f(R, T) gravity bouncing universe with cosmological parameters

Abstract

The basic aim of this manuscript is to investigate the cosmological solutions in the context of the modified f(R, T) theory of gravity, where R is the Ricci scalar and T is the trace of the energy-momentum tensor. For our current work, we consider the Friedmann–Robertson–Walker spacetime for finding the solutions of field equations. We investigate the nature of universe by considering acceleration expansion of universe, ultra-relativistic universe, sub-relativistic universe, dust universe, radiation universe, stiff universe. Moreover, we apply the power law technique by taking two different f(R, T) gravity models to observe the expanding nature of the universe. The bouncing scenario is also discussed by choosing some particular values of the model parameters and observed the energy conditions, which are satisfied for a successful bouncing model. It is also concluded that some solutions in f(R, T) theory of gravity supports the concept of exotic matter and accelerated expansion of the universe due to a large amount of negative pressure.

Appendix A

$$ \begin{aligned}&\frac{1}{(1+\gamma )[3+\gamma (13+8\gamma )]}\Big [3\alpha \lambda +7\alpha \lambda \gamma +\frac{18(\dot{a}^2+a\ddot{a})}{a^{2}}+\frac{42\gamma (\dot{a}^2+a\ddot{a})}{a^{2}} -\alpha \lambda (3+7\gamma )(1\\&\quad +\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}+\frac{1}{\lambda a^{4}} 2(8\gamma ^{2}\dot{a}^{2}-(9+\gamma (21+8\gamma ))a\ddot{a})\Big [\lambda a^{2} +12n\alpha (\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})- (24n\alpha \lambda \dot{a}(3\\&\quad (1+\gamma )(3+4\gamma )a-8\gamma ^{2}\dot{a})(-\lambda ^{2} a^{4}+36(1+2n)\dot{a}^{4}+ 72(1+2n)a\dot{a}^{2}\ddot{a}+36(1+2n)a^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-n}(-2\\&\quad \dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}))/(a(\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})^{2}) +\frac{1}{\lambda ^{5}\dot{a}^{2}}96na\gamma ^{2}(1+\frac{36(\ddot{a}+a\ddot{a})^{2}}{\lambda ^2a^{4}})^{-3-n}\Big (6\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})\\&\quad (\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a}^{2})^{2})-2a\ddot{a}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a}^{2})^{2}) -4a\dot{a}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a}^{2})^{2}) (3\dot{a}\ddot{a}+aa^{3})-288\\&\quad (1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a}^{2}))(2\dot{a}^{3} +a\dot{a}\ddot{a}+a^{2}\dddot{a})^{2}+144(1+n)a(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})(3\dot{a}\ddot{a}\\&\quad +a\dddot{a})(2\dot{a}^{3} +a\dot{a}\ddot{a}+a^{2}\dddot{a})^{2}+5184(1+n)(2+n)(\dot{a}^{2}+a\ddot{a})^{3}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a})^{2} +a^{2}[\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a}^{2})^{2}(3\ddot{a}^{2}\\&\quad +4\dot{a}\ddot{a}+a\dddot{a})-72(1+n)(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4} +36\dot{a}^{4}+72\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})[10\dot{a}^{2}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\ddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a} +a^{2}\dot{a}^{2} \\&\quad (-6\dot{a}^{2}+a\dddot{a})+a^{3}(\ddot{a}+a\dddot{a}^{2}+a\ddot{a}\dddot{a})]]\Big )\Big ]\Big ]=0, \end{aligned}$$

(A1)

