Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 72 / 2 / ii

At the present epoch, define $s=1+z$ and $F(z)=\dot{z}/H_0$. For flat matter plus a cosmological constant, neglecting radiation,In an Einstein-de Sitter universe, $E=s^{3/2}$, henceThe redshift drift is zero at $z=0$, has initial slope $F^{\prime }(0)=-1/2$, and decreases monotonically because $F^{\prime }=1-3\sqrt{s}/2<0$. Its second derivative is $-3/(4\sqrt{s})<0$, so the sketch bends downward, with $F\sim -z^{3/2}$ at large positive redshift.

A pure positive cosmological constant in a flat universe gives constant $H=H_0$, the expanding de Sitter model. ThusThe redshift drift is a straight line through the origin with positive slope. Here the source and observer expansion rates are equal; the factor $1+z$ in the observer contribution makes every positive-redshift comoving source drift toward still larger redshift.

For the specified flat matter-vacuum model,Its small-redshift behaviour is $F(z)=0.55z+O(z^2)$, since $q_0={\Omega }_{m,0}/2-{\Omega }_{\Lambda ,0}=-0.55$. At high redshift, matter dominates and $F\sim -\sqrt{0.3}{s}^{3/2}$, so the initially positive redshift drift must turn negative. To locate the crossing, square $s=\sqrt{0.3s^3+0.7}$; both sides are positive, so squaring introduces no positive-$s$ spurious root. Factorization givesBesides $z=0$, the physical nonzero redshift-drift root in flat matter-Lambda cosmology isThe redshift drift is positive for $0<z<z_{\ast }$ and negative for $z>z_{\ast }$. Its maximum satisfies $1=0.45s^2/\sqrt{0.3s^3+0.7}$ and lies near $z=0.949$, where $F\simeq 0.240$. This zero is distinct from the acceleration transition at $z_{\mathrm{acc}}=(2{\Omega }_{\Lambda ,0}/{\Omega }_{m,0}{)}^{1/3}-1\simeq 0.671$: redshift drift compares expansion at two epochs, while acceleration is local to one epoch.

For fixed cosmological density parameters, the redshift drift has the form $\dot{z}=H_{0}F(z)$. Multiplying the Hubble constant changes the vertical scale of all three sketches but leaves their zeros unchanged. Thus the drift-zero redshift is independent of $H_0$ at fixed density fractions. Every model has the trivial $z=0$ zero; only the specified mixed model has a further positive zero, $z_{\ast }\simeq 2.089$. If instead physical matter and vacuum densities are held fixed while $H_0$ changes, their density fractions change too, so that is a different family of models.