Coupled scalar fields in the late Universe: The mechanical approach and the late cosmic acceleration
Alvina Burgazli, Alexander Zhuk, João Morais, Mariam Bouhmadi-López, K. Sravan Kumar
CCoupled scalar fields in the lateUniverse: The mechanical approachand the late cosmic acceleration
Alvina Burgazli , Alexander Zhuk , Jo˜ao Morais ,Mariam Bouhmadi-L´opez , , , , K. Sravan Kumar , Astronomical Observatory, Odessa National University, Dvoryanskaya st. 2, Odessa65082, Ukraine Department of Theoretical Physics, University of the Basque Country (UPV/EHU), P.O.Box 644, 48080 Bilbao, Spain Departamento de F´ısica, Universidade da Beira Interior, Rua Marquˆes D’ ´Avila e Bolama,6201-001 Covilh˜a, Portugal Centro de Matem´atica e Aplica¸c˜oes da Universidade da Beira Interior (CMA-UBI), RuaMarquˆes D’ ´Avila e Bolama, 6201-001 Covilh˜a, Portugal IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, SpainE-mail: [email protected], [email protected], [email protected],[email protected] (On leave of absence from UPV/EHU and IKERBASQUE),[email protected]
Abstract.
In this paper, we consider the Universe at the late stage of its evolution and deepinside the cell of uniformity. At these scales, we consider the Universe to be filled with dust-like matter in the form of discretely distributed galaxies, a minimally coupled scalar fieldand radiation as matter sources. We investigate such a Universe in the mechanical approach.This means that the peculiar velocities of the inhomogeneities (in the form of galaxies) aswell as fluctuations of other perfect fluids are non-relativistic. Such fluids are designatedas coupled because they are concentrated around inhomogeneities. In the present paper weinvestigate the conditions under which a scalar field can become coupled, and show that, atthe background level, such coupled scalar field behaves as a two component perfect fluid:a network of frustrated cosmic strings with EoS parameter w = − / a r X i v : . [ g r- q c ] O c t ontents Under the assumption of homogeneity and isotropy for our Universe on its largest scales,the current observational data lead to the conclusion of the expansion of the Universe isaccelerated, as first implied by the type Ia supernova data almost twenty years ago [1, 2].However, its nature is still a great challenge for modern cosmology. This phenomena hasreceived the name dark energy (DE), which partly reflects its unclear nature. The ΛCDMmodel, where the cosmological constant Λ is responsible for the acceleration, is in very goodagreement with observations [3–5]. This model is equivalent to one with a perfect fluid withthe constant EoS parameter w = −
1. Unfortunately, this model has some puzzling andunresolved aspects such as the origin of Λ and the coincidence problem [6]. Therefore, anumber of alternatives have been proposed which try to solve these problems, with modelsusing scalar fields to explain DE being amongst the most popular ones . These are calledquintessence [9–11] whenever − < w <
0, phantom [12, 13] when w < −
1, and quintom[14] if there is w = − w . Thisimposes severe restrictions on the form of the scalar field potential [15, 16].As we mentioned above, there is a number of alternative models of DE. Therefore,it is of great importance to propose a mechanism which can verify their viability. Thetheory of perturbations is a powerful tool to investigate cosmological models [17, 18]. Suchperturbations can be considered at any stage of the Universe evolution. In our paper, weinvestigate our Universe at the late stage of its evolution and deep inside of the cell ofuniformity. At such scales the Universe looks highly inhomogeneous: galaxies, group andclusters of galaxies are already formed and can be considered as discrete sources for thegravitational potential. In our previous papers [19, 20], we have shown that in this case themechanical approach is an adequate tool to study scalar perturbations. In its turn, it