What is the DM-Radiation Relation Explained on Page 190 of Dodelson's Book?

[alejandrito29]

I am read the pag 190 of Dodelson bock, where use the following relation

$$\rho_{DM}= \rho \frac{y}{y+1}$$. where $$y= a/a_{eq}=\rho_{DM}/\rho_{rad}$$

i tried using the tipically relation $$\rho = \rho_{rad} a^-4+ \rho_{DM}$$

but, i don't understand,

healp please

 

[marcus]

I am read the pag 190 of Dodelson bock, where use the following relation

$$\rho_{DM}= \rho \frac{y}{y+1}$$. where $$y= a/a_{eq}=\rho_{DM}/\rho_{rad}$$

i tried using the tipically relation $$\rho = \rho_{rad}a^{-4}+ \rho_{DM}[[[a^{-3}]]]$$

but, i don't understand,

healp please

$$\rho = \rho_{rad}a^{-4}+ \rho_{DM}a^{-3}$$Doesn't DM mean dark matter? So density goes up with smaller volume.

You seem to be omitting ORDINARY matter in this example. Also you have no dark energy in the picture. Maybe this is some special case of Dodelson? A different universe with no DE and no ordinary matter? Only DM and rad? OK.

You had better say what you think the symbols stand for.

Back in the past,when a= 0.1 and distances were 1/10 what they are today,
the density of dark matter was 1000 times what it is today

ρ_DM(back then) = ρ_DM(now) a^-3

and the density of rad was 10000 times what it is today
ρ_rad(back then) = ρ_rad(now) a^-4

Any questions about that?

 

[alejandrito29]

$$\rho = \rho_{rad}a^{-4}+ \rho_{DM}a^{-3}$$


Doesn't DM mean dark matter? So density goes up with smaller volume.

You seem to be omitting ORDINARY matter in this example. Also you have no dark energy in the picture. Maybe this is some special case of Dodelson? A different universe with no DE and no ordinary matter? Only DM and rad? OK.

You had better say what you think the symbols stand for.

Back in the past,when a= 0.1 and distances were 1/10 what they are today,
the density of dark matter was 1000 times what it is today

ρ_DM(back then) = ρ_DM(now) a^-3

and the density of rad was 10000 times what it is today
ρ_rad(back then) = ρ_rad(now) a^-4

Any questions about that?

very thank, but, with this relation $$\rho = \rho_{rad}a^{-4}+ \rho_{DM}a^{-3}$$

i still do not understant why $$\rho_{DM}= \rho \frac{y}{y+1}$$

 

[marcus]

I suggested that you tell us what you think the symbols stand for. Does rho_rad stand for the radiation density at some particular time? Like the epoch of matter-radiation equality? Or the present?
Likewise rho_DM.

You have not explained why you are not including ordinary matter and dark energy. Is this example about the very early universe, when one might consider DM and radiation to dominate and neglect DE and ordinary matter small contribution? Since I don't have the Dodelson book, I don't know the particulars that would help make sense of the problem.

I assume that a_eq is the scale factor at matter-radiation equality,which is about 1/3400. That would be the conventional interpretation. Is that what you think it stands for in this case?

 

[Mordred]

I suggested that you tell us what you think the symbols stand for. Does rho_rad stand for the radiation density at some particular time? Like the epoch of matter-radiation equality? Or the present?
Likewise rho_DM.

You have not explained why you are not including ordinary matter and dark energy. Is this example about the very early universe, when one might consider DM and radiation to dominate and neglect DE and ordinary matter small contribution? Since I don't have the Dodelson book, I don't know the particulars that would help make sense of the problem.

I assume that a_eq is the scale factor at matter-radiation equality,which is about 1/3400. That would be the conventional interpretation. Is that what you think it stands for in this case?

I looked at my copy of Dodelson"s modern cosmology but the revision must be different. Marcus raised the key points,

Is this a single or multi-component universe example? Ie early universe prior to matter formation etc. or a De-sitter universe? the other possibility is that he is isolating the individual influences

Dodelson may show how he derived the first equation earlier on in the book, as my revision is different I have no way of knowing.

 

[cristo]

I am read the pag 190 of Dodelson bock, where use the following relation

$$\rho_{DM}= \rho \frac{y}{y+1}$$. where $$y= a/a_{eq}=\rho_{DM}/\rho_{rad}$$

i tried using the tipically relation $$\rho = \rho_{rad} a^-4+ \rho_{DM}$$

but, i don't understand,

healp please

You don't actually state what you want to understand!

That first equation is easily verified. Assuming that $\rho=\rho_{\rm rad}+\rho_{\rm DM}$, then just substitute y into your equation and you can see that it is satisfied. I also don't have the book at hand, so can't place this in context.

 

[cristo]

Having looked at the text, it's just a change of variables from t to y where y = rho_dm/rho_r. As I said above, you can verify it by direct substitution.