Scalar perturbations on Lemaître-Tolman-Bondi spacetimes

In recent years there has been growing interest in verifying the horizon-scale homogeneity of the Universe that follows from applying the Copernican principle to the observed isotropy. This program has been stimulated by the discovery that a very large void, centered near us, can explain supernova luminosity distance measurements without dark energy. It is crucial to confront such models with as wide a variety of data as possible. With this application in mind, we develop the relativistic theory of linear scalar perturbations on spherically symmetric dust (Lemaître-Tolman-Bondi) spacetimes, using the covariant $1+1+2$ formalism. We show that the evolution of perturbations is determined by a small set of new linear transfer functions. If decaying modes are ignored (to be consistent with the standard inflationary paradigm), the standard techniques of perturbation theory on homogeneous backgrounds, such as harmonic expansion, can be applied, and results closely paralleling those of familiar cosmological perturbation theory can be obtained.

Figure 1

Background LTB radial profile at the late time $t=1.0\times {10}^5$. Energy density $\rho$, volume expansion rate $\theta$, shear 2-scalar $\Sigma$, and spatial curvature ${}^{(3)}R$ are scaled as described in the text. This profile corresponds to an underdensity near the origin with effective density and curvature parameters ${\Omega }_m=0.3$ and ${\Omega }_K=0.7$.

Figure 2

Transfer functions $T_{\rho \rho }$ and $T_{\rho \Sigma }$, normalized so that $T_{\rho \rho }^{\mathrm{norm}}=1$ at all times for an Einstein-de Sitter background. The functions are presented for the times $t=46.4$ (red, dotted line), $t=2154$ (blue, dashed line), and $t=1.0\times {10}^5$ (black, solid line). Significant suppression of perturbations occurs at late times within the void, and density perturbations are also sourced by shear fluctuations at late times through $T_{\rho \Sigma }$.

Figure 3

Normalized transfer function $T_{\rho }^{\mathrm{norm}}(t,r,k,\ell )$ for $\ell =10$. The curves are shown for a point within the periphery of the LTB void, at $r/L=0.7$, as well as near the void center at $r/L=0.01$. Also shown is the local FRW approximation (LFRW) for $r/L=0.7$, together with the Einstein-de Sitter case (EdS).