enablesus to get the gravitational potential and to consider the motions of galaxies [21]. It is worthnoting that similar ideas concerning the discrete cosmology have been discussed in the recentpapers [22, 23]. The mechanical approach was applied to a number of DE models to studytheir compatibility with the theory of scalar perturbations. For example, we considered the Besides being considered as DE candidates, scalar fields can also play the role of dark matter which couldlead to the formation of gravitational structures in the earlier Universe. [7, 8]. – 1 –erfect fluid with a constant equation of state parameter [24], the model with quark-gluonnuggets [25], the CPL model [26], and the Chaplygin gas model [27].The mechanical approach works perfectly for the ΛCDM model where the peculiar veloc-ities of the inhomogeneities (e.g. galaxies) can be considered as negligibly small (as comparedwith the speed of light), and, additionally, we consider scales deep inside of the cell of unifor-mity . Then, we may drop the peculiar velocities at the first order approximation [19, 20].As we mentioned above, such an approach was generalised also to the case of cosmologicalmodels with different perfect fluids which can play the role of dark energy and dark mat-ter. Fluctuations of these additional perfect fluids also form their own inhomogeneities. Inthe mechanical approach, it is supposed that the velocities of displacement of such inhomo-geneities is of the order of the peculiar velocities of inhomogeneities of dust-like matter, i.e.they are non-relativistic. In some sense, these two types of inhomogeneities are “coupled”to each other [29]. This is an important point. This means that for the considered models,we investigate the possibility of the existence of such “coupled” fluids . They can play animportant role of dark matter and can be distributed around the baryonic inhomogeneities(e.g., galaxies) in such a way that it can solve the problem of flatness of the rotation curves[25]. It is worth noting that the generalization of the mechanical approach to the case ofnon-zero peculiar velocities and for all cosmological scales was performed in the recent papers[28, 30]. This generalization includes also the case of uncoupled perfect fluids.In the present paper, we consider a cosmological model with a scalar field minimallycoupled to gravity. The Universe is also filled with dust-like matter (in the form of discretegalaxies and group of galaxies) and radiation. We study the theory of scalar perturbationsfor such model and obtain a condition under which the inhomogeneities of dust-like mat-ter and the inhomogeneities of the scalar field can be coupled to each other (in the sensepointed above). We demonstrate that this condition imposes rather strong restrictions onthe scalar field itself. First, the coupled scalar field behaves (at the background level) asa two-component perfect fluid: a cosmological constant and a network of frustrated cosmicstrings. The latter has a parameter of EoS w = − /
3. The potential of such scalar field isvery flat at the present time (see Fig. 1). This flatness condition is a natural consequence ofthe current acceleration of the Universe, as the contribution of the term with w = − / For an estimation of the dimension of the cell of uniformity (the scale of homogeneity) from the point ofview of the gravitational interaction features at large distances, see [28]. In what follows, we shall omit quotation marks for the word “coupled”. – 2 –n Sec. 3, we investigate within the mechanical approach the scalar perturbations for theconsidered cosmological model. We define the conditions under which the scalar field satisfiesthe equations obtained. Here, we also obtain the equation for the non-relativistic gravitationalpotential and its solutions. In Sec. 4, we determine the form of the scalar field potential.The main results are summarised in the concluding Sec. 5.
