Hydrodynamical description of photons

[wphysics]

I am working on Weinberg, Cosmology book.
I am currently reading Chapter 6.

At the last paragraph of Page 257, Weinberg said these equations take a simple hydrodynamic form for cold dark matter and ..., but for calculations of high accuracy it is necessary to use the Boltzmann equations of kinetic theory ...
In addition, right below sub chapter "Photons" at Page 260, he said the density of free electrons was high enough so that photons could be described hydrodynamically : Thomson scattering gave the photons a total momentum locked to that of the baryonic plasma and a momentum distribution that was isotropic in the co-moving frame.

At this moment, I have some questions.
First one might be silly, but I have no idea about the meaning of hydrodynamical description.
The author said hydrodynamical description gives us good approximation, but I don't know what kind of assumption or situation gives us the right to use hydrodynamical description. Besides, in case of photons, high density of free electrons gives us legitimacy, but why? Could you explain in more detail physically?

Second one is why Thomson scattering gave the photons a total momentum locked to that of the baryonic plasma and a momentum distribution that was isotropic in the co-moving frame.
As far as I know, Thomson scattering is elastic one, so just energy of each particle is conserved. Why photons and baryonic plasma come to have the same total momentum and photon momentum distribution comes to isotropic?

My question becomes so messy, but I really appreciate for your answer

 

[Mordred]

In the term Hydrodynamic nature he is referring to the fluidic nature of the universe.

Energy-mass density per volume is equivelent to pressure. Realizing that should help in how the Boltzmann equations are involved. I'm not confident enough on the Thompson scattering portion without seeing the reference itself. Hopefully others are more familiar with the textbook. This forum topic has a geometry article that covers some of the fluid portion of the FLRW metric. My signature has a cosmology101 link with
numerous articles you will find of
use.

https://www.physicsforums.com/showthread.php?t=705427

 

[wphysics]

I don't think your answer is helpful for me.

So, how? What is the requirement for giving us legitimacy to use hydrodynamical description? Why can we describe photons hydrodynamically if the density of free electrons is high?

Could you explain in more detail?

 

[Mordred]

I'm not totally up on Weinbergs metrics so I can only go by your descriptive. Common forms used in Hydrodynamic representations in cosmology include Poisson equation and Euler form. In Particular the Euler pressureless form. Weinberg discusses that in this article.

http://www-astro.physics.ox.ac.uk/~pgf/B3.pdf

Perhaps if you can provide the metrics he is using along with the statement it will provide which metric system he is describing

 

[wphysics]

I appreciate for your answer
But what I am asking is not expanding equations, but the legitimacy for using hydrodynamical description. In which physical circumstances, can we use this description?

Weinberg said in case of photons, the high density of free electron is, which is I don't know why.

 

[Mordred]

In your first post where you typed "in these equations" providing us with those equations will help.
Reason being is that there are numerous alternative methods to describe a variety of hydridynamic solutions both on local and extragalactic scales. A high enough energy-mass density of photons could simply be when its pressure equivelence has influence. On that I can only conjecture.
If you need assistance on doing the latex math commands the
forum has an article

https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[Mordred]

I decided to download a copy of Cosmology by Stephen Weinberg. A couple of aspects you missed on cold dark matter is non relativistic so its energy
momentum pressure is less than relativistic energy momentum. The same applies to baryonic plasma. A key point your missing is the conservation of momentum exchange between baryonic plasma and photons during the recombination era. Which is described by his reference to thompson scattering. This interaction becomes isotropic. Assuming I am understanding his text correct. Perturbation theory is not my strong suit as Babowell will readily agree lol.

From the page numbers my copy is slightly different than yours but does have the lines you posted. It may take me a bit to understand perturbation metrics he is using. I'll plug away on it when time allows me to.

