Brane-world cosmological perturbations: a covariant approach

The standard cosmological model, based on general relativity with an inflationary era, is very effective in accounting for a broad range of observed features of the universe. However, the ongoing puzzles about the nature of dark matter and dark energy, together with the problem of a fundamental theoretical framework for inflation, indicate that cosmology may be probing the limits of validity of general relativity. The early universe provides a testing ground for theories of gravity, since gravitational dynamics can lead to characteristic imprints on the CMB and other cosmological observations. Precision cosmology is in principle a means to constrain and possibly falsify candidate quantum gravity theories like M theory. Generalized Randall-Sundrum brane-worlds provide a phenomenological means to test aspects of M theory. I outline the 1+3-covariant approach to cosmological perturbations in these brane-worlds, and its application to CMB anisotropies.Comment: Based on a talk at the Brane-world Gravity Workshop, Yukawa Institute, Jan 2002. 17 pages, 4 figure

Brane-world gravity

The observable universe could be a 1+3-surface (the "brane") embedded in a 1+3+d-dimensional spacetime (the "bulk"), with standard-model particles and fields trapped on the brane while gravity is free to access the bulk. At least one of the d extra spatial dimensions could be very large relative to the Planck scale, which lowers the fundamental gravity scale, possibly even down to the electroweak (~ TeV) level. This revolutionary picture arises in the framework of recent developments in M theory. The 1+10-dimensional M theory encompasses the known 1+9-dimensional superstring theories, and is widely considered to be a promising potential route to quantum gravity. General relativity cannot describe gravity at high enough energies and must be replaced by a quantum gravity theory, picking up significant corrections as the fundamental energy scale is approached. At low energies, gravity is localized at the brane and general relativity is recovered, but at high energies gravity "leaks" into the bulk, behaving in a truly 1+3+d-dimensional way. This introduces significant changes to gravitational dynamics and perturbations, with interesting testable implications for high-energy astrophysics, black holes and cosmology. Brane-world models offer a phenomenological way to test some of the novel predictions and corrections to general relativity that are implied by M theory. This review discusses the geometry, dynamics and perturbations of simple brane-world models for cosmology and astrophysics, mainly focusing on warped 5-dimensional brane-worlds based on the Randall-Sundrum models.Comment: Corrections, improvements, new references; to appear in Living Reviews of Relativity; 58 pages, revtex4, 13 fig

Is the Universe homogeneous?

The standard model of cosmology is based on the existence of homogeneous surfaces as the background arena for structure formation. Homogeneity underpins both general relativistic and modified gravity models and is central to the way in which we interpret observations of the CMB and the galaxy distribution. However, homogeneity cannot be directly observed in the galaxy distribution or CMB, even with perfect observations, since we observe on the past lightcone and not on spatial surfaces. We can directly observe and test for isotropy, but to link this to homogeneity, we need to assume the Copernican Principle. First, we discuss the link between isotropic observations on the past lightcone and isotropic spacetime geometry: what observations do we need to be isotropic in order to deduce spacetime isotropy? Second, we discuss what we can say with the Copernican assumption. The most powerful result is based on the CMB: the vanishing of the dipole, quadrupole and octupole of the CMB is sufficient to impose homogeneity. Real observations lead to near-isotropy on large scales - does this lead to near-homogeneity? There are important partial results, and we discuss why this remains a difficult open question. Thus we are currently unable to prove homogeneity of the Universe on large-scales, even with the Copernican Principle. However we can use observations of the CMB, galaxies and clusters to test homogeneity itself.Comment: Based on an invited talk at a Theo Murphy Meeting "Testing general relativity with cosmology". Minor corrections, references updated. To appear Phil. Trans. R. Soc.

Geometry and dynamics of the brane-world

Recent developments in string theory have led to 5-dimensional warped spacetime models in which standard-model fields are confined to a 3-brane (the observed universe), while gravity can propagate in the fifth dimension. Gravity is localized near the brane at low energies, even if the extra dimension is noncompact. A review is given of the classical geometry and dynamics of these brane-world models. The field equations on the brane modify the general relativity equations in two ways: local 5-D effects are imprinted on the brane as a result of its embedding, and are significant at high energies; nonlocal effects arise from the 5-D Weyl tensor. The Weyl tensor transmits tidal (Coulomb), gravitomagnetic and gravitational wave effects to the brane from the 5-D nonlocal gravitational field. Local high-energy effects modify the dynamics of inflation, and increase the amplitude of scalar and tensor perturbations generated by inflation. Nonlocal effects introduce new features in cosmological perturbations. They induce a non-adiabatic mode in scalar perturbations and massive modes in vector and tensor perturbations, and they can support vector perturbations even in the absence of matter vorticity. In astrophysics, local and nonlocal effects introduce fundamental changes to gravitational collapse and black hole solutions.Comment: Minor corrections to Eqs. 14, 42, 64, and reference update

