Dodelson Cosmology 6.8 Inflation Klein Gordon Equation

  • #1
StuckPhysicsStudent
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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.
 

Answers and Replies

  • #2
TSny
Homework Helper
Gold Member
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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##.
 
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  • #3
StuckPhysicsStudent
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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!
 

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