coset space coprojection admitting local sections

[[!redirects coset space coprojections admitting local sections]]

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## Idea

Given a [[topological group]] $G$ and a [[subgroup]] $H \subset G$ it is often oft interest to known that the [[coset space]] [[coprojection]] $G \to G/H$ admits [[local sections]]. For instance, these yield canonical examples of $H$-[[principal bundles]], of [[G-structure|H-structures]] and of [[equivariant bundles]].

## Recognition

\begin{prop}\label{SufficientConditionsforCosetSpaceCoprojectionsHavingLocalSections}
**(sufficient conditions for coset space coprojections having local sections)**\linebreak
Let $G$ be a [[topological group]] and $H \subset G$ a [[subgroup]].

Then sufficient conditions for the [[coset space]] [[coprojection]] $G \overset{q}{\to} G/H$ to admit [[local sections]], in that there is an [[open cover]] $\underset{i \in I}{\sqcup}U_i \to G/H$ and a [[continuous section]] $\sigma_{\mathcal{U}}$ of the [[pullback]] of $q$ to the cover,

$$
  \array{
    &&
    G_{\vert \mathcal{U}}
    &\longrightarrow&
    G
    \\
    &
    {}^{\mathllap{
      \exists \sigma
    }}
    \nearrow
    &
    \big\downarrow
    &{}^{{}_{(pb)}}&
    \big\downarrow
    \\
    \mathllap{ \exists \; }
    \underset{i \in I}{\sqcup} U_i
    &=&
    \underset{i \in I}{\sqcup} U_i
    &\longrightarrow&
    G/H
    \mathrlap{\,,}
  }
$$

include the following:

* $G$ is any [[topological group]]

  and $H$ is a [[compact Lie group]]

  ([Gleason 50, Thm. 4.1](#Gleason50))

* $G$ is a [[locally compact topological group]]

  which is moreover a [[separable metric space]] of [[finite number|finite]] [[dimension of a separable metric space|dimension]]

  and $H \subset G$ is a [[closed subgroup]].

  ([Mostert 53, Theorem 3](#Mostert53), see also [Karube 58, Theorem 2](#Karube58))

* $G$ is a ([[finite number|finite]]-[[dimension of a manifold|dimensional]]) [[Lie group]]

  and $H$ is a [[closed subgroup]]

  (e.g. [tom Dieck & Bröcker 1985, Thm. 4.3](#tomDieckBroecker85))

\end{prop}

## Examples

\begin{example}\label{UnitaryGroupOnHilbertSpaceByCircleGroup}
  For $\mathcal{H}$ an infinite-dimensional [[separable Hilbert space|separable]] [[complex numbers|complex]] [[Hilbert space]], the [[coset space]] of the [[unitary group]] [[U(ℋ)]] by its [[circle group|circle]] [[subgroup]] [[U(1)]] is the [[projective unitary group]] $\mathrm{U}(\mathcal{H})/U(1) \; \simeq$ [[PU(ℋ)]], both in their weak/strong [[operator topology]].
This coset space coprojection falls outside the applicatibility of Prop. \ref{SufficientConditionsforCosetSpaceCoprojectionsHavingLocalSections}, and yet it does admit local sections ([Simms 1970, Thm. 1](projective+unitary+group+on+a+Hilbert+space#Simms70)).
\end{example}

## Counter examples
 {#Counterexamples}

Examples of quotient coprojections $ G \to G/H$ *without* local sections are given in [Karube 58, Sec. 3](#Karube58).

## Related concepts

* [[coset space]]

* [[principal bundle]]

* [[G-structure]]

* [[equivariant bundle]]

## References

* {#Samelson41} [[Hans Samelson]], _Beiträge zur Topologie der Gruppenmannigfaltigkeiten_, Ann. of Math. 2, 42, (1941), 1091 - 1137. ([jstor:1970463](https://www.jstor.org/stable/1970463), [doi:10.2307/1970463](https://doi.org/10.2307/1970463))

* {#Gleason50} [[Andrew Gleason]], _Spaces With a Compact Lie Group of Transformations_, Proceedings of the American Mathematical Society Vol. 1, No. 1 (Feb., 1950), pp. 35-43 ([jstor:2032430](https://www.jstor.org/stable/2032430), [doi:10.2307/2032430](https://doi.org/10.2307/2032430))

* {#Mostert53} [[Paul Mostert]], _Local Cross Sections in Locally Compact Groups_, Proceedings of the American Mathematical Society, Vol. 4, No. 4 (Aug., 1953), pp.645-649 ([jstor:2032540](https://www.jstor.org/stable/2032540), [doi:10.2307/2032540](https://doi.org/10.2307/2032540))

* {#Karube58} Takashi Karube, _On the local cross-sections in locally compact groups_,  J. Math. Soc. Japan 10(4): 343-347 (October, 1958) ([doi:10.2969/jmsj/01040343](https://projecteuclid.org/journals/journal-of-the-mathematical-society-of-japan/volume-10/issue-4/On-the-local-cross-sections-in-locally-compact-groups/10.2969/jmsj/01040343.full))

* {#tomDieckBroecker85} [[Tammo tom Dieck]], [[Theodor Bröcker]], Thm. 4.3 on p. 33 of: *Representations of compact Lie groups*, Springer 1985 ([doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0))

[[!redirects coset space coprojections with local sections]]


coset space structure on n-spheres -- table

**[[coset space]]-[[structures]] on [[n-spheres]]:**

| **standard:** |   |
|---|---|
| $S^{n-1} \simeq_{diff} SO(n)/SO(n-1)$ | [this Prop.](sphere#nSphereAsCosetSpace) |
| $S^{2n-1} \simeq_{diff} SU(n)/SU(n-1)$ | [this Prop.](sphere#OddDimSphereAsSpecialUnitaryCoset) |
| $S^{4n-1} \simeq_{diff} Sp(n)/Sp(n-1)$ | [this Prop.](sphere#SphereAsSymplecticUnitaryCoset) |
|  |   |
| **[[exceptional geometry|exceptional]]:** |    |
| $S^7 \simeq_{diff} Spin(7)/G_2$ | [[Spin(7)/G2 is the 7-sphere]]  |
| $S^7 \simeq_{diff} Spin(6)/SU(3)$ | since [[Spin(6)]] $\simeq$ [[SU(4)]]   |
| $S^7 \simeq_{diff} Spin(5)/SU(2)$ | since [[Sp(2)]] is [[Spin(5)]] and [[Sp(1)]] is [[SU(2)]], see  [[Spin(5)/SU(2) is the 7-sphere]]  |
| $S^6 \simeq_{diff} G_2/SU(3)$ | [[G2/SU(3) is the 6-sphere]] |
| $S^15 \simeq_{diff} Spin(9)/Spin(7)$ | [[Spin(9)/Spin(7) is the 15-sphere]] |

see also _[[Spin(8)-subgroups and reductions -- table|Spin(8)-subgroups and reductions]]_

{#HomotopyTheoretic} [[homotopy fibers]] of [[homotopy pullbacks]] of [[classifying spaces]]:

\begin{imagefromfile}
  "file_name": "ExceptionalSpheres.jpg",
  "width": 730
\end{imagefromfile}

(from [[schreiber:Twisted Cohomotopy implies M-theory anomaly cancellation|FSS 19, 3.4]])


coset WZW model

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## Idea

The generalization of the [[WZW model]] where [[target space]] is not necessarily a [[Lie group]] $G$, but a [[coset]] space $G/H$ for some [[subgroup]] $H$.

## Related concepts

* [[gauged WZW model]]

## References

* [[Edward Witten]], _On holomorphic factorization of WZW and coset models_, Comm. Math. Phys. Volume 144, Number 1 (1992), 189-212. ([EUCLID](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.cmp/1104249222))

[[!redirects coset model]]

[[!redirects coset models]]


coshape of an (infinity,1)-topos

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## Idea

Just as the [[shape of an (∞,1)-topos]] is the functor $\infty Gpd\to \infty Gpd$ which it *corepresents* (after identifying ∞-groupoids with their presheaf (∞,1)-topoi), so the *coshape* of an (∞,1)-topos is the functor $\infty Gpd^{op}\to \infty Gpd$ which it *represents*.

Unlike the shape, which is only (co)representable (by an ∞-groupoid) when the topos is [[locally ∞-connected (∞,1)-topos|locally ∞-connected]], the coshape is *always* representable, albeit possibly by a *large* ∞-groupoid---specifically the ∞-groupoid of [[point of a topos|points]] of the (∞,1)-topos in question.

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## Definition

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###### Definition

For $\mathbf{H}$ a [[(∞,1)-topos|topos]], we say its **co-shape** $\Gamma \mathbf{H}$ is the functor $\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd$ defined by

$$
  \Gamma(\mathbf{H})(A) =
  Topos(PSh(A), \mathbf{H})
$$

=--

Let $Pt(\mathbf{H}) = Topos(*,\mathbf{H})$ denote the (possibly large) groupoid of *points* of $\mathbf{H}$, where $*$ denotes the terminal topos $Gpd$.

+-- {: .un_prop}
###### Proposition

The coshape $\Gamma(\mathbf{H})$ is represented by $Pt(\mathbf{H})$, i.e. for any (small) groupoid $A$ we have
$$ \Gamma(\mathbf{H})(A) \simeq GPD(A, Pt(\mathbf{H})). $$

=--

+-- {: .proof}
###### Proof

Recall that colimits in $Topos$ are calculated via *limits* on the level of underlying categories.  In particular, the [[copower]] of $\mathbf{K}$ by a groupoid $A$ is the topos $\mathbf{K}^A$.

Thus, in even more particular, the functor $Psh$ preserves copowers, since each groupoid is the copower of the terminal object by itself.  Therefore, we have $Topos( Psh(A), \mathbf{H} ) \simeq GPD(A, Topos( *, \mathbf{H} )) = GPD(A,Pt(\mathbf{H}))$, as desired.

=--

## In terms of Universe Enlargements

Another way of phrasing the above argument is as follows.  For the same reason cited in the proof, the embedding $Psh$ preserves all small colimits.  Therefore, since $\Gamma(\mathbf{H}) \colon Gpd^{op}\to Gpd$ is the composite of this embedding with the representable functor $Topos(-,\mathbf{H})$, it must also preserves all small limits in $Gpd^{op}$ (i.e. small colimits in $Gpd$).

Therefore, we can regard it as an object of the category $Cts(Gpd^{op},Gpd)$ of small-limit-preserving functors, also known as the [[very large (∞,1)-sheaf (∞,1)-topos]] on [[?Gpd|Gpd]] (and also the $\kappa$-[[ind-object in an (∞,1)-category|ind-objects]] of $Gpd$, for $\kappa$ the cardinality of the universe).  However, by the general theory of [[universe enlargement]] (generalized to $(\infty,1)$-categories), this category is equivalent to $GPD$, and the equivalence gives the representability theorem above.

### Enlarging the category of toposes

Instead of being content with a "large-representability" result as above, we might wish that the coshape would actually give us a right adjoint to the embedding $Psh$.  For this to be possible, we would need to enlarge $Gpd$ to $GPD$, but if we also enlarged $Topos$ to its naive enlargement $TOPOS$, we would face the same problem "one universe higher."

Thus, to get "better behavior" we can instead replace $Topos$ by its locally presentable enlargement $\Uparrow Topos$, also called the [[very large (∞,1)-sheaf (∞,1)-topos]] on $Topos$.  We can then say:

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###### Proposition

Coshape [[Yoneda extension|Yoneda-extends]] to a pair of [[adjoint (∞,1)-functor|adjoint functors]]s

$$
  GPD
  \stackrel{\overset{\Gamma}{\leftarrow}}{\underset{Codisc}{\to}}
  \Uparrow Topos.
$$

=--

+-- {: .proof}
###### Proof

By [[Higher Topos Theory|HTT, lemma 6.3.5.21]] we have a [[(∞,1)-functor|functor]]

$$
  \Uparrow Topos
  \to
  \Uparrow Grpd = GRPD
$$

that preserves $\mathcal{U}$-small colimits and finite limits and is given by sending

$$
  F : Topos^{op} \to Grpd
$$

to the composite

$$
   Grpd
   \stackrel{PSh(-)}{\to}
   (Topos/Grpd)_{et}^{op}
   \stackrel{}{\to}
   Topos^{op}
   \stackrel{F}{\to}
   Grpd
  \,,
$$

where the first step is forming [[(∞,1)-presheaf (∞,1)-topos|presheaf toposes]] which sit by their terminal [[global section]] [[(∞,1)-geometric morphism|geometric morphisms]] over [[∞Grpd|Grpd]], and the second step is the evident projection.

