Exact Cosmological Models with the Yang - Mills Fields on Lyra Manifold
V. K. Shchigolev D. N. Bezbatko


[
    Abraham Smith Paul Bendich John Harer Alex Pieloch Jay Hineman
    October 24, 2016
==================================================================





§ INTRODUCTION


The relatively  recent discovery of the present acceleration of our Universe is strictly proved by Type Ia Supernovae, Cosmic Microwave Background Radiation, and Sloan
Digital Sky Survey <cit.>-<cit.>.  This fact initiated a great number of theoretical hypothesis, and inspired researchers to a diversity of different explanations  of such an unusual behavior of the Universe. Nevertheless, mostly two distinct approaches to explain the phenomenon of accelerated expansion of the Universe have been well developed during the recent years. One of them deals with the so-called dark energy (DE) with negative pressure in General Relativity, another one acts within the frameworks of some acceptable modifications of the gravity theory on the long distance.

According to several estimations, the dark energy occupies up to 72 % of the total energy of the universe,
while dark matter gives about 24 %,  and the usual baryonic matter consists of about 4 % of the total energy.
In spite of all attempts to explain the late acceleration of our Universe, DE is still an open problem to the theoretical physicists because its nature is unknown so far. As known, there are numerous models available in the literature devoted to different descriptions of the nature of DE but none of them are fully definitive and reliable.  The simplest and the most natural candidate
among them is the cosmological constant <cit.> with the equation of state (EoS) parameter
w_Λ =-1. Although this model meets the observational data but it suffers at least from two serious problems such as fine tuning and cosmic coincidence.

Currently, there are numerous DE and modified gravity models to explain the present acceleration of the universe.  Alternatively, to explain
the decay of the density, the different forms of dynamically changing DE with an effective
equation of state (EoS) w_de = p_de/ρ_de < -1/3, were proposed instead of a constant
vacuum energy density. Other possible forms of DE include quintessence -1/3> w_q > -1, phantom (w_ph < -1)  etc.

In the last few decades there has been considerable interest in alternative theories of gravitation. One of the most interesting among them is the scalar-tensor theory proposed by G. Lyra <cit.>, and essentially developed by Sen and Dunn <cit.>. They have constructed  an analogue of the
Einstein field equation based on Lyra's geometry.
According to Halford <cit.> the
scalar-tensor treatment based on Lyra's geometry
predicts the same effects within observational
limits as in Einstein's theory.  Moreover, as it pointed out by Soleng <cit.>, the constant displacement field in
Lyra's geometry will either include a creation field
and be equal to Hoyle's creation field cosmology
or contain a special vacuum field, which together
with the gauge vector form may be considered as a
cosmological term. Subsequent investigations were
done by several authors in cosmology  within the framework of Lyra's
geometry (see, e.g., <cit.>-<cit.> and references therein).



Involving of the exotic forms of matter and fields for the explanation of the phenomenon of accelerated expansion in the context of Lyra's geometry is reasonable in the same extent as it is in Einstein's theory of gravity. Thereby the models on Lyra manifold with the quintessence, phantom field, tachyons, Chaplygin gas and some other forms of matter were studied by several authors. Nevertheless, the problem which has to be investigated in detail still remains unsolved, namely, whether the resource of Lyra's geometry in the form of the displacement field is quite enough to describe the acceleration of the universe filled by the ordinary matter and fields.

As known, the electromagnetic fields could be one of these natural sources of gravity . A number of studies in the framework of Lyra's geometry involved the electromagnetic fields as the sources of gravity in the cosmological models (see, e.g., <cit.>-<cit.> and references therein).
As far as we know, the models in Lyra's geometry  which include the Yang-Mills fields <cit.> have not  been studied yet. At the same time, a large  number of works investigated the influence and role of these fields on the behavior of the Universe in the framework of Einstein's theory (see, e.g. <cit.>-<cit.> to mention just a few).

Motivated by the above mentioned investigations, we have considered the homogeneous  cosmological model with the Yang-Mills field in Lyra geometry. The aim of this paper is to search for the exact solutions for such a model, that is to find explicit expressions for the scale factor and the displacement field of this model.  As consequence, through the time dependence of the YM content of the Universe, solutions are found
for the energy density and the pressure. Some geometrical and physical aspects of the model also have  been studied. For instance, exact solutions for the scalar field and its potential have been obtained in two specific cases regarding the effective equation of state and the exponential law of expansion.




