The quintessence scalar field in the relativistic theory of gravity. Yu.V.Chugreev, М.А. Мestvirishvili, К.А. Моdestov The variant of the quintessence theory is proposed in order to get an accelerated expansion of the Friedmannian Universe in the frameworks of relativistic theory of gravitation.The substance of quintessence is built up the scalar field of dark energy. It is shown, that function V (Φ), which factorising scalar field Lagrangian ( Φ is a scalar field) has no influence on the evolution of the Universe. Some relations, allowing to find explicit dependence Φ on time, were found, provided given function V (Φ). 1. Introduction On the edge between 20th and 21st centuries an outstanding discovery [1] had been made: it was established that expansion of the Universe presently was going with acceleration. To explain this phenomenon, it is sufficient to assume, that in the homogeneous and isotropic Universe (Friedmann Universe) there is a matter having an unusual equation of state: PQ=(− 1−ν )ρQ, (1.1) where PQ is the isotropic pressure, ρQ is the mass density, and the following restriction [3,4] for parameter ν is a consequence of general statements of relativistic theory of gravitation (RTG) [2]: 2 0<ν < . (1.2) 3 A substance having such an equation of state has been called the "quintessence". It is supposed, that the quintessence is a real scalar field. No wonder, that modeling of the perfect fluid with negative preassure is provided with a rather strange scalar field Lagrangian. Let us designate the scalar field of quintessence as Φ(xα ) and postulate, that its Lagrangian density is given as follows Lg=(−−VΦ)(I2 )q , (1.3) where gg=det µν , gµν is a metric tensor of the effective Riemannian space [2], q is a number, V ()Φ >0 is some function of the field Φ , and αβ αβ Ig=,∂αβΦ∂ Φ ≡ g∇αΦ∇βΦ (1.4) In (1.4) ∇α is a covariant derivative with respect to metric gαβ . The Lagrangian densities of the form Lg=(V)F(I), −− Φ (1.5) where FI() is an arbitrary function of I , have been considered earlier [5].1 Our choice FI()= (I2 )q follows from intention to get equation of state (1.1) in the frameworks of the field theory of quintessence, which in their turn can explain the accelerated expansion of the Friedmannian Universe in RTG [3,4]. 2. The equation of motion for the field Φ()xα . According to the variational principle, the equation of motion for the field Φ which has the Lagrangian (1.3) can be obtained from the Euler-Lagrange formula 1 Earlier references on this subject can be found in [5]. In the papers the field Φ(xα ) with the Lagrangian (1.5) is called «к-essence». Nevertheless, we shall keep the term “field theory of quintessence” for any field theory, resulting in (1.1), cause it is equation of state (1.1)which was corresponded to the term “quintessence” origin. 1 ∂LL∂ −∂µ =0. (2.1) ∂Φ ∂()∂ µ Φ After substitution of Eq. (1.3) into Eq. (2.1) we obtain dVln (Φ) (4qI−−1) 2 8q(2q−1)gαµ∂Φ(Γλ gτβ ∂∂Φ−gλβ ∂Φ∂∂Φ) − dΦ αµτλβ βλµ µτ α λµ −4(qg Γµτ ∂Φα −g ∂λ ∂Φµ )I=0. (2.2) Here we have 1 Γ∂αα=(ggβ +∂g−∂g). µτ 2 µ βτ τ βµ β µτ In terms of covariant derivatives ∇α , Eq. (2.2) has the following form dVln (Φ) (4qI−1) 2 +−8q(2q1)gαµ gλβ ∇Φ∇Φ∇∇Φ+ dΦ αβλµ λµ +∇4=qg λµ∇ΦI 0. (2.3) From the Lagrangian density (1.3) we can derive the symmetric energy-momentum tensor density of the field Φ(xα ) . According to Hilbert, δ L T =2 . µν δ g µν As L does not depend on derivatives of the metric tensor gαβ , we obtain ∂L Tg=2 = − V(Φ−)(I21)q− [gI24qI∂Φ∂Φ]. (2.4) µν ∂g µν µν µ ν 3. Equation of motion for field Φ and its energy-momentum tensor in the homogeneous and isotropic Universe. αβ For a homogeneous and isotropic Universe metric tensors gαβ and g in Cartesian coordinates are given as follows [2]: 42 gg00 =1; 11 = g22 = g33 =−amaxa, gαβ =0, α ≠ β (3.1) and 1 00 11 22 33 αβ ′ gg=1; = g= g=− 42, g=0. α ≠ β (3.1 ) aamax Here a(τ ) is the scale factor, and amax < ∞ is its maximal value. The square root −g is equal to 63 −ga=.max a (3.2) To provide that field Φ(xα ) does not break homogeneous and isotropic character of the Universe, it is necessary to assume, that Φ(xα ) depends on τ only. Then, according