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For phantom energy model we have $$H^2 = H_0^2\Omega_pa^{-3-3w_p}$$ such that $w_p < -1$ $$\dot{a}^2 = H_0^2\Omega_pa^{-1-3w_p}$$ $$\dot{a} = H_0 \sqrt{\Omega_p} a^{-1/2(1+3w_p)}$$ $$\int a^{1/2(1+3w_p)}da = \int H_0 \sqrt{\Omega_p}dt$$ Now at this part we should define the integrals upper and lower bounds. When we set time from $t → t_0$ \int_a^{...