Applied to a [[representable functor|representable]]
$F = Topos(-,\mathbf{H})$ this composite is hence $A \mapsto \Gamma(\mathbf{H})(A)$.

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## Related concepts

* [[shape of an (∞,1)-topos]]

* **coshape of an $(\infty,1)$-topos**

[[!redirects coshape of an (∞,1)-topos]]


cosheaf

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## Idea

For $C$ a [[small category]], and $PSh(C)$ its [[presheaf topos]], we have (by the discussion at _[Profunctor -- In terms of colimit preserving functors on presheaf categories](profunctor#FuncsOnPresheaves)_) that a cocontinuous,
i.e., [[colimit]]-preserving, [[functor]] $PSh(C) \to Set$ is equivalently itself a [[copresheaf]] on $C$:
$$[PSh(C), Set]_{coc} \simeq CoPSh(C).$$

If we replace in this statement presheaves with [[sheaves]], we obtain the notion of _cosheaf_ on $C$:
$$[Sh(C), Set]_{coc} \simeq CoSh(C).$$

## Definition

+-- {: .num_defn #cosheaf}
###### Definition

Let $C$ be a [[site]]. A **cosheaf** on $C$ is a [[copresheaf]]
$$F : C \to Set$$
such that it takes [[cover]]s to [[colimit]]s: for each covering family $\{U_i \to U\}$ in $C$ we have
$$
  F(U) \simeq \lim_{\to}
  \left(
     \coprod_{i j} F(U_i \times_{U} U_j)
     \stackrel{\to}{\to}
     \coprod_i F(U_i)
  \right)
$$

Write $CoSh(C) \subset CoPSh(C)$ for the full [[subcategory]] of cosheaves.

=--

## Proposition

+-- {: .num_prop}
###### Proposition

There is a [[natural transformation|natural]] [[equivalence of categories]]
$$
  CoSh(C) \simeq Func_{coc}(Sh(C), Set)
  \,,
$$
where on the left we have the category of cosheaves from def. \ref{cosheaf} and on the right we have the category of [[colimit]]-preserving functors on the [[sheaf topos]] of $C$.

Equivalently: a copresheaf is a cosheaf precisely if its [[Yoneda extension]] $PSh(C) \to Set$ factors through the [[sheafification]] functor $PSh(C) \to Sh(C)$.

=--

This is ([Bunge-Funk 06, prop. 1.4.3](BungeFunk)).

## Equivalence between cosheaves of sets and complete spreads

In complete analogy to the equivalence of categories between
[[sheaves]] of sets on a [[locale]] $X$ and [[etale maps]] $L\to X$
there is an equivalence of categories
between [[cosheaves]] of sets on a [[locale]] $X$ and [[complete spreads]] $L\to X$.
The analog of the [[etale space]] functor is the [[display locale]] functor.
The analog of the [[sheaf of sections]] functor is the [[cosheaf of connected components]] functor.

A decategorified version of this statement
was obtained by [[Marta Bunge]] and [[Jonathon Funk]] in [this article](#PLoc):
join-preserving maps $X\to\Omega$ are in bijection
with [[overt]] [[weakly closed]] [[sublocales]] of $X$.
(Here $\Omega$ is the poset of [[truth values]].)

## Examples

### In AQFT and higher AQFT

Cosheaves of [[algebras]], or notions similar to this, appear in [[AQFT]] as [[local nets of observables]]. Similar structures in [[higher category theory]] are _[[factorization algebras]]_, _[[factorization homology]]_, and _[[topological chiral homology]]_. Notably the definition of _[[factorization algebra]]_ typically explicitly involves the notion of cosheaf.

## Related concepts

* [[(0,1)-presheaf]] / [[(0,1)-sheaf]]

* [[presheaf]] /  [[sheaf]] / **cosheaf**

* [[2-sheaf]] / [[stack]]
* [[(∞,1)-sheaf]] / [[∞-stack]]

* [[Lawvere distribution]]

## References

* [[Glen E. Bredon]], _Sheaf Theory_ , Springer Heidelberg 1997. (chap. V-VI)

* [[Marta Bunge]] and [[Jonathon Funk]], _[[Singular coverings of toposes]]_ , Lecture Notes in Mathematics vol. 1890 Springer Heidelberg (2006). (sec. 1.4)

* [[Justin M. Curry]], _Abstract existence of cosheafification_ ([pdf](https://services.math.duke.edu/~curry/papers/abstract_cosheafification.pdf))

* [[Jonathon Funk]], _The Display Locale of a Cosheaf_ , Cah. Top. Géom. Diff. Cat. **36** (1995) pp.53-93.

* [[Andrei V. Prasolov]], _Cosheafification_ ([arXiv:1605.01555](https://arxiv.org/abs/1605.01555))

* [[Andrei V. Prasolov]], _Cosheaves_ ([arXiv:1804.07988](https://arxiv.org/abs/1804.07988))

* [[Steve Vickers]], _Cosheaves and Connectedness in Formal Topology_ , Ann. Pure Appl. Logic **163** no.2 (2012) pp.157-174. ([preprint](www.cs.bham.ac.uk/~sjv/ConnFT.pdf))

* {#PLoc}[[Marta Bunge]], [[Jonathon Funk]], _Constructive theory of the lower power locale_, Mathematical Structures in Computer Science, vol. 6, no. 1, pp. 69-83: [doi](https://doi.org/10.1017/s0960129500000876).

[[!redirects cosheaves]]


cosheaf of connected components

## Idea

The [[display locale]] construction defines a fully faithful functor
from [[cosheaves]] of sets on a [[locale]] $X$
to the [[slice category]] $Loc/X$ of [[locales]] over $X$.

The construction in this article describes a functor
in the opposite direction that yields
the inverse of the above functor
once restricted to the appropriate subcategory.

Thus, the functor described here
plays the same role in the equivalence between
[[cosheaves]] of sets on $X$ and [[complete spreads]] over $X$
as the sheaf of sections construction
plays in the equivalence between
[[sheaves]] of sets on $X$ and [[etale locales]] over $X$.

## Definition

Suppose $l\colon L\to X$ is a [[map of locales]],
where $L$ is a [[locally connected locale]].
Define a [[precosheaf]] $\lambda_l$ on $X$ as follows.
Send an open $U$ in $X$ to the set of [[connected components]]
of $l^* U$, which is a [[locally connected locale]] because so is $L$.
Send an inclusion of opens to the induced map on the sets of [[connected components]].

\begin{proposition}
(Funk, Proposition 2.1.)
This [[precosheaf]] is a [[cosheaf]].
\end{proposition}

## Related notions

* [[display locale]]

* [[cosheaf]]

## References

* [[Jonathon Funk]], _The display locale of a cosheaf_.


cosieve

Cosieve is a dual notion to [[sieve]]; that is, a cosieve in $C$ in a sieve in the [[opposite category]] $C^{op}$.  They can be used to define Grothendieck cotopologies, dual to [[Grothendieck topology|Grothendieck topologies]].

A __cosieve__ $R$ under an object $x$ in $C$ is a family of [[morphisms]] $f:x\to y$ in $C$ with domain $x$ closed under postcomposition with any morphism in $C$. In other words, $f\in R$ implies $h\circ f\in R$ whenever the [[composite]] $h\circ f$ exists. Cosieves under $x$ are also said to be _cosieve in_ the [[under category]] $x \backslash C$; all such cosieves for varying $x$ are said to be _cosieves on_ $C$. Cosieves may be viewed as [[subfunctor]]s of the [[representable functor|(co)representable (covariant) functors]] $h^x=C(x,-)$.

Cosieves on $C$ may be organized into a category $\mathrm{coSv}(C)$. For convenience we will note the domain $x$ of a sieve as a part of the data. Thus objects of $\mathrm{coSv}(C)$ are pairs of the form $(x,R)$ where $x\in\mathrm{Ob}(C)$ and $R$ is a cosieve in $x \backslash C$. A morphism $(x,R)\to (x',R')$ is a map $f:x\to x'$ such that the cosieve $R'\circ f = \{g\circ f | g\in R'\}$ is a [[subset]] of $R$. The usual composition of underlying morphisms in $C$ defines a composition in $\mathrm{coSv}(C)$, because $R''\circ (g\circ f)= (R''\circ g)\circ f\subset R'\circ f\subset R$ where $g:(x',R')\to (x'',R'')$.  Note that $coSv(C) = Sv(C^{op})^{op}$.


cosimplicial abelian group

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## Definition

A *cosimplicial abelian group* is, equivalently

* a [[cosimplicial object]] in [[AbelianGroups]],

* an [[abelian group|abelian]] [[group object]] [[internalization|internal]] to [[cosimplicial sets]].

## Properties

### Model structure

See at  *[[model structure on cosimplicial abelian groups]]*.

## Related concepts

* [[cosimplicial algebra]]

* [[simplicial abelian group]]

* [[monoidal Dold-Kan correspondence]]

[[!redirects cosimplicial abelian groups]]


cosimplicial algebra

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## Idea ##

A _cosimplicial algebra_ -- similarly a cosimplicial ring -- is a [[cosimplicial object]] in the category of [[algebra]]s (of [[ring]]s).

## Relation to differential graded algebras ##

Under the [[monoidal Dold–Kan correspondence]], cosimplicial algebras are essentially identified with [[differential graded algebra]]s in non-negative degree: the [[Moore complex|Moore cochain complex]] $C^\bullet(A)$ of a cosimplicial algebra $A$ is a [[differential graded algebra]] where the degreewise product on the cosimplicial algebra maps to the [[cup product]] operation that gives the [[monoid]] structure $C^\bullet(A)$.

## Model category structure ##

A standard [[model category]] structure on the category of cosimplicial rings is the following

* fibrations are the degreewise surjections

* weak equivalences are the morphisms that induce isomorphisms in [[cohomotopy]]

* cofibrations are defined by their left [[lifting property]].

For more see [[model structure on cosimplicial algebras]].

References are section 2.1 of

* [[Bertrand Toen]], _Affine stacks (Champs affines)_ ([arXiv:math/0012219](http://arxiv.org/abs/math/0012219)) .

and  [def 9.1, p. 18](http://arxiv.org/PS_cache/math/pdf/0306/0306289v3.pdf#page=18) of

* Castiglioni, Cortinas, _Cosimplicial versus dg-rings_ ([arXiv](http://arxiv.org/abs/math/0306289))

## Examples

* As cosimplcial algebras are [[duality|dual]] to simplicial spaces, each simplicial space $X$ gives rise to a cosimplicial algebra of functions on it. A list of examples is given at [[schreiber:Chevalley-Eilenberg algebra]].

## Related concepts

* [[cosimplicial abelian group]]

## References

The model category structure on cosimplicial algebras is discussed in detail in:

* [[Bertrand Toën]], Section 2.1 of: _Champs affines_, Selecta Math. new series **12** (2006), no. 1, 39-135 ([arXiv:math/0012219](https://arxiv.org/abs/math/0012219), [doi:10.1007/s00029-006-0019-z](https://doi.org/10.1007/s00029-006-0019-z))

The [[Quillen equivalence]] between cosimplicial algebras and cochain dg-algebras is discussed in

* Castiglioni, Cortinas, _Cosimplicial versus dg-rings_ ([arXiv](http://arxiv.org/abs/math/0306289))

A bit about cosimplicial algebras is in [section 7](http://atlas.mat.ub.es/personals/burgos/files/brbr.pdf#page=63) of

* José Burgos Gil, _The regulators of Beilinson and Borel_ ([pdf](http://atlas.mat.ub.es/personals/burgos/files/brbr.pdf))

This also discusses aspects of their image in [[dg-algebra]]s under the [[Moore complex]]-functor. See [[monoidal Dold-Kan correspondence]] for more on that.

