Question

Cosmology. (a) If the Universe is at critical density and matter-filled, Ω

0
​
=Ω
m
​
=1, then you will see on Worksheet 12 that we can derive the following relation between the Universe scale factor a and the age t:a=(
2
3
​
Bt)
2/3
, where B is a constant. Use this expression to find an expression for (t/t
0
​
) as a function of z. (b) The most distant quasar known to date has a mass of 1.6×10
9
M
⊙
​
and a redshift of z=7.64. Compute t/t
0
​
for this quasar in the universe considered in part (a). If t
0
​
=13.8Gyr, how much time would the quasar have had to form?

Answer 1

(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: