In $1959$ Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density $N$, which is maintained a constant. Let the charge on the proton be : 

${e_p}{\rm{ }} =  - {\rm{ }}\left( {1{\rm{ }} + {\rm{ }}y} \right)e$ where $\mathrm{e}$ is the electronic charge.

$(a)$ Find the critical value of $y$ such that expansion may start.

$(b)$ Show that the velocity of expansion is proportional to the distance from the centre.

$(a)$ Let the Universe have a radius R. Assume that the hydrogen atoms are uniformly distributed. The expansion of the universe will start if the coulomb repulsion on a hydrogen atom at $\mathrm{R}$ is larger that the gravitational attraction.

- The hydrogen atom contains one proton and one electron, charge on each hydrogen atom.

$e_{1 p}=e_{p}+e=-(1+y) e+e$

$=-e+y e+e$

$=y e$

Let $\mathrm{E}$ be electric field intensity at distance $\mathrm{R}$, on the surface of the sphere, then according to Gauss' theorem,

$\oint \overrightarrow{\mathrm{E}} \cdot d \overrightarrow{\mathrm{S}}=\frac{q}{\epsilon_{0}}$

$\therefore \mathrm{E}\left(4 \pi \mathrm{R}^{2}\right)=\frac{4}{3} \frac{\pi \mathrm{R}^{3} \mathrm{~N}|y e|}{\epsilon_{0}}$ $...(2)$ 

$\therefore \mathrm{E}=\frac{1}{3} \frac{\mathrm{N}|y e| \mathrm{R}}{\epsilon_{0}}$

Let us suppose the mass of each hydrogen atom $=m_{p}=$ mass of a proton and $\mathrm{G}_{\mathrm{R}}=$ gravitational field at distance $\mathrm{R}$ on the sphere.

$\text { Then, }-4 \pi \mathrm{R}^{2} \mathrm{G}_{\mathrm{R}}=4 \pi \mathrm{G} m_{p}\left(\frac{4}{3} \pi \mathrm{R}^{3}\right) \mathrm{N}$$...(3)$ 

$\therefore \mathrm{G}_{\mathrm{R}}=-\frac{4}{3} \pi \mathrm{G} m_{p} \mathrm{NR}$

Gravitational force on this atom is,

$\mathrm{F}_{\mathrm{G}}=m_{p} \times \mathrm{G}_{\mathrm{R}}=\frac{-4 \pi}{3} \mathrm{G} m_{p}^{2} \mathrm{NR}$

Coulomb force on hydrogen atom at $R$ is,

$\mathrm{F}_{\mathrm{C}}=(y e) \mathrm{E}=\frac{1}{3} \frac{y^{2} e^{2} \mathrm{NR}}{\epsilon_{0}}$

[From equation $(1)$]