[[!redirects cosimplicial ring]]


cosimplicial object

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## Idea

Where a [[simplicial object]] is a [[functor]] $\Delta^{op} \to \mathcal{C}$ out of the [[opposite category]] of the [[simplex category]], a cosimplicial object is a functor $\Delta \to \mathcal{C}$ out of the simplex category itself.

## Properties

### Simplicial enrichment
 {#SimplicialEnrichment}

When $\mathcal{C}$ has [[finite limits]] and finite [[colimits]], then $\mathcal{C}^{\Delta}$ is canonically a [[simplicially enriched category]] with is [[tensoring|tensored]] and [[powering|powered]] over [[sSet]]. This is called the _external simplicial structure_ in ([Quillen 67, II.1.7](#Quillen67)). Review includes ([Bousfield 03, section 2.10](#Bousfield03)).

More generally, for any $\mathcal{C}$, we can make $\mathcal{C}^{\Delta}$ into a simplicially enriched category using the [[end]] formula
$$\underline{\mathcal{C}^{\Delta}} (X, Y)_m = \int_{[n] : \Delta} (\mathcal{C} (X^n, Y^n))^{\Delta^m_n}$$
with composition inherited from $\mathcal{C}$ and $\Delta$.

## Related concepts

* [[cosimplicial abelian group]]

* [[cosimplicial algebra]]

## References

* {#Quillen67} [[Daniel Quillen]], Chapter I, _Axiomatic homotopy theory_ in: _[[Homotopical Algebra]]_, Lecture Notes in Mathematics 43, Springer 1967([doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438))

* [[Dai Tamaki]], [[Akira Kono]], Appendix A in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

* {#Bousfield03} [[Aldridge Bousfield]], _Cosimplicial resolutions and homotopy spectral sequences in model categories_ ([arXiv:0312531](http://arxiv.org/abs/math/0312531))

[[!redirects cosimplicial objects]]

[[!redirects category of cosimplicial objects]]
[[!redirects categories of cosimplicial objects]]


cosimplicial simplicial set

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## Definition

A _cosimplicial simplicial set_ is a [[cosimplicial object]] in the category of [[simplicial set]]s, hence a [[functor]]

$$
  \Delta \to sSet
$$

which equivalently is a functor

$$
  \Delta \times \Delta^{op} \to Set
  \,.
$$

There is an evident [[model category]] structure on $sSet^{\Delta}$ which models cosimplicial [[infinity groupoid]]s.

## Properties

### Model structure

There are several standard ways to equip $sSet^\Delta$ with the structure of a [[model category]]. See _[[model structure on cosimplicial simplicial sets]]_ for more.

### Homotopy and homology

The [[homotopy groups]] of the [[totalization]] of a cosimplicial homotopy type are computed by a [[Bousfield-Kan spectral sequence]]. The [[homology groups]] by an [[Eilenberg-Moore spectral sequence]].

## References

The standard reference is

* [[Paul Goerss]], [[Rick Jardine]], _Simplicial homotopy theory_

Chapter X of

* [[Aldridge Bousfield]] and [[Dan Kan]], _Homotopy limits, completions and localizations_ Springer-Verlag, Berlin, 1972. Lecture Notes in Mathematics, Vol. 304.
 {#BousfieldKan}

The homotopy [[spectral sequence]] for cosimplicial spaces is in chapter VIII.

* [[Rick Jardine]], _Cosimplicial spaces and cocycles_ ([pdf](http://www.math.uwo.ca/~jardine/papers/preprints/cosimp3.pdf))

* [[Thomas Goodwillie]], _A remark on the homology of cosimplicial spaces_ , Journal of Pure and Applied Algebra Volume 127, Issue 2, 15 May 1998, Pages 167-175

[[!redirects cosimplicial simplicial sets]]

[[!redirects cosimplicial space]]
[[!redirects cosimplicial spaces]]

[[!redirects simplicial cosimplicial set]]
[[!redirects simplicial cosimplicial sets]]

[[!redirects cosimplicial homotopy type]]
[[!redirects cosimplicial homotopy types]]

[[!redirects cosimplicial topological space]]
[[!redirects cosimplicial topological spaces]]


cosine

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## Idea

See at _[[trigonometric function]]_.

## Definitions

The _cosine function_ is the [[function]] $\cos \;\colon\; \mathbb{R} \to \mathbb{R}$ from the [[real numbers]] to themselves which is characterized by the following equivalent conditions:

1. $\cos$ is the unique [[solution]] among [[smooth functions]] to the [[differential equation]]/[[initial value problem]]

   $$
     cos'' = -cos
   $$

   (where a prime indicates the [[derivative]]) subject to the [[initial conditions]]

   $$
     \begin{aligned}
       cos(0) &= 1
       \\
       cos'(0) & = 0
       \,.
     \end{aligned}
   $$

1. ([[Euler's formula]]) $\cos$ is the [[real part]] of the [[exponential function]] with [[imaginary number|imaginary]] argument

   $$
     \begin{aligned}
       \cos(x) & = Re\left( \exp(i x) \right)
       \\
       & = \frac{1}{2}\big( \exp(i x) + \exp(- i x)\big)
     \end{aligned}
     \,.
   $$

1. $\cos$ is the unique function given by the infinite [[series]]

$$\cos(x) \coloneqq \sum_{i = 0}^{\infty} \frac{(-1)^i x^{2 i}}{(2 i)!}$$

### In arbitrary Archimedean ordered fields

In general, [[Archimedean field|Archimedean]] [[ordered field|ordered]] [[fields]] which are not [[sequentially Cauchy complete]] do not have an cosine function. Nevertheless, the cosine map is still guaranteed to be a [[partial function]], because every Archimedean ordered field is a [[Hausdorff space]] and thus a [[sequentially Hausdorff space]]. Therefore an [[axiom]] could be added to Archimedean ordered fields $F$ to ensure that the cosine partial function is actually a total function:

**Axiom of cosine function:** _For all elements $x \in F$, there exists a unique element $\cos(x) \in F$ such that for all positive elements $\epsilon \in F_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then  $-\epsilon \lt \left(\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!}\right) - \cos(x) \lt \epsilon$._

There is another axiom which uses the fact that derivatives of functions are well defined in the [[ordered local ring|ordered local]] [[local Artin algebra|Artin $F$-algebra]] $F[\epsilon]/\epsilon^2$ by the equation $f(x + \epsilon) = f(x) + f'(x) \epsilon$:

**Axiom of sine and cosine functions:** _Let $F[\epsilon]/\epsilon^2$ be the ordered local Artin $F$-algebra, with non-zero non-positive non-negative [[nilpotent element]] $\epsilon \in F[\epsilon]/\epsilon^2$ where $\epsilon^2 = 0$ and canonical $F$-algebra homomorphism $h:F \to F[x]/x^2$. There exists unique functions $\sin:F \to F$ and $\cos:F \to F$ and functions $\sin':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ and $\cos':F[\epsilon]/\epsilon^2 \to F[\epsilon]/\epsilon^2$ such that for every element $x \in F$, $h(\sin(x)) = \sin'(h(x))$, $h(\cos(x)) = \cos'(h(x))$, $\sin'(x + \epsilon) = \sin'(x) + \cos'(x) \epsilon$, $\cos'(x + \epsilon) = \cos'(x) - \sin'(x) \epsilon$, $\sin'(0) = 0$, and $\cos'(0) = 1$._

### In the dual numbers

The [[dual number]] real algebra $\mathbb{D} \coloneqq \mathbb{R}[\epsilon]/\epsilon^2$ has a notion of sine and cosine function, which are the solutions to the system of functional equations $\sin(x + \epsilon) = \sin(x) + \cos(x) \epsilon$ and $\cos(x + \epsilon) = \cos(x) - \sin(x) \epsilon$ with $\sin(0) = 1$ and $\cos(0) = 1$.

## In constructive mathematics

In [[classical mathematics]], one could prove that the [[modulated Cantor real numbers]] $\mathbb{R}_C$ are [[sequentially Cauchy complete]] and [[type equivalence|equivalent]] to the [[HoTT book real numbers]] $\mathbb{R}_H$. However, in [[constructive mathematics]], the above cannot be proven; while the HoTT book real numbers $\mathbb{R}_H$ are still sequentially Cauchy complete, the modulated Cantor real numbers $\mathbb{R}_C$ in general cannot be proven to be sequentially Cauchy complete. In particular, this means that the sequence

$$\sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!}$$

does not have a [[limit of a sequence|limit]] for all modulated Cantor real numbers $x \in \mathbb{R}_C$. However, the sequence, by definition of $\mathbb{R}_C$, does have a limit for all [[rational numbers]] $x \in \mathbb{Q}$; this means that one could restrict the [[domain]] of the cosine function to the rational numbers $\cos:\mathbb{Q} \to \mathbb{R}_C$, and define it in the usual manner:

* For all rational numbers $x \in \mathbb{Q}$, there exists a unique modulated Cantor real number $\cos(x) \in \mathbb{R}_C$ such that for all positive rational numbers $\epsilon \in \mathbb{Q}_+$, there exists a natural number $N \in \mathbb{N}$ such that for all natural numbers $n \in \mathbb{N}$, if $n \geq N$, then $-\epsilon \lt \sum_{i = 0}^{n} \frac{(-1)^i x^{2 i}}{(2 i)!} - \cos(x) \lt \epsilon$.

## Related concepts

* [[periodic function]]

* [[trigonometry]]

* [[sine]]

* [[arccos]]

* [[exponential]]

* [[exponential ring]]

* [[Euler's formula]]

* [[trigonometric identity]]

## References

Discussion in [[constructive analysis]]:

* [[Errett Bishop]], §7 in: *[[Foundations of Constructive Analysis]]*, McGraw-Hill (1967)

* [[Errett Bishop]], [[Douglas Bridges]] §7 in: *[[Constructive Analysis]]*, Grundlehren der mathematischen Wissenschaften **279**, Springer (1985) [[doi:10.1007/978-3-642-61667-9](https://doi.org/10.1007/978-3-642-61667-9)]

See also:

* Wikipedia, _[Sine and cosine](https://en.wikipedia.org/wiki/Sine_and_cosine)_

[[!redirects cos]]

[[!redirects cosine function]]
[[!redirects cosine functions]]


coskeleton > history

< [[coskeleton]]


coslice category > history

< [[coslice category]]
[[!redirects coslice category -- history]]


Cosma Shalizi

__Cosma Rohilla Shalizi__ holds Ph.D. in physics
from University of Wisconsin and is now an associate professor in Statistics Department of Carnegie Mellon University and external professor at Santa Fe Institute.

* related entries: [[information geometry]]

* University of Michigan [website](http://www.cscs.umich.edu/~crshalizi), and
[CMU website](http://www.stat.cmu.edu/~cshalizi) (at times it does not work)

* [research page](http://www.cscs.umich.edu/~crshalizi/research)

From [related faculty](http://www.ml.cmu.edu/mlpeople/relatedfaculty.html) at CMU Machine Learning:

> Dr. Shalizi has research interests in Nonparametric model discovery of state-space hidden Markov models and stochastic automata; dynamical-systems analysis of learning processes; applications of information theory, large deviations and ergodic theory in statistical inference; complex network models; heavy-tailed distributions.

category:people


Cosmas K. Zachos

* [institute page](https://www.anl.gov/profile/cosmas-k-zachos)

* [InSpire page](https://inspirehep.net/authors/982445)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Cosmas_Zachos)

## Selected writings

On [[Nambu mechanics]]:

* [[Thomas L. Curtright]], [[Cosmas K. Zachos]], _Classical and Quantum Nambu Mechanics_, Phys. Rev. D **68** 085001 (2003) [[arXiv:hep-th/0212267](http://arxiv.org/abs/hep-th/0212267), [doi:10.1103/PhysRevD.68.085001](https://doi.org/10.1103/PhysRevD.68.085001)]

On [[Weyl quantization]]:

* [[Thomas L. Curtright]], [[David B. Fairlie]], [[Cosmas K. Zachos]], *A Concise Treatise on Quantum Mechanics in Phase Space*, World Scientific (2014) [[doi:10.1142/8870](https://doi.org/10.1142/8870)]

category: people

[[!redirects Cosmas Zachos]]


cosmic censorship hypothesis

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## Idea

The _cosmic censorship hypothesis_ is the name of a pair of [[conjectures]] ([Penrose 69](#Penrose69)) saying that in the [[theory (physics)|theory]] of [[general relativity]], under physically reasonable conditions, any [[singularities]] in [[spacetime]] must lie behind an [[event horizon]].

