Isotropization of the magnetic universe in Horndeski theory with G₃(X,ϕ) and G₅(X)

$\begin{aligned} L_H=\sqrt{-g}\Big(\mathcal{L}_2+\mathcal{L}_3+\mathcal{L}_4+\mathcal{L}_5\Big) \,, \end{aligned}$

$\mathcal{L}_2 = G_2(\phi,X)\,,\, \mathcal{L}_3 = - G_3(\phi,X)\Box\phi\,,$

$\mathcal{L}_4 = G_{4}(\phi,X) R +G_{4X}(\phi,X) \left[ (\square \phi )^{2}-(\nabla_\mu \nabla_\nu \phi)^2 \right]\,,$

$\begin{aligned}[b] \mathcal{L}_5 =& G_{5}(\phi,X) G_{\mu\nu}\,\nabla^\mu \nabla^\nu \phi -\frac{1}{6}G_{5X} \left[\left( \Box \phi \right)^3\right. \\&\left.-3 \Box \phi (\nabla_\mu \nabla_\nu \phi)^2 + 2\left(\nabla_\mu \nabla_\nu \phi \right)^3 \right], \end{aligned}$

$\begin{aligned} \mathcal{L}_{F\phi}=-\frac{1}{4}[f(\phi)]^2F_{\mu\nu}F^{\mu\nu}, \end{aligned}$

$\begin{aligned} G_2=\varepsilon X-V(\phi),\,\, G_3\neq0,\,\, G_4=\frac{1}{16\pi},\,\, \varepsilon=\pm 1. \end{aligned}$

$\begin{aligned} {\rm d}s^2 = -{\rm d}t^2 + a^2_1(t){\rm d}x_1^2 + a^2_2(t){\rm d}x_2^2 + a^2_3(t){\rm d}x_3^2. \end{aligned}$

$\begin{aligned}[b] \dfrac{1}{8\pi}G^0_0= & G_2 - G_{2X}\dot{\phi}^2-3G_{3X}H\dot{\phi}^3+G_{3\phi}\dot{\phi}^2-\\& {- 5G_{5X}H_1H_2H_3\dot{\phi}^3 - G_{5XX}H_1H_2H_3\dot{\phi}^5+T^{({\rm{em}})0}_{0}\,,} \end{aligned}$

$\begin{aligned}[b] \dfrac{1}{8\pi}G^{i}_{i} =&G_2 - \dot\phi \frac{{\rm d} G_3}{{\rm d}t} -\\& -\dfrac{\rm d}{{\rm d}t}(G_{5X}\dot{\phi}^3 H_{j}H_{k})- G_{5X}\dot{\phi}^3 H_{j}H_{k}(H_{j}\\&+H_{k})+T^{({\rm{em}})i}_{i} \,, \end{aligned}$

$\begin{aligned}[b] &\dfrac{1}{a^3}\dfrac{\rm d}{{\rm d}t}\left(a^3\dot{\phi}\, \Big[ G_{2X}-2G_{3\phi} +3H\dot\phi G_{3X} +H_1H_2H_3(3G_{5X}\dot{\phi}\right. \\&\left.\qquad+G_{5XX}\dot{\phi}^3)\Big]\right)= =G_{2\phi} -\dot{\phi}^2(G_{3\phi\phi} +G_{3X\phi}\ddot \phi)\\&\qquad-\dfrac{f(\phi)f'_\phi (\phi)}{2}F_{\gamma\delta}F^{\gamma\delta}\,, \end{aligned}$

$\begin{aligned} \partial_\mu[a^3 f^2(\phi)F^{\mu\nu}]=0\,. \end{aligned}$

$\begin{aligned} G^0_0=-\left(H_1H_2 +H_2H_3 +H_3H_1\right)\,, \end{aligned}$

$\begin{aligned} G^{i}_{i}=-\left(\dot{H}_{j} +\dot{H}_{k} +H_{j}^2 +H_{k}^2 +H_{j}H_{k}\right)\,. \end{aligned}$

$\begin{aligned} T^{({\rm{em}})\mu}_{\nu}= f^2(\phi)\left(-\frac{1}{4}\delta^\mu_\nu F_{\gamma\delta}F^{\gamma\delta} + F_{\nu \beta} F^{\mu \beta}\right) \,. \end{aligned}$

$\begin{aligned} F^{03}=-F^{30}=\frac{q_e}{a^3f^2(\phi)},\,\, F_{21}=-F_{12}=q_m\,, \end{aligned}$

$\begin{aligned} E^2=F_{03}F^{30}=\frac{q^2_e}{a^2_1a^2_2f^4(\phi)}\,,\, B^2=F_{21}F^{21}=\frac{q^2_m}{a^2_1a^2_2}\,. \end{aligned}$

$\begin{aligned}[b] T^{({\rm{em}})0}_{0}&=T^{({\rm{em}})3}_{3}=-T^{({\rm{em}})1}_{1}=-T^{({\rm{em}})2}_{2}=\\&= {-\dfrac{f^2(\phi)B^2}{2}= -\dfrac{\Psi(\phi)}{2a^2_1a^2_2}\,,} \end{aligned}$

