Unified dark matter and dark energy description in a chiral cosmological model
OOctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM
Modern Physics Letters Ac (cid:13)
World Scientific Publishing Company
UNIFIED DARK MATTER AND DARK ENERGY DESCRIPTIONIN A CHIRAL COSMOLOGICAL MODEL
RENAT R. ABBYAZOV
Department of Physics, Ulyanovsk State Pedagogical University named after I.N. Ulyanov,100 years V.I. Lenin’s Birthday Square, 4, 432700 Ulyanovsk, [email protected]
SERGEY V. CHERVON
Astrophysics and Cosmology Research UnitSchool of Mathematics, Statistics and Computer Science, University of KwaZulu-NatalPrivate Bag X54 001, Durban 4000, South Africa ∗ [email protected] Received (Day Month Year)Revised (Day Month Year)We show the way of dark matter and dark energy presentation via ansatzs onthe kinetic energies of the fields in the two-component chiral cosmological model. Toconnect a kinetic interaction of dark matter and dark energy with observational datathe reconstruction procedure for the chiral metric component h and the potential of(self)interaction V has been developed. The reconstruction of h and V for the earlyand later inflation have been performed. The proposed model is confronted to Λ CDM model as well.
Keywords : Chiral cosmological model; cosmic acceleration; dark energy; dark matter.PACS Nos.: 98.80.-k, 95.36.+x
1. Introduction
The later-time cosmic acceleration of our Universe is strongly supported by obser-vational data. Namely observations of supernovae type Ia , the data from BaryonAcoustic Oscillations (BAO) and Cosmic Microwave Background (CMB) measure-ments confirm that the Universe is expending with an acceleration at the presenttime and about 70% of the energy density consists of dark energy in a wide sense ,i.e. as the substance which is responsible for an anti-gravity force.In the range with well–known ΛCDM model, which potentially provides correctdescription of the Universe evolution but suffers from fine–tuning and coincidence ∗ The permanent address: Department of Physics, Ulyanovsk State Pedagogical University namedafter I.N. Ulyanov, 100 years V.I. Lenin’s Birthday Square, 4, 432700 Ulyanovsk, Russia1 a r X i v : . [ g r- q c ] A p r ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon problems, some alternative models were proposed. We will pay attention to themodels with presence of scalar fields included in quintessence, phantom and quintom , , , models.A chiral cosmological model (CCM) as a nonlinear sigma model with a po-tential of (self)interactions has been already used extensively in various areas ofgravitation and cosmology , , and in particular for description of the very earlyUniverse , and inflation , . A CCM can be applicable as well to the late-timeUniverse with dark matter and dark energy domination as it was shown in .The purpose of this article is to put into use the two-component CCM as themodel where the dark energy content of the Universe and also the dark matter com-ponent are represented by two chiral fields with kinetic and potential interactions .By considering a target space metric in the form ds σ = h dϕ + h ( ϕ, χ ) dχ , h = const. (1.1)we prescribe a kinetic interaction between chiral fields ϕ and χ as a functional de-pendence h on the fields. The potential interaction will be included into standardpotential energy term of the action.There are no enough indications from observations about kinetic interactionsbetween dark sector fields. Therefore we always deal with the problem: what is thefunctional dependence for the chiral metric component on the fields? First idea isto attract some results from HEP, for example, to consider SO(3) symmetry (bytaking h = sin ϕ ) and/or others symmetries for a chiral space. From the otherhand one can use some testing kinetic interactions , .Thus we can state that there is no evidence for some preferable functional formof the kinetic interaction contained in the functional form of the h chiral metriccomponent. To avoid this problem we develop here the reconstruction procedurefor the chiral metric component h . We ascribe a certain desirable behavior onthe kinetic energy of the second chiral field χ and it becomes possible to determineboth the target space metric component h and a (self)interacting potential V depending on the first chiral field ϕ . So we can restore a functional dependence the h and V on the scalar field ϕ using observational data. Unfortunately it turnsout that the procedure could not be applied for the entirely Universe evolution andwe have necessity to consider separately the early and late epochs of the Universeevolution.It will be shown also that a CCM describes dark energy and dark matter inthe unified form under special restrictions on the chiral fields (ansatzs). Thereforeto include into consideration the present Universe with accelerated expansion itneeds to take into account baryonic matter and radiation in the range with a two-component CCM.Making confrontation of proposed model predictions with observational data wefound the way of a reconstruction of a kinetic interaction term h and the potential V in an exact form. This reconstruction is based on the procedure of finding thebest–fit values matching to the astrophysical observations.ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model The structure of the article is like follow. In section 2, we give the basic modelequations and discuss their properties including the exact solutions for a pure CCM(without matter and radiation). We derive the Friedmann equation for the pro-posed model with the aim to make comparison with ΛCDM in section 3. In section4, we give the details of a fitting procedure outline. We present the way of thereconstruction of the kinetic coupling and potential in section 4. The early and re-cent Universe approximations are discussed there as well. Section 6 is devoted tothe background dynamics of a CCM. Finally in section 7, we discuss the obtainedresults and consider perspectives for the future investigations.
