Comoving distance, causality volume and entropy

In summary, the conversation is about trying to find the expression for the time derivative of entropy for the CMB photon gas in a matter-dominated universe. The equations being used do not take into account the spatial curvature of the universe and the speaker is having trouble finding the correct result. They are also struggling with the integral for the volume and are looking for help with the steps to solve it.
  • #1
Xkcd
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Hello everyone !

I try to find the expression of the time derivative of the entropy for the CMB (photon gas) but I am stuck with the calculations.

We are in the matter-domination area and at the present time (Ro=1). No radiation and no vacuum, only the curvature. The different equations are:

Eqcosmo.png


The problem is that the equations are resolved regardless of the spatial curvature of the universe. For the comoving distance I used :

Eqcosmo2.png


But I cannot find the same result. I have

Eqcosmo3.png


If I use a calculator I have an equation with logarithm and I do not find the previous result andI cannot find where is my mistake.

And I have a similar problem with the integral for the volume. With a calculator I have a monster with Arcsin and I cannot find the final expression for the entropy.

If someone could help me with the different steps for the two integrals so that I can find the expression for the derivative of the entropy.

Thank you !
 

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  • #2
Why are you using equations for a horizon? There is no cosmological horizon in a matter-dominated universe.

For the time derivative of the entropy of the CMB photon gas, you just need the entropy of a photon gas (which is a function of its temperature), and how that photon gas evolves with time (which is pretty simple in a matter-dominated universe).
 

1. What is comoving distance and how is it different from regular distance?

Comoving distance is a measure of distance in the expanding universe, taking into account the expansion of space over time. It is different from regular distance because regular distance does not account for the expansion of space.

2. How does causality volume relate to comoving distance?

Causality volume is the volume of space that can be causally connected, meaning that information or signals can be exchanged between points within that volume. Comoving distance is used to calculate causality volume because it takes into account the expansion of space and the speed of light.

3. What is the relationship between causality volume and entropy?

Causality volume and entropy are related because as the universe expands, the causality volume increases and the entropy also increases. This is because the expansion of the universe leads to a decrease in energy density, resulting in an increase in entropy.

4. How is causality volume affected by the expansion of the universe?

The causality volume increases as the universe expands because the expansion of space allows for more points to be causally connected. This means that the volume of space in which information or signals can be exchanged increases over time.

5. Can causality volume and entropy be used to predict the future of the universe?

While causality volume and entropy can provide information about the past and present state of the universe, they cannot be used to accurately predict its future. The expansion and evolution of the universe is influenced by many factors, making it difficult to make precise predictions based on these measures alone.

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