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\section*{cosmological constant}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{algebraic_quantum_field_theory}{}\paragraph*{{Algebraic Quantum Field Theory}}\label{algebraic_quantum_field_theory}

[[!include AQFT and operator algebra contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{details}{Details}\dotfill \pageref*{details} \linebreak
\noindent\hyperlink{InClassicalGravity}{In classical gravity}\dotfill \pageref*{InClassicalGravity} \linebreak
\noindent\hyperlink{InPerturbativeQuantumGravity}{In perturbative quantum gravity}\dotfill \pageref*{InPerturbativeQuantumGravity} \linebreak
\noindent\hyperlink{InInhomogeneousCosmology}{In inhomogeneous cosmology}\dotfill \pageref*{InInhomogeneousCosmology} \linebreak
\noindent\hyperlink{Observation}{Observation}\dotfill \pageref*{Observation} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{observation_2}{Observation}\dotfill \pageref*{observation_2} \linebreak
\noindent\hyperlink{ReferencedInpAQFT}{In pAQFT}\dotfill \pageref*{ReferencedInpAQFT} \linebreak
\noindent\hyperlink{ReferencesFromHigherCurvatureCorrections}{From higher curvature corrections}\dotfill \pageref*{ReferencesFromHigherCurvatureCorrections} \linebreak
\noindent\hyperlink{from_inhomogeneous_cosmology}{From inhomogeneous cosmology}\dotfill \pageref*{from_inhomogeneous_cosmology} \linebreak
\noindent\hyperlink{CCProblemReferences}{The ``cosmological constant problem''}\dotfill \pageref*{CCProblemReferences} \linebreak
\noindent\hyperlink{ReferencesInStringTheory}{In string theory}\dotfill \pageref*{ReferencesInStringTheory} \linebreak
\noindent\hyperlink{ReferencesStringTheoryVanishingCC}{For fundamentally vanishing cc (``Witten's Dark Fantasy'')}\dotfill \pageref*{ReferencesStringTheoryVanishingCC} \linebreak
\noindent\hyperlink{for_fundamentally_nonvanishing_cc}{For fundamentally non-vanishing cc}\dotfill \pageref*{for_fundamentally_nonvanishing_cc} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In theories of [[gravity]] and [[cosmology]] a \emph{cosmological constant} refers to an [[energy]] contribution associated with the [[vacuum]] itself.

This is called a \emph{cosmological constant} because the simplest way to see this effect in [[theory (physics)|theory]] is by a summand to the [[Einstein-Hilbert action]] which is proportional to the [[volume]] of [[spacetime]], with proportionality factor some ``constant'' $\lambda$, as in \eqref{EinsteinHilbertWithCosmologicalConstant} below.

See below

\begin{itemize}%
\item \emph{\hyperlink{InClassicalGravity}{In classical gravity}}

\end{itemize}
More generally, almost-constant contributions to matter fields may effectively have the same kind of effect.

Specifically in [[perturbative quantum field theory]] [[AQFT on curved spacetimes|on curved spacetimes]] the [[vacuum expectation value]] of the [[stress-energy tensor]] of the various [[matter]]-fields receives constant or essentially constant contributions, notably from [[renormalization]] freedom (e.g. \hyperlink{Hack15}{Hack 15, section 3.2.1}).

See below

\begin{itemize}%
\item \emph{\hyperlink{InPerturbativeQuantumGravity}{In perturbative quantum gravity}}

\end{itemize}
In [[perturbative string theory]] the [[string perturbation series]] associated with a [[2d SCFT]] is (supposedly) UV-finite and hence has ``chosen its own [[renormalization]]'' already, hence here the cosmological constant may in principle be read off from a choice of [[perturbative string theory vacuum]]. See below

\begin{itemize}%
\item \emph{\hyperlink{ReferencesInStringTheory}{In string theory}}

\end{itemize}
Observation shows that the effective cosmological constant of the [[observable universe]] is comparatively small but [[positive real number|positive]], see

\begin{itemize}%
\item \emph{\hyperlink{Observation}{Observation}}.

