We propose a new test of the critical earthquake model based on the hypothesis that precursory earthquakes are ``actors that create fluctuations in the stress field which exhibit an increasing correlation length as the critical large event becomes imminent. Our approach constitutes an attempt to build a more physically-based cumulative function in the spirit of but improving on the cumulative Benioff strain used in previous works documenting the phenomenon of accelerated seismicity. Using a space and time dependent visco-elastic Green function in a two-layer model of the Earth lithosphere, we compute the spatio-temporal stress fluctuations induced by every earthquake precursor and estimate, through an appropriate wavelet transform, the contribution of each event to the correlation properties of the stress field around the location of the main shock at different scales. Our physically-based definition of the cumulative stress function adding up the contribution of stress loads by all earthquakes preceding a main shock seems to be unable to reproduce an acceleration of the cumulative stress nor an increase of the stress correlation length similar to those observed previously for the cumulative Benioff strain. Either earthquakes are ``witnesses of large scale tectonic organization and/or the triggering Green function requires much more than just visco-elastic stress transfers.
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