The time expression of the matter-raditation equality era

[Magister]

1. The problem statement

I am asked to find the time of the matter-radiation equality.
I have already find the red shift of this era, so I just need to find the expression of time in terms of red shift.
From the Friedmann equation I can derive 2 relations between time and the red shift, one to the matter dominated era and another to the radiation dominated era.
I suppose that I will have to use the second one because from the beginning until the equality of the densities it was a radiation dominated universe.
My problem is that I saw, in the Kolb and Turner book, that they use the matter dominated expression. What am I missing??

 

[Dick]

I would guess that they are then computing the time between now and the equality time. But that's basically the age of the universe. The more interesting number would be time from big bang to equality time. Do you have a chapter page reference from Kolb and Turner?

 

[Magister]

The $t_{eq}$ is in page 89 (chapter 3.5) and the expression they use are in page 65 (chapter 3.2) (for $\Omega_0 = 1$).
I understand why do you say that they must be computing the age from the equality time until now but the they get a $t_{eq}=2.4 X 10^3$ years so it must be from the Big Bang until the equility time...
Thanks for your help!

 

[Dick]

Ok, my page numbers are a little different, but the equation they refer to (3.45) looks to be a more exact expression, valid for both matter and radiation dominated periods. I think the reason for the matter dominated form in the approximate solution is because it's based on H_0, the current Hubble rate. So to project back to get temperature etc at equality time, it's appropriate to use the matter dominated form.