment τ = τ ∗ . Near this point the function H ( τ ) canbe presented as H ( τ ) ≃ h ( τ ∗ − τ )+ 12 ω ( τ ∗ − τ ) + ... (82)Depending on the guiding parameters σ , ξ , ρ , andon the initial data Z (1) and Z ′ (1), the quantity h can vanish or be positive.(1) h = 0.In the vicinity of the moment τ ∗ one obtains that τ ∗ − τ = h ( t − t ∗ ) , and thus a ( t ) ∝ exp (cid:2) − h ( t − t ∗ ) (cid:3) .(2) h =0.The decomposition (82) starts with ( τ ∗ − τ ) , we ob-tain that a ( t ) ∝ exp h − e − √ ω ( t − t ∗ ) i .In both cases the Universe has finite size and itslife-time is finite.
4. Resume
To summarize the results of the analysis let us in-dicate the domains on the plane of the parameters ξ and σ , for which the Big Rip scenaria can be re-alized: these domains are displayed on Fig.1. First,let us focus on the case ξ =0. According to the re-sults of Subsection IIA, the Big Rip can be realizedon the interval − < σ < ξ =0.It is the ”classical” Big Rip domain, which is char-acterized by the condition ρ +Π= σσ ρ <
0. When0 < ξ < this ”classical” Big Rip zone contractsalong the line σ to the interval − < σ < ξ − √ ξ ,and disappears at all when ξ ≥ (see the domainI on Fig.1). In this sense, the simplest rheologicalproperty of the DE, namely, the retardation of theDE response to the Universe expansion, can avoidthe Big Rip, which was the fate of the Universe at − < σ <
0, if the relaxation parameter ξ exceedsthe critical value ξ = . In other words, when the DErelaxation time τ ( t ) ≡ ξH ( t ) is bigger than H = ,where Θ ≡ ∇ k U k is the Universe expansion scalar,the regime of the Big Rip can not be supported bythe DE with such relaxation time. If ξ is negative,the interval − < σ < ρ can notavoid the Big Rip scenario, but it predetermined thetype of the Big Rip: the power-law type, hyperbolicor trigonometric ones.When σ >
0, there was no classical Big Rip at ξ =0. The same fact can be indicated at ξ > ξ is negative, the Big Ripscenario is possible for σ >
0, since instead of re-tardation the DE displays the acceleration of theresponse to the Universe expansion (see the domainIII on Fig.1).When σ < −
1, there was no ”classical” Big Ripat ξ =0. However, the Big Rip becomes possible at σ < − ξ = 0,but ξ →
0, the roots of the mentioned characteristicequation can be estimated as s → − σ ) σ ≡ α , s → − σξ . (83)The first root does not depend on ξ and gives usthe ”classical” Big Rip, when − < σ <
0. Thesecond root is positive and thus describes the ”non-classical” Big Rip, when either σ < ξ >
0, or σ > ξ <
0. This explains the appearance oftwo new domains IV and V on Fig.1. For small val-ues of ξ the second characteristic number s → − σξ is respectively big; from the physical point of view,we can speak about instability of the DE responseto the Universe expansion near the bifurcation line ξ =0. IV. BIG RIP AVOIDANCE AT ν ∗ = 0 , ξ =0 In this model we suggest that the dark energy isdescribed by the equation of state ρ = ρ + σ Π, butnow the Archimedean-type coupling is switched on.In other words, there are two coupled energy reser-voirs in the Universe: the dark energy and dark mat-ter, and an effective energy redistribution betweenthem is possible in the course of the Universe accel-erated expansion.
A. Asymptotic behavior at Π → −∞ Let the moment t (or equivalently, x =1) be cho-sen so that the value Π(1) is large and negative. The1 FIG. 1. The domains on the plane of the parameters ξ and σ , for which the Universe evolution follows theBig Rip scenario in the model of the asymptotic darkenergy domination. The horizontal stripe − < σ < σ =3 ξ − √ ξ (0 ≤ ξ < ), by thevertical line ξ =0 and horizontal line σ = −
1; the seconddomain is situated at ξ <
0. Two rectangular sectors ξ > σ < − ξ < σ > term J (see (17)) takes the form J → E ∗ ν ∗ I ∗ x e ν ∗ Π(1) Π ′ ( x ) e − ν ∗ Π( x ) , (84)where we use the new parameter I ∗ ≡ Z ∞ q dqe − λ ∗ √ q = 2 λ ∗ e − λ ∗ (cid:18)
1+ 3 λ ∗ + 3 λ ∗ (cid:19) . (85)Here and below the symbols with asterisk: I ∗ , E ∗ , ν ∗ , relate to the parameters of the leading DM com-ponent. The key equation (16) transforms into non-linear differential equation of the first order x Π ′ (cid:20) E ∗ ν ∗ I ∗ x e ν ∗ [Π(1) − Π] − σ (cid:21) =3 [ ρ +(1+ σ )Π] , (86)and its structure prompts us the following transfor-mations.
