Statistical properties and probability densities of correlation functions

• Applicant: Professor Dr. Peter Schneider
• Subject Area: Astrophysics and Astronomy
• Term: from 2010 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 178182033

• Final Report Year: 2015

Final Report Abstract

In 2009, we discovered that correlation functions of statistical processes have to obey a set of inequalities and that, as a consequence, the probability density of measured correlation functions cannot be a multi-variate Gaussian, as almost always assumed when comparing observations with model predictions to obtain parameter estimates. This effect is severe and affects the results of the analysis of at least some recent cosmological surveys – in those cases, the Gaussian is not even a reasonable approximation. Within this project we have developed a significantly improved approach to describe the probability density of correlation functions, and obtained exact results in some limiting cases. We have tested our new approximation using simulations and could demonstrate its superiority over the ‘standard’ Gaussian approximation; furthermore, we have shown that this makes a marked difference in the estimate of model parameters and their uncertainty ranges. Therefore, our finding are relevant for an accurate estimate of cosmological parameters and their confidence regions in ongoing and future surveys.

Publications

• 2011, “Constrained probability distributions of correlation functions”, A&A 534, A76 (13 pages)
  Keitel, D. & Schneider, P.
  
• 2012, “The bispectrum covariance beyond Gaussianity. A log-normal approach”, A&A 540, A9 (9 pages)
  Martin, S., Schneider, P. & Simon, P.
  
• 2013, “A quasi-Gaussian approximation for the probability distribution of correlation functions”, A&A 556, A70 (16 pages)
  Wilking, P. & Schneider, P.
  
• 2015, “Constrained correlation functions from the Millennium Simulation”, A&A 582, A107 (11 pages)
  Wilking, P., Röseler, R. & Schneider, P.