Flat Universe with Hyperbolic Voids

The properties of geodesics flow are studied in a Friedmann-Robertson-Walker metric perturbed due to the inhomogeneities of matter. The basic, averaged Jacobi equation is derived, which reveals that the low density regions (voids) are able to induce hyperbolicity, even if the global curvature of the Universe is zero or slightly positive. It is shown that the energy independence is a characteristic property of these geometric effects. The importance of these conclusions is determined by the temperature independent ellipticity of excursion sets and regions of different randomness found in Kolmogorov CMB maps. Introduction. – The increase in the accuracy of Cosmic Microwave Background (CMB) radiation measurements has not only largely revealed the set of the basic cosmological parameters [1], but has also stipulated the importance of the search of non-gaussian effects. The list of already reported non-gaussianities includes the low multipole alignment anomaly, the Cold Spot, etc (e.g. [2] and references therein). The CMB distortions due to the cumulative contribution of inhomogeneous matter distribution in the Universe are among the discussed effects, which on the one hand, can influence the primordial nongaussianities, on the other hand, will enable the link with the observed filaments (e.g. [3]). Another empirical fact for potential interest here is the temperature independent ellipticity of anisotropies found in CMB sky maps [4]. For example, it was noticed [5] that, if the nature of the dark energy is due to the matter inhomogeneities, then it might be linked with those ellipticities in the CMB maps. For further consideration of the backreaction of the inhomogeneities see [6]. Here we consider within a geometric framework the CMB distortions both due to the matter distribution and the overall geometry. This approach is based on the concept of the instability of geodesic flows and involves the Hamiltonian formalism, known by its efficiency in the problems of instability of dynamical systems [7]. It is shown that the temperature independence of the distortions of CMB excursion sets is a generic feature of the geometric nature of the effect, as studied initially in [8]. Namely, whatever is the agent causing the hyperbolicity itself, one has an empirical indication of a geometric effect, thus supporting such a nature for temperature threshold independence of ellipticities [4]. Then, we show that the low density regions, voids, can induce hyperbolicity even in conditions of globally flat or positively curved Universe. To have possibly self-consistent account, we start with a brief account of the reduction of the null geodesics from (3+1) to 3-spaces, then move to the averaged Jacobi equation and the conditions for the instability of the flows. Anosov flows. – The basic steps can be separated as: (a) to find out the conditions when the geodesics in (3+1)D can be projected as geodesics in 3D Riemannian manifold, (b) the Hamiltonian representation of the flows, (c) the time correlations functions of Anosov flows. Let M be a smooth d dimensional manifold with a smooth Riemannian metric g, τ : TM → M be the tangent bundle over M , and K : TE → E the linear connection on a vector bundle π : E → M over M , such that for each m ∈ M and X ∈ Em = π (m) the restriction K : TXE → Em is linear and surjective. Then, the following diagram is commutative TE K −→ E τE ↓ ↓ π E π −→ M.