To start with, we consider the background equations which describe the homogeneous andisotropic Universe. The background metric is the Friedmann-Lemaˆıtre-Ro-bertson-Walkerone ds = a ( η ) (cid:16) dη − γ αβ dx α dx β (cid:17) , (2.1)where the exact form of the metric γ αβ is defined by the topology of the Universe. Forgenerality, we shall consider all three possible topologies with scalar curvatures K = − , , +1for open, flat and closed Universes, respectively. The Universe is filled with a scalar fieldminimally coupled to gravity. Such a field is described by the action S φ = (cid:90) d x √− g (cid:20) g µν ∂ µ φ ∂ ν φ − V ( φ ) (cid:21) (2.2)and has the energy-momentum tensor: T µν ( φ ) = g µλ ∂ ν φ∂ λ φ − δ µν (cid:20) g λρ ∂ λ φ ∂ ρ φ − V ( φ ) (cid:21) . (2.3)The equation of motion reads1 √− g ∂ µ (cid:0) √− gg µν ∂ ν φ (cid:1) + dVdφ ( φ ) = 0 . (2.4)For the homogeneous and isotropic Universe, the scalar field depends only on time. Let φ c ( η ) describes such a background scalar field. Then, for the background energy density andpressure we get ¯ T ≡ ¯ ε ϕ = 12 a ( φ (cid:48) c ) + V ( φ c ) , (2.5) − ¯ T ii ≡ ¯ p ϕ = 12 a ( φ (cid:48) c ) − V ( φ c ) , (2.6)where the prime denotes the derivative with respect to the conformal time η . The backgroundequation of motion is φ (cid:48)(cid:48) c + 2 H φ (cid:48) c + a dVdφ ( φ c ) = 0 , (2.7)where H = a (cid:48) /a .As matter sources, we also include dust-like matter (baryonic and CDM) and radiation.The background (i.e. average) energy density of the dust-like matter takes the form ¯ ε dust =¯ ρc /a , where ¯ ρ = const is the average comoving rest mass density [20]. As usual, forradiation, we have the EoS ¯ p rad = (1 / ε rad and ε rad ∼ /a .– 3 –or the above described cosmological model, the Friedmann and Raychaudhuri equa-tions take, respectively, the form H = κa (cid:20) ¯ ε dust + ¯ ε rad + 12 ( φ (cid:48) c ) /a + V ( φ c ) (cid:21) − K (2.8)and H (cid:48) = 13 a κ (cid:20) − ¯ ε rad −
12 ¯ ε dust − ( φ (cid:48) c ) /a + V ( φ c ) (cid:21) , (2.9)where κ ≡ πG N /c ( c is the speed of light and G N is the Newtonian gravitational constant)and we have used Eqs. (2.5) and (2.6). Let us turn now to the scalar perturbations. Then, the metrics reads [17, 18]: ds = a ( η ) (cid:104) (1 + 2Φ) dη − (1 − γ αβ dx α dx β (cid:105) . (3.1)The perturbations of the scalar field energy-momentum tensor are [18]: δT ≡ δε ϕ = − a (cid:0) φ (cid:48) c (cid:1) Φ + 1 a φ (cid:48) c ϕ (cid:48) + dVdφ ( φ c ) ϕ , (3.2) δT i = 1 a φ (cid:48) c ∂ i ϕ , (3.3) δT ij ≡ − δ ij δp ϕ ,δp ϕ = − a (cid:0) φ (cid:48) c (cid:1) Φ + 1 a φ (cid:48) c ϕ (cid:48) − dVdφ ( φ c ) ϕ , (3.4)where we split the scalar field into its background part φ c ( η ) and its fluctuations part ϕ ( η, (cid:126)r ): φ = φ c + ϕ . (3.5)For the considered model, the Einstein equations are reduced (after linearising thesystem of 3 equations) to:∆Φ − H (Φ (cid:48) + H Φ) + 3 K Φ = κ a ( δε dust + δε rad ) − κ (cid:20) ( φ (cid:48) c ) Φ − φ (cid:48) c ϕ (cid:48) − a dVdφ ( φ c ) ϕ (cid:21) , (3.6) ∂ i Φ (cid:48) + H ∂ i Φ = κ φ (cid:48) c ∂ i ϕ (3.7)and 2 a (cid:20) Φ (cid:48)(cid:48) + 3 H Φ (cid:48) + Φ (cid:18) a (cid:48)(cid:48) a − H − K (cid:19)(cid:21) = κ (cid:20) δp rad − a (cid:0) φ (cid:48) c (cid:1) Φ + 1 a φ (cid:48) c ϕ (cid:48) − dVdφ ( φ c ) ϕ (cid:21) . (3.8)Here, according to the mechanical approach (see details in [19, 20]), we drop the termscontaining the peculiar velocities of the inhomogeneities and radiation as these are negligible– 4 –hen compared with their respective energy density