Covariant velocity and density perturbations in quasi-Newtonian cosmologies

Recently a covariant approach to cold matter universes in the zero-shear hypersurfaces (or longitudinal) gauge has been developed. This approach reveals the existence of an integrability condition, which does not appear in standard non-covariant treatments. A simple derivation and generalization of the integrability condition is given, based on showing that the quasi-Newtonian models are a sub-class of the linearized `silent' models. The solution of the integrability condition implies a propagation equation for the acceleration. It is shown how the velocity and density perturbations are then obtained via this propagation equation. The density perturbations acquire a small relative-velocity correction on all scales, arising from the fully covariant general relativistic analysis.Comment: 11 pages Revtex; to appear Phys. Rev.

Detecting extra dimensions with gravity wave spectroscopy: the black string brane-world

Using the black string between two branes as a model of a brane-world black hole, we compute the gravity wave perturbations and identify the features arising from the additional polarizations of the graviton. The standard four-dimensional gravitational wave signal acquires late-time oscillations due to massive modes of the graviton. The Fourier transform of these oscillations shows a series of spikes associated with the masses of the Kaluza-Klein modes, providing in principle a spectroscopic signature of extra dimensions.Comment: 4 pages, 5 figures. Comments on the frequency and detectability of the massive mode signal added, and title modified. Version accepted for publication in Phys. Rev. Let

Stress effects in structure formation

Residual velocity dispersion in cold dark matter induces stresses which lead to effects that are absent in the idealized dust model. A previous Newtonian analysis showed how this approach can provide a theoretical foundation for the phenomenological adhesion model. We develop a relativistic kinetic theory generalization which also incorporates the anisotropic velocity dispersion that will typically be present. In addition to density perturbations, we consider the rotational and shape distortion properties of clustering. These quantities together characterize the linear development of density inhomogeneity, and we find exact solutions for their evolution. As expected, the corrections are small and arise only in the decaying modes, but their effect is interesting. One of the modes for density perturbations decays less rapidly than the standard decaying mode. The new rotational mode generates precession of the axis of rotation. The new shape modes produce additional distortion that remains frozen in during the subsequent (linear) evolution, despite the rapid decay of the terms that caused it.Comment: significantly improved discussion of kinetic theory of CDM velocity dispersion; to appear Phys. Rev.

Asymmetric brane-worlds with induced gravity

The Randall-Sundrum scenario, with a 1+3-dimensional brane in a 5-dimensional bulk spacetime, can be generalized in various ways. We consider the case where the Z2-symmetry at the brane is relaxed, and in addition the gravitational action is generalized to include an induced gravity term on the brane. We derive the complete set of equations governing the gravitational dynamics for a general brane and bulk, and identify how the asymmetry and the induced gravity act as effective source terms in the projected field equations on the brane. For a Friedmann brane in an anti de Sitter bulk, the solution of the Friedmann equation is given by the solution of a quartic equation. We find the perturbative solutions for small asymmetry, which has an effect at late times.Comment: references added; version to appear Phys Rev

Cosmological perturbations in Horava-Lifshitz theory without detailed balance

In the Horava-Lifshitz theory of quantum gravity, two conditions -- detailed balance and projectability -- are usually assumed. The breaking of projectability simplifies the theory, but it leads to serious problems with the theory. The breaking of detailed balance leads to a more complicated form of the theory, but it appears to resolve some of the problems. Sotiriou, Visser and Weinfurtner formulated the most general theory of Horava-Lifshitz type without detailed balance. We compute the linear scalar perturbations of the FRW model in this form of HL theory. We show that the higher-order curvature terms in the action lead to a gravitational effective anisotropic stress on small scales. Specializing to a Minkowski background, we study the spin-0 scalar mode of the graviton, using a gauge-invariant analysis, and find that it is stable in both the infrared and ultraviolet regimes for $0 \le \xi \le 2/3$. However, in this parameter range the scalar mode is a ghost.Comment: Some typos were corrected. Verison appeared in Phys. Rev. D81, 024009 (2010