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I'm trying to solve the inflation problem for exponential potential. $$v(\phi) = v_0 \exp(-\alpha \phi)$$ (it's known as barrow or power law inflation) we have two main equations: $$H^2 = 8π G / 3 (1/2 (\dot{\phi})^2 + v(\phi))$$ $$\ddot{\phi} + 3H \dot{\phi} + v(\phi)'=0$$ I must solve these two equations and find $\phi(t)$ & H(Hubble). in the book of cosmology by Weinberg has written,it is easy.many articles have mentioned it;but I can't do it.it has exact solution.answer is: $$\phi(t)=\phi_0 \ln t/ \ln t_0$$ and $$R(t)=R_0 (t/t_0)^b$$ with suitable constant b,$\phi_0$,$R_0$ .

Should I use slow-roll condition for solve that? ($\dot{\phi}^2 \ll v(\phi)$ and $\ddot{\phi} \ll v(\phi)'$) or without that I can?

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This is probably too late to be useful, however yes you should assume slow roll and that the inflaton's potential energy is dominating the energy density of the Universe.

So solve:

$3H\dot{\phi} + V'(\phi)=0$

with:

$H^2 = 8\pi G V(\phi) /3$

This slow roll approximation is valid if the slow roll parameters are $\ll 1$ which requires $\alpha \ll 1/m_p$.

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