and pressure fluctuations. However,such comparison with respect to the scalar field is not evident since the quantity treatedas the peculiar velocity of the scalar field is proportional to the scalar field perturbation ϕ .Therefore, in our analysis we propose the following strategy. First, we preserve the scalar fieldperturbation in (3.7) since we keep it in Eqs. (3.6) and (3.8). Then, a subsequent analysis ofthe equations must show whether or not we can equate to zero the right hand side (r.h.s.) ofEq. (3.7). In what follows, we shall demonstrate that for the coupled scalar field the r.h.s. ofthis equation can indeed be set to zero in a consistent way within the mechanical approach asit usually happens for the coupled fluids [29]. We also applied the standard reasoning to putΨ = − Φ [18], i.e., we have assumed absence of anisotropies. In Eq. (3.6), ∆ is the Laplaceoperator with respect to the metric γ αβ . From Eq. (3.7), we get the following relation:Φ (cid:48) + H Φ = κ φ (cid:48) c ϕ . (3.9)Let us consider now Eq. (3.8) in more detail. The substitution of Φ (cid:48) from (3.9) into(3.8) givesΦ (cid:20) H (cid:48) − H − K + κ
12 ( φ (cid:48) c ) (cid:21) = ϕ (cid:20) − κ φ (cid:48)(cid:48) c − H κφ (cid:48) c − a κ dVdφ ( φ c ) (cid:21) + κ a δp rad , (3.10)where we have also used the relation 2 a (cid:48)(cid:48) /a = 2( H (cid:48) + H ). With the help of the Eqs. (2.8)and (2.9), this equation finally takes the formΦ (cid:20) − a κ ¯ ε rad − a κ ¯ ε dust (cid:21) = κ a δp rad , (3.11)where we have taken into account the equation of motion (2.7). Because, ¯ ε rad ∼ /a and¯ ε dust ∼ /a , we can drop the first term in the brackets in the left-hand-side of this equationand obtain δp rad = − Φ¯ ε dust = − Φ ¯ ρc a = 13 δε rad , (3.12)similar to the expression (4.19) in [19]. Therefore, the spatial distribution of δε rad is definedby the gravitational potential Φ derived below in Eq. (3.32).Now, we turn to Eq. (3.6). Since (see [20]) δε dust = δρc a + 3 ¯ ρ Φ a , (3.13)where δρ is the difference between real and average rest mass densities for the dust-likematter: δρ = ρ − ¯ ρ (3.14)and taking into account (3.12), this equation reads∆Φ − H (Φ (cid:48) + H Φ) + 3 K Φ = κ δρc a − κ (cid:20) ( φ (cid:48) c ) Φ − φ (cid:48) c ϕ (cid:48) − a dVdφ ( φ c ) ϕ (cid:21) . (3.15)From (3.9) we get ϕ = Φ (cid:48) + H Φ κ φ (cid:48) c , (3.16) ϕ (cid:48) = Φ (cid:48)(cid:48) + H (cid:48) Φ + H Φ (cid:48) κ φ (cid:48) c − Φ (cid:48) + H Φ κ ( φ (cid:48) c ) φ (cid:48)(cid:48) c . (3.17)– 5 –he substitution of (3.16) and (3.17) into (3.15) gives:∆Φ − κ δρc a = Φ (cid:20) H − K − κ φ (cid:48) c ) + H (cid:48) − H φ (cid:48)(cid:48) c φ (cid:48) c + a dVdφ ( φ c ) 1 φ (cid:48) c H (cid:21) + Φ (cid:48) (cid:20) H − φ (cid:48)(cid:48) c φ (cid:48) c + a dVdφ ( φ c ) 1 φ (cid:48) c (cid:21) + Φ (cid:48)(cid:48) . (3.18)Since Φ (cid:48) = d Φ da a H , Φ (cid:48)(cid:48) = d Φ da a H + d Φ da a H (cid:48) + d Φ da a H , (3.19)then Eq. (3.18) can be written in the form∆Φ − κ δρc a = Φ (cid:20) H − K − κ φ (cid:48) c ) + H (cid:48) − H φ (cid:48)(cid:48) c φ (cid:48) c + a dVdφ ( φ c ) 1 φ (cid:48) c H (cid:21) + d Φ da a (cid:20) H + H (cid:48) − H φ (cid:48)(cid:48) c φ (cid:48) c + a dVdφ ( φ c ) 1 φ (cid:48) c H (cid:21) + d Φ da H a , (3.20)which after substitution Φ = Ω /a , where Ω is a function of a and the spatial coordinates,reads ∆Ω a − κ δρc a = − Ω a (cid:104) K + κ φ (cid:48) c ) (cid:105) + d Ω da (cid:20) H + H (cid:48) − H φ (cid:48)(cid:48) c φ (cid:48) c + a dVdφ ( φ c ) 1 φ (cid:48) c H (cid:21) + d Ω da a