Plot of the evolution of the size of the universe (scale parameter <i>a</i>) over time (in billion years, Gyr). Different models are shown, which are all solutions to the <a href="https://en.wikipedia.org/wiki/Friedmann_equations#Density_parameter" class="extiw" title="w:Friedmann equations">Friedmann equations</a> with different parameters. The evolution is governed by the equation
<dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}=H_{0}^{2}\left({\frac {\Omega _{r}}{a^{4}}}+{\frac {\Omega _{m}}{a^{3}}}+{\frac {\Omega _{k}}{a^{2}}}+\Omega _{\Lambda }\right)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mrow><mo>(</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow class="MJX-TeXAtom-ORD"><mover><mi>a</mi><mo>˙<!-- ˙ --></mo></mover></mrow><mi>a</mi></mfrac></mrow><mo>)</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>=</mo><msubsup><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mrow class="MJX-TeXAtom-ORD"><mfrac><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi></mrow></msub><msup><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup></mfrac></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub><msup><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup></mfrac></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msub><msup><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup></mfrac></mrow><mo>+</mo><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Λ<!-- Λ --></mi></mrow></msub></mrow><mo>)</mo></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}=H_{0}^{2}\left({\frac {\Omega _{r}}{a^{4}}}+{\frac {\Omega _{m}}{a^{3}}}+{\frac {\Omega _{k}}{a^{2}}}+\Omega _{\Lambda }\right)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/419eaae071f4b1c490f22ef2ebcf5d043474a66c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:39.458ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\dot {a}}{a}}\right)^{2}=H_{0}^{2}\left({\frac {\Omega _{r}}{a^{4}}}+{\frac {\Omega _{m}}{a^{3}}}+{\frac {\Omega _{k}}{a^{2}}}+\Omega _{\Lambda }\right)}"></span> .</dd></dl><p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{r}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{r}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db9489e0c004be602dd7422bb6c6cd15ba116241" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.652ex; height:2.509ex;" alt="{\displaystyle \Omega _{r}}"></span> is the radiation density, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{m}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{m}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07efe3d3121e264d3736a69159f92ced3f8e26d8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.353ex; height:2.509ex;" alt="{\displaystyle \Omega _{m}}"></span> the matter density, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{k}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{k}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b18029871270775ecb905d24b7fc1a0016930965" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.767ex; height:2.509ex;" alt="{\displaystyle \Omega _{k}}"></span> the curvature parameter and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\Lambda }}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Λ<!-- Λ --></mi></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{\Lambda }}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f6a4cef7abd563d3e0b2f2ee6fff9f8560be454" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.051ex; height:2.509ex;" alt="{\displaystyle \Omega _{\Lambda }}"></span> the dark energy, all normalized such that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{r}+\Omega _{m}+\Omega _{k}+\Omega _{\Lambda }=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>r</mi></mrow></msub><mo>+</mo><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub><mo>+</mo><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msub><mo>+</mo><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Λ<!-- Λ --></mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{r}+\Omega _{m}+\Omega _{k}+\Omega _{\Lambda }=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ddc0c637f00bc2b7f7fd2467589714099cf0da4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:24.605ex; height:2.509ex;" alt="{\displaystyle \Omega _{r}+\Omega _{m}+\Omega _{k}+\Omega _{\Lambda }=1}"></span> represents the fact that today's expansion rate is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{0}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>H</mi><mrow class="MJX-TeXAtom-ORD"><mn>0</mn></mrow></msub></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle H_{0}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43910602a221b7a4c373791f94793e3008622070" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:2.985ex; height:2.509ex;" alt="{\displaystyle H_{0}}"></span>.<br>
Plotted parameter sets:
</p>
<ul><li><a href="https://en.wikipedia.org/wiki/De_Sitter_universe" class="extiw" title="w:De Sitter universe">De Sitter universe</a>: Only dark energy: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\Lambda }=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Λ<!-- Λ --></mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{\Lambda }=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56a71e9acfc18a0f299bad9df2b6c8bd31017d1f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.312ex; height:2.509ex;" alt="{\displaystyle \Omega _{\Lambda }=1}"></span></li>
<li><a href="https://en.wikipedia.org/wiki/Lambda-CDM_model" class="extiw" title="w:Lambda-CDM model">Lambda-CDM model</a>: The model that fits the observations best: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{m}=0.3}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub><mo>=</mo><mn>0.3</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{m}=0.3}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e965b519448db113f72933d31fddccf8a8ae2bab" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.423ex; height:2.509ex;" alt="{\displaystyle \Omega _{m}=0.3}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{\Lambda }=0.7}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">Λ<!-- Λ --></mi></mrow></msub><mo>=</mo><mn>0.7</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{\Lambda }=0.7}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef52578973070f262d55fc38aca4690a906bafb8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:9.121ex; height:2.509ex;" alt="{\displaystyle \Omega _{\Lambda }=0.7}"></span></li>
<li>An empty universe (no relevant contributions of matter, radiation, dark energy) with negative curvature: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{k}=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{k}=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/126197f056447dacda02c7d0cab669f1fbe91e20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.028ex; height:2.509ex;" alt="{\displaystyle \Omega _{k}=1}"></span></li>
<li><a href="https://en.wikipedia.org/wiki/Einstein%E2%80%93de_Sitter_universe" class="extiw" title="w:Einstein–de Sitter universe">Einstein–de Sitter universe</a>: A flat universe dominated by cold matter: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{m}=1}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub><mo>=</mo><mn>1</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{m}=1}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/158dc3c4168c36b5d82a0779a7acd82aa186f06c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.614ex; height:2.509ex;" alt="{\displaystyle \Omega _{m}=1}"></span></li>
<li>A closed Friedmann model: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{m}=6}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>m</mi></mrow></msub><mo>=</mo><mn>6</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{m}=6}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340dc04c1c23b95940f3098755f8ca56419924d2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.614ex; height:2.509ex;" alt="{\displaystyle \Omega _{m}=6}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{k}=-5}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi mathvariant="normal">Ω<!-- Ω --></mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi></mrow></msub><mo>=</mo><mo>−<!-- − --></mo><mn>5</mn></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \Omega _{k}=-5}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9e23f3f7182ad4511b63eafd99f4a7194295a71" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:8.836ex; height:2.509ex;" alt="{\displaystyle \Omega _{k}=-5}"></span></li></ul>
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