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Register a new userThe perturbed FLRW metric on all scales: Newtonian limit and top-hat collapse
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Andreas Finke
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The applicability of a linearized perturbed FLRW metric to the late, lumpy universe has been subject to debate. We consider in an elementary way the Newtonian limit of the Einstein equations with this ansatz for the case of structure formation in late-time cosmology, on small and large scales, and argue that linearizing the Einstein tensor produces only a small error down to arbitrarily small, decoupled scales (e.g. Solar system scales). On subhorizon patches, the metric scale factor becomes a coordinate choice equivalent to choosing the spatial curvature, and not a sign that the FLRW metric cannot perturbatively accommodate very different local physical expansion rates of matter; we distinguish these concepts, and show that they merge on large scales for the Newtonian limit to be globally valid. Furthermore, on subhorizon scales, a perturbed FLRW metric ansatz does not already imply assumptions on isotropy, and effects beyond an FLRW background, including those potentially caused by non-linearities of general relativity (GR), may be encoded into non-trivial boundary conditions. The corresponding cosmologies have already been developed in a Newtonian setting by Heckmann and Schucking and none of these boundary conditions can explain the accelerated expansion of the universe. Our analysis of the field equations is confirmed on the level of solutions by an example of pedagogical value, comparing a collapsing top-hat overdensity (embedded into a cosmological background) treated in such perturbative manner to the corresponding exact solution of GR, where we find good agreement even in the regimes of strong density contrast.
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Based on an examination of the solutions to the Killing Vector equations for the FLRW-metric in co moving coordinates , it is conjectured and proved that the components(in these coordinates) of Killing Vectors, when suitably scaled by functions, are emph{zero modes} of the corresponding emph{scalar} Laplacian. The complete such set of zero modes(infinitely many) are explicitly constructed for the two-sphere. They are parametrised by an integer n. For $n,ge,2$, all the solutions are emph{irregular} (in the sense that they are neither well defined everywhere nor are emph{square-integrable}). The associated 2-d vectors are also emph{not normalisable}. The $n=0$ solutions being constants (these correspond to the zero angular momentum solutions) are regular and normalizable. Not all of the $n=1$ solutions are regular but the associated vectors are normalizable. Of course, the action of scalar Laplacian coordinate independent significance only when acting on scalars. However, our conclusions have an unambiguous meaning as long as one works in this coordinate system. As an intermediate step, the covariant Laplacians(vector Laplacians) of Killing vectors are worked out for four-manifolds in two different ways, both of which have the novelty of not explicitly needing the connections. It is further shown that for certain maximally symmetric sub-manifolds(hypersurfaces of one or more constant comoving coordinates) of the FLRW-spaces also, the scaled Killing vector components are zero modes of their corresponding scalar Laplacians. The Killing vectors for the maximally symmetric four-manifolds are worked out using the elegant embedding formalism originally due to Schrodinger . Some consequences of our results are worked out. Relevance to some very recent works on zero modes in AdS/CFT correspondences , as well as on braneworld scenarios is briefly commented upon.
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