H . (3.21)We can use this equation to determine the unknown function Ω and, consequently, the grav-itational potential Φ.In what follows, the dust like matter component is considered in the form of discretedistributed inhomogeneities (e.g. galaxies and their group and clusters). Then, we arelooking for solutions of (3.21) which have a Newtonian limit near gravitating masses. Suchan asymptotic behaviour will take place if we impose Ω = Ω( (cid:126)r ) (see solutions (3.32) and(3.33) below). Moreover, such a choice is in agreement with the transition to the astrophysicalapproach (in other words, the Minkowski background limit) where a → const ⇒ H → ρ = 0 and φ (cid:48) c = 0), andwe should select the flat topology K = 0. In this limit, if the dust-like matter is described bythe discrete distributed gravitating sources (e.g. galaxies) with masses m i and the rest-massdensity ρ = (cid:88) i m i δ ( (cid:126)r − (cid:126)r i ) , (3.22)the gravitational potential Φ isΦ = − G N c a (cid:88) i m i | (cid:126)r − (cid:126)r i | = − G N c (cid:88) i m i | (cid:126)R − (cid:126)R i | , (3.23)as it should be [31]. In Eq. (3.23), we took into account the relations between the physicaland comoving radius vectors: (cid:126)R = a(cid:126)r . This equation also demonstrates that Φ ∼ /a . It can be easily seen that the condition Ω = Ω( (cid:126)r ) demands also ( φ (cid:48) c ) = const and vice versa – 6 –ow, we analyse the case Ω = Ω( (cid:126)r ) ⇒ Φ ∼ /a and ( φ (cid:48) c ) = const in more details. Letus denote φ (cid:48) c = β = const. Then we get φ c = βη + γ, γ = const . (3.24)The substitution of (3.24) into the equation of motion (2.7) gives:2 a (cid:48) a β + a dVdφ ( φ c ) = 2 a (cid:48) a β + a V (cid:48) β = 0 , ⇒ V = β a + V ∞ , V ∞ = const . (3.25)Obviously, V ∞ plays the role of the cosmological constant. It appears as a solution of theequations of motion. To achieve the accelerated expansion of the late Universe, there isno need to include the cosmological constant in the model by hand. This is an importantconsequence of our approach. To get the late cosmic acceleration, we must demand that V ∞ >
0. Using the data of the Planck mission in combination with other experiments [5],we obtain that the potential V is very flat at the present time (see Fig. 1 below). Therefore,the coupled scalar field can provide the late cosmic acceleration.For the background energy density and pressure (see Eqs. (2.5) and (2.6)) we obtain:¯ ε ϕ = 32 β a + V ∞ , ¯ p ϕ = − β a − V ∞ . (3.26)It can be easily seen that the considered scalar field behaves (at the background level) as a two-component perfect fluid: a cosmological constant and a network of frustrated cosmic strings[16, 24, 32]. The latter has the parameter of EoS w = − /
3. This behaviour is a consequenceof imposing the coupling between the scalar field perturbations and the inhomogeneities. Inother words, we derived a specific form for the time dependence of the background scalarfield and for its potential (see Eqs. (3.24) and (3.25), respectively) so that the scalar fieldis consistent with such a coupling to the inhomogeneities. Obviously, in the general case,when the coupling condition is not imposed, the scalar field is not limited to these specificsolutions.There is also another very important feature of the considered scalar field. Since Φ ∼ /a , then we find Φ (cid:48) + H Φ = 0. Hence, it follows from Eqs. (3.16) and (3.17) that fluctuationsof the scalar field are absent: ϕ = ϕ (cid:48) = 0. The physical reason of this is that the “coupling”between the