$\begin{aligned} \Psi(\phi)\equiv q_m^2f^2(\phi)>0. \end{aligned}$

$\begin{aligned} a_1=ae^{\beta_++\sqrt{3}\beta_-},\,\, a_2=ae^{\beta_+-\sqrt{3}\beta_-},\,\, a_3=ae^{-2\beta_+}. \end{aligned}$

$\begin{aligned} H_1=H+\dot{\beta}_++\sqrt{3}\dot{\beta}_-\,,\, H_2=H+\dot{\beta}_+-\sqrt{3}\dot{\beta}_-\,, \, H_3=H-2\dot{\beta}_+\,. \end{aligned}$

$\begin{aligned}[b] \dfrac{3}{8\pi}\big(H^2-\sigma^2\big) =&\dfrac{\Psi(\phi)}{2a^4{\rm e}^{4\beta_+}}-G_2 +\dot{\phi}^2G_{2X}+3G_{3X}H\dot{\phi}^3 \\&-G_{3\phi}\dot{\phi}^2+ +\dot{\phi}^3(5G_{5X} +G_{5XX}\dot{\phi}^2) \\& \times (H - 2\dot{\beta}_+) \big[(H+\dot{\beta}_+)^2 -3\dot{\beta}_-^2\big]\,, \\ \sigma^2 \equiv & \,\dot{\beta}^2_+ + \dot{\beta}^2_-, \end{aligned}$

$\begin{aligned}[b] \dfrac{1}{8\pi}\big(2\dot{H}+3H^2 +& 3\sigma^2\big)=-\dfrac{\Psi(\phi)}{6a^4{\rm e}^{4\beta_+}}-G_2+G_{3\phi}\dot{\phi}^2 \\&+G_{3X}\dot{\phi}^2\ddot{\phi}+ +\dfrac{\rm d}{{\rm d}t}\left[G_{5X}\dot{\phi}^3 \left(H^2-\sigma^2\right)\right] \\&+2G_{5X}\dot{\phi}^3\left(H^3+\dot{\beta}_+^3-3\dot{\beta}_+\dot{\beta}_-^2 \right)\,, \end{aligned}$

$\begin{aligned} \frac{\dot{\beta}_-}{8\pi} +G_{5X}\dot\phi^3 \left(2\dot\beta_+\dot\beta_- -H\dot\beta_-\right) = \frac{C_-}{{ a}^3}\,, \end{aligned}$

$\begin{aligned} \frac{\dot{\beta}_+}{8\pi} +G_{5X}\dot\phi^3\left(\dot\beta_-^2-\dot\beta_+^2 -H\dot\beta_+\right)=\frac{1}{3a^3}\int\frac{\Psi(\phi){\rm d}t}{a{\rm e}^{4\beta_+}}+\frac{C_+}{a^3}, \end{aligned}$

$\begin{aligned}[b] &{\dot{\phi}\, \Big\{ G_{2X}-2G_{3\phi}+3H\dot\phi G_{3X} +(3G_{5X}\dot{\phi} +G_{5XX}\dot{\phi}^3)(H - 2\dot{\beta}_+)}\\&\qquad \times\big[(H+\dot{\beta}_+)^2 -3\dot{\beta}_-^2\big]\Big\}= \dfrac{1}{a^3}\displaystyle{\int} \Bigg(G_{2\phi}-\dot{\phi}^2(G_{3\phi\phi} \\&\qquad +G_{3X\phi}\ddot \phi)-\dfrac{\Psi'_\phi}{2a^4{\rm e}^{4\beta_+}}\Bigg)a^3{\rm d}t+\dfrac{C_\phi}{a^3}\,, \end{aligned}$

$\begin{aligned} \frac{a_i}{a}\rightarrow \text{const} \,\,\, \text{as} \,\,\, v \rightarrow +\infty, \end{aligned}$

$\begin{aligned} \beta_+=0. \end{aligned}$

$\begin{aligned} H_3=H. \end{aligned}$

$\begin{aligned} B=\frac{\text{const}}{a^2}. \end{aligned}$

$\begin{aligned} G_5=-\frac{1}{32\pi \gamma X}\,, \end{aligned}$

$\begin{aligned} C_-=0\,. \end{aligned}$

$\begin{aligned} H=\gamma\dot{\phi}, \end{aligned}$

$\begin{aligned} \dot\beta_-=\sqrt{\frac{8\pi\gamma\dot{\phi}}{3a^{3}} \left[\int {\rm d}t\, \frac{\Psi(\phi)}{a}+3C_+\right]}\,\,. \end{aligned}$

$\begin{aligned} \sigma^2=\frac{8\pi\gamma\dot{\phi}}{3a^{3}} \left[\int {\rm d}t\, \frac{\Psi(\phi)}{a}+3C_+\right], \end{aligned}$

