Light-like Geodesic - What are the limits of integration?

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Homework Statement

Consider the following geodesic of a massless particle where $\alpha$ is a constant:

$$\dot r = \frac{\alpha}{a(t)^2}$$
$$c^2 \dot t^2 = \frac{\alpha^2}{a^2(t)}$$

Homework Equations

The Attempt at a Solution

Part (a)
$$c \frac{dt}{d\lambda} = \frac{\alpha}{a}$$
$$a dt = \frac{\alpha}{c} d\lambda$$
$$\frac{1}{H} a = \frac{\alpha}{c} \lambda + \epsilon^{'}$$
$$a = \frac{H}{c} \left( \alpha \lambda + \epsilon \right)$$

Similarly,
$$r = \frac{c^2}{H^2} \left[ -\frac{1}{\alpha \lambda + \epsilon} + \delta \right]$$

Part(b)
I'm confused as to what the limits of integration are. I'm not sure if this is right:

At $t = t_0$, $a(t_0) = 1 = \frac{H}{c} \left(\alpha \lambda + \epsilon \right)$.

At $r = r_e$ what happens to $\delta$?

 

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How do I get the final expression?

 

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bumpp

 

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bumpp anyone?

 

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bumping on light-like, limits.

 

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bumpp

 

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bumpp

 

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bumpp

 

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bumpp on last part

 

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bumpp

 

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bumpp

 

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bump