Both the weak and the string form of the conjecture concern solutions of [[Einstein's equations]] with physically realistic matter and generic compact or asymptotically flat initial data.

The **strong cosmic censorship conjecture**, proposed by Penrose in 1986, states that the maximal Cauchy development of such a solution is locally inextendible as a regular Lorentzian manifold.

The **weak cosmic censorship hypothesis**, proposed by him in 1988, asserts that for generic initial data, the maximal [[Cauchy surface|Cauchy development]] of a solution of Einstein's equations with realistic matter possesses a complete future null infinity.

Despite their names, the strong version does not necessarily imply the weak version.

Without the 'genericity' condition, which needs to be carefully stated, both strong and weak cosmic censorship fail.  That is, with carefully fine-tuned [[initial value problem|initial data]] one can find counterexamples.

## Properties

### Relation to the weak gravity conjecture

The weak cosmic censorship conjecture has many counterexamples in [[dimension]] $d \geq 5$.  [[Cumrun Vafa]] has argued that the _[[weak gravity conjecture]]_ will save the cosmic censorship conjecture. The weak gravity conjecture says that in consistent theories of [[quantum gravity]] the gravitational force exerted by any object (e.g. a [[black hole]]) has to be weaker, in suitable units, than any other force. As discussed in ([Crisford-Santos 17](#CrisfordSantos17)) it seems plausible that this constraint would indeed rule out the counterexample constructed there:

> Subsequent calculations by Santos and Crisford supported Vafa's hunch; the simulations they're running now could verify that naked singularities become cloaked in black holes right at the point where gravity becomes the weakest force. ([Wolchover, June 20 2017](https://www.quantamagazine.org/where-gravity-is-weak-and-naked-singularities-are-verboten-20170620/))

The [[weak gravity conjecture]] was motivated from [[string theory]], where there are various plausibility arguments that it holds.

## Related concepts

* [[Penrose singularity theorem]]

* [[weak gravity conjecture]]

## References

### General

The original article:

* {#Penrose69} [[Roger Penrose]],  _Gravitational collapse: The role of general relativity_, Riv. Nuovo Cim. 1 (1969) 252-276, Gen. Rel. Grav. 34 (2002) 1141-1165 ([doi:10.1023/A:1016578408204](https://doi.org/10.1023/A:1016578408204))

Review:

* [[Roger Penrose]], _The Question of Cosmic Censorship_, Journal of Astrophysics and Astronomy, Vol. 20, p.233, 1999 ([doi:10.1007/BF02702355](https://doi.org/10.1007/BF02702355), [web](http://adsabs.harvard.edu/full/1999JApA...20..233P))

* [[Klaas Landsman]], _Singularities, black holes, and cosmic censorship: A tribute to Roger Penrose_ ([arXiv:2101.02687](https://arxiv.org/abs/2101.02687))

See also:

* Wikipedia, _[Cosmic censorship hypothesis](http://en.wikipedia.org/wiki/Cosmic_censorship_hypothesis)_

### Strong cosmic censorship

For strong cosmic censorship, see:

* Mihalis Dafermos, _Strong cosmic censorship_ ([web](http://web.math.princeton.edu/~dafermos/research/structure-of-singularities/strong-cosmic-censorship.html))

In 2017 Dafermos and Luk found a counterexample to strong cosmic censorship without the requirement that the initial data be generic:

* Mihalis Dafermos and Jonathan Luk, _The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon_ ([arXiv:1710.01722](https://arxiv.org/abs/1710.01722))

Discussion in relation to [[computability in physics]] and [[Malament–Hogarth spacetimes]]:

* {#Etesi12} [[Gabor Etesi]], _A proof of the Geroch-Horowitz-Penrose formulation of the strong cosmic censor conjecture motivated by computability theory_ ([arXiv:1205.4550](http://arxiv.org/abs/1205.4550))

In 2020, Hod claimed a "remarkably compact proof" of strong cosmic censorship:

* Shahar Hod, _A proof of the strong cosmic censorship conjecture_, International Journal of Modern Physics D 29, 2042003 (2020) ([arXiv:2012.01449](https://arxiv.org/abs/2012.01449), [doi:10.1142/S0218271820420031](https://doi.org/10.1142/S0218271820420031))

### Weak cosmic censorship

Review of weak cosmic censorship:

* [[Robert Wald]], _Gravitational collapse and cosmic censorship_,  In: Iyer B.R., Bhawal B. (eds) _Black Holes, Gravitational Radiation and the Universe_, Fundamental Theories of Physics, vol 100. Springer 1999 ([arXiv:gr-qc/9710068](https://arxiv.org/abs/gr-qc/9710068), [doi:10.1007/978-94-017-0934-7_5](https://doi.org/10.1007/978-94-017-0934-7_5))

In 1993 Choptuik found a counterexample to weak cosmic censorship without the requirement that the initial data be generic, by considering a spherically symmetric solution of gravity coupled to a scalar field right on the brink of the formation of a black hole.   This and subsequent work is reviewed here:

* Carsten Gundlach, _Critical phenomena in gravitational collapse_, Living Rev. Relativ. 10, 5 (2007)   ([arXiv:gr-qc/0001046](https://arxiv.org/abs/gr-qc/0001046), [doi:10.12942/lrr-2007-5](https://doi.org/10.12942/lrr-2007-5))

For a proof of weak cosmic censorship in the spherically symmetric case, see:

*  Demetrios Christodoulou, The instability of naked singularities in the gravitational collapse of a scalar field, _Ann. Math._ 149 (1999), 183-217.  ([arxiv](http://arxiv.org/abs/math/9901147))

A counterexample in 4-dimensional [[anti-de Sitter spacetime]]:

* {#CrisfordSantos17} Toby Crisford, [[Jorge Santos]], _Violating weak cosmic censorship in $AdS_4$_, Phys. Rev. Lett. 118, 181101 (2017) ([arXiv:1702.05490](https://arxiv.org/abs/1702.05490))

A counterexample in 6-dimensional spacetime:

* {#AndradeFiguerasSperhake20} Tomas Andrade, Pau Figueras, Ulrich Sperhake, _Violations of Weak Cosmic Censorship in black hole collisions_ ([arXiv:2011.03049](https://arxiv.org/abs/2011.03049))

### More references

For [[BTZ black holes]]:

* [[Roberto Emparan]], Marija Tomašević, _Strong cosmic censorship in the BTZ black hole_, JHEP 06 (2020) 038 ([arXiv:2002.02083](https://arxiv.org/abs/2002.02083))

Relation to the [[weak gravity conjecture]]:

* [[Gary Horowitz]], [[Jorge Santos]], _Further evidence for the weak gravity - cosmic censorship connection_ ([arXiv:1901.11096](https://arxiv.org/abs/1901.11096))

Relation to [[higher curvature corrections]]:

* Akash K Mishra, Sumanta Chakraborty, _Strong Cosmic Censorship in higher curvature gravity_, Phys. Rev. D 101, 064041 (2020) ([arXiv:1911.09855](https://arxiv.org/abs/1911.09855))

[[!redirects cosmic censorship hypotheses]]

[[!redirects cosmic censorship]]

[[!redirects weak cosmic censorship conjecture]]
[[!redirects weak cosmic censorship conjectures]]


cosmic cube

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## Idea

The **Cosmic Cube of [[higher category theory]]** is the name for a  [[diagram]] whose vertices correspond to special types of [[n-categories]].   The cube looks like this:

[[cosmiccube.jpg:pic]]

We may take $n = \infty$ here as well, and we may also consider a version for [[(n,r)-categories]].  The three axes correspond to:

* making $n$-categories 'groupoidal' --- that is, making morphisms invertible, thus passing from general $n$-categories to _[[n-groupoids]]_;

* making $n$-categories strict, thus passing from general $n$-categories to _[[strict n-categories|strict]]_ $n$-categories;

* making $n$-categories symmetric monoidal or 'stable', thus passing from general $n$-categories to _[[symmetric monoidal category|symmetric monoidal]]_ $n$-categories.

### In terms of homotopy theory {#HomotopyTheory}

Each vertex of the cube can also be understood as corresponding to a version of [[homotopy theory]]:

$\infty$-groupoids yield ordinary [[homotopy theory]], symmetric monoidal and groupal $\infty$-groupoids correspond to [[stable homotopy theory]], strictly abelian strict $\infty$-groupoids correspond to [[homological algebra]]. $\infty$-Categories that are not $\infty$-groupoids correspond to [[directed homotopy theory]].

## Vertices of the cube

Here we list the 8 vertices of the cube in the case of $\infty$-categories.

### Strict $\infty$-categories

* [[strict ∞-category]]

### Strict $\infty$-groupoids

* [[strict ∞-groupoid]]

* [[crossed complex]]

### Stably monoidal $\infty$-categories

* [[k-tuply monoidal n-category]]

* [[stabilization hypothesis]]

* [[periodic table]]

### Stably monoidal $\infty$-groupoids

* [[∞-group]]

* [[infinite loop space]], [[spectrum]]

### Strictly stably monoidal strict $\infty$-groupoids

* [[chain complex]], [[Dold-Kan correspondence]]

### Etc.

(...)

## Edges of the cube

### Strict $\infty$-groupoids in all $\infty$-groupoids

A [[strict ∞-groupoid]] is modeled by a [[crossed complex]]. Under [[∞-nerve]] it embeds into all [[∞-groupoid]]s, modeled as [[Kan complex]]es.

$$
  \array{
    CrsCplx &\stackrel{N^\Delta}{\hookrightarrow}& KanCplx
    \\
    \downarrow && \downarrow
    \\
    Str \infty Grpd &\hookrightarrow& \infty Grpd
  }
  \,.
$$

### Strictly stable strict $\infty$-groupoids in strict $\infty$-groupoids

A strictly stable strict ∞-groupoid is modeled by a bounded-below [[chain complex]] of abelian groups. Under the embedding $\Theta$ of complexes into [[crossed complex]]es it embeds into [[strict ∞-groupoid]]s.

$$
  \array{
    ChnCplx &\stackrel{\Theta}{\hookrightarrow}& CrsCplx
    \\
    \downarrow && \downarrow
    \\
    StrAb Str \infty Grpd &\hookrightarrow& Str \infty Grpd
  }
  \,.
$$

For the definition of $\Theta$ see _[[Nonabelian Algebraic Topology]]_ , section [Crossed complexes from chain complexes](#http://ncatlab.org/nlab/show/Nonabelian+Algebraic+Topology#CrsFromCh).

### Strictly stable strict $\infty$-groupoids in all $\infty$-groupoids

Combining the above inclusions

$$
  \array{
    ChainCplx &\stackrel{\Theta}{\hookrightarrow}&
    CrossedCplx
    &\stackrel{N^\Delta}{\hookrightarrow}&
    KanCplx
    \\
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    &&
    \downarrow^{\mathrlap{\simeq}}
    \\
    StrAb Str\infty Grpd
    &\hookrightarrow&
    Str \infty Grpd
    &\hookrightarrow&
    \infty Grpd
  }
$$

yields in total the map $ChnCplx \to sAb$ from [[chain complex]]es to [[simplicial abelian group]]s (followed by the forgetful $sAb \to KanCpx$) of the [[Dold-Kan correspondence]].

### Strictly stable strict $\infty$-groupoids in strictly stable $\infty$-groupoids

A strictly stable strict ∞-groupoid is modeled by a bounded-below [[chain complex]] of abelian groups. Under [[∞-nerve]] it embeds into all ([[connective spectrum|connective]]) [[spectra]]s, modeled as [[spectrum object]]s in [[Kan complex]]es.

$$
  \array{
    ChnCplx^+
    &\stackrel{\Sigma^\infty \N^\Delta \Theta}{\hookrightarrow}& Sp(KanCplx)
    \\
    \downarrow && \downarrow
    \\
    StrAb Str \infty Grpd &\hookrightarrow& Sp(\infty Grpd)
  }
  \,.
$$

### Strictly stable $\infty$-groupoids in all $\infty$-groupoids

A strictly stable ∞-groupoid is modeled by a connective [[spectrum]].  The forgetful functor to [[∞-groupoid]]s is also called $\Omega^\infty$ or the "zeroth-space functor."

