Is the Friedmann Paradox Solved by the Concordance Model?

[meteor]

Please show me how to solve this paradox
cos the Friedmann equation is:
$$H^{2}= \frac{8*pi*G*rho}{3}-\frac{k*c^{2}}{R^{2}}$$
but the concordance model says that k=0, so we can eliminate the last term of the equation
then we isolate rho, the density of energy:
$$rho=\frac{3*H^{2}}{8*pi*G}$$
but however, the formula for the critical energy density is
$$rho_{crit}=\frac{3*H^2*c^2}{8*pi*G}$$
but the concordance model says that rho=rho_crit
but you see that the 2 formulae are not equal, there's an extra c² in the formula for rho_crit

I can't figure where is the mistake

 

[Stingray]

rho is mass density. rho_crit is energy density (as you wrote it). Mass and energy have different units, and you use c^2 to convert between them.

Most people like to use units such that c=1, and then energy and mass are interchangable. This convention is so common that books are sometimes careless about distinguishing the two.

 

[meteor]

please, take a look to this thread
https://www.physicsforums.com/showthread.php?t=2864
Here's jeff, and I think that he is a knowledgeable person, and says that rho includes all kinds of energy, not only mass

It's not rhocrit but rho, which does in fact - as marcus pointed out - include all forms of energy so there's no need for the lamda term (review the derivation of the FRW equation)

 

[Stingray]

please, take a look to this thread
https://www.physicsforums.com/showthread.php?t=2864
Here's jeff, and I think that he is a knowledgeable person, and says that rho includes all kinds of energy, not only mass

That's true. I was being a little sloppy myself. rho includes everything, but the way you wrote it, it has units of mass/volume, whereas rho_crit has units of energy/volume. Wherever you're quoting rho_crit from has a slightly different form for Friedmann's equation than you do, so their answer is different by c^2. This is just a convention. You can choose either mass units or energy units for rho as long as H works out as 1/time in the end.

 

[Chronos]

use the conversion c^2=m/e