§ THE MODEL EQUATIONS



The Einstein's gravitational equations in Lyra's geometry  in normal gauge (see, e.g. <cit.> and <cit.>) can be written as

    R_ik- 1/2 g_ik R - Λ g_ik +  3/2ϕ_i ϕ_k - 3/4g_ikϕ^j ϕ_j = T^(tot)_ik,

where ϕ_i is a displacement vector, and  Λ is a time-varying cosmological term. For simplicity, we assume  the gravitational constant 8π G=1. All other symbols have their usual meanings in the Riemannian geometry, and T^(tot)_ik stands for the total energy-momentum tensor (EMT) of all kinds of matter in the Universe.
The Friedmann-Robertson-Walker (FRW) line element can be represented by

    ds^2 = d t^2- a^2 (t)[d r^2+ξ^2(r)d Ω^2],

where a(t) is a scale factor of the Universe, and

    ξ_(r)={[  sin(r),   k =+1,;       r,    k =0,; sinh(r),   k =-1.;          ].

in accordance with the  sign of the spatial  curvature k.

In the case of minimal coupling of the displacement field with matter, the energy-momentum tensor of matter can be derived in a usual manner from the Lagrangian of matter. Therefore, the EMT of the YM fields  is followed from the Lagrangian density of SO_3 YM field <cit.> as

    L_YM=-1/16πF_ik^aF^a ik,

where F_ik^a=∂_iW_k^a-∂_kW_i^a+
eε_abcW_i^bW_k^c is the stress tensor of YM fields W^a_i.
Varying this Lagrangian with respect to metric, one can easily get
the following EMT of YM fields

    T_i^k=-1/4πF_ij^a
    F^akj+1/16πδ_i^k F_mn^a
    F^amn.

At the same time, the variation of Lagrangian (<ref>) with respect to YM fields W^a_i yields the following YM equation

    D_i(√(-g)F^aik)=0,

where D_i stands for the covariant derivative.

Let us suppose further that the remaining content of the universe can be described as a perfect fluid with the EMT of the following form

    𝒯_ik= (ρ_m +p_m)u_i u_k -p_m  g_ik,

where  u_i = (1,0,0,0) is  4-velocity of the co-moving observer, satisfying u_i u^i = 1, ρ_m and p_m are the energy density and pressure of matter, consequently. Hence, the total EMT equals T^(tot)_ik=T_ik+𝒯_ik.





§ EXACT SOLUTION FOR THE YM EQUATION






As well-known, the generalized Wu-Yang ansatz for the SO_3 YM
fields can be written as <cit.>

    W^a_μ =ε_μ abx^bK(r,t)-1/gr^2+(δ^a_μ-x^ax_μ/r^2)S(r,t)/gr, 
    
    W^a_0 = x^a W(r,t)/gr,

where μ=1,2,3. The example of exact solution for the YM equation (<ref>) in metric (<ref>) has been obtained in <cit.> with the help of the following substitution

    W = d α/d t,   K = P(r)cosα(t),   S = P(r)sinα(t),

where α is an arbitrary function of time.

As a result of simple calculation on the base of (<ref>) and (<ref>), we have the following expressions for the YM tensor components

    {[   F_12=g^-1 P'(r)( m  cosα + 
    l sinα),; F_13=g^-1 P'(r)sinθ( m  sinα-
    l cosα),;             F_23=g^-1sinθ(P^2(r)-1) n,;                    F_01= F_02= F_03=0,;                                        ].

where

    n= (sinθcosϕ, sinθsinϕ,
    cosθ),        
     l = (cosθcosϕ, cosθsinϕ, -sinθ),     
     m= (-sinϕ, cosϕ, 0)

are the orthonormalized isoframe vectors, and the prime means a
derivative with respect to r. As it is noted in <cit.>, the YM fields (<ref>) have only magnetic components.


Here and below an overdot denotes the partial derivative with respect to time t,
and a prime denotes the partial derivative with respect to r.
Taking into account the Wu-Yang ansatz (<ref>), the energy-momentum tensor (<ref>) can be written down in the form

    T_0^0 = 1/8π g^2 a^4(a^2 W'^2+a^2 Z + X + Y),                
    
    T_1^1 =1/8π g^2 a^4(a^2 W'^2-a^2 Z - X + Y),                
    
    T_2^2 = T_3^3 = - 1/8π g^2 a^4(a^2 W'^2+ Y), T_0^1=1/2π g^2 a^2J,

where

    X=2(K'^2+S'^2)/ξ^2(r), Y=(K^2-1+S^2)^2/ξ^4(r),
    
    Z=2[(K̇+WS)^2+(Ṡ-WK)^2]/ξ^2(r),         
    
    J=K'(K̇+WS)+S'(Ṡ-WK)/ξ^2(r).


The set of equations for the YM fields (<ref>) can be written down with the help of
ansatz (<ref>) as follows

    (ξ ^2 W')'
    +2[(Ṡ-WK )K-(K̇+WS )S]=0,  
    
    a∂/∂ t[a(Ṡ-WK)]-
    S”                                
    
    +[(K^2-1+S^2 )S/ξ^2
    -(K̇+WS )W]=0,
    
    a∂/∂ t[a(K̇+WS)]-
    K”                                
    
    +[(K^2-1+S^2 )K/ξ^2
    +(Ṡ-WK )W]=0.



