- #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.