2. The model equations and their properties
Recently we proposed a CCM coupling to a perfect fluid with the aim to inves-tigate chiral fields interaction with CDM. For the sake of shortness we termed thismodel as σCDM to stress its difference from Λ CDM, QCDM and others mod-els. σCDM model presents a generalization of a single scalar field model coupledto CDM in the form of a perfect fluid . The model is described by the actionfunctional S = (cid:90) d x √− g (cid:18) − g µν h AB ∂ µ ϕ A ∂ ν ϕ B − V ( ϕ C ) (cid:19) + S ( pf ) . (2.1)Here S ( pf ) stands for the perfect fluid part of the action, h AB = h AB ( ϕ C ) are thetarget space metric components depending on the scalar fields ϕ C . The line elementof a target (chiral) space is ds σ = h AB ( ϕ C ) dϕ A dϕ B . (2.2)We use shortened notations for the partial derivatives with respect to the space-time coordinates: ∂ϕ A ∂x µ = ∂ µ ϕ A . As usual g µν ( x α ) denotes a space-time metric asa function on the space-time coordinates, so Greek indices α, µ, ... vary in a rangefrom 0 to 3, Latin capital letters A , B, ... – take values from 1 to N where N isevidently corresponding to the chiral fields number.The space-time of homogeneous and isotropic Universe is described by aspatially-flat Friedmann – Robertson – Walker (FRW) metric ds = − dt + a ( t ) (cid:0) dr + r (cid:0) dθ + sin θdφ (cid:1)(cid:1) . (2.3)The two-component CCM has a target space metric simplified to ds σ = h dϕ + h ( ϕ ) dχ , h = const. (2.4)The σCDM (2.1) with internal space metric (2.4) includes the models proposedearlier: cold dark matter and cosmological constant (ΛCDM, when h = h =0 , V = const = Λ) model , , quintessence model (QCDM, when h = 1 , h = 0),phantom model (PhCDM, when h = − , h = 0), quintom model (qCDM, whenctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon h = 1 , h = − , , , . Thus the model under consideration is a generalizationof the models investigated earlier and mentioned above.As a first step of our study we consider the system of equations of the two-component CCM without a perfect fluid. Using assumptions h = const and h = h ( ϕ ) expressed in (2.4) one can obtain the system of Einstein and chiral fieldequations H = 8 πG (cid:20) h ˙ ϕ + 12 h ˙ χ + V ( ϕ, χ ) (cid:21) , (2.5)˙ H = − πG (cid:20) h ˙ ϕ + 12 h ˙ χ (cid:21) , (2.6)¨ ϕ + 3 H ˙ ϕ − h dh dϕ ˙ χ + 1 h ∂V∂ϕ = 0 , (2.7)¨ χ + 3 H ˙ χ + 1 h dh dϕ ˙ ϕ ˙ χ + 1 h ∂V∂χ = 0 . (2.8)Here H = ˙ aa , (˙) = ddt .When first inflationary models were analyzed it was much attention to a verysimple case when an inflationary potential V ( φ ) equals to the constant . Moreoverthis regime is very important because it leads to an exponential expansion of theUniverse. Note that a scalar field is equal to a constant value as well in this regime.Let us consider for a minute the case of V = const for the model under consid-eration (2.5)-(2.8). From (2.8) one can obtain , ˙ χ = 2 Ch a . (2.9)Combining (2.5) and (2.6) one can obtain the well-known solution of a de SitterUniverse with Hubble parameter and scale factor H = (cid:114) Λ3 tanh( √ t ) , a = a ∗ [cosh( √ t )] / . (2.10)This solution with some approximation corresponds to the inflationary stage of theUniverse evolution. But our intention is to proceed further in time therefore we needto include into consideration radiation and matter to describe the present epoch ofthe Universe.The method of the exact solutions construction for a CCM (2.5)-(2.8) is basedon exploiting an additional degree of freedom (see, for ex. discussion in ). Namelyeven we fix the potential V ( φ, χ ) there are still four equations with four unknownfunctions H, ϕ, χ, h ( h can be set equal to ± ).Nevertheless the equation (2.5) can be obtained from the linear combination of thechiral field equations (2.7)-(2.8), so the equation (2.5) doesn’t independent one.Therefore one may insert the symmetry on the target space or can suggest a testingctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model interaction between chiral (dark sector) fields , . Essentially new approach to thisissue we propose here as a reconstruction both h and V from observational data.Let us remind that for the scalar field cosmology