$\begin{aligned} H_{1,2}=H\pm\sqrt{\frac{8\pi\gamma\dot{\phi}}{a^{3}} \left[\int {\rm d}t\, \frac{\Psi(\phi)}{a}+3C_+\right]}\,\,. \end{aligned}$

$\begin{aligned} \phi=\frac{1}{\gamma}\ln\left(\frac{a}{c_1}\right),\;\;\, \, \text{or}\,\, a=c_1{\rm e}^{\gamma\phi}. \end{aligned}$

$\begin{aligned}[b]3\gamma\dot{\phi}^2\left\{G_{3X}\dot{\phi}^2-\frac{\gamma}{8\pi}\right\}=&G_2-\dot{\phi}^2G_{2X}+G_{3\phi}\dot{\phi}^2 \\& -\frac{\gamma^2\dot{\phi}^2}{8\pi}-\frac{\Psi(\phi)}{2a^4}\, ,\end{aligned}$

$\begin{aligned}[b] 3\gamma\dot{\phi}^2\left\{G_{3X}\dot{\phi}^2-\dfrac{\gamma}{8\pi}\right\} =&-\dot{\phi}^2G_{2X}+2G_{3\phi}\dot{\phi}^2-\dfrac{2\gamma^2\dot{\phi}^2}{8\pi}\\&+\dfrac{(C_\phi-3\gamma C_+)\dot{\phi}}{a^3}+ \dfrac{\dot{\phi}}{a^3}\displaystyle{\int} {\rm d}t\, a^3\\& \times \Bigg[G_{2\phi}\left.-\dot{\phi}^2(G_{3\phi\phi} +G_{3X\phi}\ddot \phi) \right.\\&\left.-\dfrac{1}{2a^4}\left(\Psi'_\phi+2\gamma\Psi\right)\right]. \end{aligned}$

$\begin{aligned} -G_{2\phi}+\dot{\phi}^2(G_{3\phi\phi} +G_{3X\phi}\ddot \phi)+\frac{c_0{\rm e}^{2\gamma\phi}}{2a^6}\left(\Psi_\phi+2\gamma\Psi\right)=H\frac{U'_a(a)}{a^2}\,. \end{aligned}$

$\begin{aligned}[b] 3\gamma\dot{\phi}^2\left\{G_{3X}\dot{\phi}^2-\frac{\gamma}{8\pi}\right\} =&-\dot{\phi}^2G_{2X}+2G_{3\phi}\dot{\phi}^2-\frac{2\gamma^2\dot{\phi}^2}{8\pi}\\&+\frac{ \dot{\phi}}{a^3}\left(C_\phi-3\gamma C_+-U(a)\right) . \end{aligned}$

$\begin{aligned} G_3=\frac{\eta}{2}\ln\frac{X}{C}+\chi\sqrt{2X}+G(\phi),\,\, G_2=\varepsilon X-V(\phi). \end{aligned}$

$\begin{aligned} \dot{\phi}\left[\frac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta-3\gamma\chi|\dot{\phi}|+2G'_\phi\right]=\frac{ U(a)-C_\phi+3\gamma C_+}{a^3}, \end{aligned}$

$\begin{aligned} V+\frac{\Psi(\phi)}{2a^4}=\dot{\phi}^2\left[\frac{2\gamma^2}{8\pi}-\frac{\varepsilon}{2}-3\gamma\eta-3\gamma\chi|\dot{\phi}| +G'_\phi\right], \end{aligned}$

$\begin{aligned} V'_{\phi}+\frac{1}{2a^4}\left(\Psi'_\phi+2\gamma\Psi\right)=\gamma\dot{\phi}\frac{U'_a}{a^2}-\dot{\phi}^2G''_{\phi\phi}. \end{aligned}$

$\begin{aligned} V(\phi(a))+\frac{\Psi(\phi(a))}{2a^4}=S(a)> 0, \end{aligned}$

$\begin{aligned} V'_{a}+\frac{1}{2a^4}\left(\Psi'_a+2\frac{\Psi}{a}\right)=N(a), \end{aligned}$

$\begin{aligned} S(a)\equiv \dot{\phi}^2\left[\frac{2\gamma^2}{8\pi}-\frac{\varepsilon}{2}-3\gamma\eta-3\gamma\chi|\dot{\phi}| +G'_\phi\right], \end{aligned}$

$\begin{aligned} N(a)\equiv\frac{1}{\gamma a}\left[\gamma\dot{\phi}\frac{U'_a}{a^2}-\dot{\phi}^2G''_{\phi\phi}\right]. \end{aligned}$

$\begin{aligned} \Psi=\frac{a^5}{3}(N-S'_a), \end{aligned}$

$\begin{aligned} V=S+\frac{a}{6}(S'_a-N). \end{aligned}$

$\begin{aligned} U(a)=0,\,\, G=\xi {\rm e}^{-k\phi}, \,\, \xi>0,\,\,k>0, \,\, C_\phi=C_+=0. \end{aligned}$

$\begin{aligned}[b] |\dot{\phi}|&=\dfrac{1}{3\gamma\chi}\left[\dfrac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta-2\xi k {\rm e}^{-k\phi}\right]\\& =\dfrac{1}{3\gamma\chi}\left[\dfrac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta-2\xi k \left(\dfrac{c_1}{a}\right)^{k/\gamma}\right], \end{aligned}$

$\begin{aligned}[b] S(a)=&\dfrac{1}{9\gamma^2\chi^2}\left[\dfrac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta-2\xi k \left(\dfrac{c_1}{a}\right)^{k/\gamma}\right]^2\\& \times\left[\dfrac{\gamma^2}{8\pi}+\dfrac{\varepsilon}{2}+\xi k \left(\dfrac{c_1}{a}\right)^{k/\gamma}\right], \end{aligned}$

$\begin{aligned}[b] N(a)=&-\dfrac{\xi k^2}{9\gamma^3\chi^2 a}\left(\dfrac{c_1}{a}\right)^{k/\gamma} \\& \times\left[\dfrac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta-2\xi k \left(\dfrac{c_1}{a}\right)^{k/\gamma}\right]^2. \end{aligned}$

$\begin{aligned}[b] \Psi=&\dfrac{2c_1^4 k^2\xi {\rm e}^{(4\gamma-k)\phi}}{27\chi^2\gamma^3}\cdot\left[\varepsilon+\dfrac{2\gamma^2}{8\pi}+2\xi k {\rm e}^{-k\phi}\right]\\& {\times \left[\varepsilon-\dfrac{\gamma^2}{8\pi}+3\gamma\eta+2\xi k {\rm e}^{-k\phi}\right],} \end{aligned}$

$\begin{aligned}[b] V=&\dfrac{1}{18\gamma^2\chi^2}\left[\varepsilon+\dfrac{2\gamma^2}{8\pi}+2\xi k {\rm e}^{-k\phi}\right]\\& {\times \left[\varepsilon-\dfrac{\gamma^2}{8\pi}+3\gamma\eta+2\xi k {\rm e}^{-k\phi}\right]}\\& {\times\left[\varepsilon-\dfrac{\gamma^2}{8\pi}+3\gamma\eta+2\xi k {\rm e}^{-k\phi}\left(1-\dfrac{k}{3\gamma}\right)\right].} \end{aligned}$

$\begin{aligned} \varepsilon+\frac{2\gamma^2}{8\pi}\geqslant0,\,\,\varepsilon-\frac{\gamma^2}{8\pi}+3\gamma\eta\geqslant0,\,\,k<3\gamma,\,\, \gamma>0,\,\,\chi<0 \end{aligned}$

$\begin{aligned} H=H_3=\frac{1}{3|\chi|}\left[\varepsilon-\frac{\gamma^2}{8\pi}+3\gamma\eta\right.\left.+2\xi k\left(\frac{c_1}{a}\right)^{k/\gamma}\right], \end{aligned}$

$\begin{aligned} H_{1,2}=H\left[1\pm\sqrt{\dfrac{16\pi \xi k^2}{3\gamma^3}\left(\dfrac{c_1}{a}\right)^{k/\gamma}\cdot\dfrac{\dfrac{\varepsilon+\dfrac{2\gamma^2}{8\pi}}{3-k/\gamma}+\dfrac{2\xi k}{3-2k/\gamma}\left(\dfrac{c_1}{a}\right)^{k/\gamma}}{\varepsilon-\dfrac{\gamma^2}{8\pi}+3\gamma\eta+2\xi k\left(\dfrac{c_1}{a}\right)^{k/\gamma}}}\,\right], ~~ k<3\gamma/2. \end{aligned}$

$\begin{aligned}[b] \dfrac{\sigma^2}{H^2}=&\dfrac{8\pi}{3a^{3}H} \displaystyle{\int} {\rm d}t\, \dfrac{\Psi(\phi)}{a} =\dfrac{16\pi \xi k^2}{9\gamma^3}\left(\dfrac{c_1}{a}\right)^{k/\gamma}\\&\cdot\dfrac{\dfrac{\varepsilon+\dfrac{2\gamma^2}{8\pi}}{3-k/\gamma}+\dfrac{2\xi k}{3-2k/\gamma}\left(\dfrac{c_1}{a}\right)^{k/\gamma}}{\varepsilon-\dfrac{\gamma^2}{8\pi}+3\gamma\eta+2\xi k\left(\dfrac{c_1}{a}\right)^{k/\gamma}}. \end{aligned}$

$\begin{aligned} a=a_3=c_1\left\{\frac{2\xi k}{3|\chi|H_\infty}\left[\exp\left(\frac{kH_\infty t}{\gamma}\right)-1\right]\right\}^{\gamma/k},\,\, t\geq 0, \end{aligned}$