$$ \begin{aligned}&\frac{1}{2\lambda ^{5}(1+\gamma )[3+\gamma (13+8\gamma )]\dot{a}^{2}}\gamma ((-1+\gamma )(\frac{1}{(-1+\gamma )\gamma }\lambda ^{4}a^{8} (2(2(-1+\gamma )\gamma \dot{a}^{2}-(9+\gamma (19+8\gamma ))a\ddot{a})(\lambda a^{2}\\&\quad +12n\alpha a^{2}(6(\dot{a}^{2}+a\ddot{a})+\alpha \lambda a^{2}(1-(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}))-(24\alpha \lambda ^{2}a^{3}\dot{a}(9(1+\gamma )^{2}a-2(-1+ \gamma )\gamma \dot{a})(-\lambda ^{2}a^{4}\\&\quad +36(1+2n)\dot{a}^{4}+72(1+2n)a\dot{a}^{2}\ddot{a}+36(1+2n)a^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n} (-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}))/(\lambda ^{2}a^{4}\\&\quad +36\dot{a}^{4}+72a\dot{a}\ddot{a}+36a^{2}\ddot{a}^{2})^{2})+24n \alpha (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(6\dot{a}(\dot{a}^{2}+a\ddot{a})^{2})^{2}-2a\ddot{a}(\dot{a}^{2}+a\ddot{a}) (\lambda ^{2}a^{4}+36(\dot{a}^{2}\\&\quad +a\ddot{a})^{2})^{2}(3\dot{a}\ddot{a}+a\dddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2}) (2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+144(1+n)a(\dot{a}^{2}\\&\quad +a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})(3\dot{a}\ddot{a}+ a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+5184(1+n)(2+n)(\dot{a}^{2}+a\ddot{a})^{3}(-2\dot{a}^{3}-a\dot{a}\ddot{a}\\&\quad +a^{2}\dddot{a})^{2}+a^{2}(\lambda ^{2}a^{4}+36(\dot{a}+a\ddot{a})^{2})^{2}(3\ddot{a}+4\dot{a}\ddot{a}+a\ddot{a})-72(1+n)(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+ 36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\\&\quad \ddot{a}^{2})(10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}^{2}+a\dddot{a}) +a^{3}(\ddot{a}^{3}+a\dddot{a}^{2}+a\ddot{a}\dddot{a}))))+4\Big [72n\alpha \lambda ^{4}a^{8}\dot{a}^{2}\\&\quad (\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n} -24n\alpha \lambda ^{4}a\lambda ^{9}\ddot{a}(\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}+24\lambda ^{4}a^{8}(\dot{a}^{2}-a\ddot{a})\\&\quad (\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n})\lambda ^{4}\gamma a^{8}(6\lambda a^{2}(\dot{a}^{2}+a\ddot{a})^{2}-6\dot{a}(\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})\\&\quad (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n})+\alpha \lambda ^{2} a^{4}(1-(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n})-48n\alpha \lambda ^{4}a^{9}\dot{a}(\dot{a}^{2}+a\ddot{a})^{2} (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n})\\&\quad (3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^(3)-a\dot{a}\ddot{a}-a^{2}\dddot{a})+ 1728n(1+n)\alpha \lambda ^{2}a^{5}(\dot{a}^{2}+a\ddot{a})^{2}(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n}(3\dot{a}\ddot{a}+a\dddot{a})\\&\quad (2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})-(12n\alpha \lambda ^{6}a^{11}\dot{a}(3(1+\gamma )a-2(1+3\gamma )\dot{a})(-\lambda a^{4}+36(1+2n)\dot{a}^{4}+72(1+2n)+2a\dot{a}^{2}\ddot{a}\\&\quad +36(1+2n)a^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n} (-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}))/(\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})^{2}+62208n\\&\quad (1+n)(2+n)\alpha (\dot{a}^{2}+a\ddot{a})^{3}(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})(-2\dot{a}^{3}+a\dot{a}\ddot{a}-a^{2}\dddot{a})+12n\alpha \lambda ^{4}a^{10} (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}\\&\quad (3\ddot{a}^{2}+4\dot{a}\ddot{a}+a\dddot{a})-846n(1+n)\alpha \lambda ^{2}a^{4} (\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n}(10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3} +2a^{3}\dot{a}\ddot{a}\dddot{a}\\&\quad +a\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}^{2}+a\ddddot{a})+a^{3}(\ddot{a}^{3}+a\dddot{a}+a\dot{a}\ddot{a}))+36n\alpha \gamma (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}}^{-3-n})\Big (6\dot{a}^{2}(36(\dot{a}^{2}+a\ddot{a})^{2})\\&\quad (\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2} -a\dot{a}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}(3\dot{a}\ddot{a}+a\ddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}[\lambda ^{2}a^{4}+ (36(\dot{a}^{2}\\&\quad +a\ddot{a})^{2})(2\dot{a}^{3}+a\dot{a}\ddot{a}-a^{2}\dddot{a})+144(1+n)a(\dot{a}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2}) (3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}+a\dot{a}\ddot{a}-a^{2}\dddot{a})\\&\quad +5184(1+n)(2+n)(\dot{a}^{2}+a\ddot{a})^{3}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}) +[10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\dddot{a}+2a^{2}\dot{a}\ddot{a}\dddot{a}+a^{3}(\ddot{a}\\&\quad +a\dddot{a}^{2}+a\ddot{a}\ddddot{a})]]\Big )\Big ]=0. \end{aligned} $$

