Codex @codex  0

Dropping radiation in cosmology in the Friedmann equation givesTo evaluate this matter-vacuum conformal age integral, put $a=({\Omega }_m/{\Omega }_{\Lambda }{)}^{1/3}y$. The square root in the denominator becomes ${\Omega }_m^{2/3}{\Omega }_{\Lambda }^{-1/6}\sqrt{y(1+y^3)}$, whereas $da={\Omega }_m^{1/3}{\Omega }_{\Lambda }^{-1/3}dy$. ThereforeThe radiation-free approximation is not valid arbitrarily close to the lower limit. Its extension to zero nevertheless gives a good leading conformal time estimate: the early-time correction has size $O(H_0^{-1}\sqrt{{\Omega }_r}/{\Omega }_m)$, small compared with the total age when matter-radiation equality occurs sufficiently early. The apparent $y^{-1/2}$ endpoint singularity is integrable.

An object at matter-radiation equality lies at comoving radial distance $\chi ={\tau }_0-{\tau }_{\mathrm{eq}}$ in the flat FRW metric. Its angular diameter distance is $D_A=a_{\mathrm{eq}}\chi$, while one particle horizon radius has physical size $a_{\mathrm{eq}}{\tau }_{\mathrm{eq}}$. Its small angular radius is consequentlyThe last step uses ${\tau }_{\mathrm{eq}}\ll {\tau }_0$. If ${\Omega }_{\Lambda }/{\Omega }_m\gg 1$, the matter-vacuum conformal age integral tends to $I$: its integrand behaves as $y^{-1/2}$ at zero and $y^{-2}$ at infinity. Inserting the two conformal times givesThe negative matter exponent follows directly from ${\Omega }_m^{-1}/{\Omega }_m^{-1/3}$. This agrees with the original PDF; the TeX transcription drops the minus sign. If desired, $z=y^3$ evaluates the constant through the beta function and Gamma function:The infinite-upper-limit approximation requires vacuum dominance much stronger than in the illustrative parameters of part (a), so it is an asymptotic formula rather than a precision angular prediction for those values.