arXiv subjects

E. Ahmadi-Azar

Publications and source records attributed to E. Ahmadi-Azar.

4 recordsLinked to original sources

arXiv2026

Exact solutions of the FLRW cosmological model via invariants of the Hamilton-Jacobi method

In this study, we proceed to solve the field equations of the spatially flat Friedman-Lemaitre-Robertson-Walker (FLRW) cosmological model in the presence of the cosmological constant $\Lambda$ by making use of the Invariants of Hamilton-Jacobi method (IHJM). This method enables us to extract systematically two independent first integrals such as $l_{\rm HJ,1}(a,\dot{a})=c_{1}$ and $l_{\rm HJ,2}(t,a,\dot{a})=c_{2}$ associated to the transformations group keeping the form of the Hamilton's canonical equations (HCEs) of the cosmological model invariant. Extracting these invariants means not only finding the general solution of the field equations of the model, but also obtaining the Lagrangian and Hamiltonian functions for the model whose dynamics acts like the dynamics of a single particle in a one-dimensional mini-super space $\mathbb{Q}=(a)$. In addition, to obtain the general solution of the model, the IHJM have also solved the inverse problem of calculus of variation (IPCV) without resorting to Helmholtz conditions and whether the necessary conditions for the existence of the Lagrangian function are hold or not. The main part of the IHJM is to find the generating function of the canonical transformation (CT) and then extract two independent invariants for the desired model by using the Hamilton-Jacobi equation (HJE). This study shows that there is a close relationship between the group of the CTs of the Hamiltonian function of the particle and the one-parameter Lie group of transformations keeping invariant the Einstein-Friedmann dynamical equation (EFDE) $\ddot{a}=F(a,\dot{a})$, so that both of them lead to the same result. In this way, having both the IHJM and the invariants of the symmetry groups method (ISGM), a comprehensive integration theory by unifying them can be achieved for studying the desired models.

gr-qc↗

arXiv2024

Linear independence of field equations in the Brans-Dicke theory

In solving the Brans-Dicke (BD) equations in the BD theory of gravity, their linear independence is important. This is due to fact that in solving these equations in cosmology, if the number of unknown quantities is equal to the number of independent equations, then the unknowns can be uniquely determined. In the BD theory, the tensor field $g_{\mu \nu}$ and the BD scalar field $\varphi$ are not two separate fields, but they are coupled together. The reason behind this is a corollary that proposed by V. B. Johri and D. Kalyani in cosmology, which states that the cosmic scale factor of the universe, $a$, and the BD scalar field $\varphi$ are related by a power law. Therefore, when the principle of least action is used to derive the BD equations, the variations $\delta g^{\mu \nu}$ and $\delta \varphi$ should not be considered as two independent dynamical variables. So, there is a constraint on $\delta g^{\mu \nu}$ and $\delta \varphi$ that causes the number of independent BD equations to decrease by one unit, in such a way that in the equations that have been known as BD equations, one of them is redundant. In this paper, we prove this issue, that is, we show that one of these equations, which we choose as the modified Klein-Gordon equation, is not an independent equation, but a result establishing other BD equations, the law of conservation of energy-momentum of matter and Bianchi's identity. Therefore, we should not look at the modified Klein-Gordon equation as an independent field equation in the BD theory, but rather it is included in the other BD equations and should not be mentioned separately as one of the BD equations once again.

gr-qc↗

arXiv2024

Cosmological solutions in the Brans-Dicke theory via invariants of symmetry groups

We proceed to obtain an exact analytical solution of the Brans-Dicke (BD) equations for the spatially flat ($k=0$) Friedmann-Lamaitre-Robertson-Walker (FLRW) cosmological model in both cases of the absence and presence of the cosmological constant. The solution method that we use to solve the field equations of the BD equations is called the "invariants of symmetry groups method" (ISG-method). This method is based on the extended Prelle-Singer (PS) method and it employs the Lie point symmetry, $λ$-symmetry, and Darboux polynomials (DPs). Indeed, the ISG-method tries to provide two independent first-order invariants associated to the one-parameter Lie groups of transformations keeping ordinary differential equations (ODEs) invariant, as solutions. It should be noted for integrable ODEs, the ISG-method guarantees the extraction of these two invariants. In this work for the BD equations in FLRW cosmological model, we find the Lie point symmetries, $λ$-symmetries and DPs, and obtain the basic quantities of the extended PS method (which are the null forms and the integrating factors). By making use of the extended PS method we find two independent first-order invariants, in such a way appropriate cosmological solutions from solving these invariants as a system of algebraic equations are simultaneously obtained. These solutions are wealthy so that they include many known special solutions, such as O'Hanlon-Tupper vacuum solutions, Nariai's solutions, Brans-Dicke dust solutions, inflationary solutions, and etc.

gr-qc↗

arXiv2024

Exact FLRW cosmological solutions via invariants of the symmetry groups

Until now, various methods have been demonstrated to solve the Friedmann-Lama\'ıtre-Robertson-Walker (FLRW) equations in the spatially flat $(k=0)$ cosmological model. In this study, in order to solve the field equations of the spatially flat FLRW cosmological model in the presence of $Λ$, a new method based on the invariants of the symmetry groups which we called ISG-method, is presented. This method is based on the extended Prelle-Singer (PS) method and it uses the Lie point symmetry, $λ$-symmetry and Darboux polynomials (DPs). We employ this method to extract systematically the two independent first integrals (or invariants) such as $I_1 (a,\dot{a})=c_1$ and $I_2 (t,a,\dot{a})=c_2$ associated to the group of the Lie point transformations keeping the Friedmann-Einstein dynamical equation (DE), $\ddot{a}=ϕ(t,a,\dot{a})$, invariant. The obtained solutions from solving the DEs of the FLRW cosmological model by the ISG-method are explicitly written, so that they are suitable for cosmological applications. Finally, as an application of the solutions we look at the age of universe in the presence of cosmological constant where the dominant matter of the universe is considered to be fluid with the state parameter $w$. In this regard, we calculate the age of universe when the dominant matter is dust.

gr-qc↗