Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 53 / 1 / ii

For an expanding universe, cosmic inflation requires $\ddot{a}>0$. Including any vacuum contribution in the total density and pressure, the acceleration equation gives $\boxed{\rho +3P<0}$, or $w=P/\rho <-1/3$ for positive density. Equivalently the comoving Hubble radius $(aH{)}^{-1}={\dot{a}}^{-1}$ decreases. Since $|{\Omega }_{\mathrm{total}}-1|=|k|/(aH{)}^2$, prolonged acceleration suppresses curvature and addresses the flatness problem. A sufficiently long inflationary phase also stretches a region that was initially causally connected to encompass the observed region, addressing the horizon problem. The condition on acceleration must last long enough; an arbitrarily short interval is insufficient.

For a homogeneous canonical scalar field $\Phi$ with potential $V(\Phi )$,Thus $V>{\dot{\Phi }}^2$ is sufficient for inflation. A suitably flat positive potential gives slow-roll inflation: Hubble damping in $\ddot{\Phi }+3H\dot{\Phi }+V_{,\Phi }=0$ keeps the kinetic energy small compared with $V$, so $P_{\Phi }\simeq -{\rho }_{\Phi }$.

A physical positive scale factor here requires $t>1/C$. In that domain $\dot{a}=1/t>0$ but $\ddot{a}=-1/t^2<0$. There is no inflationary expansion. The range $0<t<1/C$ has a negative printed scale factor and cannot supply a positive expanding branch of this model.

For $t>0$, $\dot{a}=n{t}^{n-1}>0$ and $\ddot{a}=n(n-1)t^{n-2}$. Consequently inflation occurs exactly when $\boxed{n>1}$. The case $n=1$ is coasting, and $0<n<1$ is decelerating. The Hubble slow-roll parameter is ${ϵ}_H=-\dot{H}/H^2=1/n$, giving the same acceleration criterion ${ϵ}_H<1$.

The Hubble parameter is $H=An{t}^{n-1}>0$, andFor $n=1$ this is positive at every $t>0$, so expansion is exponential and inflationary throughout. For $0<n<1$, acceleration begins only afterThe interval $0<t<t_{\ast }$ is decelerating, while $t=t_{\ast }$ is the transition. Thus the model admits inflation, but not arbitrarily near its specified time origin when $n<1$; choosing parameters can place the transition early. Equivalently ${ϵ}_H=(1-n)/(An{t}^n)$ becomes smaller than one at the same time.