Exponential potential for inflation

[shooride]

hi
I want to solve inflation problem for exponential potential.
$$v(\phi) = v_0 exp(-\alpha \phi)$$
(it's known as barrow or pawer law inflation )
we have 2 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 this 2 equ and find $\phi(t)$ and H(Hubble).
in the book of cosmology by weinberg has written,it is easy but i can't do it.can anyone help me?
best

 

[Chalnoth]

hi
I want to solve inflation problem for exponential potential.
$$v(\phi) = v_0 exp(-\alpha \phi)$$
(it's known as barrow or pawer law inflation )
we have 2 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 this 2 equ and find $\phi(t)$ and H(Hubble).
in the book of cosmology by weinberg has written,it is easy but i can't do it.can anyone help me?
best

The assumption of slow-roll inflation is that $\ddot{\phi}$ is small compared to the "friction" term $3H\dot{\phi}$, and thus can be neglected.

So your job is basically two-fold:
1. Solve the equations in the slow-roll regime.
2. Show the parameter values for which the slow-roll regime is valid.

 

[Chronos]

You made this problem way too easy to solve, Chalnoth.

 

[shooride]

The assumption of slow-roll inflation is that $\ddot{\phi}$ is small compared to the "friction" term $3H\dot{\phi}$, and thus can be neglected.

So your job is basically two-fold:
1. Solve the equations in the slow-roll regime.
2. Show the parameter values for which the slow-roll regime is valid.

thanks,but I think that it has exact solution.without slow-roll condition..