Editing Randall–Sundrum model

{{Short description|Extra-dimensional model of the universe}}
{{Redirect|Warp factor|the concept in [[Star Trek]]|Warp drive (Star Trek)}}
In [[physics]], '''Randall–Sundrum models''' (RS) (also called '''5-dimensional warped geometry theory''') are [[Mathematical model|model]]s that describe the world in terms of a [[warped geometry|warped-geometry]] higher-dimensional [[universe]], or more concretely as a 5-dimensional [[anti-de Sitter space]] where the [[elementary particle]]s (except the [[graviton]]) are localized on a (3&nbsp;+&nbsp;1)-[[dimension]]al [[Membrane (M-Theory)|brane]] or branes.

The two models were proposed in two articles in 1999 by [[Lisa Randall]] and [[Raman Sundrum]] because they were dissatisfied with the [[Universal extra dimension|universal extra-dimensional]] models then in vogue. Such models require two fine tunings; one for the value of the [[Brane cosmology|bulk]] [[cosmological constant]] and the other for the brane [[tension (physics)|tensions]]. Later, while studying RS models in the context of the [[AdS/CFT correspondence|anti-de Sitter / conformal field theory (AdS/CFT) correspondence]], they showed how it can be dual to [[Technicolor (physics)|technicolor model]]s.

The first of the two models, called '''RS1''', has a finite size for the extra dimension with two branes, one at each end.<ref>{{cite journal
 |last=Randall |first=Lisa |author2=Sundrum, Raman
 |year=1999
 |title=Large Mass Hierarchy from a Small Extra Dimension
 |journal=[[Physical Review Letters]]
 |volume=83 |issue=17 |pages=3370&ndash;3373
 |doi=10.1103/PhysRevLett.83.3370 |bibcode=1999PhRvL..83.3370R
 |arxiv = hep-ph/9905221 }}</ref> The second, '''RS2''', is similar to the first, but one brane has been placed infinitely far away, so that there is only one brane left in the model.<ref>{{cite journal
 |last=Randall |first=Lisa |author2=Sundrum, Raman
 |year=1999
 |title=An Alternative to Compactification
 |journal=Physical Review Letters |volume=83 |issue=23 |pages=4690–4693 |arxiv = hep-th/9906064 |bibcode=1999PhRvL..83.4690R|doi=10.1103/PhysRevLett.83.4690|s2cid=18530420 }}</ref>

==Overview==
The model is a [[braneworld]] theory developed while trying to solve the [[hierarchy problem]] of the [[Standard Model]]. It involves a finite five-dimensional [[Brane cosmology|bulk]] that is extremely warped and contains two [[branes]]: the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the [[Electronvolt|Tevbrane]] (our home with the Standard Model particles; also called "Weakbrane"). In this model, the two branes are separated in the not-necessarily large fifth dimension by approximately 16 units (the units based on the brane and bulk energies). The Planckbrane has positive brane energy, and the Tevbrane has negative brane energy. These energies are the cause of the extremely warped [[spacetime]].

==Graviton probability function==
In this warped spacetime that is ''only'' warped along the fifth dimension, the [[graviton]]'s [[wave function|probability function]] is extremely high at the Planckbrane, but it drops exponentially as it moves closer towards the Tevbrane. In this, gravity would be much weaker on the Tevbrane than on the Planckbrane.

==RS1 model==
The RS1 model attempts to address the [[hierarchy problem]]. The warping of the extra dimension is analogous to the warping of [[spacetime]] in the vicinity of a massive object, such as a [[black hole]]. This warping, or red-shifting, generates a large ratio of energy scales, so that the natural energy scale at one end of the extra dimension is much larger than at the other end:

:<math>\mathrm{d}s^2 = \frac{1}{k^2 y^2}(\mathrm{d}y^2 + \eta_{\mu\nu}\,\mathrm{d}x^\mu\,\mathrm{d}x^\nu),</math>

where ''k'' is some constant, and η has "−+++" [[metric signature]]. This space has [[Boundary (topology)|boundaries]] at ''y'' = 1/''k'' and ''y'' = 1/(''Wk''), with <math>0 \le 1/k \le 1/(Wk)</math>, where ''k'' is around the [[Planck scale]], ''W'' is the warp factor, and ''Wk'' is around a [[TeV]]. The boundary at ''y'' = 1/''k'' is called the '''Planck brane''', and the boundary at ''y'' = 1/(''Wk'') is called the '''TeV brane'''. The particles of the [[Standard Model]] reside on the TeV brane. The distance between both branes is only −ln(''W'')/''k'', though.

