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--> --> --> $ \widetilde{R}_{\ c}^{a}-\frac{1}{2}\delta_{\ c}^{a}\widetilde{R} = 8\pi G\left( T+T_{\Lambda}\right)_{\ c}^{a} \;, $ | (1) |
$ \rho_{\Lambda} = -\dfrac{c^4}{8\pi G}(3{\cal{K}}^2+6{\cal{K}}\dfrac{\dot{a}}{a}-\Lambda_0), $ | (2) |
$ p_{\Lambda} = -\dfrac{c^4}{8\pi G}({\cal{K}}^2+4{\cal{K}}\dfrac{\dot{a}}{a}+2\dot{{\cal{K}}}-\Lambda_0), $ | (3) |
$ {\cal{K}}^2 + 2{\cal{K}}\dfrac{\dot{a}}{a} +\left( \dfrac{\dot{a}}{a}\right) ^2 = \dfrac{1}{3}\left( \rho+\Lambda_{0}\right), $ | (4) |
$ \ddot{a} = -\frac{a}{2}\left( p+\dfrac{\rho}{3}\right) +\dfrac{1}{3}a\Lambda_{0}-\dfrac{\rm d}{{\rm d}t}\left( a{\cal{K}}\right), $ | (5) |
$\begin{split} \dot{H}(t)&+\dot{{\cal{K}}}(t)+H(t)\left[ H(t)+{\cal{K}}(t)\right]\\&+\dfrac{3w+1}{2}\left[ H(t)+{\cal{K}}(t)\right] ^2-\dfrac{w+1}{2}\Lambda_{0} = 0, \end{split} $ | (6) |
As discussed in reference [7], the equations of motion for
The evolution of
$ \dot{{\cal{K}}} = \dfrac 13\Lambda_0-\dfrac 13\Lambda-H{\cal{K}}\;\;, $ | (7) |
$ \dot{{\cal{K}}}+(3w+2)H{\cal{K}}+\dfrac {3w+1}2{\cal{K}}^2 = \frac {w+1}2(\Lambda_0-\Lambda), $ | (8) |
$ (3w_0+1){\cal{K}}^2+(6w_0+4)H{\cal{K}}+2\dot{{\cal{K}}} = (w_0+1)\Lambda_0, $ | (9) |
The initial value of
$ {\cal{K}}(t_0) = H_0\left( \pm\sqrt{1-\dfrac{\Lambda-\Lambda_{0}}{3H_0}}-1\right), $ | (10) |
$ \Lambda_{0}\geqslant-(3H_0-\Lambda)\approx-\dfrac{2}{5}\Lambda \;. $ | (11) |
$ \Lambda_{\rm eff} = \Lambda_0-3\left({\cal{K}}^2+2{\cal{K}}\dfrac{\dot{a}}{a}\right)\;\;, $ | (12) |
Figure3. (color online) The transition of
Figure4. (color online) The transition of
$\begin{split} S =& \int {\rm d}^4x\sqrt{-g}\left[\frac 12M^2_{pl}R\left( 1+\xi\phi^2\right)\right. \\&\left.-\frac 12g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right]+S_m \;, \end{split}$ | (13) |
$ \left( 1+\xi\phi^2\right) \left( R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R\right) = 8\pi GT_{\mu\nu}, $ | (14) |
$ V_{,\phi}+3H\dot{\phi}+\ddot{\phi}-M_{pl}^2R\xi\phi = 0, $ | (15) |
$ \Lambda_{\rm eff} = \rho_{\phi} = \frac {\dot{\phi}^2+2V(\phi)}{2+2\xi\phi^2}, $ | (16) |
$ \dot{\Lambda}_{\rm eff}\left( 1+\xi\phi^2\right)+2\Lambda_{\rm eff}\xi\phi\dot{\phi}+3H\dot{\phi}^2-M_{pl}^2R\xi\phi\dot{\phi} = 0, $ | (17) |
$\begin{split} \dot{\phi} =& M_{pl}^2\xi\phi\left( \dfrac{\dot{H}}{H}+2H\right)-\dfrac{\Lambda_{\rm eff}\xi\phi}{3H}\\&\pm\sqrt{\left[ M_{pl}^2\xi\phi\left( \dfrac{\dot{H}}{H}+2H\right)-\dfrac{\Lambda_{\rm eff}\xi\phi}{3H}\right]^2-\dfrac{\dot{\Lambda}_{\rm eff}}{3H}\left( 1+\xi\phi^2\right) } \;, \end{split}$ | (18) |
$ \left[3M_{pl}^2\left(\dot{H}+2H^2\right)-\Lambda_{\rm eff}\right]^2\phi^2\xi^2-3H\dot{\Lambda}_{\rm eff}\phi^2\xi-3H\dot{\Lambda}_{\rm eff}\geqslant 0 \;. $ | (19) |
Figure5. (color online) The cases of initial value
Figure6. (color online) The cases of initial value
The numerical calculation of Eq. (18) with plus sign gives the evolution of
Figure7. (color online) The monotonic evolution of
