Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 14A (Cosmology)

The Flatness problem is that $|\Omega -1|$ grows during ordinary decelerating expansion, so its small present value requires the early density to have been extraordinarily close to critical. The Horizon problem is the near uniformity of regions whose past light cones do not meet in a noninflationary hot Big Bang model.

Inflation requires accelerated expansion,For a constant equation of state $P=w\rho$, this means $w<-1/3$; representative scale factors are $a\propto t^p$ with $p>1$ and $a\propto e^{Ht}$. Equivalently, the comoving Hubble radius $(aH{)}^{-1}$ decreases. Sincemany e-folds drive the universe toward spatial flatness. A small region that was initially in causal contact is stretched to a size far larger than the later Hubble radius, allowing the observed universe to originate from one equilibrated patch.

For the inflaton,The continuity equation therefore becomesAway from an isolated turning point division by $\dot{\varphi }$ gives, and continuity extends through such points,

The Friedmann acceleration equation givesFor the scalar field,and henceAn inflationary phase occurs precisely when potential energy dominates enough that

The slow-roll approximation assumesThe first condition makes the energy density potential dominated, and the second neglects inertial acceleration relative to Hubble friction. The scalar and Friedmann equations reduce to

For $V=\lambda {\varphi }^4/4$ and positive $\varphi$, the slow-roll Friedmann equation givesThe scalar equation then reduces toWith $\varphi (t_0)={\varphi }_0$,Furthermore,Thus the quartic-potential slow-roll solution is