One can readily verify that the substitution  (<ref>) into equations (<ref>) yields the only equation for unknown function P(r), namely

    P”-(P^2 -1)P/ξ^2= 0.

At the same time, inserting (<ref>) into  (<ref>), one can obtain the following components of EMT for the YM field

    T_0^0 = 1/8π g^2 a^4 ξ^2( 2 P'^2+(P^2
    -1)^2/ξ^2),       
    
    T_1^1 =1/8π g^2 a^4 ξ^2(- 2 P'^2+(P^2
    -1)^2/ξ^2),    
    
    T_2^2 = T_3^3 = - 1/8π g^2 a^4(P^2
    -1)^2/ξ^4,

and T_0^1=0.
Assuming  the displacement field ϕ_i(t) is a homogeneous  vector field, and considering the homogeneity and isotropy of the line element (<ref>), it is necessary to require the independence of the EMT components (<ref>) on the radial coordinate r. Thus, we get

    {[ 2 P'^2+(P^2
    -1)^2/ξ^2=A ξ^2,;                             ; 2 P'^2-(P^2
    -1)^2/ξ^2=B ξ^2,;                             ;         (P^2
    -1)^2/ξ^2=Cξ^2,;                              ].

where A, B and C are some constants. A simple analysis of equations (<ref>) and (<ref>) allows to find out that A=3C and B=C. Therefore, the set of equations (<ref>) is reduced to the following two equations

    P'^2=C ξ^2,     
    P^2=1+ϵ√(C)ξ^2,

where ϵ = ± 1. The direct consequence of (<ref>) is as follows

    ξ'^2=1+ϵ√(C)ξ^2.

Taking into account (<ref>) and (<ref>), we can get C=|k|, ϵ=-k and

    P(r)=±√(1- k ξ^2).

Hence, due to (<ref>), the latter leads to
the nontrivial solution for equation (<ref>) given by

    P(r)=±ξ'(r)={[  ±cos r,   k =+1,;     ± 1,    k =0,; ±cosh r,   k =-1.;          ].


Interpreting EMT of the YM gauge field as the EMT of a perfect fluid, that is

    T_ik= (ρ_g +p_g)u_i u_k -p_g  g_ik,

we can get from (<ref>) and (<ref>) that

    T_0^0=ρ_g=3|k|/8π g^2 a^4, T_1^1=T_2^2=T_3^3=-p_g=- |k|/8π g^2 a^4,

Thus, the effective EoS of the given YM field is the same one as  for the pure radiation, that is p_g=ρ_g/3, when k=±1.

It is useful to mention that the conservation equation for YM field, T^k_i;k=0, for EMT of YM field in form (<ref>), namely

    ρ̇_g+3H(ρ_g+p_g)=0,

where H=ȧ/a is the Hubble parameter, is satisfied identically by our solution (<ref>).
As can be seen, the reason of such satisfaction of the continuity equation in the case k=0 is nothing more than the equality of ρ_g and p_g to zero according to (<ref>).
That is why we will consider further only two types of the non-flat models with k = ± 1.






§ COSMOLOGICAL EQUATIONS



Let ϕ_i be a time-like displacement vector field,

    ϕ_i = (2/√(3) β,0,0,0),

where β = β(t) is a function of time only, and the factor 2/√(3) is used in order to simplify the writing of all the following equations.
Given the line element (<ref>) and  taking the total EMT as a sum of (<ref>) and (<ref>), one can reduce the main equations of the model (<ref>) to the following set of equations:

    3H^2 + 3 k/a^2 - β^2 = ρ_m + ρ_g +Λ,            
    
    2 Ḣ + 3H^2 + k/a^2 + β^2 = -  p_m -p_g +Λ.



The continuity equation follows from Eqs. (<ref>) and (<ref>) as

    ρ̇_m + ρ̇_g + Λ̇+ 2 ββ̇+ 3 H (ρ_m+ρ_g + p_m +p_g+ 2β^2 )=0.


Obviously, the displacement vector field can be treated through the EMT in the same way as in the most studies on the Lyra geometry in cosmology. In such approach, the conservation of the energy-momentum should be accounted for the sum of EMTs <cit.>.  As emphasized in <cit.>, the displacement vector field can give rise to an effective cosmological term Λ_eff. It means that Λ(t) is not an independent dynamical parameter of the model, and it should be removed from the system of equations (<ref>) and (<ref>).


Moreover,  we are going to preserve the continuity equation for matter in its standard form,

    ρ̇_m + 3 H (ρ_m + p_m )=0,

which follows from the conservation equation, 𝒯^k_i;k=0, and (<ref>).
Taking into account equations (<ref>) and (<ref>), the continuity equation  (<ref>) becomes

    Λ̇+ 2 ββ̇+ 6 H β^2 =0.