(A2)

$$ \begin{aligned}&\frac{1}{3\lambda ^{5}(1+\gamma )[3+\gamma (13+8\gamma )]a^{12}}(-\lambda ^{4}a^{8}(2(2(1-\gamma )\gamma \dot{a}^{2}-(9+\gamma (19+8\gamma ))a\ddot{a})(\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}\lambda (3+\gamma )(1+2\gamma )a^{2} (6(\dot{a}^{2}+a\ddot{a})+\alpha \lambda a^{2}(1-(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n})-(24n\alpha \lambda ^{2}a^{3}\dot{a}(9(1+\gamma )^{2}\\&\quad a-2(-1+\gamma )\gamma \dot{a}) (-\lambda ^{2}a^{4}+36(1+2n)a^{4}+72(1+2n)aa^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}(-2\dot{a}+a\dot{a}\ddot{a}+a^{2}\dddot{a})/\lambda ^{2}a^{4}\\&\quad +36\dot{a}^{4}+72a\dot{a}\ddot{a}+36a^{2}\ddot{a}^{2})^{2})+24n\alpha (1-\gamma )\gamma (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(6\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4} +36(\dot{a}^{2}+a\ddot{a})^{2})^{2}\\&\quad -2a\ddot{a}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}-4a\dot{a}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2} (3\dot{a}\ddot{a}+a\dddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4}\\&\quad +36(\dot{a}^{2}+a\ddot{a})^{2})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+144(1+n)a(\dot{a}^{2}+a\ddot{a}) (\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})+(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})\\&\quad +5184(1+n)(2+n)(\dot{a}^{2}+a\ddot{a})^{3}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2} \dddot{a})^{2}+a^{2}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}(3\ddot{a}^{2}+4\dot{a}\ddot{a}+a\dddot{a})-72(1+n)\\&\quad (\dot{a}^{2}+a\ddot{a})\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2} (10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}^{2}+a\dddot{a})+a^{3}(\ddot{a}^{3}+a\dddot{a}\\&\quad +a\ddot{a}\dddot{a})))+ 6\gamma (72n\alpha \lambda ^{4}a^{8}\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}+2\lambda ^{4}a^{8}(\dot{a}^{2}-a\ddot{a})\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}-\lambda ^{4}\gamma a^{8}(6\lambda a^{2}(\dot{a}^{2}+a\ddot{a})-6\dot{a}^{2}(\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}+\alpha \lambda ^{2}a^{4}(1-(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}))-48n\alpha \lambda ^{4} a^{9}\dot{a}(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}(36\dot{a}\ddot{a}+a\dddot{a})-3456n(1+n)a\lambda ^{2} a^{4}\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{2}(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+1728n(1+n) a\lambda ^{2}a^{5}(\dot{a}^{2}+a\ddot{a}) (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n}(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}\\&\quad -a^{2}\dddot{a})-(12n\alpha \lambda ^{6}a^{11}\dot{a}\Big (3(1+\gamma )a-2(1+3\gamma )\dot{a}\Big [-\lambda ^{2}a^{4}+36(1+2n)\dot{a}^{4} +72(1+2n)\dot{a}^{2}\ddot{a}+36(1+2n)a^{2}\dot{a}^{2}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}) /(\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a} +36a^{2}\ddot{a})^{2}+62208n(1+n)(2+n)\alpha (\dot{a}^{2}+a\ddot{a})^{3}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(-2\dot{a}^{3}+a\dot{a} \ddot{a}+a^{2}\dddot{a})+12n\alpha \lambda ^{4}a^{10} (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{1-n}(3\ddot{a}^{2}+4\dot{a}\dddot{a}+a\ddddot{a})-864n(1+n)\alpha \\&\quad \lambda ^{2}a^{4}(\dot{a}^{2} +a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n} (10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}dot{a}^{3}\dddot{a}+2a^{2}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}+a\ddddot{a})+a^{3}(\ddot{a}^{3}+ a\dddot{a}^{2}\\&\quad +a\ddot{a}\ddddot{a}))+36n\alpha \gamma (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(6\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2} -2a\ddot{a}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}\\&\quad +a\ddot{a})^{2})^{2} -4a\dot{a}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}(3\dot{a}\ddot{a}+a\dddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4}+36( \dot{a}^{2}+a\ddot{a})^{2})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})\\&\quad +144(1+n) a(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+5184(1+n)(2+n) (\dot{a}^{2}+a\ddot{a})^{2})\\&\quad (3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})^{2}(3\ddot{a}^{2}+4\dot{a}\dddot{a}+a\ddddot{a})-72(1+n) (\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36\dot{a}^{4} +72\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})[10\dot{a}^{6}\\&\quad -6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2} (-6\ddot{a}^{2}+a\ddddot{a})+a^{3}(\ddot{a}^{3}+a\dddot{a}^{2}+a\ddot{a}\dddot{a})]\Big ]\Big )=0. \end{aligned} $$

