## The strongest gravitational lenses

### III. The order statistics of the largest Einstein radii

^{1}
Blue Yonder GmbH, Karlsruher Strasse 88,
76139
Karlsruhe,
Germany

e-mail:
jean-claude.waizmann@blue-yonder.com

^{2}
Dipartimento di Fisica e Astronomia, Università di
Bologna, viale Berti Pichat
6/2, 40127
Bologna,
Italy

^{3}
INAF – Osservatorio Astronomico di Bologna, via Ranzani 1,
40127
Bologna,
Italy

^{4}
INFN, Sezione di Bologna, viale Berti Pichat 6/2,
40127
Bologna,
Italy

^{5}
Zentrum für Astronomie der Universität Heidelberg, Institut für
Theoretische Astrophysik , Albert-Ueberle-Str. 2, 69120
Heidelberg,
Germany

^{6}
Jet Propulsion Laboratory , 4800 Oak Grove Drive, Pasadena
CA
91109,
USA

Received:
10
November
2013

Accepted:
12
February
2014

*Context. *The Einstein radius of a gravitational lens is a key
characteristic. It encodes information about decisive quantities such as halo mass,
concentration, triaxiality, and orientation with respect to the observer. Therefore, the
largest Einstein radii can potentially be utilised to test the predictions of the
ΛCDM model.

*Aims. *Hitherto, studies have focussed on the single largest observed
Einstein radius. We extend those studies by employing order statistics to formulate
exclusion criteria based on the *n* largest Einstein radii and apply these criteria
to the strong lensing analysis of 12 MACS clusters at *z*>
0.5.

*Methods. *We obtain the order statistics of Einstein radii by a Monte
Carlo approach, based on the semi-analytic modelling of the halo population on the past
lightcone. After sampling the order statistics, we fit a general extreme value
distribution to the first-order distribution, which allows us to derive analytic relations
for the order statistics of the Einstein radii.

*Results. *We find that the Einstein radii of the 12 MACS clusters are not
in conflict with the ΛCDM
expectations. Our exclusion criteria indicate that, in order to exhibit tension with the
concordance model, one would need to observe approximately twenty Einstein radii with
*θ*_{eff} ≳
30″, ten with *θ*_{eff} ≳ 35″, five with *θ*_{eff} ≳ 42″, or
one with *θ*_{eff} ≳
74″ in the redshift range 0.5 ≤ *z* ≤ 1.0 on the full sky (assuming a source
redshift of *z*_{s} =
2). Furthermore, we find that, with increasing order, the haloes with
the largest Einstein radii are on average less aligned along the line-of-sight and less
triaxial. In general, the cumulative distribution functions steepen for higher orders,
giving them better constraining power.

*Conclusions. *A framework that allows the individual and joint order
distributions of the *n*-largest Einstein radii to be derived is presented.
From a statistical point of view, we do not see any evidence of an Einstein ring problem
even for the largest Einstein radii of the studied MACS sample. This conclusion is
consolidated by the large uncertainties that enter the lens modelling and to which the
largest Einstein radii are particularly sensitive.

Key words: gravitational lensing: strong / methods: statistical / galaxies: clusters: general / cosmology: miscellaneous

*© ESO, 2014*