Since Λ =constant, equation (<ref>) supposes the displacement field to be the so called stiff fluid. The assumption of a non-vanishing and time-varying Λ term gives us some new possibilities <cit.>. Therefore, we assume that  Λ≠ constant. Moreover, we suppose that  (<ref>) can be satisfied by some Λ(t), which can be found as the result of formal integration of equation (<ref>),

    Λ(t) =λ_0 -β^2-6∫ H β^2 d t,

where λ_0 is a constant of integration. Substituting this expression along with ρ_g, p_g from (<ref>) into equations (<ref>) and (<ref>), one can get the basic equations of the model as follows

    3H^2 + 3k/a^2 = 3/8π g^2 a^4+ρ_eff,             
    
    2 Ḣ -2k/a^2 = -1/2π g^2 a^4 - (ρ_eff + p_eff),

where we have introduced the effective cosmological term

    Λ_eff(t)=λ_0 - 6 ∫ H β^2 d t,

and the effective energy density and pressure

    ρ_eff=ρ_m +Λ_eff,    
    p_eff=p_m -Λ_eff - Λ̇_eff/3 H.




One can readily verify that due to  equations (<ref>) and (<ref>), and the continuity equation (<ref>), the effective energy density and pressure (<ref>) also satisfy the continuity equation in its usual form:

    ρ̇_eff + 3 H (ρ_eff + p_eff )=0.


At the same time, the deceleration parameter in Lyra's geometry is defined just as in the standard cosmology, that is

    q = -a^2 ä/ȧ^2 = -1-Ḣ/H^2.

It should be noted that the set of dynamical equations (<ref>), (<ref>) and (<ref>) consists of two independent equations, and fully determines the dynamics of our model. However, to determine three parameters, say a(t), ρ_eff and p_eff,  one more condition should be set. For example,  an effective EoS w_eff=p_eff/ρ_eff can play the role of such  additional equation.
Assuming the matter also obeys  a barotropic EoS  p_m = w_m ρ_m, we also can obtain the full set  of equations but only if we make some assumption about Λ_eff(t) (see, e.g. <cit.>). Let us now consider one interesting example of exact solution for our model.







§ SOLUTION FOR THE EFFECTIVE VACUUM MODEL



In the following, we provide the exact solution
for our model in one simple case. Let us suppose

    p_eff=-ρ_eff,

that is the effective vacuum EoS w_eff=-1. From equation (<ref>), we get

    ρ_eff = Λ_0=constant>0.

Hence, the only independent equation, which has to be solved, follows from (<ref>) as

    3H^2 + 3k/a^2 = 3/8π g^2 a^4+Λ_0.

Equation (<ref>) can be readily solved for all possible relations between its constant parameters (see, e.g., <cit.> and references therein). In particular, for a special choice of the constants of integration, these solutions can be given by a=a_k,δ(t),

    a={[   (√(A)sinh[2√(Λ_0/3)(t+t_ks)]+3k/2Λ_0)^1/2, δ=+1,;                                                   ; a_0 exp[√(Λ_0/3)t],                           δ=0,;                                                   ;   (√(A)cosh[2√(Λ_0/3)(t+t_kc)]+3k/2Λ_0)^1/2, δ=-1,;                                                    ].

where

    A=3/4Λ_0|1/2π g^2-3/Λ_0|,  δ=sgn(1/2π g^2-3/Λ_0),

and the constants of integration  are supposed to be

    t_ks=(1-k)/4√(3/Λ_0)arsinh(3/2√(A)Λ_0)

and

    t_kc=(1-k)/4√(3/Λ_0)arcosh(3/2√(A)Λ_0).




Therefore, we can obtain from (<ref>) the following expression for the Hubble parameter H(t)=H_k,δ(t),

    H_k,δ(t)={[ H_0cosh[2H_0(t+t_ks)]/sinh[2H_0(t+t_ks)]+B_k,  δ=+1,;                                                     ; H_0sinh[2H_0(t+t_ks)]/cosh[2H_0(t+t_ks)]+B_k,  δ=-1,;                                                      ] .

where H_0=√(Λ_0/3), B_k=k/2√(A)H_0^2, and H(t)=H_0 for the case of δ=0. As one can see, at any case, H_0 = lim_t →∞H(t).



Taking into account these equations and (<ref>), one can get the deceleration parameter q=q(t) as follows

    q={[ B_k^2+1-(sinh[2H_0(t+t_ks)]+ B_k)^2/cosh^2[2H_0(t+t_ks)], δ=+1,;                                                                ; B_k^2-1-(cosh[2H_0(t+t_ks)]+ B_k)^2/sinh^2[2H_0(t+t_ks)], δ=-1.;                                                                 ] .


The behavior of the  Hubble parameter (<ref>) and the deceleration parameter (<ref>) with time for k=+1 and different values of δ is shown in Fig. 1. To be specific, we have chosen A=1. Then, one can see the black lines for H_0=1 and δ=+1, the blue lines for H_0=1 and δ=-1, the red lines for H_0=1.25 and δ=+1, and the green lines for H_0=1.25 and δ=-1.