inhomogeneities of the dust-like matter and of the scalar field imposes a strongrestriction on the scalar field. Nevertheless, the above analysis demonstrates that such ascalar field can exist, i.e. such scalar field does not contradict the above equations both atthe background and at the linear perturbation levels. On the other hand, the fluctuations ofthe energy density and pressure of the scalar field are non-zero (see Eqs. (3.2) and (3.4)): δε ϕ = δp ϕ = − a (cid:0) φ (cid:48) c (cid:1) Φ = − β a Ω( (cid:126)r ) (cid:54) = 0 . (3.27)These fluctuations arise due to the interaction between the scalar field background and thegravitational potential. It is well known that the energy densities and pressure but notfields are measurable values. Eq. (3.27) shows that δε ϕ ∼ /a in analogy also with thefluctuations of the energy density for a perfect fluid with the constant equation of stateparameter ω = − / δε ϕ = δp ϕ and the EoS parameter forthe fluctuations is δp ϕ /δε ϕ = 1 whereas for the fluctuations of the perfect fluid the EoS– 7 –arameter is still -1/3 as for the background matter. Therefore, these two models are notcompletely equivalent to each other.Additionally, the fluctuations of the energy density of the scalar field contribute to thegravitational potential. To prove it, we can rewrite Eq. (3.21) as follows (for the consideredcase Ω = Ω( (cid:126)r )): ∆Ω + 3 K Ω = κ (cid:16) δρc + (cid:103) δε ϕ (cid:17) , (3.28)where we have introduced the comoving fluctuations of the energy density of scalar field: (cid:103) δε ϕ = a δε ϕ = − (cid:0) φ (cid:48) c (cid:1) Ω = − β Ω . (3.29)Eq. (3.28) can also be written in the form∆ f − λ f = 4 πG N δρ , (3.30)where f ( (cid:126)r ) = c Ω( (cid:126)r ) and λ = − κ β − K . (3.31)This equation can be solved for any spatial topology [24]. For example, in the case of spatiallyflat ( K = 0) and hyperbolic ( K = −
1) geometries we get, respectively: f = − G N (cid:88) i m i | r − r i | exp ( − λ | r − r i | ) + 4 πG N ¯ ρλ , (3.32) f = − G N (cid:88) i m i exp( − l i √ λ + 1 )sinh l i + 4 πG N ρλ , < l i < + ∞ , (3.33)where l i denotes the geodesic distance between the i -th mass m i and the point of observa-tion. To obtain these physically reasonable solutions (i.e. solutions which have the correctNewtonian limit near the inhomogeneities and converge at spatial infinity), we impose that λ >
0. In the case of spatially flat topology, this means that β <
0. Eq. (3.26) showsthat the scalar field background energy density can be negative if V ∞ < − β /a . To avoid apossible problem with a ghost-like instability, we can impose the condition V ∞ > − β /a .Moreover, for a hyperbolic space, λ can acquire positive values if β is positive and suchpossible problem is absent for sure.In the case of spherical spatial topology ( K = +1) and for the physical reasonable values β >
0, we get λ <
0. Then, the solution of Eq. (3.30) is [24]: f = − G N (cid:88) i m i sin (cid:104) ( π − l i ) √ − λ (cid:105) sin (cid:16) π √ − λ (cid:17) sin l i + 4 πG N ρλ , < l i ≤ π . (3.34)For √ − λ (cid:54) = 2 , , . . . , this formula is finite at any point l i ∈ (0 , π ] and has the Newtonianlimit for l i → the fluctuations of the density of the scalar field are concen-trated around the inhomogeneities of the dust-like matter (i.e. around the galaxies) which In the flat case this condition should be replaced by λ > This condition defines a minimum scale factor a min = | β | / √ V ∞ such that for a > a min the energy densityof the scalar field is positive. – 8 – s in full agreement with the coupling