Study of Noether symmetry analysis for a cosmological model with variable G and (cid:2) gravity theory

In this present model we have discussed about the solution of the matter-dominated cosmological equations using the expression of (cid:2) = (cid:2)( G ) . Using Noether symmetry analysis we have analyzed both the classical and quantum cosmology. We have made a suitable point transformation in order to make one variable cyclic. As a result the evolution equations become simpler to solve. Wheeler–DeWitt (WD) equation has been constructed for this cosmological model and using conserved charge we have found the solution of WD equation. Finally the solutions have been analyzed in the cosmological point of view.


Introduction
Based on the series of observational data for the last fifteen to twenty years cosmologists have a unanimous opinion about concordance cosmological model [1]-a cosmological paradigm based on general Relativity with a cosmological constant . This model not only describe the early formation of large-scale structures but also the present era of accelerated expansion. At present two different types of unknown matter are well known from observational point of view and they constitute more than 95% of the matter energy around us. These matter components are termed as dark matter (DM) and dark energy (DE). Dark matter is invisible but attractive in nature as visible matter. It is speculated that it is present in between the galaxies and successfully describes rotational curves in spiral galaxies [2] (for an alternative view point see Ref. [3]). Dark energy, ( being the simplest choice) on the other hand is supposed to be the major ingredient of the cosmic matter [4] to account for the present accelerating era. a e-mail: roshnibhaumik1995@gmail.com b e-mail: sduttaju@gmail.com c e-mail: schakraborty.math@gmail.com (corresponding author) Although there are lot of physical dark energy models in the literature, still none of them is suitable both from theoretical as well as observational point of view.
As there is no strong basis (both theoretical and experimental) for these hidden matter parts so there are several alternative ways to accommodate the above Cosmological and Astrophysical issues. One such possible physical theory deals with variable cosmological constant as well as variable gravitational coupling. Such physical theory describes cosmological dynamics by analyzing renormalization group induced quantum version [5][6][7][8][9][10][11][12][13] of the theory with nonperturbative renormalization due to non-Gaussianity -the quantum Einstein Gravity [14]. In cosmological context, the inherent infrared divergence in the above quantum Einstein Gravity implies dynamical nature of the cosmological constant [15].
The basic aim of the present work is to determine analytic solutions of the above variable G, cosmological model using the Noether symmetries of the field equations. The idea of using Noether symmetry to cosmological models is not at all new, rather there are lot of works in the literature [16,17]. The key idea of this approach is geometric [18,19] and in particular the homothetic vectors of the kinetic metric for a first order Lagrangian are the Noether point symmetries, i.e, determination of Noether point symmetries reduces to a problem of differential geometry. Also mathematically, the first integral in Noether symmetry can be considered as a tool to simplify a system of differential equations or to determine the integrability of the system.
In the context of quantum cosmology, symmetry analysis has a great role to determine the solution of the Wheeler-Dewitt (WD) equation. Noether symmetries play the role of a bridge between quantum cosmology and classically observable Universe. In fact Noether symmetries provide a subset of the general solutions of the WD equation, giving oscillatory behaviours with suitable physical meaning [20][21][22]. More-over, the criterion due to Hartle are associated with Noether symmetries to identify typical classical trajectories satisfying the cosmological evolution equations [22,23]. The plan of the paper is as follows: The basic equations are presented in Sect. 2. Noether point symmetry has been used in the present model and classical cosmological solutions are obtained in Sect. 3. The quantum cosmology has been formulated (and solution of the WD equation is evaluated) in Sect. 4. Finally, the paper ends with a brief summary of the whole work in Sect. 5.

