Dodelson Cosmology 6.8 Inflation Klein Gordon Equation

[StuckPhysicsStudent]

Homework Statement

Show that Eq. (6.33) follows from Eq. (6.32) by changing variables from t to $\eta$.

Homework Equations

(6.32) $$\frac{d^2\phi^{(0)}}{dt^2}+3H\frac{d\phi^{(0)}}{dt}+V'=0$$
(6.33) $$\ddot{\phi^{(0)}}+2aH\dot{\phi}^{(0)}+a^2V'=0$$

The Attempt at a Solution

So $\frac{1}{a}\frac{d}{d\eta}=\frac{d}{dt}$ and so just doing a simple substitution I got

$$\frac{1}{a^2}\ddot{\phi^{(0)}}+\frac{3}{a}H\dot{\phi}^{(0)}+V'=0$$

Where $H=\frac{\dot{a}}{a^2}$ where the over dots are the derivative with respect to conformal time (but he doesn't do anything with H here so I just leave it as is).

Multiplying through by $a^2$ I get

$$\ddot{\phi^{(0)}}+3aH\dot{\phi}^{(0)}+a^2V'=0$$

And I get a 3 instead of a 2.

I am sure there are other ways to do this but I am trying to do it the way I think the problem is asking. The problem is copied verbatim from the book.

 

[TSny]

Hello. Welcome to PF!

Homework Equations

(6.32) $$\frac{d^2\phi^{(0)}}{dt^2}+3H\frac{d\phi^{(0)}}{dt}+V'=0$$
(6.33) $$\ddot{\phi^{(0)}}+2aH\dot{\phi}^{(0)}+a^2V'=0$$

The Attempt at a Solution

So $\frac{1}{a}\frac{d}{d\eta}=\frac{d}{dt}$ and so just doing a simple substitution I got

$$\frac{1}{a^2}\ddot{\phi^{(0)}}+\frac{3}{a}H\dot{\phi}^{(0)}+V'=0$$

The mistake occurs in how you handled the term $\frac{d^2\phi^{(0)}}{dt^2}$. This does not go over to simply $\frac{1}{a^2}\ddot {\phi^{(0)}}$. You need to take into account that $a$ is a function of $\eta$.

 

[StuckPhysicsStudent]

The mistake occurs in how you handled the term d2ϕ(0)dt2d2ϕ(0)dt2\frac{d^2\phi^{(0)}}{dt^2}. This does not go over to simply 1a2¨ϕ(0)1a2ϕ(0)¨\frac{1}{a^2}\ddot {\phi^{(0)}}. You need to take into account that aaa is a function of ηη\eta.

I got it now, thanks!