We solve analytically and numerically the generalized Einstein equations in scalar-tensor cosmologies to obtain the evolution of dark energy and matter linear perturbations .
We compare our results with the corresponding results for minimally coupled quintessence perturbations .
Our results for natural ( O ( 1 ) ) values of parameters in the Lagrangian which lead to a background expansion similar to \Lambda CDM are summarized as follows : 1 .
Scalar-Tensor dark energy density perturbations are amplified by a factor of about 10 ^ { 4 } compared to minimally coupled quintessence perturbations on scales less than about 1000 { h ^ { -1 } Mpc } ( sub-Hubble scales ) .
This amplification factor becomes even larger ( \gtrsim 10 ^ { 6 } ) for scales less than 100 { h ^ { -1 } Mpc } .
On these scales dark energy perturbations constitute a fraction of about 10 \% compared to matter density perturbations .
2 .
Scalar-Tensor dark energy density perturbations are anti-correlated with matter linear perturbations on sub-Hubble scales .
Thus clusters of galaxies are predicted to overlap with voids of dark energy .
3 .
This anti-correlation of matter with negative pressure perturbations induces a mild amplification of matter perturbations by about 10 \% on sub-Hubble scales .
4 .
The evolution of scalar field perturbations on sub-Hubble scales , is scale independent and therefore it corresponds to a vanishing effective speed of sound ( c _ { s \Phi } = 0 ) .
It also involves large oscillations at early times induced by the amplified effective mass of the field .
This mass amplification is due to the non-minimal coupling of the field to the Ricci curvature scalar and ( therefore ) to matter .
No such oscillations are present in minimally coupled quintessence perturbations which are suppressed on sub-Hubble scales ( c _ { s \Phi } = 1 ) .
We briefly discuss the observational implications of our results which may include predictions for galaxy and cluster halo profiles which are modified compared to \Lambda CDM .
The observed properties of these profiles are known to be in some tension with the predictions of \Lambda CDM .