Astronomy - Distance with different scale factors

[tosv]

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?

 

[Redbelly98]

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.

 

[tosv]

Thanks for your reply, now I was able to solve the problem.

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

 

[Redbelly98]

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.

 

[rwisz]

I would first check the units of the two constants to make sure they are consistent. C1 has units of meters per second, while C2 has units of meters per second squared. This means that they cannot be equal to each other, as they represent different physical quantities.

To find C2, we can use the given information that the scale factor for a radiation-dominated universe (t < 200,000 years) is R(t)=C2√t. We know that the scale factor at t=3.787x10^17 seconds is 9.461x10^24 meters. Plugging this into the equation, we get:

9.461x10^24 = C2√(3.787x10^17)

Solving for C2, we get C2=6.21x10^6 meters per second. This means that the distance between two sites one year after the Big Bang would be:

R(1 year)=6.21x10^6√(3.1536x10^16)

=1.425x10^20 meters

This is a very large distance, which is expected given the rapid expansion of the universe during its early stages. However, it is important to note that this calculation assumes a flat universe, which is still a topic of ongoing research and debate in the field of astronomy. Further studies and observations may provide more accurate values for the scale factors and distances in the early universe.