(A3)

$$ \begin{aligned}&\frac{1}{4\lambda ^{5}(1+\gamma )[3+\gamma (13+8\gamma )]a^{12}}(-\lambda ^{4}a^{8}(2(2(1-\gamma )\gamma \dot{a}^{2}-(9+\gamma (19+8\gamma ))a\ddot{a})(\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}\lambda (3+\gamma )(1+2\gamma )a^{2} (6(\dot{a}^{2}+a\ddot{a})+\alpha \lambda a^{2}(1-(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n})-(24n\alpha \lambda ^{2}a^{3}\dot{a}(9(1+\gamma )^{2}\\&\quad a-2(-1+\gamma )\gamma \dot{a}) (-\lambda ^{2}a^{4}+36(1+2n)a^{4}+72(1+2n)aa^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}(-2\dot{a}+a\dot{a}\ddot{a}+a^{2} \dddot{a})/\lambda ^{2}a^{4}\\&\quad +36\dot{a}^{4}+72a\dot{a}\ddot{a}+36a^{2}\ddot{a}^{2})^{2})+24n\alpha (1-\gamma )\gamma (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(6\dot{a}^{2} (\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4} +36(\dot{a}^{2}+a\ddot{a})^{2})^{2}\\&\quad -2a\ddot{a}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}-4a\dot{a}(\lambda ^{2}a^{4}+36 (\dot{a}^{2}+a\ddot{a})^{2})^{2} (3\dot{a}\ddot{a}+a\dddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4}\\&\quad +36(\dot{a}^{2}+a\ddot{a})^{2})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2} \dddot{a})+144(1+n)a(\dot{a}^{2}+a\ddot{a}) (\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})+(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})\\&\quad +5184(1+n)(2+n)(\dot{a}^{2} +a\ddot{a})^{3}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2} \dddot{a})^{2}+a^{2}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}(3\ddot{a}^{2}+4\dot{a}\ddot{a}+a\dddot{a})-72(1+n)\\&\quad (\dot{a}^{2}+a\ddot{a})\lambda ^{2} a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2} (10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}^{2}+a\dddot{a})+a^{3}(\ddot{a}^{3} +a\dddot{a}\\&\quad +a\ddot{a}\dddot{a})))+ 6\gamma (72n\alpha \lambda ^{4}a^{8}\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}+2\lambda ^{4}a^{8}(\dot{a}^{2} -a\ddot{a})\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}-\lambda ^{4}\gamma a^{8}(6\lambda a^{2}(\dot{a}^{2}+a\ddot{a})-6\dot{a}^{2}(\lambda a^{2}+12n\alpha (\dot{a}^{2}+a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}+\alpha \lambda ^{2}a^{4}(1-(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}))-48n\alpha \lambda ^{4} a^{9}\dot{a}(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n}(36\dot{a}\ddot{a}+a\dddot{a})-3456n(1+n)a\lambda ^{2} a^{4}\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{2}(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+1728n(1+n) a\lambda ^{2}a^{5}(\dot{a}^{2}+a\ddot{a}) (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n}(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}\\&\quad -a^{2}\dddot{a})-(12n\alpha \lambda ^{6} a^{11}\dot{a}\Big (3(1+\gamma )a-2(1+3\gamma )\dot{a}[-\lambda ^{2}a^{4}+36(1+2n)\dot{a}^{4} +72(1+2n)\dot{a}^{2}\ddot{a}+36(1+2n)a^{2}\dot{a}^{2}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2} \dddot{a})/(\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a} +36a^{2}\ddot{a})^{2}+62208n(1+n)(2+n)\alpha (\dot{a}^{2}+a\ddot{a})^{3}(1\\&\quad +\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(-2\dot{a}^{3} +a\dot{a}\ddot{a}+a^{2}\dddot{a})+12n\alpha \lambda ^{4}a^{10} (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{1-n}(3\ddot{a}^{2}+4\dot{a}\dddot{a}+a\ddddot{a})-864n(1+n)\alpha \\&\quad \lambda ^{2}a^{4}(\dot{a}^{2} +a\ddot{a})(1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-2-n} (10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}dot{a}^{3}\dddot{a}+2a^{2}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\ddot{a}+a\ddddot{a})+a^{3}(\ddot{a}^{3} +a\dddot{a}^{2}\\&\quad +a\ddot{a}\ddddot{a}))+36n\alpha \gamma (1+\frac{36(\dot{a}^{2}+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-3-n}(6\dot{a}^{2}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}- 2a\ddot{a}(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}\\&\quad +a\ddot{a})^{2})^{2} -4a\dot{a}(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})^{2}(3\dot{a}\ddot{a}+a\dddot{a})-288(1+n)\dot{a}(\dot{a}^{2}+a\ddot{a})^{2}(\lambda ^{2}a^{4} +36(\dot{a}^{2}+a\ddot{a})^{2})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})\\&\quad +144(1+n) a(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36(\dot{a}^{2}+a\ddot{a})^{2})(3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})+5184(1+n)(2+n) (\dot{a}^{2}+a\ddot{a})^{2})\\&\quad (3\dot{a}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})^{2}(3\ddot{a}^{2}+4\dot{a}\dddot{a}+a\ddddot{a}) -72(1+n)(\dot{a}^{2}+a\ddot{a})(\lambda ^{2}a^{4}+36\dot{a}^{4} +72\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})[10\dot{a}^{6}\\&\quad -6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2} (-6\ddot{a}^{2}+a\ddddot{a})+a^{3}(\ddot{a}^{3}+a\dddot{a}^{2}+a\ddot{a}\dddot{a})]\} \Big )=0. \end{aligned} $$