$H_\infty\equiv\frac{1}{3|\chi|}\left(\varepsilon-\frac{\gamma^2}{8\pi}+3\gamma\eta\right).$

$\begin{aligned} a=a_3\propto t^{\gamma/k } \,\, \text{as}\,\, \, H_\infty t\ll 1; \,\,\, a=a_3\propto {\rm e}^{H_\infty t} \,\, \text{as}\,\, \, H_\infty t\gg 1. \end{aligned}$

$\begin{aligned} \phi=\frac{1}{k}\ln\left\{\frac{2\xi k}{3|\chi|H_\infty}\left[\exp\left(\frac{kH_\infty t}{\gamma}\right)-1\right]\right\}. \end{aligned}$

$\begin{aligned} \varepsilon+\frac{2\gamma^2}{8\pi}>0,\,\,\,\varepsilon-\frac{\gamma^2}{8\pi}+3\gamma\eta>0. \end{aligned}$

$\begin{aligned} H_{1,2}\approx H_\infty \left[1\pm \sqrt{\frac{8\pi k\left(\varepsilon+\dfrac{2\gamma^2}{8\pi}\right)}{3\gamma^3\left(3-k/\gamma\right)}}\cdot {\rm e}^{-\tfrac{kH_\infty t}{2\gamma}}\right] \end{aligned}$

$\begin{aligned} a_{1,2}\approx s_{1,2}\cdot {\rm e}^{H_\infty t}\cdot\exp\left\{\mp 4\sqrt{\dfrac{2\pi \left(\varepsilon+\dfrac{2\gamma^2}{8\pi}\right)}{3k\gamma\left(3-k/\gamma\right)}}\cdot {\rm e}^{-\tfrac{kH_\infty t}{2\gamma}}\right\}. \end{aligned}$

$\begin{aligned}[b] \frac{a_{1,2}}{a}\sim &\exp\left\{\mp 4\sqrt{\frac{2\pi \left(\varepsilon+\frac{2\gamma^2}{8\pi}\right)}{3k\gamma\left(3-k/\gamma\right)}}\cdot {\rm e}^{-\tfrac{kH_\infty t}{2\gamma}}\right\}\\&\rightarrow \text{const}\,\, \,\text{as}\,\, \, H_\infty t\rightarrow +\infty. \end{aligned}$

$\begin{aligned} H_{1,2}\approx\pm\sqrt{\frac{8\pi |\chi|}{ k^2(3-2k/\gamma)}}\cdot \frac{1}{t^{3/2}}+\frac{\gamma}{k}\cdot\frac{1}{t} \end{aligned}$

$\begin{aligned} a_{1,2}\approx s_{1,2}\cdot t^{\gamma/k}\cdot\exp\left\{\mp4\sqrt{\frac{2\pi|\chi|}{k^2(3-2k/\gamma)}}\cdot\frac{1}{t^{1/2}}\right\}. \end{aligned}$

$\begin{aligned} \frac{1}{3-2k/\gamma}\left(\varepsilon-\frac{\gamma^2}{8\pi}+3\gamma\eta\right)= \left(\varepsilon+\frac{2\gamma^2}{8\pi}\right)\frac{1}{3-k/\gamma}>0, \end{aligned}$

$\begin{aligned} a_{1,2}=s_{1,2}\cdot a\cdot\exp \left\{\mp8\cdot\sqrt{\frac{\pi \xi}{3\gamma(3-2k/\gamma)}}\cdot\left(\frac{c_1}{a}\right)^{k/(2\gamma)}\,\right\}. \end{aligned}$

$\begin{aligned} a_{2\min}=c_1\cdot\left(\frac{16\pi \xi k^2}{3\gamma^3(3-2k/\gamma)}\right)^{\gamma/k}, \end{aligned}$

$\begin{aligned} \varepsilon=-1, \,\, \gamma=2\sqrt{\pi}, \,\, \eta>\frac{1}{4\sqrt{\pi}}, \,\, k<3\sqrt{\pi}, \end{aligned}$

$\begin{aligned}[b] a_{1,2}=&s_{1,2}\cdot a\times \exp\left\{\mp4\sqrt{\dfrac{\pi}{3k(3\sqrt{\pi}-k)}}\right.\\&\left.\cdot\sqrt{-3/2+6\eta\sqrt{\pi}+2\xi k\left(\dfrac{c_1}{a}\right)^{k/(2\sqrt{\pi})}}\right\} \end{aligned}$

$\begin{aligned} a=c_1\left\{\frac{2\xi k}{3|\chi|H_\infty}\left[\exp\left(\frac{kH_\infty t}{2\sqrt{\pi}}\right)-1\right]\right\}^{2\sqrt{\pi}/k },\,\, t\geq 0, \end{aligned}$

