The probability distribution of cluster formation times and implied Einstein radii

Sharon Sadeh*, Yoel Rephaeli

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We provide a quantitative assessment of the probability distribution function of the concentration parameter of galaxy clusters. We do so by using the probability distribution function of halo formation times, calculated by means of the excursion set formalism, and a formation redshift-concentration scaling derived from results of N-body simulations. Our results suggest that the observed high concentrations of several clusters are quite unlikely in the standard Λ cold dark matter (ΛCDM) cosmological model, but that due to various inherent uncertainties, the statistical range of the predicted distribution may be significantly wider than commonly acknowledged. In addition, the probability distribution function of the Einstein radius of A1689 is evaluated, confirming that the observed value of ∼45 ± 5 arcsec is very improbable in the currently favoured cosmological model. If, however, a variance of ∼20 per cent in the theoretically predicted value of the virial radius is assumed, then the discrepancy is much weaker. The measurement of similarly large Einstein radii in several other clusters would pose a difficulty to the standard model. If so, earlier formation of the large-scale structure would be required, in accord with predictions of some quintessence models. We have indeed verified that in a viable early dark energy model large Einstein radii are predicted in as many as a few tens of high-mass clusters.

Original languageEnglish
Pages (from-to)1759-1765
Number of pages7
JournalMonthly Notices of the Royal Astronomical Society
Issue number4
StatePublished - Aug 2008


  • Galaxies: clusters: general
  • Gravitational lensing
  • Large-scale structure of Universe


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