by introducing the selfinteract-ing potential V ( φ ) we have two equations with two unknown functions. (The samesituation will be if we set the dependence a scalar field on time or if we know thescale factor of the Universe as a function on time ).To solve the system of a CCM interacting with a perfect fluid (or matter) inexplicit form is a very difficult task. Therefore we will use an additional freedomconnecting with the chiral metric components h and h as a part of a kineticenergy.An interesting approach for a two-fields model with a cross interaction wasproposed in the work . To describe a dark matter component it was constructedthe special ansatzs for the time derivatives of the scalar fields. In our approach wewill use instead some constraints on the kinetic parts of the chiral fields (ansatzs)to obtain a correct description of the present Universe.Now let us turn our attention to a study of the model equations (2.5)-(2.8). Itis easy to check that the solution (2.9) for the constant potential will be valid forthe case when V = V ( ϕ ) only. By extracting from (2.9) the kinetic energy term forthe field χ one can obtain 12 h ˙ χ = Ch a . (2.11)We can ascribe by suggestion h ∼ a − dust matter like behavior to the kineticenergy of the field χ . Using the behavior h ∼ a − it is easy to see that the secondfield can be related to the dark matter term provided the restriction to the kineticenergy of the second field χ (ansatz)12 h ˙ χ = Ca − . (2.12)Let us mention here, that more simple ansatz h ˙ χ = Λ ψ = const has been ana-lyzed in and gave possibility to obtain the exact solutions for the two-componentCCM. For the kinetic energy of the first field ϕ we can form the ansatz by a simpleway 12 h ˙ ϕ = B = const. (2.13)Further we will show that this relation is associated with the dark energy componentin the present Universe.For convenience let us represent the ansatzs (2.11), (2.13) in the general forms:12 h ˙ ϕ = f ( a ) , (2.14)12 h ˙ χ = g ( a ) . (2.15)ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon
Thus we have f ( a ) = B = const, g ( a ) = Ca − in (2.14)-(2.15) and the chiral metriccomponent h = a − . (2.16)Let us note that the suggested restrictions above give rise to the exact solutionfor the CCM describing by equations (2.5)-(2.8). Indeed from ansatzs we can findthe solutions for the chiral fields ϕ = (cid:114) Bh t + ϕ , χ = √ Ct + χ . (2.17)Then from Einstein equations (2.5)-(2.6) we can define the potential V ( a ) = − B ln a + Ca − + V ∗ . (2.18)The solution for the scale factor can be obtained from the equation H = C ∗ a + 2 κ (cid:18) B − B ln a + C a + V ∗ (cid:19) . It is difficult to find the scale factor in exact view from this general equation,but for the special case assuming C ∗ = 0 and C = 0 (under this assumption thesecond field χ becomes a constant), we found that the Universe is in the stage withan exponential expansion with a ∝ exp( Bt ).
3. A CCM coupling to barion matter and radiation.Friedmann equation of the model
Our following task is to connect the energy densities of various species of the Uni-verse to the Hubble parameter. To this end we need to include into Friedmannequation (2.5) the energy density of barion matter ρ b and radiation ρ r . Thus (2.5)for the recent Universe takes the form H = 8 πG ρ σ + ρ b + ρ r ] (3.1)where ρ σ = h ˙ ϕ + h ˙ χ + V . Introducing the ”pressure” of chiral fields p σ = h ˙ ϕ + h ˙ χ − V and using ansatzs (2.14) and (2.15) we can obtain ρ σ = f + g + V, p σ = f + g − V. Using (2.18) and extracting the cosmological parameter Λ from V ∗ the energydensity, potential and pressure of the two-component CCM can be expressed as ρ σ = Λ − B ln a + 2 Ca − , Λ = B + V ∗ (3.2) V = Λ − B ln a + Ca − − B, (3.3) p σ = 2 B − Λ + 6 B ln a. (3.4)ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model By standard way (see, for ex. ) one can define a critical density ρ c = H πG , where H is the Hubble parameter of today expansion H = ˙ aa ( t ) . Herefrom thesubscript ”0” is related to the present time t when the scale factor a ( t ) = a = 1 . Also we will use the density parameter Ω = ρρ c ( t ) and the individual rationsΩ i = ρ i ρ c ( t ) for chiral fields, barion