Horndeski's theory

Horndeski's theory is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion.{{clarify|date=November 2017|reason=Second order in what?}} The theory was first proposed by Gregory Horndeski in 1974{{Cite journal|last=Horndeski|first=Gregory Walter|date=1974-09-01|title=Second-order scalar-tensor field equations in a four-dimensional space|journal=International Journal of Theoretical Physics|language=en|volume=10|issue=6|pages=363–384|doi=10.1007/BF01807638|issn=0020-7748|bibcode=1974IJTP...10..363H|s2cid=122346086}} and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy.{{Cite journal|last1=Clifton|first1=Timothy|last2=Ferreira|first2=Pedro G.|last3=Padilla|first3=Antonio|last4=Skordis|first4=Constantinos|date=March 2012|title=Modified Gravity and Cosmology|journal=Physics Reports|volume=513|issue=1–3|pages=1–189|doi=10.1016/j.physrep.2012.01.001|arxiv=1106.2476|bibcode=2012PhR...513....1C|s2cid=119258154}} Horndeski's theory contains many theories of gravity, including general relativity, Brans–Dicke theory, quintessence, dilaton, chameleon particle and covariant Galileon{{Cite journal|last1=Deffayet|first1=C.|last2=Esposito-Farese|first2=G.|last3=Vikman|first3=A.|date=2009-04-03|title=Covariant Galileon|journal=Physical Review D|volume=79|issue=8|pages=084003|doi=10.1103/PhysRevD.79.084003|issn=1550-7998|arxiv=0901.1314|bibcode=2009PhRvD..79h4003D|s2cid=118855364}} as special cases.

Action

Horndeski's theory can be written in terms of an action as{{Cite journal|last1=Kobayashi|first1=Tsutomu|last2=Yamaguchi|first2=Masahide|last3=Yokoyama|first3=Jun'ichi|date=2011-09-01|title=Generalized G-inflation: Inflation with the most general second-order field equations|journal=Progress of Theoretical Physics|volume=126|issue=3|pages=511–529|doi=10.1143/PTP.126.511|issn=0033-068X|arxiv=1105.5723|bibcode=2011PThPh.126..511K|s2cid=118587117}}

$S[g_{\mu\nu},\phi] = \int\mathrm{d}^{4}x\,\sqrt{-g}\left[\sum_{i=2}^{5}\frac{1}{8\pi G_{\text{N}}}\mathcal{L}_{i}[g_{\mu\nu},\phi]\,+\mathcal{L}_{\text{m}}[g_{\mu\nu},\psi_{M}]\right]$

with the Lagrangian densities

$\mathcal{L}_{2} = G_{2}(\phi,\, X)$

$\mathcal{L}_{3} = G_{3}(\phi,\,X)\Box\phi$

$\mathcal{L}_{4} = G_{4}(\phi,\,X)R+G_{4,X}(\phi,\,X)\left[\left(\Box\phi\right)^{2}-\phi_{;\mu\nu}\phi^{;\mu\nu}\right]$

$\mathcal{L}_{5} = G_{5}(\phi,\,X)G_{\mu\nu}\phi^{;\mu\nu}-\frac{1}{6}G_{5,X}(\phi,\,X)\left[\left(\Box\phi\right)^{3}+2{\phi_{;\mu}}^{\nu}{\phi_{;\nu}}^{\alpha}{\phi_{;\alpha}}^{\mu}-3\phi_{;\mu\nu}\phi^{;\mu\nu}\Box\phi\right]$

Here $G_N$ is Newton's constant, $\mathcal{L}_m$ represents the matter Lagrangian, $G_2$ to $G_5$ are generic functions of $\phi$ and $X$ , $R,G_{\mu\nu}$ are the Ricci scalar and Einstein tensor, $g_{\mu\nu}$ is the Jordan frame metric, semicolon indicates covariant derivatives, commas indicate partial derivatives, $\Box\phi \equiv g^{\mu\nu}\phi_{;\mu\nu}$ , $X\equiv -1/2g^{\mu\nu}\phi_{;\mu}\phi_{;\nu}$ and repeated indices are summed over following Einstein's convention.

Constraints on parameters

Many of the free parameters of the theory have been constrained, $\mathcal{L}_{1}$ from the coupling of the scalar field to the top field and $\mathcal{L}_{2}$ via coupling to jets down to low coupling values with proton collisions at the ATLAS experiment.{{Cite journal|last=ATLAS Collaboration|date=2019-03-04|title=Constraints on mediator-based dark matter and scalar dark energy models using $\sqrt{s}=13$ TeV $pp$ collision data collected by the ATLAS detector |journal=Jhep|volume=05|page=142|arxiv=1903.01400|doi=10.1007/JHEP05(2019)142|s2cid=119182921}} $\mathcal{L}_{4}$ and $\mathcal{L}_{5}$ , are strongly constrained by the direct measurement of the speed of gravitational waves following GW170817.{{Cite journal|last1=Lombriser|first1=Lucas|last2=Taylor|first2=Andy|date=2016-03-16|title=Breaking a Dark Degeneracy with Gravitational Waves|journal=Journal of Cosmology and Astroparticle Physics|volume=2016|issue=3|pages=031|doi=10.1088/1475-7516/2016/03/031|issn=1475-7516|arxiv=1509.08458|bibcode=2016JCAP...03..031L|s2cid=73517974}}{{Cite journal|last1=Bettoni|first1=Dario|last2=Ezquiaga|first2=Jose María|last3=Hinterbichler|first3=Kurt|last4=Zumalacárregui|first4=Miguel|date=2017-04-14|title=Speed of Gravitational Waves and the Fate of Scalar-Tensor Gravity|journal=Physical Review D|volume=95|issue=8|pages=084029|doi=10.1103/PhysRevD.95.084029|issn=2470-0010|arxiv=1608.01982|bibcode=2017PhRvD..95h4029B|s2cid=119186001}}{{cite journal|last1=Creminelli|first1=Paolo|last2=Vernizzi|first2=Filippo|date=2017-10-16|title=Dark Energy after GW170817|journal=Physical Review Letters|volume=119|issue=25|pages=251302|arxiv=1710.05877|doi=10.1103/PhysRevLett.119.251302|pmid=29303308|bibcode=2017PhRvL.119y1302C|s2cid=206304918}}{{cite journal|last1=Sakstein|first1=Jeremy|last2=Jain|first2=Bhuvnesh|date=2017-10-16|title=Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories|journal=Physical Review Letters|volume=119|issue=25|pages=251303|arxiv=1710.05893|doi=10.1103/PhysRevLett.119.251303|pmid=29303345|bibcode=2017PhRvL.119y1303S|s2cid=39068360}}{{Cite journal|last1=Ezquiaga|first1=Jose María|last2=Zumalacárregui|first2=Miguel|date=2017-12-18|title=Dark Energy After GW170817: Dead Ends and the Road Ahead|journal=Physical Review Letters|volume=119|issue=25|pages=251304|arxiv=1710.05901|doi=10.1103/PhysRevLett.119.251304|pmid=29303304|bibcode=2017PhRvL.119y1304E|s2cid=38618360}}{{Cite news|url=https://www.sciencenews.org/article/what-detecting-gravitational-waves-means-expansion-universe|title=What detecting gravitational waves means for the expansion of the universe|last=Grossman|first=Lisa|date=2017-10-24|work=Science News|access-date=2017-11-08|language=en}}

References

Category:General relativity

Horndeski's theory

Action

Constraints on parameters

See also

References