## References

* [[John Baez]], _[What $n$-Categories Should Be Like](http://math.ucr.edu/home/baez/n_categories/what.pdf)_.

* Some blog [discussion](http://golem.ph.utexas.edu/category/2008/03/kim_on_fundamental_groups_in_n.html#c015463)

[[!redirects cosmic cube]]
[[!redirects Cosmic Cube]]
[[!redirects cosmic cube of higher category theory]]


cosmic Galois group

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## Idea

What is called the _cosmic Galois group_ is a _[[motivic Galois group]]_ that naturally [[action|acts]] on structures in [[renormalization]] in [[quantum field theory]]. The actual [[renormalization group]] is a 1-parameter [[subgroup]] of the cosmic Galois group.

In more detail, in ([Connes-Marcolli 04](#ConnesMarcolli04)) the authors consider a [[differential equation]] satisfied by divergences appearing in the [[Hopf algebra]] formulation of [[renormalization]] (see there).  This leads to a [[category]] of "equisingular flat connections" that turns out to be a [[Tannakian category]], meaning that it is equivalent to the [[category of modules]] over some pro-algebraic group $G$. The authors observe that $G$ [[action|acts]] on any renormalizable [[theory (physics)|theory]] in a nice way. Due to this property [[Pierre Cartier]] referred to this as the _cosmic Galois group_:

> La parenté de plus en plus manifeste entre le [[groupe de Grothendieck-Teichmüller]] d'une part, et le groupe de renormalisation de la Théorie Quantique des Champs n'est sans doute que la première manifestation d'un groupe de symétrie des constantes fondamentales de la physique, une espèce de groupe de Galois cosmique!" ([[Pierre Cartier]] according to [Connes 12](#Connes12), see also the end of [Cartier 01](#Cartier01)).

This cosmic Galois group $G$ is (non-canonically) isomorphic to some [[motivic Galois group]].

In ([Kitchloo-Morava 12](#KitchlooMorava12)) the cosmic Galois group is related to the motivic/[[Tannaka duality|Tannakian group]] of a motivic stabilization of the [[symplectic category]] of [[symplectic manifolds]] and [[Lagrangian correspondences]] between them, the _[[stable symplectic category]]_ ([Kitchloo 12](#Kitchloo12)).

## Related concepts

* [[motivic Galois group]]

* [[Grothendieck-Teichmüller group]]

## References

The "cosmic Galois group" in [[renormalization]] theory was introduced in

* {#ConnesMarcolli04} [[Alain Connes]], [[Matilde Marcolli]], _Renormalization and motivic Galois theory_ ([arXiv:math/0409306](http://arxiv.org/abs/math/0409306))

The name originates in

* {#Cartier01} [[Pierre Cartier]], _A mad day's work: from Grothendieck to Connes and Kontsevich The evolution of concepts of space and symmetry_, Bull. Amer. Math. Soc. __38__ (2001), 389-408, [pdf](http://www.ams.org/bull/2001-38-04/S0273-0979-01-00913-2/S0273-0979-01-00913-2.pdf)

A similar statement about a [[motivic Galois group]] acting on the space of [[deformation quantizations]] of a [[free field theory]] appeared earlier in

* [[Maxim Kontsevich]], _Operads and Motives in Deformation Quantization_, Lett. Math. Phys.48:35-72,1999 ([arXiv:math/9904055](http://arxiv.org/abs/math/9904055))

Review and exposition includes

* {#Connes12} [[Alain Connes]], talk at [Galois 200th birthday conference](http://www.galois.ihp.fr/manifestations/colloque/) January (2012) [pdf slides](http://www.alainconnes.org/docs/slidesgaloisihp1.pdf)

* {#Brown15} [[Francis Brown]], _Feynman Amplitudes and Cosmic Galois group_ ([arXiv:1512.06409](https://arxiv.org/abs/1512.06409))

* [[Francis Brown]], _Periods and Feynman amplitudes_ ([arXiv:1512.09265](https://arxiv.org/abs/1512.09265))

A technical review of aspects of this is in

* [[Jack Morava]], _The motivic Thom isomorphism_ ([web](http://www.math.uiuc.edu/K-theory/0654/),  [pdf](http://www.math.uiuc.edu/K-theory/0654/gamma2.pdf))

See also

* [[Paul Goerss]], _Toward a Kozmik Galois group_, talk at Northwester, April 2010 ([pdf](http://www.math.northwestern.edu/~pgoerss/midwest10/MidW010.pdf))

* {#Morava11} [[Jack Morava]], _The Cosmic Galois group as Koszul dual to Waldhausen's A(pt)_ ([arXiv:1108.4627](http://arxiv.org/abs/1108.4627))

A review of cosmic Galois/Grothendieck-Teichmüller group and a discussion from the perspective of the [[motivic symplectic category]] is section 5 of

* {#KitchlooMorava12} [[Nitu Kitchloo]], [[Jack Morava]], _The Grothendieck--Teichmüller group and the stable symplectic category_, 2012.  ([arxiv:1212.6905](http://arxiv.org/abs/1212.6905))

based on

* [[Nitu Kitchloo]], _The Stable Symplectic Category and Geometric Quantization_ ([arXiv:1204.5720](http://arxiv.org/abs/1204.5720v1))
 {#Kitchloo12}

* [MO discussion](http://mathoverflow.net/questions/94954/connes-kreimer-hopf-algebra-and-cosmic-galois-group)

[[!redirects cosmic Galois groups]]


cosmic inflation

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## Idea

In the context of [[cosmology]], _cosmic inflation_ is a [[model (in theoretical physics)]] that can explain certain large-scale features of the [[observable universe]] (flatness, horizon problem, CMB anisotropy) by assuming a finite period of drastic expansion of the universe shortly after the [[big bang]]. Cosmic inflation is part of the [[standard model of cosmology]].

The typical [[model (physics)|model]] of cosmic inflation adds to a standard [[FRW model]] simply a [[scalar field]] $\phi$ --  then called the **inflaton field** -- with standard kinetic term and some potential term. If the potential term is chosen suitably one can obtain solutions to [[Einstein's equations]] of this simple homogenous and isotropic model which exhibit "slow roll behaviour" for $\phi$, meaning that $\phi$ (homogeneous in space) starts out in the vicinity of the [[big bang]] with some finite value and then slowly "rolls down" its potential well (where one speaks in the analogy with the [[model (in theoretical physics)|model]] describing a single [[particle]] on the [[real line]] in the given potential, which has the same kind of [[action functional]]). Therefore in this "slow roll" period the contribution of $\phi$ to the [[FRW model]] is essentially that of a [[cosmological constant]] and so this drives the expansion of the "universe" in this model. But since $\phi$ is only approximately constant it eventually reaches the minimum of its potential well. Again, if the potential parameters of the model are chosen suitably one can arrange that it stays there (called the "graceful exit property" of the inflationary model) and so it stops driving the expansion of the "universe". In conclusion this yields variants of the [[FRW model]] that exhibit pronounced  expansion shortly after the initial singularity and then asymptote to the behaviour of the plain FRW model. This is what is called _cosmic inflation_.

Simple as it is, this model has proven to successfully match the observations that it was designed to match (the large-scale homogeneity of the [[observable universe]], notably). But of course people are trying all kinds of variants, too. A central conceptual problem of most of these models is that it is unclear what the field $\phi$ should be in terms of [[particle physics]] or other known phyisics. In some variants it is identified with the [[Higgs field]], in other it is a scalar moduli field of some [[Kaluza-Klein compactification]], but all of this is speculative.

The experimental data ([PlanckCollaboration 13](#PlanckCollaboration13), [BICEP-Keck-Planck 15](#BICEPKeckPlanck15), [PlanckCollaboration 15](#PlanckCollaboration15)) strongly favors the [[Starobinsky model of cosmic inflation]].

## Variants

### Old inflation

([Guth 91](#Guth), [Sato 81](#Sato))

### New inflation

([Linde 82](#Linde), [Albrecht-Steinhardt 82](#AlbrechtSteinhardt))

### Eternal inflation

* [[eternal inflation]]

### Chaotic inflation

[[chaotic inflation]]

([Linde 83](#Linde83))

## Candidates for the inflaton field

### Higgs inflation
 {#HiggsInflation}

The idea that the inflaton field in cosmology might be the [[Higgs field]] from the [[standard model of particle physics]] is as old as the idea of inflation itself, but at least in the naive versions it seems to be ruled out by data. However, with the experimental detection of the previously hypothesized Higgs field itself, the topic is gaining interest again and various variations are being proposed to solve the problems with the naive idea, for instance a small non-[[minimal coupling]] of the Higgs field to [[gravity]] (see e.g. [Atkins 12](#Atkins12), [Kamada 12](#Kamada12), [Kehagias 12](#Kehagias12)).

In particular, the near-criticality of the Higgs potential (see [there](Higgs+field#Curiously)) has been argued to be just the right condition to make Higgs inflation viable ([Jegerlehner 13](#Jegerlehner13), [Jegerlehner 14](#Jegerlehner14), [Jegerlehner 15](#Jegerlehner15), [Jegerlehner 18](#Jegerlehner18)), for review see also [Rubio 18 ](#Rubio18).

### Axion inflation

see _[[axion inflation]]_

### Higher curvature inflation (Starobinsky model)

It is possible that instead of the inflaton being a fundamental [[scalar field]], it is an effective result of [[higher curvature corrections]] to [[gravity]].

The first such $R^2$ correction leads to the _[[Starobinsky model of cosmic inflation]]_, which sits right in the middle of the parameter space preferred by the PLANCK satellite data.

Discussion of inflationary effects of ever higher curvature corrections includes [Arciniega-Edelstein-Jaime 18](#ArciniegaEdelsteinJaime18), [ABCEHJ 18](#ABCEHJ18).

### Ekpyrotic cosmology

See _[[ekpyrotic cosmology]]_.

## Related concepts

* [[eternal cosmic inflation]]

* [[cosmology]]

* [[FRW model]]

* [[big bang]], [[steady state model]]

[[!include fields and quanta - table]]

## References

### Reviews

* [[Andrei Linde]], _Particle Physics and Inflationary Cosmology_, Harwood, Chur (1990).

* A. R. Liddle, D. H. Lyth, _Cosmological inflation and large-scale structure_, Cambridge University Press (2000).

* Shinji Tsujikawa, _Introductory review of cosmic inflation_,  lecture notes given at The Second Tah Poe School on Cosmology _Modern Cosmology_, Naresuan (2003) ([arXiv:hep-ph/0304257](http://arxiv.org/abs/hep-ph/0304257)).

* Jerome Martin, Christophe Ringeval, Vincent Vennin, _Encyclopaedia Inflationaris_ ([arXiv:1303.3787](http://arxiv.org/abs/1303.3787))

* Jerome Martin, _The Theory of Inflation_ ([arXiv:1807.11075](https://arxiv.org/abs/1807.11075))

* Debika Chowdhury, Jerome Martin, Christophe Ringeval, Vincent Vennin, _Inflation after Planck: Judgment Day_ ([arXiv:1902.03951](https://arxiv.org/abs/1902.03951))

* Jerome Martin, Christophe Ringeval, Vincent Vennin, *Encyclopaedia Inflationaris*, Phys. Dark Univ. **5**-**6** (2014) 75-235 [[arXiv:1303.3787](https://arxiv.org/abs/1303.3787), [doi:10.1016/j.dark.2014.01.003](https://doi.org/10.1016/j.dark.2014.01.003)]

With emphasis on the [[Schwinger effect]]:

* Jerome Martin, _Inflationary Perturbations: the Cosmological Schwinger Effect_, Lect. Notes Phys. 738:193-241, 2008 ([arXiv:0704.3540](https://arxiv.org/abs/0704.3540), [doi:10.1007/978-3-540-74353-8_6](https://doi.org/10.1007/978-3-540-74353-8_6))

See also:

* Wikipedia, _[Inflation (cosmology)](http://en.wikipedia.org/wiki/Inflation_%28cosmology%29)_

### Original articles

* Demosthenes Kazanas, _Dynamics of the universe and spontaneous symmetry breaking_, Astrophysical Journal, Part 2 - Letters to the Editor, **241** (Oct. 15, 1980) L59-L63 [[doi:10.1086/183361](https://ui.adsabs.harvard.edu/link_gateway/1980ApJ...241L..59K/doi:10.1086/183361)]

* {#Starobisnky80} [[Aleksei Starobinsky]], *A new type of isotropic cosmological models without singularity*, Phys. Lett. B **91**  (1980) 99-102 \[doi:10.1016/0370-2693(80)90670-X\]

  > (cf. [[Starobinsky model of cosmic inflation]])

* {#Guth} [[Alan Guth]], Phys. Rev. D 23, 347 (1981).