\end{itemize}
\hypertarget{details}{}\subsection*{{Details}}\label{details}

\hypertarget{InClassicalGravity}{}\subsubsection*{{In classical gravity}}\label{InClassicalGravity}

In an [[action functional]] on a space of [[pseudo-Riemannian manifolds]] -- such as the [[Einstein-Hilbert action]] functional for [[gravity]] -- a \textbf{cosmological constant} is a term proportional to the [[volume]]

\begin{displaymath}
S_{cc} \;\colon\;  (X,g) \mapsto \lambda \int_X dvol_g
  \,,
\end{displaymath}
where $\lambda \in \mathbb{R}$ is the \emph{cosmological constant} .

For instance, pure Einstein-Hilbert gravity with cosmological constant (and no other fields) is given by the functional

\begin{equation}
S_{EH} + S_{cc} :  (X,g) \mapsto \int_X R\, dvol + \lambda \int_X d vol_g
  \,,
\label{EinsteinHilbertWithCosmologicalConstant}\end{equation}
Generically it happens that one considers action functionals where $\lambda$ is in fact not a constant, but a function of other fields $\phi$ on $X$.

\begin{displaymath}
S \;\colon\; (X,g,\phi) \mapsto \int_X \lambda(\phi) dvol_g
  \,.
\end{displaymath}
In this context those solutions to the [[Euler-Lagrange equation]]s are of interest in which $\lambda(\phi)$ happens to be exactly or approximately constant. Many such models of not-really-constant-but-effectively-constant terms proportional to the volume are being proposed and considered in attempts to explain observed or speculated dynamics of the cosmos.

See in particular at \emph{[[FRW model]]} for the role of the cosmological constant in homogeneous and isotropic models as in the [[standard model of cosmology]]. In that context the cosmological constant is also called the \emph{dark energy} (density), which makes up about 70\% of the energy density of the [[observable universe]] (the rest being [[dark matter]]) and a comparatively little bit of [[baryon|baryonic]] [[matter]].

\hypertarget{InPerturbativeQuantumGravity}{}\subsubsection*{{In perturbative quantum gravity}}\label{InPerturbativeQuantumGravity}

In [[perturbative quantum field theory]] [[QFT on curved spacetime|on curved spacetimes]] the cosmological constant receives contributions from the [[vacuum expectation value]] of the [[stress-energy tensor]] of the [[matter]] [[field (physics)|fields]]. There is [[renormalization]]-freedom in this contribution (\hyperlink{Wald78}{Wald 78}, \hyperlink{TichyFlanagan98}{Tichy-Flanagan 98} \hyperlink{Moretti01}{Moretti 01}).

Explicitly for [[FRW models]] this is discussed in (\hyperlink{DappiaggiFredenhagenPinamonti08}{Dappiaggi-Fredenhagen-Pinamonti 08}, \hyperlink{DappiagiHackMollerPinamonti10}{Dappiagi-Hack-Moeller-Pinamonti 10}). Specifically for the [[standard model of cosmology]] see (\hyperlink{Hack13}{Hack 13, around (4)}).

A useful review is in (\hyperlink{Hack15}{Hack 15, section 3.2.1}).

This means that apart from the freedom of choosing a classical comsological constant in the [[Einstein-Hilbert action]] as above, its \href{A+first+idea+of+quantum+field+theory#InteractingQuantumFields}{perturbative quantization} ([[perturbative quantum gravity]]) introduces [[renormalization]] freedom to the value of the cosmological constant.

The folklore discussion of the ``cosmological constant problem'' (see the references \hyperlink{CCProblemReferences}{below}) tends not to take this freedom in the theory into account (see the discussion at \emph{[[naturalness]]}).

