The recent discovery of a diffuse cosmic neutrino flux extending up to PeV energies raises the question of which astrophysical sources generate this signal .
One class of extragalactic sources which may produce such high-energy neutrinos are blazars .
We present a likelihood analysis searching for cumulative neutrino emission from blazars in the 2nd Fermi-LAT AGN catalogue ( 2LAC ) using an IceCube neutrino dataset 2009-12 which was optimised for the detection of individual sources .
In contrast to previous searches with IceCube , the populations investigated contain up to hundreds of sources , the largest one being the entire blazar sample in the 2LAC catalogue .
No significant excess is observed and upper limits for the cumulative flux from these populations are obtained .
These constrain the maximum contribution of the 2LAC blazars to the observed astrophysical neutrino flux to be 27 \% or less between around 10 TeV and 2 PeV , assuming equipartition of flavours at Earth and a single power-law spectrum with a spectral index of -2.5 .
We can still exclude that the 2LAC blazars ( and sub-populations ) emit more than 50 \% of the observed neutrinos up to a spectral index as hard as -2.2 in the same energy range .
Our result takes into account that the neutrino source count distribution is unknown , and it does not assume strict proportionality of the neutrino flux to the measured 2LAC \gamma -ray signal for each source .
Additionally , we constrain recent models for neutrino emission by blazars .