(A4)

$$ \begin{aligned}&\frac{-1}{3+\gamma (13+8\gamma )}[3\alpha \lambda +\frac{18(\dot{a}^2+a\ddot{a})}{a^{2}} -\alpha \lambda (3+4\gamma )(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-n}-\frac{1}{\lambda a^{3}} (18\ddot{a}(\lambda a^{2} +12n\alpha (\dot{a}^2+a\ddot{a})\\&\quad (1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-1-n}))- 8\gamma (-\dot{a}^2-2a\ddot{a})(\lambda \alpha ^{2}+12n\alpha (\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2}a^{4}})^{-1-n})- (24n\alpha \lambda \dot{a}((9+6\gamma )a\\&\quad +4\gamma \dot{a}) (-\lambda ^{2} a^{4}+36(1+2n)\dot{a}^{4}++72(1+2n)a\dot{a}^{2}\ddot{a}+36(1+2n)a^{2}\ddot{a}^{2})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-n}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a}))/\\&\quad (a(\lambda ^{2}a^{4}+36\dot{a}^{4}+72a\dot{a}^{2}\ddot{a}+36a^{2}\ddot{a}^{2})^{2}) +\frac{1}{\lambda ^{5} a^{12}}(\alpha \lambda ^{6}a^{12}+6\lambda ^{5}a^{10}(\dot{a}^2+a\ddot{a})-72\alpha \lambda ^{4}a^{8}\dot{a}(\dot{a}^2+a\ddot{a})a(1\\&\quad +\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-1-n}+24n\alpha \lambda ^{4}a^{9}\ddot{a}(\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-1-n}+48n\alpha \lambda ^{4}a^{9}\dot{a}(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-1-n}(3\dot{a}^{3}\ddot{a}\\&\quad +a\dddot{a})+3456n(1+n)\alpha \lambda ^{2}a^{4}\dot{a}(\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-2-n}(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})-\Big (2\dot{a}^{3}-1728n(1+n)\alpha \lambda ^{2}a^{5}\\&\quad (\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a} )^{2}}{\lambda ^{2} a^{4}})^{-2-n}(3\dot{a}^{3}\ddot{a}+a\dddot{a})(2\dot{a}^{3}-a\dot{a}\ddot{a}-a^{2}\dddot{a})-62280n(1+n)(2+n)\alpha [\dot{a}^2+a\ddot{a}(1\\&\quad +\frac{36(\dot{a}^2+ a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-3-n}(-2\dot{a}^{3}+a\dot{a}\ddot{a}+a^{2}\dddot{a})-12n\alpha \lambda ^{4}a^{10}(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-2-n}(3\ddot{a}^{2}+4\dot{a}\ddot{a}+a\dddot{a})+864n(1+n)\alpha \\&\quad \lambda ^{2}a^{4}(\dot{a}^2+a\ddot{a})(1+\frac{36(\dot{a}^2+a\ddot{a})^{2}}{\lambda ^{2} a^{4}})^{-2-n} [10\dot{a}^{6}-6a\dot{a}^{4}\ddot{a}-4a^{2}\dot{a}^{3}\dddot{a}+2a^{3}\dot{a}\ddot{a}\dddot{a}+a^{2}\dot{a}^{2}(-6\dot{a}^{2}+a\dddot{a})+a^{3} (\ddot{a}+a\dddot{a}^{2}\\&\quad +a\ddot{a}\dddot{a})]]\Big )=0, \end{aligned} $$

(A5)