\hypertarget{InInhomogeneousCosmology}{}\subsubsection*{{In inhomogeneous cosmology}}\label{InInhomogeneousCosmology}

It has been suggested the observed cosmological constant/dark energy may be but an artifact of the overly idealistic approximation of cosmic homogeneity, and that a more accurate [[inhomogeneous cosmology]] would not need to assume any dark energy (e.g. \hyperlink{Buchert07}{Buchert 07}, \hyperlink{Buchert11}{Buchert 11}, \hyperlink{BucherRasanen11}{Buchert-Rasanen 11}, \hyperlink{Scharf13}{Scharf 13}).

A seminal argument that it \emph{is} consistent to neglect cosmic inhomogeneity due to (\hyperlink{GreenWald10}{Green-Wald 10}, \hyperlink{GreenWald13}{Green-Wald 13}), has been called into question in \hyperlink{BuchertEtAl15}{Buchert et al. 15}, where it is concluded that the question is more subtle and remains open. Recent review is in \hyperlink{BelejkoKorzynski16}{Belejko-Korzyński 16}.

If the apparent small positive [[cosmological constant]] were but an artifact of neglecting backreaction of inhomegeneities, some theoretical puzzlements regarding [[quantum gravity]] on [[de Sitter spacetimes]] would disappear (see \href{de+Sitter+spacetime#Rajaraman16}{Rajaraman 16} for general discussion and \hyperlink{DanielssonVanRiet18}{Danielsson-VanRiet 18, p. 27} for discussion of [[perturbative string theory vacua]]).

\hypertarget{Observation}{}\subsubsection*{{Observation}}\label{Observation}



(e.g. \hyperlink{Einasto09}{Einasto 09, fig 17})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[FRW model]]


\item [[cosmology]]


\item [[energy]]

\begin{itemize}%
\item [[matter]], [[radiation]], \textbf{cosmological constant}

\end{itemize}


\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{observation_2}{}\subsubsection*{{Observation}}\label{observation_2}

Cautioning against the interpretation of type Ia [[supernovae]] as indicative of a small positive cosmoplogical constant ([[de Sitter spacetime]]) as in the current [[standard model of cosmology]] includes the following (see also at [[inhomogeneous cosmology]]):

\begin{itemize}%
\item [[Subir Sarkar]], \emph{Is the evidence for dark energy secure?}, Gen. Rel. Grav. 40:269-284, 2008 (\href{https://arxiv.org/abs/0710.5307}{arXiv:0710.5307})


\item J. T. Nielsen, A. Guffanti \& [[Subir Sarkar]], \emph{Marginal evidence for cosmic acceleration from Type Ia supernovae}, Nature Scientific Reports volume 6, Article number: 35596 (2016) (\href{https://arxiv.org/abs/1506.01354}{arXiv:1506.01354}, \href{https://4gravitons.wordpress.com/2016/11/11/a-response-from-nielsen-guffanti-and-sarkar/}{web discussion})


\item Koushik Dutta, Ruchika, Anirban Roy, Anjan A. Sen, M.M. Sheikh-Jabbari, \emph{Negative Cosmological Constant is Consistent with Cosmological Data} (\href{https://arxiv.org/abs/1808.06623}{arXiv:1808.06623})


\item Rui-Yun Guo, Jing-Fei Zhang, Xin Zhang, \emph{Can the $H_0$ tension be resolved in extensions to ΛCDM cosmology?} (\href{https://arxiv.org/abs/1809.02340}{arXiv:1809.02340})



\end{itemize}
\hypertarget{ReferencedInpAQFT}{}\subsubsection*{{In pAQFT}}\label{ReferencedInpAQFT}

Discussion of the cosmological constant in the rigorous formulation of [[perturbative AQFT]] [[AQFT on curved spacetimes|on curved spacetimes]] includes the following.

