Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 66 / 1

Write $H=\dot{a}/a$ for the Hubble parameter. Subtracting the Friedmann equation from the second equation gives the Friedmann acceleration equation,Since $\dot{H}=\ddot{a}/a-H^2$, eliminating $H^2$ with the first equation yieldsDifferentiating the first equation, with constant cosmological constant and curvature parameter, now givesThus the cosmological perfect-fluid continuity equation is $\dot{\rho }+3H(\rho +p/c^2)=0$. Multiplying by $a^3$ and using $d(a^3)/dt=3H{a}^3$ provesThis is local conservation of energy: expansion changes the energy in a comoving volume through pressure work.

With constant equation-of-state parameter $w$, the cosmological perfect-fluid continuity equation becomes $\dot{\rho }+3(1+w)H\rho =0$. ConsequentlyThe constant-equation-of-state density scaling includes $\rho \propto a^{-3}$ for pressureless matter and $\rho \propto a^{-4}$ for radiation in cosmology.

For $w=-1$, constant-equation-of-state density scaling makes ${\rho }_{\mathrm{DE}}$ constant. In the first Friedmann equation, its contribution combines with the cosmological constant asIn the second equation, $-8\pi G{p}_{\mathrm{DE}}/c^2=8\pi G{\rho }_{\mathrm{DE}}$ produces precisely the same shift of $\Lambda c^2$. Therefore homogeneous expansion measures the sum of these contributions; a constant-density fluid with $p=-\rho c^2$ is dynamically indistinguishable from cosmological constant energy in these equations.

Define the present critical density and cosmological density parameter bySpatial flatness with $\Lambda =0$ implies ${\Omega }_{M,0}=1-{\Omega }_{\mathrm{DE},0}$. Separately conserved cosmological fluids and constant-equation-of-state density scaling give ${\rho }_M={\rho }_{M,0}{a}^{-3}$ and ${\rho }_{\mathrm{DE}}={\rho }_{\mathrm{DE},0}{a}^{-3(1+w)}$. Inserting both into the Friedmann equation yieldsThis is the constant-$w$ case of the Hubble parameter for matter and dynamical dark energy.

Assume both densities are nonnegative and ${\Omega }_{\mathrm{DE},0}<1$, so a nonzero pressureless matter component exists. Their ratio obeysThus matter domination givesAt fixed $H_0$, the early expansion rate does depend on the present dark energy fraction: increasing it reduces ${\rho }_{M,0}=(1-{\Omega }_{\mathrm{DE},0})3H_0^2/(8\pi G)$. There is no contradiction with negligible early dark energy, since the leading expression is simply $H^2\sim 8\pi G{\rho }_M/3$. At fixed physical matter density, that leading rate is independent of the negligible component.

The endpoint ${\Omega }_{\mathrm{DE},0}=1$ has no matter, so the ratio argument does not apply. Its Hubble parameter is $H=H_0{a}^{-3(1+w)/2}$: it diverges for $-1<w<0$, is constant for $w=-1$, and tends to zero for $w<-1$.