Narrow Results

We introduce $D_{(0)}^{\nu }{W}_{\nu \mu \rho \sigma }=h_{\mu \rho \sigma }$, the inhomogeneous equation suitable to describe the solution of the linearized version of the conformal part of the Riemann tensor connected to the perturbations of the Kerr spacetime far from the origin. Then we find the right decays we have to impose to the source term $h_{\mu \rho \sigma }$ to obtain the peeling decays for this linearized solution. We explain how these decays are compatible with the ones we need to attack the full nonlinear problem following the Christodoulou–Klainerman approach, see [7, 11]. This result is expressed explicitly in Theorem 1.1 and requires some new detailed estimates for the connection coefficients related to the null cone foliation in Kerr, see [10], which could be considered as a useful result by itself.

In this paper, we investigate the linear perturbations of the spherically symmetric space–times with kinematic self-similarity of the second kind. The massless scalar field equations are solved which yield the background and an exact solutions for the perturbed equations. We discuss the boundary conditions of the resulting perturbed solutions. The possible perturbation modes turn out to be stable as well as unstable. The analysis leads to the conclusion that there does not exist any critical solution.

We study the stability properties of static, spherically symmetric configurations in $k$-essence theories with the Lagrangians of the form $F(X)$, $X\equiv {\varphi }_{,\alpha }{\varphi }^{,\alpha }$. The instability under spherically symmetric perturbations is proved for the two recently obtained exact solutions for $F(X)=F_0{X}^{1/3}$ and for $F(X)=F_0{X}^{1/2}-2\mathrm{\Lambda }$, where $F_0$ and $\mathrm{\Lambda }$ are constants. The first solution describes a black hole in an asymptotically singular spacetime, the second one contains two horizons of infinite area connected by a wormhole. It is argued that spherically symmetric $k$-essence configurations with $n<1/2$ are generically unstable because the perturbation equation is not of hyperbolic type.

These notes are based on 4 lectures given at Theoretical Advanced Study Institute in 2009 on the large scale structure of the universe. They provide a pedagogical introduction to the temporal evolution of linear density perturbations in the universe and a discussion of how density perturbations on small scales depend on the particle properties of dark matter. The notes assume the reader is familiar with the concepts and mathematics required to describe isotropic and homogeneous cosmology.