File:LookBackFromRedshiftEqns.png

Equations for calculating the look-back time from redshift, for 2853: Redshift, by ChatGPT. See Talk:2853: Redshift#Calculation.

I was unable to resist the urge to include the Sydney Harris cartoon quote to emphasize how amazing it is that a closed-form solution of that integral exists.

LaTeX source:

\text{ageAtRedshift}(z) = \int_z^{\infty} \frac{1}{(1 + z') \cdot \sqrt{\Omega_{\Lambda} + \Omega_{m} \cdot (1 + z')^3}} \, dz' \cdot \frac{977.8}{H_0}
\text{then a miracle occurs... [apologies to Sydney Harris]}
= {}_2F_1\left(\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; -\frac{\Omega_{\Lambda}}{\Omega_{m} \cdot (1 + z)^3}\right) \cdot \frac{2 \cdot 977.8}{3 \cdot \sqrt{\Omega_{m}} \cdot (1 + z)^{3/2} \cdot H_0} \, \text{Gyr}.
\text{lookBackTime}(z) = \text{ageAtRedshift}(0) - \text{ageAtRedshift}(z)

And as Unicode plain text:

ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀

    then a miracle occurs... [apologies to Sydney Harris]

  = ₂F₁(1/2, 1/2; 3/2; -ΩΛ / (Ωm * (1 + z)^3)) * (2 * 977.8) / (3 * √(Ωm) * (1 + z)^(3/2) * H₀) Gyr.

lookBackTime(z) = ageAtRedshift(0) - ageAtRedshift(z)

Liv2splain (talk) 19:44, 14 November 2023 (UTC)