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| Equations for calculating the look-back time from redshift, for [[2853: Redshift]], by ChatGPT. See [[Talk:2853: Redshift#Calculation]]. | | Equations for calculating the look-back time from redshift, for [[2853: Redshift]], by ChatGPT. See [[Talk:2853: Redshift#Calculation]]. |
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− | I was unable to resist the urge to include the [https://www.researchgate.net/profile/Michael-Wade-5/publication/302632920/figure/fig2/AS:751645805789184@1556217733527/Then-a-Miracle-Occurs-Copyrighted-artwork-by-Sydney-Harris-Inc-All-materials-used-with_W640.jpg Sydney Harris cartoon] quote to emphasize how amazing it is that a closed-form solution of that integral exists.
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− | LaTeX source:
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− | <tt><pre><nowiki>
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− | \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}
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− | \text{then a miracle occurs... [apologies to Sydney Harris]}
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− | = {}_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}.
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− | \text{lookBackTime}(z) = \text{ageAtRedshift}(0) - \text{ageAtRedshift}(z)
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− | </nowiki></pre></tt>
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− | And as Unicode plain text:
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− | :<nowiki>ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀</nowiki>
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− | : then a miracle occurs... [apologies to Sydney Harris]
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− | : <nowiki>= ₂F₁(1/2, 1/2; 3/2; -ΩΛ / (Ωm * (1 + z)^3)) * (2 * 977.8) / (3 * √(Ωm) * (1 + z)^(3/2) * H₀) Gyr.</nowiki>
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− | :lookBackTime(z) = ageAtRedshift(0) - ageAtRedshift(z)
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− | [[User:Liv2splain|Liv2splain]] ([[User talk:Liv2splain|talk]]) 19:44, 14 November 2023 (UTC)
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