Randall-Sundrum Cosmology and Dp-Brane Dynamics

Let me work in the large-warped Randall-Sundrum D-brane scenario and consider how an {S^1}/{Z_2} orbifolding leads to a non-singular cosmic bounce on the brane thus alleviating the rip of the big-bang singularity. The Randall-Sundrum 5-D braneworld action is:

    \[\begin{array}{*{20}{c}}{S = \int {{d^5}} x\sqrt { - {g_{\left( 5 \right)}}} \left[ {{M^3}{R_{\left( 5 \right)}} - \Lambda } \right]}\\{ + \int {{d^4}x} \sqrt { - {g_{\left( 4 \right)}}} {\rm{ \tilde L}}_{Brane}^{{\rm{large}}}}\end{array}\]

M the 5-D Planck mass, {M^3} = 1/\left( {16\pi {G_5}} \right), and \Lambda the cosmological constant in the bulk, yielding the metric on the brane:

    \[\begin{array}{c}d{s^2} = {e^{ - 2\left| y \right|/L}}\left[ {{\eta _{\mu \nu }} + {h_{\mu \nu }}\left( {x,y} \right)} \right] \cdot \\d{x^\mu }d{x^\nu } + d{y^2}\end{array}\]

with L being the radius of AdS, defined by:

    \[R_{MNPQ}^{\left( 5 \right)} = - \frac{1}{{{L^2}}}\left( {g_{MP}^{\left( 5 \right)}g_{NQ}^{\left( 5 \right)} - g_{MQ}^{\left( 5 \right)}g_{NP}^{\left( 5 \right)}} \right)\]

thus allowing us to derive the CFT-brane relation:

    \[\begin{array}{l}{h_{\mu \nu }}\left( p \right) = - \frac{2}{{L{M^3}}}{\Delta _3}\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{2}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\\ - \frac{1}{{{M^3}}}{\Delta _{KK}}\left( p \right)\left[ {{T_{\mu \nu }}\left( p \right) - \frac{1}{3}{\eta _{\mu \nu }}{T^\alpha }_\alpha \left( p \right)} \right]\end{array}\]

where the dynamics of N-parallel topologically intersecting Dp-branes with gauge group U\left( N \right) is:

    \[\begin{array}{l}S = - \frac{{{T_p}{g_s}{{\left( {2\pi \alpha '} \right)}^2}}}{4}\int {{{\rm{d}}^{p + 1}}} \xi {\rm{tr}}\left( {{F_{ab}}{F^{ab}} + } \right.\\2{D_a}{\Phi ^m}{D^a}{\Phi _m} + \sum\limits_{m \ne n} {{{\left[ {{\Phi ^m},{\Phi ^n}} \right]}^2} + \left. {{\rm{fermions}}} \right)} \end{array}\]

with the Yang-Mills potential being:

    \[V\left( \Phi \right) = {\sum\limits_{m \ne n} {\left[ {{\Phi ^m},{\Phi ^n}} \right]} ^2}\]

and we have:

    \[{F_{ab}} = {D_a}{\Phi ^m} = {\psi ^\alpha } = 0\]

    \[V\left( \Phi \right) = 0\]

The Einstein field equations in the RS theory are derived for both, the 2-stack 3-branes as well as inter-brane separation. Our 5-D ADS spacetime geometry is an

    \[{S^1}/{Z_2}\]

orbifolding and our branes are localized at orbifolded fixed points:

    \[\left\{ {\begin{array}{*{20}{c}}{\varphi = 0}\\{\varphi = \pi }\end{array}} \right.\]

with \varphi the Planck brane. The action is hence given by:

    \[\begin{array}{l}S = \frac{1}{{2{\kappa ^2}}}\int {{d^4}} xd\varphi \sqrt { - G} \left[ {{R^{\left( 5 \right)}} + \left( {12/{l^2}} \right)} \right]\\ - \int {{d^4}} x\left[ {\sqrt { - {g_{hid}}} {V_{hid}} + \sqrt { - {g_{vis}}} {V_{vis}}} \right]\end{array}\]