Figure8. (color online) The monotonic evolution of
Figure9. (color online)
Figure10. (color online)
It is a universal conclusion that the minimal Brans-Dicke coupling is non-trivial, i.e.,
$ \dot{\Lambda}_{\rm eff} = \dfrac{\dot{\phi}\left( \ddot{\phi}+V_{,\phi}-2\xi\phi\Lambda_{\rm eff}\right) }{1+\xi\phi^2} = \dfrac{\dot{\phi}\left[ \left( M^2_{pl}R-2\Lambda_{\rm eff}\right)\xi\phi-3H\dot{\phi} \right] }{1+\xi\phi^2}, $ | (20) |
$ \left( M^2_{pl}R-2\Lambda_{\rm eff}\right)\xi\phi-3H\dot{\phi} \geqslant 0 \;. $ | (21) |
$ M^2_{pl}R-2\Lambda_{\rm eff} > 0, $ | (22) |
$ \xi\geqslant \dfrac{3H\dot{\phi}}{\left(M^2_{pl}R-2\Lambda_{\rm eff}\right)\phi} \;. $ | (23) |
The discussion on the non-triviality of
x=?0.1 | x=?0.05 | x=0 | x=0.1 | x=?0.2 | x=?0.186 | x=?0.1 | x=0.1 | ||
0 | 0 | 0.135 | 0.249 | 0 | 0.003 | 0.111 | 0.391 | ||
0 | 0.029 | 0.142 | 0.255 | 0 | 0 | 0.114 | 0.401 | ||
0 | 0.041 | 0.152 | 0.261 | 0 | 0 | 0.115 | 0.412 | ||
0 | 0.05 | 0.155 | 0.268 | 0 | 0 | 0.114 | 0.422 | ||
0 | 0.057 | 0.161 | 0.274 | 0 | 0 | 0.109 | 0.432 | ||
0 | 0.063 | 0.167 | 0.28 | 0 | 0 | 0.099 | 0.442 | ||
0 | 0.069 | 0.173 | 0.284 | 0 | 0 | 0.076 | 0.452 | ||
0 | 0.074 | 0.178 | 0.291 | 0 | 0 | 0 | 0.461 | ||
0 | 0.078 | 0.183 | 0.297 | 0 | 0 | 0 | 0.471 |
Table1.
x=?0.1 | x=?0.066 | x=0 | x=0.1 | x=?0.25 | x=?0.2 | x=?0.1 | x=0.1 | ||
0 | 0 | 0.136 | 0.229 | 0 | 0.032 | 0.12 | 0.371 | ||
0 | 0.034 | 0.144 | 0.234 | 0 | 0.017 | 0.127 | 0.379 | ||
0 | 0.05 | 0.151 | 0.24 | 0 | 0 | 0.13 | 0.388 | ||
0 | 0.063 | 0.158 | 0.245 | 0 | 0 | 0.13 | 0.396 | ||
0 | 0.073 | 0.165 | 0.25 | 0 | 0 | 0.129 | 0.404 | ||
0 | 0.082 | 0.171 | 0.255 | 0 | 0 | 0.126 | 0.411 | ||
0 | 0.09 | 0.178 | 0.261 | 0 | 0 | 0.12 | 0.418 | ||
0 | 0.098 | 0.184 | 0.266 | 0 | 0 | 0.111 | 0.423 | ||
0 | 0.105 | 0.19 | 0.271 | 0 | 0 | 0.097 | 0.428 |
Table2.
x=?0.1 | x=?0.001 | x=0.1 | x=0.2 | x=?0.1 | x=0.001 | x=0.1 | x=0.2 | ||
0 | 0 | 0.318 | 0.428 | 0 | 0.034 | 0.445 | 0.75 | ||
0 | 0 | 0.32 | 0.431 | 0 | 0.031 | 0.448 | 0.75 | ||
0 | 0 | 0.322 | 0.433 | 0 | 0.028 | 0.452 | 0.751 | ||
0 | 0 | 0.325 | 0.435 | 0 | 0.015 | 0.458 | 0.755 | ||
0 | 0 | 0.327 | 0.438 | 0 | 0 | 0.465 | 0.759 | ||
0 | 0 | 0.33 | 0.44 | 0 | 0 | 0.473 | 0.765 | ||
0 | 0 | 0.332 | 0.442 | 0 | 0 | 0.483 | 0.772 | ||
0 | 0 | 0.335 | 0.444 | 0 | 0 | 0.495 | 0.78 | ||
0 | 0 | 0.337 | 0.447 | 0 | 0 | 0.51 | 0.789 |
Table3.
x=?0.1 | x=0.119 | x=0.15 | x=0.2 | x=?0.1 | x=0.075 | x=0.1 | x=0.2 | ||
0 | 0 | 0.088 | 0.151 | 0 | 0 | 0.212 | 0.635 | ||
0 | 0.03 | 0.096 | 0.157 | 0 | 0.108 | 0.248 | 0.655 | ||
0 | 0.044 | 0.104 | 0.163 | 0 | 0.161 | 0.282 | 0.675 | ||
0 | 0.055 | 0.112 | 0.168 | 0 | 0.205 | 0.314 | 0.696 | ||
0 | 0.064 | 0.118 | 0.173 | 0 | 0.246 | 0.347 | 0.718 | ||
0 | 0.073 | 0.124 | 0.178 | 0 | 0.287 | 0.381 | 0.741 | ||
0 | 0.08 | 0.13 | 0.183 | 0 | 0.332 | 0.417 | 0.765 | ||
0 | 0.087 | 0.135 | 0.187 | 0 | 0.384 | 0.458 | 0.789 | ||
0 | 0.093 | 0.14 | 0.191 | 0 | 0.447 | 0.504 | 0.815 |
Table4.