Cosmic voids and induced hyperbolicity
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Cosmic voids and induced hyperbolicity M. Samsonyan1, A. A. Kocharyan2 , A. Stepanian1, V. G. Gurzadyan1,3,a 1 Center for Cosmology and Astrophysics, Alikhanian National Laboratory and Yerevan State University,
Yerevan, Armenia
2 School of Physics and Astronomy, Monash University, Clayton, Australia 3 SIA, Sapienza Universita di Roma, Rome, Italy
Received: 2 November 2020 / Accepted: 23 November 2020 © Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021
Abstract Cosmic voids—the low density regions in the Universe—as characteristic features of the large-scale matter distribution, are known for their hyperbolic properties. The latter implies the deviation of photon beams due to their underdensity, thus mimicing the negative curvature. We now show that the hyperbolicity can be induced not only by negative curvature or underdensity but also depends on the anisotropy of the photon beams.
1 Introduction The low density regions in the large scale Universe—voids—are among actively studied phenomena, see [1–4] and references therein. Cosmic voids are acting as probes for modified gravity theories, evolution of cosmological density perturbations, etc. Various observational surveys aim to reveal the characteristics of the voids (e.g., [5–7]), the distributions of their spatial scales, underdensity parameter, as their knowledge is of particular importance for the reconstruction of the spectrum of the density perturbations and the formation of the large scale Universe. Cosmic Microwave Background (CMB) provided another window to trace the presence of the voids [8–11], along with the traditional galaxy surveys. For example, the Cold Spot, a remarkable non-Gaussian feature known in the CMB sky was shown to reveal properties of a void [12], as supported also with galactic survey [13]. At the study of the Cold Spot, the hyperbolicity property of voids was used, namely, the deviation of the photon trajectories, i.e., of null geodesics due to the underdensity of the void. The deviation of geodesic flows is known to be a property of negatively curved spaces as studied in theory of dynamical systems [14,15]. Regarding the voids, it was shown that the low-density spatial regions can induce hyperbolicity even in conditions of globally flat or positively curved Universe [16,17]. The voids as divergent lenses were considered also in [18]. Below, we show that the hyperbolicity of geodesic flows can be caused not only by the underdensity parameter of a void but also will depend on the anisotropy of the photon beams.
a e-mail: [email protected] (corresponding author)
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Eur. Phys. J. Plus
(2020) 135:946
The property of hyperbolicity can be defined by means of the equation of deviation of close geodesics defined in a d-dim Riemannian manifold M, known as Jacobi equation [14,16] ∇u2 n + u (n) = 0,
(1)
where the deviation vector n = nˆ ˆ n) ˆ = 1, g(n, ˆ u) = 0. is orthogonal to the velocity vector u and g(n, n) = 2 , g(n, Jacobi equation can be wr
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