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..

 

[sardar]

As a fellow scientist, I understand your interest in solving the inflation problem with an exponential potential. This is a challenging problem, but with determination and the right approach, it is certainly achievable.

Firstly, it's important to note that the equations you have provided are the Friedmann equations for a single scalar field model of inflation. These equations describe the evolution of the universe during inflation and can be solved numerically using computer simulations.

To solve these equations, you will need to choose values for the parameters v_0 and \alpha and then use numerical methods to solve for \phi(t) and H. This can be done using software such as Mathematica or Python, which have built-in functions for solving differential equations.

Another approach is to use analytical methods, which may be what the book by Weinberg is referring to. This involves using mathematical techniques to find an exact solution to the equations. However, this can be quite challenging and may require a strong background in mathematical physics.

In either case, it's important to carefully check your calculations and make sure they are consistent with known physical laws and observations. Additionally, it may be helpful to consult with other experts in the field or seek guidance from a mentor or supervisor.

Overall, solving the inflation problem with an exponential potential is a complex and ongoing research topic, but with dedication and hard work, I am confident you can make progress in this area. Best of luck in your studies.