Density Parameter for radiation dominated universe

[ibysaiyan]

Greetings everyone ,
Can anyone point me into the right direction on how to come out with a value/ expression for the density parameter of a radiation dominated universe.

Things that I know of/ can recall are:

Friedmann equation :
$$8/3 \pi G ρ R^2 -kc^2$$

Also when radiation dominates matter then we get ρ = 1/R^4 ( I think)

Density parameter :
So what am I missing to give me the density parameter ? $\Omega_{m}$ = $\rho / \rho_{critical}$
where
$\rho_{critical}= (3H^2 q / 4Pi G)$
Any sort of help is appreciated.

-ibysaiyan

 

[cristo]

Your post doesn't make much sense. Your Friedmann 'equation' is not an equation, since you do not have an equals sign. You don't define R, or any of your other variables. But it's not really clear what you want to find -- an expression for $\Omega_R$ in terms of what?

 

[ibysaiyan]

Your post doesn't make much sense. Your Friedmann 'equation' is not an equation, since you do not have an equals sign. You don't define R, or any of your other variables. But it's not really clear what you want to find -- an expression for $\Omega_R$ in terms of what?

I am sorry about that , the LHS should have (dR/dT)^2 for this to become the Friedmann equation anyways.. what I want is the density parameter of a radiation dominated universe.
I.e density at that state / critical density.

 

[ibysaiyan]

Oh i have figured it out that the deceleration parameter at the radiation dominated epoch is equivalent to the ratio of density at that time / critical density ( Omega) .

 

[sardar]

Hello ibysaiyan,

To calculate the density parameter for a radiation dominated universe, we can use the Friedmann equation, which relates the expansion rate of the universe to its energy density and curvature. In this case, we can set the matter density to zero since radiation dominates, and use the equation you mentioned: ρ = 1/R^4.

To find the density parameter, we can use the definition you provided: Ωm = ρ/ρcritical, where ρcritical is the critical density for the universe. This critical density is defined as the density at which the universe is flat (meaning the curvature term in the Friedmann equation is zero).

To find the critical density for a radiation dominated universe, we can use the Friedmann equation and set the expansion rate (H) to the critical value, which is when q = 1. This gives us: ρcritical = (3H^2 q / 8πG). Plugging in the value for q, we get ρcritical = 3H^2 / 8πG.

Now, we can plug this value for ρcritical into the density parameter equation to get: Ωm = ρ / (3H^2 / 8πG). Since we already know that ρ = 1/R^4, we can substitute this in to get the final expression for the density parameter: Ωm = 8πG / (3H^2 R^4).

I hope this helps guide you in the right direction. However, please keep in mind that this is just one possible approach and there may be other ways to calculate the density parameter for a radiation dominated universe. It's always important to double check your calculations and assumptions when working on scientific problems. Good luck!