Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 1 / 15B (Cosmology)

Since $H=\dot{a}/a$, differentiation gives the Hubble parameter identitySubtracting the given Friedmann equation from the Raychaudhuri equation therefore yieldsOn the other hand, differentiating the Friedmann equation givesSubstitute $\dot{H}=-4\pi G{\dot{\varphi }}^2$ and divide by $(8\pi G/3)\dot{\varphi }$, which is permitted by the assumption $\dot{\varphi }\ne 0$. The result is the homogeneous inflaton equation of motionThe middle term is Hubble friction.

For $V(\varphi )=\lambda {\varphi }^4$, the slow-roll approximation givesTaking the expanding branch $H>0$ and using $H=\sqrt{8\pi G\lambda /3}{\varphi }^2$ gives the first-order linear differential equationThus the quartic-potential slow-roll solution with the normalization used in this question isIf ${\varphi }_0<0$, the field remains negative and $\dot{\varphi }>0$, so it rolls toward the potential minimum at zero.

The number of e-folds $N=\log a$ satisfiesWhen $\varphi$ is monotone, the chain rule gives $dt=d\varphi /\dot{\varphi }$, and henceUsing the two slow-roll equations, $3H\dot{\varphi }\simeq -V^{\prime }(\varphi )$ and $H^2\simeq 8\pi GV/3$, eliminates both $H$ and time:This is the differential form of the slow-roll e-fold count.

For the quartic potential,Integrating the result of part (c) along the trajectory from ${\varphi }_i$ to ${\varphi }_f$ givesBecause the negative field rolls toward zero, ${\varphi }_i<{\varphi }_f<0$ and $|{\varphi }_i|>|{\varphi }_f|$, so this monomial slow-roll e-fold count is positive.

In the decelerating hot Big Bang model, the particle horizon at recombination encloses only a finite comoving region. Widely separated parts of the Cosmic microwave background therefore had no overlapping past light cones in that model, although they have almost the same temperature. This is the Horizon problem. During cosmic inflation, accelerated expansion makes the comoving Hubble radius $(aH{)}^{-1}$ shrink. A small region that was initially in causal contact can then expand to contain the whole presently observable universe.

The Flatness problem follows fromDuring ordinary radiation- or matter-dominated decelerating expansion this quantity grows, so the observed closeness of the cosmological density parameter $\Omega$ to one requires extreme tuning at early times. Inflation makes $aH$ grow nearly exponentially and therefore drives $|\Omega -1|$ rapidly toward zero. A sufficiently large number of e-folds makes spatial flatness a dynamical outcome rather than a finely tuned initial condition.