The freedom of [[renormalization]] of the [[vacuum expectation value]] of any [[stress-energy tensor]], hence of the cosmological constant, was discussed in

\begin{itemize}%
\item [[Robert Wald]], \emph{Trace anomaly of a conformally invariant quantum field in curved spacetime}, Phys. Rev. D 17, 1477 1978 (\href{https://doi.org/10.1103/PhysRevD.17.1477}{doi:10.1103/PhysRevD.17.1477})

\end{itemize}
Notice that (\hyperlink{Wald78}{Wald 78}) is based on

\begin{itemize}%
\item [[Robert Wald]], \emph{The back reaction effect in particle creation in curved spacetime}, Commun. Math. Phys. (1977) 54: 1. (\href{https://doi.org/10.1007/BF01609833}{doi:10.1007/BF01609833})

\end{itemize}
which claimed \emph{no} freedom of renormalization, but \hyperlink{Wald78}{Wald 78} explains that this was due to a mistake inherited from a citation.

Further development of this includes

\begin{itemize}%
\item Wolfgang Tichy, Eanna E. Flanagan, \emph{How unique is the expected stress-energy tensor of a massive scalar field?}, Phys.Rev. D58 (1998) 124007 (\href{https://arxiv.org/abs/gr-qc/9807015}{arXiv:gr-qc/9807015})


\item [[Valter Moretti]], \emph{Comments on the Stress-Energy Tensor Operator in Curved Spacetime}, Commun. Math. Phys. 232 (2003) 189-221 (\href{https://arxiv.org/abs/gr-qc/0109048}{arXiv:gr-qc/0109048})



\end{itemize}
A useful review is in

\begin{itemize}%
\item [[Thomas-Paul Hack]], section 3.2.1, \emph{Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes}, Springer 2016 (\href{https://arxiv.org/abs/1506.01869}{arXiv:1506.01869}, \href{https://doi.org/10.1007/978-3-319-21894-6}{doi:10.1007/978-3-319-21894-6})

\end{itemize}
Realization of renormalization of stress-energy/cosmological constant in concrete [[FRW models]] is discussed in

\begin{itemize}%
\item [[Claudio Dappiaggi]], [[Klaus Fredenhagen]], [[Nicola Pinamonti]], \emph{Stable cosmological models driven by a free quantum scalar field}, Phys. Rev. D77:104015, 2008 (\href{https://arxiv.org/abs/0801.2850}{arXiv:0801.2850})


\item [[Claudio Dappiaggi]], [[Thomas-Paul Hack]], Jan Möller, [[Nicola Pinamonti]], \emph{Dark Energy from Quantum Matter} (\href{https://arxiv.org/abs/1007.5009}{arXiv:1007.5009})



\end{itemize}
Discussion specifically for the [[standard model of cosmology]] is in

\begin{itemize}%
\item [[Thomas-Paul Hack]], \emph{The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation} (\href{https://arxiv.org/abs/1306.3074}{arXiv:1306.3074})

\end{itemize}
\hypertarget{ReferencesFromHigherCurvatureCorrections}{}\subsubsection*{{From higher curvature corrections}}\label{ReferencesFromHigherCurvatureCorrections}

Effective dark energy from [[higher curvature corrections]], as in the [[Starobinsky model of cosmic inflation]]:

\begin{itemize}%
\item Michal Artymowski, Zygmunt Lalak, \emph{Inflation and dark energy from $f(R)$ gravity}, JCAP09(2014)036 (\href{https://arxiv.org/abs/1405.7818}{arXiv:1405.7818})


\item Michal Artymowski, Zygmunt Lalak, Marek Lewicki, \emph{Inflation and dark energy from $f(R)$ gravity} (\href{https://arxiv.org/abs/1510.04864}{arXiv:1510.04864})



\end{itemize}
\hypertarget{from_inhomogeneous_cosmology}{}\subsubsection*{{From inhomogeneous cosmology}}\label{from_inhomogeneous_cosmology}