Basic equations of the model
In Quantum Einstein gravity, for homogeneous and isotropic space-time geometry if G is treated as independent dynamical variable then at classical level it reduces to metric-scalar gravity. On the other hand for both G and to be independent variables leads to a pathological situation: the momentum conjugate to vanishes. Then the preservation of this primary constraint leads to vanishing Lapse function i.e, a collapse of space-time geometry. Hence it is reasonable to assume a generic functional dependence = (G). In the present model the point like Lagrangian has the explicit form as [24] where dot indicates the derivative with respect to the cosmic time t, G is a function of t and = (G) while μ is a non-vanishing interaction parameter. L m , the Lagrangian for the matter field, is chosen for a perfect fluid as −Da −3(γ −1) . Where γ (index of state parameter) is a constant and D is a suitable integration constant. Here lapse function N = 1 and shift vector N i = 0.
So the Euler-Lagrange equations for a and G take the form as: The Hamiltonian constraint is given bẏ This is equivalent to the constraint on the energy function associated with the Lagrangian.
The kinetic metric corresponding to the Lagrangian (1) is given by with effective potential, It is to be noted that the Noether point symmetries of the system are generated by the elements of the homothetic group of the above kinetic metric as the Lagrangian is in the form of a point particle.
Further, the above kinetic metric can be rewritten as Hence the kinetic metric is conformal to the flat 2D Minkowskian geometry, having four dimensional homothetic Lie Algebra with elements (a) the gradient homothetic vector l 1 = a∂ a (b) three killing vectors which span the E(2) group.

Noether symmetry and classical cosmological solutions
Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. For solving the field equations we use Noether symmetry approach. In the present model the point like Lagrangian is given by (1). Now the existence of Noether symmetry demands that there exist a vector valued function F(t, a, G) such that where X [1] is the first prolongation vector defined by (9) and the vector field X is defined as D t , the total derivative operator, is given by From (8) using (1), (9), (10) and (11) we get the following set of partial differential equations.
Now if one consider ζ as a function of t only then ζ takes the form ζ(t) = c 1 t + c 2 and α, β are independent of t.
For solving the set of differential equations we use the method of separation of variables i.e, where α 0 and β 0 are arbitrary constants. We also get μ = 6; and the value of γ is either 1 or 0. Putting γ = 1 in (12) we get where 0 is a strictly positive integration constant. Similarly putting γ = 0 in (12) we get where is positive integration constant. Now we want to make a point transformation (a, G) → (u, v) in such a way that u becomes a cyclic coordinate. So, the symmetry vector X should satisfy the following equations where i X is the inner product operator of X . Solving these equations we get and v = ln a G (26) The transformed Lagrangian takes the form Using Euler-Lagrangian equation we get and e 2vü = 3 Solving Eqs. (28) and (29) we obtain and where A, B, C and E are arbitrary constants. Hence the explicit cosmological solutions are of the form, where a 1 , t 1 , c 1 and d 1 are (arbitrary) constants constructed out of the integration constants A, B, C and E. Here t = t 1 is the big-bang singularity (assuming c 1 t 1 + d 1 > 0). So the above solution describes an early era of evolution and it is supported by the choice γ = 1 i.e, stiff fluid. The Figs. 1, 2, 3 shows the graphical representation of the evolution history. From the figure, it is found that the scale factor gradually increases from the big-bang singularity and it becomes enormously large as t → ∞. The Hubble parameter gradually decreases and the Universe is in accelerated era of expansion throughout the evolution. The gravitational parameter blows up both at the big-bang and as t → ∞ and it reaches a finite minimum at some finite time. The Eq. (21) shows that the variable cosmological constant has zero value at the big-bang, then it increases to a maximum and finally it approaches to zero again. However, throughout the evolution G 2 remains a finite constant.
Case II: γ = 0 The transformed Lagrangian takes the form Using Euler-Lagrangian equation we get where A , B , C and E are arbitrary constants. As γ = 0 corresponds to dust era of evolution so by choosing the above constants suitable one may write the explicit cosmological solution as where the evolution has started from a time t > −t 0 (with −c 0 t 0 + d 0 > 0) with a finite value of the scale factor and the scale factor blows up at t → ∞. The Figs. 4, 5, 6 show the diagrammatic representation of the evolution of the Universe. The solution represents the evolution of the Universe from matter dominated era to the present late time accelerated evolution. Thus the present model may be considered as an alternative to the dark energy model. Now due to Eq. (22) both and G have similar behaviour namely both of them have finite value at the beginning and then they gradually blows up to infinity. Further, the above expressions for the scale factor show that the improper integral in the expression for the particle horizon does not converge. Hence the particle horizon does not exist for the present model both for γ = 0, 1.

