Question

Assuming that the universe currently is well described by a density parameter Ω0 = 1, there is no dark energy and the current temperature of the universe is 2.73 K, compute how long from the present it will take for the universe to cool down by 0.2 K. Remember that the temperature of the universe is inversely proportional to its radius (the scale factor).

Answer 1

We know that , the temperature,T of the universe is inversely proportional to its scale factor,R, Hence we can write,

Here T0 is the intial temperature and z is the redshift value. Differentiating we get,

We know that,

(H is hubble constant, and R is scale factor)

Substituting,

Now, let us assume that the universe cools down by 0.2K (dT/dt will be negative) in t seconds, Therefore we can say,

Substituting values, (Hubble constant is taken as 70 km/s/Mpc and scale factor is calculated for z = 0.15)

Therefore it will take 891996.44 years for the temperature to drop by 0.2 K