Applying (<ref>) for (<ref>) and (<ref>), we get

    p_m+ρ_m=Λ̇_eff/3 H,

and, taking into account (<ref>) and (<ref>),

    ρ_m=Λ_0-Λ_eff,    p_m=-Λ_0+Λ_eff +Λ̇_eff/3 H.

The last means that the matter EoS is

    w_m=p_m/ρ_m=-1+Λ̇_eff/3 H(Λ_0-Λ_eff).


Since

    Λ̇_eff= - 6 H β^2

in accordance to (<ref>), than equation (<ref>) leads to

    p_m+ρ_m=-2β^2.

Let us suppose hat the matter in our model exists in the form of a scalar field ϕ(t) with a potential V(ϕ), that is

    ρ_m=1/2ϵϕ̇^2 + V(ϕ),     p_m=1/2ϵϕ̇^2 - V(ϕ),

where ϵ=±1 for the quintessential and phantom fields, respectively.
Due to the real character of the displacement vector and non-negativity of the matter density,  we obtain  w_m ≤ -1. It means that the matter should be of the phantom nature. For example, it can be a phantom scalar field ϕ(t), for which ϵ=-1, that is

    ρ_m+p_m=-ϕ̇^2,   ρ_m-p_m= 2 V(ϕ)

From these expressions and equations (<ref>) - (<ref>), we have

    ϕ̇^2=2β^2,

and

    V =  Λ_0+β^2-Λ_eff.

The further investigation of this model involves the imposition of some additional conditions.
For example, we can supplement the model by the following simplest assumption regarding the displacement field

    β=β_0=constant.

which is often discussed in the literature.
Taking into account equations (<ref>) and (<ref>), we obtain

    ϕ(t)=√(2)β_0 t+ϕ_0,

where ϕ_0 is a constant of integration. In view of (<ref>), the integration of equation (<ref>) yields the following effective cosmological term

    Λ_eff= λ_0- 6 β_0^2ln[a(t)/a_0],

where λ_0 is a constant of integration. Substituting (<ref>) into (<ref>) alon with (<ref>) and (<ref>), we get the following potential of the scalar field

    V(ϕ)= V_0 + 6 β_0^2ln|a(ϕ-ϕ_0/√(2)β_0)|,

where V_0 = Λ_0+β_0^2-λ_0- 6 β_0^2ln a_0.
Substituting equation (<ref>) for different δ and k=±1 in (<ref>), we can obtain the scalar potential V(ϕ)=V_k,δ(ϕ) as follows

    V_k,+1(ϕ)= V_0 + 3 β_0^2ln|sinh(1/β_0√(2Λ_0/3) ϕ)+3k/2√(A)Λ_0|,

for δ=+1 and ϕ_0=√(2)β_0 t_ks,

    V_k,-1(ϕ)= V_0 + 3 β_0^2ln|cosh(1/β_0√(2Λ_0/3) ϕ)+3k/2√(A)Λ_0|,

for δ=-1 and ϕ_0=√(2)β_0 t_kc, and

    V(ϕ)= V_0 + √(6Λ_0) β_0 ϕ,

when δ=+1. Here V_0 is a constant.

For the specific values of constants, the scalar potentials (<ref>) and (<ref>) are shown in Fig.2.

Substituting (<ref>) in equation (<ref>), we get the following equation for the scalar field EoS

    w_m(t)=-1-1/R+3ln[a(t)/a_0],

where R=(Λ_0-λ_0)(2β_0^2)^-1, and a(t) is defined by equation (<ref>).
As can be seen from equation (<ref>), the EoS of matterw_m could remain less than -1 all the time but tends to -1 as t→∞.





§ COSMOLOGICAL MODEL WITH A CONSTANT HUBBLE PARAMETER



Now we consider the case of inflationary model, when H=H_0 = constant, and hence

    a(t)=a_0 exp(H_0 t).

Then from equation (<ref>), one can obtain

    Λ_eff(t)=λ_0 - 6H_0∫β^2(t) d t .


According to equations (<ref>), (<ref>) and (<ref>), it is easy to get the following energy density and pressure of matter


    ρ_m=3k/a^2-3/8π g^2a^4-λ_0+3H_0^2+6H_0∫β^2 d t,      
    
    p_m =-k/a^2-1/8π g^2a^4+λ_0-3H_0^2-6H_0∫β^2 d t -2 β^2.

By summarizing these equations, one can get that

    ρ_m + p_m = 2k/a^2-1/2π g^2a^4-2β^2.


Once again, the further investigation of this model requires to apply some extra conditions.
For example, we can supplement the model by the following simplest assumption regarding the displacement field

    β(t)= β_0/a^2(t)=β_0/a_0^2exp(-2H_0 t),

where β_0 is a constant. Taking into account (<ref>), (<ref>) and  (<ref>), one can get the following equation for the EoS of matter w_m=p_m/ρ_m

    w_m =-k/a_0^2 e^-2H_0 t+1/2a_0^4(1/4π g^2+β_0^2) e^-4H_0 t+ρ_0/3k/a_0^2 e^-2H_0 t-3/2a_0^4(1/4π g^2+β_0^2) e^-4H_0 t+ρ_0,

where and further ρ_0=3H_0^2-λ_0 is the asymptotic (at t→∞) value of the energy density of matter. It is interesting to note that this equation asymptotically leads to the constant quintessential EoS  w_m=-1/3 when and only when ρ_0=0, that is λ_0=3H_0^2, for any sign of the curvature, and for any numeric values of other parameters of the model.

In general, the detailed behavior of w_m(t) depends on the numeric values of parameters k, a_0, β_0 and g in equation (<ref>), and also depends on the relation between them. If we introduce the following notation

    C_1=2/a_0^2,   C_2=2/a_0^4(1/4π g^2+β_0^2),

we can find that the EoS parameter (<ref>) behaves itself  in  different manners depending to the relative relation between C_1 and C_2. The dependence of EoS w_m(τ)  on the dimensionless time τ=H_0 t for  C_1=0.8, S_2=1.6 and k=+1 (in blue), k=-1 (in red) is shown in Fig. 3 for several values of ρ_0.

Let us assume again that the matter exists in the form of a scalar field ϕ(t) with a potential V(ϕ). Therefore, it follows from equation (<ref>) that

    ρ_m+p_m=ϵϕ̇^2,   ρ_m-p_m= 2 V(ϕ),

From these equations and (<ref>)-(<ref>),  we can obtain

    ϵ H_0^2 (ϕ')^2 =kC_1-C_2x^2,

and

    V(x) =  kC_1x^2-1/4C_2x^4 + ρ_0,

where x=exp(-H_0 t)∈ (0,1] and ϕ'=dϕ/dx.
The exact solutions for equation (<ref>) can be readily obtained as follows

    ϕ(x) =±1/2H_0[x√(C_1-C_2x^2)                    
    
    +C_1/√(C_2)arcsin(√(C_2/C_1)x)]+ϕ_0,

for k=+1, C_1≥ C_2, ϵ=+1, or  C_1<C_2, x∈ [0,x_0] where x_0=√(C_1/C_2),

    ϕ(x) =±1/2H_0[x√(C_2x^2-C_1)                    
    
    -C_1/√(C_2)arcosh(√(C_2/C_1)x)]+ϕ_0,

for k=+1, C_1<C_2, ϵ=-1, x∈ [x_0,1], and

    ϕ(x) =±1/2H_0[x√(C_1+C_2x^2)                    
    
    +C_1/√(C_2)arsinh(√(C_2/C_1)x)]+ϕ_0,

for k=-1, ϵ=-1.

Notably that almost the same result could be obtained from the assumption of the other rate of changing for the displacement field, namely

    β(t)= β_0/a(t)=β_0/a_0exp(-H_0 t).

In this case, equation (<ref>) for the EoS of matter is rewritten as follows

    w_m =-1/a_0^2(k-β_0^2) e^-2H_0 t+1/8π g^2a_0^4 e^-4H_0 t+ρ_0/3/a_0^2(k-β_0^2) e^-2H_0 t-1/8π g^2a_0^4 e^-4H_0 t+ρ_0,

and both equations (<ref>) and (<ref>) are valid again but with re-defined constants C_1, C_2 by

    C_1=2(1-kβ_0^2)/a_0^2,   C_2=1/2π g^2a_0^4.

Therefore, we can obtain the same solutions (<ref>)-(<ref>) with new constants (<ref>).

Putting aside a detailed discussion of a generalization of this model in the spirit of Ref.<cit.>, let us note that the consequences  of such generalization for the cosmic acceleration could be more radical than those obtained here.




§ CONCLUSION


In this paper, we have investigated some cosmological non-flat FRW models with YM fields on Lyra manifold. By using the exact solution of the YM equations on the FRW cosmological background, the exact solutions in Lyra's geometry have been obtained with  the certain propositions regarding the effective equation of state and the rate of expansion.

As a result, we have obtained the explicit expressions for the EoS of matter. The scalar field potential  has been reconstructed assuming that the matter  consists of the quintessential or phantom scalar field.
The results of this paper may also be extended without
much effort to the case of coupling the YM fields with the ordinary matter. In that case the accelerated regime of expansion could be reached by means of energy exchange between different forms of matter.

Finally, we can conclude that these models still contain a great potential for the study of cosmological expansion within Lyra geometry based on the exact solutions with participation of the Yang-Mills fields.


