energy density of photons and matter

indie452

hi

im doing a question at the moment and am having issues understanding the question, and i cant ask my lecturer as he is stuck abroad with no internet.

the question asks to calculate the ratio of current energy density of CMBR photons to that of baryonic matter.
the present density of baryonic matter in the universe is p_m,0=2.56x10^-27kg/m³
CMBR Temp = 2.725K

ok so i calculated energy density of the CMBR = 4/c * [sigma]T⁴ = 0.417x10^-13J/m³
sigma=stefans constant
i'm not quite sure how to turn the p_m,0 of baryonic matter into an energy density. but i can see it needs to be multiplied by dimensions (m/s)^2

thanks for any hints

also what does p_m,0 mean?

zachzach

The problem states what [tex] \rho_{m,0} [/tex] means:
The present density of baryonic matter in the universe. This is the measured mass of baryons in our universe per cubic meter.

zachzach

Hint: Where does the matter get its energy from??

indie452

i am thinking that i times the baryonic mass density by c^2
this would give 1.44Gev/m^3
so the ratio would be 0.261Mev/m3 / 1.44Gev/m^3 = 1.8125x10^-4

zachzach

But why? I know the units work and that is what I got as well. Where does energy from mass come form?

indie452

oh from the E=mc^2 relation of mass to energy.

zachzach

Yep and since energy density [tex] \epsilon = \frac{E}{V} [/tex]


[tex] \epsilon = \frac{mc^2}{\frac{m}{\rho_0}} = \rho_0c^2 [/tex]

indie452

ok thanks for that the reasoning behind that makes more sense now

btw the next part says to calculate th redshift at which the energy density of matter = that of radiation (i.e. the CMBR photons)

i tried:

e = energy density

e[rad] / e[matter] = 1 when equal and this is also proportional to a^-4/a^-3 where a is cosmological scale factor. and i am told a ~ (z+1)

this would mean however that 1= 1/a = 1/(z+1) and so z=0
i know this is wrong

i think it should be ~3600 this value i found many times in my reading but didnt understand how it was found

indie452

ok i just tried

z+1 ~ (1.8x10^-4)^-1
so z is ~5526

i know this doesnt take into account 3 types of neutrinos
do i times 1.8x10^-4 by 1.68 to take them into account?

zachzach

[tex] \epsilon_m = \epsilon_{m,0}(1+z)^3 [/tex] and

[tex] \epsilon_{rad} = \epsilon_{rad, 0}(1+z)^4 [/tex]

indie452

ok so e_m0/e_rad0 = 1+z

z= (1.8x10-4)^-1 - 1 = 5526.0066027

this gives a temp of the CMBR => T=1/(z+1) = 1.809x10-4 K
this doesnt seem right. surely temp should be larger than now?

zachzach

No, [tex]

\frac{\epsilon_{rad}}{\epsilon_{m}} = \frac{\epsilon_{rad,0}}{\epsilon_{m,0}}(1+z) [/tex]

indie452

but surely
e_rad/e_m = 1 [cause im looking for equality]
and so
e_rad/e_m = 1 = e_rad0(z+1)/e_m0
so
e_rad0/e_m0 = 1/(z+1)
therefore
z+1 = e_m0/e_rad0
which is what i put

zachzach

O right, my bad.

indie452

thats ok, its just that it doesnt seem right.
is it correct to get the temp by saying T~1/(z+1)?
cause this just gives values i would have thought as being too small. or is this the overall temp of radiation and matter? and i only want the CMBR temp