Analyzing stability of five-dimensional Einstein Chern–Simons gravity through dynamical systems

Cosmic Evolution of Five‐dimensional Einstein Chern‐Simons Gravity Through Dark Energy Model

“…The S-expansion method and Lie-algebra concepts are utilized to derive the gravitational Lagrangian for this theory. The total action of this theory can be given as [19,44]…”

“…[19] The first condition about torsionless implies the result 𝛿L M 𝛿k ab = 0 from which we can say that both conditions are consistent with each other. For the spinless matter, we get 𝛿L M 𝛿𝜔 ab = 0, the above equations take the form [19,44] 𝜀 abcde (2R ab e c e d ± l…”

“…The S ‐expansion method and Lie‐algebra concepts are utilized to derive the gravitational Lagrangian for this theory. The total action of this theory can be given as [ 19,44 ] $$$$\begin{equation} S = S^{(5)}_{EChS}+S_M, \end{equation}$$$$where $$S_M$$ indicates an action for matter distribution which is a function of $$(e^m, h^m, \omega ^{mn}, k^{mn},\ldots..)$$ and $$S^{(5)} _{EChS}$$ is the action form Lagrangian which is given as $$$$\begin{eqnarray} \nonumber L^{(5)}_{EChS} &=& \alpha _1(\pm l^2)\varepsilon _{abcde}e^aR^{bc}R^{de} + \alpha _3\varepsilon _{abcde}\\ &&\times{\left(\frac{2}{3}R^{ab}e^c e^d e^e + (\pm l^2)R^{ab}R^{cd}h^e + 2(\pm l^2)...$$…”

“…[ 19 ] The first condition about torsionless implies the result $$\frac{\delta L_M}{\delta k^{ab}} = 0$$ from which we can say that both conditions are consistent with each other. For the spinless matter, we get $$\frac{\delta L_M}{\delta \omega ^{ab}}=0$$, the above equations take the form [ 19,44 ] $$$$\begin{eqnarray} \varepsilon _{abcde}(2R^{ab}e^ce^d \pm l^2R^{ab}R^{cd}) &=& -8\kappa \frac{\delta L_M}{\delta e^e}, \end{eqnarray}$$$$ $$$$\begin{eqnarray} \pm l^2\varepsilon _{abcde}R^{ab}R^{cd}&=& -8\kappa \frac{\delta L_M}{\delta h^e}, \end{eqnarray}$$$$ $$$$\begin{eqnarray} \pm l^2 \varepsilon _{abcde} R^{cd}D h^e &=& 0, \end{eqnarray}$$$$where κ corresponds to effective Newton constant and known as a coupling const...…”