How to Calculate the Age of the Universe

In order to calculate the age of the universe, one must determine the distances of the furthest objects.  If an object is x light years distant, its light must have taken x years to reach the earth, so x must be the age of the object. To do this one must measure the redshift of the spectral lines of the object.  The redshift is the percentage by which the lines of the spectrum are shifted towards the red end of the spectrum.  If the wavelength of the light undergoing redshift is Λ₀ (lambda) when it is at rest and the observed wavelength when redshifted is Λ, caused by its speed of recession, the redshift is given by .  This can be expressed as a percentage of the speed of light, which in the case of the spectral lines of the quasar 3C273 is 0,16 (16%).

Einstein's formula linking redshift with the speed of recession, is   where c is the speed of light, v is the speed of recession of the radiating body and z is the redshift.  In the case of 3C273,

 ,  i.e.  

or    

or    

∴   

i.e.  .

∴   .

The speed of recession of quasar 3C273 is thus 14,7% of the speed of light.  To find the distance of 3C273 we must apply Hubble's law which states: “the speed of recession of a distant object is proportional to its distance in megaparsecs”.  This means that the farther away an object is, the greater its speed of recession is. 

i.e. V = H_o D where D is the distance in parsecs,  ∴ D = V ÷ H_o .  V is the speed of recession and H_o is the Hubble constant.  Today’s latest measurements give a value for H_o of 71 km per sec per megaparsec.

In the case of 3C273 V = 0,147c = 0,147 x 300 000 km per sec.

∴   light years.

We multiply by 3,26 to convert parsecs into light years and divide by 1000 to get milliards (thousand millions) and 300 000 is the speed of light in kilometres per second.  3C273 is thus 2,0248 milliard light years distant.  Distance divided by speed of recession equals time or age.

At its speed of recession of 0,147 times the speed of light, the light from the quasar has been travelling for 2,0248 - 0,147 = 13,77 milliard years (13,77 thousand millions).  So the universe must be at least 13,77 milliard years old. 

Let us try it on another quasar, 3C466 which has a redshift of 1,4:

   so that   

i.e.    so that  5,76c - 5,76v = c + v

or  5,76c - c = v + 4,76c = 6,76v.

This gives:   for the speed of recession of 3C466.  

By Hubble's law:

 milliard light years. The time taken to reach the Earth is 9,697 ÷ 0,704 = 13,77 milliard years - the same value as for the other quasar.

So we can confidently say the age of the universe is 13,8 milliard years (13,8 x 10⁹ years).

Try it out on quasar PHL 957 which has a redshift of 2,69.  A prize is offered for the first correct solution received.

Jan Eben van Zyl