with the metric being:

    \[d{s^2} = {\tilde b^2}\left( x \right)d{\varphi ^2} + {e^{ - 2A\left( {\varphi ,x} \right)}}{h_{\mu \nu }}\left( x \right)d{x^\mu }d{x^\nu }\]

with

    \[A\left( {\varphi ,x} \right)\]

the spacetime warped brane factor along the extra dimensions. From this, one gets the Einstein field equations as:

    \[\begin{array}{l}\frac{{{e^{ - 2\xi }}}}{{\tilde b}}{\left( {K_\nu ^\mu } \right)_{,\varphi }} - {e^{ - 2\xi }}KK_\nu ^\mu + R_\nu ^{\left( 4 \right)}\left( h \right)\\ - {\nabla ^\mu }{\nabla _\nu }\xi - {\nabla ^\mu }\xi {\nabla _\nu }\xi = - \frac{4}{{{l^2}}}\delta _\nu ^\mu + {\kappa ^2}\left( {\frac{1}{3}\delta _\nu ^\mu {V_{hid}}} \right)\frac{{{e^{ - \xi }}}}{{\tilde b}}\delta \left( \varphi \right)\\ + {\kappa ^2}\left( {\frac{1}{3}\delta _\nu ^\mu {V_{vis}}} \right)\frac{{{e^{ - \xi }}}}{{\tilde b}}\delta \left( {\varphi - \pi } \right)\end{array}\]

 

    \[\begin{array}{l}\frac{{{e^{ - 2\xi }}}}{{\tilde b}}{K_{,\varphi }} - {e^{ - 2\xi }}{K^{\mu \nu }}{K_{\mu \nu }} - {\nabla ^\mu }{\nabla _\mu }\xi - {\nabla ^\mu }\xi {\nabla _\mu }\xi \\ = - \frac{4}{{{l^2}}} + \frac{{4{\kappa ^2}}}{3}{V_{hid}}\frac{{{e^{ - \xi }}}}{{\tilde b}}\delta \left( \varphi \right)\\ - \frac{{4{\kappa ^2}}}{3}{V_{vis}}\frac{{{e^{ - \xi }}}}{{\tilde b}}\delta \left( {\varphi - \pi } \right)\end{array}\]

with the constraint:

    \[{\nabla _\nu }\left( {{e^{ - \xi }}K_\nu ^\mu } \right) - {\nabla _\mu }\left( {{e^{ - \xi }}K} \right) = 0\]

We also impose the condition that the brane curvature radius L is much larger than the bulk curvature l:

    \[\varepsilon = {\left( {\frac{l}{L}} \right)^2} \ll 1\]

Visually, we get the following model:

First, let us recall the remarkable way in which the Randall-Sundrum scenario solves the hierarchy problem. I will make it easy by simply quoting Graham D. Kribs:

Randall and Sundrum (RS) proposed a fascinating solution to the hierarchy problem. The setup involves two 4D surfaces (“branes”) bounding a slice of 5D compact AdS space taken to be on an S1/Z2 orbifold. Gravity is effectively localized one brane, while the Standard Model (SM) fields are assumed to be localized on the other. The wavefunction overlap of the graviton with the SM brane is exponentially suppressed, causing the masses of all fields localized on the SM brane to be exponentially rescaled. The hierarchy problem can be solved by assuming all fields initially have masses near the 4D Planck scale, and arranging that the exponential suppression rescales the Planck mass to a TeV on the SM brane. This requires stabilizing the size of the extra dimension to be about thirty-five times larger than the AdS radius. Goldberger and Wise proposed adding a massive bulk scalar field with suitable brane potentials causing it to acquire a vev with a nontrivial x5-dependent profile. The desired exponential suppression could be obtained without any large fine-tuning of parameters. Fluctuations about the stabilized RS model include both tensor and scalar modes