Discussion of the cosmological constant as an artefact of [[inhomogeneous cosmology]] (see there for more) includes the following

\begin{itemize}%
\item [[Thomas Buchert]], \emph{Dark Energy from structure: a status report}, Gen.Rel.Grav.40:467-527, 2008 (\href{http://xxx.lanl.gov/abs/0707.2153}{arXiv:0707.2153})


\item Stephen R. Green, [[Robert Wald]], \emph{A new framework for analyzing the effects of small scale inhomogeneities in cosmology}, Phys.Rev.D83:084020, 2011 (\href{https://arxiv.org/abs/1011.4920}{arXiv:1011.4920})


\item [[Thomas Buchert]], \emph{Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects}, Class.Quant.Grav.28:164007, 2011 (\href{https://arxiv.org/abs/1103.2016}{arXiv:1103.2016})


\item [[Thomas Buchert]], Syksy Rasanen, \emph{Backreaction in late-time cosmology}, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 (\href{https://arxiv.org/abs/1112.5335}{arXiv:1112.5335})


\item Stephen Green, [[Robert Wald]], \emph{Examples of backreaction of small scale inhomogeneities in cosmology}, Phys.Rev.D87:124037, 2013 (\href{https://arxiv.org/abs/1304.2318}{arxiv:1304.2318})


\item [[Günter Scharf]], \emph{Inhomogeneous cosmology in the cosmic rest frame without dark stuff}, chapter 6 in the latest edition of \emph{[[Quantum Gauge Theories -- A True Ghost Story]]}, Wiley 2001 (\href{https://arxiv.org/abs/1312.2695}{arXiv:1312.2695})


\item [[Thomas Buchert]] et. al, \emph{Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?}, Class. Quantum Grav. 32 215021, 2015 (\href{https://arxiv.org/abs/1505.07800}{arXiv:1505.07800})

exposition in \emph{\href{https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/}{The Universe is inhomogeneous. Does it matter?}} CQG+, 2016


\item Krzysztof Bolejko, Mikołaj Korzyński, \emph{Inhomogeneous cosmology and backreaction: Current status and future prospects}, Int. J. Mod. Phys. D 26, 1730011 (2017) (\href{https://arxiv.org/abs/1612.08222}{arXiv:1612.08222})


\item [[Ulf Danielsson]], Thomas Van Riet, \emph{What if string theory has no de Sitter vacua?} (\href{https://arxiv.org/abs/1804.01120}{arXiv:1804.01120})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Inhomogeneous_cosmology}{Inhomogeneous cosmology}}

\end{itemize}
\hypertarget{CCProblemReferences}{}\subsubsection*{{The ``cosmological constant problem''}}\label{CCProblemReferences}

Discussion of the experimentally observed tiny cosmological constant and the folklore theoretical problem with that includes the following

\begin{itemize}%
\item Subir Sarkar, \emph{New results in cosmology} (\href{https://arxiv.org/abs/hep-ph/0201140}{arXiv:hep-ph/0201140})


\item Stefanus Nobbenhuis, \emph{The cosmological constant problem -- an inspiration for new physics} PhD thesis (2006) (\href{http://igitur-archive.library.uu.nl/dissertations/2006-0615-200938/}{web} \href{http://igitur-archive.library.uu.nl/dissertations/2006-0615-200938/c1.pdf}{pdf})


\item Joan Sola, \emph{Cosmological constant and vacuum energy: old and new ideas}, J.Phys.Conf.Ser. 453 (2013) 012015 (\href{https://arxiv.org/abs/1306.1527}{arXiv:1306.1527})


\item Jaan Einasto, \emph{Dark matter} (\href{https://arxiv.org/abs/0901.0632}{arXiv:0901.0632}) 2009



\end{itemize}
\hypertarget{ReferencesInStringTheory}{}\subsubsection*{{In string theory}}\label{ReferencesInStringTheory}

Discussion from the point of view of [[perturbative string theory]], where the cosmological constant is fixed by the choice of [[perturbative string theory vacuum]].

