Cosmological dynamics of a non-minimally coupled bulk scalar field in DGP setup

Abstract

We consider cosmological dynamics of a canonical bulk scalar field, which is coupled non-minimally to 5-dimensional Ricci scalar in a DGP setup. We show that presence of this non-minimally coupled bulk scalar field affects the jump conditions of the original DGP model significantly. Within a superpotential approach, we perform some numerical analysis of the model parameter space and consider bulk-brane energy exchange in this setup. Also we show that the normal, ghost-free branch of the DGP solutions in this case has the potential to realize a self-consistent phantom-like behavior and therefore explains late time acceleration of the universe in a consistent way.

Appendices

Appendix A

$$\begin{aligned} H^{2} =&\frac{1}{ (-3\kappa_{5}^{2}-16\xi^{2}\phi_{0}^{2}\kappa _{5}^{2}+3\xi\phi_{0}^{2}\kappa_{5}^{2}- 24\xi^{2}\phi_{0}^{2}\kappa_{4}^{2}+24\xi^{3}\phi_{0}^{4}\kappa _{4}^{2} )^{2}} \bigl\{384\xi^{2} \phi^{2} \kappa_{4}^{4}+2048\xi^{4} \phi^{4} \kappa_{4}^{4} \\&{} +2048\xi^{6}\phi^{8}\kappa_{4}^{4}-72 \kappa_{4}^{4}\xi^{3}\phi^{6}-4096 \xi^{5}\phi^{6}\kappa_{4}^{4}-1152 \xi^{3}\phi^{4} \kappa_{4}^{4}-384 \kappa_{4}^{4}\xi^{5}\phi^{8}-1152 \xi^{5}\phi^{6}\kappa_{4}^{4}m \\&{} +48\xi^{3}\phi^{4}\kappa_{5}^{4}m-48 \xi^{2}\phi^{2} \kappa_{5}^{4}m+576 \xi^{6}\phi^{8}\kappa_{4}^{4}m+576 \xi^{4}\phi^ {4}\kappa_{4}^{4}m-256 \xi^{4}\phi^{4}\kappa_{5}^{4}m+108\kappa _{4}^{4}\xi^{2}\phi^{4} \\&{} +3\rho^{(b)} \kappa_{5}^{4} \kappa_{4}^{2}+1152 \xi^{4}\phi^{6} \kappa_{4}^{4}+18\kappa_{4}^{4} \xi^{4}\phi^{8}-72 \kappa_{4}^{4}\xi \phi^ {2}+18\kappa_{4}^{4}+192 \xi^{4} \phi^{4}\rho^{(b)}\kappa_{5}^{2} \kappa_{4}^{4}- 6\xi\phi^{2}\kappa_{5}^{4} \rho^{(b)}\kappa_{4}^{2} \\&{} -144\xi^{3}{\phi}^{4}\kappa_{4}^{2}m \kappa_{5}^{2}+96\xi^{4}\phi^{5} \kappa_{4}^{4}g\kappa_{5}^{2}+ 72 \xi^{2}\phi^{2}\kappa_{4}^{2}m \kappa_{5}^{2}-40\xi^{3}\phi^{4} \rho^{(b)}\kappa_{5}^{4}\kappa_{4}^{2}-48 \xi^{5}\phi^{7}\kappa_{4}^{4}g \kappa_{5}^{2} \\&{} -48\xi^{3}\phi^{3}\kappa_{4}^{4}g \kappa_{5}^{2}+128\xi^{4}\phi^{4}p \kappa_{5}^{4}\kappa_{4}^{2} +72 \xi^{4}\phi^{6}\kappa_{4}^{2}m \kappa_{5}^{2} -192\xi^{5}\phi^{6} \kappa_{4}^{4}\rho^{(b)}\kappa_{5}^{2}+3 \xi^{2}\phi^{4}\kappa_{5}^{4} \rho^{(b)}\kappa_{4}^{2} \\&{} +12\xi^{2}\phi^{3}\kappa_{4}^{2}g \kappa_{5}^{4}-32\xi^{3}\phi^{3} \kappa_{5}^{4}\kappa_{4}^{2}g-24\xi ^{3}\phi^{4}p\kappa_{5}^{4} \kappa_{4}^{2} +40\xi^{2}\phi^{2} \rho^{(b)}\kappa_{5}^{4}\kappa_{4}^{2}-6 \xi\phi\kappa_{4}^{2}g\kappa_{5}^{4} +24 \xi^{4}\phi^{6}\kappa_{4}^{4} \rho^{(b)}\kappa_{5}^{2} \\&{} -6\xi^{3}\phi^{5}\kappa_{4}^{2} g \kappa_{5}^{4}+192\xi^{4}\phi^{4} \kappa_{4}^{4}p\kappa_{5}^{2}-48\xi ^{3}\phi^{4}\kappa_{4}^{4} \rho^{(b)}\kappa_{5}^{2} +24\xi^{2} \phi^{2}\kappa_{4}^{4}\rho^{(b)} \kappa_{5}^{2}-192\xi^{5}\phi^{6} \kappa_{4}^{4}p\kappa_{5}^{2} \\&{} +128\xi^{4}\phi^{4}\rho^{(b)} \kappa^{4} \kappa_{4}^{2}+24\xi^{2} \phi^{2}p \kappa_{5}^{4}\kappa_{4}^{2} +32 \xi^{4}\phi^{5}\kappa_{4}^{2}g \kappa_{5}^{4} \\&{} \pm2 \bigl[\kappa_{4}^{4} \bigl(\xi\phi^{2}-1 \bigr)^{2} \bigl(32\xi^{2}\phi^{2}-3\xi \phi^{2}+3 \bigr)^{2} \bigl(192\xi^{2} \phi^{2} \kappa_{4}^{4}+1024\xi^{4} \phi^{4}\kappa_{4}^{4}+1024\xi^{6} \phi^{8} \kappa_{4}^{4} -36\kappa_{4}^{4} \xi^{3}\phi^{6} \\&{} -2048\xi^{5}\phi^{6}\kappa_{4}^{4}-576 \xi^{3}\phi^{4}\kappa_{4}^{4} -192 \kappa_{4}^{4}\xi^{5}\phi^{8} -96 \kappa_{5}^{4}{\mathcal{F}}\xi^{3} \phi^{4}+256\xi^{4}\phi^{4}\kappa _{5}^{4}{\mathcal{F}}+576\xi^{6} \phi^{8} \kappa_{4}^{4}{\mathcal{F}} \\&{} +576\xi^{4}\phi^{4}\kappa_{4}^{4}{ \mathcal{F}}-18\kappa_{5}^{4}{\mathcal{F}}\xi \phi^{2}+9\kappa_{5}^{4}{\mathcal{F}} \xi^{2}\phi^{4} +96\kappa_{5}^{4}{ \mathcal{F}}\xi^{2}\phi^{2}-1152\xi^{5} \phi^{6}\kappa_{4}^{4}{\mathcal{F}}-1152 \xi^{5}\phi^{6}\kappa_{4}^{4}m \\&{} +48\xi^{3}\phi^{4}\kappa_{5}^{4}m-48 \xi^{2}\phi^{2}\kappa_{5}^{4}m+576 \xi^{6}\phi^{8}\kappa_{4}^{4}m +576 \xi^{4}\phi^{4}\kappa_{4}^{4}m -256 \xi^{4}\phi^{4}\kappa_{5}^{4}m+54 \kappa_{4}^{4}\xi^{2}\phi^{4} \\&{} +3\rho^{(b)} \kappa_{5}^{4} \kappa_{4}^{2}+576 \xi^{4}\phi^{6} \kappa_{4}^{4}+9\kappa_{4}^{4} \xi^{4}\phi^{8}-36 \kappa_{4}^{4}\xi \phi^{2}+ 9\kappa_{4}^{4}+9 \kappa_{5}^{4}{ \mathcal{F}} +192\xi^{4} \phi^{4}\rho^{(b)} \kappa_{5}^{2} \kappa_{4}^{4}-6\xi \phi^{2}\kappa_{5}^{4} \rho^{(b)} \kappa_{4}^{2} \\&{} -144\xi^{3}\phi^{4}\kappa_{4}^{2}m \kappa_{5}^{2}+96\xi^{4}\phi^{5}\kappa _{4}^{4}g\kappa_{5}^{2} +72 \xi^{2}\phi^{2}\kappa_{4}^{2}m \kappa_{5}^{2}-40\xi^{3}\phi^{4}\rho ^{(b)}\kappa_{5}^{4}\kappa_{4}^{2} -48\xi^{5}\phi^{7}\kappa_{4}^{4}g \kappa_{5}^{2}-48\xi^{3}\phi^{3}\kappa _{4}^{4}g\kappa_{5}^{2} \\&{} -288\xi^{3}\phi^{4}\kappa_{4}^{2} \kappa_{5}^{2}{\mathcal{F}}+144\xi^{2} \phi^{2}\kappa_{4}^{2}\kappa_{5}^{2}{ \mathcal{F}} +128\xi^{4}\phi^{4}p\kappa_{5}^{4} \kappa_{4}^{2}+72\xi^{4}\phi^{6}\kappa _{4}^{2}m\kappa_{5}^{2} -192 \xi^{5}\phi^{6}\kappa_{4}^{4} \rho^{(b)}\kappa_{5}^{2} \\&{} +3\xi^{2}\phi^{4}\kappa_{5}^{4} \rho^{(b)}\kappa_{4}^{2}- 768\kappa_{4}^{2} \xi^{5}\phi^{6}\kappa_{5}^{2}{ \mathcal{F}}+144\xi^{4}\phi^{6}\kappa_{4}^{2} \kappa_{5}^{2}{\mathcal{F}} +768\kappa_{4}^{2} \xi^{4}\phi^{4}\kappa_{5}^{2}{ \mathcal{F}} +12\xi^{2}\phi^{3}\kappa_{4}^{2}g \kappa^{4} \\&{} -32\xi^{3}\phi^{3}\kappa_{5}^{4} \kappa_{4}^{2}g-24\xi^{3}\phi^{4}p\kappa _{5}^{4}\kappa_{4}^{2}+ 40 \xi^{2}\phi^{2}\rho^{(b)}\kappa_{5}^{4} \kappa_{4}^{2}-6\xi\phi\kappa_{4}^{2}g \kappa_{5}^{4} +24\xi^{4}\phi^{6} \kappa_{4}^{4}\rho^{(b)}\kappa_{5}^{2}-6 \xi^{3}\phi^{5}\kappa_{4}^{2}g \kappa_{5}^{4} \\&{} +192\xi^{4}\phi^{4}\kappa_{4}^{4}p \kappa_{5}^{2}-48\xi^{3}\phi^{4}\kappa _{4}^{4}\rho^{(b)}\kappa_{5}^{2} +24\xi^{2}\phi^{2}\kappa_{4}^{4} \rho^{(b)}\kappa_{5}^{2}-192\xi^{5}\phi ^{6}\kappa_{4}^{4}p\kappa_{5}^{2} +128\xi^{4}\phi^{4}\rho^{(b)}\kappa_{5}^{4} \kappa_{4}^{2} \\&{} +24\xi^{2}\phi^{2}p\kappa_{5}^{4} \kappa_{4}^{2}+32\xi^{4}\phi^{5}\kappa _{4}^{2}g\kappa_{5}^{4} \bigr) \bigr]^{\frac{1}{2}} \bigr\}_{0}. \end{aligned}$$

Appendix B

