Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 2

Vary the scalar field and integrate by parts, taking the variation to vanish at the boundary:Thus the curved-spacetime Klein-Gordon equation is $\square \varphi -V_{,\varphi }=0$. In the flat FRW metric, $\sqrt{-g}=a^3$ and $g^{00}=-1$. A spatially homogeneous field therefore hasConsequentlyDots now denote cosmic time derivatives. The $3H\dot{\varphi }$ term is Hubble friction caused by expansion of the comoving volume.

For a variation of the covariant metric tensor, $\delta g^{\alpha \beta }=-g^{\alpha \mu }{g}^{\beta \nu }\delta g_{\mu \nu }$ and $\delta \sqrt{-g}=\frac{1}{2}\sqrt{-g}{g}^{\mu \nu }\delta g_{\mu \nu }$. Writing $\mathcal{L}=-\frac{1}{2}{g}^{\alpha \beta }{\partial }_{\alpha }\varphi {\partial }_{\beta }\varphi -V$, one obtainsThe specified metric-variation convention therefore givesFor $\varphi =\varphi (t)$, its invariant gradient square is $-{\dot{\varphi }}^2$. Hence $T^{00}={\dot{\varphi }}^2-({\dot{\varphi }}^2/2-V)={\dot{\varphi }}^2/2+V$, while $T^{ij}=a^{-2}{\delta }^{ij}({\dot{\varphi }}^2/2-V)$ and $T^{0i}=0$. Lowering one index makes $T^{\mu }{}_{\nu }=\mathrm{diag}(-\rho ,P,P,P)$, withPotential energy has negative pressure, whereas scalar kinetic energy contributes positive pressure. This sign distinction is what permits accelerated expansion.

With spatial flatness, the scalar as the only source, and $M_{\mathrm{Pl}}=(8\pi G{)}^{-1/2}$, the Friedmann equation isThe scalar equation gives $\dot{\rho }=\dot{\varphi }(\ddot{\varphi }+V_{,\varphi })=-3H{\dot{\varphi }}^2$, which is the energy-conservation equation $\dot{\rho }+3H(\rho +P)=0$. Differentiating the Friedmann equation on the expanding branch yields $6M_{\mathrm{Pl}}^{2}H\dot{H}=-3H{\dot{\varphi }}^2$, so $\dot{H}=-{\dot{\varphi }}^2/(2M_{\mathrm{Pl}}^2)$. Combining this with $\ddot{a}/a=H^2+\dot{H}$ gives the Friedmann acceleration equation:Thus cosmic inflation occurs exactly when $V(\varphi )>{\dot{\varphi }}^2$ in this canonical scalar model. Equivalently $P<-\rho /3$, or the first Hubble slow-roll parameter ${ϵ}_H=-\dot{H}/H^2$ is smaller than one.

Assume $V>0$ and that its derivatives do not change substantially over the field interval traversed in the estimate. With initially negligible kinetic energy, $H^2\simeq V/(3M_{\mathrm{Pl}}^2)$. The force $-V_{,\varphi }$ changes the field velocity on the Hubble friction time. Keeping $H$ and $V_{,\varphi }$ approximately constant for that local estimate, and starting at rest, solves the field equation explicitly:After one Hubble time,Writing $A=M_{\mathrm{Pl}}^2{V}_{,\varphi }^2/V^2$, the condition $A<1$ keeps this kinetic-to-potential ratio below about $0.31$, comfortably inside the inflationary bound. Integrating once more gives the fractional potential change estimateover the same interval. Thus the characteristic time for an order-one potential change is $H^{-1}/A$, up to constants of order one. The requested local Hubble-timescale criterion isUsing a strict inequality with one as the order-one threshold is conventional; its coefficient is not an exact universal end-of-inflation bound. Much smaller $A$ gives the controlled slow-roll regime and many Hubble times of cosmic inflation if the bounds persist.

After the transient in relaxation to the slow-roll attractor, $3H\dot{\varphi }\simeq -V_{,\varphi }$. Differentiating this approximate relation givesTherefore the standard sufficient conditions for negligible acceleration throughout slow roll are ${ϵ}_V\ll 1$ andThe absolute value matters: large negative curvature also drives rapid evolution. The more precise local acceleration criterion is $|{\eta }_V-{ϵ}_V|\ll 1$, with the usual separate smallness conditions avoiding a tuned cancellation. If a field starts exactly at rest on a nonzero slope, its initial acceleration is not negligible; the displayed exponential transient describes its approach to the attractor.

These are local timescale and slow-roll conditions, rather than a global duration theorem from derivatives at a single point. A smooth potential can have a narrow flat shoulder followed by a steep drop, ending cosmic inflation soon despite small starting derivatives. The conditions must hold over the traversed interval. This is the additional smooth-evolution assumption implicit in the estimate.