$H_\infty=\frac{4\eta\sqrt{\pi}-1}{2|\chi|}.$

$\begin{aligned}[b] a_{1,2}=&s_{1,2}\cdot a\cdot\left[ \sqrt{1 +\dfrac{\nu^2 }{A^2_0}\left(\dfrac{a}{c_1}\right)^{k/\gamma}}+ \sqrt{\dfrac{\nu^2}{A^2_0}\left(\dfrac{a}{c_1}\right)^{k/\gamma}}\,\right]^{\pm\tfrac{2|\nu|\gamma}{k}}\\& \times\exp \left\{\mp\dfrac{2\gamma}{k}\cdot\sqrt{\nu^2+A^2_0\left(\dfrac{c_1}{a}\right)^{k/\gamma}}\,\right\}, \end{aligned}$

$\begin{aligned} \nu^2\equiv\dfrac{8\pi k}{3\gamma^3}\cdot \dfrac{\varepsilon+\dfrac{2\gamma^2}{8\pi}}{3-k/\gamma}\,,\,\, A_0^2\equiv\dfrac{16\pi \xi k^2}{3\gamma^3(3-2k/\gamma)}. \end{aligned}$

$\begin{aligned} \frac{\sigma^2}{H^2}=\frac{\nu^2}{3}+\frac{A^2_0}{3}\left(\frac{c_1}{a}\right)^{k/\gamma}. \end{aligned}$

$\begin{aligned} \frac{a_{1,2}}{a}\sim a^{\pm |\nu|}\,\,\, \text{as} \,\,\, a \rightarrow +\infty. \end{aligned}$

$\begin{aligned} \left|\frac{\sigma}{H}\right|\rightarrow \frac{|\nu|}{\sqrt{3}}\ll 1\,\,\, \text{as} \,\,\, a \rightarrow +\infty. \end{aligned}$

$\begin{aligned} H=\frac{2\xi k}{3|\chi|}\left(\frac{c_1}{a}\right)^{k/\gamma}, \end{aligned}$

$\begin{aligned} a=c_1\left[\frac{2\xi k^2}{3|\chi|\gamma}\cdot t\right]^{\gamma/k},\,\, t\geqslant 0. \end{aligned}$

$\begin{aligned} \phi=\frac{1}{k}\ln\left(\frac{2\xi k^2 t}{3|\chi|\gamma}\right). \end{aligned}$

$\begin{aligned} \chi=0,\,\, G=\xi\cdot\phi. \end{aligned}$

$\begin{aligned} \dot{\phi}=\frac{\widetilde{U}}{\alpha a^3},\,\, \widetilde{U}\equiv U(a)-C_\phi+3\gamma C_+, \end{aligned}$

$\begin{aligned} S(a)= \frac{\beta \widetilde{U}^2}{\alpha^2a^6},\,\, N(a)=\frac{\widetilde{U}\widetilde{U}'_a}{\alpha a^6}\,, \end{aligned}$

$\begin{aligned} \alpha\equiv\frac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta+2\xi,\,\,\beta\equiv\frac{2\gamma^2}{8\pi}-\frac{\varepsilon}{2}-3\gamma\eta +\xi>0. \end{aligned}$

$\begin{aligned}[b] \Psi&=\dfrac{a^{-1}}{3\alpha^2}\left[\left(\dfrac{\alpha}{2}-\beta\right)(\widetilde{U}^2)'_a+\dfrac{6\beta \widetilde{U}^2}{a}\right]\\&\; {=\dfrac{{\rm e}^{-2\gamma\phi}}{3c_1^2\alpha^2}\left[\left(\dfrac{\alpha}{2}-\beta\right)\gamma^{-1} [\widetilde{U}^2(a(\phi))]'_\phi+6\beta \widetilde{U}^2(a(\phi))\right],} \end{aligned}$

$\begin{aligned}[b] V&=\dfrac{1}{6\alpha^2}\left(\beta-\dfrac{\alpha}{2}\right)\dfrac{(\widetilde{U}^2)'_a}{a^5}\\&\; {=\dfrac{\gamma^{-1}{\rm e}^{-6\gamma\phi}}{6\alpha^2 c_1^6}\left(\beta-\dfrac{\alpha}{2}\right)[\widetilde{U}^2(a(\phi))]'_\phi\,.} \end{aligned}$

$\begin{aligned} H=H_3=\frac{\gamma \widetilde{U}(a)}{\alpha a^3}, \end{aligned}$

$\begin{aligned} H_{1,2}=H\left[ 1 \pm \sqrt{\frac{8\pi}{3\gamma^2}\left( \alpha -2\beta + \frac{6\beta}{\widetilde{U}}\int\frac{{\rm d}a \cdot \widetilde{U}}{a} + \frac{9\alpha \gamma C_+}{\widetilde{U}} \right)}\,\right], \end{aligned}$

$\begin{aligned} \frac{\sigma^2}{H^2}=\frac{8\pi}{9\gamma^2}\left(\alpha-2\beta+ \frac{6\beta}{\widetilde{U}}\int\frac{{\rm d}a \cdot \widetilde{U}}{a}+\frac{9\alpha \gamma C_+}{\widetilde{U}}\right). \end{aligned}$

