Difference between revisions of "File:LookBackFromRedshiftEqns.png"
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− | :<nowiki>ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀ = ₂F₁(1/2, 1/2; 3/2; -ΩΛ / (Ωm * (1 + z)^3)) * (2 * 977.8) / (3 * √(Ωm) * (1 + z)^(3/2) * H₀) Gyr</nowiki> | + | :<nowiki>ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀</nowiki> |
+ | :<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> | ||
[[User:Liv2splain|Liv2splain]] ([[User talk:Liv2splain|talk]]) 19:44, 14 November 2023 (UTC) | [[User:Liv2splain|Liv2splain]] ([[User talk:Liv2splain|talk]]) 19:44, 14 November 2023 (UTC) |
Revision as of 21:07, 14 November 2023
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
- = ₂F₁(1/2, 1/2; 3/2; -ΩΛ / (Ωm * (1 + z)^3)) * (2 * 977.8) / (3 * √(Ωm) * (1 + z)^(3/2) * H₀) Gyr
Liv2splain (talk) 19:44, 14 November 2023 (UTC)
File history
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 19:00, 14 November 2023 | 740 × 269 (73 KB) | Liv2splain (talk | contribs) | Humorous version with Sydney Harris cartoon quote | |
18:23, 14 November 2023 | 738 × 217 (59 KB) | Liv2splain (talk | contribs) | More common representation of Gaussian hypergeometric function 2F1 | ||
18:08, 14 November 2023 | 742 × 218 (59 KB) | Liv2splain (talk | contribs) | Save horizontal space for legibility in thumbnail | ||
17:55, 14 November 2023 | 788 × 216 (50 KB) | Liv2splain (talk | contribs) | Label units | ||
10:54, 13 November 2023 | 747 × 224 (59 KB) | Liv2splain (talk | contribs) | Equations for calculating the look-back time from redshift, for 2853: Redshift, by ChatGPT. |
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