Astronomy - Distance with different scale factors

AI Thread Summary
The discussion focuses on calculating the scale factor R(t) for a flat universe and determining the distance between two sites one year after the Big Bang. The user initially expressed the distance in meters and time in seconds, leading to confusion over the constants C1 and C2 due to differing units. They realized that using years and light-years would simplify the calculations. After receiving advice, they successfully solved the problem, highlighting the importance of unit consistency in cosmological calculations. The conversation emphasizes the utility of adapting units to fit the problem context for easier problem-solving.
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Homework Statement
Expansion of the universe is described by the scale factor R(t), where t is the time after the Big Bang. For a flat universe the scale factor is today
R(t)=C_{1}\cdot t^{\frac{2}{3}}

When the Universe was radiation dominated, for t <200,000 years, the scale factor was
R(t)=C_{2}\cdot \sqrt{t}

Today, about 12 billion years after the Big Bang, we measure the distance to another superhop to 1 billion light years. How big would the distance between these sites one year after the Big Bang, if we assume that we live in a flat universe.

The attempt at a solution
I express the distance in meters
R(t)=9.461\cdot 10^{24}\, m

Then I express the time in seconds
t=3.787\cdot 10^{17}\, s

I found that the constant is
C_{1}=1.8075\cdot 10^{13}

I need to know the constant C_{2}, but I'm not sure how I'll be able to calculate it.

I have tested just to put C_{1}=C_{2}, but the result for the distance was wrong. I then realized that the two constants have different units.

Does anyone have a suggestion how I may proceed?
 
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For t = 200,000 years, both expressions for R(t) should give the same result.

To make your life easier, consider using units of years and light-years rather than seconds and meters.
 
Thanks for your reply, now I was able to solve the problem.

Thanks for your advice, I'll keep it in mind.
 
Glad it worked out.

I know we get SI units drilled into our heads in intro physics classes, but they are not always the most convenient. If the problem statement uses alternate units, that's a hint for you to consider it too. :smile:
 
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