Editing Cosmological constant (section)

== Equation ==
[[File:121236 NewPieChart320.png|thumb|Estimated ratios of [[dark matter]] and dark energy (which may be the cosmological constant) in the universe. This image is made by NASA using the 9 year WMAP data. This is the final WMAP release.]]
The cosmological constant {{math|Λ}} appears in the [[Einstein field equations]] in the form
<math display=block>R_{\mu \nu} - \tfrac{1}{2} R\, g_{\mu \nu} + \Lambda g_{\mu \nu} = \kappa T_{\mu \nu} ,</math>
where [[Ricci tensor|the Ricci tensor]] {{math|''R<sub>μν</sub>''}}, Ricci scalar {{math|''R''}} and the [[metric tensor (general relativity)|metric tensor]] {{math|''g<sub>μν</sub>''}} describe the structure of [[spacetime]], the [[stress–energy tensor]] {{math|''T<sub>μν</sub>''}} describes the energy density, momentum density and stress at that point in spacetime, and {{math|1=''κ'' = 8''πG''/''c''{{sup|4}}}}. The [[gravitational constant]] {{mvar|G}} and the [[speed of light]] {{mvar|c}} are universal constants. When {{math|Λ}} is zero, this reduces to the field equation of general relativity usually used in the 20th&nbsp;century. When {{math|''T<sub>μν</sub>''}} is zero, the field equation describes empty space (a [[vacuum]]).

The cosmological constant has the same effect as an intrinsic [[energy density]] of the vacuum, ''{{mvar|ρ}}''{{sub|vac}} (and an associated [[pressure]]). In this context, it is commonly moved to the right-hand side of the equation using {{math|1=Λ = ''κρ''{{sub|vac}}}}. It is common to quote values of energy density directly, though still using the name "cosmological constant". The dimension of {{math|Λ}} is generally understood as length{{sup|−2}}.

Using the Planck units, and the value evaluated in 2025 for the [[Hubble's law|Hubble constant]] ''{{mvar|H}}''<sub>0</sub> = {{val|76.5|2.2|u=(km/s)/Mpc}} = {{val|2.48|0.07|u=s-1|e=-18}},<ref>{{Cite journal |last1=Scolnic |first1=Daniel |last2=Riess |first2=Adam G. |last3=Murakami |first3=Yukei S. |last4=Peterson |first4=Erik R. |last5=Brout |first5=Dillon |last6=Acevedo |first6=Maria |last7=Carreres |first7=Bastien |last8=Jones |first8=David O. |last9=Said |first9=Khaled |last10=Howlett |first10=Cullan |last11=Anand |first11=Gagandeep S. |date=2025-01-15 |title=The Hubble Tension in Our Own Backyard: DESI and the Nearness of the Coma Cluster |journal=The Astrophysical Journal Letters |volume=979 |issue=1 |pages=L9 |doi=10.3847/2041-8213/ada0bd |doi-access=free |arxiv=2409.14546 |bibcode=2025ApJ...979L...9S |issn=2041-8205}}</ref> {{math|Λ}} has the value of <math display="block">\begin{align}
\Lambda = 3\, \left( \frac{\,H_0\,}{c} \right)^2 \Omega_\Lambda &= 1.4657 \times 10^{-52}\ \text{m}^{-2} \\
&= 3.827 \times 10^{-123} \,l_{\text{P}}^{-2}
\end{align}</math> where <math display="inline">l_{\text{P}}</math> is the [[Planck length]]. A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, and vice versa. If the energy density is positive, the associated negative pressure will drive an accelerated expansion of the universe, as observed. (See ''[[Dark energy]]'' and ''[[Cosmic inflation]]'' for details.)

=== Density parameter Ω ===
The dimensionless density parameter Ω represents the ratio of the actual density of a component of the universe to the critical density. The total density parameter for a flat universe (concluded by [https://wmap.gsfc.nasa.gov/universe/uni_shape.html WMAP]), can be expressed as:

<math display="block">\Omega_\text{total} = \Omega_m + \Omega_{\Lambda} + \Omega_k = 1</math>

Where:

* Ω<sub>''m''</sub> is the matter density parameter (including both baryonic and dark matter);
* Ω<sub>Λ</sub> is the density parameter for dark energy (cosmological constant); and
* Ω<sub>''k''</sub> describes the curvature of the universe which is 0 in a flat universe.

Instead of the cosmological constant itself, cosmologists often refer to the ratio between the energy density due to the cosmological constant and the [[Friedmann equations#Density parameter|critical density]] of the universe, the tipping point for a sufficient density to stop the universe from expanding forever (the dark energy density parameter). This ratio is estimated to be 0.714, according to results published by the [[Planck Collaboration]] in 2018 and clearly mentioned on [https://wmap.gsfc.nasa.gov/universe/uni_matter.html WMAP].<ref name="Planck-20182">{{harvp|The Planck Collaboration|2020}}.</ref> More intuitively, this parameter could be described as the fraction of the universe that is made up of dark energy. Note that this value changes over time: The critical density changes with [[cosmological time]] but the energy density due to the cosmological constant remains unchanged throughout the history of the universe, because the amount of dark energy increases as the universe grows but the amount of matter does not.<ref>{{Cite web |last=Siegel |first=Ethan |title=Dark Energy May Not Be A Constant, Which Would Lead To A Revolution In Physics |url=https://www.forbes.com/sites/startswithabang/2019/01/31/dark-energy-may-not-be-a-constant-which-would-lead-to-a-revolution-in-physics/ |access-date=2023-09-10 |website=Forbes |language=en}}</ref>{{sfnp|Peebles|Ratra|2003}}<ref>{{Cite book |last=Davies |first=Paul |title=What's Eating the Universe?: And Other Cosmic Questions |publisher=Penguin Books Australia |year=2021 |isbn=9780141993720 |language=en}}</ref>

=== Equation of state ===
Another ratio that is used by scientists is the [[Equation of state (cosmology)|equation of state]], usually denoted {{mvar|w}}, which is the ratio of pressure that dark energy puts on the universe to the energy per unit volume.<ref>{{harvp|Brumfiel|2007|p=246}}.</ref> This ratio is {{nowrap|{{mvar|w}} {{=}} −1}} for the cosmological constant used in the Einstein equations; alternative time-varying forms of vacuum energy such as [[quintessence (physics)|quintessence]] generally use a different value. The value {{mvar|w}} = {{val|-1.028|0.032}}, measured by the Planck Collaboration (2018)<ref name="Planck-2018">{{harvp|The Planck Collaboration|2020}}.</ref> is consistent with {{val|-1}}, assuming {{mvar|w}} does not change over cosmic time.