Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 310 / 1

A perfect fluid in general relativity has stress-energy tensorwhere $u^{\mu }{u}_{\mu }=-1$. Besides the metric tensor, one needs the energy density $\rho$, the pressure $P$, and three independent components of the normalized four-velocity $u^{\mu }$: five scalar functions of spacetime in total.

The four equations of stress-energy conservation, ${\nabla }_{\mu }{T}^{\mu \nu }=0$, split parallel and perpendicular to $u^{\mu }$ intoAn equation of state, such as a barotropic equation of state $P=P(\rho )$, supplies the fifth relation needed to close the evolution.

In cosmic time, all three Friedmann-Lemaître-Robertson-Walker metrics can be writtenwhereThe closed spatial slice is a three-sphere of radius $a(t)$, soThe complete flat and open spatial slices are noncompact and have

For a homogeneous fluid in the Spatially flat FLRW metric, the equations in part (a) reduce to the cosmological perfect-fluid continuity equationwhile homogeneity makes the spatial relativistic Euler equation automatic. The Einstein field equations give the flat Friedmann equation and Friedmann acceleration equation,equivalently $\ddot{a}/a=-4\pi G(\rho +3P)/3$.

Normalize the present scale factor to $a(t_0)=1$. In a flat universe containing radiation in cosmology, pressureless matter, and a cosmological constant, the Friedmann equation iswhere ${\Omega }_{r,0}+{\Omega }_{m,0}+{\Omega }_{\Lambda ,0}=1$. Since $dt=da/(aH)$, the age of an FLRW universe isFor a matter-only flat Einstein-de Sitter universe,Taking the present value $H_0\simeq 70\mathrm{km}{\mathrm{s}}^{-1}\mathrm{Mp}{\mathrm{c}}^{-1}$, for which the Hubble time is $H_0^{-1}\simeq 14.0\mathrm{Gyr}$, gives

The identityshows that accelerating expansion has $H>0$ and $ϵ<1$, whereas accelerated contraction has $H<0$, $\ddot{a}<0$, and $ϵ>1$.

For the monomial inflation potential $V=\lambda {\varphi }^p$, the two potential slow-roll parameters areThus ${ϵ}_V<1$ requires $|\varphi |>p{M}_{\mathrm{Pl}}/\sqrt{2}$, while $|{\eta }_V|<1$ requires $|\varphi |>\sqrt{p|p-1|}{M}_{\mathrm{Pl}}$.

At ${\varphi }_{\ast }$, the specified velocity gives kinetic energyHence ${\rho }_{\varphi }=3V$, $P_{\varphi }=V$, and the first Hubble slow-roll parameter isThe universe is therefore decelerating, despite the flatness of the potential: the chosen initial kinetic energy violates the slow-roll approximation.