1. New dynamic variables
Let us introduce new variables X , Y and param-eter a ∗ . For the case σ > − X ≡ x a ∗ , Y ≡ − ν ∗ (cid:20) Π + ρ σ (cid:21) , (87) a ∗ ≡ E ∗ ν ∗ I ∗ σ ) e − Y (1) . (88)When σ < −
1, we replace formally 1+ σ by | σ | and X by − X . For the case σ = − X ( Y ): Y dXdY = 4 αX + e Y . (89)The energy density of the DE and DM in these newterms can be written as ρ = ρ σ − σν ∗ Y , E = − σ αν ∗ X e Y , (90)thus, the square of the Hubble function takes theform H = 8 πG (cid:20) ρ σ − σν ∗ (cid:18) Y + 14 αX e Y (cid:19)(cid:21) , (91)and the rate of the Hubble function evolution ˙ H isdescribed by the formula˙ H = 4 πG (1+ σ ) ν ∗ (cid:20) Y − X e Y (cid:21) . (92)The function a ( t ) can be extracted from the equation r πG t − t )= Z Y ( t ) Y (1) dY h α + e Y X ( Y ) i Y q ρ σ − σν ∗ (cid:0) Y + αX e Y (cid:1) . (93)Here we have to put X ( Y ) as the solution of (89)and to use the inverse function Y ( t ) ≡ Y [ X ( t )] = Y "(cid:18) a ( t ) a ( t ) a ∗ (cid:19) (94)on the upper limit of the integral.2
2. Qualitative analysis: divergence of the basic integral
The solution to the equation (89) is X ( Y ) = X (1) (cid:20) YY (1) (cid:21) α + Y α Z YY (1) duu α e u . (95)Here the constants X (1) and Y (1) are defined usingthe initial data as follows: X (1) = 1 a ∗ , Y (1) = − ν ∗ (cid:20) Π(1) + ρ σ (cid:21) . (96)Whatever the parameter α is, when Π → −∞ (i.e., Y → + ∞ ) the leading order terms in the asymptoticdecomposition of the function X ( Y ) X ( Y ) → Y e Y (cid:20)
1+ 1+4 αY + (1+4 α )(2+4 α ) Y + ... (cid:21) (97)come from the generalized integral exponential E ( α ) ( Y ) ≡ Z YY (1) duu α e u , (98)which appears in the right-hand side of (95). Thus,when X → Y e Y , the integrand in (93) behaves at Y → ∞ as s ν ∗ (3 − σ ) dY √ Y . (99)This asymptotic estimation is valid when σ <
3, i.e.,in both interesting cases: − < σ < σ < − Y → ∞ . In our casethe integral diverges at Y → ∞ as √ Y , and Big Ripcan not be realized in the scenario with ξ =0 and ν ∗ = 0. In the asymptotic limit Y → ∞ the formula(91) for the H yields H → πG (cid:20) ρ σ + σ − ν ∗ + (3 − σ )4 ν ∗ Y (cid:21) . (100)In other words, when σ < H → q (3 − σ )4 ν ∗ Y . As for ˙ H , (92)yields ˙ H → πG (3 − σ )3 ν ∗ (cid:20) Y (cid:21) , (101)i.e., the rate of the Hubble function growth tends tothe positive constant ˙ H ∞ = πG (3 − σ )3 ν ∗ .
