Answer:
(a) In a matter-filled universe with critical density (Ω0 = Ωm = 1), the relation between the scale factor a and the age t is given by the equation:
a = (2/3 * B * t)^(2/3)
Here, B is a constant. We want to find an expression for (t/t0) as a function of z, where t0 represents the current age of the universe.
To relate a and z, we use the equation:
1 + z = 1 / a
Solving for a, we find:
a = 1 / (1 + z)
Substituting this expression for a into the first equation, we have:
1 / (1 + z) = (2/3 * B * t)^(2/3)
To isolate (t/t0), we divide both sides of the equation by (2/3 * B * t0)^(2/3):
1 / [(2/3 * B * t0)^(2/3) * (1 + z)] = t / t0
Therefore, the expression for (t/t0) as a function of z is:
(t/t0) = 1 / [(2/3 * B * t0)^(2/3) * (1 + z)]
(b) To compute (t/t0) for the quasar with a mass of 1.6×10^9 M⊙ and a redshift of z=7.64 in the universe considered in part (a), we use the given value of t0 = 13.8 Gyr.
Substituting the values, we have:
(t/t0) = 1 / [(2/3 * B * t0)^(2/3) * (1 + 7.64)]
However, without knowing the specific value of the constant B, we cannot calculate the exact value of (t/t0). To determine the time the quasar would have had to form, we need the exact value of t, which we don't have.
Step-by-step explanation: