Randall–Sundrum model ($d{s}^2=e^{-2k|y|}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }+d{y}^2$)

A warped five-dimensional geometry redshifts masses of fields localized on the distant brane: $m_{\mathrm{IR}}=e^{-kL}{m}_0$. For the usual doubled orbifold interval, integration of the Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_5^3(1-e^{-2kL})/k$. Exponential redshift can generate a large separation of observable scales with a moderately sized dimensionless distance $kL$. Stabilizing the radion remains a dynamical requirement.

// Target: physics.bigb

For a spatially flat, tuned one-brane background without energy transfer to the bulk, the Israel junction conditions produce the displayed modified Friedmann equation. Positive brane tension $\lambda$ gives ordinary expansion when $\rho \ll \lambda$, while the quadratic density correction enhances expansion at high energy. Big Bang nucleosynthesis, early acoustic evolution and primordial spectra can constrain this change. Spatial curvature, a residual cosmological constant and bulk energy exchange require their additional terms rather than this special formula.

Projected bulk Weyl tensor can contribute a radiation-scaling term to the brane Friedmann equation, even without ordinary brane photons. For an anti-de Sitter black-hole bulk the integration constant is related to its mass. This dark radiation on a cosmological brane changes the expansion rate tested by Big Bang nucleosynthesis and Cosmic microwave background anisotropy; it should not automatically be identified with a freely specifiable thermal particle species.

In a one-brane warped background, the proper extra dimension can be infinite while the four-dimensional graviton zero mode has finite norm. The Einstein-Hilbert action weighs its four-dimensional curvature term by the warp factor integral shown above, finite for $k>0$. Thus the proper length is not the effective gravitational volume. This differs from an unwarped noncompact direction, whose volume integral diverges. It also differs from the two-brane hierarchy mechanism, where finite brane separation redshifts masses on the distant brane. The noncompact construction is established in the original single-brane warped-gravity paper.

An inertial observer of a fixed-$y$ flat brane in metric $d{y}^2+e^{-2y/L}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ is accelerated relative to the higher-dimensional bulk. Normalization and ${\Gamma }_{\mu \nu }^y=e^{-2y/L}{\eta }_{\mu \nu }/L$ give $a^y=-1/L$. This is its invariant bulk proper acceleration, not a nonzero intrinsic four-dimensional acceleration on the flat induced brane metric.