matter and radiation.Let us remember that equations of state for radiation and baryons are p r = 13 ρ r , p b = 0 , . The energy densities and the contribution to the critical density can be representedas ρ r = ρ r a − = Ω r ρ c a − , ρ b = ρ b a − = Ω b ρ c a − , ρ c = 3 H πG . Taking into account (3.2) Friedmann equation (3.1) can be transformed to thenormalised Hubble parameter form H H = 1 ρ c (cid:0) Λ − B ln a + 2 Ca − (cid:1) + Ω b a − + Ω r a − . Making renormalization of the constants we finally obtain the normalised Hubblerate in the form which is suitable for further confronting with observational data˜ H = H H = ˜Λ − B ln a + 2 ˜ Ca − + Ω b a − + Ω r a − , (3.5)where ˜ B = Bρ c , ˜ C = Cρ c , ˜Λ = Λ ρ c , ˜ H = H H . (3.6)We need to find ˜Λ at a = a = 1 with the help of Friedmann equation. Colddark matter (CDM) is included in the model as the kinetic ansatz (2.11)˜Λ = 1 − C − Ω b − Ω r = Ω σ Λ0 , Ω σcdm = 2 ˜ C, Ω m = Ω σcdm + Ω b . (3.7)Summing up the notations above we display the final form of the normalised Hubbleparameter ˜ H = Ω σ Λ0 − B ln a + Ω σcdm a − + Ω b a − + Ω r a − . (3.8)We propose here the σ CDM model containing the dark energy with variableequation of state. The model is an alternative to ΛCDM model with the cosmologicalconstant and CDM. Let us note that generally speaking the presence of ˜ B in (3.8)may change the values of Ω m and Ω Λ0 , thus they can be distinctive from thecorresponding quantities in ΛCDM model. Nevertheless to find the exact values ofthis distinction we need to perform comparison with the experimental data.ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon
4. Comparison with experimental data
From the very beginning , supernovae Ia type observations directly indicated anaccelerated expansion of the Universe. Observing supernovae luminosity distance d L as a function of a redshift one can infer about an expansion history of the Universe.Here we use one of the most recent compilation of the supernovae sets Union 2.1 . The procedure of confronting cosmological model predictions with observationsconsists of minimizing quantity procedure and calculation as a result best–fit valuesof the model parameters for χ SN = N (cid:88) i =1 [ µ obs ( z i ) − µ ( z i )] σ i ( z i ) . (4.1)Here as usual in the supernovae experimental analysis module distance µ ( z i ) isused. The dependence on the luminosity distance is µ ( z i ) = 5 log [ D L ( z i )] + µ , D L = H d L , ˜ H = H/H . (4.2) µ = 5 log (cid:20) H − M pc (cid:21) + 25 = 42 . − h, H = h − . (4.3)In order to find more accurate parameter values and to reduce errors significantlyit is necessary to supplement the supernovae observations with information aboutbaryonic acoustic oscillations (BAO) and cosmic microwave background (CMB) . BAO χ function is defined as χ BAO = (cid:18) D V ( z = 0 . /D V ( z = 0 . − . . (cid:19) , (4.4)where D V ≡ (cid:20) (1 + z ) D A ( z ) zH ( z ) (cid:21) / (4.5)is an effective distance measure, while D A = (1 + z ) − d L ( z ) (4.6)is the angular diameter distance .Function χ for CMB is χ CMB = ( x thi − x obsi )( C − ) ij ( x thj − x obsj ) , (4.7)where x i = ( l A , R, z ∗ ) — the vector of quantities which characterizes the cosmologi-cal model and ( C − ) ij — WMAP7 covariance matrix . Here we use acoustic scale,ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model from which first acoustic peak of CMB power spectrum is depending on l A ≡ (1 + z ∗ ) πD A ( z ∗ ) r s ( z ∗ ) , (4.8)which has been taken at the moment z ∗ of decoupling of radiation from matter, andon the sound horizon r s ( z ) = 1 √ (cid:90) / (1+ z )0 daa H ( a ) (cid:112) b / γ ) a . (4.9)We will use the fitting formula z ∗ = 1048[1 + 0 . b h ) − . ][1 + g (Ω m h ) g ] , (4.10) g = 0 . b h ) − . . b h ) . , g = 0 . . b h ) . , (4.11)for decoupling moment . Shift parameter R is defined as R ( z ∗ ) = (cid:113) Ω m H (1 + z ∗ ) D A ( z ∗ ) . (4.12)Minimizing the sum χ joint = χ SN + χ BAO + χ CMB one can find the best–fit ˜ B and ˜ C values. We also keep fixed the radiation and baryonic contributionsto the critical density today Ω γ = 2 . · − h − , Ω b = 0 . · − , h =0 . r = (1 + N eff )Ω γ , where N eff = 3 .