condition. The presence of these fluctuations leads tothe screening of the gravitational potential as it follows from Eqs. (3.32)-(3.34). It was alsoshown in [24] that for all topologies of the space the solutions found for the gravitationalpotential satisfy the important condition that the total gravitational potential averaged overthe whole Universe is equal to zero: ¯ f = 0 ⇒ Ω = 0. This demand results in another physi-cally reasonable condition: δε ϕ = 0 (see Eq. (3.27)). It is obvious that the averaged value ofthe fluctuations should be equal of zero. In the previous section we have obtained the dependence of the scalar field potential com-patible with the mechanical approach (see Eq. (3.25)). We now seek to obtain the shape of V ( φ c ) as a function of the scalar field itself. In order to do this, we begin by writing theFriedmann equation as H = H c a (cid:20) Ω V ∞ + (Ω K + Ω β ) (cid:16) a a (cid:17) + Ω dust (cid:16) a a (cid:17) + Ω rad (cid:16) a a (cid:17) (cid:21) , (4.1)where Ω V ∞ ≡ κc V ∞ H , Ω K ≡ − K c H a , Ω β ≡ κc β H a , Ω dust ≡ κc ¯ ε dust , H = κc ¯ ρ c H a , Ω rad ≡ κc ¯ ε rad , H . (4.2)After some algebra and using Eq. (3.24) to replace the conformal time η by φ c , we canintegrate Eq. (4.1) from some initial value a i till a f and obtain φ c ( a f ) − φ c ( a i ) = ± A (cid:90) afa aia d ( a/a ) (cid:114)(cid:16) aa (cid:17) + c (cid:16) aa (cid:17) + c aa + c . (4.3)Here the dimensionless coefficients c i ’s are defined as c ≡ (Ω K + Ω β ) / Ω V ∞ , c ≡ Ω dust / Ω V ∞ , c ≡ Ω rad / Ω V ∞ , and the constant A is A ≡ (cid:115) κ Ω β Ω V ∞ . (4.4)It follows from current observations [5] that at the present time the main contributionsto the total energy density come from the cosmological constant and dust, i.e., | c | (cid:28) c < c > c >
0, and the integrand on the r. h. s. of the equation– 9 – - ϕ c / Δϕ c V ( ϕ c / A ) / V ∞ Figure 1 . The full scalar field potential obtained from Eq. (4.5) (blue full curve) and the late timequadratic approximation (red dashed curve) as functions of φ c / ∆ φ c . If initially at a value φ c = − ∆ φ c ,the scalar field rolls down the potential, as indicated by the black arrows, until it reaches the minimumof the potential at φ c = 0. Since this point corresponds to the distant future ( a → + ∞ ) it acts as awall, marked by the vertical line, that separates the regions with negative and positive values of φ c .The blue point indicates the values of the scalar field and the potential at the present time. behaves as 1 / ( a/a ) when a/a (cid:29)
1, we find that the integral is well defined and finite forall values of a i,f ∈ [0 , + ∞ ), as well as in the limit of a f → + ∞ . This means that fromthe distant past, a = 0, till the distant future, a → + ∞ , the variation of the scalar field,∆ φ c = | φ c (+ ∞ ) − φ c (0) | , is finite.Using the freedom in the definition of the integration constants, we can take the limit a f → + ∞ , set φ c (+ ∞ ) = 0, and rewrite Eq. (4.3) as φ c ( a ) = ± A (cid:90) + ∞ aa d ( a/a ) (cid:114)(cid:16) aa (cid:17) + c (cid:16) aa (cid:17) + c aa + c . (4.5)The solution with a + ( − ) sign can then be identified with the case where the scalar field rollsdown the potential to the left (right) of the limiting value φ c (+ ∞ ), i.e. with negative (posi-tive) values of φ c . The relation (4.5) can be inverted numerically and inserted in Eq. (3.25)in order to obtain V ( φ c ). In the future, when a/a (cid:29)