99


Riess A. G. Riess , et al. ,  "Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant",  Astronomical Journal,  Vol. 116, (1998), 1009. http://dx.doi.org/10.1086/300499

Perlmutter S. Perlmutter, et al.,  "Measurements of Omega and Lambda from 42 High-Redshift Supernovae", Astrophysical Journal, Vol. 517, (1999), 565. http://dx.doi.org/10.1086/307221


Netterfield C. B. Netterfield, et al., "A Measurement by Boomerang of
Multiple Peaks in the Angular Power Spectrum of the Cosmic Microwave
Background", Astrophysical Journal, 571 (2002), 604-614.
http://dx.doi.org/10.1086/340118

Halverson N. W. Halverson, et al., "Degree Angular Scale Interferometer
First Results: A Measurement of the Cosmic Microwave Background
Angular Power Spectrum", Astrophysical Journal, 568 (2002), 38-
45. http://dx.doi.org/10.1086/338879

Spergel D. N. Spergel, et al., "First-Year Wilkinson Microwave Anisotropy
Probe (WMAP) Observations: Determination of Cosmological
Parameters", Astrophysical Jounal Supplement Series, 148 (2003), 175-
194. http://dx.doi.org/10.1086/377226

Tegmark M. Tegmark, M. A. Strauss, et al., "Cosmological parameters
from SDSS and WMAP", Physical Review D, 69 (2004), 103501.
http://dx.doi.org/10.1103/PhysRevD.69.103501

Allen S. W. Allen, R. W. Schmidt, H. Ebeling, et al. , "Constraints on
dark energy from Chandra observations of the largest relaxed galaxy
clusters", Monthly Notices of the Royal Astronomical Society,
353 (2004), 457-467. http://dx.doi.org/10.1111/j.1365-2966.2004.08080.x

Peebles  P. J. E. Peebles, B. Ratra, "The cosmological constant and dark
energy", Reviews of Modern Physics, 75 (2003), 559.
http://dx.doi.org/10.1103/RevModPhys.75.559

Lyra G. Lyra, "Über eine modifikation der riemannschen geometrie", Mathematische Zeitschrift 54 (1951), 52-64. http://dx.doi.org/10.1007/BF01175135



Sen  D. K. Sen  and   K. A. Dunn,  "A Scalar-Tensor Theory of Gravitation in a Modified Riemannian Manifold",
 Journal of Mathematical Physics,  12 (1971), 578. http://dx.doi.org/10.1063/1.1665623

Halford	W. D. Halford, "Scalar-tensor theory of gravitation in a Lyra manifold," Journal of Mathematical Physics, 13(11) (1972), 1699-1703. http://dx.doi.org/10.1063/1.1665894

Soleng H. H. Soleng, "Cosmologies based on Lyra's geometry," General Relativity and Gravitation, 19(12) (1987),  1213-1216. http://dx.doi.org/10.1007/BF00759100



Shchigolev1 V. K. Shchigolev,  "Cosmology with an Effective Λ-term in Lyra manifold", Chinese Physics Letters, 30
(2013), 119801. http://dx.doi.org/10.1088/0256-307X/30/11/119801

Shchigolev10 V. K. Shchigolev,  "Cosmological Models with a Varying Λ -Term in Lyra's Geometry", Modern Physics Letters A,
27(29) (2012), 1250164. http://dx.doi.org/10.1142/S0217732312501647

Matyjasek2 J. Matyjasek, "Junction conditions and static fluid cylinders in Lyra's geometry",  International Journal of Theoretical Physics,  Vol.33, Issue 4, (1994), pp.  967982. http://dx.doi.org/10.1007/BF00672827

SerpaNilo Sylvio Costa Serpa, "A New Approach on Quantum Gravity in Lyra Geometry", Journal of Physical Science and Application, 6(5), (2016), 1-7.
http://dx.doi.org/10.17265/2159-5348/2016.05.001

HovaHoavo Hova, "A Dark Energy Model in Lyra Manifold", Journal of Geometry and Physics, 64, (2013), 146-154.
http://dx.doi.org/10.1016/j.geomphys.2012.08.004

ZhiHaizhao Zhi, Mengjiao Shi, Xinhe Meng, Lianzhong Zhang, "A new global 1-form in Lyra geometric cosmos model", International Journal of Theoretical Physics,  Vol.53, Issue 11, (2014), pp 4002-4011.
http://dx.doi.org/10.1007/s10773-014-2151-4

Khurshudyan1M. Khurshudyan, J. Sadeghi, R. Myrzakulov, A. Pasqua, H. Farahani, "Interacting quintessence dark energy models in Lyra manifold", Advances in High Energy Physics, 2014, (2014), 878092.
http://dx.doi.org/10.1155/2014/878092


Shchigolev2  V. K. Shchigolev, E. A. Semenova, "Scalar field cosmology in Lyra's geometry", International Journal of Advanced Astronomy, Vol. 3, No. 2, (2015) p. 117-122. http://dx.doi.org/10.14419/ijaa.v3i2.5401