$\begin{aligned} U(a)=Aa^n, \,\, n>0 \end{aligned}$

$\begin{aligned} H=\frac{\gamma A}{\alpha a^{3-n}}. \end{aligned}$

$\begin{aligned} \frac{\gamma A}{\alpha}>0 \end{aligned}$

$\begin{aligned} a=\left[\frac{\gamma A(3-n)}{\alpha}\cdot t\right]^{1/(3-n)}. \end{aligned}$

$\begin{aligned} 2<n<3, \end{aligned}$

$\begin{aligned} n>3, \end{aligned}$

$\begin{aligned} a=\left[\frac{\alpha}{\gamma A(n-3)}\right]^{1/(n-3)}\cdot\left(\frac{1}{-t}\right)^{1/(n-3)},\,\, a(0)=+\infty. \end{aligned}$

$\begin{aligned} \phi=\frac{1}{\gamma}\ln\left\{\frac{1}{c_1} \left[\frac{\gamma A(3-n)}{\alpha}\cdot t\right]^{1/(3-n)}\right\} . \end{aligned}$

$\begin{aligned} \Psi={\rm e}^{2(n-1)\gamma\phi}\cdot\frac{c_1^{2(n-1)}A^2}{\alpha^2} \left[2\beta+\frac{n}{3}(\alpha-2\beta)\right], \end{aligned}$

$\begin{aligned} V={\rm e}^{(2n-6)\gamma\phi}\cdot \frac{nc_1^{2n-6}A^2\left(2\beta-\alpha\right)}{6\alpha^2}. \end{aligned}$

$\begin{aligned} 2\beta+\frac{n}{3}(\alpha-2\beta)>0,\,\,2\beta-\alpha>0 \end{aligned}$

$\begin{aligned} \frac{\sigma^2}{H^2}= \frac{8\pi}{3n\gamma^2}\left[2\beta+\frac{n}{3}(\alpha-2\beta)\right]=\text{const}, \end{aligned}$

$\begin{aligned} \frac{8\pi}{3n\gamma^2}\left[2\beta+\frac{n}{3}(\alpha-2\beta)\right]>0. \end{aligned}$

$\begin{aligned} \left|\frac{8\pi}{3n\gamma^2}\left[2\beta+\frac{n}{3}(\alpha-2\beta)\right]\right|\ll 1. \end{aligned}$

$\begin{aligned} H_{1,2}=H\left(1\pm \sqrt{\frac{8\pi}{n\gamma^2}\left[2\beta+\frac{n}{3}(\alpha-2\beta)\right]}\,\right)\,. \end{aligned}$

$\begin{aligned} a_{1,2}=s^{\pm}_{0}\cdot a^{1\pm \sqrt{\tfrac{8\pi}{n\gamma^2}\left[2\beta+\tfrac{n}{3}(\alpha-2\beta)\right]}\,},\,\, a_3=a. \end{aligned}$

$\begin{aligned} \frac{a_{1,2}}{a}=s^{\pm}_{0}\cdot a^{\pm \sqrt{\tfrac{8\pi}{n\gamma^2}\left[2\beta+\tfrac{n}{3}(\alpha-2\beta)\right]}\,}\approx \text{const}. \end{aligned}$

$\begin{aligned} \frac{2\gamma^2}{8\pi}-3\eta\gamma-\frac{\varepsilon}{2}+\xi>0, \end{aligned}$

$\begin{aligned} \frac{\gamma A}{{\gamma^2}/({8\pi})-3\gamma\eta-\varepsilon+2\xi}>0, \end{aligned}$

$\begin{aligned} \frac{\gamma^2}{8\pi}-\gamma\eta>0, \end{aligned}$

$\begin{aligned} \frac{\gamma^2}{8\pi}(4-n)+\eta\gamma(n-6)-\varepsilon+2\xi>0, \end{aligned}$

$\begin{aligned} \frac{8\pi}{n\gamma^2}\left[\frac{\gamma^2}{8\pi}(4-n)+\eta\gamma(n-6)-\varepsilon+2\xi\right]\ll 1. \end{aligned}$

$\begin{aligned} \eta=0,~~ \varepsilon=1,~~ \xi=0,~~ n>2. \end{aligned}$

$\begin{aligned} 4-n-\frac{8\pi}{\gamma^2}=\mu^2>0, \end{aligned}$

$\begin{aligned} \frac{2\gamma^2}{8\pi}-\frac{1}{2}>0,~~\frac{\gamma A}{\dfrac{\gamma^2}{8\pi}-1}>0. \end{aligned}$

$\begin{aligned} \chi\neq0,\,\, G=\xi\cdot\phi, \,\, C_\phi=C_+=0. \end{aligned}$

$\frac{\gamma^2}{8\pi}-\varepsilon-3\gamma\eta+2\xi=0,$

$\gamma>0,\,\, \dot{\phi}>0,\,\,\chi<0,\,\,U>0,$

$\begin{aligned} \dot{\phi}=\left(\frac{1}{3\gamma|\chi|}\cdot\frac{U}{a^3}\right)^{1/2}, \end{aligned}$