04 — the effectiveneutrino number . Our results for the best–fit from χ joint minimization are ˜ B =0 . , Ω σm = Ω b + 2 ˜ C = Ω b + Ω σcdm = 0 . m = 0 .
27 and Ω Λ0 = 1 − Ω m − Ω r . To avoid confusion betweenΩ cdm and Ω m in σ CDM and ΛCDM models we put additional index σ in Ω above.
5. The reconstruction of the metric component h andthe potential V In order to learn more about kinetic and potential interactions between DM andDE we must extend the standard reconstruction of the expansion history of theUniverse to a restoration of a functional dependence on the scalar field ϕ for thetarget space metric component h = h ( ϕ ) and the potential V = V ( ϕ ) of σ CDM.Let us transform the ansatz (2.14) (setting h = 1)12 ˙ ϕ = B = f, (cid:18) dϕdt (cid:19) = B, (5.1)ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon . . . . . . µ z Union2.1 σ CDM
Fig. 1. Supernovae Union 2.1 data and prediction from σCDM model.
Changing variables from t to a (cid:18) dϕda (cid:19) = (cid:18) dtda (cid:19) B, (5.2)and introducing already known for us (3.8) Hubble parameter one can obtain12 ϕ (cid:48) = BH a ˜ H . (5.3)Our next goal is to find the dependence ϕ = ϕ ( a ), so we fix limits of integrationfrom some early epoch a i up to desired time moment, corresponded to a (cid:90) aa i H ϕ (cid:48) da = √ (cid:90) aa i √ Bdaa ˜ H . (5.4)The integral written here cannot be taken in an explicit form. Nevertheless thereis a possibility to use some approximations based on a behavior of the differentenergy densities components with a scale factor. Therefore following the idea of we consider the early a (cid:28) a ≈ a (cid:28)
1. The Universe is known to be radiation dom-inated at the very early times. This means that all components contribution inHubble parameter is negligible in comparison with the radiation term. So such ob-servation makes possible to do integration inctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM
Unified dark matter and dark energy description in a chiral cosmological model . . . . . .
21 0 .
22 0 .
23 0 .
24 0 .
25 0 . ˜ B Ω m σ σ best − fit Fig. 2. Contour plots corresponding to 1 σ (68%) 1 σ and 2 σ (95%) likelihood levels for σ CDMmodel parameters. H ( ϕ ( a ) − ϕ ( a i )) = (cid:90) aa i √ Ba (cid:113) Ω r a da = √ B √ r (cid:0) a − a i (cid:1) . We can normalize the scalar field on today’s critical density ϕ → ϕ √ ρ c , whichgives us ˜ B instead of B in (5.3). Here ϕ ( a i ) = ϕ ( a = a i ), where a i is the fixed valueof a scale factor which will be taken equal to 10 − and normalized to current value a . It will be helpful for our analysis to introduce ˜ ϕ early = ϕ ( a i ) − √ ˜ BH √ r a i = ϕ ( a i ) + const and ϕ early = ϕ ( a i ).Now we have to invert ϕ = ϕ ( a ) dependence H ( ϕ − ˜ ϕ early ) = (cid:112) ˜ B √ r a to get the a = a ( ϕ ) dependence a = (cid:115) H √ r (cid:112) ˜ B ( ϕ − ˜ ϕ early ) . If we know a we can write down the chiral metric component h as a functionon ϕ ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon . . . . . . . . . e −
05 0 . .
001 0 .
01 0 . Ω a Ω r Ω σr Ω b Ω σb Ω m Ω σm Ω de Ω σde Ω σ Fig. 3. Evolution of contributions to critical density of various components in ΛCDM and σCDM models h = a − = (cid:32) H √ r (cid:112) ˜ B ( ϕ − ˜ ϕ early ) (cid:33) − / Taking the constant V = Λ − B in (3.3) one can obtain V = V − B ln (cid:34) H √ r (cid:112) ˜ B ( ϕ − ˜ ϕ early ) (cid:35) + C (cid:34) H √ r (cid:112) ˜ B ( ϕ − ˜ ϕ early ) (cid:35) − / . Thus we have finished the procedure of reconstruction of h and V for the veryearly epoch of the Universe evolution when a (cid:28) a ≈ H ( a ) = 1 + Ω r a (1 − a ) + Ω m a (cid:0) − a (cid:1) − B ln a, we obtain˜ H ( a ) = 1 + Ω r (1 − (1 − a )) − Ω r + Ω m (1 − (1 − a )) − Ω m − B ln(1 − (1 − a )) . (5.5)ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model − − . − . − . − . . . . . e −
05 0 . .