* {#Sato} K. Sato, Mon. Not. R. Astron. Soc. 195, 467 (1981); Phys. Lett. 99B, 66 (1981)

* {#Linde} [[Andrei Linde]], Phys. Lett. 108B, 389 (1982)

* {#AlbrechtSteinhardt} A. Albrecht and [[Paul Steinhardt]], Phys. Rev. Lett. 48, 1220 (1982)

* {#Linde83} [[Andrei Linde]], Phys. Lett. 129B, 177 (1983).

On [[structure formation]] during inflation by [[inhomogeneous cosmology|inhomogeneous]] [[quantum cosmology]]:

* [[Martin Bojowald]], Ding Ding, *Canonical description of cosmological backreaction*, JCAP **03** (2021) 083
([arXiv:2011.03018](https://arxiv.org/abs/2011.03018))

In [[supersymmetric quantum cosmology]]:

* N.E. Martínez-Pérez, C. Ramírez, V.M. Vázquez Báez, *Phenomenological inflationary model in Supersymmetric Quantum Cosmology* [[arXiv:2208.04412](https://arxiv.org/abs/2208.04412)]

### Experimental evidence
 {#ReferencesExperimentalEvidence}

* C. L. Bennett et al. _First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results_, Astrophys.J.Suppl.148:1 (2003) ([arXiv:astro-ph/0302207](http://arxiv.org/abs/astro-ph/0302207))

* H .V. Peiris et al, _First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for Inflation_, Astrophys.J.Suppl.148:213,2003 ([arXiv:astro-ph/0302225](http://arxiv.org/abs/astro-ph/0302225))

* {#PlanckCollaboration13} [[Planck Collaboration]], _Planck 2013 results. XXII. Constraints on inflation_ ([arXiv:1303.5082](http://arxiv.org/abs/1303.5082))

  * Resonaances, _[Planck about inflation](http://resonaances.blogspot.de/2013/04/planck-about-inflation.html)_

  * {#Linde14} [[Andrei Linde]], _Inflationary Cosmology after Planck 2013_ ([arXiv:1402.0526](http://arxiv.org/abs/1402.0526))

* {#BICEPKeckPlanck15} _A Joint Analysis of BICEP2/Keck Array and Planck Data_ ([arXiv:1502.00612](http://arxiv.org/abs/1502.00612))

* {#PlanckCollaboration15} [[Planck Collaboration]], _Planck 2015, Overview of results_ ([pdf](http://www.cosmos.esa.int/documents/387566/522789/Planck_2015_Results_I_Overview_Products_Results.pdf/14d90465-03c2-4e3c-b8d1-73181248673d))

* Debika Chowdhury, Jerome Martin, Christophe Ringeval, Vincent Vennin, _Inflation after Planck: Judgment Day_ ([arxiv:1902.03951](https://arxiv.org/abs/1902.03951))

### Inflation from higher curvature corrections

Besides the references at _[[Starobinsky model of cosmic inflation]]_ the following discuss inflation driven by [[higher curvature corrections]]:

* {#ArciniegaEdelsteinJaime18} Gustavo Arciniega, Jose D. Edelstein, Luisa G. Jaime, _Towards purely geometric inflation and late time acceleration_ ([arXiv:1810.08166](https://arxiv.org/abs/1810.08166))

* {#ABCEHJ18} Gustavo Arciniega, Pablo Bueno, Pablo A. Cano, Jose D. Edelstein, Robie A. Hennigar, Luisa G. Jaimem, _Geometric Inflation_ ([arXiv:1812.11187](https://arxiv.org/abs/1812.11187))

### Higgs field inflation
 {#ReferencesHiggsInflation}

Literature discussing whether or how the [[Higgs field]] might be identified as the inflaton field includes

* {#Atkins12} Michael Atkins, _Could the Higgs boson be the inflaton?_, talk (March 2012) ([pdf](http://indico.cern.ch/getFile.py/access?resId=0&materialId=slides&confId=180122))

* {#Kamada12} Kohei Kamada, _Generalized Higgs inflation models_, talk at PLANCK 2012 (May 2012)([pdf](http://planck12.fuw.edu.pl/talks/kamada.pdf))

* {#Kehagias12} [[Alex Kehagias]], _New Higgs inflation_, talk (September 2012) ([pdf](http://www.physics.ntua.gr/corfu2010/Talks/kehagias@central_ntua_gr_01.pdf))

* Takehiro Nabeshima, _A model for Higgs inflation and its testability at the ILC_, talk (October 2012) ([pdf](http://ilcagenda.linearcollider.org/getFile.py/access?contribId=64&sessionId=8&resId=0&materialId=slides&confId=5468))

* {#Rubio18} Javier Rubio, _Higgs inflation_, Front. Astron. Space Sci. 5:50 (2019) ([arXiv:1807.02376](https://arxiv.org/abs/1807.02376))

A popular account in the context of the 2013 [[Plack Collaboration]] results is in

* [[Matthew Strassler]], _[Cosmic Conflation: The Higgs, The Inflaton, and Spin](http://profmattstrassler.com/2013/03/26/cosmic-conflation-the-higgs-the-inflaton-and-spin/)_

Discussion of Higgs inflation with emphasis on relation to the [near-criticality of the Higgs field](Higgs+field#Curiously):

* {#Jegerlehner13} [[Fred Jegerlehner]], _The hierarchy problem of the electroweak Standard Model revisited_ ([arXiv:1305.6652](https://arxiv.org/abs/1305.6652))

* {#Jegerlehner14} [[Fred Jegerlehner]], _Higgs inflation and the cosmological constant_, Acta Phys.Polon. B45 (2014) 1215-1254 ([arXiv:1402.3738](https://arxiv.org/abs/1402.3738))

* [[Fred Jegerlehner]], _About the role of the Higgs boson in the evolution of the early universe_ ([arXiv:1406.3658](https://arxiv.org/abs/1406.3658))

* {#Jegerlehner15} [[Fred Jegerlehner]], _The hierarchy problem and the cosmological constant problem in the Standard Model_ ([arXiv:1503.00809](https://arxiv.org/abs/1503.00809))

* {#Jegerlehner18} [[Fred Jegerlehner]], _The Hierarchy Problem and the Cosmological Constant Problem Revisited -- A new view on the SM of particle physics_ ([arXiv:1812.03863](https://arxiv.org/abs/1812.03863))

See also

* Tommi Tenkanen, Eemeli Tomberg, _Initial conditions for plateau inflation_ ([arXiv:2002.02420](https://arxiv.org/abs/2002.02420))

### Gauge field inflation
 {#ReferencesHiggsInflation}

Literature discussing whether or how [[gauge field]] might be identified as the inflaton field include

* A. Maleknejad, M. M. Sheikh-Jabbari, J. Soda, _Gauge Fields and Inflation_ ([arXiv:1212.2921](http://arxiv.org/abs/1212.2921))

### String modeled inflation

In [[string theory]] the [[inflaton]] field can be modeled by various effects, such as

* [[open string]] stretching between [[D-brane]]-[[anti D-brane]] pairs.

For review and further pointers to the literature see

* [[Cliff Burgess]], M. Cicoli, F. Quevedo, _String Inflation After Planck 2013_ ([arXiv:1306.3512](http://arxiv.org/abs/1306.3512))

* [[Daniel Baumann]], [[Liam McAllister]], _Inflation and String Theory_, Cambridge University Press (2015) [[arXiv:1404.2601](https://arxiv.org/abs/1404.2601), [doi:10.1017/CBO9781316105733]( https://doi.org/10.1017/CBO9781316105733)]

See also at _[[string phenomenology]]_.

[[!redirects inflation]]

[[!redirects inflaton]]
[[!redirects inflatons]]
[[!redirects inflaton field]]
[[!redirects inflaton fields]]

[[!redirects Higgs inflation]]


cosmic microwave background radiation

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## Idea

(...)

## Properties

### Hydrogen line

It has been argued ([Higan-Rees 79](#HiganRees79)) that observation of a absorption at the 21cm [[hydrogen line]] in the cosmic microwave background would be a signal from the time when the first [[stars]] formed and ignited, a cosmological epoch not otherwise accessible to observation. A claim of detection of this signal by the [[EDGES]] [[experiment]] appeared in [BRMMM 18](#BRMMM18).

## Related concepts

* [[cosmology]], [[standard model of cosmology]]

* [[inhomogeneous cosmology]]

* [[dark radiation]]

* [[hydrogen line]]

## References

### General

Lecture notes includes

* [[Daniel Baumann]], from section 3 on in _TASI Lectures on Primordial Cosmology_ ([arXiv:1807.03098](https://arxiv.org/abs/1807.03098))

See also the references at _[[cosmic inflation]]_.

See also

* Wikipedia, _[Cosmic microwave backround radiation](http://en.wikipedia.org/wiki/Cosmic_microwave_background_radiation)_

### On the hydron line

* {#HiganRees79} C. J. Hogan, [[Martin Rees]], _Spectral appearance of non-uniform gas at high $z$_, Monthly Notices of the Royal Astronomical Society, Volume 188, Issue 4, 1 October 1979, Pages 791–798 ([doi:10.1093/mnras/188.4.791]( https://doi.org/10.1093/mnras/188.4.791))

A claim of detection of the cosmological signal by the [[EDGES]] [[experiment]] is due to

* {#BRMMM18} Judd D. Bowman, Alan E. E. Rogers, Raul A. Monsalve, Thomas J. Mozdzen,  Nivedita Mahesh, _An absorption profile centred at 78 megahertz in the sky-averaged spectrum_, Nature volume 555, pages 67–70 (01 March 2018) ([doi:10.1038/nature25792](https://www.nature.com/articles/nature25792))

Discussion of this claim incluces

* {#Falkowski18} [[Adam Falkowski]], _[21cm to dark matter](http://resonaances.blogspot.fr/2018/03/21cm-to-dark-matter.html)_, March 2018

Applying [[topological data analysis]]/[[persistent homology]] to the CMB in search for signatures of [[inhomogeneous cosmology]]:

* Pratyush Pranav, [[Thomas Buchert]], *Homology reveals significant anisotropy in the cosmic microwave background* ([arXiv:2308.10738](https://arxiv.org/abs/2308.10738))

[[!redirects cosmic microwave background]]

[[!redirects CMB]]


cosmic reionization

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## Idea

(...)

## Related concepts

* [[cosmology]]

## References

See also

* Wikipedia, [Reionization](https://en.wikipedia.org/wiki/Reionization)

Simulation of [[cosmic structure formation]]/[[galaxy formation]] in feedback with cosmic reionization:

* [[Pierre Ocvirk]] et al. , _Cosmic Dawn (CoDa): the First Radiation-Hydrodynamics Simulation of Reionization and Galaxy Formation in the Local Universe_ ([arXiv:1511.00011](https://arxiv.org/abs/1511.00011))

* [[Pierre Ocvirk]], et al., _Cosmic Dawn II (CoDa II): a new radiation-hydrodynamics simulation of the self-consistent coupling of galaxy formation and reionization_ ([arXiv:1811.11192](https://arxiv.org/abs/1811.11192))

[[!redirects reionization]]


cosmic string

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## Idea

In a [[gauge theory]] with a degenerate [[vacuum]] (such as when a [[Higgs mechanism]] applies), the [[moduli space]] of [[vacua]] is the [[quotient]] $G/H$ (the [[coset]]) of the [[gauge group]] $G$ by the [[stabilizer subgroup]] $H \hookrightarrow G$ of any of these vacua ([[spontaneous symmetry breaking]]).

