Dodelson Cosmology 6.8 Inflation Klein Gordon Equation

In summary, the conversation discusses the process of proving that Eq. (6.33) can be obtained from Eq. (6.32) by changing variables from t to ##\eta##. The mistake in the attempted solution is identified, where the term ##\frac{d^2\phi^{(0)}}{dt^2}## should be taken into account as a function of ##\eta##, leading to the correct substitution of ##\frac{1}{a^2}\ddot{\phi^{(0)}}##.
  • #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.
 
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  • #2
Hello. Welcome to PF!
StuckPhysicsStudent said:

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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Likes StuckPhysicsStudent
  • #3
TSny said:
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!
 

1. What is Dodelson Cosmology 6.8?

Dodelson Cosmology 6.8 is a cosmological model proposed by physicist Scott Dodelson in 2003. It is based on the inflationary theory of the universe, which suggests that the universe underwent a rapid period of expansion in its early stages.

2. What is the Inflationary Theory of the Universe?

The inflationary theory proposes that the universe experienced a brief period of exponential expansion in its early stages. This theory helps to explain why the universe appears to be flat and uniform on a large scale, among other observed phenomena.

3. What is the Klein Gordon Equation?

The Klein Gordon Equation is a relativistic wave equation that describes the behavior of particles with zero spin, such as scalar particles. It is used in cosmology to describe the behavior of the inflaton field during inflation.

4. How does the Klein Gordon Equation relate to Dodelson Cosmology 6.8?

In Dodelson Cosmology 6.8, the Klein Gordon Equation is used to model the behavior of the inflaton field during the inflationary period. This helps to explain how the universe underwent rapid expansion and how it reached its current state.

5. What are the implications of Dodelson Cosmology 6.8 for our understanding of the universe?

Dodelson Cosmology 6.8 provides a framework for understanding the early stages of the universe and how it evolved into its current state. It also helps to explain various observed phenomena, such as the uniformity and flatness of the universe, and provides insights into the nature of dark matter and dark energy.

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