We use semi-analytic models of structure formation to interpret gravitational lensing measurements of substructure in galaxy cluster cores ( R { \leq } 250 { \mathrel { h ^ { -1 } { kpc } } } ) at z = 0.2 .
The dynamic range of the lensing-based substructure fraction measurements is well matched to the theoretical predictions , both spanning \mathrel { f _ { sub } } \sim 0.05 - 0.65 .
The structure formation model predicts that \mathrel { f _ { sub } } is correlated with cluster assembly history .
We use simple fitting formulae to parameterize the predicted correlations : \Delta _ { 90 } = \tau _ { 90 } + \alpha _ { 90 } \log ( \mathrel { f _ { sub } } ) and \Delta _ { 50 } = \tau _ { 50 } + \alpha _ { 50 } \log ( \mathrel { f _ { sub } } ) , where \Delta _ { 90 } and \Delta _ { 50 } are the predicted lookback times from z = 0.2 to when each theoretical cluster had acquired 90 % and 50 % respectively of the mass it had at z = 0.2 .
The best-fit parameter values are : \alpha _ { 90 } = ( -1.34 \pm 0.79 ) { Gyr } , \tau _ { 90 } = ( 0.31 \pm 0.56 ) { Gyr } and \alpha _ { 50 } = ( -2.77 \pm 1.66 ) { Gyr } , \tau _ { 50 } = ( 0.99 \pm 1.18 ) { Gyr } .
Therefore ( i ) observed clusters with \mathrel { f _ { sub } } \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $% \sim$ } } } \hbox { $ < $ } } } 0.1 ( e.g .
A 383 , A 1835 ) are interpreted , on average , to have formed at z \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } } \hbox { $ > $ } } } 0.8 and to have suffered \leq 10 \% mass growth since z \simeq 0.4 , ( ii ) observed clusters with \mathrel { f _ { sub } } \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $% \sim$ } } } \hbox { $ > $ } } } 0.4 ( e.g .
A 68 , A 773 ) are interpreted as , on average , forming since z \simeq 0.4 and suffering > 10 \% mass growth in the \sim 500 { Myr } preceding z = 0.2 , i.e .
since z = 0.25 .
In summary , observational measurements of \mathrel { f _ { sub } } can be combined with structure formation models to estimate the age and assembly history of observed clusters .
The ability to “ age-date ” approximately clusters in this way has numerous applications to the large clusters samples that are becoming available .