Question

Imagine an alternative universe where the characteristic decay time of neutrons is 3 min instead of 15 min. All other properties remain unchanged from what was discussed in class.

(a) Show that the maximum possible value of the primordial helium fraction is: Ymax = 2f 1 + f (1) where f = nn/np ≥ 1 is the neutron-to-proton ratio at the time of nucleosynthesis. Prove that this assertion is true.
(b) Estimate the maximum fraction of the baryonic matter in the form of helium today. You may assume all He nuclei exist in the form of helium 4

Answer 1

(a)
`[Y_(p) ]_(max) = (2f)/(1+f)`

(b)
`f_(new) = 0.013`;
`[Y_(p) ]_(max)` = 0.026

Step-by-step explanation:

Since the neutron-to-proton ratio at the time of nucleosynthesis is given:

`f = (n_(n) )/(n_(p) )`

Therefore:

`n_(n) = f*n_(p)`

Then, to determine the maximum ⁴He fraction if all the available
`n_(n)` neutrons bind to all the protons. Since, there are 2 protons and 2 neutrons in a ⁴He nucleus, it shows that there would be
`n_(n)/2` nuclei of ⁴He.

In addition, a ⁴He nucleus has a mass of
`4m_(p)`, where
`m_(p)` is the mass of one proton. Thus,
`n_(n)/2` nuclei of such nuclei will have a mass of
`n_(n)/2`*
`4m_(p)`.

Assuming that
`m_(p)=m_(n)`, there would be a total of
`(n_(n)+n_(p))` protons and neutrons with a total mass of
`(n_(n)+n_(p))*m_(p)`.

Thus:
`[Y_(p) ]_(max) = (2f)/(1+f)`

(b) Given:

`t_(nuc) = 200 s`; τ
`_(n)` = 3*60s = 180 s

`f_(new) = (n_(nf) )/(n_(pf) ) = (exp (-200/180))/(5 +[1- exp(-200/180)]) =(0.077)/(5.923) = 0.013`

`[Y_(p) ]_(max) = (2f)/(1+f)` = (2*0.013)/(1+0.013) = 0.026