3. Search for the scale factor
Keeping in mind (93) and (99) we can reconstructnow the function a ( t ) using the function Y ( t ) → πG (3 − σ )3 ν ∗ ( t − t ) , (102)for the case 0 ≤ α < ∞ . When Y → ∞ , the behav-ior of the function X ( Y ) is predetermined by thesecond term in (95), since the generalized integralexponent (98) gives the exponential leading orderterm X ∝ Y e Y (see (97)). The inverse function Y ( X ) is of the logarithmic type, which can be pre-sented by the following iteration procedure Y → log XY → log [ X log[ X log[ X... ]]] , (103)zero-order estimations yielding Y → log XY (1) . (104)The scale factor a ( t ) → a ( t ) a ∗ Y (1) exp (cid:26) πG (3 − σ )3 ν ∗ ( t − t ) (cid:27) (105)is described by the anti-Gaussian function, whichwas obtained in [37] as the exact solution of masterequations of the Archimedean-type model. Since thecorresponding Hubble function H ( t ) → πG (3 − σ )3 ν ∗ ( t − t ) (106)grows linearly with time, the obtained solution canbe classified as the Little Rip with H → ∞ and˙ H → const .To complete the analysis let us consider two spe-cial cases. (i) Special case σ = − α = ∞ , or equivalently, σ = −
1, weshould consider the solutions in more details. Thekey equation x Π ′ (cid:20) E ∗ ν ∗ I ∗ x e ν ∗ [Π(1) − Π] +1 (cid:21) =3 ρ (107)can be now transformed into d ˜ Xd ˜ Y = − ρ ν ∗ (cid:16) ˜ X + e ˜ Y (cid:17) , (108)using the modified replacements˜ X ≡ x a ∗∗ , a ∗∗ ≡ E ∗ ν ∗ I ∗ , (109)3˜ Y ≡ − ν ∗ [Π − Π(1)] , ˜ Y (1) = 0 . (110)The solution to (108)˜ X ( ˜ Y ) = " ˜ X (1)+ 11+ ρ ν ∗ e − ρ ν ∗ ˜ Y − ρ ν ∗ e ˜ Y , (111)shows that ˜ X ( ˜ Y ) takes zero value at some ˜ Y = ˜ Y ∗ and then changes the sign. Since ˜ X is a positivelydefined function, we can conclude that the regime a → ∞ with Π → −∞ can not be realized, when σ = −
1. In other words, the phantom-crossing value σ = −
1, which corresponds to α = ∞ is the criticalvalue: when − < σ ≤
0, the Little Rip regimewith a ( t ) → ∞ and Π( t ) → −∞ is possible; when σ = −
1, the scale factor can not reach infinite value. (ii) Special case σ =3The special case: α = − , or equivalently, σ =3, re-lates to the ultrarelativistic DE, when ρ =0, since ρ =3Π. The functions X , H and ˙ H take now theform X = 1 Y (cid:0) K + e Y (cid:1) , (112) H = 2 πG (cid:20) ρ − KYν ∗ ( K + eY ) (cid:21) , (113)˙ H = 16 πGKν ∗ ( K + e Y ) Y , (114)where the new constant K is introduced K ≡ X (1) Y (1) − e Y (1) . (115)When Y → ∞ , H → q πGρ , ˙ H →
0, thus, we dealwith the de Sitter asymptote.
V. BIG RIP AVOIDANCE IN GENERALCASE: ξ = 0 AND ν ∗ = 0 In terms of the dynamic variables (87) the keyequation for the DE pressure takes the form16 ξ σ ) X Y ′′ + Y ′ (cid:20) σ +7 ξ )3(1+ σ ) X − e Y (cid:21) + Y = 0 . (116)At ξ =0 it coincides with (89), as it should be. Thefunction ˙ H and the square of the Hubble functioncan be presented as follows˙ H = 4 πGν ∗ (cid:20) (1 + σ ) (cid:18) Y − e Y X (cid:19) + 4 ξX dYdX (cid:21) , (117) H = 8 πG (cid:20) ρ σ − σν ∗ (cid:18) Y + 14 αX e Y (cid:19) − ξν ∗ X dYdX (cid:21) . (118)Let us, first, analyze these functions in terms of thevariable Y . A. Qualitative analysis
1. The model with < ξ < and − < σ < Let us assume that the asymptotic solution of(116) at Y → ∞ differs insignificantly from the so-lution to (89), and let us consider the function X ( Y ) → Y e Y (cid:20) B ( ξ ) Y (cid:21) , (119)in which the parameter B ( ξ ) is unknown, but shouldsatisfy the condition B (0) = 1 + 4 α . (120)This means that we consider the solution in the form(97) and restrict ourselves by the term Y in theparentheses. With mentioned accuracy the function X ( Y ) (119) satisfies (116) with (120) , when B ( ξ ) = 1 − σ + 3 ξ )3(1 + σ ) . (121)From the qualitative point of view the behavior ofthe DE with ξ = 0 at Π → −∞ is analogous to thatof the DE with ξ =0 and effective parameter σ ∗ equalto σ ∗ ≡ σ + 3 ξ − ξ . (122)It is interesting that σ ∗ = −
1, when σ = −
1, i.e.,these two parameters coincide for arbitrary ξ at thephantom-crossing point σ = σ ∗ = − X dYdX behaves as (1+ Y ), when Y → ∞ . Thus, the con-tributions of the last terms in (117) and (118) (pro-portional to the parameter ξ ) can be neglected at Y → ∞ . Thus, the main conclusion that theArchimedean-type interaction avoids the Big Rip re-mains valid at 0 < ξ < , − < σ <
2. The model with ξ > and σ < − The key equation for this case can be obtainedfrom (116) by the formal replacement X → − ˜ X and Y ′ ( X ) → − Y ′ ( ˜ X ) with positively defined ˜ X . Nu-merical calculations show that in this case the regime4 Y → ∞ does not exist. To illustrate qualitativelysuch a behavior one can mention the following: theregime with Y → ∞ would require that the lead-ing order terms in the equation (116) are linked by Y ′ e Y + Y =0, which is in contradiction with the re-quirement ˜ X >
0. Then two scenaria are available:first, with Y → const , second with Y → −∞ . Inthe first scenario there is no Big Rip, and, as a rule,we deal with the Pseudo Rip. In the second scenariothe quantity Y with positive initial value Y (1) > H = − πGν ∗ (cid:20) | σ | (cid:18) Y + e Y ˜ X (cid:19) + 4 ξX (cid:12)(cid:12)(cid:12)(cid:12) dYdX (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (123)showing explicitly that the derivative of the Hub-ble function is negative. This means that at somemoment t ∗ (finite or infinite) the Hubble functionwill take zero value providing the scale factor a ( t )to reach its maximal value a max . In other words, inthis scenario the Big Rip can nor appear in contrastto the model with ν ∗ =0.
3. The model with ξ < and σ > − When ξ is negative, the procedure similar to theone used in the case 0 < ξ < , − < σ < H ( Y → ∞ ) → πG ν ∗ (3 − σ ) , (124) H ( Y → ∞ ) → πG (3 − σ )3 ν ∗ Y , (125)which do not contain the parameter ξ in the leadingorder term. However, now we consider the parame-ter 1+ σ to be positive, and can conclude that we dealwith the Little Rip, when − < σ <
3, and the solu-tion with finite | a ( t ) | , when σ >
3. In any case theArchimedean-type interaction avoids the Big Rip.
4. The special model σ = − As in the general case described in [37, 38], asymp-totic model admits the solution of the anti-Gaussiantype, when σ = −
1. Indeed, the corresponding keyequation ξx Π ′′ +(4 ξ − x Π ′ +3 ρ = E ∗ ν ∗ I ∗ x Π ′ e − ν ∗ [Π( x ) − Π(1)] (126) is satisfied with the solutionΠ( x ) = Π(1) − ν ∗ log x , (127)if the guiding parameters are coupled by the equality E ∗ I ∗ = 3 ξ − ν ∗ − ρ . (128)Since E ∗ I ∗ is positive, one requires that ξ > + ρ ν ∗ , i.e., this equality can not be satisfied when ξ =0. When the right-hand side of (128) is posi-tive, the logarithmic solution can appear as a resultof some ”fine tuning”: one can satisfy (128), e.g.,by varying the initial temperature T ∗ of the DM,which enters the parameter λ ∗ = m ∗ c k B T ∗ , the argumentof the function I ∗ ( λ ∗ ) (see (85)). The asymptoticDE energy-density is now ρ ( x ) = ρ (1) + 4 ν ∗ log x , (129)where the relationΠ ′ (1) = 1 ξ [ ρ (1) − ρ + Π(1)] (130)is used. The sum Π( x )+ ρ ( x ) remains constant ρ ( x ) + Π( x ) = ρ (1) + Π(1) = ρ − ξν ∗ . (131)The DM energy density E ( x ) is now constant, i.e., E ( x ) = E ∗ I ∗ , (132)and the Hubble function H ( x ) can be found fromthe equation H ( x ) = 8 πG (cid:20) ρ (1) + E ∗ I ∗ + 4 ν ∗ log x (cid:21) . (133)The corresponding scale factor a ( t ) can be writtenin the form a ( t ) = a ∗ exp (cid:26) πG ν ∗ ( t − t ∗ ) (cid:27) , (134)where the parameters with asterisks are defined asfollows a ∗ ≡ a ( t ) exp n − ν ∗ ρ (1) + E (0) ] o ,t ∗ ≡ t − ν ∗ r πG [ ρ (1) + E (0) ] . (135)The acceleration parameter − q ( t ) for the anti-Gaussian expansion is positive and exceeds theunity: − q ( t ) ≡ ¨ aaH = 1 + 3 ν ∗ πG ( t − t ∗ ) ≥ . (136)Clearly, this solution illustrates one of the versionsof the Little Rip.5
5. Numerical analysis