This means that gauge equivalence classes of vaccum configurations on a [[spacetime]] $X$ are given by [[homotopy]] classes of maps $X \to \Pi(G/H)$ (where the notation on the right denotes the underlying [[homotopy type]] of the [[coset]] space, $\Pi$ is the [[shape modality]]).

If [[spacetime]] is locally to be taken of the form $\mathbb{R} \times (\mathbb{R}^3 - D^2 \times \mathbb{R}^1)$, hence with a 1-dimensional ("string"-like) piece taken out, then homotopy classes of maps $X \to \Pi(G/H)$ are classified by the [[fundamental group]] $\pi_1(G/H)$. For a given nontrivial element here the correponding [[vacuum]] is said to contain a **cosmic string defect**. ("Cosmic" just because this effect is thought to be most relevant on scales of [[cosmology]].)

In other words this means that the vacuum strcture changes continuously as one moves around the string, but has a [[singularity]] on the locus of the string itself.

For more see at _[[QFT with defects]]_ the section _[Topological defects from spontaneously broken symmetry](http://ncatlab.org/nlab/show/QFT+with+defects#DefectsFromBrokenSymmetry)_.

## Related concepts

* [[domain wall]], [[monopole]]

* [[vortex]], [[vortex string]]

* [[defect brane]]

There is *no* direct relation to _[[strings]]_ in the sense of [[perturbative string theory]]; maybe to [[D1-branes]].

## References

* {#VilenkinShellard94} [[Alexander Vilenkin]], E. P. S. Shellard, *Cosmic strings and other topological defects*, Cambridge University Press (1994) [[ISBN:9780521654760](https://www.cambridge.org/ae/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/cosmic-strings-and-other-topological-defects?format=PB&isbn=9780521654760), [spire:1384873](https://inspirehep.net/literature/1384873)]

[[!redirects cosmic strings]]


cosmological constant

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## Idea

In theories of [[gravity]] and [[cosmology]] a _cosmological constant_ refers to an [[energy]] contribution associated with the [[vacuum]] itself.

This is called a _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  (eq:EinsteinHilbertWithCosmologicalConstant) below.

See below

* _[In classical gravity](#InClassicalGravity)_

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. [Hack 15, section 3.2.1](#Hack15)).

See below

* _[In perturbative quantum gravity](#InPerturbativeQuantumGravity)_

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

* _[In string theory](#ReferencesInStringTheory)_

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

* _[Observation](#Observation)_.

## Details

### In classical gravity
 {#InClassicalGravity}

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

$$
  S_{cc} \;\colon\;  (X,g) \mapsto \lambda \int_X dvol_g
  \,,
$$

where $\lambda \in \mathbb{R}$ is the _cosmological constant_ .

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

$$
  \label{EinsteinHilbertWithCosmologicalConstant}
  S_{EH} + S_{cc} :  (X,g) \mapsto \int_X R\, dvol + \lambda \int_X d vol_g
  \,,
$$

Generically it happens that one considers action functionals where $\lambda$ is in fact not a constant, but a function of other fields $\phi$ on $X$.

$$
  S \;\colon\; (X,g,\phi) \mapsto \int_X \lambda(\phi) dvol_g
  \,.
$$

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 _[[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 _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]].

### In perturbative quantum gravity
 {#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  ([Wald 78](#Wald78), [Tichy-Flanagan 98](#TichyFlanagan98) [Moretti 01](#Moretti01)).

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

A useful review is in ([Hack 15, section 3.2.1](#Hack15)).

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

The folklore discussion of the "cosmological constant problem" (see the references [below](#CCProblemReferences)) tends not to take this freedom in the theory into account (see the discussion at _[[naturalness]]_).

### In inhomogeneous cosmology
 {#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. [Buchert 07](#Buchert07), [Buchert 11](#Buchert11), [Buchert-Rasanen 11](#BucherRasanen11), [Scharf 13](#Scharf13)).

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

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

### Observation
 {#Observation}

<img src="http://ncatlab.org/nlab/files/DarkEnergyConcordance.gif" width="500">

(e.g. [Einasto 09, fig 17](#Einasto09))

## Related concepts

* [[phantom dark energy]]

* [[quintessence]]

* [[FRW model]]

* [[cosmology]]

* [[inhomogeneous cosmology]]

* [[Witten's Dark Fantasy]]

* [[energy]]

  * [[matter]], [[radiation]], **cosmological constant**

## References

### General

Introduction and review:

* Konstantinos Xenos, _An Introduction to FRW Cosmology and dark energy models_ ([arXiv:2101.06135](https://arxiv.org/abs/2101.06135))

Careful discussion in [[perturbative quantum gravity]]:

* [[John Donoghue]], *The cosmological constant and the use of cutoffs* ([arXiv:2009.00728](https://arxiv.org/abs/2009.00728))

### Tests

Tests to constrain the value of the cosmological constant are developed in:

* David Benisty, Anne-Christine Davis, N. Wyn Evans, *Constraining Dark Energy from Local Group dynamics*, The Astrophysical Journal Letters, Volume 953, Number 1, August 8, 2023. ([doi:10.3847/2041-8213/ace90b](https://doi.org/10.3847/2041-8213/ace90b), [arXiv:2306.14963](https://arxiv.org/abs/2306.14963))

* David Benisty, Jenny Wagner, Denitsa Staicova, *Dark Energy as a Critical Period in Binary Motion: Bounds from Multi-scale Binaries* ([arXiv:2310.11488](https://arxiv.org/abs/2310.11488))

### Observation

(...)

### Non-Observation
 {#NonObservation}

Cautioning against the interpretation of type Ia [[supernovae]] as indicative of a small positive cosmological constant ([[de Sitter spacetime]]) as in the current [[standard model of cosmology]] includes the following (see also at _[[inhomogeneous cosmology]]_ and at _[[Witten's Dark Fantasy]]_):

* [[Subir Sarkar]], _Is the evidence for dark energy secure?_, Gen. Rel. Grav. 40:269-284, 2008 ([arXiv:0710.5307](https://arxiv.org/abs/0710.5307))

* {#NielsenGuffantiSarkar16} J. T. Nielsen, A. Guffanti & [[Subir Sarkar]], _Marginal evidence for cosmic acceleration from Type Ia supernovae_, Nature Scientific Reports volume 6, Article number: 35596 (2016) ([arXiv:1506.01354](https://arxiv.org/abs/1506.01354), [web discussion](https://4gravitons.wordpress.com/2016/11/11/a-response-from-nielsen-guffanti-and-sarkar/))

* Koushik Dutta, Ruchika, Anirban Roy, Anjan A. Sen, M.M. Sheikh-Jabbari, _Negative Cosmological Constant is Consistent with Cosmological Data_ ([arXiv:1808.06623](https://arxiv.org/abs/1808.06623))

* Rui-Yun Guo, Jing-Fei Zhang, Xin Zhang, _Can the $H_0$ tension be resolved in extensions to ΛCDM cosmology?_ ([arXiv:1809.02340](https://arxiv.org/abs/1809.02340))

* Yijung Kang, Young-Wook Lee, Young-Lo Kim, Chul Chung, Chang Hee Ree, _Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology_, Astrophysical Journal  ([arXiv:1912.04903](https://arxiv.org/abs/1912.04903))

  exposition:

  _[New evidence shows that the key assumption made in the discovery of dark energy is in error](https://phys.org/news/2020-01-evidence-key-assumption-discovery-dark.amp?__twitter_impression=true)_

* Eleonora Di Valentino, Alessandro Melchiorri, Joseph Silk, _Cosmic Discordance: Planck and luminosity distance data exclude LCDM_ ([arXiv:2003.04935](https://arxiv.org/abs/2003.04935))

  (preferring [[phantom dark energy]])

* {#MRS21} [[Roya Mohayaee]], [[Mohamed Rameez]], [[Subir Sarkar]], *Do supernovae indicate an accelerating universe?* ([arXiv:2106.03119](https://arxiv.org/abs/2106.03119))

### In pAQFT
 {#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

* {#Wald78} [[Robert Wald]], _Trace anomaly of a conformally invariant quantum field in curved spacetime_, Phys. Rev. D 17, 1477 1978 ([doi:10.1103/PhysRevD.17.1477](https://doi.org/10.1103/PhysRevD.17.1477))

Notice that ([Wald 78](#Wald78)) is based on

* {#Wald77} [[Robert Wald]], _The back reaction effect in particle creation in curved spacetime_, Commun. Math. Phys. (1977) 54: 1. ([doi:10.1007/BF01609833](https://doi.org/10.1007/BF01609833))

which claimed _no_ freedom of renormalization, but [Wald 78](#Wald78) explains that this was due to a mistake inherited from a citation.

Further development of this includes

* {#TichyFlanagan98} Wolfgang Tichy, Eanna E. Flanagan, _How unique is the expected stress-energy tensor of a massive scalar field?_, Phys.Rev. D58 (1998) 124007 ([arXiv:gr-qc/9807015](https://arxiv.org/abs/gr-qc/9807015))

* {#Moretti01} [[Valter Moretti]], _Comments on the Stress-Energy Tensor Operator in Curved Spacetime_, Commun. Math. Phys. 232 (2003) 189-221 ([arXiv:gr-qc/0109048](https://arxiv.org/abs/gr-qc/0109048))

A useful review is in

* {#Hack15} [[Thomas-Paul Hack]], section 3.2.1, _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6))

Realization of renormalization of stress-energy/cosmological constant in concrete [[FRW models]] is discussed in

* {#DappiaggiFredenhagenPinamonti08} [[Claudio Dappiaggi]], [[Klaus Fredenhagen]], [[Nicola Pinamonti]], _Stable cosmological models driven by a free quantum scalar field_, Phys. Rev. D77:104015, 2008 ([arXiv:0801.2850](https://arxiv.org/abs/0801.2850))

* {#DappiagiHackMollerPinamonti10} [[Claudio Dappiaggi]], [[Thomas-Paul Hack]], Jan Möller, [[Nicola Pinamonti]], _Dark Energy from Quantum Matter_ ([arXiv:1007.5009](https://arxiv.org/abs/1007.5009))

Discussion specifically for the [[standard model of cosmology]] is in

* {#Hack13} [[Thomas-Paul Hack]], _The Lambda CDM-model in quantum field theory on curved spacetime and Dark Radiation_ ([arXiv:1306.3074](https://arxiv.org/abs/1306.3074))

More on subtleties of [[renormalization]] of the cosmological constant:

* [[John Donoghue]], _The cosmological constant and the use of cutoffs_ ([arXiv:2009.00728](https://arxiv.org/abs/2009.00728))

### From higher curvature corrections
 {#ReferencesFromHigherCurvatureCorrections}

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

* Michal Artymowski, Zygmunt Lalak, _Inflation and dark energy from $f(R)$ gravity_, JCAP09(2014)036 ([arXiv:1405.7818](https://arxiv.org/abs/1405.7818))

* Michal Artymowski, Zygmunt Lalak, Marek Lewicki, _Inflation and dark energy from $f(R)$ gravity_ ([arXiv:1510.04864](https://arxiv.org/abs/1510.04864))

### From inhomogeneous cosmology

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

* {#Buchert07} [[Thomas Buchert]], _Dark Energy from structure: a status report_, Gen.Rel.Grav.40:467-527, 2008 ([arXiv:0707.2153](http://xxx.lanl.gov/abs/0707.2153))

* {#GreenWald10} Stephen R. Green, [[Robert Wald]], _A new framework for analyzing the effects of small scale inhomogeneities in cosmology_, Phys.Rev.D83:084020, 2011 ([arXiv:1011.4920](https://arxiv.org/abs/1011.4920))

* {#Buchert11} [[Thomas Buchert]], _Toward physical cosmology: focus on inhomogeneous geometry and its non-perturbative effects_, Class.Quant.Grav.28:164007, 2011 ([arXiv:1103.2016](https://arxiv.org/abs/1103.2016))

* {#BuchertRasanen11} [[Thomas Buchert]], Syksy Rasanen, _Backreaction in late-time cosmology_, Annual Review of Nuclear and Particle Science 62 (2012) 57-79 ([arXiv:1112.5335](https://arxiv.org/abs/1112.5335))

* {#GreenWald13} Stephen Green, [[Robert Wald]], _Examples of backreaction of small scale inhomogeneities in cosmology_, Phys.Rev.D87:124037, 2013 ([arxiv:1304.2318](https://arxiv.org/abs/1304.2318))

* {#Scharf13} [[Günter Scharf]], _Inhomogeneous cosmology in the cosmic rest frame without dark stuff_, chapter 6 in the latest edition of _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001 ([arXiv:1312.2695](https://arxiv.org/abs/1312.2695))

* {#BuchertEtAl15} [[Thomas Buchert]] et. al, _Is there proof that backreaction of inhomogeneities is irrelevant in cosmology?_, Class. Quantum Grav. 32 215021, 2015 ([arXiv:1505.07800](https://arxiv.org/abs/1505.07800))

  exposition in _[The Universe is inhomogeneous. Does it matter?](https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/)_ CQG+, 2016

* {#BelejkoKorzynski16} Krzysztof Bolejko, Mikołaj Korzyński, _Inhomogeneous cosmology and backreaction: Current status and future prospects_, Int. J. Mod. Phys. D 26, 1730011 (2017) ([arXiv:1612.08222](https://arxiv.org/abs/1612.08222))

* {#DanielssonVanRiet18} [[Ulf Danielsson]], Thomas Van Riet,  _What if string theory has no de Sitter vacua?_ ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120))

See also at _[[inhomogeneous cosmology]]_.