1, we can take the approximation thatonly the quartic term inside the square root contributes to the integral in Eq. (4.5). In thiscase we find that the potential is given by V (cid:39) V ∞ + β a (cid:18) φ c A (cid:19) = V ∞ (cid:34) β λ (cid:18) φ c A (cid:19) (cid:35) . (4.6)In Fig. 1 we present the shape of the full potential obtained, as well as the approximationfor large a/a (see Eq. (4.6)). We mark the present time values of the potential and scalarfield by a blue point on the curve of the potential. From the results of the Planck mission[5] we obtained the values Ω V ∞ = 0 . dust = 0 . rad = 9 . × − . The value– 10 –f Ω K + Ω β was set at a conservative value of 10 − . We would like to remind the readerthat the observational constraints on the curvature term Ω K cannot be applied to a generalterm whose contribution to the energy density evolves as 1 /a . Therefore, Ω β remains a freeparameter constrained only by the requirement that Ω β (cid:28) Ω dust , Ω V ∞ . In our paper, we have considered the late stage of the Universe evolution in the case whenour Universe is filled with a minimal scalar field. We have also included dust-like matter(in the form of discrete distributed galaxies and groups of galaxies) and radiation as mattersources. To study the motion of galaxies in this Universe, we should know the distributionof the gravitational potential, which can be found from the theory of scalar perturbations.We considered this theory in the mechanical approach [19, 20]. In this approach, all typesof inhomogeneities, e.g. galaxies as well as inhomogeneities associated with the fluctuationsof other form of matter, have non-relativistic velocities. In this case, different types ofinhomogeneities do not run away considerably during the Universe evolution. Moreover,fluctuations of the energy density of such perfect fluids are usually concentrated around theinhomogeneities of dust-like matter (i.e. galaxies). From this point, we call those perfectfluids ”coupled” [29]. They can screen the gravitational potential of galaxies [24] and canalso play the role of dark matter flattening the rotation curves of dwarf galaxies [25].In the present paper, we have investigated the possibility for a scalar field to be coupledwith galaxies in the late Universe. For such scalar fields to exist, we have shown that theyhave to meet certain conditions. First, at the background level, such scalar field behaves as atwo component perfect fluid: a network of frustrated cosmic strings with the EoS parameter w = − / Acknowledgements
A.Zh. acknowledges the hospitality of UBI during the completion of a part of this work. Theauthors would like to thank Maxim Eingorn for his contribution during the initial phase ofwork and for the useful discussion of the obtained results. J.M. is thankful to UPV/EHU– 11 –or a PhD fellowship and UBI for the hospitality during the completion of part of this workand acknowledges the support from the Basque government Grant No. IT592-13 (Spain) andFondos FEDER, under grant FIS2014-57956-P (Spanish Government). The work of MBL issupported by the Portuguese Agency “Funda¸c˜ao para a Ciˆencia e Tecnologia” through anInvestigador FCT Research contract, with reference IF/01442/2013/CP1196/CT0001. Shealso wishes to acknowledge the support from the Basque government Grant No. IT592-13(Spain) and Fondos FEDER, under grant FIS2014-57956-P (Spanish Government). K.S.K.is grateful for the support of the Grant SFRH/BD/51980/2012 from the Portuguese Agency“Funda¸c˜ao para a Ciˆencia e Tecnologia”. This research work was supported by the Portuguesegrant UID/MAT/00212/2013.
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