$\begin{aligned} H=\left(\frac{\gamma}{3|\chi|}\cdot\frac{U}{a^3}\right)^{1/2}>0, \end{aligned}$

$\begin{aligned} \frac{2\gamma^2}{8\pi}-\frac{\varepsilon}{2}-3\gamma\eta+\xi=0, \end{aligned}$

$\begin{aligned} S(a)=\frac{1}{(3\gamma|\chi|)^{1/2}}\cdot\left(\frac{U}{a^3}\right)^{3/2}, \end{aligned}$

$\begin{aligned} N(a)=\left(\frac{1}{3\gamma|\chi|}\cdot\frac{U}{a^3}\right)^{1/2}\frac{U'_a}{a^3}, \end{aligned}$

$\begin{aligned} \Psi=\frac{a^5}{3}\cdot\left(\frac{1}{3\gamma|\chi|}\cdot\frac{U}{a^3}\right)^{1/2}\cdot \left[\frac{U'_a}{a^3}-\frac{3}{2}\left(\frac{U}{a^3}\right)'_a\right], \end{aligned}$

$\begin{aligned} V=\left(\frac{1}{3\gamma|\chi|}\cdot\frac{U}{a^3}\right)^{1/2} \cdot\left[\frac{U}{a^3}-\frac{a}{6}\left(\frac{U'_a}{a^3}-\frac{3}{2}\left(\frac{U}{a^3}\right)'_a\right)\right]. \end{aligned}$

$\begin{aligned} U=A^2a^{-2n+3},\,\, A>0, \end{aligned}$

$\begin{aligned} H=\left(\frac{\gamma }{3|\chi|}\right)^{1/2}\cdot \frac{A}{a^n}. \end{aligned}$

$\begin{aligned} a=\left(\sqrt{\frac{\gamma}{3|\chi|}}\cdot nA\cdot t\right)^{1/n}. \end{aligned}$

$\begin{aligned} a=\left[\sqrt{\frac{\gamma}{3|\chi|}}\cdot |n|A\right]^{-1/|n|}\cdot\left(\frac{1}{-t}\right)^{1/|n|},\,\, a(0)=+\infty. \end{aligned}$

$\begin{aligned} \phi=\frac{1}{\gamma}\ln\left\{\frac{1}{c_1} \left(\sqrt{\frac{\gamma}{3|\chi|}}\cdot nA\cdot t\right)^{1/n}\right\} . \end{aligned}$

$\begin{aligned} V=\frac{(3-n)A^{3}}{6(3\gamma|\chi|)^{1/2}a^{3n}}=\frac{(3-n)A^{3}}{6(3\gamma|\chi|)^{1/2}c_1^{3n}}\cdot {\rm e}^{-3n\gamma\phi}, \end{aligned}$

$\begin{aligned} \Psi=\frac{\left(3+n\right)A^{3}a^{4-3n}}{3(3\gamma|\chi|)^{1/2}}= \frac{\left(3+n\right)A^{3}c_1^{4-3n}}{3(3\gamma|\chi|)^{1/2}}\cdot {\rm e}^{(4-3n)\gamma\phi}. \end{aligned}$

$\begin{aligned} -3<n<3 \end{aligned}$

$\begin{aligned} H_{1,2}=H\left[1\pm\sqrt{\frac{8\pi |\chi|^{1/2}A(3+n)}{3^{1/2}\gamma^{3/2}(3-2n)}}\cdot\frac{1}{a^{n/2}}\,\right], \end{aligned}$

$\begin{aligned} a_{1,2}=s_{1,2}\cdot a\cdot\exp\left[\mp\frac{2}{na^{n/2}} \cdot\sqrt{\frac{8\pi |\chi|^{1/2}A(3+n)}{3^{1/2}\gamma^{3/2}(3-2n)}}\,\right]. \end{aligned}$

$\begin{aligned}[b] \frac{a_{1,2}}{a}=&s_{1,2}\cdot\exp\left[\mp\frac{2}{na^{n/2}} \cdot\sqrt{\frac{8\pi |\chi|^{1/2}A(3+n)}{3^{1/2}\gamma^{3/2}(3-2n)}}\,\right]\\&\rightarrow \text{const} \,\,\, \text{as} \,\,\, a \rightarrow +\infty. \end{aligned}$

$\begin{aligned} a_{2{\rm min}}=\left(\frac{8\pi |\chi|^{1/2}A(3+n)}{3^{1/2}\gamma^{3/2}(3-2n)}\right)^{1/n} \end{aligned}$

$\begin{aligned}[b] G_4&=1/(16\pi),\,\,G_2=\varepsilon X-V(\phi),\\ G_5&=-\dfrac{1}{32\pi \gamma X},\,\, G_3=\dfrac{\eta}{2}\ln\dfrac{X}{C}+\chi\sqrt{2X}+G(\phi). \end{aligned}$