In order to confirm the qualitative conclusionsmade for the model with ξ = 0, ν ∗ = 0, we ana-lyzed numerically the models for the parameters ξ and σ belonging to the domains I,II,III,IV displayedon Fig.1. These calculations are illustrated by Fig.2and Fig.3. These figures contain four panels: thefirst one displays the Hubble function H ( x ), the sec-ond panel demonstrates the behavior of the scalefactor a ( t ), the third panel describes the rate of thegrowth of the Hubble function ˙ H , the fourth paneldisplays the function ρ ( x ). FIG. 2. The illustration to the model with σ = − . ξ = − . ν ∗ =1 (i.e., for the case − < σ < ξ < σ +3 ξ <
0, which refers to the Big Rip scenario at ν ∗ =0 according to Fig.1). Clearly, when ν ∗ = 0, i.e., theArchimedean-type coupling is switched on, the modelsolution is of the Little Rip type with a → ∞ , H → ∞ ,˙ H → const , ρ → ∞ , Π → −∞ . VI. DISCUSSION
When the dark matter is effectively coupled tothe dark energy by the Archimedean-type interac-tion, the late-time evolution of the Universe withnegative DE pressure is protected from the Big Ripsingularity, and the Little Rip becomes a typical fate
FIG. 3. The illustration to the model with σ = − . ξ = 0 . ν ∗ =1 (i.e., for the case σ < − ξ > σ +3 ξ >
0, which refers to the Big Rip scenario at ν ∗ =0according to Fig.1). When ν ∗ = 0 the model solution isof the Little Rip type with a → ∞ , H → ∞ , ˙ H → const , ρ → ∞ , Π → −∞ . of the Universe. This is the main conclusion of thepaper. More detailed discussion includes the follow-ing three items.1. In the presented model of Archimedean-type cou-pling between the dark matter and dark energy theDE pressure, Π, is the key element of modeling. Allthe model solutions can be divided into three classeswith respect to asymptotic behavior of the statefunction Π( t → t ∞ )=Π ∞ (we consider both cases: t ∞ = t s (future finite time singularity) and t ∞ = ∞ ).The first class is characterized by finite Π ∞ ; the sec-ond class relates to Π ∞ =+ ∞ ; we deal with the solu-tions of the third class, when Π ∞ = −∞ . In [38, 39]we focused on the solutions of the first class, andhave shown qualitatively that the solutions of thesecond and third classes exist for some values of theeffective guiding model parameters ξ , σ , ρ and ν ∗ .Three parameters ξ , σ and ρ describe the equationof state of the dark energy (see (12)), ν ∗ is the ef-fective Archimedean-type coupling constant. Thispaper is devoted to the analysis of the solutions ofthe second and third classes. When Π ∞ is finite orΠ ∞ =+ ∞ , the dark matter is asymptotically decou-6pled from the dark energy and its energy density be-comes negligible in comparison with the DE energydensity. We indicated this asymptotic situation asthe DE domination (the coupling constant ν ∗ is notequal to zero, but it becomes a hidden parameter ofthe model). In this case only the appropriate choiceof the constitutive parameters ξ , σ and ρ can pro-tect the Universe from the Big Rip singularity. Tosummarize the analytical results for the DE domi-nation epochs we prepared Fig.1.When ξ =0, the Big Rip can be realized on the in-terval − < σ < w = σ < − < ξ < this ”classical” BigRip zone contracts along the line σ to the interval − < σ < ξ − √ ξ , and disappears at all when ξ ≥ . In this sense, the retardation of the DEresponse to the Universe expansion, can avoid theBig Rip, if the relaxation parameter ξ exceeds thecritical value ξ = . In other words, when the DErelaxation time τ ( t ) ≡ ξH ( t ) is bigger than H = ,where Θ ≡ ∇ k U k is the Universe expansion scalar,the regime of the Big Rip can not be supported. If ξ is negative, the interval − < σ < ρ can not provide the avoid-ance of the Big Rip scenario, but it predeterminedthe type of the Big Rip: the power-law type, hyper-bolic or trigonometric ones. When σ >
0, there wasno ”classical” Big Rip at ξ =0. The same fact can beindicated at ξ >