 

[Mordred]

Ok I think I have the gist of what he is doing in chapter 6. To fully understand it you must understand chapter 5 and look at the Boltzmann equations in appendix H.
In essence he is describing the necessary conditions with a chosen gauge described in 5.3.32 that allow us to describe individual components such as non interacting cold dark matter and baryonic matter as a perfect fluid

http://en.m.wikipedia.org/wiki/Perfect_fluid

In chapter 5 he explains why a gauge must be chosen. He also covers the stress energy tensers, rotational vortice and vector tensers involved in obtaining an unperturbed perfect fluid.
If I'm correct then the statement "the photon density is high enough" is when the interactions between photons and baryonic matter is sufficiently high enough to maintain thermal equilibriam and can be described as an unperturbed perfect fluid.
In particular he is describing how to derive that during the recombination era whose relics form the CMB.

Hope that helps

 

[wphysics]

I think our point is getting close. First of all, thank you so much for your willingness to answer my silly question.

I have further question.

As you said, if the free electron density is high enough, we can think that electrons and photons are in thermal equilibrium. Than, does it mean that mixture of electrons and photons can be considered as a single fluid?
If so, as a single fluid, it has a perfect fluid form in comoving coordinate, right?

At this moment, I am still confused why Thomson scattering takes position here.
I see that during thermal equilibrium era, they usually interact by Thomson scattering. But, why does this make the momentum of each component same?

The last question is if hydrodynamic description is valid, all equations should be expressed as relativistic extension of Euler equations. But, is this true in case of cold dark matter (Eq 6.1.3 of Weinberg) ?

Again, I really appreciate.

 

[Mordred]

I think our point is getting close. First of all, thank you so much for your willingness to answer my silly question.

I have further question.

As you said, if the free electron density is high enough, we can think that electrons and photons are in thermal equilibrium. Than, does it mean that mixture of electrons and photons can be considered as a single fluid?
If so, as a single fluid, it has a perfect fluid form in comoving coordinate, right?

At this moment, I am still confused why Thomson scattering takes position here.
I see that during thermal equilibrium era, they usually interact by Thomson scattering. But, why does this make the momentum of each component same?

The last question is if hydrodynamic description is valid, all equations should be expressed as relativistic extension of Euler equations. But, is this true in case of cold dark matter (Eq 6.1.3 of Weinberg) ?

Again, I really appreciate.

working from my phone so its too much effort to cut and paste portions of your reply.

You are correct in the first paragraph on thermal equilibrium and considered a single perfect fluid in a commoving frame.

For Thompson scattering this is a low energy interaction whose frequency and kinetic energy remains the same before and after. As opposed to Compton scattering which has a frequency and kinetic energy change after. The uniformity of the CMB is an indication of which was involved.
In the latter case the CMB
wouldn't be uniform.

The CDM is a slow non relativistic and non interactive particle. So Weinberg states it can be treated as a scalar pressure only.
Keep in mind this section is dealing with small scale perturbations such as dust distributions. Dust is considered
as stars, galaxies baryonic matter.

 

[wphysics]

In CDM case, I think it can be treated as a scalar density perturbation only.
What I asked was that the form of equation describing dark matter perturbation is really the form of hydrodynamics form?

 

[Mordred]

http://www.mpia.de/homes/dullemon/lectures/fluiddynamics08/chap_1_hydroeq.pdf

any perfect fluid can be treated in hydrodynamical form. From my understanding from the above link. This includes gas, plasma, CDM and baryonic matter. I've seen treatments of relativistic matter as a perfect fluid so it can also be treated as such.

 

[Mordred]

Just FYI I am enjoying this textbook. Though I do prefer Barbera Rydens "Introductory to Cosmology" Weinbergs textbook however provides good supportive material to Scott Dodelsons "Modern Cosmology"

A dedicated free textbook covering perturbation and related fields which you will find of great use is "Feilds". If you can relate to Weinbergs textbook this lengthy book should provide you with numerous useful insights.

http://arxiv.org/abs/hepth/9912205