001 0 .
01 0 . q a − . . . . . q σ q ΛCDM a σ acc = 0 . a ΛCDM acc = 0 . Fig. 4. Evolution of the decelaration parameter in ΛCDM and σCDM models
Let us apply the Taylor expansion about (1 − a ) ≈ H ( a ) = 1 + (cid:16) r + 3Ω σm + 6 ˜ B (cid:17) (1 − a ) , Ω σm = Ω b + Ω σcdm . (5.6)From this moment we are able to fulfill the reconstruction procedure. Dividingscalar field on √ ρ c once again we come to H ( ϕ ( a ) − ϕ ( a i )) = (cid:90) aa i (cid:112) Bdaa (cid:114) (cid:16) m + 4Ω r + 6 ˜ B (cid:17) (1 − a ) = (cid:90) aa i (cid:112) Bdaa √ α − βa , (5.7)where β = 3Ω m + 4Ω r + 6 ˜ B = 3 · .
23 + 4 · · − · (1 + 0 .
6) + 6 · . > ,α = 1 + β > . It is essential for the subsequent analysis that the best–fit value of ˜ B is known,so we have an opportunity to make calculation of the integral (5.7). Such a type ofan integral is calculated by (cid:90) dx √ x ( x − b ) = − √ b ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) √ x + √ b √ x − √ b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , b > . ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon − − . . e −
05 0 . .
001 0 .
01 0 . ω a − . − . . . . . ω σeff ω ΛCDM eff ω ΛCDM de a σ acc = 0 . a ΛCDM acc = 0 . Fig. 5. Evolution of the effective equation of state parameter in ΛCDM and σCDM models
One more issue is about a transition through a = 1. This scale factor valueshould be explicitly presented in the expression for ϕH ( ϕ − ϕ ( a = a i )) = (cid:112) B (cid:26)(cid:90) a i daa ˜ H + (cid:90) a daa ˜ H (cid:27) . (5.8)With the help of˜ ϕ recent = ϕ ( a = a i ) + (cid:112) BH √ α ln (cid:12)(cid:12)(cid:12)(cid:12) √ α − βa i + √ α √ α − βa i − √ α (cid:12)(cid:12)(cid:12)(cid:12) , and ϕ recent = ϕ ( a = a i ) + (cid:112) BH √ α (cid:20) − ln (cid:12)(cid:12)(cid:12)(cid:12) √ α − β + √ α √ α − β − √ α (cid:12)(cid:12)(cid:12)(cid:12) + ln (cid:12)(cid:12)(cid:12)(cid:12) √ α − βa i + √ α √ α − βa i − √ α (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , one can carry out computaions further in more compact form. Plausibility ofthe early and recent approximations can be deduced from the comparison of H ( ϕ − ϕ early ) and H ( ϕ − ϕ recent ) for the exact (3.8) and approximate (5.6) Hubbleparameter expressions and the time limits (for which corresponding approximationsare hold on) will be extracted graphically.ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model e − e − . e − e − . e − e −
05 3 e −
05 5 e −
05 7 e −
05 9 e − H ( ϕ − ϕ e a r l y ) aapproximated ˜ Hexact ˜ H Fig. 6. Early Universe approximation.
The reconstruction is performed as usual when a = a ( ϕ ) is obtained. Usingnotation of the ˜ ϕ recent in (5.8) we have H ( ϕ − ˜ ϕ recent ) = − (cid:112) B √ α ln (cid:12)(cid:12)(cid:12) √ α − βa + √ α √ α − βa − √ α (cid:12)(cid:12)(cid:12) , and a = α ( α −
1) cosh ( A ( ϕ )) , A ( ϕ ) = − √ α (cid:112) B H ( ϕ − ˜ ϕ recent ) . We can substitiute this result to h (2.16) and V (3.3) to get h = (cid:18) α − α (cid:19) cosh − ( A ( ϕ )) ,V = V − B ln (cid:20) α ( α −
1) cosh ( A ( ϕ )) (cid:21) + C (cid:20) α ( α −
1) cosh ( A ( ϕ )) (cid:21) − .