Khurshudyan2  M. Khurshudyan, "Interacting extended Chaplygin gas cosmology in Lyra manifold", Astrophysics and Space Science,  360, (2015), 44.
http://dx.doi.org/10.1007/s10509-015-2557-3

Khurshudyan3 M. Khurshudyan, A. Pasqua, J. Sadeghi, H. Farahani, "Quintessence Cosmology with an Effective Λ-Term in Lyra Manifold", Chinese Physics Letters Vol. 32, No. 10 (2015) 109501. http://dx.doi.org/10.1088/0256-307X/32/10/109501


Khurshudyan5M. Khurshudyan, J. Sadeghi, A. Pasqua, S. Chattopadhyay, R. Myrzakulov, H. Farahani,
"Interacting Ricci dark energy models with an effective Λ-term in Lyra manifold", International Journal of Theoretical Physics, 54, (2015), 749.
http://dx.doi.org/10.1007/s10773-014-2266-7

Khurshudyan4 M. Khurshudyan, "Toy Models of Universe with an Effective Varying
Λ-Term in Lyra Manifold", Advances in High Energy Physics,
Volume 2015, Article ID 796168, 10 pages.
http://dx.doi.org/10.1155/2015/796168

Shchigolev6  V. K. Shchigolev, "Inhomogeneous cosmology with quasi-vacuum effective equation of state on Lyra manifold", International Journal of Physical Research,  4(1) (2016),  15-19. doi:http://dx.doi.org/10.14419/ijpr.v4i1.5902


PradhanAnirudh Pradhan, Priya Mathur, "Inhomogeneous Perfect Fluid Universe with Electromagnetic Field in Lyra Geometry", 	Fizika B, Vol.18, (2009), pp.243-264.
http://dx.doi.org/10.1007/s10509-009-0015-9

Casana1R. Casana, C. A. M. de Melo, and B. M. Pimentel, "Electromagnetic Field in Lyra Manifold: A First Order Approach",  Brazilian Journal of Physics, Vol.35, No.4B, (2005), pp.1151-1154. http://dx.doi.org/10.1590/S0103-97332005000700040

Casana2R. Casana, C. A. M. de Melo and B. M. Pimentel, "A New Class of Inhomogeneous Cosmological Models
with Electromagnetic Field in Normal Gauge for Lyras
Manifold",  Int J Theor Phys,   50 (2011), 5669.
http://dx.doi.org/10.1007/s10773-010-0493-0

AbdelM. Abdel-Megied, Ragab M. Gad, E. A. Hegazy, "Inhomogeneous Bianchi Type I Cosmological Model with Electromagnetic Field in Lyra Geometry", Assiut Univ. J. of Mathematics and Computer Science, 38 (2009), 1-10. 	arXiv:1411.5978 [gr-qc]



AliAhmad T. Ali, F. Rahaman, A. Mallick, "Invariant Solutions of Inhomogeneous Universe
with Electromagnetic Field in Lyra Geometry", International Journal of Theoretical Physics,
53(12) (2014), 4197-4210. http://dx.doi.org/10.1007/s10773-014-2171-0

Yang C.N. Yang , R.L. Mills , "Conservation of isotopic spin and isotopic gauge invariance", Physical Review, 96, 1 (1954), pp. 191 - 195. http://dx.doi.org/10.1103/PhysRev.96.191


Shchigolev3  V.K. Shchigolev, "Modelling Cosmic Acceleration in Modified Yang-Mills Theory", Gravitation and Cosmology, Vol. 17, No. 3 (2011) 272-275. http://dx.doi.org/10.1134/S0202289311030078

Shchigolev4 V. K. Shchigolev, G. N. Orekhova, "Non-Minimal Cosmological Model in modified Yang-Mills Theory", Modern Physics Letters A, Vol. 56, No. 2 (2011) 389-396. http://dx.doi.org/10.1142/S0217732311036462

Gal D.V. Gal'tsov, E.A. Davydov, "Yang-Mills Condensates in Cosmology, International Journal of Modern Physics: Conference Series, 14 (2012) 316-325.
http://dx.doi.org/10.1142/S201019451200743X

Matyjasek1 J. Matyjasek, "On Sen equations in arbitrary gauge",
 Astrophysics and Space Science, 207 (1993) 313-316. http://dx.doi.org/10.1007/BF00627248

Samaroo V. K. Shchigolev, K. Samaroo, "Generalized Exact Cosmologies with Interacting Yang-Mills and Nonlinear Scalar Fields", General Relativity and Gravitation, 36(7) (2004),  1661. http://dx.doi.org/10.1023/B:GERG.0000032158.16161.1b

Matyjasek3 J. Matyjasek, "Cosmolagical models with a time-dependent Λ term", Physical Review D, 51(8) (1995), 4154. https://doi.org/10.1103/PhysRevD.51.4154

Stephani H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt. Exact Solutions of Einsteins Field Equations, 2nd Edition, Cambridge University Press, 2003.





