Bulk gravitational field and cosmological perturbations on the brane

Abstract

We investigate the effect of the bulk gravitational field on the cosmological perturbations on a brane embedded in the 5D Anti-de Sitter (AdS) spacetime. The effective 4D Einstein equations for the scalar cosmological perturbations on the brane are obtained by solving the perturbations in the bulk. Then the behaviour of the corrections induced by the bulk gravitational field to the conventional 4D Einstein equation are determined. Two types of the corrections are found. First we investigate the corrections which become significant at scales below the AdS curvature scales and in the high energy universe with the energy density larger than the tension of the brane. The evolution equation for the perturbations on the brane is found and solved. Another type of the corrections is induced on the brane if we consider the bulk perturbations which do not contribute to the metric perturbations but do contribute to the matter perturbations. At low energies, they have imaginary mass m 2 = −(2/3)k 2 in the bulk where k is the 3D comoving wave number of the perturbations. They diverge at the horizon of the AdS spacetime. The induced density perturbations behave as sound waves with sound velocity 1/ √ 3 in the low energy universe. At large scales, they are homogeneous perturbations that depend only on time and decay like radiation. They can be identified as the perturbations of the dark radiation. They produce isocurvature perturbations in the matter dominated era. Their effects can be observed as the shifts of the location and the height of the acoustic peak in the CMB spectrum.

Closed system of equations on a brane

“…(20)-(23) below], without the necessity of solving field equations in the bulk. A similar observation in the cosmological context was previously made in [2] and recently noted in [3] for the linearized equations in the case of Z 2 symmetry.…”

Mass spectrum of primordial black holes from inflationary perturbation in the Randall–Sundrum braneworld: a limit on blue spectra

“…Four-dimensional inflationary theories predict that these values are determined by the inflaton expectation-value fluctuation and are almost constant outside the cosmological horizon [47]. On the other hand, in the five-dimensional Randall-Sundrum cosmology, a non-adiabatic Kaluza-Klein mode contributes to ζ and generally it varies with time [32,48,49,50], which can modify the resulting matter spectrum. In addition, the initial value measured at the exit of horizon during inflation is also modified by the higher-dimensional effect [51,52,53].…”

“…We focus on super-horizon regime and, for the reasons described in the previous subsection, omit anisotropic stress, dark radiation including its perturbation, and isocurvature. Then it follows that the gauge-invariant curvature perturbation ζ conserves [48,49,50], which leads to, in the longitudinal gauge [59], metric perturbation Φ (now equals to −Ψ) is also constant [49]:…”

Holographic View of Cosmological Perturbations

“…In the brane world cosmology, in general, we must consider the initial and boundary conditions at the same time (Koyama and Soda, 2000;Koyama and Soda, 2002). For this reason, it is difficult to deduce the predictions such as CMB fluctuations.…”