6. Background dynamics of the model
One of the most important cosmological parameter used for the description of abackground evolution is a contribution to a critical density of the Universe. Thectober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM R. R. Abbyazov, S. V. Chervon − . − . − . − . − . . . . . . . H ( ϕ − ϕ r e c e n t ) aapproximated ˜ Hexact ˜ H Fig. 7. Recent Universe approximation. last is defined as Ω = ρρ c . For the chiral fields sector we haveΩ σ = ρ σ ρ c = ρ σ H ˜ H πG . Using (3.2) and (3.8) one can obtainΩ σ = ˜Λ − B ln a + 2 ˜ Ca − Ω σ Λ0 + Ω σcdm a − + Ω b a − + Ω r a − − B ln a . In order to understand a general picture of the Universe evolution and to analyzeperiods of domination by various species of the Universe we represent the residualcomponents of σ CDM modelΩ σr = Ω r a − ˜ H , Ω σb = Ω b a − ˜ H , Ω σm = Ω σm a − ˜ H , Ω σde = Ω σ Λ0 − B ln a ˜ H , where ˜ H comes from (3.8). The latter quantity is responsible for the late acceleratedexpansion of the Universe, supported by σ CDM model.In the ΛCDM we haveΩ r = Ω r a − ˜ H , Ω b = Ω b a − ˜ H , Ω m = Ω m a − ˜ H , Ω de = Ω Λ ˜ H . ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM Unified dark matter and dark energy description in a chiral cosmological model Here ˜ H is given by˜ H = Ω Λ0 + Ω m a − + Ω b a − + Ω r a − . (6.1)Let us turn our attention to the effective equation of the state parameter ω eff = (cid:80) α p α (cid:80) α ρ α . It is necessary to take into account expressions for densities and pressures of thechiral fields (3.2), (3.4) and the other components, together with (3.8) ρ r = ρ r a − = Ω r ρ c a − , ρ b = ρ b a − = Ω b ρ c a − , p r = 13 ρ r , p b = 0 . Then in the σ CDM model we will have ω σ ( eff ) = p r + p b + p σ ρ r + ρ b + ρ σ = Ω r a − + (cid:16) − Ω σ Λ0 + 6 ˜ B ln a + 2 ˜ B (cid:17) Ω r a − + Ω σm a − + Ω σ Λ0 − B ln a . At the same time for ΛCDM model the effective equation of state parameter is ω ΛCDM( eff ) = 12 2Ω r a − + Ω m a − + ( − Λ0 )Ω Λ0 + Ω m a − + Ω b a − + Ω r a − . Using the definition of the deceleration parameter q broadly used for backgrounddynamics studies we can obtain q = − ¨ aa ˙ a = πG ( (cid:80) α ρ α + 3 p α ) πG (cid:80) α ρ α . Presence in the q the second derivative of the scale factor gives us evidence ofthe transition from decelaration to acceleration epoch at the time when q = 0. Thedeceleration parameter for chiral sector takes the view q σ = 12 Ω σm a − + 2Ω r a − + 12 ˜ B ln a − σ Λ0 + 6 ˜ B ˜ H , where ˜ H is defined in (3.8). For ΛCDM model the expression for decelerationparameter looks like q ΛCDM = 12 (cid:0) Ω b a − + 2Ω r a − + Ω cdm a − − Λ0 (cid:1) ˜ H with Hubble parameter taken from (6.1). The evident differences both in numeratorand denominator in q σ and q ΛCDM inevitably lead to distinctive evolution of theUniverse if it is supported by σ CDM or ΛCDM models.It is known feature of ω eff that a moment of time of deceleration/accelarationtransition corresponds to value − / q analysis. We would like to pointout here that in ΛCDM model equation of state of the dark energy parameter isequal ω ΛCDM de = − R. R. Abbyazov, S. V. Chervon
7. Discussion
Fig.1 shows good agreement of supernovae data with σ CDM model taken with thebest–fit parameters values. This fact confirm the validity of proposed model, i.e., σ CDM model does not contradict to observational data and may serve as a gooddynamical alternative to ΛCDM.In fig. 2 the confidence contours are depicted. We keep only positive values ofparameter ˜ B in order to prevent a crossing of the phantom divide.One can see from the evolution of the individual densities Ω i , deceleration pa-rameters q and effective equation of state parameters ω eff (figs. 3, 4 and 5) thataccelerated expansion takes place earlier in the Universe supported by σ CDM. Alsoone may notice that radiation/matter domination transition occurs earlier in ΛCDMmodel. These observations are in the full agreement with a smaller total matteramount including cold dark and baryonic components in σ CDM model in compar-ison to ΛCDM model.The graphical comparison (see fig. 4 and fig. 5) of the a ΛCDM acc and a σacc (takenfrom the scale factor values corresponding to − / q and ω eff values and has been alreadymentioned above.From fig. 6 one can conclude that the early Universe approximation holds for a = 10 − up a = 5 · − scale factor values. The recent Universe approximationdepicted on fig. 7 is true from a = 0 . a = 1 . H ( ϕ − ϕ early )and H ( ϕ − ϕ recent ). The deviation for approximated and exact ˜ H is associatedwith the lost of domination of DE for the early times.In conclusion it needs to stress that we first time reconstructed from observationsthe kinetic interaction between DM and DE in the form of chiral metric component h for σ CDM. Also we have hope that the reconstruction techniques presentedhere may be useful for exact solution construction because of obtaining h fromobservational data. Acknowledgments
SVC is thankful to the University of KwaZulu-Natal, the University of Zululandand the NRF for financial support and warm hospitality during his visit in 2012 toSouth Africa where the part of the work was done. RRA is grateful to participantsof the scientific seminars headed by Melnikov V.N.(Institute of Gravitation andCosmology, Moscow), Rybakov Yu.P. (PFUR, Moscow) and Sushkov S.V. (KFU,Kazan) for valuable comments and criticize.
