Question on the Derivation of Temperature/Scale Factor Relation

In summary, Barbara Ryden argues that the first law of thermodynamics can be used to derive the relation T(t) ∝ a(t)-1. This relation is used to calculate the rate of expansion of the universe.
  • #1
TRB8985
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In 'Introduction to Cosmology' by Barbara Ryden, there is an argument made using the first law of thermodynamics to derive the relation T(t) ∝ a(t)-1 on pages 29 and 30.
MENTOR NOTE: removed copyrighted material.
I've been able to work out all the omitted details up to 2.37, which gives the following relation:

$$ \frac{1}{T} \frac{dT}{dt} = - \frac{1}{3V} \frac{dV}{dt}$$

She informs the reader that since V ∝ a(t)3, the rate of change of the photons' temperature is related to the rate of expansion of the universe by the following relation (2.38):

$$ \frac{d}{dt} (ln\,T) = - \frac{d}{dt} (ln\,a) $$

I'm running into a little difficulty seeing how she arrives at 2.38. Here's my attempt getting there from 2.37:

$$ \frac{d}{dt} (\frac{dT}{T}) = -\frac{1}{3V} (\frac{d}{dt}\, a(t)^3)$$

$$ \frac{d}{dt} (\frac{dT}{T}) = -\frac{1}{3a(t)^3} ( 3a(t)^2)$$

$$ \frac{d}{dt} (\frac{dT}{T}) = -\frac{1}{a(t)}$$

$$ \frac{d}{dt} (\int \frac{dT}{T}) = -\frac{1}{a(t)}$$

$$ \frac{d}{dt} (ln\,T) = -\frac{1}{a(t)}$$

I can't seem to get the d/dt piece on the right-hand side like at the top of my post. I thought maybe I could pull out the d/dt from the expression, like the following:

$$- \frac{1}{3V} \frac{dV}{dt} = -\frac{1}{3} \frac{d}{dt}(\frac{dV}{V})$$

But that doesn't seem to net me the a(t)-1 result like my original path did. Is there some simple mistake I'm making in my calculations here? Additionally, is the negative sign "absorbed" into the proportionality expression? It doesn't seem to make much sense that I'm left with a negative sign in what I'm doing, since the scale factor never has a negative value.

Thank you very much for your time and any assistance you may be able to provide.
 
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  • #2
You left out the chain rule on the second step.
TRB8985 said:
$$ \frac{d}{dt} (\frac{dT}{T}) = -\frac{1}{3V} (\frac{d}{dt}\, a(t)^3)$$

The far right term simplifies to:

$${d \over dt} a(t)^3 = 3 a(t)^2 {da \over dt}$$

Leaving the chain rule expansion in place should make the rest of the derivation fall into place nicely.
 
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  • #3
##V \propto a(t)^3 \iff V = k a(t)^3## for some constant ##k \iff \ln V = \ln k + 3 \ln a##
So $$\frac {d}{dt} (\ln V) = 3 \frac {d}{dt} (\ln a)$$
Equation (2.37) says
$$\frac{1}{T} \frac{dT}{dt} = -\frac{1}{3V}\frac{dV}{dt}$$
which is equivalent to
$$\frac{d}{dt} (\ln T) = -\frac{1}{3} \frac{d}{dt} (\ln V) = -\left ( \frac{1}{3}\right ) 3 \frac {d}{dt} (\ln a) = -\frac{d}{dt} (\ln a)$$
 
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You two are awesome, I can't believe I missed that. Thank you very much!
 

FAQ: Question on the Derivation of Temperature/Scale Factor Relation

1. How is temperature related to the scale factor?

The temperature of a system is directly proportional to the scale factor, which represents the expansion or contraction of the system's dimensions.

2. Can you explain the derivation of the temperature/scale factor relation?

The derivation involves using the ideal gas law, which relates the pressure, volume, and temperature of a gas. By rearranging the equation and incorporating the scale factor, we can derive the relationship between temperature and scale factor.

3. What is the significance of the temperature/scale factor relation in thermodynamics?

The temperature/scale factor relation is important in understanding the behavior of gases and other systems undergoing expansion or contraction. It also helps in predicting the effects of changes in temperature on the dimensions of a system.

4. How does the temperature/scale factor relation apply to real-world scenarios?

The temperature/scale factor relation is applicable in various fields such as engineering, meteorology, and astrophysics. For example, it is used in designing structures to withstand temperature changes, predicting weather patterns, and understanding the expansion of the universe.

5. Are there any limitations to the temperature/scale factor relation?

The temperature/scale factor relation is based on the ideal gas law, which assumes certain conditions such as a constant pressure and the absence of intermolecular forces. In real-world scenarios, these assumptions may not hold true, leading to deviations from the predicted relationship.

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