$$\begin{aligned} \omega_{eff} =&-\frac{1}{3} \bigl\{ \bigl[-96\xi^{2} \phi\kappa_{5}^{4}m \dot{\phi} -48\xi^{2} \phi^{2}\kappa_{5}^{4}\dot{m}+144\xi^{2} \phi\kappa_{4}^{2}m \kappa_{5}^{2}\dot{ \phi}+72\xi^{2} \phi^{2}\kappa_{4}^{2} \dot{m}\kappa_{5}^{2}-12\xi\phi\kappa_{5}^{4} \rho^{(b)}\kappa_{4}^{2} \dot{\phi}-\dot{ \rho}_{m} \\&{} +48\xi^{2}\phi\kappa_{4}^{4} \rho^{(b)} \kappa_{5}^{2}\dot{\phi}+24 \xi^{2} \phi^{2}{\kappa_{4}}^{4} \dot{ \rho}^{(b)} \kappa_{5}^{2}+48\xi^{2}\phi p \kappa_{5}^{4} \kappa_{4}^{2}\dot{\phi} +24\xi^{2} \phi^{2}\dot{p}\kappa_{5}^{4} \kappa_{4}^{2}+36 \xi^{2}\phi^{2} \kappa_{4}^{2}g \kappa_{5}^{4}\dot{\phi} +12\xi^{2} \phi^{3}\kappa_{4}^{2}\dot{g} \kappa_{5}^{4} \\&{} -6\xi\phi^{2}\kappa_{5}^{4}\dot{ \rho}^{(b)} \kappa_{4}^{2}+80\xi^{2}\phi \rho^{(b)}\kappa_{5}^{4}\kappa_{4}^{2} \dot{\phi}+40\xi^{2}\phi^{2} \dot{\rho}^{(b)} \kappa_{5}^{4}\kappa_{4}^{2}-6\xi\dot{ \phi}\kappa_{4}^{2}g \kappa_{5}^{4} -6\xi \phi\kappa_{4}^{2}\dot{g}\kappa_{5}^{4}+12 \xi^{2}\phi^{3}\kappa_{5}^{4} \rho^{(b)} \kappa_{4}^{2}\dot{\phi} \\&{} +3\xi^{2}\phi^{4}\kappa_{5}^{4} \dot{ \rho}^{(b)}\kappa_{4}^{2} +3\dot{ \rho}^{(b)} \kappa_{5}^{4}\kappa_{4}^{2}+432 \kappa_{4}^{4}\xi^{2}\phi^{3}\dot{\phi} +768\xi^{2}\phi\kappa_{4}^{4}\dot{\phi}-144 \kappa_{4}^{4}\xi\phi\dot{\phi} -3\kappa_{4}^{8} \bigl(3\dot{\rho}^{(b)}\kappa_{5}^{4}\kappa _{4}^{2}+48\xi^{2}\phi\kappa_{4}^{4} \rho^{(b)}\kappa_{5}^{2}\dot{\phi} \\&{} +48\xi^{2}\phi p\kappa_{5}^{4} \kappa_{4}^{2}\dot{\phi}+108\xi^{2} \phi^{2}\kappa_{4}^{2}g \kappa_{5}^{4} \dot{\phi}+208\xi^{2}\phi\rho^{(b)} \kappa_{5}^{4} \kappa_{4}^{2}\dot{\phi}+180\xi^{2} \phi^{3}\kappa_{5}^{4}\rho^{(b)} \kappa_{4}^{2}\dot{\phi}+288\xi^{2}\phi \kappa_{4}^{2}\kappa_{5}^{2}{\mathcal{F}} \dot{\phi} +9\kappa_{5}^{4}\dot{{\mathcal{F}}} \\&{} -12\xi\phi\kappa_{4}^{2}g\kappa_{5}^{4} \dot{\phi}-96\xi^{2}\phi\kappa_{5}^{4}m\dot{ \phi}+72\xi^{2}\phi^{2}\kappa_{4}^{2} \dot{m}\kappa_{5}^{2}-18\xi\phi^{2} \kappa_{5}^{4} \dot{\rho}^{(b)}\kappa_{4}^{2} +24\xi^{2}\phi^{2}\kappa_{4}^{4}\dot{ \rho}^{(b)}\kappa_{5}^{2}+24\xi^{2} \phi^{2} \dot{p}\kappa_{5}^{4}\kappa_{4}^{2} \\&{} +36\xi^{2}\phi^{3}\kappa_{4}^{2} \dot{g} \kappa_{5}^{4} +104\xi^{2} \phi^{2}\dot{ \rho}^{(b)}\kappa_{5}^{4} \kappa_{4}^{2}+45 \xi^{2}\phi^{4} \kappa_{5}^{4} \dot{ \rho}^{(b)}\kappa_{4}^{2}-6 \xi\phi^{2}{ \kappa_{4}}^{2}\dot{g} \kappa_{5}^{4} +144 \xi^{2}\phi^{2} \kappa_{4}^{2} \kappa_{5}^{2}\dot{{ \mathcal{F}}}+576\kappa_{5}^{4}{ \mathcal{F}} \xi^{2}\phi\dot{\phi} \\&{} +540\kappa_{5}^{4}{\mathcal{F}}\xi^{2} \phi^{3}\dot{\phi}-108\kappa_{5}^{4}{\mathcal{F}} \xi\phi\dot{\phi}-36\xi\phi\kappa_{5}^{4}\rho^{(b)} \kappa_{4}^{2}\dot{\phi}+144\xi^{2}\phi \kappa_{4}^{2}m \kappa_{5}^{2}\dot{\phi} -48\xi^{2}\phi^{2}\kappa_{5}^{4}\dot{m} +1008\kappa_{4}^{4}\xi^{2}\phi^{3}\dot{ \phi} \\&{} -144\kappa_{4}^{4}\xi\phi\dot{\phi}+288 \kappa_{5}^{4} \dot{{\mathcal{F}}}\xi^{2} \phi^{2}+135\kappa_{5}^{4} \dot{{\mathcal{F}}} \xi^{2} \phi^{4}-54\kappa_{5}^{4} \dot{{ \mathcal{F}}}\xi\phi^{2} \bigr) \bigl(24\xi^{2} \phi^{2} \kappa_{4}^{4}\rho^{(b)} \kappa_{5}^{2}-72 \kappa_{4}^{4}\xi \phi^{2}+9\kappa_{4}^{4} \\&{} +252\kappa_{4}^{4}\xi^{2} \phi^{4}+72 \xi^{2}\phi^{2}\kappa_{4}^{2}m \kappa_{5}^{2}-18\xi\phi^{2}\kappa_{5}^{4} \rho^{(b)} \kappa_{4}^{2}-6\xi\phi^{2} \kappa_{4}^{2}g \kappa_{5}^{4}+24 \xi^{2}\phi^{2}p\kappa_{5}^{4} \kappa_{4}^{2}+36\xi^{2}\phi^{3} \kappa_{4}^{2}g \kappa_{5}^{4} \\&{} +104\xi^{2}\phi^{2}\rho^{(b)} \kappa_{5}^{4} \kappa_{4}^{2}+45 \xi^{2} \phi^{4} \kappa_{5}^{4} \rho^{(b)}\kappa_{4}^{2}+144 \xi^{2} \phi^{2}\kappa_{4}^{2} \kappa_{5}^{2}{ \mathcal{F}} +3\rho^{(b)} \kappa_{5}^{4} \kappa_{4}^{2} +9 \kappa_{5}^{4}{ \mathcal{F}} -48\xi^{2} \phi^{2}\kappa_{5}^{4}m +288\kappa_{5}^{4}{ \mathcal{F}}\xi^{2} \phi^{2} \\&{} +135\kappa_{5}^{4}{\mathcal{F}} \xi^{2} \phi^{4}-54\kappa_{5}^{4}{\mathcal{F}} \xi \phi^{2} \bigr)^{-1/2} \bigr] \bigl(3\kappa_{5}^{2}+16 \xi^{2}\phi^{2}\kappa_{5}^{2}-3\xi\phi ^{2}\kappa_{5}^{2}+24\xi^{2} \phi^{2}\kappa_{4}^{2} \bigr)^{-4}H^{-3} \\&{} -2H^{-1} \bigl(3\kappa_{5}^{2}+16 \xi^{2} \phi^{2}\kappa_{5}^{2}-3\xi \phi^{2} \kappa_{5}^{2} +24\xi^{2} \phi^{2} \kappa_{4}^{2} \bigr)^{-1} \bigl(32\xi^{2} \phi\kappa_{5}^{2}\dot{\phi}-6\xi \phi\kappa_{5}^{2} \dot{\phi} +48\xi^{2}\phi \kappa_{4}^{2}\dot{\phi} \bigr) \bigr\}_{0}-1. \end{aligned}$$

Appendix C

