Structure on the largest scales: constraints from the isotropy of radio source counts

We show how the size of the largest scales of cellular structure in the Universe may be deduced from the anisotropy of the sky distribution of extragalactic radio sources. Using a Voronoi tessellation to represent the skeleton structure of the galaxy distribution, together with an evolving luminosity function for the mean space density of extragalactic radio sources, we have calculated the source-count anisotropy expected as a function of tessellation scale ρ^-1/3, radio-survey depth and survey angular diameter. Deep pencil-beam surveys of opening angle ≲0.1 rad provide strong limits on ρ. We find, for example, that, if the density parameter Ω = 1 and ρ^-1/3 ∼ 100 h^-1 Mpc, significant ($$\gt\sqrt{N}$$) area-to-area fluctuations are expected for counts from pencil-beam surveys slightly deeper than the deepest yet achieved, S_{1 GHz}<0.1 mJy for a $$0_\cdot^\circ 5$$-diameter survey, S_{1 GHz}<3 mJy for a 5°-diameter survey. The analysis demonstrates (a) the way in which deep radio-survey data can be used to constrain large-scale structure, and (b) the care which must be exercised in defining cosmologically representative samples of faint radio sources. We review the isotropy (or otherwise) of (1) large-area radio surveys, typically covering several steradians to ∼1 Jy, and (2) pencil-beam radio surveys to mJy and μJy levels. Large-area surveys limit $$\rho^{-1/3}\lesssim 250 \enspace h^{-1} \textrm {Mpc}$$ (if Ω = 1). The counts from the 5C pencil-beam surveys (3000 sources in 10 fields to S_{408 MHz}>10 mJy) limit $$\rho^{-1/3}\lesssim 150 \enspace h^{-1} \textrm {Mpc}$$ (for Ω = 1). A similar $$\rho^{-1/3}\lesssim 100 \enspace h^{-1} \textrm {Mpc}$$ can be inferred from the isotropy (or marginal anisotropy) of deep VLA and WSRT radio surveys. We show how spectroscopy of low-redshift (z<0.4) optical identifications from deep pencil-beam surveys can be used to trace large-scale structure directly, and we illustrate this with some results from the 5C12 surveys.