Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 1 / 15D (Cosmology)

The radial equation isMultiplying by $\dot{r}$ and integrating givesSince $\rho$ is an energy density,Set $r=a{x}_0$, divide the energy equation by $r^2/2$, and defineThen

Differentiate the Friedmann equation and useAfter cancelling the curvature term with the original Friedmann equation, one obtains the Friedmann acceleration equation

The Newtonian derivation does not explain why pressure gravitates through the $3P$ term; that input enters only through a relativistic-looking continuity law. It also treats space and time as Euclidean and absolute, so it cannot supply the spacetime meaning of curvature $k$, horizons, or the global geometry of the cosmological model.

Radiation conservation givesConsequentlyEvaluating at $t_0$, where $a=1$, shows thatPutThen $\dot{b}=2\sqrt{A+B{b}^2}$. Choosing the big bang as $t=0$ and integrating givesThereforewhere

At early times, $\sinh (\beta t)\sim \beta t$, so $a\propto t^{1/2}$, as in a radiation-dominated universe. At late times,the expected de Sitter expansion with $H\to \sqrt{\Lambda c^2/3}$.

Acceleration begins when the radiation deceleration and cosmological-constant acceleration balance:Thus $a=\alpha$, which in the exact solution means $\sinh (\beta t_{\Lambda })=1$. HenceThis is the radiation-to-cosmological-constant transition.