### The "cosmological constant problem"
 {#CCProblemReferences}

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

* Subir Sarkar, _New results in cosmology_ ([arXiv:hep-ph/0201140](https://arxiv.org/abs/hep-ph/0201140))

* Stefanus Nobbenhuis, _The cosmological constant problem -- an inspiration for new physics_ PhD thesis (2006) ([web](http://igitur-archive.library.uu.nl/dissertations/2006-0615-200938/) [pdf](http://igitur-archive.library.uu.nl/dissertations/2006-0615-200938/c1.pdf))

* Joan Sola, _Cosmological constant and vacuum energy: old and new ideas_, J.Phys.Conf.Ser. 453 (2013) 012015 ([arXiv:1306.1527](https://arxiv.org/abs/1306.1527))

* Ioanna Kourkoulou, Alberto Nicolis, Guanhao Sun, _A technical analog of the cosmological constant problem and a solution thereof_ ([arXiv:2101.11620](https://arxiv.org/abs/2101.11620), [talk rec](https://t.co/0jV2L9R79q?amp=1))

### In string theory
  {#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]].

#### For fundamentally vanishing cc ("Witten's Dark Fantasy")
{#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]]):

* {#Witten00} [[Edward Witten]], _The Cosmological Constant From The Viewpoint Of String Theory_, lecture at [DM2000](http://inspirehep.net/record/972507) ([arXiv:hep-ph/0002297](https://arxiv.org/abs/hep-ph/0002297))

  (see p. 7)

* [[Edward Witten]], _Strong coupling and the cosmological constant_, Mod. Phys. Lett. A 10:2153-2156, 1995 ([arXiv:hep-th/9506101](https://arxiv.org/abs/hep-th/9506101))

* [[Edward Witten]], Section 3 of _Some Comments On String Dynamics_, talk at [Strings95](https://cds.cern.ch/record/305869) ([arXiv:hep-th/9507121](http://arxiv.org/abs/hep-th/9507121))

This suggestion was called _[[Witten's Dark Fantasy]]_ in [Heckmann-Lawrie-Lin-Zoccarato 19, Section 8](#HeckmannLawrieLinZoccarato19),
where a concrete realization of this scenario in [[F-theory]] is claimed:

* [[Cumrun Vafa]], Section 4.3 of: _Evidence for F-Theory_, Nucl. Phys. B469:403-418, 1996 ([arxiv:hep-th/9602022](https://arxiv.org/abs/hep-th/9602022))

* {#HeckmannLawrieLinZoccarato19} [[Jonathan Heckman]], Craig Lawrie, Ling Lin, [[Gianluca Zoccarato]], _F-theory and Dark Energy_, Fortschritte der Physik  ([arXiv:1811.01959](https://arxiv.org/abs/1811.01959), [doi:10.1002/prop.201900057]( https://doi.org/10.1002/prop.201900057))

* [[Jonathan Heckman]], Craig Lawrie, Ling Lin, Jeremy Sakstein, [[Gianluca Zoccarato]], _Pixelated Dark Energy_ ([arXiv:1901.10489](https://arxiv.org/abs/1901.10489))

#### For fundamentally non-vanishing 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:

* {#KachruKalloshLindeTrivedi03} [[Shamit Kachru]], [[Renata Kallosh]], [[Andrei Linde]], [[Sandip Trivedi]], _de Sitter Vacua in String Theory_, Phys.Rev.D68:046005, 2003 ([arXiv:hep-th/0301240](https://arxiv.org/abs/hep-th/0301240))

This observation led to the discussion of the "[[landscape of string theory vacua]]".

For review see

* {#Kallosh05} [[Renata Kallosh]], section 3 of _de Sitter vacua and the landscape of string theory_, J. Phys. Conf. Ser. 24 (2005) 87-110 ([spire](http://inspirehep.net/record/701033/))

However, consistency problems of these arguments are raised in

* {#DanielssonVanRiet18} [[Ulf Danielsson]], Thomas Van Riet,  _What if string theory has no de Sitter vacua?_ ([arXiv:1804.01120](https://arxiv.org/abs/1804.01120))

[[!redirects dark energy]]


cosmology

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Cosmology
+-- {: .hide}
[[!include cosmology -- contents]]
=--
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

_Cosmology_ is the study of the [[observable universe]] on its largest [[length scale|scales]].

## Related entries

[[cosmology -- contents]]

* [[redshift]], [[comoving time]]

* [[gravity]], [[standard model of particle physics]]

* [[observable universe]]

  * [[hydrogen line]]

* [[big bang]], [[steady state model of cosmology]]

* [[FRW model]], [[standard model of cosmology]]

* [[QCD cosmology]]

* [[structure formation]]

* [[cosmological constant]], [[dark energy]]

  * [[inhomogeneous cosmology]]

  * [[Witten's Dark Fantasy]]

* [[dark matter]], [[fuzzy dark matter]]

* [[cosmic inflation]]

  * [[chaotic inflation]], [[multiverse]]

* [[cosmic microwave background radiation]]

* [[quantum cosmology]], [[quantum gravity]]

* [[Kaluza-Klein compactification]]

* [[Randall-Sundrum model]]

* [intersecting D-brane model cosmology](intersecting+D-brane+model#Cosmology)

* [[brane gas cosmology]]

* [[ekpyrotic cosmology]]

* [string theory FAQ -- Does string theory tell us anything about cosmology?](string+theory+FAQ#InsightsIntoCosmology)

## References

See also the references at *[[standard model of cosmology]]*.

### General
 {#ReferencesGeneral}

Brief review:

* Jorge L. Cervantes-Cota, George Smoot, *Cosmology today -- A brief review*, AIP Conf. Proc. **1396** (2011) 28-52 [[arXiv:1107.1789](http://arxiv.org/abs/1107.1789), [doi:10.1063/1.3647524](https://doi.org/10.1063/1.3647524)]

Textbook accounts:

* [[P. J. E. Peebles]], *Principles of Physical Cosmology*, Princeton University Press (1993, 2020) [[ISBN:9780691209814](https://press.princeton.edu/books/paperback/9780691209814/principles-of-physical-cosmology), [pdf](http://fma.if.usp.br/~mlima/teaching/PGF5292_2021/Peebles_PPC.pdf)]

* [[John Peacock]], _Cosmological Physics_, Cambridge University Press (1998, 2012) [[doi:10.1017/CBO9780511804533](https://doi.org/10.1017/CBO9780511804533), [book webpage](http://www.roe.ac.uk/~jap/book.html)]

* [[Pietro Fré]], *Gravity, a Geometrical Course*, Volume 2: *Black Holes, Cosmology and Introduction to Supergravity*, Springer (2013) [[doi:10.1007/978-94-007-5443-0](https://doi.org/10.1007/978-94-007-5443-0)]

Lecture notes:

* [[Matthias Blau]], Part F of: *Lecture notes on general relativity* ([web](http://www.blau.itp.unibe.ch/GRLecturenotes.html))

* Oliver F. Piattella, *Lecture Notes in Cosmology*, UNITEXT for Physics, Springer (2018)  [[arXiv:1803.00070](https://arxiv.org/abs/1803.00070), [doi:10.1007/978-3-319-95570-4](https://doi.org/10.1007/978-3-319-95570-4)]

* [[Daniel Baumann]], _TASI Lectures on Primordial Cosmology_ ([arXiv:1807.03098](https://arxiv.org/abs/1807.03098))

* Gianluca Calcagni, Maria Grazia Di Luca, Tomáš Fodran, Lecture at [Corfu2021](http://www.physics.ntua.gr/corfu2021/)*Lectures on classical and quantum cosmology* [[arXiv:2202.13875](https://arxiv.org/abs/2202.13875), [spire:2039729](https://inspirehep.net/literature/2039729)]

See also:

* Ned Wright, _[Cosmology tutorial](http://www.astro.ucla.edu/~wright/cosmolog.htm)_

* Wikipedia, *[Cosmology](https://en.wikipedia.org/wiki/Cosmology)*

Emphasis on [[non-perturbative effects]]:

* Anna Ijjas, Frans Pretorius, [[Paul Steinhardt]], _Stability and the gauge problem in non-perturbative cosmology_,  Journal of Cosmology and Astroparticle Physics, Volume 2019, January 2019 ([arXiv:1809.07010](https://arxiv.org/abs/1809.07010), [doi:10.1088/1475-7516/2019/01/015](https://iopscience.iop.org/article/10.1088/1475-7516/2019/01/015))

### In terms of AQFT on curved spacetimes

Discussion of cosmology in terms of [[perturbative AQFT]] [[AQFT on curved spacetimes]] includes the following:

* {#FredenhagenHack13} [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], _Quantum field theory on curved spacetime and the standard cosmological model_ ([arXiv:1308.6773](https://arxiv.org/abs/1308.6773))

* {#BrunettiFredenhagenHackPinamontoRejzner16} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Thomas-Paul Hack]], [[Nicola Pinamonti]], [[Katarzyna Rejzner]], _Cosmological perturbation theory and quantum gravity_ ([arXiv:1605.02573](https://arxiv.org/abs/1605.02573))

* {#Hack15} [[Thomas-Paul Hack]], _Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes_, Springer 2016 ([arXiv:1506.01869](https://arxiv.org/abs/1506.01869), [doi:10.1007/978-3-319-21894-6](https://doi.org/10.1007/978-3-319-21894-6))

[[!redirects cosmology]]
[[!redirects cosmological]]


cosmology -- contents

**[[cosmology]]**

* [[gravity]]

* [[standard model of cosmology]]

* [[FRW model]]

* [[redshift]], [[comoving time]]

* [[big bang]], [[steady state model of cosmology]]

* [[inhomogeneous cosmology]]

* [[observable universe]]

  * [[hydrogen line]]

* [[QCD cosmology]]

* [[cosmic reionization]]

* [[cosmic structure formation]]

* [[cosmological constant]], [[dark energy]]

  * [[Witten's Dark Fantasy]]

* [[cosmic inflation]]

  * [[Starobinsky model of cosmic inflation]]

  * [[eternal inflation]]

    * [[false vacuum eternal inflation]]

  * [[chaotic inflation]]

  * [[multiverse]]

* [[dark matter]], [[fuzzy dark matter]]

* [[cosmic microwave background radiation]]

* [[quantum cosmology]], [[quantum gravity]]

* [[Randall-Sundrum model]]

* [intersecting D-brane model cosmology](intersecting+D-brane+model#Cosmology)

* [[brane gas cosmology]]

* [[ekpyrotic cosmology]]

* [string theory FAQ -- Does string theory tell us anything about cosmology?](string+theory+FAQ#InsightsIntoCosmology)