References
1. N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah et al. , Astrophys.J. ,85 (2012). ctober 16, 2018 10:52 WSPC/INSTRUCTION FILEabbyazov˙chervon˙sCDM
Unified dark matter and dark energy description in a chiral cosmological model
2. W. J. Percival et al. , Mon.Not.Roy.Astron.Soc. , 2148 (2010).3. E. Komatsu et al. , Astrophys.J.Suppl. , 18 (2011).4. S. Tsujikawa, astro-ph/1004.1493, (2010).5. E. J. Copeland, M. Sami and S. Tsujikawa,
Int.J.Mod.Phys.
D15 , 1753 (2006).6. M. Li, X.-D. Li, S. Wang and Y. Wang,
Commun.Theor.Phys. , 525 (2011).7. T. Padmanabhan, Phys.Rept. , 235 (2003).8. Y.-F. Cai, E. N. Saridakis, M. R. Setare and J.-Q. Xia,
Phys.Rept. , 1 (2010).9. Chervon S. V.,
Quantum Matter. , 1 (2013).10. Chervon S. V., Grav.Cosmol. , 145 (1997).11. Chervon S. V., Grav.Cosmol. , 32 (2002).12. K. Bronnikov, S. Chervon and S. Sushkov, Grav.Cosmol. , 241 (2009).13. Beesham A., Chervon S. V., Maharaj S. D., Kubasov A. S., Quantum Matter. ,(2013).14. A. Beesham, S. Chervon and S. Maharaj, Class.Quant.Grav. , 075017 (2009).15. Chervon S. V. , Zhuravlev V. M. and Shchigolev V. K., Phys. Lett. B 398 , 269 (1997).16. S. V. Chervon, N. A. Koshelev,
Grav.Cosmol. , 196 (2003).17. Panina O. G., Chervon S. V. in On the pre-inflationary dark sector fields influence onthe cosmological perturbations , Proc. of Sci. The XXth International Workshop HEPand QFT, Sept.-24 – Oct.1, 2011; http://pos.sissa.it, (2011).18. R. R. Abbyazov, S. V. Chervon,
Grav.Cosmol. , 262 (2012).19. S. Sur, astro-ph/0902.1186, (2009).20. L. P. Chimento, M. I. Forte, R. Lazkoz and M. G. Richarte, Phys.Rev.
D79 , 043502(2009).21. E. N. Saridakis and J. M. Weller,
Phys.Rev.
D81 , 123523 (2010).22. C. van de Bruck and J. M. Weller,
Phys.Rev.
D80 , 123014 (2009).23. A. D. Linde,
Particle Physics and Inflationary Cosmology
Harwood Acad. Publ.,Paris–New York, (1990) [Russ. original, Nauka, Moscow, 1990].24. S. V. Chervon
J. Astrophys. Astron., Suppl. , , 65 (1995).25. S. Chervon and O. Panina, Journal ”Vestnik RUDN” , 121 (2010).26. S. Chervon and O. Panina, Vestnik SamGU, Estestvennonauchnaya seriya
No.8/1(67) , 611 (2008).27. Garcia-Bellido J., astro-ph/0502139, (2005).28. S. Perlmutter et al. , Astrophys.J. , 565 (1999).29. A. G. Riess et al. , Astron.J. , 1009 (1998).30. M. Li, X. Li and X. Zhang,
Sci.China Phys.Mech.Astron. , 1631 (2010).31. W. Hu and N. Sugiyama, Astrophys.J. , 542 (1996).32. Sahni V., Starobinsky A., Int.J.Mod.Phys.
D15 , 2105 (2006).33. S. Chervon,