Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 70 / 2

Normalize $a_0=1$ and use the open dust model, with no cosmological constant and negligible present radiation. The Friedmann equation at the present epoch reads $H_0^2=H_0^2{\Omega }_m-k{c}^2$. Since $k=-r_{\mathrm{curv}}^{-2}$, the curvature radius of an open FLRW universe isIn the metric's $c=1$ convention the numerator is one. With $a_0=1$ this is also the present physical curvature radius; at scale factor $a$ the physical radius is $a{r}_{\mathrm{curv}}$. Inferring curvature from ${\Omega }_m$ alone would require this matter-only assumption; an additional dark-energy density would also enter the present Friedmann equation.

At fixed conformal time and fixed radial coordinate, the metric gives transverse physical separation $D=a({\tau }_1)r_1\theta$ to first order in the angle. Thus the angular diameter distance isThe coordinate $r_1$ is the transverse comoving radius appearing in the angular metric coefficient. It is not the radial geodesic distance $r_{\mathrm{curv}}\mathrm{arsinh}(r_1/r_{\mathrm{curv}})$; replacing one by the other would omit the curvature dependence of angular projection.

Adiabatic photon cooling gives $a_{\ast }\simeq 1/1200$. During matter domination, $H(a)\simeq H_0\sqrt{{\Omega }_m}{a}^{-3/2}$, so the comoving sound distance traveled before last scattering isMultiplication by $a_{\ast }$ gives the physical sound horizon radius:This is a radius, or one-way sound-propagation length, rather than its diameter. The integral treats the early radiation interval as negligible under the specified late-equality approximation; it also neglects curvature at last scattering, requiring ${\Omega }_m{a}_{\ast }^{-3}\gg (1-{\Omega }_m)a_{\ast }^{-2}$. A very small matter fraction would violate those approximations.

A radial null ray obeys $cd\tau =dr/\sqrt{1+r^2/r_{\mathrm{curv}}^2}$. HenceFor the open matter solution, let $\beta =\sqrt{1-{\Omega }_m}$. The Friedmann equation gives $a^{\prime }=H_0\sqrt{{\Omega }_{m}a+{\beta }^2{a}^2}$, and the age in conformal time isThusHere the stated approximation ${\tau }_{\ast }\ll {\tau }_0$ was used only in the last distance estimate. The flat limit is smooth: ${\tau }_0\to 2/H_0$ and $r_{\ast }\to 2c/H_0$. Retaining ${\tau }_{\ast }$ in the preceding null-ray formula gives the finite-emission-time correction.

Divide the physical sound radius by the angular diameter distance $a_{\ast }{r}_{\ast }$. The sound-horizon angle in an open matter universe isThe characteristic acoustic multipole is approximately half a wavelength across this angle:Thus negative curvature moves the acoustic feature to smaller angular scales and larger multipoles. The first temperature peak is of order $200/\sqrt{{\Omega }_m}$ in this approximation; precise peak positions include the sound-speed history, radiation effects and acoustic phase shifts, so the acoustic-scale multipole is not an exact peak-location formula.

For instantaneous complete hydrogen reionization, take $n_{e0}={\Omega }_b{\rho }_{\mathrm{crit},0}/(m_p{c}^2)=3{\Omega }_b{H}_0^2/(8\pi G{m}_p)$ and $n_e(z)=n_{e0}(1+z{)}^3$. Helium or incomplete ionization would multiply this by the appropriate free-electron count per baryon. In the matter-plus-curvature model,The reionization optical depth in an open matter universe is thereforeSet $u=\sqrt{1+{\Omega }_{m}z}$ and ${\beta }^2=1-{\Omega }_m$. The integral is $2{\Omega }_m^{-2}[u^3/3-{\beta }^{2}u{]}_1^{u_{\mathrm{re}}}$, givingAt high redshift where matter dominates over curvature, this approaches $\kappa \simeq {\Omega }_b{C}_H(1+z_{\mathrm{re}}{)}^{3/2}/\sqrt{{\Omega }_m}$. The exact expression should be retained when ${\Omega }_m{z}_{\mathrm{re}}$ is not large.

For ${\Omega }_m=1$ the preceding result reduces to $\kappa ={\Omega }_b{C}_H[(1+z_{\mathrm{re}}{)}^{3/2}-1]$. Using $C_H=0.032$ and ${\Omega }_b=0.05$ givesAt fixed $H_0$ and ${\Omega }_b$, a lower matter density lowers $H(z)$ relative to the flat dust value at positive redshift. The same electron density then persists for a longer scattering time, increasing the optical depth. Hence the reionization redshift giving unit depth is lower in an open universe.

In the high-redshift matter regime the scaling is $1+z_{\mathrm{re}}\propto {\Omega }_m^{1/3}$ at fixed optical depth; for ${\Omega }_m=0.3$ this suggests a redshift near fifty rather than seventy. This comparison holds the baryon density fixed, not the baryon fraction ${\Omega }_b/{\Omega }_m$, and is not an estimate of the actual observed reionization epoch.