Quantifying cosmic variance

We determine an expression for the cosmic variance of any 'normal' galaxy survey based on examination of M* +/- 1 mag galaxies in the Sloan Digital Sky Survey (SDSS) Data Release 7 (DR7) data cube. We find that cosmic variance will depend on a number of factors principally: total survey volume, survey aspect ratio and whether the area surveyed is contiguous or comprising independent sightlines. As a rule of thumb cosmic variance falls below 10 per cent once a volume of 107 h-3(0.7) Mpc3 is surveyed for a single contiguous region with a 1:1 aspect ratio. Cosmic variance will be lower for higher aspect ratios and/or non-contiguous surveys. Extrapolating outside our test region we infer that cosmic variance in the entire SDSS DR7 main survey region is similar to 7 per cent to z < 0.1.

The equation obtained from the SDSS DR7 region can be generalized to estimate the cosmic variance for any density measurement determined from normal galaxies (e.g. luminosity densities, stellar mass densities and cosmic star formation rates) within the volume range 103-107 h-3(0.7) Mpc3.

We apply our equation to show that two sightlines are required to ensure that cosmic variance is < 10 per cent in any ASKAP galaxy survey (divided into delta z similar to 0.1 intervals, i.e. similar to 1 Gyr intervals for z < 0.5). Likewise 10 MeerKAT sightlines will be required to meet the same conditions. GAMA, VVDS and zCOSMOS all suffer less than 10 per cent cosmic variance (similar to 3-8 per cent) in delta z intervals of 0.1, 0.25 and 0.5, respectively. Finally we show that cosmic variance is potentially at the 50-70 per cent level, or greater, in the Hubble Space Telescope (HST) Ultra Deep Field depending on assumptions as to the evolution of clustering. 100 or 10 independent sightlines will be required to reduce cosmic variance to a manageable level (< 10 per cent) for HST ACS or HST WFC3 surveys, respectively (in delta z similar to 1 intervals). Cosmic variance is therefore a significant factor in the z > 6 HST studies currently underway.