Assuming a simple form for the growth index \gamma ( z ) depending on two parameters \gamma _ { 0 } \equiv \gamma ( z = 0 ) and \gamma _ { 1 } \equiv \gamma ^ { \prime } ( z = 0 ) , we show that these parameters can be constrained using background expansion data .
We explore systematically the preferred region in this parameter space .
Inside General Relativity we obtain that models with a quasi-static growth index and \gamma _ { 1 } \approx - 0.02 are favoured .
We find further the lower bounds \gamma _ { 0 } \gtrsim 0.53 and \gamma _ { 1 } \gtrsim - 0.15 for models inside GR .
Models outside GR having the same background expansion as \Lambda CDM and arbitrary \gamma ( z ) with \gamma _ { 0 } = \gamma _ { 0 } ^ { \Lambda CDM } , satisfy G _ { { eff } , 0 } > G for \gamma _ { 1 } > \gamma _ { 1 } ^ { \Lambda CDM } , and G _ { { eff } , 0 } < G for \gamma _ { 1 } < \gamma _ { 1 } ^ { \Lambda CDM } .
The first models will cross downwards the value G _ { { eff } } = G on very low redshifts z < 0.3 , while the second models will cross upwards G _ { { eff } } = G in the same redshift range .
This makes the realization of such modified gravity models even more problematic .