Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 1 / 15E (Cosmology)

Differentiate the Friedmann equation and use cosmological energy conservation:For an expanding universe $H>0$, division by $2H$ and substitution of the Friedmann equation give the Friedmann acceleration equationThe strong energy condition makes $\ddot{a}\le 0$. Since $aH=\dot{a}$ and $\dot{a}>0$,Thus the quantity is nonincreasing, with strict decrease when the energy condition is strict. Dividing the Friedmann equation by $H^2$ yieldsFor fixed nonzero curvature, the magnitude of $\Omega -1$ grows as expansion decelerates. The observed closeness to one therefore requires much tighter early-time closeness to one in the standard decelerating model: this is the flatness problem. Accelerated expansion during inflation reverses this evolution and suppresses the curvature contribution.

Normalize the scale factor by $a(t_0)=1$. Separate conservation for a constant equation-of-state parameter $w$ gives $d\log {\rho }_i=-3(1+w_i)d\log a$. Thus radiation has ${\rho }_R={\rho }_{R0}{a}^{-4}$ while the dark energy component with $w=-1$ has constant density. Flatness gives ${\Omega }_{R0}+{\Omega }_{\Lambda 0}=1$, soHere $\rho$ has mass-density units in the stated Friedmann equation; the corresponding energy density is $\rho c^2$.

Assume both density parameters are positive and choose the Big Bang at $t=0$. The expanding branch satisfies $\dot{a}=H_0\sqrt{{\Omega }_{R0}{a}^{-2}+{\Omega }_{\Lambda 0}{a}^2}$. Set $u=a^2$ and integrate:ConsequentlyThe factor $H_0$ is required because $\beta$ has inverse-time units. As $t\downarrow 0$, $a(t)\sim (2H_0\sqrt{{\Omega }_{R0}}t{)}^{1/2}$, the expected radiation-dominated power law. As $t\to \infty$, $a(t)\sim \alpha e^{H_0\sqrt{{\Omega }_{\Lambda 0}}t}/\sqrt{2}$, the exponential expansion driven by a cosmological constant.