We perform population synthesis calculations of present-day post common envelope binaries ( PCEBs ) and zero-age cataclysmic variables ( ZACVs ) using a common envelope efficiency parameter , \alpha _ { CE } , that is a function of secondary mass , M _ { s } .
We investigate three basic possibilities : ( 1 ) a standard constant \alpha _ { CE } prescription , with \alpha _ { CE } = 1.0 , 0.6 , 0.3 , 0.2 , 0.1 and 0.05 , to provide a baseline for comparison , ( 2 ) a power law dependence , \alpha _ { CE } = ( M _ { s } ) ^ { n } , with n = 0.5 , 1.0 and 2.0 , and ( 3 ) a dependence in which \alpha _ { CE } approaches 1 for large secondary masses and \alpha _ { CE } = 0 below some assumed cutoff mass , \alpha _ { CE } = 1 - M _ { cut } / M _ { s } , where M _ { cut } is the cutoff mass and is equal to 0.0375 , 0.075 and 0.15 M _ { \odot } .
For each population , we compute orbital period , orbital separation , white dwarf mass and secondary mass distributions .

We find that if \alpha _ { CE } \lesssim 0.2 in our constant \alpha _ { CE } sequence , the predicted present-day ZACV population is significantly modified compared with our standard model ( \alpha _ { CE } = 1.0 ) .
All prior population synthesis calculations of the formation of CVs only considered values of \alpha _ { CE } \geq 0.3 and found that their model populations were not strongly dependent upon the value of \alpha _ { CE } .
Our results indicate that a much wider range of values for \alpha _ { CE } , including very low values , must be considered in order for a dependence to be seen .
In our variable \alpha _ { CE } sequences for ZACVs , we find that for models in which \alpha _ { CE } decreases very rapidly for small secondary masses , the orbital period distribution below the period gap differs significantly from our standard model .
These differences are most evident in our power law sequence model with n = 2 and in our cutoff mass sequence model with M _ { cut } = 0.15 M _ { \odot } .
In these two models , the fraction of CVs forming with orbital periods below the gap is reduced significantly , the fraction forming in the gap is increased significantly , and both the short-period peak and the minimum period in the ZACV orbital period distribution shift to considerably longer orbital periods compared with our standard model .
We suggest that the observed scarcity of CVs with P < 77 min may possibly provide evidence that progenitor binaries with very low mass secondaries ( M _ { s } \lesssim 0.10 M _ { \odot } ) are unable to avoid merger within the common envelope .
We also suggest that if \alpha _ { CE } decreases rapidly for small secondary masses , as in our power law sequence with n \gtrsim 1 , it is possible that the lower edge of the period gap could be , in part , an imprint of the ZACV population .
Such an imprint could be important in recently-proposed non-standard scenarios of CV secular evolution , such as circumbinary disks , which have difficulty in reproducing the sharpness of the lower edge of the gap .

In our constant \alpha _ { CE } model sequence for present-day PCEBs , we find that for all values of \alpha _ { CE } , the majority of the systems contain secondaries with masses > 0.375 M _ { \odot } , orbital periods > 1 day and orbital separations > 0.025 AU , with most having periods of \sim 3 days and separations of \sim 0.05 AU .
These models further predict that the present-day population of PCEBs should contain roughly an equal number of systems with He and CO WDs if \alpha _ { CE } is globally near unity , but should be clearly dominated by systems containing CO WDs if \alpha _ { CE } is globally small ( \lesssim 0.30 ) .
In our model sequences of present-day PCEBs in which \alpha _ { CE } is a function of secondary mass , the only distribution that varies significantly is the secondary mass distribution .
In the power law model sequence , as n is increased from 0.5 to 2.0 , the fraction of PCEBs with M _ { s } < 0.375 M _ { \odot } decreases by a factor of \sim 4 from 0.26 to 0.06 .
In the n = 2.0 model , there are no present-day PCEBs with secondary masses less than 0.10 M _ { \odot } .
In the cutoff mass model sequence , significant changes only occur near the cutoff mass and the distributions are nearly identical for M _ { s } \gtrsim 0.375 M _ { \odot } .

Based on the results of this investigation , we suggest to theorists who perform detailed hydrodynamical calculations of common envelope evolution that a sequence of models with a fixed mass giant and very low mass secondaries , ranging from 0.3 M _ { \odot } down to 0.05 M _ { \odot } , should be given some priority .
We further suggest to observers that a well-defined , statistically complete sample of PCEBs , particularly with regard to very low mass secondaries , is crucially needed to provide tests of detailed models of common envelope evolution .