We present here the Randall-Sundrum model (RS model) which is a more succesfull extension of the Arkani-Hamed-Dimopoulos-Dvali model (VDD model). These modes are inspired by string theory and they aim to solve among other things the hierarchy problem, i.e. why gravity is so weak compared to the other forces. The most important ingredients of these models are 1) extra dimensions and 2) Dp-branes. My own interest in these models is the possibility that Dark Matter can be explained by the massive Kaluza-Klein gravitons obtained by compactification of the extra dimensions.
We consider one extra dimension labelled by $z$ given by the orbifold ${\bf S}^1/{\bf Z}_2$ and NOT by the circle ${\bf S}^1$.
In other words, we impose the two compactification requirements:
-Periodicity $y\longrightarrow y+2y_c$, $y_c=\pi r_c$. The range of the extra dimension is $[0,2\pi r_c]$.
-Orbifold symmetry $y\longrightarrow -y$. In other words, we identify $(x,y)=(x,-y)$ and this allows the range to be extended to $[-\pi r_c,+\pi r_c]$.
Now, we place two four-dimensional 3-branes located at the two fixed points of the orbifold $y=y_c$ (the IR-brane where the Standard Model and Dark Matter live) and $y=0$ (the UV-brane where Quantum gravity lives).
We use $\varphi=y/r_c$.
The five-dimensional spacetime (the bulk) is then bounded between these two branes. The action is given by
\[S=S_{\rm bulk}+S_{\rm IR}+S_{\rm UV}.\]
\[S_{\rm bulk}=\frac{1}{2}M_5^3\int d^4x \int_{-\pi}^{+\pi} d\varphi \sqrt{-G}(R-2\Lambda_{\rm bulk}).\]
\[S_{\rm IR}=\int d^4x\int_{-\pi}^{+\pi}d\varphi \sqrt{-g_{\rm IR}}(-V_{\rm IR}+{\cal L}_{\rm IR})\delta(\varphi-\pi).\]
\[S_{\rm UV}=\int d^4x\int_{-\pi}^{+\pi}d\varphi \sqrt{-g_{\rm UV}}(-V_{\rm UV}+{\cal L}_{\rm UV})\delta(\varphi).\]
The Dark Matter fields on the IR-brane are denoted by $\phi$ and they can only interact gravitationlly with the fields of the Standard Model. They are both contained in the Lagrangian density ${\cal L}_{\rm IR}$. This is the basic idea.
Randall-Sundrum model
The Einstein equations of motion derived from the above action can be solved by a a four-dimensional Poincare invariant background given by
\[ds^2=\exp(-2\sigma(\varphi))g_{\mu\nu}dx^{\mu}dx^{\nu}+dy^2.\]
The warp factor $\exp(-2\sigma)$ is given in terms of the warping parameter $\sigma$ which is such that
\[\sigma=|y|\sqrt{\frac{-\Lambda_{\rm bulk}}{6}}.\]
The vacuum energies on the branes must be related by
\[V_{\rm UV}=-V_{\rm IR}=6M_5^3 k~,~k=\frac{-\Lambda_{\rm bulk}}{6}.\]
The background solution is a five-dimensional anti-de Sitter spacetime ${\bf AdS}^5$ with radius $R=1/k$.
We substitute the solution in the action $S_{\rm bulk}$ and assuming that the fields do not depend on the extra dimension $y$ to obtain the effective action
\[S_{\rm bulk}=\int d^4x \int_{-y_c}^{+y_c} dy\frac{1}{2}M_5^3 e^{-4k|y|}\sqrt{-g^{(4)}}e^{2k|y|}R^{(4)}=\frac{M_5^3}{2k}(1-e^{-2k|y_c|})\int d^4x \sqrt{-g^{(4)}}R^{(4)}.\]
Thus, the four-dimensional Planck mass is given by
\[M_{\rm Pl}^2=\frac{M_5^3}{k}(1-e^{-2k|y_c|}).\]
The same considerations on the IR-brane gives the result that the effective gravitational coupling is suppressed by the warp factor, viz
\[\Lambda_{\rm eff}=\frac{1}{\sqrt{G}}=M_{\rm Pl}e^{-k|y_c|}=M_{\rm Pl}e^{-\pi kr_c}.\]
In fact all VEV (vacuum expectation values) are suppressed or more precisely redshifted on the IR-brane.
This is the core idea behind brane world scenarios or brane cosmology.
The Planck mass $M_{\rm Pl}$ is obviously at the Planck scale which is of order $10^{19}$ GeV. The hierarchy problem can then be solved by choosing $M_5$, $k=-\Lambda_{\rm bulk}/6$ and $y_c$ appropriately. But in this case (as opposed to ADD model) the size of the extra dimensions has no real impact on the ratio $M_{\rm Pl}/M_5$.
However, we can choose $\Lambda_{\rm eff}<<M_{\rm Pl}$ even for moderate choices of $kr_c$. For example, for $kr_c\sim 10$ the RS model can solve the hierarchy problem, i.e. $\Lambda_{\rm eff}$ is at the TeV scale with the choice $M_5\sim M_{\rm Pl}\sim k$.
The gravitational content of the Randall-Sundrum model is obtained by expanding the five-dimensional metric as follows
\[G_{MN}\longrightarrow G_{MN}+\kappa h_{MN}~,~\kappa=2/M_5^{3/2}.\]
The field content is a 1) a spin two tensor field (graviton), 2) a spin one vector field (can be made to vanish) and 3) a spin zero scalar field (radion). The graviton and the radion can also be made to decouple. Explicitly, we write
\[G_ {\mu\nu}=e^{-2k|y|-2\hat{u}}(\eta_{\mu\nu}+\kappa \hat{h}_{\mu\nu}) ~,~G_{55}=-(1+2\hat{u})^2~,~G_{\mu 5}=G_{5\mu}=0.\]
We will not discuss the radion field which measures the width of the extra dimension.
These fields are five-dimensional fields.
By integrating the extra dimension we get four-dimensional fields. This is Kaluza-Klein reduction which is given for the metric by the field expansion
\[\hat{h}_{\mu\nu}(x,y)=\sum_{n=0}^{\infty}\frac{1}{\sqrt{r_c}}h_{\mu\nu}^{(n)}(x)\psi_n(\varphi).\]
The massless single 5D-graviton is transformed into a tower of massive 4D-gravitons (Kaluza-Klein modes). These are the particles of the Dark Matter. The mode $n=0$ is the precisely the massless graviton of the theory of general relativity. The wave functions $\psi_n$ are determined in terms of the mass $m_n$ of the $n-$th graviton by the equation
\[\frac{d}{d\varphi}\bigg(e^{-4kr_c|\varphi|}\frac{d\psi_n}{d\varphi}\bigg)=-r_c^2m_n^2e^{-2kr_c|\varphi|}\psi_n.\]
The equation of motion of the $n-$th massive graviton is precisely the Pauli-Fierz equation of massive gravity given by
\[(\eta_{\mu\nu}\partial^{\mu}\partial^{\nu}+m_n^2)h_{\mu\nu}^{(n)}(x)=0.\]
The masses of the KK-graviton modes are given by
\[m_n=kx_ne^{-\pi kr_c}~,~J_1(x_n)=0.\]
In other words, the masses $m_n$ (or more precisely $x_n$) are the roots of the Bessel function $J_1$.
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References:
1907.04340
2012.09672
2107.14548
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