Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 1

Use metric signature $(-,+,+,+)$ and units $c=1$. Write ${\dot{x}}^{\mu }=d{x}^{\mu }/d\lambda$ in this part. Varying the worldline einbein, without first fixing it, givesThe canonical four-momentum is $p_{\mu }=m{g}_{\mu \nu }{\dot{x}}^{\nu }/e$, so the same mass shell constraint is the mass shell $g^{\mu \nu }{p}_{\mu }{p}_{\nu }=-m^2$.

The Euler-Lagrange equation for the trajectory isMultiplying by $e$ and then by the inverse metric, and collecting the metric tensor derivatives into the Christoffel symbols, givesThis is a geodesic equation in a possibly nonaffine parameter.

Under a change of parameter, $e^{\prime }({\lambda }^{\prime })=e(\lambda )d\lambda /d{\lambda }^{\prime }$, so $ed\lambda$ is invariant. On the positive branch, the mass shell constraint identifies this invariant with the proper time increment $ds=\sqrt{-g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}=ed\lambda$. Choosing ${\lambda }^{\prime }=s$ therefore sets $e^{\prime }=1$. The trajectory equation is then the affinely parametrized geodesic equation and its tangent has norm minus one. This is proper-time gauge for a massive worldline einbein. The mass shell constraint must still be imposed: fixing $e$ in the action before varying would lose it. A fixed parameter interval can retain a proper-time modulus; here changing to the actual proper-time interval is permitted.

In proper-time gauge the particle Lagrangian is $L=m(-{\dot{t}}^2+a^2{\dot{\mathbf{x}}}^2-1)/2$, now with dots referring to proper time $s$. The canonical spatial momentum isThe FRW metric is independent of all spatial coordinates. Hence $\partial L/\partial x^i=0$ and the Euler-Lagrange equation gives $d{p}_i^C/ds=0$. This conserved quantity is the comoving momentum.

The mass-shell mass shell constraint gives $(dt/ds{)}^2-a^2|d\mathbf{x}/ds{|}^2=1$. For future-directed motion and proper peculiar velocity $\mathbf{v}=ad\mathbf{x}/dt$, it follows that $dt/ds=(1-v^2{)}^{-1/2}=\gamma$. ThereforeThe kinetic momentum here is the physical momentum in an FRW universe, measured in the orthonormal frame of a comoving observer. It redshifts as $a^{-1}$ irrespective of whether the particle is relativistic or nonrelativistic.

The mass shell constraint and the conserved comoving momentum giveDividing yieldsThe comoving direction is constant because ${\mathbf{p}}^C$ is constant. With $a(t_0)=1$ and physical momentum magnitude $p_0>0$ at $t_0$, we have $p^C=p_0$. Integrating the speed along that fixed direction givesThis is a distance travelled, rather than a light-ray distance: the extra factor is the particle's proper peculiar speed $v=p^K/\sqrt{m^2+(p^K{)}^2}$. For a particle at rest, $p_0=0$, the distance is zero, understood separately or as the limit. The formula assumes collisionless motion over the interval. Its existence at the lower endpoint depends on the early scale factor; in radiation domination the integral is finite.

Let $t_{\ast }$ be the observation time and normalize $a(t_{\ast })=1$ temporarily. In pure radiation domination, $a(t)=(t/t_{\ast }{)}^{1/2}$ and $H_{\ast }=1/(2t_{\ast })$. With $p(t)=T_r(t)=T_{\ast }/a(t)$, the proper speed is $v=[1+(ma/T_{\ast }{)}^2{]}^{-1/2}$. Put $u=\sqrt{t/t_{\ast }}=a(t)$ in the preceding integral. Since $dt=2t_{\ast }udu$,The small-$m/T_{\ast }$ limit is $H_{\ast }^{-1}$, the radiation-era particle horizon. For $m/T_{\ast }\gg 1$, the ratio is approximately $(T_{\ast }/m)\log (2m/T_{\ast })$, retaining the distance accumulated while the particle was faster.

There is a normalization qualification: in arbitrary fixed comoving coordinates the distance is $\chi =H_{\ast }^{-1}\mathrm{arsinh}(y)/(a_{\ast }y)$, $y=m/T_{\ast }$. The expression just derived is also the physical distance $L_{\ast }=a_{\ast }\chi$ at observation. Thus it equals a comoving distance only when $a_{\ast }=1$ is used, as in the preceding part. To express it in today's convention $a_0=1$, multiply $L_{\ast }$ by $1+z_{\ast }$. This is the accumulated free-streaming distance in a radiation-dominated universe.

Write ${\omega }_M={\Omega }_M{h}^2$, where $h$ here is the dimensionless present Hubble constant. The supplied equality redshift gives $1+z_{\mathrm{eq}}\simeq 2.5\times 10^4{\omega }_M$, and $T_{\mathrm{eq}}=T_0(1+z_{\mathrm{eq}})$. For $m=30\mathrm{eV}$ the requested dimensionless ratio isThe Hubble rate in this formula is the radiation-era extrapolation used in deriving it. A unique numerical ratio cannot be chosen without ${\omega }_M$.

The printed CMB temperature contains a factor-of-ten error. Its $2.35\mathrm{meV}$ means $2.35\times 10^{-3}\mathrm{eV}$, whereas the cosmic microwave background temperature in energy units is about $2.35\times 10^{-4}\mathrm{eV}=0.235\mathrm{meV}$, corresponding to $2.725\mathrm{K}$. The temperature is supported by NASA's CMB measurements. Using the printed value gives $y_{\mathrm{eq}}\simeq 0.5106/{\omega }_M$; using the corrected value gives $y_{\mathrm{eq}}\simeq 5.106/{\omega }_M$. For the illustrative choice ${\Omega }_M=1,h=0.5$, so ${\omega }_M=0.25$, these give respectively $T_{\mathrm{eq}}\simeq 14.7$ and $1.47\mathrm{eV}$, and distance-to-Hubble-radius ratios about $0.716$ and $0.182$. This benchmark is an assumption for the estimate, not an extra parameter supplied by the paper.

For the present-day comoving length, use $H_r(z_{\mathrm{eq}})=H_0\sqrt{{\Omega }_M}(1+z_{\mathrm{eq}}{)}^{3/2}$, the radiation contribution extrapolated to equality. Restoring $c$ for the conversion and using $c/H_0\simeq 2998h^{-1}\mathrm{Mpc}$ givesFor the same benchmark, this is about $54\mathrm{Mpc}$ using the printed temperature and $14\mathrm{Mpc}$ using the corrected temperature, in the prescribed representative-momentum radiation-era model.

At equality matter is no longer negligible, so these are transition-era estimates. The actual total $H_{\mathrm{eq}}$ is $\sqrt{2}{H}_r(z_{\mathrm{eq}})$; inserting it in the simple normalization would lower the length by $\sqrt{2}$, but would not constitute an exact integration through equality. An exact matter-plus-radiation treatment instead replaces the radiation integral bywith $p_0=T_0$ in the assumed model. This makes the order-one equality correction explicit. Real neutrino physical momenta in an FRW universe are distributed and their temperature need not equal the photon temperature; the calculation above follows the specified $p=T_r$ idealization.