Randall–Sundrum model

The model is a braneworld theory developed while trying to solve the hierarchy problem of the Standard Model. It involves a finite five-dimensional bulk that is extremely warped and contains two branes: the Planckbrane (where gravity is a relatively strong force; also called "Gravitybrane") and the 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.

In this warped spacetime that is only warped along the fifth dimension, the graviton's 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.

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:

${\displaystyle \mathrm {d} s^{2}={\frac {1}{k^{2}y^{2}}}(\mathrm {d} y^{2}+\eta _{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }),}$

where k is some constant, and η has "−+++" metric signature. This space has boundaries at y = 1/k and y = 1/(Wk), with ${\displaystyle 0\leq 1/k\leq 1/(Wk)}$, 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,

${\displaystyle \varphi \ {\stackrel {\mathrm {def} }{=}}\ -{\frac {\pi \ln(ky)}{\ln(W)}},}$

so that

${\displaystyle 0\leq \varphi \leq \pi ,}$

and

${\displaystyle \mathrm {d} s^{2}=\left({\frac {\ln(W)}{\pi k}}\right)^{2}\,\mathrm {d} \varphi ^{2}+e^{\frac {2\ln(W)\varphi }{\pi }}\eta _{\mu \nu }\,\mathrm {d} x^{\mu }\,\mathrm {d} x^{\nu }.}$

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.

In 1998/99 Merab Gogberashvili published on arXiv a number of articles on a very similar theme.[3][4][5] 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 requirement, since the extra component of the Einstein field equations giving the localized solution for matter fields coincides with the one of the conditions of stability.

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.[6]

• DGP model
• Goldberger–Wise mechanism
• Kaluza–Klein theory
• ADD model
• Scientific importance of GW170817, a neutron-merger

• Randall, Lisa (2005). Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions. New York: HarperCollins. ISBN 978-0-06-053108-9.

• Lisa Randall's web page at Harvard University Archived 2013-04-13 at the Wayback Machine
• Raman Sundrum's web page at the University of Maryland