Now, integrating over the dimensions yields the effective 4-D action:

    \[\begin{array}{l}{S_{eff}} = \frac{l}{{2{\kappa ^2}}}\int {{d^4}} x\sqrt { - f} \left[ {\Phi \left( x \right)} \right.{R^{\left( 4 \right)}}\left( f \right)\\ + \frac{3}{{2\left( {1 + \Phi } \right)}}{h^{\mu \nu }}{\partial _\mu }\Phi \left. {{\partial _\nu }\Phi } \right]\end{array}\]

with:

    \[\Phi \left( x \right) = \left[ {{e^{2\pi \frac{{\tilde b\left( x \right)}}{l}}} - 1} \right]\]

and since:

    \[{R^{\left( 4 \right)}}\left( f \right)\]

is the induced RS visible-sector brane Ricci scalar, we have a Brans-Dicke type theory. Our metric equation:

    \[d{s^2} = {\tilde b^2}\left( x \right)d{\varphi ^2} + {e^{ - 2A\left( {\varphi ,x} \right)}}{h_{\mu \nu }}\left( x \right)d{x^\mu }d{x^\nu }\]

splits the hidden and the visible RS sectors along a path as such:

    \[d\left( x \right) = \int_0^\pi {d\varphi \tilde b} \left( x \right) = \pi \tilde b\left( x \right)\]

and d\left( x \right) is our 4-D modulus field and is identical to the field \Phi \left( x \right) occurring in:

    \[\begin{array}{l}{S_{eff}} = \frac{l}{{2{\kappa ^2}}}\int {{d^4}} x\sqrt { - f} \left[ {\Phi \left( x \right)} \right.{R^{\left( 4 \right)}}\left( f \right)\\ + \frac{3}{{2\left( {1 + \Phi } \right)}}{h^{\mu \nu }}{\partial _\mu }\Phi \left. {{\partial _\nu }\Phi } \right]\end{array}\]

From the effective action, one can derive the scalar and gravitational equations of motion:

    \[\begin{array}{l}\Phi {E_{\mu \nu }} + {f_{\mu \nu }}\left[ {{\diamondsuit ^{d'Ale}}\Phi + \frac{3}{{4\left( {1 + \Phi } \right)}}{\nabla _\alpha }\Phi {\nabla ^\alpha }\Phi } \right]\\ - {\nabla _\mu }\nabla \nu \Phi - \frac{3}{{2\left( {1 + \Phi } \right)}}{\nabla _\mu }\Phi {\nabla _\nu }\Phi = 0\end{array}\]

 

    \[\frac{3}{{\left( {1 + \Phi } \right)}}{\diamondsuit ^{d'Ale}}\Phi - \frac{3}{{2{{\left( {1 + \Phi } \right)}^2}}}{\nabla _\mu }\Phi {\nabla ^\mu }\Phi = 0\]

where {E_{\mu \nu }} is the Einstein tensor and the corresponding covariant derivatives emerge on the visible-sector brane metric {f_{\mu \nu }}.

Visually:

Now take the Friedmann–Robertson–Walker brane-metric:

    \[ds_{\left( 4 \right)}^2 = {f_{\mu \nu }}d{x^\mu }d{x^\nu } = - d{t^2} + {a^2}\left( t \right)\left[ {\frac{{d{r^2}}}{{\left( {1 + {r^2}} \right)}} + {r^2}d{\Omega ^2}} \right]\]

with a\left( t \right) the cosmic scale factor and we switched to polar coordinates. Given the ansatz defined above by our metric, our field equations reduce to:

    \[{H^2} = \frac{1}{{{a^2}}} - H\frac{{\dot \Phi }}{\Phi } + \frac{{{{\left( {\dot \Phi } \right)}^2}}}{{4\Phi \left( {1 + \Phi } \right)}}\]

with overdot being {\partial _t} and H = \dot a/a is the Hubble parameter and \Phi is the 4-D RS modulus. We introduce conformal time defined via:

    \[ad\eta = dt\]

Hence, we …