We investigate the influence of dark energy on structure formation , within five different cosmological models , namely a concordance \Lambda CDM model , two models with dynamical dark energy , viewed as a quintessence scalar field ( using a RP and a SUGRA potential form ) and two extended quintessence models ( EQp and EQn ) where the quintessence scalar field interacts non-minimally with gravity ( scalar-tensor theories ) .
We adopted for all models the normalization of the matter power spectrum \sigma _ { 8 } to match the CMB data .
In the models with dynamical dark energy and quintessence , we describe the equation of state with w _ { 0 } \approx - 0.9 , still within the range allowed by observations .
For each model , we have performed hydrodynamical simulations in a cosmological box of ( 300 { Mpc } h ^ { -1 } ) ^ { 3 } including baryons and allowing for cooling and star formation .
The contemporary presence of evolving dark energy and baryon physics allows us to investigate the interplay between the different background cosmology and the evolution of the luminous matter .
Since cluster baryon fraction can be used to constrain other cosmological parameters such as \Omega _ { m } , we also analyse how dark energy influences the baryon content of galaxy clusters .
We find that , in models with dynamical dark energy , the evolving cosmological background leads to different star formation rates and different formation histories of galaxy clusters , but the baryon physics is not affected in a relevant way .
We investigate several proxies for the cluster mass function based on X-ray observables like temperature , luminosity , M _ { gas } , and Y _ { X } .
We conclude that the X-ray temperature and M _ { gas } functions are better diagnostic to disentangle the growth of structures among different dark energy models .
We also evaluate the cosmological volumes needed to distinguish the dark energy models here investigated using the cluster number counts ( in terms of the mass function and the X-ray luminosity and temperature functions ) .
Relaxed , massive clusters , when studied in regions sufficiently far from from the centre , are built up in a very similar way despite the different dark energy models here considered .
We confirm that the overall baryon fraction is almost independent of the dark energy models at a few percent level .
The same is true for the gas fraction .
This evidence reinforces the use of galaxy clusters as cosmological probe of the matter and energy content of the Universe .