to (1.4) and (3.1 '), 2 I =(∂0Φ) and for Tµν taken from Eq. (2.4), in view of Eq. (3.1), we obtain 63 22q Tq00 =(1−Φ4 )amax aV( )[(∂0Φ) ] , (3.3) 10 5 2 2q TT11 ==22 T33 =−Φamax aV()[(∂0Φ)], (3.4) 2 Tµν =0, µ ≠ν. It is standardly assumed that in the Friedman Universe the matter can be described by the energy-momentum tensor density of some perfect fluid. This tensor is written as follows Tgµν =[−+(ρ p)UµUν −gµν p]. (3.5) As for the homogeneous and isotropic Universe 4-velocity U µ has the form: UU0 =1, k =0, we find from Eq. (3.5), in view of Eq. (3.1), 63 Ta00 =,max aρ (3.6) 10 5 TT11 ==22 T33 =amaxap,Tµν =0,µ ≠ν. (3.7) If we extend this hypothesis of an opportunity to represent the energy-momentum tensor density of matter by the ideal fluid energy-momentum tensor density, onto the quintessence field, then after comparison Eq. (3.3) and Eq. (3.4) with Eq. (3.6) and Eq. (3.7) accordingly, we obtain 22q ρQ = (1−Φ4qV) ( )[(∂0Φ) ] , (3.8) 22q PVQ =(−Φ)[(∂0Φ)]. (3.9) Hence we have 1 P =.− ρ (3.10) QQ(1− 4q) By putting 1 14− q = , (3.11) 1−ν we see, that Eq. (3.10) transforms into Eq. (1.1). It gives a ground to treat field Φ with Lagrangian (1.3) as quintessence field, if only ν q =.− (3.12) 4(1−ν ) Considering restriction (1.2) for parameter ν , we obtain from Eq. (3.12) 1 − <<q 0. (3.13) 2 Thus, the Lagrangian density for quintessence has the following form ν − Lg=(−−VΦ)(I2 )4(1−ν ) , (3.14) and the density and pressure for this field in the homogeneous and isotropic Universe will be given as follows ν 1 − ρ =(V Φ∂)[(Φ)2 ]2(1−ν ) , (3.15) Q 1−ν 0 ν − 2 2(1−ν ) PVQ = −Φ( )[(∂0Φ) ] . (3.16) To obtain explicit form of (3.15) and (3.16), it is necessary to solve equation of motion (2.2). As for the Friedmann Universe we have Christoffel symbols of the form: 3 ΓΓ00=0, =aa4∂aδ ; gµτ Γ0=−∂a, 00 ik max 0 ik µτ a 0 then from Eq. (2.2), in view of Eq. (3.12) we find 3 1ldVn(Φ) 3ν 1 −∂()Φ43−∂a(∂Φ)+ 1(−Φννda001−)0 ν +∂()Φ22∂Φ=0. (1−ν )2 00 1 By dividing this expression over −(∂Φ)3 and considering identity (1−ν )2 0 dVln (Φ) ∂ Φ≡∂ lnV (Φ), dΦ 00 we obtain ∂Φln ⎡⎤(Va( ))13−−νν∂Φ−ν (1ν)= 0, 00⎣⎦ and then ν − Λ V ()Φ∂Φ1−ν = ,Λ=const >0. (3.17) 0 a3ν By substituting this expression into Eq. (3.15) and Eq. (3.16), we finally get Λ 1 ρ =, (3.18) Q 1−ν a3ν Λ P =.− (3.19) Q a3ν It is well-know [2,4], that for equation of state (1.1) the covariant conservation law µν µν ∇µT =0, where T , looks like (3.5), in the Friedmann Universe leads to formulas BB ρν==and P −−(1),where B =const >0. (3.20) QQaa33νν After a redefinition of constant Λ as follows Λ =(1−ν )B we can see, that Eqs. (3.18) - (3.19) transform into (3.20), that means the equivalence of the µν equation of motion (2.3) to the covariant conservation law ∇µT , when the last one is supplemented by Eq. (1.1). However, it is important to note, that for deriving basic formulas (3.18) and (3.19), describing the evolution of homogeneous and isotropic Universe, the explicit form of V (Φ) is not required. Let’s denote the values of the scaling factor a(τ ) and the scalar field Φ(τ ) at the start time of expansion of the Universe τ = 0 as amin and Φ (0) correspondingly. Then Eq. (3.17) can be rewritten in the form Φ(τ ) (1−ν ) τ dΦ − =,Λ ν ad3(1−ν ) τ (3.21) ∫∫1−ν Φ()00V ()Φ ν The Friedmannian Universe in the RTG performs the cyclic motion, with the scaling factor a(τ )changing in the domain 0 <≤aamin (τ ) ≤amax <∞. (3.22) Therefore the integral 4 τ J()τ = ∫ a31()−ν dτ (3.23) 0 is a finite one. Consequently, the l.h.s. of the Eq.(3.21) should be finite as well. It imposes a (very weak) limit on the functionV (Φ). With the help of RTG results one can express the integral (3.23) through the observables in order to lighten the further analisis. Toward this end it is possible to rewright the Eq.(3.23) as follows a(τ ) 23− ν ⎛dτ ⎞ Ja()τ = ⎜a⎟da. (3.24) ∫ da amin ⎝⎠ According RTG [2,4] for the expansion stage of the Universe evolution we have 1 − dGτπ⎡⎤8ρm2 ⎛31⎞2 a =−⎢⎜1−42+6⎟⎥, (3.25) da ⎣⎦36⎝⎠2β a 2a 2 ⎛⎞mc2 where β = a , m2 = g , m is the rest mass of the graviton ( m ≅⋅3.6 10−66 г [4]), and ρ is max ⎜⎟g g ⎝⎠= a total mass density in the Universe: ρ = ρρrm++ρQ.