Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 2 / 15D (Cosmology)

Differentiate $E=\rho c^2{a}^3{V}_0$ and use $dE=-PdV$, with $V=a^3{V}_0$. Division by $c^2{a}^3{V}_0$ givesThis is homogeneous adiabatic expansion without entropy production or exchange across the comoving element.

The two scalings give $\rho ={\rho }_{r0}{a}^{-4}+{\rho }_{s0}{a}^{-2}$. Flatness and the Friedmann equation give $H^2=8\pi G\rho /3$, with $H_0^2=(8\pi G/3)({\rho }_{r0}+{\rho }_{s0})$. ThusOn the expanding branch, putting the Big Bang at zero, separation givesIts positive square root is the required solution. If $\beta >0$, its early behaviour is $a\propto t^{1/2}$, as for radiation. The values $a=1$ and $a^2=\beta$ giveFor $\beta =0$ there is only the string component, with linear expansion and no radiation era.

For radiation $P_r/c^2={\rho }_r/3$, so $d\log {\rho }_r=-4d\log a$. Normalization at $a=1$ gives ${\rho }_r={\rho }_{r0}{a}^{-4}$.

For strings $P_s/c^2=-{\rho }_s/3$, so $d\log {\rho }_s=-2d\log a$. Thus ${\rho }_s={\rho }_{s0}{a}^{-2}$. These separate scalings assume no energy transfer between components.