In another [[coordinate system]],

:<math>\varphi\ \stackrel{\mathrm{def}}{=}\ -\frac{\pi \ln(ky)}{\ln(W)},</math>

so that

:<math>0 \le \varphi \le \pi,</math>

and

:<math>\mathrm{d}s^2 = \left(\frac{\ln(W)}{\pi k}\right)^2\, \mathrm{d}\varphi^2 + e^\frac{2\ln(W)\varphi}{\pi} \eta_{\mu\nu}\,\mathrm{d}x^\mu\, \mathrm{d}x^\nu.</math>

==RS2 model==
The RS2 model uses the same geometry as RS1, but there is no TeV brane. The particles of the standard model are presumed to be on the Planck brane. This model was originally of interest because it represented an infinite 5-dimensional model, which, in many respects, behaved as a 4-dimensional model. This setup may also be of interest for studies of the [[AdS/CFT]] conjecture.

==Prior models==
In 1998/99 Merab Gogberashvili published on [[arXiv]] a number of articles on a very similar theme.<ref>M. Gogberashvili, "Hierarchy problem in the shell universe model", [https://arxiv.org/abs/hep-ph/9812296 arXiv:hep-ph/9812296].</ref><ref>M. Gogberashvili, "Our world as an expanding shell", [https://arxiv.org/abs/hep-ph/9812365 arXiv:hep-ph/9812365].</ref><ref>M. Gogberashvili, "Four dimensionality in noncompact Kaluza-Klein model", [https://arxiv.org/abs/hep-ph/9904383 arXiv:hep-ph/9904383].</ref> He showed that if the Universe is considered as a thin shell (a mathematical [[synonym]] for "brane") expanding in 5-dimensional space, then there is a possibility to obtain one scale for particle theory corresponding to the 5-dimensional [[cosmological constant]] and Universe thickness, and thus to solve the hierarchy problem. It was also shown that four-dimensionality of the Universe is the result of [[Stability theory|stability]] requirement, since the extra component of the [[Einstein field equations]] giving the localized solution for [[matter]] fields coincides with one of the conditions of stability.

==Experimental results==
In August 2016, experimental results from the LHC excluded RS gravitons with masses below 3.85 and 4.45 TeV for ˜k = 0.1 and 0.2 respectively and for ˜k = 0.01, graviton masses below 1.95 TeV, except for the region between 1.75 TeV and 1.85 TeV. Currently, the most stringent limits on RS graviton production.{{clarify|date=August 2016}}<ref>[[Compact Muon Solenoid|CMS]] Collaboration. [https://cds.cern.ch/record/2205245/files/EXO-16-027-pas.pdf "CMS Physics Analysis Summary"]. Accessed: August 4, 2016.</ref>

==See also==
{{Portal|Physics}}
*[[DGP model]]
*[[Goldberger–Wise mechanism]]
*[[Kaluza–Klein theory]]
*[[Large extra dimension|ADD model]]
*[[GW170817#Scientific importance|Scientific importance of GW170817]], a neutron star merger

==References==
{{Reflist}}

==Sources==
*{{cite book
 |last=Randall
 |first=Lisa
 |author-link=Lisa Randall
 |year=2005
 |title=Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions
 |publisher=[[HarperCollins]]
 |location=New York
 |isbn=978-0-06-053108-9
 |url-access=registration
 |url=https://archive.org/details/warpedpassagesun00rand_1
 }}

==External links==
*[http://www.physics.harvard.edu/people/facpages/randall Lisa Randall's web page at Harvard University] {{Webarchive|url=https://web.archive.org/web/20130413031711/http://www.physics.harvard.edu/people/facpages/randall.html |date=2013-04-13 }}
*[https://umdphysics.umd.edu/people/faculty/current/item/518-raman.html Raman Sundrum's web page at the University of Maryland]

{{DEFAULTSORT:Randall-Sundrum Model}}
[[Category:Physical cosmology]]
[[Category:Particle physics]]
[[Category:String theory]]
[[Category:Quantum gravity]]