$$\begin{aligned} q =&-1-\frac{1}{2H} \bigl\{ \bigl[768\kappa_{4}^{4} \xi^{2}\phi\dot{\phi}-144\kappa_{4}^{4}\xi\phi\dot{ \phi} +432\kappa_{4}^{4}\xi^{2}\phi^{3} \dot{\phi}-96\xi^{2}\phi\kappa_{5}^{4}m\dot{\phi} -48\xi^{2}\phi^{2}\kappa_{5}^{4}\dot{m}-6 \xi\phi\kappa_{4}^{2}\dot{g}\kappa_{5}^{4} \\&{} -12\xi\phi\kappa_{5}^{4}\rho\kappa_{4}^{2} \dot{\phi}-6\xi{\phi}^{2}\kappa_{5}^{4}\dot{ \rho}^{(b)}\kappa_{4}^{2} -6\xi\dot{\phi} \kappa_{4}^{2}g\kappa_{5}^{4}+80 \xi^{2}\phi\rho\kappa_{5}^{4}\kappa_{4}^{2} \dot{\phi} +40\xi^{2}\phi^{2}\dot{\rho}^{(b)} \kappa_{5}^{4}\kappa_{4}^{2}+48\kappa _{4}^{4}\xi^{2}\phi\rho\kappa_{5}^{2} \dot{\phi} \\&{} +24\kappa_{4}^{4}\xi^{2}\phi^{2} \dot{ \rho}^{(b)}\kappa_{5}^{2}+144\xi^{2} \phi^{2}\kappa_{4}^{2}m\kappa_{5}^{2} \dot{\phi} +72{\xi}^{2}\phi^{2}\kappa_{4}^{2} \dot{m}\kappa_{5}^{2}+48\xi^{2}\phi\, p \kappa_{5}^{4}\kappa_{4}^{2}\dot{\phi}+24 \xi^{2}\phi^{2}\dot{p}^{(b)}\kappa_{5}^{4} \kappa_{4}^{2} \\&{} +12\xi^{2}\phi^{3}\kappa_{5}^{4} \rho\kappa_{4}^{2}\dot{\phi}+3\xi^{2} \phi^{4} \kappa_{5}^{4}\dot{\rho}^{(b)} \kappa_{4}^{2} +3\dot{\rho}^{(b)}\kappa_{5}^{4} \kappa_{4}^{2}-3 \bigl\{9\kappa_{5}^{4} \dot{{\mathcal{F}}}\kappa_{4}^{4} +3\kappa_{4}^{6} \dot{\rho}^{(b)} \kappa_{5}^{4}+104 \kappa_{4}^{6}\xi^{2}\phi^{2}\dot{ \rho}^{(b)}\kappa_{5}^{4} \\&{} +45\kappa_{4}^{6}\xi^{2}\phi^{4} \kappa_{5}^{4}\dot{\rho}^{(b)}+36\kappa_{4}^{6} \xi^{2}\phi^{3}\dot{g}\kappa_{5}^{4}-48 \kappa_{4}^{4}\xi^{2}\phi^{2} \kappa_{5}^{4}\dot{m} -6\kappa_{4}^{6}\xi \dot{\phi}g\kappa_{5}^{4}-6 \kappa_{4}^{6} \xi\phi\dot{g}\kappa_{5}^{4}+24\kappa_{4}^{8} \xi^{2}\phi^{2}\dot{\rho}^{(b)}\kappa_{5}^{2} \\&{} +72\kappa_{4}^{6}\xi^{2}\phi^{2} \dot{m} \kappa_{5}^{2}+24\kappa_{4}^{6} \xi^{2}\phi^{2}\dot{p}^{(b)}\kappa_{5}^{4}+ 144\xi^{2}\phi^{2}\kappa_{5}^{2} \kappa_{4}^{6}\dot{{\mathcal{F}}}+288\kappa_{5}^{4} \dot{{\mathcal{F}}}\kappa_{4}^{4}\xi^{4} \phi^{6} -54\kappa_{5}^{4}\dot{{\mathcal{F}}} \kappa_{4}^{4}\xi\phi^{2} \\&{} +135\kappa_{5}^{4}\dot{{\mathcal{F}}} \kappa_{4}^{4} \xi^{2} \phi^{4}+208 \kappa_{4}^{6}{ \xi}^{2}\phi\rho \kappa_{5}^{4}\dot{\phi}+ 180 \kappa_{4}^{6} \xi^{2}\phi^{3} \kappa_{5}^{4}\rho\dot{ \phi}+108\kappa_{4}^{6} \xi^{2}\phi^{2}g \kappa_{5}^{4}\dot{\phi} -96\kappa_{4}^{4} \xi^{2}\phi\kappa_{5}^{4}m \dot{\phi} \\&{} +48\kappa_{4}^{8}\xi^{2}\phi\rho \kappa_{5}^{2}\dot{\phi}+144\kappa_{4}^{6} \xi^{2}\phi m\kappa_{5}^{2}\dot{\phi}+48 \kappa_{4}^{6}\xi^{2}\phi p\kappa_{5}^{4} \dot{\phi}+288\xi^{2}\phi\kappa_{5}^{2}\kappa _{4}^{6}{\mathcal{F}}\dot{\phi} +1728\kappa_{5}^{4}{ \mathcal{F}}\kappa_{4}^{4}\xi^{4}\phi^{5} \dot{\phi} \\&{} -108\kappa_{5}^{4}{\mathcal{F}}\kappa_{4}^{4} \xi\phi\dot{\phi} +540\kappa_{5}^{4}{\mathcal{F}} \kappa_{4}^{4}\xi^{2}\phi^{3}\dot{\phi}+ 1008\kappa_{4}^{8}\xi^{2}\phi^{3}\dot{ \phi}-144\kappa_{4}^{8}\xi\phi\dot{\phi}+768 \kappa_{4}^{8}\xi^{2}\phi\dot{\phi} \bigr\} \bigl\{ \bigl(9 \kappa_{5}^{4}{\mathcal{F}}\kappa_{4}^{4} \\&{} +384\kappa_{4}^{8}\xi^{2} \phi^{2}+252 \kappa_{4}^{8}\xi^{2} \phi^{4}-72 \kappa_{4}^{8}\xi\phi^{2} +3 \kappa_{4}^{6} \rho\kappa_{5}^{4}+104 \kappa_{4}^{6} \xi^{2}\phi^{2}\rho \kappa_{5}^{4}+45 \kappa_{4}^{6} \xi^{2}\phi^{4} \kappa_{5}^{4}\rho-6 \kappa_{4}^{6}\xi\phi g\kappa_{5}^{4} \\&{} +36\kappa_{4}^{6}\xi^{2}\phi^{3}g \kappa_{5}^{4}+9\kappa_{4}^{8}-48 \kappa_{4}^{4}\xi^{2}\phi^{2}\kappa _{5}^{4}m+24\kappa_{4}^{8} \xi^{2}\phi^{2} \rho\kappa_{5}^{2}+72 \kappa_{4}^{6}\xi^{2}\phi^{2}m \kappa_{5}^{2}+24 \kappa_{4}^{6} \xi^{2}\phi^{2}p\kappa_{5}^{4} \\&{} +144\xi^{2}\phi^{2}\kappa_{5}^{2} \kappa_{4}^{6}{\mathcal{F}}+288\kappa_{5}^{4}{ \mathcal{F}}\kappa_{4}^{4}\xi^{4} \phi^{6}- 54\kappa_{5}^{4}{\mathcal{F}} \kappa_{4}^{4}\xi\phi^{2}+135\kappa _{5}^{4}{\mathcal{F}}\kappa_{4}^{4} \xi^{2}\phi^{4} \bigr)^{\frac{1}{2}} \bigr\}^{-1} \bigr] \bigl(144 \xi^{2}\phi^{2}\kappa_{5}^{2} \kappa_{4}^{2}+9\kappa_{5}^{4} \\&{} +96\xi^{2}\phi^{2}\kappa_{5}^{4}+9 \xi^{2}\phi^{4}\kappa_{5}^{4}-18\xi \phi^{2}\kappa_{5}^{4} \bigr)^{-1} -{H}^{2} \bigl(288\xi^{2}\phi\kappa_{5}^{2} \kappa_{4}^{2}\dot{\phi}+192\xi^{2}\phi \kappa_{5}^{4}\dot{\phi}+36\xi^{2} \phi^{3}\kappa_{5}^{4}\dot{\phi}-36\xi\phi \kappa_{5}^{4}\dot{\phi} \bigr) \bigr\}_{0}. \end{aligned}$$