0. However, if the parameter ξ isnegative, the Big Rip scenario is possible for σ > σ < − − < w < ξ =0, however,the Big Rip becomes possible at σ < − ξ > t → t ∞ ) → −∞ , for which the Archimedean-typecoupling leads to the effective heating of the darkmatter component of the dark fluid. Due to the forceproportional to the four-gradient of the DE pressurethe DM becomes effectively ultrarelativistic and thusplays an active role in the energy redistribution pro-cesses inside the dark fluid. The reviving of this sec-ond player in the late-time scenario of the Universeevolution change essentially the character of expan-sion: at ν ∗ = 0 the Big Rip scenaria happen to be avoided and instead of them the Little Rip scenariabecome typical for the Universe late-time evolution.This avoidance is typical both for the cases ξ =0 and ξ = 0.We would like to emphasize that the so-called anti- Gaussian type solutions for the scale factor a ( t ) ap-pearing as some specific exact solution in [37], be-come typical asymptotic solutions in the case underdiscussion (see, e.g., (105) and (134)). These anti-Gaussian type solutions correspond to the Little Ripscenario [29, 32], since a ( t → ∞ ) → ∞ , H ( t →∞ ) → ∞ , | Π( t → ∞ ) | → ∞ and ρ ( t → ∞ ) → ∞ .In other words, in this model the life-time of theUniverse is infinite, the Hubble function and scalefactor tend to infinity without vertical asymptotes.Discussing the mechanism of the Big Rip avoid-ance in the presence of the Archimedean-type in-teraction, we would like to attract the attention tothe following feature. When the coupling constant ν ∗ vanishes, the DE pressure behaves as the power-law function of the scale factor. In terms of dimen-sionless variables Y and X (see (87)) this power-lawfunction is predetermined by the first term in (95).When ν ∗ = 0, and the DM contribution to the to-tal energy-density gains the same order as the DEenergy-density, the DE pressure behaves accordingto the super-logarithmic law (103), which is typi-cal for the Little Rip. Super-logarithmic law for Y means that the DE pressure grows more slowly thanin the case of ν ∗ =0. In other words, since the DEtransfers energy to the DM due to the Archimedean-type interaction, the rate of the DE pressure growthdecreases, thus protecting the Universe from the BigRip singularity.3. The model under discussion displays that thereare three critical values of the guiding parameters.The first one is the value σ = −
1, which correspondsto the well-known phantom-crossing value w = σ = − w . Thesymptom of criticality is that the term (1+ σ ) − sys-tematically appears in the key equations and expres-sions (see, e.g., (21) and (38)). The manifestation ofthe criticality is that the solutions with σ → − σ ≡ −
1, so thatwe considered the last case as the special one. Inthis sense, one can see some analogy between thismodel and the critical behavior of the magnetic andelectric fields affected by the gravitational wave: weobtained principally different solutions for the case,when the refractive index tends to one, n →
1, andfor the case, when this parameter is equal to oneidentically, n ≡
1, since the term n − ν ∗ =0. Again, the param-eter ν ∗ appears in the denominators in the functions a ( t ) and H ( t ) (see, e.g., (105), (106), as the symp-tom of criticality, and the solutions with ν ∗ =0 (wemean the solutions of the Big Rip type) differ prin-cipally from the solutions with ν ∗ = 0 (the solutionsof the Little Rip type). The third critical value is7connected with the parameter ξ . The value ξ =0 canbe considered as the critical one, since crossing thevertical line ξ =0 on Fig.1, we see the structural rear-rangement of the domains corresponding to the BigRip type solutions. In addition, according to (83),this parameter appears in the denominator, indicat-ing that the situation with ξ → ξ ≡ ACKNOWLEDGMENTS
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