The minisuperspace approach in quantum cosmology
Usually, in superspace the symmetries are characterized by the metric and matter field. On the other hand, restrictions of geometrodynamics of the superspace is termed as minisuperspaces. In cosmological context, both physically relevant and interesting models are normally defined over minisuper-  space. The simplest minisuperspace model obeys the cosmological principle and one has the homogeneous and isotropic metrics with homogeneous matter fields. As a consequence, the lapse function is a function of 't alone and shift vector vanishes i.e, the line element takes the form Using this 3 + 1-decomposition, the Einstein-Hilbert action has the explicit form Here k ab stands for the extrinsic curvature, k = k ab q ab is termed as trace of the extrinsic curvature, (3) R is the usual 3space curvature and represents the cosmological constant. Now, due to homogeneity of the three space, the metric q ab corresponds to a finite number (n) of functions q α (t), α = 0, 1, . . . , n − 1 and consequently (42) becomes with μ αβ , the metric on the minisuperspace. The above action is familiar to a relativistic point particle having self interacting potential V (q) and the particle is moving in nD spacetime with metric μ αβ . Thus the equation of motion of the particle (obtained by variation of the metric f αβ ) is the following 2nd order differential equation The motion of the point particle is constrained by (due to variation of the action by the lapse function) the differential equation 1 2N 2 f αβq αqβ + V (q) = 0 (45) (Note that the particle motion is described by (2n − 1) free parameters) In Hamiltonian formulation, the canonical momenta corresponding to the dynamical variables are defined as So the canonical Hamiltonian can be written as It is to be noted that by writing the constraint equation (45) in terms of the momenta variables then one has In course of canonical quantization one has to make the transformation p α → −i ∂ ∂q α so that the Hamiltonian constraint (48) becomes This is known as Wheeler Dewitt equation. The time independent function ψ(q α ) is termed as wave function of the Universe. In construction of the WD equation there is an ambiguity related to factor ordering with may be avoided by demanding the quantization in minisuperspace is of covariant nature (i.e, invariant under the transformation q α → q α (q α )). As the above WD equation is a second order hyperbolic partial differential equation so for probability measure in quantum cosmology one may consider the conserved probability current Thus the appropriate probability measure on the minisuperspace takes the form with dV , a volume element on minisuperspace. Further, using WKB approximation one may write the wave function as ψ(x k ) ∼ e is(x k ) and consequently, the WD equation transforms to first order Hamilton-Jacobi equation. In the following we shall consider the two cases (for γ = 0, 1) separately. Case I: γ = 1 From (27), the momentum associated with the variables u and v can be written as Then the Hamiltonian takes the form Thus the WD equation takes the form The operator version of the conserved momentum can be written as with ψ 0 , a constant of integration. Now from the WD equation we get where k 1 = 3D i4π p 0 and k 2 = 3 0 i32π 2 p 0 .
Solving Eq. (57), we get, with ψ 01 , a constant. From the solutions (32) and (33) we see that as the Universe approaches the big-bang singularity a G → 0 while aG → a finite non-zero constant. So near the big-bang singularity the above wave function has purely oscillatory in nature with finite amplitude and frequency. Thus there is finite probability to have the big-bang singularity for the present model with stiff fluid. The graphical representation of the wave function has been shown in Fig. 9.
As the probability is constant near the Big-Bang singularity so no boundary proposal will not be valid there. Further, it is to be noted that there is no (curvature) divergence even at the singularity due to the running of the Newton's constant.
Case II: γ = 0 From (34), the momentum can be written as Then the Hamiltonian takes the form Thus the WD equation takes the form The operator version of the conserved momentum can be written as with φ 0 , a constant of integration. Now from the WD equation we get where k 3 = 3 i32π 2 p 1 .
Solving Eq. (65), we get, φ 0 is a constant of integration. The wave function of the Universe has similar form as in the previous case. As in this case the model does not correspond to early epoch of evolution so quantum cosmology will not be interesting.

Brief summary
The present work deals with a complicated cosmological model where Newtonian gravitational coupling is no longer a constant and the cosmological constant is a function of this coupling parameter. Due to non-linear coupled field equations it is hard to infer any physical prediction from the model. As a result, Noether symmetry analysis has been used to the model. By obtaining Noether symmetry vector, it is possible to have a transformation in the augmented space so that   does not exist a fixed point for which G vanishes at small 't' and diverges but the product G approaches a constant. Further, the Noether symmetry analysis has a very crucial role in studying quantum cosmology. The associated conserved charge in Noether symmetry identifies is the oscillatory part of the wave function and consequently the WD equation simplifies to a great extend. The graphical representation of the wave function (shown in Fig. 9) shows that the quantum cosmological description favours the occurrence of big-bang singularity at the beginning.

Data Availability Statement
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