\hypertarget{ReferencesStringTheoryVanishingCC}{}\paragraph*{{For fundamentally vanishing cc (``Witten's Dark Fantasy'')}}\label{ReferencesStringTheoryVanishingCC}

An argument for [[non-perturbative effect|non-perturbative]] non-[[supersymmetry|supersymmetric]] 4d [[string phenomenology]] with fundamentally vanishing [[cosmological constant]], based on 3d [[M-theory on 8-manifolds]] decompactified at strong coupling to 4d via [[duality between M-theory and type IIA string theory]] (recall the [[super 2-brane in 4d]]):

\begin{itemize}%
\item [[Edward Witten]], \emph{The Cosmological Constant From The Viewpoint Of String Theory}, lecture at \href{http://inspirehep.net/record/972507}{DM2000} (\href{https://arxiv.org/abs/hep-ph/0002297}{arXiv:hep-ph/0002297})

(see p. 7)


\item [[Edward Witten]], \emph{Strong coupling and the cosmological constant}, Mod. Phys. Lett. A 10:2153-2156, 1995 (\href{https://arxiv.org/abs/hep-th/9506101}{arXiv:hep-th/9506101})


\item [[Edward Witten]], Section 3 of \emph{Some Comments On String Dynamics}, talk at \href{https://cds.cern.ch/record/305869}{Strings95} (\href{http://arxiv.org/abs/hep-th/9507121}{arXiv:hep-th/9507121})



\end{itemize}
This suggestion was called \emph{Witten's Dark Fantasy} in \hyperlink{HeckmannLawrieLinZoccarato19}{Heckmann-Lawrie-Lin-Zoccarato 19, Section 8}, where a concrete realization of this scenario in [[F-theory]] is claimed:

\begin{itemize}%
\item [[Cumrun Vafa]], Section 4.3 of: \emph{Evidence for F-Theory}, Nucl. Phys. B469:403-418, 1996 (\href{https://arxiv.org/abs/hep-th/9602022}{arxiv:hep-th/9602022})


\item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Gianluca Zoccarato, \emph{F-theory and Dark Energy}, Fortschritte der Physik (\href{https://arxiv.org/abs/1811.01959}{arXiv:1811.01959}, \href{https://doi.org/10.1002/prop.201900057}{doi:10.1002/prop.201900057})


\item [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, Gianluca Zoccarato, \emph{Pixelated Dark Energy} (\href{https://arxiv.org/abs/1901.10489}{arXiv:1901.10489})



\end{itemize}
\hypertarget{for_fundamentally_nonvanishing_cc}{}\paragraph*{{For fundamentally non-vanishing cc}}\label{for_fundamentally_nonvanishing_cc}

But a) there might be a large space of [[perturbative string theory vacua]] and b) the [[de Sitter spacetime|de Sitter vacua]] that seem to correspond to observation tend to exist (only) as metastable vacua:

\begin{itemize}%
\item [[Shamit Kachru]], [[Renata Kallosh]], [[Andrei Linde]], [[Sandip Trivedi]], \emph{de Sitter Vacua in String Theory}, Phys.Rev.D68:046005, 2003 (\href{https://arxiv.org/abs/hep-th/0301240}{arXiv:hep-th/0301240})

\end{itemize}
This observation led to the discussion of the ``[[landscape of string theory vacua]]''.

For review see

\begin{itemize}%
\item [[Renata Kallosh]], section 3 of \emph{de Sitter vacua and the landscape of string theory}, J. Phys. Conf. Ser. 24 (2005) 87-110 (\href{http://inspirehep.net/record/701033/}{spire})

\end{itemize}
However, consistency problems of these arguments are raised in

\begin{itemize}%
\item [[Ulf Danielsson]], Thomas Van Riet, \emph{What if string theory has no de Sitter vacua?} (\href{https://arxiv.org/abs/1804.01120}{arXiv:1804.01120})

\end{itemize}
[[!redirects dark energy]]



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