Past exam of the general relativity and cosmology course of the University of Oxford / 2013 / 3

Use mass density units for the $\rho$ appearing in the Friedmann equation; an actual energy density is $\rho c^2$. Conservation, $\dot{\rho }+3H(\rho +P/c^2)=0$, gives $\boxed{{\rho }_M={\rho }_{M,0}{a}^{-3},{\rho }_R={\rho }_{R,0}{a}^{-4}}$ for pressureless matter and radiation in cosmology, respectively, with $a_0=1$. The first scaling counts dilution of particle number, while the second includes the cosmological redshift of each photon's energy.

Define the present cosmological density parameters by ${\Omega }_{X,0}=8\pi G{\rho }_{X,0}/(3H_0^2)$. ThenFor $k=0$, conformal time gives $da/d\eta =a^{2}H$, so the expanding branch obeys $(da/d\eta {)}^2=H_0^2({\Omega }_{M,0}a+{\Omega }_{R,0})$. Set $\eta =0$ at the Big Bang. Integrating givesand hence the flat matter-radiation conformal solutionThe limiting pure-radiation and pure-matter cases follow continuously.

At matter-radiation equality, ${\rho }_{M,0}{a}^{-3}={\rho }_{R,0}{a}^{-4}$, so $a_{\mathrm{eq}}={\Omega }_{R,0}/{\Omega }_{M,0}$. Evaluating the integrated square-root relation givesThis assumes both present densities are positive and uses the same Big Bang origin for conformal time.

For dust alone, write ${\dot{a}}^2=c^2(A/a-k)$, whereSet $\theta =\sqrt{k}c\eta$ and $a=(A/k){\sin }^2(\theta /2)$. Differentiation gives $\dot{a}=c\sqrt{k}\cot (\theta /2)$ once $dt=ad\eta$ is used, verifying the Friedmann equation. Integrating the time relation from the Big Bang yields the closed matter-dominated Friedmann solutionThe Hubble parameter and the cosmological density parameter areThe first decreases from positive infinity through zero at maximum expansion to negative infinity at the Big Crunch. The second approaches one near either singular endpoint and diverges at maximum expansion. This is a density-parameter divergence at a cosmological turnaround: $H=0$ makes the critical density vanish even though the physical dust density there is finite.

At maximum expansion, $a_{\max }=A/k$. With $a_0=1$, the present Friedmann equation gives $k{c}^2=H_0^2({\Omega }_{M,0}-1)$, so positive curvature requires ${\Omega }_{M,0}>1$. Taking $H_0>0$ for the presently expanding branch givesThe lifetime of a closed dust universe follows by setting $\theta =2\pi$. The deceleration parameter follows from the Friedmann acceleration equation, $\ddot{a}/a=-4\pi G{\rho }_M/3$. Thus this model always decelerates while expanding. Its total lifetime includes the elapsed age; it is not the remaining lifetime.

The supernova discovery of accelerating expansion does not support this dust-only recollapse model as a description of the observed expansion history. Already the 1998 and 1999 Type Ia supernova analyses favoured accelerating expansion, requiring ingredients beyond this zero-cosmological constant, positive-curvature dust model within the usual FLRW metric interpretation.^[ref][ref]