Seeing patterns in noise: Gigaparsec-scale `structures' that do not violate homogeneity

Clowes et al. (2013) have recently reported the discovery of a Large Quasar Group (LQG), dubbed the Huge-LQG, at redshift z~1.3 in the DR7 quasar catalogue of the Sloan Digital Sky Survey. On the basis of its characteristic size ~500 Mpc and longest dimension >1 Gpc, it is claimed that this structure is incompatible with large-scale homogeneity and the cosmological principle. If true, this would represent a serious challenge to the standard cosmological model. However, the homogeneity scale is an average property which is not necessarily affected by the discovery of a single large structure. I clarify this point and provide the first fractal dimension analysis of the DR7 quasar catalogue to demonstrate that it is in fact homogeneous above scales of at most 130 Mpc/h, which is much less than the upper limit for \Lambda CDM. In addition, I show that the algorithm used to identify the Huge-LQG regularly finds even larger clusters of points, extending over Gpc scales, in explicitly homogeneous simulations of a Poisson point process with the same density as the quasar catalogue. This provides a simple null test to be applied to any cluster thus found in a real catalogue, and suggests that the interpretation of LQGs as `structures' is misleading.Comment: 9 pages, 6 figures. MNRAS published online. v2: minor typo corrected, added one missing referenc

Weyl curvature and the Euler characteristic in dimension four

We give lower bounds, in terms of the Euler characteristic, for the $L^2$-norm of the Weyl curvature of closed Riemannian 4-manifolds. The same bounds were obtained by Gursky, in the case of positive scalar curvature metrics.Comment: 6 page

Evidence for strong extragalactic magnetic fields from Fermi observations of TeV blazars

Magnetic fields in galaxies are produced via the amplification of seed magnetic fields of unknown nature. The seed fields, which might exist in their initial form in the intergalactic medium, were never detected. We report a lower bound $B\ge 3\times 10^{-16}$~gauss on the strength of intergalactic magnetic fields, which stems from the nonobservation of GeV gamma-ray emission from electromagnetic cascade initiated by tera-electron volt gamma-ray in intergalactic medium. The bound improves as $\lambda_B^{-1/2}$ if magnetic field correlation length, $\lambda_B$, is much smaller than a megaparsec. This lower bound constrains models for the origin of cosmic magnetic fields.Comment: 14 pages, 3 figure