File:LookBackFromRedshiftEqns.png - Revision history

42.book.addict at 17:56, 11 February 2026

2026-02-11T17:56:40Z

2025-12-05T20:11:30Z

2023-11-14T21:12:12Z

Indent

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Cartoon quote

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Formatting

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third line

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line break

2023-11-14T20:23:48Z

semicolons were correct

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text

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LaTeX source

← Older revision		Revision as of 17:56, 11 February 2026
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	[[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)
−	[[Category:Helper comic images]	+
		+	[[Category:Helper comic images]]

← Older revision		Revision as of 20:11, 5 December 2025
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	[[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)
		+	[[Category:Helper comic images]

@@ Line 13: / Line 13: @@
 And as Unicode plain text:
 :<nowiki>ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀</nowiki>
-:then a miracle occurs... [apologies to Sydney Harris]
+:&nbsp;&nbsp;&nbsp;&nbsp;then a miracle occurs... [apologies to Sydney Harris]
 :&nbsp;&nbsp;<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>
 :lookBackTime(z) = ageAtRedshift(0) - ageAtRedshift(z)

@@ Line 13: / Line 13: @@
 And as Unicode plain text:
 :<nowiki>ageAtRedshift(z) = ∫[z to ∞] (1 / ((1 + z') * √(ΩΛ + Ωm * (1 + z')^3))) dz' * 977.8 / H₀</nowiki>
 :&nbsp;&nbsp;<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>
 :lookBackTime(z) = ageAtRedshift(0) - ageAtRedshift(z)

@@ Line 13: / Line 13: @@
 And as Unicode plain text:
 :<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>
+:&nbsp;&nbsp;<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>
 :lookBackTime(z) = ageAtRedshift(0) - ageAtRedshift(z)
 [[User:Liv2splain|Liv2splain]] ([[User talk:Liv2splain|talk]]) 19:44, 14 November 2023 (UTC)

@@ Line 7: / Line 7: @@
 \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}.
+= {}_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)
 </nowiki></pre></tt>
 And as Unicode plain text:
-:<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 = 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)

← Older revision		Revision as of 19:44, 14 November 2023
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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.		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.
		+
		+	LaTeX source:
		+	<tt><pre><nowiki>
		+	\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)
		+	</nowiki></pre></tt>
		+
		+	[[User:Liv2splain\|Liv2splain]] ([[User talk:Liv2splain\|talk]]) 19:44, 14 November 2023 (UTC)